The underlying philosophy of the book is that all the results that are stated should be proved or at least explained in detail. This implies a somewhat different organization than existing books.

*Daron Acemoglu*

1248 Pages

73360 Reads

74 Downloads

English

PDF Format

7.26 MB

- Daron Acemoglu
- 1248 Pages
- 18 Feb 2015

##### Page - 1

##### Page - 2

Introduction to ModernEconomic Growth: Parts 1-5Daron AcemogluDepartment of Economics,Massachusetts Institute of Technology read more..

##### Page - 3

ContentsPrefacexiPart 1.Introduction1Chapter 1.Economic Growth and Economic Development:The Questions31.1.Cross-Country Income Diﬀerences31.2.Income and Welfare61.3.Economic Growth and Income Diﬀerences101.4.Origins of Today’s Income Diﬀerences and World Economic Growth121.5.Conditional Convergence161.6.Correlates of Economic Growth201.7.From Correlates to Fundamental Causes221.8.The Agenda251.9.References and Literature27Chapter 2.The Solow Growth Model312.1.The Economic Environment of read more..

##### Page - 4

Introduction to Modern Economic Growth4.2.Economies of Scale, Population, Technology and World Growth1334.3.The Four Fundamental Causes1364.4.The Eﬀect of Institutions on Economic Growth1474.5.What Types of Institutions?1644.6.Disease and Development1674.7.Political Economy of Institutions: First Thoughts1704.8.Taking Stock1714.9.References and Literature1714.10.Exercises174Part 2.Towards Neoclassical Growth177Chapter 5.Foundations of Neoclassical Growth1795.1.Preliminaries1795.2.The read more..

##### Page - 5

Introduction to Modern Economic Growth7.10.References and Literature2997.11.Exercises301Part 3.Neoclassical Growth307Chapter 8.The Neoclassical Growth Model3098.1.Preferences, Technology and Demographics3098.2.Characterization of Equilibrium3138.3.Optimal Growth3178.4.Steady-State Equilibrium3188.5.Transitional Dynamics3208.6.Technological Change and the Canonical Neoclassical Model3238.7.Comparative Dynamics3298.8.The Role of Policy3308.9.A Quantitative read more..

##### Page - 6

Introduction to Modern Economic Growth11.3.The Two-Sector AK Model42611.4.Growth with Externalities43011.5.Taking Stock43411.6.References and Literature43511.7.Exercises436Part 4.Endogenous Technological Change443Chapter 12.Modeling Technological Change44512.1.Diﬀerent Conceptions of Technology44512.2.Science and Proﬁts44912.3.The Value of Innovation in Partial Equilibrium45212.4.The Dixit-Stiglitz Model and “Aggregate Demand Externalities”45912.5.Individual R&D Uncertainty and the read more..

##### Page - 7

Introduction to Modern Economic GrowthChapter 16.Stochastic Dynamic Programming61316.1.Dynamic Programming with Expectations61316.2.Proofs of the Stochastic Dynamic Programming Theorems*62116.3.Stochastic Euler Equations62616.4.Generalization to Markov Processes*62916.5.Applications of Stochastic Dynamic Programming63116.6.Taking Stock63916.7.References and Literature64016.8.Exercises641Chapter 17.Stochastic Growth Models64517.1.The Brock-Mirman Model64617.2.Equilibrium Growth under read more..

##### Page - 8

Introduction to Modern Economic Growth20.3.Agricultural Productivity and Industrialization83520.4.Taking Stock84120.5.References and Literature84320.6.Exercises844Chapter 21.Structural Transformations and Market Failures in Development84921.1.Financial Development85121.2.Fertility, Mortality and the Demographic Transition85721.3.Migration, Urbanization and The Dual Economy86521.4.Distance to the Frontier and Changes in the Organization of Production87621.5.Multiple Equilibria From Aggregate read more..

##### Page - 9

Introduction to Modern Economic GrowthChapter A.Odds and Ends in Real Analysis and Applications to Optimization1117A.1.Distances and Metric Spaces1118A.2.Mappings, Functions, Sequences, and Continuity1121A.3.A Minimal Amount of Topology: Continuity and Compactness1126A.4.The Product Topology1131A.5.Correspondences and Berge’s Maximum Theorem1134A.6.Convexity, Concavity, Quasi-Concavity and Fixed Points1138A.7.Diﬀerentiation, Taylor Series and the Mean Value Theorem1142A.8.Functions of read more..

##### Page - 10

PrefaceThis book is intended to serve two purposes:(1) First and foremost, this is a book about economic growth and long-run economicdevelopment. The process of economic growth and the sources of diﬀerences ineconomic performance across nations are some of the most interesting, importantand challenging areas in modern social science. The primary purpose of this book isto introduce graduate students to these major questions and to the theoretical toolsnecessary for studying them. The book read more..

##### Page - 11

Introduction to Modern Economic GrowthChapters 5-7 provide the conceptual and mathematical foundations of modern macroeco-nomic analysis. Chapter 5 provides the microfoundations for much of the rest of the book(and for much of modern macroeconomics), while Chapters 6 and 7 provide a quick but rel-atively rigorous introduction to dynamic optimization. Most books on macroeconomics oreconomic growth use either continuous time or discrete time exclusively. I believe that a se-rious study of both read more..

##### Page - 12

Introduction to Modern Economic Growthfor the analysis are often taken for granted or developed in appendices. In contrast, I havestrived to provide simple proofs of almost all results stated in this book. It turns out thatonce unnecessary generality is removed, most results can be stated and proved in a way thatis easily accessible to graduate students. In fact, I believe that even somewhat long proofsare much easier to understand than general statements made without proof, which leave read more..

##### Page - 13

Introduction to Modern Economic Growthmany more errors. I also thank Lauren Fahey for editorial suggestions and help with thereferences. I would also like to thank George-Marios Angeletos, Olivier Blanchard, FrancescoCaselli, Melissa Dell, Peter Funk, Oded Galor, Hugo Hopenhayn, Simon Johnson, Chad Jones,Ismail Saglam, Jesse Zinn for useful suggestions and corrections on individual chapters, andespecially Pol Antras, Kiminori Matsuyama, James Robinson, Jesus Fernandez-Villaverdeand Pierre Yared read more..

##### Page - 14

Part 1Introduction read more..

##### Page - 15

We start with a quick look at the stylized facts of economic growth and the most basicmodel of growth, the Solow growth model. The purpose is both to prepare us for the analysisof more modern models of economic growth with forward-looking behavior, explicit capitalaccumulation and endogenous technological progress, and also to give us a way of mapping thesimplest model to data. We will also discuss diﬀerences between proximate and fundamentalcauses of economic growth and economic development. read more..

##### Page - 16

CHAPTER 1Economic Growth and Economic Development:The Questions1.1. Cross-Country Income DiﬀerencesThere are very large diﬀerences in income per capita and output per worker across coun-tries today. Countries at the top of the world income distribution are more than thirty timesas rich as those at the bottom. For example, in 2000, GDP (or income) per capita in theUnited States was over $34000. In contrast, income per capita is much lower in many othercountries: about $8000 in Mexico, about read more..

##### Page - 17

Introduction to Modern Economic Growth1960198020000.00005.0001.00015.0002.00025Density of coutries01000020000300004000050000gdp per capitaFigure 1.1. Estimates of the distribution of countries according to PPP-adjusted GDP per capita in 1960, 1980 and 2000.time, x1 (t) − x2 (t) will also grow, while log x1 (t) − log x2 (t) will remain constant). Figure1.2 shows a similar pattern, but now the spreading-out is more limited. This reﬂects thefact that while the absolute gap between rich and read more..

##### Page - 18

Introduction to Modern Economic Growth1960198020000.1.2.3.4Density of coutries67891011log gdp per capitaFigure 1.2. Estimates of the distribution of countries according to log GDPper capita (PPP-adjusted) in 1960, 1980 and 2000.United States, and Russia receive greater weight because they have larger populations. Thepicture that emerges in this case is quite diﬀerent. In fact, the 2000 distribution looks lessspread out, with thinner left tail than the 1960 distribution. This reﬂects the fact read more..

##### Page - 19

Introduction to Modern Economic Growth19601980200001.000e+092.000e+093.000e+09 Density of coutries weighted by population67891011log gdp per capitaFigure 1.3. Estimates of the population-weighted distribution of countriesaccording to log GDP per capita (PPP-adjusted) in 1960, 1980 and 2000.are not available for a large number of countries, “workers” here refer to the total economi-cally active population (according to the deﬁnition of the International Labour Organization).Figure 1.4 read more..

##### Page - 20

Introduction to Modern Economic Growth1960198020000.1.2.3.4Density of coutries681012log gdp per workerFigure 1.4. Estimates of the distribution of countries according to log GDPper worker (PPP-adjusted) in 1960, 1980 and 2000.when one compares an advanced, rich country with a less-developed one, there are strikingdiﬀerences in the quality of life, standards of living and health.Figures 1.5 and 1.6 give a glimpse of these diﬀerences and depict the relationship betweenincome per capita in 2000 read more..

##### Page - 21

Introduction to Modern Economic GrowthESPTJKMDGTZAECUARGTUNMACBFALUXHKGBOLLCAIRNURYMYSPERFRAMWIETHRWATCDZAFGHABLZPANNORGTMMARCOLCANMDAKNANZLDZAKAZATGLVAZMBGEOSENAUTGBRCIVSLVVCTMKDGNQCHNEGYHNDBENSYCKORBGRFINTGOLBNNERGABCRITTOMUSUSABRBCHEGMBZWEGRDHRVRUSGERSVKBDICOMPRYMEXCHLIRLIDNARMBLRPRTKGZBELROMNLDGRCBGDCZEPOLCPVDMAALBUKRNICNGAVENLKAGINCOGTHAISLAUSLSOCMRDOMPAKSWZPHLBRANPLKENYEMJPNSTPHUNINDESTDNKMLIGNBSYRITAJAMMOZTURSWEISRLTUSVNJORAZEUGA5678910log consumption per capita 200067891011log gdp per read more..

##### Page - 22

Introduction to Modern Economic GrowthALBARGARMAUSAUTAZEBDIBELBENBFABGDBGRBLRBLZBOLBRABRBCANCHECHLCHNCIVCMRCOGCOLCOMCPVCRICZEDNKDOMDZAECUEGYESPESTFINFRAGABGBRGEOGHAGINGMBGNBGNQGRCGTMHKGHNDHRVHUNIDNINDIRLIRNISLISRITAJAMJORJPNKAZKENKGZKORLBNLCALKALSOLTULUXLVAMACMARMDAMDGMEXMKDMLIMOZMUSMWIMYSNERNGANICNLDNORNPLNZLPAKPANPERPHLPOLPRTPRYROMRUSRWASENSLVSVKSVNSWESWZSYRTCDTGOTHATJKTTOTUNTURTZAUGAUKRURYUSAVCTVENYEMZAFZMBZWEETH405060708090life expectancy 200067891011log gdp per capita 2000Figure 1.6. The read more..

##### Page - 23

Introduction to Modern Economic Growth19601980200005101520Density of coutries−.1−.050.05.1average growth ratesFigure 1.7. Estimates of the distribution of countries according to thegrowth rate of GDP per worker (PPP-adjusted) in 1960, 1980 and 2000.Africa was not beneﬁcial. South Africa is still one of the richest countries in sub-SaharanAfrica. Nevertheless, this observation alerts us to other aspects of the economy and alsounderlines the potential conﬂicts inherent in the growth read more..

##### Page - 24

Introduction to Modern Economic GrowthSpainSouth KoreaIndiaBrazilUSASingaporeNigeriaGuatemalaUKBotswana678910log gdp per capita19601970198019902000yearFigure 1.8. The evolution of income per capita in the United States, UnitedKingdom, Spain, Singapore, Brazil, Guatemala, South Korea, Botswana,Nigeria and India, 1960-2000.richer than Nigeria because it has grown steadily over an extended period of time, whileNigeria has not (and we will see that there is a lot of truth to this simple calculation; read more..

##### Page - 25

Introduction to Modern Economic Growthincreasing at a steady pace, with a slightly faster growth in the United States, so that thelog (“proportional”) gap between the two countries is larger in 2000 than it is in 1960. Spainstarts much poorer than the United States and the UK in 1960 but grows very rapidly between1960 and the mid-1970s, thus closing the gap between itself and the United States and theUK. The three countries that show very rapid growth in this ﬁgure are Singapore, read more..

##### Page - 26

Introduction to Modern Economic Growthalready a very large gap between the United States on the one hand and India and Nigeriaon the other.DZAARGAUSAUTBRBBELBENBOLBRABFABDICMRCANCPVTCDCHLCHNCOLCOMCOGCRICIVDNKDOMECUEGYSLVGNQETHFINFRAGABGMBGHAGRCGTMGINGNBHNDHKGISLINDIDNIRNIRLISRITAJAMJPNJORKENKORLSOLUXMDGMWIMYSMLIMUSMEXMARMOZNPLNLDNZLNICNERNGANORPAKPANPRYPERPHLPRTROMRWASENSGPZAFESPLKASWECHESYRTZATHATGOTTOTURUGAGBRUSAURYVENZMBZWE.6.7.8.911.1log GDP per worker relative to the US in 2000.6.7.8.91log read more..

##### Page - 27

Introduction to Modern Economic GrowthWestern OffshootsWestern EuropeAfricaLatin AmericaAsia678910log gdp per capita18001850190019502000yearFigure 1.10. The evolution of average GDP per capita in Western Oﬀshoots,Western Europe, Latin America, Asia and Africa, 1820-2000.does not include observations for all countries going back to 1820. Finally, while these datainclude a correction for PPP, this is less reliable than the price comparisons used to constructthe price indices in the Penn World read more..

##### Page - 28

Introduction to Modern Economic Growthof the current book, we will later discuss the relationship between economic crises and growthas well as potential sources of volatility in economic growth.A variety of other evidence suggests that diﬀerences in income per capita were evensmaller once we go back further than 1820. Maddison also has estimates for average incomefor the same groups of countries going back to 1000 AD or even earlier. We extend Figure1.10 using these data; the results are shown read more..

##### Page - 29

Introduction to Modern Economic GrowthWestern OffshootsWestern EuropeAfricaLatinAmericaAsia678910log gdp per capita100012001400160018002000yearFigure 1.11. The evolution of average GDP per capita in Western Oﬀshoots,Western Europe, Latin America, Asia and Africa, 1000-2000.Figure 1.12 shows the evolution of income per capita for United States, Britain, Spain,Brazil, China, India and Ghana. This ﬁgure conﬁrms the patterns shown in Figure 1.10for averages, with the United States Britain and read more..

##### Page - 30

Introduction to Modern Economic GrowthUSABritainSpainGhanaBrazilChinaIndia678910log gdp per capita18001850190019502000yearFigure 1.12. The evolution of income per capita in the United States,Britain, Spain, Brazil, China, India and Ghana, 1820-2000.since the early 1800s. The analysis focused on the “unconditional” distribution of incomeper capita (or per worker). In particular, we looked at whether the income gap betweentwo countries increases or decreases irrespective of these countries’ read more..

##### Page - 31

Introduction to Modern Economic Growthwhere gt,t−1 is the annual growth rate between dates t − 1 and t, yt−1 is output per worker(or income per capita) at date t − 1,and Xt−1 is a vector of variables that the regressionis conditioning on with coeﬃcient vector α. These variables are included because they arepotential determinants of steady state income and/or growth. First note that without co-variates equation (1.1) is quite similar to the relationship shown in Figure 1.9 above. read more..

##### Page - 32

Introduction to Modern Economic GrowthWhile there is no convergence for the entire world, when we look among the “OECD”nations,1 we see a diﬀerent pattern. Figure 1.14 shows that there is a strong negative re-lationship between log GDP per worker in 1960 and the annual growth rate between 1960and 2000 among the OECD countries. What distinguishes this sample from the entire worldsample is the relative homogeneity of the OECD countries, which have much more similarinstitutions, policies and read more..

##### Page - 33

Introduction to Modern Economic Growthincomes across nations), there is some evidence for conditional convergence, meaning thatthe income gap between countries that are similar in observable characteristics appears tonarrow over time. This last observation is relevant both for understanding among whichcountries the economic divergence has occurred and for determining what types of models wemight want to consider for understanding the process of economic growth and diﬀerences ineconomic read more..

##### Page - 34

Introduction to Modern Economic GrowthARGAUSAUTBELBENBOLBRABFACANCHLCHNCOLCRIDNKDOMECUEGYSLVETHFINFRAGHAGRCGTMGINHNDISLINDIRNIRLISRITAJAMJPNJORKENKORLUXMWIMYSMUSMEXMARNLDNZLNICNGANORPAKPANPRYPERPHLPRTZAFESPLKASWECHETWNTHATTOTURUGAGBRUSAURYVENZMBZWE0.02.04.06.08Average growth rate of GDP per capita 1960−2000−.04−.020.02.04Average growth of investment ratio 1960−2000Figure 1.15. The relationship between average growth of GDP per capitaand average growth of investments to GDP ratio, read more..

##### Page - 35

Introduction to Modern Economic GrowthARGAUSAUTBDIBELBENBGDBOLBRABRBCANCHECHLCHNCMRCOGCOLCRIDNKDOMDZAECUEGYESPFINFRAGBRGHAGMBGRCGTMHKGHNDIDNINDIRLIRNISLISRITAJAMJORJPNKENKORLKALSOMEXMLIMOZMUSMWIMYSNERNICNLDNORNPLNZLPAKPANPERPHLPRTPRYRWASENSLVSWESYRTGOTHATTOTURUGAURYUSAVENZAFZMBZWE−.020.02.04.06average growth gdp per capita 1960−20000246810average schooling 1960−2000Figure 1.16world-wide process of economic growth and cross-country diﬀerences. The role of technologyin economic growth will read more..

##### Page - 36

Introduction to Modern Economic Growthand Singapore by looking at the correlates of economic growth–or at the proximate causesof economic growth. We can conclude, as many have done, that rapid capital accumulationhas been a major cause of these growth miracles, and debate the role of human capital andtechnology. We can blame the failure of Nigeria to grow on its inability to accumulate capitaland to improve its technology. These answers are undoubtedly informative for understandingthe read more..

##### Page - 37

Introduction to Modern Economic Growthincentives to businesses to invest and upgrade their technologies. It therefore seems necessaryto look for fundamental causes of economic growth that make contact with these facts andthen provide coherent explanations for the divergent paths of these countries. Jumping aheada little, it will already appear implausible that luck can be the major explanation. There werealready signiﬁcant diﬀerences between South Korea, Singapore and Nigeria at the read more..

##### Page - 38

Introduction to Modern Economic Growthtechnology. Therefore, an understanding of the mechanics of economic growth is essential forevaluating whether candidate fundamental causes of economic growth could indeed play therole that they are sometimes ascribed. Growth empirics plays an equally important role indistinguishing among competing fundamental causes of cross-country income diﬀerences. Itis only by formulating parsimonious models of economic growth and confronting them withdata that we can read more..

##### Page - 39

Introduction to Modern Economic Growth• Some of the other patterns we encountered in this chapter will inform us about thetypes of models that have the most promise in explaining economic growth and cross-country diﬀerences in income. For example, we have seen that cross-country incomediﬀerences can only be accounted for by understanding why some countries havegrown rapidly over the past 200 years, while others have not. Therefore, we needmodels that can explain how some countries can go read more..

##### Page - 40

Introduction to Modern Economic Growthdo they have diﬀerent market structures? Why do some societies adopt policiesthat encourage economic growth while others put up barriers against technologicalchange? These questions are central to a study of economic growth, and can only beanswered by developing systematic models of the political economy of developmentand looking at the historical process of economic growth to generate data that canshed light on these fundamental causes.Our next task is to read more..

##### Page - 41

Introduction to Modern Economic Growthacross countries. Without PPP adjustment, diﬀerences in income per capita across countriescan be computed using the current exchange rate or some fundamental exchange-rate. Thereare many problems with such exchange-rate-based measures. The most important one is thatthey do not make an allowance for the fact that relative prices and even the overall pricelevel diﬀer markedly across countries. PPP-adjustment brings us much closer to diﬀerencesin “real read more..

##### Page - 42

Introduction to Modern Economic Growthto a pattern in which there was already an income gap between Western Europe and Chinaby the end of the 18th century.The term takeoﬀ I used in Section 1.4 is introduced in Walter Rostow’s famous bookStages of Economic Growth (1960) and has a broader connotation than the term “IndustrialRevolution,” which economic historians typically use to refer to the process that startedin Britain at the end of the 18th century (e.g., Ashton, 1968). Mokyr (1990) read more..

##### Page - 43

Introduction to Modern Economic Growthkernel. The speciﬁc details of the kernel estimates do not change the general shape of thedensities. Quah was also the ﬁrst to emphasize the stratiﬁcation in the world income distrib-ution and the possible shift towards a “bi-modal” distribution, which is visible in Figure 1.3.He dubbed this the “Twin Peaks” phenomenon (see also Durlauf and Quah, 1994). Barro(1991) and Barro and Sala-i-Martin (1992) emphasize the presence and importance of read more..

##### Page - 44

CHAPTER 2The Solow Growth ModelThe previous chapter introduced a number of basic facts and posed the main questionsconcerning the sources of economic growth over time and the causes of diﬀerences in economicperformance across countries. These questions are central not only for growth theory but alsofor macroeconomics and social sciences more generally. Our next task is to develop a simpleframework that can help us think about the proximate causes and the mechanics of the processof economic read more..

##### Page - 45

Introduction to Modern Economic GrowthAt ﬁrst, it may appear too simple or too abstract. After all, to do justice to the process ofgrowth or macroeconomic equilibrium, we have to think of many diﬀerent individuals withdiﬀerent tastes, abilities, incomes and roles in society, many diﬀerent sectors and multiplesocial interactions. Instead, the Solow model cuts through these complications by construct-ing a simple one-good economy, with little reference to individual decisions. Therefore, read more..

##### Page - 46

Introduction to Modern Economic Growtheconomy can be represented as if it resulted from the behavior of a single household. We willreturn to what the representative household assumption entails in Chapter 5 and see that itis not totally innocuous. But that is for later.What do we need to know about households in this economy? The answer is: not much.We do not yet endow households with preferences (utility functions). Instead, for now, wesimply assume that they save a constant exogenous fraction read more..

##### Page - 47

Introduction to Modern Economic Growthconsumption and as an input, as “seed”, for the production of more corn tomorrow. Capitalthen corresponds to the amount of corn used as seeds for further production.Technology, on the other hand, has no natural unit. This means that A (t),for us,is a shifter of the production function (2.1). For mathematical convenience, we will oftenrepresent A (t) in terms of a number, but it is useful to bear in mind that, at the end ofthe day, it is a representation read more..

##### Page - 48

Introduction to Modern Economic GrowthAssumption 1. (Continuity, Diﬀerentiability, Positive and Diminishing Mar-ginal Products, and Constant Returns to Scale) The production function F : R3+→ R+is twice continuously diﬀerentiable in K and L,and satisﬁesFK(K, L, A) ≡∂F (K, L, A)∂K> 0,FL(K, L, A) ≡∂F (K, L, A)∂L> 0,FKK(K, L, A) ≡∂2F(K, L, A)∂K2< 0,FLL(K, L, A) ≡∂2F(K, L, A)∂L2< 0.Moreover, F exhibits constant returns to scale in K and L.All of the read more..

##### Page - 49

Introduction to Modern Economic GrowthLinearly homogeneous (constant returns to scale) production functions are particularlyuseful because of the following theorem:Theorem 2.1. (Euler’s Theorem) Suppose that g : RK+2 → R is continuously diﬀer-entiable in x ∈ R and y ∈ R, with partial derivatives denoted by gx and gy and is homogeneousof degree m in x and y.Thenmg (x, y, z)= gx (x, y, z) x + gy (x, y, z) y for all x ∈ R, y ∈ R and z ∈ RK .Moreover, gx (x, y, z) and gy (x, y, z) read more..

##### Page - 50

Introduction to Modern Economic Growthlet the wage rate (or the rental price of labor) at time t be w (t), then the labor marketclearing condition takes the form L (t) ≤¯L (t), w (t) ≥ 0 and¡L(t)−¯L(t)¢w(t)=0.Thecomplementary slackness formulation makes sure that labor market clearing does not happenat a negative wage–or that if labor demand happens to be low enough, employment could bebelow ¯L (t) at zero wage. However, this will not be an issue in most of the models studied inthis read more..

##### Page - 51

Introduction to Modern Economic GrowthThis discussion should already alert you to a central fact: you should think of all of themodels we discuss in this book as general equilibrium economies, where diﬀerent commodi-ties correspond to the same good at diﬀerent dates. Recall from basic general equilibriumtheory that the same good at diﬀerent dates (or in diﬀerent states or localities) is a diﬀerentcommodity. Therefore, in almost all of the models that we will study in this book, there read more..

##### Page - 52

Introduction to Modern Economic Growthstatic program(2.4)maxL(t)≥0,K(t)≥0F [K(t),L(t),A(t)] − w (t) L (t) − R (t) K (t) .When there are irreversible investments or costs of adjustments, as discussed in Section 7.8in Chapter 7, we would need to consider the dynamic optimization problems of ﬁrms. But inthe absence of these features, the production side can be represented as a static maximizationproblem.A couple of additional feature are worth noting:(1) The maximization problem is set up read more..

##### Page - 53

Introduction to Modern Economic GrowthF(K, L, A)K0K0Panel APanel BF(K, L, A)Figure 2.1. Production functions and the marginal product of capital. Theexample in Panel A satisﬁes the Inada conditions in Assumption 2, while theexample in Panel B does not.This result is both important and convenient; it implies that ﬁrms make no proﬁts, soin contrast to the basic general equilibrium theory with strictly convex production sets, theownership of ﬁrms does not need to be speciﬁed. All we need read more..

##### Page - 54

Introduction to Modern Economic Growth2.2.1. Fundamental Law of Motion of the Solow Model.Recall that K depreciatesexponentially at the rate δ, so that the law of motion of the capital stock is given by(2.7)K (t +1) = (1 − δ) K (t)+ I (t) ,where I (t) is investment at time t.From national income accounting for a closed economy, we have that the total amount ofﬁnal goods in the economy must be either consumed or invested, thus(2.8)Y (t)= C (t)+ I (t) ,where C (t) is consumption.1 Using read more..

##### Page - 55

Introduction to Modern Economic Growth2.2.2. Deﬁnition of Equilibrium.The Solow model is a mixture of an old-style Key-nesian model and a modern dynamic macroeconomic model. Households do not optimizewhen it comes to their savings/consumption decisions. Instead, their behavior is capturedby a behavioral rule. Nevertheless, ﬁrms still maximize and factor markets clear. Thus it isuseful to start deﬁning equilibria in the way that is customary in modern dynamic macromodels. Since L (t)= ¯L read more..

##### Page - 56

Introduction to Modern Economic Growthprogress thus A is constant and can be normalized to A =1.2 From Theorem 2.1, we canalso express the marginal products of capital and labor (and thus their rental prices) asR (t)= f0 (k (t)) > 0 andw (t)= f (k (t)) − k (t) f0 (k (t)) > 0.(2.14)The fact that both of these factor prices are positive follows from Assumption 1, whichimposed that the ﬁrst derivatives of F with respect to capital and labor are always positive.Example 2.1. (The read more..

##### Page - 57

Introduction to Modern Economic Growthwhich is equal to the previous expression and thus veriﬁes the form of the marginal productgiven in equation (2.14). Similarly, from (2.14),w (t)= Ak (t)α − αAk (t)−(1−α) × k (t)=(1 − α) AK (t)α L (t)−α ,which veriﬁes the alternative expression for the wage rate in (2.5).Returning to the analysis with the general production function, the per capita represen-tation of the aggregate production function enables us to divide both sides of read more..

##### Page - 58

Introduction to Modern Economic Growthk(t+1)k(t)45°sf(k(t))+(1–δ)k(t)k*k*0Figure 2.2. Determination of the steady-state capital-labor ratio in theSolow model without population growth and technological change.which corresponds to the case where capital is an essential factor, meaning that if K (t)= 0,then output is equal to zero irrespective of the amount of labor and the level of technology.However, if capital is not essential, f (0) will be positive and k =0 will cease to be a steadystate read more..

##### Page - 59

Introduction to Modern Economic Growthk(t+1)k(t)45°k*k*εsf(k(t))+(1−δ)k(t)0Figure 2.3. Unique steady state in the basic Solow model when f (0) = ε> 0.thesetwo termsmakeup f (k∗). Second, Figure 2.4 also emphasizes that the steady-stateequilibrium in the Solow model essentially sets investment, sf (k), equal to the amount ofcapital that needs to be “replenished”, δk. This interpretation will be particularly usefulwhen we incorporate population growth and technological change read more..

##### Page - 60

Introduction to Modern Economic Growthoutputk(t)f(k*)k*δk(t)f(k(t))sf(k*)sf(k(t))consumptioninvestment0Figure 2.4. Investment and consumption in the steady-state equilibrium.(see Theorem A.3 in Appendix Chapter A) there exists k∗ such that (2.17) is satisﬁed. Tosee uniqueness, diﬀerentiate f (k) /k with respect to k,which gives(2.20)∂ [f (k) /k]∂k=f0 (k) k − f (k)k2= −wk2< 0,where the last equality uses (2.14). Since f (k) /k is everywhere (strictly) decreasing, therecan only read more..

##### Page - 61

Introduction to Modern Economic Growthk(t+1)k(t)45°sf(k(t))+(1–δ)k(t)0k(t+1)k(t)45°sf(k(t))+(1–δ)k(t)0k(t+1)k(t)45°sf(k(t))+(1–δ)k(t)0Panel APanel BPanel CFigure 2.5. Examples of nonexistence and nonuniqueness of interior steadystates when Assumptions 1 and 2 are not satisﬁed.simple way, and assume thatf (k)= a ˜f (k) ,where a> 0,so that a is a shift parameter, with greater values corresponding to greaterproductivity of factors. This type of productivity is referred to as read more..

##### Page - 62

Introduction to Modern Economic Growth2.3 are intuitive, and start giving us a sense of some of the potential determinants of thecapital-labor ratios and output levels across countries.The same comparative statics with respect to a and δ immediately apply to c∗ as well.However, it is straightforward to see that c∗ will not be monotone in the saving rate (think, forexample, of the extreme case where s =1), and in fact, there will exist a speciﬁc level of thesaving rate, sgold, referred to read more..

##### Page - 63

Introduction to Modern Economic Growthconsumptionsavings rate(1–s)f(k*gold)s*gold10Figure 2.6. The “golden rule” level of savings rate, which maximizes steady-state consumption.In other words, there exists a unique saving rate, sgold, and also unique correspondingcapital-labor ratio, k∗gold, which maximize the level of steady-state consumption. When theeconomy is below k∗gold, the higher saving rate will increase consumption, whereas when theeconomy is above k∗gold, steady-state read more..

##### Page - 64

Introduction to Modern Economic Growthis an important point to bear in mind, especially since the term “equilibrium” is used diﬀer-ently in economics than in physical sciences. Typically, in engineering and physical sciences,an equilibrium refers to a point of rest of a dynamical system, thus to what we have so farreferred to as the steady state equilibrium. One may then be tempted to say that the systemis in “disequilibrium” when it is away from the steady state. However, in read more..

##### Page - 65

Introduction to Modern Economic GrowthThe next theorem provides the main results on the stability properties of systems of lineardiﬀerence equations. The Chapter B in the Appendix contains an overview of eigenvaluesand some other properties of diﬀerential equations, which are also relevant for diﬀerenceequations. Deﬁnitions and certain elementary results the matrix of partial derivatives (theJacobian), which we will use below, are provided in Appendix Chapter A. The followingtheorems are read more..

##### Page - 66

Introduction to Modern Economic Growthx∗, is locally asymptotically stable if |g0 (x∗)| < 1. Moreover, if |g0 (x)| < 1 for all x ∈ R,thenx∗ is globally asymptotically stable.Proof. The ﬁrst part follows immediately from Theorem 2.2. The local stability of g inthe second part follows from Theorem 2.3. Global stability follows since|x(t +1) − x∗| = |g(x(t)) − g(x∗)|=¯¯¯¯¯Zx(t)x∗g0(x)dx¯¯¯¯¯< |x(t) − x∗|,where the last inequality follows from the read more..

##### Page - 67

Introduction to Modern Economic Growthwhere the ﬁrst line follows by subtracting (2.27) from (2.26), the second line uses the funda-mental theorem of calculus, and the third line follows from the observation that g0 (k) > 0for all k. Next, (2.16) also impliesk (t +1) − k (t)k (t)= sf (k (t))k (t)− δ>sf (k∗)k∗ − δ=0,where the second line uses the fact that f (k) /k is decreasing in k (from (2.28) above) andthe lastlineusesthe deﬁnition of k∗. These two arguments together read more..

##### Page - 68

Introduction to Modern Economic Growth45°k*k*k(0)k’(0)0k(t+1)k(t)Figure 2.7. Transitional dynamics in the basic Solow model.an end. We will see in Section 2.6 that the Solow model can incorporate economic growthby allowing exogenous technological change. Before doing this, it is useful to look at therelationship between the discrete-time and continuous-time formulations.2.4. The Solow Model in Continuous Time2.4.1. From Diﬀerence to Diﬀerential Equations.Recall that the time periodscould read more..

##### Page - 69

Introduction to Modern Economic GrowthThis equation states that between time t and t +1, the absolute growth in x is given byg (x (t)). Let us now consider the following approximationx (t + ∆t) − x (t) ' ∆t · g (x (t)) ,for any ∆t ∈ [0, 1].When ∆t =0, this equation is just an identity. When ∆t =1,it gives(2.29). In-between it is a linear approximation, which should not be too bad if the distancebetween t and t +1 is not very large, so that g (x) ' g (x (t)) for all x ∈ [x (t) ,x read more..

##### Page - 70

Introduction to Modern Economic Growthwhich implies that˙k (t)k (t)=˙K (t)K (t)−˙L (t)L (t),=˙K (t)K (t)− n.From the limiting argument leading to equation (2.30) in the previous subsection, the law ofmotion of the capital stock is given by˙K (t)= sF [K (t) ,L (t) ,A(t)] − δK (t) .Now using the deﬁnition of k (t) as the capital-labor ratio and the constant returns to scaleproperties of the production function, we obtain the fundamental law of motion of the Solowmodel in continuous read more..

##### Page - 71

Introduction to Modern Economic GrowthWe still have a fraction δ of the capital stock depreciating. In addition, the capital stock ofthe economy also has to increase as population grows in order to maintain the capital-laborratio constant. The amount of capital that needs to be replenished is therefore (n + δ) k.outputk(t)f(k*)k*f(k(t))sf(k*)sf(k(t))consumptioninvestment0(δ+n)k(t)Figure 2.8. Investment and consumption in the state-state equilibrium withpopulation growth.Therefore we read more..

##### Page - 72

Introduction to Modern Economic GrowthProposition 2.8. Suppose Assumptions 1 and 2 hold and f (k)= a ˜f (k). Denote thesteady-state equilibrium level of the capital-labor ratio by k∗ (a, s, δ, n) and the steady-statelevel of output by y∗ (a, s, δ, n) when the underlying parameters are given by a, s and δ.Thenwe have∂k∗ (a, s, δ, n)∂a> 0,∂k∗ (a, s, δ, n)∂s> 0,∂k∗ (a, s, δ, n)∂δ< 0 and∂k∗ (a, s, δ, n)∂n< 0∂y∗ (a, s, δ, n)∂a> 0,∂y∗ (a, read more..

##### Page - 73

Introduction to Modern Economic Growthexists an open neighborhood of x∗, B(x∗) ⊂ Rn such that starting from any x(0) ∈ B(x∗),we have x(t) → x∗.Once again an immediate corollary is:Corollary 2.2. Let x (t) ∈ R, then the steady state of the linear diﬀerence equation˙x (t)= ax (t) is globally asymptotically stable (in the sense that x (t) → 0)if a< 0.Let g : R→ R be continuous and diﬀerentiable at x∗ where g (x∗)= 0. Then, the steadystate of the nonlinear read more..

##### Page - 74

Introduction to Modern Economic Growthk(t)f(k(t))k(t)k(t)s–(δ+g+n)k*0k(t)Figure 2.9. Dynamics of the capital-labor ratio in the basic Solow model.As noted above, the Cobb-Douglas production function is special, mainly because it has anelasticity of substitution between capital and labor equal to 1. Recall that for a homotheticproduction function F (K, L), the elasticity of substitution is deﬁned by(2.36)σ ≡−∙∂ln(FK/FL)∂ ln (K/L)¸−1,where FK and FL denote the marginal products read more..

##### Page - 75

Introduction to Modern Economic GrowthSimilarly, the share of labor is αL (t)=1 − α. The reason for this is that with an elasticityof substitution equal to 1, as capital increases, its marginal product decreases proportionally,leaving the capital share (the amount of capital times its marginal product) constant.Recall that with the Cobb-Douglas technology, the per capita production function takesthe form f (k)= Akα, so the steady state is given again from (2.33) (with population growthat read more..

##### Page - 76

Introduction to Modern Economic GrowthExample 2.3. (The Constant Elasticity of Substitution Production Function)Theprevious example introduced the Cobb-Douglas production function, which featured an elas-ticity of substitution equal to 1. The Cobb-Douglas production function is a special case ofthe constant elasticity of substitution (CES) production function, ﬁrst introduced by Arrow,Chenery, Minhas and Solow (1961). This production function imposes a constant elasticity,σ, not necessarily read more..

##### Page - 77

Introduction to Modern Economic Growthprovides an alternative derivation of this production function along the lines of the originalarticle by Arrow, Chenery, Minhas and Solow (1961).2.5.1. A First Look at Sustained Growth.Can the Solow model generate sustainedgrowth without technological progress? The answer is yes, but only if we relax some of theassumptions we have imposed so far.The Cobb-Douglas example above already showed that when α is close to 1, adjustment ofthe capital-labor ratio read more..

##### Page - 78

Introduction to Modern Economic GrowthThis proposition not only establishes the possibility of endogenous growth, but also showsthat in this simplest form, there are no transitional dynamics. The economy always growsat a constant rate sA − δ − n, irrespective of what level of capital-labor ratio it starts from.Figure 2.10 shows this equilibrium diagrammatically.45°(A−δ−n)k(t)k(t+1)k(0)0k(t)Figure 2.10. Sustained growth with the linear AK technology with sA −δ−n> 0.Does the AK read more..

##### Page - 79

Introduction to Modern Economic Growth2.6. Solow Model with Technological Progress2.6.1. Balanced Growth.The models analyzed so far did not feature technologicalprogress. We now introduce changes in A (t) to capture improvements in the technologicalknow-how of the economy. There is little doubt that today human societies know how toproduce many more goods than before and they can do so much more eﬃciently than inthe past. In other words, the productive knowledge of the human society has read more..

##### Page - 80

Introduction to Modern Economic Growth0%10%20%30%40%50%60%70%80%90%100%19291934193919441949195419591964196919741979198419891994Labor and capital share in total value addedLaborCapitalFigure 2.11. Capital and Labor Share in the U.S. GDP.For us, the most important reason to start with balanced growth is that it is much easierto handle than non-balanced growth, since the equations describing the law of motion of theeconomy can be represented by diﬀerence or diﬀerential equations with read more..

##### Page - 81

Introduction to Modern Economic GrowthK0LYYK0LYY0LYYKFigure 2.12. Hicks-neutral, Solow-neutral and Harrod-neutral shifts in isoquants.2.6.2. Types of Neutral Technological Progress.What are some convenient specialforms of the general production function F [K (t) ,L (t) ,A (t)]?First we could haveF [K (t) ,L (t) ,A (t)] = A (t) ˜F [K (t) ,L (t)] ,for some constant returns to scale function ˜F . This functional form implies that the tech-nology term A (t) is simply a multiplicative constant in read more..

##### Page - 82

Introduction to Modern Economic GrowthThis functional form implies that an increase in technology A (t) increases output as if theeconomy had more labor. Equivalently, the slope of the isoquants are constant along rayswith constant capital-output ratio, and the approximate shape of the isoquants are plottedin the third panel of Figure 2.12.Of course, in practice technological change can be a mixture of these, so we could have avector valued index of technology A(t)=(AH (t) ,AK (t) ,AL (t)) and a read more..

##### Page - 83

Introduction to Modern Economic Growthwith ˜A (t) representing technology at time t and aggregate resource constraint˙K (t)= Y (t) − C (t) − δK (t) .Suppose that there is a constant growth rate of population, i.e., L (t)=exp (nt) L (0) andthat there exists an asymptotic path where output, capital and consumption grow at constantrates, i.e., ˙Y (t) /Y (t)= gY , ˙K (t) /K (t)= gK and ˙C (t) /C (t)= gC.Suppose ﬁnally thatgK + δ> 0. Then, along the growth path we have(1) gY = gK = read more..

##### Page - 84

Introduction to Modern Economic GrowthMoreover,thisequationistrue for t irrespective of the initial τ ,thusY (t)=˜F [K (t) , exp ((t − τ )(gY − n)) L (t)] ,=˜F [K (t) ,A (t) L (t)] ,with˙A (t)A (t)= gY − nestablishing the second part of the theorem.¤A remarkable feature of this result is that it was stated and proved without any referenceto equilibrium behavior or market clearing. Also, contrary to Uzawa’s original theorem, itis not stated for a balanced growth path (meaning an read more..

##### Page - 85

Introduction to Modern Economic Growththe previous paragraph also works to make technology endogenously more labor-augmentingthan capital-augmenting.Notice also that this proposition does not state that technological change has to belabor-augmenting all the time. Instead, it requires that technological change has to be labor-augmenting asymptotically, i.e., along the balanced growth path. This is exactly the patternthat certain classes of endogenous-technology models will generate.Finally, it is read more..

##### Page - 86

Introduction to Modern Economic Growth2.7 and that ˜F exhibits constant returns to scale and thus its derivative is homogeneous ofdegree 0.¤This corollary, together with Theorem 2.7, implies that any asymptotic path with constantgrowth rates for output, capital and consumption must be a balanced growth path and canonly be generated from an aggregate production function asymptotically featuring Harrod-neutral technological change.In light of this corollary, we can provide further intuition for read more..

##### Page - 87

Introduction to Modern Economic GrowthDiﬀerentiating this expression with respect to time, we obtain(2.44)˙k (t)k (t)=˙K (t)K (t)− g − n.The quantity of output per unit of eﬀective labor can be written asˆy (t) ≡Y (t)A (t) L (t)= F∙ K (t)A (t) L (t), 1¸≡ f (k (t)) .Income per capita is y (t) ≡ Y (t) /L (t), i.e.,y (t)= A (t)ˆy (t)(2.45)= A (t) f (k (t)) .It should be clear that if ˆy (t) is constant, income per capita, y (t), will grow over time,since A (t) is growing. This read more..

##### Page - 88

Introduction to Modern Economic GrowthProposition 2.11. Consider the basic Solow growth model in continuous time, withHarrod-neutral technological progress at the rate g and population growth at the rate n.Sup-pose that Assumptions 1 and 2 hold, and deﬁne the eﬀective capital-labor ratio as in (2.43).Then there exists a unique steady state (balanced growth path) equilibrium where the eﬀectivecapital-labor ratio is equal to k∗ ∈ (0, ∞) and is given by(2.47)f (k∗)k∗=δ + g + ns.Per read more..

##### Page - 89

Introduction to Modern Economic GrowthProposition 2.13. Suppose that Assumptions 1 and 2 hold, then the Solow growth modelwith Harrod-neutral technological progress and population growth in continuous time is as-ymptotically stable, i.e., starting from any k (0) > 0,the eﬀective capital-labor ratio convergesto a steady-state value k∗ (k (t) → k∗).Proof. See Exercise 2.19.¤Therefore, the comparative statics and dynamics are very similar to the model withouttechnological progress. The read more..

##### Page - 90

Introduction to Modern Economic Growthsaving rate is realized, the capital stock remains unchanged (since it is a state variable).After this point, it follows the dashed arrows on the horizontal axis.0k(t)f(k(t))k(t)k(t)sk*k(t)k**f(k(t))k(t)s’–(δ+g+n)–(δ+g+n)Figure 2.13. Dynamics following an increase in the savings rate from s tos0. The solid arrows show the dynamics for the initial steady state, while thedashed arrows show the dynamics for the new steady state.The comparative dynamics read more..

##### Page - 91

Introduction to Modern Economic Growth2.8. Taking StockWhat have we learned from the Solow model? At some level, a lot. We now have asimple and tractable framework, which allows us to discuss capital accumulation and theimplications of technological progress. As we will see in the next chapter, this framework isalready quite useful in helping us think about the data.However, at some other level, we have learned relatively little. The questions that Chapter1 posed are related to why some read more..

##### Page - 92

Introduction to Modern Economic Growth2.9. References and LiteratureThe model analyzed in this chapter was ﬁrst developed in Solow (1956) and Swan (1956).Solow (1970) gives a nice and accessible treatment, with historical references. Barro andSala-i-Martin’s (2004, Chapter 1) textbook presents a more up-to-date treatment of thebasic Solow model at the graduate level, while Jones (1998, Chapter 2) presents an excellentundergraduate treatment.The treatment in the chapter made frequent read more..

##### Page - 93

Introduction to Modern Economic Growthoptimality property, since it is not derived from well-deﬁned preferences. Optimal savingpolicies will be discussed in greater detail in Chapter 8.The balanced growth facts were ﬁrst noted by Kaldor (1963). Figure 2.11 uses data fromPiketty and Saez (2004). Homer and Sylla (1991) discuss the history of interest rates overmany centuries and across diﬀerent societies; they show that there is no notable upward ordownward trend in interest rate. read more..

##### Page - 94

Introduction to Modern Economic GrowthExercise 2.5. Let us introduce government spending in the basic Solow model. Considerthe basic model without technological change. In particular, suppose that (2.8) takes theformY (t)= C (t)+ I (t)+ G (t) ,with G (t) denoting government spending at time t. Imagine that government spending isgiven by G (t)= σY (t).(1) Discuss how the relationship between income and consumption should be changed?Is it reasonable to assume that C (t)= sY (t)?(2) Suppose that read more..

##### Page - 95

Introduction to Modern Economic Growth(1) First suppose that there is no population growth. Find the steady-state capital-laborratio and output level. Prove that the steady state is unique and globally stable.(2) Now suppose that there is population growth at the rate n,i.e., ˙L/L = n.Whathappens to the capital-labor ratio and output level as t →∞? What happens toreturn to land and the wage rate as t →∞?(3) Would you expect the population growth rate n or the saving rate s to change read more..

##### Page - 96

Introduction to Modern Economic GrowthSolow model given by f (k∗) /k∗ = δ/s. Characterize the dynamic equilibrium pathof this economy starting withsome amount of physical capital K (0) > 0.(2) Next consider a diﬀerent form of labor market imperfection, whereby workers receivea fraction β> 0 of output in each ﬁrm as their wage income. Characterize a dynamicequilibrium path in this case. [Hint: recall that the saving rate is still equal to s].Exercise 2.14. Consider the read more..

##### Page - 97

Introduction to Modern Economic Growth(1) Determine conditions under which this production function satisﬁes Assumptions 1and 2.(2) Characterize the unique steady-state equilibrium when Assumptions 1 and 2 hold.(3) Now suppose that σ is suﬃciently high so that Assumption 2 does not hold. Showthat in this case equilibrium behavior can be similar to that in Exercise 2.15 withsustained growth in the long run. Interpret this result.(4) Now suppose that σ → 0, so that the production function read more..

##### Page - 98

Introduction to Modern Economic Growthtime and suppose that A (t)= A, so that there is no technological progress of the usualkind. However, assume that the relationship between investment and capital accumulationmodiﬁed toK (t +1) = (1 − δ) K (t)+ q (t) I (t) ,where [q (t)]∞t=0 is an exogenously given time-varying process. Intuitively, when q (t) is high,the same investment expenditure translates into a greater increase in the capital stock. There-fore, we can think of q (t) as the read more..

##### Page - 99

##### Page - 100

CHAPTER 3The Solow Model and the DataIn this chapter, we will see how the Solow model or its simple extensions can be used tointerpret both economic growth over time and cross-country output diﬀerences. Our focus ison proximate causes of economic growth, that is, the factors such as investment or capitalaccumulation highlighted by the basic Solow model, as well as technology and human capitaldiﬀerences. What lies underneath these proximate causes is the topic of the next chapter.There are read more..

##### Page - 101

Introduction to Modern Economic Growthits arguments by FA, FK and FL,thisyields(3.1)˙YY=FAAY˙AA+FKKY˙KK+FLLY˙LL.Now denote the growth rates of output, capital stock and labor by g ≡˙Y/Y , gK ≡˙K/Kand gL≡˙L/L, and also deﬁnex ≡FAAY˙AAas the contribution of technology to growth. Next, recall from the previous chapter that withcompetitive factor markets, we have w = FL and R = FK (equations (2.5) and (2.6)) anddeﬁne the factor shares as αK ≡ RK/Y and αL ≡ wL/Y . Putting read more..

##### Page - 102

Introduction to Modern Economic Growththe time horizon in question, factor shares can change; should we use beginning-of-period orend-of-period values of αK and αL? It can be shown that the use of either beginning-of-periodor end-of-period values might lead to seriously biased estimates of the contribution of TFP tooutput growth, ˆx. This is particularly likely when the distance between the two time periodsis large(seeExercise 3.1). Thebestway of avoiding such biases is to use as read more..

##### Page - 103

Introduction to Modern Economic Growthof ˆx. And in fact there are good reasons for suspecting that Solow’s estimates and eventhe higher-quality estimates that came later may be mismeasuring the growth of inputs. Themost obvious reason for this is that what matters is not labor hours, but eﬀective labor hours,so it is important–though diﬃcult–to make adjustments for changes in the human capitalof workers. We will discuss issues related to human capital in Section 3.3 below and then read more..

##### Page - 104

Introduction to Modern Economic GrowthwhereA (t)isthelabor-augmenting(Harrod-neutral)technologyterm,k (t)≡K (t) / (A (t) L (t)) is the eﬀective capital labor ratio and f (·) is the per capita productionfunction. Equation (3.6) follows from the constant technological progress and constant pop-ulation growth assumptions, i.e., ˙A (t) /A (t)= g and ˙L (t) /L (t)= n.Now diﬀerentiating(3.5) with respect to time and dividing both sides by y (t),weobtain(3.7)˙y (t)y (t)= g + εf (k (t))˙k read more..

##### Page - 105

Introduction to Modern Economic GrowthCombining this with the previous equation, we obtain the following “convergence equation”:(3.8)˙y (t)y (t)' g − (1 − εf (k∗)) (δ + g + n)(log y (t) − log y∗ (t)) .Equation (3.8) makes it clear that, in the Solow model, there are two sources of growthin output per capita: the ﬁrst is g, the rate of technological progress, and the second is“convergence”. This latter source of growth results from the negative impact of the gapbetween the read more..

##### Page - 106

Introduction to Modern Economic Growthbe around 0.054 (' 0.67 × 0.08). This is a very rapid rate of convergence and would implythat income gaps between two similar countries that have the same technology, the samedepreciation rate and the same rate of population growth should narrow rather quickly. Forexample, it can be computed that with these numbers, the gap of income between two similarcountries should be halved in little more than 10 years (see Exercise 3.4). This is clearly atodds with read more..

##### Page - 107

Introduction to Modern Economic Growthcharacteristics, such as institutional factors, human capital (see next section), or even theinvestment rate.If the true equation is (3.10), in the sense that the Solow model applies but certaindeterminants of economic growth diﬀer across countries, equation (3.9) would not be a goodﬁt to the data. Put diﬀerently, there is no guarantee that the estimates of b1 resulting fromthis equation will be negative. In particular, it is natural to expect that read more..

##### Page - 108

Introduction to Modern Economic GrowthNevertheless, there are several problematic features with regressions of this form. Theseinclude:(1)Most, if notall, ofthe variablesin Xi,t as well as log yi,t−1,are econometricallyendogenous in the sense that they are jointly determined with the rate of economicgrowth between dates t−1 and t. For example, the same factors that make the coun-try relatively poor in 1950, thus reducing log yi,t−1, should also aﬀect its growth rateafter 1950. Or the read more..

##### Page - 109

Introduction to Modern Economic Growthwe can end up with a negative estimate of b1, even when there is no conditionalconvergence.(3) The interpretation of regression equations like (3.11) is not always straightforwardeither. Many of the regressions used in the literature include the investment rate aspart of the vector Xi,t (and all of them include the schooling rate). However, in theSolow model, diﬀerences in investment rates are the channel via which convergencewill take place–in the sense read more..

##### Page - 110

Introduction to Modern Economic Growth(5) Finally, the motivating equation for the growth regression, (3.8), is derived for aclosed Solow economy. When we look at cross-country income diﬀerences or growthexperiences, the use of this equation imposes the assumption that “each country isan island”. In other words, we are representing the world as a collection of non-interacting closed economies. In practice, countries trade goods, exchange ideas andborrow and lend in international ﬁnancial read more..

##### Page - 111

Introduction to Modern Economic Growthall omitted variable biases and econometric endogeneity problems. Simultaneity bias oftenresults from time-varying inﬂuences, which cannot be removed by including ﬁxed eﬀects.Moreover, to the extent that some of the variables in the vector Xi,t are slowly-varyingthemselves, the inclusion of country ﬁxed eﬀects will make it diﬃcult to uncover the statisticalrelationship between these variables and income per capita.This discussion highlights that read more..

##### Page - 112

Introduction to Modern Economic Growthenable us to embed all three of the main proximate sources of income diﬀerences; physicalcapital, human capital and technology.For the purposes of this section, let us focus on the continuous time economy and supposethat the aggregate production function of the economy is given by a variant of equation (2.1):(3.13)Y = F (K, H, AL) ,where H denotes “human capital”. How this is measured in the data will be discussed below.As usual, we assume throughout read more..

##### Page - 113

Introduction to Modern Economic Growthand using the constant returns to scale feature in Assumption 10, output per eﬀective unit oflabor can be written asˆy (t) ≡Y (t)A (t) L (t)= Fµ K (t)A (t) L (t),H (t)A (t) L (t), 1¶≡ f (k (t) ,h (t)) .With the same steps as in Chapter 2, the law of motion of k (t) and h (t) can then be obtainedas:˙k (t)= skf (k (t) ,h (t)) − (δk + g + n) k (t) ,˙h (t)= shf (k (t) ,h (t)) − (δh + g + n) h (t) .A steady-state equilibrium is now deﬁned not read more..

##### Page - 114

Introduction to Modern Economic Growthhk0k=0h=0k*h*Figure 3.1. Steady-state equilibrium in the Solow model with human capital.where fk≡ ∂f /∂k. Rewriting (3.14), we have skf (k∗,h∗) /k∗ −(δk + g + n)=0.Now recallthat since f is strictly concave in k in view of Assumption 10 and f (0,h∗) ≥ 0,wehavef (k∗,h∗) >fk (k∗,h∗) k∗ + f (0,h∗)>fk (k∗,h∗) k∗.Therefore, (δk + g + n) − skfk (k∗,h∗) > 0, and (3.16) is strictly positive.Similarly, deﬁning fh read more..

##### Page - 115

Introduction to Modern Economic GrowthNext, we prove that (3.16) is steeper than (3.17) whenever (3.14) and (3.15) hold, so thatcan it most be one intersection. First, observe thatdhdk¯¯¯¯˙h=0 <dhdk¯¯¯¯˙k=0mshfk (k∗,h∗)(δh + g + n) − shfh (k∗,h∗)<(δk + g + n) − skfk (k∗,h∗)skfh (k∗,h∗)mskshfkfh <skshfkfh +(δh + g + n)(δk + g + n)−(δh + g + n) skfk − (δk + g + n) shfh.Now using (3.14) and (3.15) and substituting for (δk + g + n)= skf (k∗,h∗) read more..

##### Page - 116

Introduction to Modern Economic Growththe two curves representing the loci for ˙k =0 and ˙h =0, respectively, (3.14) and (3.15).When we are to the right of the (3.14) curve, there is too much physical capital relative tothe amount of labor and human capital, and consequently, ˙k< 0. When we are to its left,we are in the converse situation and ˙k> 0. Similarly, when we are above the (3.15) curve,there is too little human capital relative to the amount of labor and physical capital, read more..

##### Page - 117

Introduction to Modern Economic Growthwith thesamedeﬁnition of ˆy (t), k (t) and h (t) as above. Using this functional form, (3.14)and (3.15) give the unique steady-state equilibrium ask∗ =Ãµ skn + g + δk¶1−βµ shn + g + δh¶β! 11−α−β(3.19)h∗ =Ãµ skn + g + δk¶αµ shn + g + δh¶1−α! 11−α−β,which shows that higher saving rate in physical capital not only increases k∗, but also h∗.The same applies for a higher saving rate in human capital. This reﬂects the read more..

##### Page - 118

Introduction to Modern Economic GrowthThis production function nests the basic Solow model without human capital when α =0.First, assume that countries diﬀer in terms of their saving rates, sk,j and sh,j, populationgrowth rates, nj, and technology growth rates ˙Aj (t) /Aj (t)= gj.As usual, deﬁne kj ≡Kj/AjLj and hj ≡ Hj/AjLj.Since our main interest here is cross-country income diﬀerences, rather than studying thedynamics of a particular country over time, let us focus on a world in read more..

##### Page - 119

Introduction to Modern Economic GrowthSince technological progress is taken as exogenous in the Solow model, it is, in manyways, more appropriate for the Solow model to assume a common rate of technical progress.Motivated by this, Mankiw, Romer and Weil (1992) make the following assumption:Common technology advances assumption:Aj (t)= ¯Aj exp (gt) .That is, countries diﬀer according to their technology level, in particular, according totheir initial level of technology, ¯Aj, but they share read more..

##### Page - 120

Introduction to Modern Economic GrowthUnder the orthogonal technology assumption, ln ¯Aj, which is part of the error term,is orthogonal to the key right-hand side variables and equation (3.22) can be estimatedconsistently.3.4.2. Mankiw, Romer and Weil Estimation Results.Mankiw, Romer and Weilﬁrst estimate equation (3.22) without the human capital term (i.e., imposing α =0)for thecross-sectional sample of non-oil producing countries. In particular, their estimating equationis:ln y∗j = read more..

##### Page - 121

Introduction to Modern Economic GrowthTable 3.1Estimates of the Basic Solow ModelMRW Updated data198519852000ln(sk)1.421.011.22(.14)(.11)(.13)ln(n + g + δ)-1.97-1.12-1.31(.56)(.55)(.36)Adj R2.59.49.49Implied α.59.50.55No. of observations9898107The most natural reason for the high implied values of the parameter α in Table 3.1is that εj is correlated with ln (sk,j), either because the orthogonal technology assumptionis not a good approximation to reality or because there are also human read more..

##### Page - 122

Introduction to Modern Economic GrowthTable 3.2Estimates of the Augmented Solow ModelMRWUpdated data198519852000ln(sk).69.65.96(.13)(.11)(.13)ln(n + g + δ)-1.73-1.02-1.06(.41)(.45)(.33)ln(sh).66.47.70(.07)(.07)(.13)Adj R2.78.65.60Implied α.30.31.36Implied β.28.22.26No. of observations9898107To the extent that these regression results are reliable, they give a big boost to theaugmented Solow model. In particular, the estimate of Adjusted R2suggests that over (orclose to) three quarters of read more..

##### Page - 123

Introduction to Modern Economic Growthtwo reasons. The ﬁrst is a version of the omitted variable bias problem; as we will discussin detail later in the book, technology diﬀerences are also outcomes of investment decisions.Thus societies with high levels of ¯Aj will be those that have invested more in technology forvarious reasons; it is then natural to expect the same reasons to induce greater investment inphysical and human capital as well. Second, even ignoring the omitted variable bias read more..

##### Page - 124

Introduction to Modern Economic Growthto education, measuring the proportional increase in earnings resulting from one more yearof schooling. The microeconometrics literature suggests that equation (3.24) provides a goodapproximation to the data and estimates φ to be between 0.06 and 0.10, implying that aworker with one more year of schooling earns about 6 to 10 percent more than a comparableworker with one less year of schooling. If labor markets are competitive, or at the very least,if wages read more..

##### Page - 125

Introduction to Modern Economic GrowthThis implies that all ﬁrms ought to function at the same physical to human capital ratio, andconsequently, all workers, irrespective of their level of schooling, ought to work at the samephysical to human capital ratio. Another direct implication of competitive labor markets isthat in country j,wages perunitofhuman capitalwillbeequal towj =(1 − α) αα/(1−α)AjR−α/(1−α)j.Consequently, a worker with human capital hi will receive a wage income of read more..

##### Page - 126

Introduction to Modern Economic Growththe coeﬃcient α is, in turn, most likely related to the possible correlation between the errorterm εj andthe keyright-handsideregressors in equation (3.23).To recap, the comparison between the parameter estimates from the regression of (3.23)and the microeconometric Mincerian rates of return estimates to schooling imply that cross-country regression analysis is not necessarily giving us an accurate picture of the productivitydiﬀerences and thus the read more..

##### Page - 127

Introduction to Modern Economic Growthforces, but in the quality of schooling and the amount of post-schooling human capital. Third,the rate of return to schooling may vary systematically across countries. As we will see ingreater detail below, the rate of return to schooling may be lower in countries with a greaterabundance of human capital. It is possible to deal with each of these problems to some extentby constructing better estimates of the stocks of human capital.Following Hall and Jones, read more..

##### Page - 128

Introduction to Modern Economic GrowthK1/3US (AUSHUS)2/3. Throughout, time indices are dropped. In the Hall and Jones exercise,all values refer to 1985.Once a series for ˆYj has been constructed, it can be compared to the actual output series.The gap between the two series represents the contribution of technology. Alternatively, wecouldexplicitlybackout country-speciﬁc technology terms (relative to the United States) asAjAUS=µ YjYUS¶3/2µKUSKj¶1/2µHUSHj¶.Figures 3.3-3.4 show the results read more..

##### Page - 129

Introduction to Modern Economic GrowthARGAUSAUTBELBENBGDBOLBRABRBBWACAFCANCHECHLCHNCMRCOGCOLCRICYPDNKDOMECUEGYESPFINFJIFRAGBRGHAGMBGRCGTMGUYHKGHNDHUNIDNINDIRLIRNISLISRITAJAMJORJPNKENKORLKALSOMEXMLIMOZMUSMWIMYSNERNICNLDNORNPLNZLPAKPANPERPHLPNGPRTPRYRWASENSGPSLESLVSWESYRTGOTHATTOTUNTURUGAURYUSAVENZAFZMBZWE0.511.5Predicted relative technology7891011log gdp per worker read more..

##### Page - 130

Introduction to Modern Economic GrowthAs a byproduct of investigating this question, we will see that the calibration approach isin fact a close cousin of the growth-accounting exercise (and for this reason, it is sometimesreferred to as “levels accounting”).Recall equation (3.4), where we constructed TFP estimates from a general constant re-turns to scale production function (under competitive labor markets) by using average factorshares. Now instead imagine that the production function read more..

##### Page - 131

Introduction to Modern Economic GrowthTo sum up, the approach of calibrating productivity diﬀerences across countries is a usefulalternative to the regression analysis, but has to rely on a range of stringent assumptions onthe form of the production function and can also lead to biased estimates of technologydiﬀerences when factors are mismeasured.3.6. Estimating Productivity DiﬀerencesIn the previous section, productivity/technology diﬀerences are obtained as “residuals”from a read more..

##### Page - 132

Introduction to Modern Economic Growth3.6.2. Learning from International Trade*.We will discuss models of growth intrade in Chapter 19. Even without a detailed discussion of international trade theory, wecan use data from international trade ﬂows and some simple principles of international tradetheory to obtain another way of estimating productivity diﬀerences across countries.Letusfollowanimportant paper byTreﬂer (1993), which uses an augmented version of thestandard Heckscher-Ohlin read more..

##### Page - 133

Introduction to Modern Economic Growthcalculated (suﬃce it to say that as with all things empirical, the devil is in the detail andthese calculations are far from straightforward and require a range of assumptions). Then,the absence of trading frictions across countries and identical homothetic preferences implythatXKj= Akj Kj − csjNXi=1Aki Ki(3.29)XHj= Ahj Hj − csjNXi=1Ahi Hiwhere csj is the share of country j in world consumption (the value of this country’s consump-tion divided by read more..

##### Page - 134

Introduction to Modern Economic GrowthAUTBGDBELCANCOLDNKFINFRAGRCHKGIDNIRLISRITAJPNNLDNZLNORPAKPANPRTSGPESPLKASWECHETHATTOGBRUSAURYDEUYUG.2.4.6.81Capital productivity0.2.4.6.81Labor productivityFigure 3.5. Comparison of labor-productivity and capital-productivity dif-ferences across countries.very similar, so there appears to be some validity to this approach. Given this validation, wecan presume that there is some information in the numbers that Treﬂer obtains.Figure 3.5 shows Treﬂer’s read more..

##### Page - 135

Introduction to Modern Economic GrowthAUTBGDBELCANCOLDNKFINFRAGRCHKGIDNIRLISRITAJPNNLDNZLNORPAKPANPRTSGPESPLKASWECHETHAURY0.511.5Calibrated productivity differences 19880.2.4.6.81Estimated labor−productivity differencesFigure 3.6. Comparison of the labor productivity estimates from the Treﬂerapproach with the calibrated productivity diﬀerences from the Hall-Jones ap-proach.Figure 3.7 shows that the relationship between the calibrated productivity diﬀerences andthe capital-productivity read more..

##### Page - 136

Introduction to Modern Economic GrowthAUTBGDBELCANCOLDNKFINFRAGRCHKGIDNIRLISRITAJPNNLDNZLNORPAKPANPRTSGPESPLKASWECHETHAURY0.511.5Calibrated productivity differences 19880.2.4.6.81Estimated capital−productivity differencesFigure 3.7. Comparison of the capital productivity estimates from the Tre-ﬂer approach with the calibrated productivity diﬀerences from the Hall-Jonesapproach.caution. Nevertheless, this approach is important in showing how diﬀerent data and addi-tional theory can be read more..

##### Page - 137

Introduction to Modern Economic GrowthMoreover, each of these diﬀerent methods gives us some idea about the sources of economicgrowth over time and of income diﬀerences across countries.On the negative side, however, no single approach is entirely convincing. Each relies ona range of stringent auxiliary assumptions. Consequently, no ﬁrm conclusions can be drawn.The simplest applications of the Solow accounting framework suggest that technology isthe main source of economic growth over read more..

##### Page - 138

Introduction to Modern Economic Growthhow to incentivize diﬀerent agents in the economy. This again shapes our agenda for therest of the book, especially paving the way for our investigation of endogenous technologicalchange in Part 4 and of diﬀerences in technology and productive eﬃciency across countriesin Parts 6 and 7.There is one more sense in which what we have learned in this chapter is limited. Whatthe Solow model makes us focus on, physical capital, human capital and technology, read more..

##### Page - 139

Introduction to Modern Economic Growthfocusing on issues of convergence. Wooldridge (2002) contains an excellent discussion of is-sues of omitted variable bias and how diﬀerent approaches can be used (see, for example,Chapters4,5, 8,9and10). The diﬃculties involved in estimating models with ﬁxed eﬀectsand lagged dependent variables are discussed in Chapter 11.The augmented Solow model with human capital is a generalization of the model presentedin Mankiw, Romer and Weil (1992). As noted read more..

##### Page - 140

Introduction to Modern Economic GrowthExplain the importance of diﬀerences in factor proportions (capital-labor ratio) between thebeginning and end dates in these results.Exercise 3.2. Consider the economy with labor market imperfections as in the second partof Exercise 2.13 from the previous chapter, where workers were paid a fraction β> 0 ofoutput. Show that in this economy the fundamental growth accounting equation leads tobiased estimates of TFP.Exercise 3.3. For the Cobb-Douglas read more..

##### Page - 141

Introduction to Modern Economic GrowthAj is labor-augmenting technology. Prove that if Kj/Yj = Kj0/Yj0 in two diﬀerent countriesj and j0, than the rental rates of capital in the two countries, Rj and Rj0 will also be equal.Exercise 3.11. Imagine you have a cross-section of countries, i =1,..., N ,and for eachcountry, at a single point in time, you observe labor Li, capital Ki, total output Yi,and theshare of capital in national income, σKi . Assume that all countries have access to a read more..

##### Page - 142

CHAPTER 4Fundamental Determinants of Diﬀerences in EconomicPerformance4.1. Proximate Versus Fundamental Causes“...the factors we have listed (innovation, economies of scale, education, capitalaccumulation etc.) are not causes of growth; they are growth." (North andThomas, 1973, p. 2, italics in original).The previous chapter illustrate how the Solow growth model can be used to understandcross-country income diﬀerences and the process of economic growth. In the context of theSolow read more..

##### Page - 143

Introduction to Modern Economic Growththese fundamental causes. Second, if part of our study of economic growth is motivatedby improving the growth performance of certain nations and the living standards of theircitizens, understanding fundamental causes is central, since attempting to increase growthjust focusing on proximate causes would be tantamount to dealing with symptoms of diseaseswithout understanding what the diseases themselves are. While such attacks on symptomscan sometimes be read more..

##### Page - 144

Introduction to Modern Economic Growth(4) The institutions hypothesis.By luck, we refer to the set of fundamental causes that explain divergent paths of economicperformance among otherwise-identical countries, either because some small uncertainty orheterogeneity between them have led to diﬀerent choices with far-ranging consequences, orbecause of diﬀerent selection among multiple equilibria. By multiple equilibria, we refer todiﬀerent equilibrium conﬁgurations that may be possible for read more..

##### Page - 145

Introduction to Modern Economic Growthcultural diﬀerences that may aﬀect economic behavior. Diﬀerences in preferences, for exam-ple, regarding how important wealth is relative to other status-generating activities and howpatient individuals should be, might be as important as or even more important than luck,geography and institutions in aﬀecting economic performance. Broadly speaking, culture canaﬀect economic outcomes through two major channels. First, it can aﬀect the willingness read more..

##### Page - 146

Introduction to Modern Economic GrowthIt is also important to emphasize that institutions themselves, even if they are a funda-mental cause of economic growth and income diﬀerences across countries, are endogenous.They are equilibrium choices made either by the society at large or by some powerful groupsin society. One can then argue that at some level luck, geography or culture should be moreimportant, because they can be “more exogenous” in the sense that they are not equilibriumchoices read more..

##### Page - 147

Introduction to Modern Economic Growthbeen going on in the background needs to reach a critical threshold for the process of growthto begin. These stories are quite plausible. World population has indeed increased tremen-dously over the past one million years and the world’s inhabitants today have access to apool of knowledge and technology unimaginable to our ancestors. Could these long-run de-velopments of the world economy also account for cross-country diﬀerences? Is the increasein world read more..

##### Page - 148

Introduction to Modern Economic GrowthThis shows how a model of economies of scale (increasing returns) in population can generatea steady increase in technology. It is also straightforward to verify that(4.5)Y (t)= φα1−α A (t) ,so that aggregate income also grows at the constant level λφ1/(1−α). Such a model wouldgenerate steady growth but no acceleration. Simon and Kremer, instead, assume that thereare stronger externalities to population than in (4.1). They impose the following read more..

##### Page - 149

Introduction to Modern Economic Growthpart of the proximate causes of the growth process (for example, the part lying in the blackbox of technology). But this discussion suggests that these models need to be augmentedby fundamental causes in order to explain why, when and where the takeoﬀ occurred. Thisfurther motivates our investigation of the fundamental causes.4.3. The Four Fundamental Causes4.3.1. Luck and Multiple Equilibria.In Chapter 21, we will see a number of modelsin which multiple read more..

##### Page - 150

Introduction to Modern Economic Growthup in diﬀerent equilibria, it could help in explaining very large diﬀerences in income percapita. Second, the two equilibria in this game are also “Pareto-ranked”–all individuals arebetter-oﬀ in the equilibrium in which everybody chooses high investment.In addition to models of multiple equilibria, we will also study models in which therealization of stochastic variables determine when a particular economy transitions from low-productivity to read more..

##### Page - 151

Introduction to Modern Economic Growthit were possible for Nigerians to change their behavior and for all individuals in the nation tobecome better-oﬀ (say by switching from low to high investment in terms of the game above),it is very diﬃcult to believe that for 200 years they have not been able to coordinate on sucha better action. Most readers will be aware that Nigerian history is shaped by religious andethnic conﬂict, by thecivil warthatravagedthe nation, and is still adversely read more..

##### Page - 152

Introduction to Modern Economic GrowthAdiﬀerent, and perhaps more promising, argument about the importance of luck can bemade by emphasizing the role of leaders. Perhaps it was Mao who held back China, and hisdeath and the identity, beliefs and policies of his successor were at the root of its subsequentgrowth. Perhaps the identity of the leader of a country can thus be viewed as a stochasticevent, shaping economic performance. This point of view probably has a lot of merit. Recentempirical read more..

##### Page - 153

Introduction to Modern Economic Growthhuman attitudes and eﬀort, and via this channel, aﬀected both economic and social outcomes.He wroteinhis classicbook The Spirit of the Laws:“The heat of the climate can be so excessive that the body there will beabsolutely without strength. So, prostration will pass even to the spirit; nocuriosity, no noble enterprise, no generous sentiment; inclinations will all bepassive there; laziness there will be happiness,”“People are ... more vigorous in read more..

##### Page - 154

Introduction to Modern Economic GrowthThere are a number of reasons for questioning this second, and more widely-held view,of geographic determinism as well. Most of the technological diﬀerences emphasized by theseauthors refer to agriculture. But as we have seen in Chapter 1 and will encounter againbelow, the origins of diﬀerential economic growth across countries goes back to the age ofindustrialization. Modern economic growth came with industry, and it is the countries thathave failed to read more..

##### Page - 155

Introduction to Modern Economic GrowthThe fact that the burden of disease is heavier in poor nations today is as much a consequenceas a cause of poverty. European nations in the 18th and even 19th centuries were plaguedby many diseases. The process of economic development enabled them to eradicate thesediseases and create healthier environments for living. The fact that many poor countrieshave unhealthy environments is, at least in part, a consequence of their failure to read more..

##### Page - 156

Introduction to Modern Economic Growthoverstated and will be the topic of analysis in Part 8 of this book. But this is not where wewill begin.A more natural starting point for the study of the fundamental causes of income dif-ferences across countries is with economic institutions, which comprise such things as thestructure of property rights, the presence and (well or ill) functioning of markets, and thecontractual opportunities available to individuals and ﬁrms. Economic institutions are read more..

##### Page - 157

Introduction to Modern Economic Growthdegree of equality of opportunity. Economic institutions that only protect a rich elite or thealready-privileged will not achieve such equality of opportunity and will often create otherdistortions, potentially retarding economic growth. We will also see in Chapter 14 that theprocess of Schumpeterian creative destruction, where new ﬁrms improve over and destroyincumbents, is an essential element of economic growth. Schumpeterian creative read more..

##### Page - 158

Introduction to Modern Economic Growthbut we have to postpone their discussion to Part 8. For now, however, we can note that theendogeneity of institutions has another important implication; endogeneity of institutionsmakes empirical work on assessing the role of institutions more challenging, because it impliesthat the standard “simultaneity” biases in econometrics will be present when we look at theeﬀect of institutions on economic outcomes.4In this chapter, we will focus on the read more..

##### Page - 159

Introduction to Modern Economic GrowthThe most famous link between culture and economic development is that proposed byWeber (1930), who argued that the origins of industrialization in Western Europe could betraced to a cultural factor–the Protestant reformation and particularly the rise of Calvinism.Interestingly, Weber provided a clear summary of his views as a comment on Montesquieu’sarguments:“Montesquieu says of the English that they ‘had progressed the farthest of allpeoples of the read more..

##### Page - 160

Introduction to Modern Economic Growthit becomes diﬃcult to explain why these Asian values did not lead to growth before. Whydo these values not spur economic growth in North Korea? If Asian values are important forChinese growth today, why did they not lead to a better economic performance under Mao’sdictatorship? Both of these challenges are, in principle, surmountable. One may be able todevelop models of culture, with better mapping to data, and also with an associated theory ofwhen and read more..

##### Page - 161

Introduction to Modern Economic GrowthPolitical Risk Services, a private company which assesses the risk that foreign investmentswill be expropriated in diﬀerent countries. These data are not perfect. They reﬂect thesubjective assessment of some analysts about how secure property rights are. Nevertheless,they are useful for our purposes. First, they emphasize the security of property rights, whichis an essential aspect of economic institutions, especially in regards to their eﬀect on read more..

##### Page - 162

Introduction to Modern Economic Growth.Log GDP per capita, PPP, in 1995Latitude0.2.4.6.86810AFGDZAAGOARGARMAUSAUTAZEBHSBHRBGDBRBBLRBELBLZBENBTNBOLBIHBWABRABGRBFABDICMRCANCPVCAFTCDCHLCHNCOLCOMZARCOGCRICIVHRVCZEDNKDJIDMADOMECUEGYSLVESTETHFJIFINFRAGABGMBGEODEUGHAGRCGRD read more..

##### Page - 163

Introduction to Modern Economic GrowthKim Il Sung. U.S. authorities therefore supported the inﬂuential nationalist leader SyngmanRhee, who was in favor of separation rather than a united communist Korea. Elections in theSouth were held in May 1948, amidst a widespread boycott by Koreans opposed to separation.The newly elected representatives proceeded to draft a new constitution and established theRepublic of Korea to the south of the 38th parallel. The North became the DemocraticPeople’s read more..

##### Page - 164

Introduction to Modern Economic Growth02000400060008000100001200014000195019601970198019901998South KoreaNorth KoreaFigure 4.3. Evolution of income per capita North and South Korea after the separation.institutions). Nevertheless, the perspective of “luck” is unlikely to be particularly useful inthis context, since what is remarkable is the persistence of the dysfunctional North Koreaninstitutions. Despite convincing evidence that the North Korean system has been generatingpoverty and read more..

##### Page - 165

Introduction to Modern Economic Growthnations. The colonization experience transformed the institutions in many diverse landsconquered or controlled by Europeans. Most importantly, Europeans imposed very diﬀerentsets of institutions in diﬀerent parts of their global empire, as exempliﬁed most sharply by thecontrast of the institutional structure that developed in the Northeastern United States, basedon small-holder private property and democracy, versus the institutions in the read more..

##### Page - 166

Introduction to Modern Economic Growthcan sustain large urban centers and a dense population. Figure 4.4 shows the relationshipbetween income per capita and urbanization (fraction of the population living in urban centerswith greater than 5,000 inhabitants) today, and demonstrates that even today, long afterindustrialization, there is a signiﬁcant relationship between urbanization and prosperity.Naturally, high rates of urbanization do not mean that the majority of the populationlived in read more..

##### Page - 167

Introduction to Modern Economic Growth.Log GDP per capita, PPP, in 1995Log Population Density in 1500-505678910AGOARGAUSBDIBENBFABGDBHSBLZBOLBRABRBBWACAFCANCHLCIVCMRCOGCOLCOMCPVCRIDMADOMDZAECUEGYERIGABGHAGINGMBGRDGTMGUYHKGHNDHTIIDNINDJAMKENKNALAOLCALKALSOMARMDGMEXMLIMOZMRTMWIMYSNAMNERNGANICNPLNZLPAKPANPERPHLPRYRWASDNSENSGPSLESLVSURSWZTCDTGOTTOTUNTZAUGAURYUSAVCTVENVNMZAFZARZMBZWEFigure 4.6. Reversal of Fortune: population density in 1500 versus incomeper capita in 1995 them on the former European read more..

##### Page - 168

Introduction to Modern Economic Growthtypes of institutions they imposed on in the diﬀerent colonies.5 These institutional diﬀerencesamong the former colonies are likely at the root of the reversal in economic fortunes.To bolster this case, let us look at the timing and the nature of the reversal a little moreclosely. When did the reversal occur? One possibility is that it arose shortly after the conquestof societies by Europeans but Figure 4.7 shows that the previously-poor colonies read more..

##### Page - 169

Introduction to Modern Economic Growthtoday it is the reverse. This makes it implausible to base a theory of relative prosperity on theintrinsic poverty of the tropics, climate, disease environments or other ﬁxed characteristics.Nevertheless, following Diamond (1997), one could propose what Acemoglu, Johnson andRobinson (2002a) call a “sophisticated geography hypothesis,” which claims that geographymatters but in a time varying way. For example, Europeans created “latitude read more..

##### Page - 170

Introduction to Modern Economic GrowthIn particular, the evidence points out that, others things equal, the higher the initial pop-ulation density or the greater initial urbanization, the worse were subsequent institutions,including both institutions right after independence and also institutions today. Figures 4.8and 4.9 illustrate these relationships using the same measure of current economic institutionsused in Figure 4.1, protection against expropriation risk today. They document that the read more..

##### Page - 171

Introduction to Modern Economic GrowthEuropeans to have institutions facilitating the extraction of resources, without any respect forthe property rights of the majority of the populace. In contrast, in the sparsely-settled areasit was in their interests to develop institutions protecting property rights. These incentivesled to an institutional reversal.The institutional reversal, combined with the institutions hypothesis, predicts the Rever-sal of Fortune: relatively rich places ended up with read more..

##### Page - 172

Introduction to Modern Economic GrowthThe explanation for the reversal that emerges from the discussion so far is one in which theeconomic institutions in various colonies were shaped by Europeans to serve their own (eco-nomic) interests. Moreover, because conditions and endowments diﬀered between colonies,Europeans consciously created diﬀerent economic institutions, which persisted and con-tinue to shape economic performance. Why did Europeans introduce better institutionsin previously-poor read more..

##### Page - 173

Introduction to Modern Economic Growthcurrent institutions. From an econometric point of view, this will correspond to a validinstrument to estimate the casual eﬀect of economic institutions on prosperity. Althoughmortality rates of potential European settlers could be correlated with indigenous mortality,which may determine income today, in practice local populations had developed much greaterimmunity to malaria and yellow fever. Acemoglu, Johnson and Robinson (2001) present avariety of read more..

##### Page - 174

Introduction to Modern Economic Growththat life expectancy in Australia and New Zealand was in fact greater than in Britain. Incontrast, all Europeans faced extremely high mortality rates in Africa, India and South-EastAsia. Diﬀerential mortality was largely due to tropical diseases such as malaria and yellowfever and at the time it was not understood how these diseases arose nor how they could beprevented or cured..Avg. Protect. Against Risk ExpropriationLog Settler read more..

##### Page - 175

Introduction to Modern Economic Growth.Log GDP per capita, PPP, in 1995Log Settler Mortality24686810AGOARGAUSBFABGDBHSBOLBRACANCHLCIVCMRCOGCOLCRIDOMDZAECUEGYGABGHAGINGMBGTMGUYHKGHNDHTIIDNINDJAMKENLKAMARMDGMEXMLIMYSNERNGANICNZLPAKPANPERPRYSDNSENSGPSLESLVSURTGOTTOTUNTZAUGAURYUSAVENVNMZAFZARFigure 4.11. The relationship between mortality of potential European set-tlers and GDP per capita in 1995.countries are a major determinant of their economic fortunes, while geographic diﬀerencesare much less read more..

##### Page - 176

Introduction to Modern Economic Growththe Spanish may have condemned Latin America by endowing it with an Iberian culture.Second, Europeans may have had a culture, work ethic or set of beliefs that were conduciveto prosperity. Finally, Europeans also brought diﬀerent religions with diﬀerent implicationsfor prosperity. Many of these hypotheses have been suggested as explanations for why LatinAmerica, with its Roman Catholic religion and Iberian culture, is poor relative to the Anglo-Saxon read more..

##### Page - 177

Introduction to Modern Economic GrowthJust British ColoniesLog GDP per capita, PPP, 1995Log Population Density in 1500-505678910AUSBGDBHSBLZBRBBWACANDMAEGYGHAGMBGRDGUYHKGINDJAMKENKNALCALKALSOMWIMYSNAMNGANPLNZLPAKSDNSGPSLESWZTTOUGAUSAVCTZAFZMBZWEFigure 4.12. The Reversal of Fortune among British Colonies: populationdensity in 1500 versus GDP per capita in 1995 among former British colonies.those of European descent in 1975 is less than 5 percent of the population–thus a sample ofcountries in read more..

##### Page - 178

Introduction to Modern Economic GrowthFormer Colonies, Percent European Descent in 1975 <5%Log GDP per capita, PPP, 1995Log Population Density in 1500-20246678910AGOBDIBENBFABGDBWACAFCIVCMRCOGDZAEGYERIETHGABGHAGINGMBGUYHKGHTIIDNINDKENLAOLKALSOMARMDGMLIMOZMRTMWIMYSNAMNERNGANPLPAKPHLRWASDNSENSGPSLESWZTCDTGOTUNTZAUGAVNMZARZMBZWEFigure 4.13. The Reversal of Fortune among former European colonies withtwo current European inhabitants.There is relatively little work on “unbundling” the broad read more..

##### Page - 179

Introduction to Modern Economic Growththe legal system imposed by colonial powers appears to have a strong eﬀect on contractinginstitutions, but little impact on the available measures of property rights institutions. Atthe same time, both mortality rates for potential European settlers and population density in1500, which we have seen above as important determinants of European colonization strategy,have a large eﬀect on current property rights institutions, and no impact on contracting read more..

##### Page - 180

Introduction to Modern Economic Growthsmall aﬀects, so that they are hard to detect when we look at countries with thirty-fold dif-ferences in income per capita. Therefore, this evidence should be interpreted as suggestingthat contracting institutions are less important in generating the large diﬀerences in eco-nomic development than the property rights institutions, not necessarily as suggesting thatcontracting institutions do not matter for economic outcomes.4.6. Disease and DevelopmentThe read more..

##### Page - 181

Introduction to Modern Economic Growth33.23.43.63.844.24.44.619301940195019601970198019902000Initially RichInitially Middle IncomeInitially PoorFigure 4.14. Evolution of life expectancy at birth among initially-poor,initially-middle-income and initially-rich countries, 1940-2000.new chemicals and drugs. More important for the purposes of understanding the eﬀect ofdisease on economic growth, these health improvements were by and large exogenous fromthe viewpoint of individual nations. Moreover, read more..

##### Page - 182

Introduction to Modern Economic GrowthThis reasoning suggests that potentially-exogenous variation in the health conditions of thecountry can be measured by calculating a measure of predicted mortality, driven by the inter-action of baseline cross-country disease prevalence with global intervention dates for speciﬁcdiseases. Acemoglu and Johnson (2006) show that such measures of predicted mortality havea large and robust eﬀect on changes in life expectancy starting in 1940, but have no read more..

##### Page - 183

Introduction to Modern Economic Growthis later compensated by higher output as more people enter the labor force. However, thereis no reason to expect either a complete oﬀset of the initial decline in income per capita or alarge signiﬁcant increase, especially when many of the eﬀect countries are heavily vested inagriculture, so that land-to-labor ratios may change permanently. Consequently, small ben-eﬁcial eﬀects of health on productivity may not be suﬃcient to oﬀset or reverse read more..

##### Page - 184

Introduction to Modern Economic Growth4.8. Taking StockThis chapter has emphasized the diﬀerences between the proximate causes of economicgrowth, related to physical capital accumulation, human capital and technology, and thefundamental causes, which inﬂuence the incentives to invest in these factors of production.We have argued that many of the questions motivating our study of economic growth mustlead us to an investigation of the fundamental causes. But an understanding of read more..

##### Page - 185

Introduction to Modern Economic Growthignoring fundamental cause of economic growth. Diamond (1997) also draws a distinctionbetween proximate and fundamental explanations.The importance of population in generating economies of scale was ﬁrst articulated byJulian Simon (1990). The model presented in Section 4.2 draws on Simon’s work and workby Michael Kremer (1993). Kremer (1993) argues for the importance of economies of scaleand increasing returns to population based on the acceleration in read more..

##### Page - 186

Introduction to Modern Economic Growthand Rios-Rull (1999), and Bourguignon and Verdier (2000). There is a much smaller liter-ature on endogenous institutions and the eﬀect of these institutions on economic outcomes.Surveys of this work can be found in Acemoglu (2007) and Acemoglu and Robinson (2006).Theliteratureonthe eﬀect of economic institutions on economic growth is summarized anddiscussed in greater detail in Acemoglu, Johnson and Robinson (2006), which also providesan overview of the read more..

##### Page - 187

Introduction to Modern Economic Growthand Hall and Jones (1999) present the ﬁrst instrumental-variable estimates on the eﬀect ofinstitutions (or corruption) on long-run economic development.The evidence reported here, which exploits diﬀerences in colonial experience to create aninstrumental-variables strategy, is based on Acemoglu, Johnson and Robinson (2001, 2002).The urbanization and population density data used here are from Acemoglu, Johnson andRobinson (2002), which compiled these read more..

##### Page - 188

Introduction to Modern Economic Growth(2) Modify equation (4.2) toL (t)= φY (t)β ,for some β ∈ (0, 1). Justify this equation and derive the law of motion technologyand income per capita under the two scenarios considered in the text. Are theimplications of this model more reasonable than those considered in the text?Exercise 4.4. In his paper “Tropical Underdevelopment”, Jeﬀ Sachs notes that diﬀerencesin income per capita between tropical and temperate zones have widened over the read more..

##### Page - 189

##### Page - 190

Part 2Towards Neoclassical Growth read more..

##### Page - 191

This part of the book is a preparation for what is going to come next. In some sense,it can be viewed as the “preliminaries” for the rest of the book. Our ultimate purposeis to enrich the basic Solow model by introducing well-deﬁned consumer preferences andconsumer optimization, and in the process, clarify the relationship between growth theoryand general equilibrium theory. This will enable us to open the blackbox of savings and capitalaccumulation, turning these decisions into read more..

##### Page - 192

CHAPTER 5Foundations of Neoclassical GrowthThe Solow growth model is predicated on a constant saving rate. Instead, it wouldbe much more satisfactory to specify the preference orderings of individuals, as in standardgeneral equilibrium theory, and derive their decisions from these preferences. This will enableus both to have a better understanding of the factors that aﬀect savings decisions and also todiscuss the “optimality” of equilibria–in other words, to pose and answer questions read more..

##### Page - 193

Introduction to Modern Economic GrowthAs in basic general equilibrium theory, we make enough assumptions on preference or-derings (in particular, reﬂexivity, completeness and transitivity) so that these preferenceorderings can be represented by utility functions. In particular, suppose that each householdi has an instantaneous utility function given byui (ci (t)) ,where ui : R+→ R is increasing and concave and ci (t) is the consumption of household i.Here and throughout, we take the domain read more..

##### Page - 194

Introduction to Modern Economic Growthwith and satisfy all of the standard axioms of rational decision-making. Although time-inconsistent preferences may be useful in the modeling of certain behaviors we observe inpractice, such as problems of addiction or self-control, time-consistent preferences are idealfor the focus in this book, since they are tractable, relatively ﬂexible and provide a goodapproximation to reality in the context of aggregative models. It is also worth noting thatmany read more..

##### Page - 195

Introduction to Modern Economic Growth5.2. The Representative HouseholdWhen we say that an economy admits a representative household, this means that the pref-erence (demand) side of the economy can be represented as if there were a single householdmaking the aggregate consumption and saving decisions (and also the labor supply decisionswhen these are endogenized) subject to a single budget constraint. The major convenience ofthe representative household assumption is that instead of thinking of read more..

##### Page - 196

Introduction to Modern Economic Growththis is not without any costs. First, in this case, the representative household will havepositive meaning, but not always a normative meaning (see below). Second, it is not in facttrue that most models with heterogeneity lead to a behavior that can be represented as if itwere generated by a representative household.In fact most models do not admit a representative household. To illustrate this, letus consider a simple exchange economy with a ﬁnite number read more..

##### Page - 197

Introduction to Modern Economic GrowthTo prepare for this theorem, consider an economy with a ﬁnite number N of commoditiesand recall that an indirect utility function for household i, vi¡p,yi¢,speciﬁesthehousehold’s(ordinal) utility as a function of the price vector p =(p1,...,pN ) and the household’s incomeyi. Naturally, any indirect utility function vi¡p,yi¢hastobehomogeneousofdegree0inpand y.Theorem 5.2. (Gorman’s Aggregation Theorem) Consider an economy with a ﬁnitenumber read more..

##### Page - 198

Introduction to Modern Economic GrowthAnother attractive feature of Gorman preferences for our purposes is that they containsome commonly-used preferences in macroeconomics. To illustrate this, let us start with thefollowing example:Example 5.1. (Constant Elasticity of Substitution Preferences)Averycommonclassof preferences used in industrial organization and macroeconomics are the constant elasticityof substitution (CES) preferences, also referred to as Dixit-Stiglitz preferences after the read more..

##### Page - 199

Introduction to Modern Economic GrowthWe will see below that preferences closely related to the CES preferences will play aspecial role not only in aggregation but also in ensuring balanced growth in neoclassicalgrowth models.It is also possible to prove the converse to Theorem 5.2. Since this is not central toour focus, we state this result in the text rather than stating and proving it formally. Theessence of this converse is that unless we put some restrictions on the distribution of read more..

##### Page - 200

Introduction to Modern Economic Growththe economy is equal to the value of the endowments. The third set of constraints requiresthat all prices are nonnegative.Now compare the above maximization problem to the following problem:maxXi∈Hai (p)+ b (p) ysubject to the same set of constraints. The only diﬀerence between the two problems is thatin the latter each household has been assigned the same weight.Let (p∗,y∗) be a solution to the second problem. By deﬁnition it is also a solution to read more..

##### Page - 201

Introduction to Modern Economic Growth(5.8) impliesαM b (p∗∗α )Xi∈H(y∗∗α )i≥ αM b (p∗)Xi∈H(y∗)ib (p∗∗α )(y∗∗α ) ≥ b (p∗)(y∗) ,which contradicts equation (5.8), and establishes that, under the stated assumptions, anyPareto optimal allocation maximizes the utility of the representative household.¤5.3. Inﬁnite Planning HorizonAnother important microfoundation for the standard preferences used in growth theoryand macroeconomics concerns the planning horizon read more..

##### Page - 202

Introduction to Modern Economic Growtharguments imply that this individual would have an expected utility at time t =0 given byU (0) = u (c (0)) + ˆβ (1 − ν) u (c (0)) + ˆβνu (0)+ˆβ2(1 − ν)2 u (c (1)) + ˆβ2(1 − ν) νu (0) + ...=∞Xt=0³ˆβ(1−ν)´tu(c(t))=∞Xt=0βtu (c (t)) ,(5.9)where the second line collects terms and uses u (0) = 0, while the third line deﬁnesβ ≡ˆβ (1 − ν) as the “eﬀective discount factor” of the individual. With this formulation, read more..

##### Page - 203

Introduction to Modern Economic Growth(including accidental bequests like the individual facing random death probability just dis-cussed). Nevertheless, a natural benchmark might be one in which the individual is “purelyaltruistic” so that he cares about the utility of his oﬀspring (with some discount factor).1 Letthe discount factor apply between generations be β. Also assume that the oﬀspring will havean income of w without the bequest. Then the utility of the individual can be read more..

##### Page - 204

Introduction to Modern Economic Growthadmit a representative household, the standard assumptions we adopt in general equilibriumtheory or a dynamic general equilibrium analysis (in particular no production externalitiesand competitive markets) are suﬃcient to ensure that the formulation with a representativeﬁrm is without loss of any generality.This result is stated in the next theorem.Theorem 5.4. (The Representative Firm Theorem) Consider a competitive produc-tion economy with N ∈ read more..

##### Page - 205

Introduction to Modern Economic Growthfor some yf ∈ Y f for each f ∈ F.Let ˆyf ∈ˆY f (p). Then,Xf ∈Fp · yf ≤Xf ∈Fp · ˆyf ,which implies that(5.10)p · ˆy ≤ p ·Xf ∈Fˆyf .Since, by hypothesis,Pf∈Fˆyf ∈ Y and ˆy ∈ˆY (p),wealsohavep · ˆy ≥ p ·Xf ∈Fˆyf .Therefore, inequality (5.10) must hold with equality, so thatp · yf = p · ˆyf ,for each f ∈ F, and thus yf ∈ˆY f (p). This completes the proof of the theorem.¤This theorem implies that, given the read more..

##### Page - 206

Introduction to Modern Economic Growthwith a discount factor of β ∈ (0, 1).In continuous time, this utility function of the representative household becomes(5.12)Z ∞0exp (−ρt)u(c(t))dtwhere ρ> 0 is now the discount rate of the individuals.Where does the exponential form of the discounting in (5.12) come from? At some level,we called discounting in the discrete time case also “exponential”, so the link should beapparent.To see it more precisely, imagine we are trying to calculate read more..

##### Page - 207

Introduction to Modern Economic Growththeorems and develop the relevant connections between the theory of economic growth anddynamic general equilibrium models.Let us start with models that have a ﬁnite number of consumers, so that in terms of thenotation above, the set H is ﬁnite. However, we allow an inﬁnite number of commodities,since in dynamic growth models, we are ultimately interested in economies that have aninﬁnite number of time periods, thus an inﬁnite number of commodities. read more..

##### Page - 208

Introduction to Modern Economic Growthyf ≡nyfjo∞j=0is a feasible production plan for ﬁrm f if yf ∈ Y f . For example, if there wereonly two commodities, labor and a ﬁnal good, Y f would include pairs (−l,y) such that withlabor input l (hence a negative sign), the ﬁrm can produce at most as much as y.As isusual in general equilibrium theory, let us take each Y f to be a cone, so that if y ∈ Y f ,thenλy ∈ Y f for any λ ∈ R+. This implies two important features: ﬁrst, 0 ∈ read more..

##### Page - 209

Introduction to Modern Economic Growth(2) For every ﬁrm f ∈ F, yf∗ maximizes proﬁts, i.e.,p∗ · yf ∗ ≤ p∗ · y for all y ∈ Yf .(3) For every consumer i ∈ H, xi∗ maximizes utility, i.e.,ui¡xi∗¢≥ui(x) for all x such that x ∈ Xi and p∗ · x ≤ p∗ ·⎛⎝ωi+Xf ∈Fθif yf⎞⎠.A major focus of general equilibrium theory is to establish the existence of a competitiveequilibrium under reasonable assumptions. When there is a ﬁnite number of commodities read more..

##### Page - 210

Introduction to Modern Economic GrowthTheorem 5.5. (First Welfare Theorem I) Suppose that (x∗, y∗,p∗) is a competitiveequilibrium of economy E ≡ (H,F,u,ω,Y,X,θ) with H ﬁnite. Assume that all householdsare locally non-satiated at x∗.Then (x∗, y∗) is Pareto optimal.Proof. To obtain a contradiction, suppose that there exists a feasible (ˆx, ˆy) such thatui¡ˆxi¢≥ui¡xi¢foralli∈Handui¡ˆxi¢>ui¡xi¢foralli∈H0,whereH0 is a non-emptysubset of H.Since (x∗, y∗,p∗) read more..

##### Page - 211

Introduction to Modern Economic Growthis proﬁt-maximizing at prices p∗,wehavethat(5.16)p∗ ·Xf ∈Fyf ∗ ≥ p∗ ·Xf ∈Fyf for anynyfof ∈Fwith yf ∈ Yf for all f ∈ F.However, by feasibility of ˆxi(Deﬁnition 5.1, part 1), we haveXi∈Hˆxij≤Xi∈Hωij +Xf ∈Fˆyfj ,and therefore, by multiplying both sides by p∗ and exploiting (5.16), we havep∗ ·Xi∈Hˆxij≤ p∗ ·⎛⎝Xi∈Hωij +Xf ∈Fˆyfj⎞⎠≤ p∗ ·⎛⎝Xi∈Hωij +Xf ∈Fyf ∗j⎞⎠,which contradicts read more..

##### Page - 212

Introduction to Modern Economic Growthover an inﬁnite number of households. However, since endowments are ﬁnite, the assumptionthatP∞j=0p∗j<∞ensuresthatthesumsin(5.15)areindeedﬁniteandtherestoftheproofgoes through exactly as in the proof of Theorem 5.5.¤Theorem 5.6 will be particularly useful when we discuss overlapping generation models.We next brieﬂy discuss the Second Welfare Theorem, which is the converse of the FirstWelfare Theorem. It answers the question of whether a read more..

##### Page - 213

Introduction to Modern Economic Growth(3) for all i ∈ H,if xi ∈ Xi involves ui¡xi¢>ui¡xi∗∗¢,thenp∗∗·xi≥p∗∗·wi∗∗,where wi∗∗ ≡ ωi∗∗ +Pf∈Fθi∗∗fyf∗∗.Moreover, if p∗∗ · w∗∗ > 0 [i.e., p∗∗ · wi∗∗ > 0 for each i ∈ H], then economy E has acompetitive equilibrium (x∗∗, y∗∗,p∗∗).The proof of this theorem involves the application of the Geometric Hahn-Banach Theo-rem, Theorem A.25, from Appendix Chapter A. It is read more..

##### Page - 214

Introduction to Modern Economic Growthallocation can be decentralized as a competitive equilibrium, a competitive equilibrium mustexist (at least for the endowments leading to Pareto optimal allocations).The Second Welfare Theorem motivates many macroeconomists to look for the set ofPareto optimal allocations instead of explicitly characterizing competitive equilibria. Thisis especially useful in dynamic models where sometimes competitive equilibria can be quitediﬃcult to characterize or even read more..

##### Page - 215

Introduction to Modern Economic Growthdeﬁnition of the set P , there would exist©˜xiªi∈H such thatPi∈H˜xi =˜y , ˜xi ∈ Xi,andui¡˜xi¢≥ui¡xi∗∗¢ for all i ∈ H with at least one strict inequality). This would contradictthe hypothesis that (x∗∗,y∗∗) is a Pareto optimum.Since Y0 has an interior point, P and Y0 are convex, and P ∩ Y0 = ∅, Theorem A.25implies that there exists a nonzero continuous linear functional φ such that(5.17)φ (y) ≤ φ (x∗∗) ≤ φ read more..

##### Page - 216

Introduction to Modern Economic GrowthThis also implies that¯φ (x)= limT →∞φ (x [T ]) =limT →∞TXJ=0¯φJ (xJ )=limT →∞TXJ=0p∗∗J· xJ ,so that ¯φ is a continuous linear functional within inner productive presentation.To complete this part of the proof, we only need to show that ¯φ (x)=P∞j=0 ¯φj (xj) canbe used instead of φ as the continuous linear functional in (5.17). This follows immediatelyfrom the hypothesis that 0 ∈ Xi for each i ∈ H,that for each x, x0 ∈ Xi read more..

##### Page - 217

Introduction to Modern Economic Growthtrade or production in the economy. This may be a good approximation to reality whendiﬀerent commodities correspond to diﬀerent goods. However, when diﬀerent commoditiescorrespond to the same good in diﬀerenttimeperiods or in diﬀerent states of nature, tradingonce and for all at a single point is much less reasonable. In models of economic growth,we typically assume that trading takes place at diﬀerent points in time. For example, inthe Solow read more..

##### Page - 218

Introduction to Modern Economic Growthsuﬃcient for household h to have an income ofPNi=1pi,t0xhi,t0 and know that it can purchaseas many units of each commodity as it wishes at time t0 at the price vector¡p1,t0,...,pN,t0¢.This result can be stated in a slightly more formal manner. Let us consider a dynamicexchange economy running across periods t =0, 1,...,T ,possibly with T = ∞.3 Nothing heredepends on the assumption that we are focusing on an exchange economy, but suppressingproduction read more..

##### Page - 219

Introduction to Modern Economic Growthhousehold h is denoted by bht ∈ R, and since each bond is in zero net supply, market clearingrequires thatXh∈Hbht =0 for each t =0, 1, ..., T .Notice that in this speciﬁcation we have assumed the presence of only T bonds (Arrowsecurities). More generally, we could have allowed additional bonds, for example bondstraded at time t> 0 for delivery of good 1 at time t0 >t. This restriction to only T bonds iswithout loss of any generality (see Exercise read more..

##### Page - 220

Introduction to Modern Economic Growthgrowth models, the price of the consumption good in each date will be normalized to 1 andthe interest rates will directly give the intertemporal relative prices. This is the justiﬁcationfor focusing on interest rates as the key relative prices in macroeconomic (economic growth)models.5.9. Optimal Growth in Discrete TimeMotivated by the discussion in the previous section let us start with an economy char-acterized by an aggregate production function, and a read more..

##### Page - 221

Introduction to Modern Economic Growthrepresentative household, we only need to look at the maximization problem of this consumer.Assuming that the representative household has one unit of labor supplied inelastically, thisproblem can be written as:max{c(t),k(t)}∞t=0∞Xt=0βtu (c (t))subject to some given a (0) and(5.21)a (t +1) = r (t)[a (t) − c (t)+ w (t)] ,where a (t) denotes the assets of the representative household at time t, r (t) is the rate ofreturn on assets and w (t) is the read more..

##### Page - 222

Introduction to Modern Economic Growth5.11. Taking StockThis chapter introduced the preliminaries necessary for an in-depth study of equilibriumand optimal growth theory. At some level it can be thought of as an “odds and ends” chapter,introducing the reader to the notions of representative household, dynamic optimization,welfare theorems and optimal growth. However, what we have seen is more than odds andends, since a good understanding of the general equilibrium foundations on economic read more..

##### Page - 223

Introduction to Modern Economic Growth5.12. References and LiteratureThis chapter covered a lot of ground and in most cases, many details were omitted forbrevity. Most readers will be familiar with much of the material in this chapter. Mas-Colell,Winston and Green (1995) have an excellent discussion of issues of aggregation and what typesof models admit representative households. They also have a version of the Debreu-Mantel-Sonnenschein theorem, with a sketch proof. The representative ﬁrm read more..

##### Page - 224

Introduction to Modern Economic Growthversion of Theorem 5.6 is presented in Bewley (2006), which contains an excellent discussionof overlapping generations models.5.13. ExercisesExercise 5.1. Recall that a solution {x(t)}Tt=0 to a dynamic optimization problem is time-consistent if the following is true: whenever {x(t)}Tt=0 is an optimal solution starting at timet =0, {x(t)}Tt=t0 is an optimal solution to the continuation dynamic optimization problemstarting from time t = t0 ∈ [0,T ].(1) read more..

##### Page - 225

Introduction to Modern Economic Growththe problemmax{x(t)}Tt=1u (x (1)) + δTXt=2βt−1u (x (t))subject tox (t) ∈ [0, ¯x]G (x∗ (0) ,..., x (T )) ≤ 0.Provethatthe solution from t =1 onwards, {x∗∗ (t)}Tt=1 is not necessarily the sameas {x∗ (t)}Tt=1.(4) Explain which standard axioms of preferences in basic general equilibrium theoryare violated by those in parts 2 and 3 of this exercise.Exercise 5.2. This exercise asks you to work through an example that illustrates the dif-ference read more..

##### Page - 226

Introduction to Modern Economic Growth(2) Show that maximization of (5.6) leads to the indirect utility function correspondingto the representative household.(3) Now suppose that U i¡xi1,...,xiN¢ =PNj=1³xij−ξij´σ−1σ. Repeat the same com-putations and verify that the resulting indirect utility function is homogeneous ofdegree 0 in p and y, but does not satisfy the Gorman form. Show, however, that amonotonic transformation of the indirect utility function satisﬁes the Gorman form.Is read more..

##### Page - 227

Introduction to Modern Economic GrowthExercise 5.12. Consider an economy consisting of N households each with utility functionat time t =0 given by∞Xt=0βtu¡ci(t)¢,with β ∈ (0, 1),where ci(t) denotes the consumption of household i at time t. The economystarts with an endowment of Y units of the ﬁnal good and has access to no productiontechnology. This endowment can be saved without depreciating or gaining interest ratebetween periods.(1) What are the Arrow-Debreu commodities in this read more..

##### Page - 228

CHAPTER 6Dynamic Programming and Optimal GrowthThis chapter will provide a brief introduction to inﬁnite horizon optimization in discretetime, focusing particularly on stationary dynamic programming problems under certainty.The main purpose of the chapter is to introduce the reader to dynamic programming tech-niques, which will be used in the rest of the book. Since dynamic programming has becomean important tool in many areas of economics and especially in macroeconomics, a goodunderstanding read more..

##### Page - 229

Introduction to Modern Economic Growth6.1. Brief Review of Dynamic ProgrammingUsing abstract but simple notation, the canonical dynamic optimization program in dis-crete time can be written asProblem A1:V∗ (x (0)) =sup{x(t+1)}∞t=0∞Xt=0βtU(x (t) ,x (t +1))subject tox (t +1)∈ G(x (t)),for all t ≥ 0x (0) given.where β ∈ (0, 1),and x (t) is a vector of variables, or more formally, x (t) ∈ X ⊂ RK for someK ≥ 1. G(x) is a set-valued mapping, or a correspondence, also written asG : read more..

##### Page - 230

Introduction to Modern Economic GrowthAgain the added generality in this case is not particularly useful for most of the problems weare interested in, and the discounted objective function ensures time-consistency as discussedin the previous chapter.Of particular importance for us in this chapter is the function V∗ (x (0)),which can bethought of as the value function, meaning the value of pursuing the optimal strategy startingwith initial state x (0).Problem A1 is somewhat abstract. However, read more..

##### Page - 231

Introduction to Modern Economic Growthsequence. The relevant functional equation can be written as follows:Problem A2:V (x)=supy∈G(x){U(x,y)+ βV (y}), for all x ∈ X,(6.1)where V : X → R is a real-valued function. Intuitively, instead of explicitly choosing thesequence {x(t)}∞t=0,in (6.1),we choose a policy, which determines what the control vectorx (t +1) should be for a given value of the state vector x (t). Since instantaneous payoﬀfunction U (·,·) does not depend on time, there read more..

##### Page - 232

Introduction to Modern Economic Growthhave thatV∗ (x (0)) =∞Xt=0βtU(x∗ (t) ,x∗ (t +1))= U (x (0) ,x∗ (1)) + β∞Xs=0βsU(x∗ (s +1) ,x∗ (s +2))= U (x (0) ,x∗ (1)) + βV∗ (x∗ (1)) .This equation encapsulates the basic idea of dynamic programming: the Principle of Opti-mality, and it is stated more formally in Theorem 6.2.Essentially, an optimal plan can be broken into two parts, what is optimal to do today, andthe optimal continuation path. Dynamic programming exploits this read more..

##### Page - 233

Introduction to Modern Economic Growthresults will ﬁrst be stated informally, without going into the technical details. Section 6.3will then present these results in greater formality and provide their proofs.6.2. Dynamic Programming TheoremsLet us start with a number of assumptions on Problem A1. Since these assumptions areonly relevant for this section, we number them separately from the main assumptions usedthroughout the book. Consider ﬁrst a sequence {x∗ (t)}∞t=0 which attains the read more..

##### Page - 234

Introduction to Modern Economic GrowthThis assumption is also natural. We need to impose that G (x) is compact-valued, sinceoptimization problems with choices from non—compact sets are not well behaved (see Ap-pendix Chapter A). In addition, the assumption that U is continuous leads to little loss ofgenerality for most economic applications. In all the models we will encounter in this book,U will be continuous. The most restrictive assumption here is that the state variable lies ina compact read more..

##### Page - 235

Introduction to Modern Economic GrowthOur next assumption puts some more structure on the objective function, in particular itensures that the objective function is increasing in the state variables (its ﬁrst K arguments),and that greater levels of the state variables are also attractive from the viewpoint of relaxingthe constraints; i.e., a greater x means more choice.Assumption 6.4. For each y ∈ X, U (·,y) is strictly increasing in each of its ﬁrst Karguments, and G is monotone in the read more..

##### Page - 236

Introduction to Modern Economic Growthstarting from the state vector from tomorrow onwards, x∗ (t +1). In view of the fact thatV∗ in Problem A1 and V in Problem A2 are identical from Theorem 6.1, (6.3) also impliesV (x∗ (t)) = U (x∗ (t) ,x∗ (t +1)) + βV (x∗ (t +1)).Notice also that the second part of Theorem 6.2 is equally important. It states that ifany feasible plan x∗ starting with x (0),thatis, x∗ ∈Φ(x(0)),satisﬁes (6.3), then x∗ attainsV∗ (x (0)).Therefore, this read more..

##### Page - 237

Introduction to Modern Economic Growththere exists a vector of parameters zcontinuously aﬀecting either the constraint correspon-dence Φ or the instantaneous payoﬀ function U , then the same argument establishes that πis also continuous in this vector of parameters. This feature will enable qualitative analysisof dynamic macroeconomic models under a variety of circumstances.Our next result shows that under Assumption 6.4, we can also establish that the valuefunction V is strictly read more..

##### Page - 238

Introduction to Modern Economic Growthand the next section are useful for a good understanding of foundations of dynamic program-ming and should enable the reader to achieve a better understanding of these methods. Thereader may also wish to consult Appendix Chapter A before reading this section.Recall from Appendix Chapter A that (S, d) is a metric space, if S is a space and d isametricdeﬁned over this space with the usual properties. The metric is referred to as “d”since it loosely read more..

##### Page - 239

Introduction to Modern Economic GrowthSince T is a contraction, we have thatd(z2,z1)= d(Tz1,T z0) ≤ βd(z1,z0).Repeating this argument(6.5)d(zn+1,zn)≤ βnd(z1,z0),n =1, 2,...Hence, for any m>n,d(zm,zn) ≤ d(zm,zm−1)+ ... + d(zn+2,zn+1)+ d(zn+1,zn)(6.6)≤¡βm−1+...+βn+1+βn¢d(z1,z0)= βn¡βm−n−1+...+β+1¢d(z1,z0)≤βn1 − βd(z1,z0),where the ﬁrst inequality uses the triangle inequality (which is true for any metric d,seeAppendix Chapter A). The second inequality uses read more..

##### Page - 240

Introduction to Modern Economic GrowthExample 6.3. Consider the following one-dimensional diﬀerential equation(6.7)˙x (t)= f (x (t)) ,with a boundary condition x (0) = c ∈ R. Suppose that f : R→ R is Lipschitz continuousin the sense that it is continuous and also for some M< ∞,it satisﬁes the following bound-edness condition, |f (x00) − f (x0)| ≤ M |x00 − x0| for all x0,x00 ∈ R. The Contraction MappingTheorem, Theorem 6.7, can be used to prove the existence of a continuous read more..

##### Page - 241

Introduction to Modern Economic GrowthProof. Take z0 ∈ S0, and construct the sequence {Tnz0}∞n=0. Each element of thissequence is in S0 by thefactthat T (S0) ⊂ S0. Theorem 6.7 implies that T nz0 → ˆz.Since S0is closed, ˆz ∈ S0, proving part 1 in the theorem.We know that ˆz ∈ S0.Then the fact that T (S0) ⊂ S00 ⊂ S0 implies that ˆz = T ˆz ∈ T (S0) ⊂S00, establishing part 2.¤The second part of this theorem is very important to prove results such as strict concavityor that read more..

##### Page - 242

Introduction to Modern Economic Growthconverse argument,g (x) ≤ f (x)+ kg − f kfor any x ∈ X,(Tg)(x) ≤ T [f + kg − f k](x)for any x ∈ X,(Tg)(x) ≤ (Tf )(x)+ β kg − f k for any x ∈ X.Combining the last two inequalities implieskTf − Tgk ≤ β kf − gk,proving that T is a contraction.¤We will see that Blackwell’s suﬃcient conditions are straightforward to check in manyeconomic applications, including the models of optimal or equilibrium growth.6.4. Proofs of the Main read more..

##### Page - 243

Introduction to Modern Economic Growthus start with Problem A1. Using the notation introduced in this section, we can write thatfor any x (0) ∈ X,V∗(x (0)) =supx∈Φ(x(0))¯U(x).In view of Assumption 6.1, which ensures that all values are bounded, this immediatelyimplies(6.9)V∗(x (0)) ≥¯U(x) for all x∈ Φ(x (0)),since no other feasible sequence of choices can give higher value than the supremum,V∗ (x (0)). However, if some function ˜V satisﬁes condition (6.9), so will α ˜V for read more..

##### Page - 244

Introduction to Modern Economic GrowthNext,takeanarbitraryε>0.By(6.10),thereexistsx0ε=(x (0) ,x0ε (1) ,x0ε (2) ,...) ∈Φ(x(0)) such that¯U¡x0ε¢≥V∗(x(0))−ε.Now since x00ε=(x0ε (1) ,x0ε (2) ,...) ∈Φ(x0ε (1)) and V∗ (x0ε (1)) is the supremum in Problem A1starting with x0ε (1), Lemma 6.1 impliesU¡x(0),x0ε(1)¢+β¯U¡x00ε¢ ≥ V∗ (x (0)) − εU¡x(0),x0ε(1)¢+βV∗¡x0ε(1)¢ ≥ V∗ (x (0)) − ε,The last inequality veriﬁes (6.12) since x0ε (1) ∈ G (x read more..

##### Page - 245

Introduction to Modern Economic GrowthProofofTheorem 6.2. By hypothesis x∗ ≡ (x (0) ,x∗ (1) ,x∗ (2) ,...) is a solutionto Problem A1, i.e., it attains the supremum, V∗ (x (0)) starting from x (0).Let x∗t ≡(x∗ (t) ,x∗ (t +1) ,...).We ﬁrst show that for any t ≥ 0, x∗t attains the supremum starting from x∗ (t),sothat(6.13)¯U(x∗t )= V∗ (x (t)) .The proof is by induction. The base step of induction, for t =0, is straightforward, since, bydeﬁnition, x∗0 = x∗ read more..

##### Page - 246

Introduction to Modern Economic Growththus x∗ attains the optimal value in Problem A1, completing the proof of the second part ofthe theorem.¤We have therefore established that under Assumptions 6.1 and 6.2, we can freely inter-change Problems A1 and A2. Our next task is to prove that a policy achieving the optimalpath exists for both problems. We will provide two alternative proofs for this to show howthis conclusion can be reached either by looking at Problem A1 or at Problem A2, andthen read more..

##### Page - 247

Introduction to Modern Economic GrowthThese two proofs illustrate how diﬀerent approaches can be used to reach the same con-clusion, once the equivalences in Theorems 6.1 and 6.2 have been established.An additional result that follows from the second version of the theorem (which can alsobe derived from version 1, but would require more work), concerns the properties of thecorrespondence of maximizing valuesΠ : X⇒ X.An immediate application of the Theorem of the Maximum, Theorem A.13 in read more..

##### Page - 248

Introduction to Modern Economic Growthconcave, thus T [C0(X)] ⊂ C00(X). Then Theorem 6.8 implies that the unique ﬁxed point V∗is in C00 (X), and hence it is strictly concave.¤proofofCorollary 6.1. Assumption 6.3 implies that U (x, y) is concave in y,andunder this assumption, Theorem 6.4 established that V (y) is strictly concave in y.The sum ofa concave function and a strictly concave function is strictly concave, thus the right-hand sideof Problem A2 is strictly concave in y. Therefore, read more..

##### Page - 249

Introduction to Modern Economic Growthsubgradient p of −V is also a subgradient of −W.Since W is diﬀerentiable at x (0),itssubgradient p must be unique, and another standard result in convex analysis implies thatany convex function with a unique subgradient at an interior point x (0) is diﬀerentiable atx (0). This establishes that −V (·),thus V (·),isdiﬀerentiable as desired.The expression for the gradient (6.4) is derived in detail in the next section.¤6.5. Fundamentals of read more..

##### Page - 250

Introduction to Modern Economic GrowthNow using the notation y∗ = π (x) to denote the optimal policy function (which is single-valued in view of Assumption 6.3) and the fact that DxV (y)= DxV (π (x)),we can combinethese two equations to write(6.21)DyU (x, π (x)) + βDxU (π (x) ,π (π (x))) = 0,where DxU represents the gradient vector of U with respect to its ﬁrst K arguments, andDyU represents its gradient with respect to the second set of K arguments. Notice that(6.21) is a functional read more..

##### Page - 251

Introduction to Modern Economic Growthof choice variables. In contrast, in ﬁnite-dimensional problems, there is no need for such acondition, since the ﬁrst-order conditions are suﬃcient to rule out possible gains when wechange many or all of the control variables at the same time. The role that the transversal-ity condition plays in inﬁnite-dimensional optimization problems will become more apparentafter we see Theorem 6.10 and after the discussion in the next subsection.In the general read more..

##### Page - 252

Introduction to Modern Economic Growthfor any x∈Φ(x(0)). Using the fact that x∗ (0) = x (0) and rearranging terms, we obtain∆x≥limT →∞TXt=0βt [DyU (x∗ (t) ,x∗ (t +1)) + βDxU (x∗ (t +1) ,x∗ (t +2))] · (x∗ (t +1) − x (t +1))− limT →∞βT DxU (x∗ (T ) ,x∗ (T +1)) · x∗ (T +1)+limT →∞βT DxU (x∗ (T ) ,x∗ (T +1)) · x (T +1)).Since x∗ satisﬁes (6.21), the terms in ﬁrstlineare allequal to zero. Moreover, sinceitsatisﬁes(6.25), the second line is read more..

##### Page - 253

Introduction to Modern Economic Growthis diﬀerentiable, the Euler equation for the one-dimensional case, (6.22), implies1xα − y= βV0 (y) .The envelope condition, (6.23), gives:V0 (x)=αxα−1xα − y.Thus using the notation y = π (x) and combining these two equations, we have1xα − π (x)= βαπ (x)α−1π (x)α − π (π (x))for all x,which is a functional equation in a single function, π (x). There are no straightforward waysof solving functional equations, but in most cases read more..

##### Page - 254

Introduction to Modern Economic Growthincome stream of {w (t)}∞t=0, and moreover starts life with a given amount of assets a (0).Hereceives a constant net rate of interest r> 0 on his asset holdings (so that the gross rate ofreturn is 1+ r). To start with, let us suppose that wages are constant, that is, w (t)= w.Then, the utility maximization problem of the individual can be written asmax{c(t),a(t)}∞t=0∞Xt=0βtu (c (t))subject to the ﬂow budget constrainta (t +1) = (1 + r)(a (t)+ w read more..

##### Page - 255

Introduction to Modern Economic Growthdoes not become “too negative” at inﬁnity. However, here we do not need this additionalrestriction, since we have already imposed that a (t) ≥ 0 for all t.Let us also focus on the case where the wealth of the individual is ﬁnite, so a (0) < ∞and w/r < ∞. With these assumptions, let us now write the recursive formulation of theindividual’s maximization problem. The state variable is a (t), and consumption can beexpressed asc (t)= a (t)+ read more..

##### Page - 256

Introduction to Modern Economic Growthby the product of the discount factor and the gross rate of return. Since we have assumed thatβ and (1 + r) are constant, the relationship between today’s and tomorrow’s consumptionnever changes. In particular, since u (·) is assumed to be continuously diﬀerentiable andstrictly concave, u0 (·) always exists and is strictly decreasing. Therefore, the intertemporalconsumption maximization problem implies the following simple rule:(6.31)if r = β−1 read more..

##### Page - 257

Introduction to Modern Economic Growthsequence of labor income {w (t)}∞t=0. It also shows that the exact shape of this labor incomesequence has no eﬀect on the slope or level of the consumption proﬁle.6.5.2. Dynamic Programming Versus the Sequence Problem.To get more in-sights into dynamic programming, let us return to the sequence problem. Also, let us supposethat x is one dimensional and that there is a ﬁnite horizon T . Then the problem becomesmax{x(t+1)}Tt=0TXt=0βtU(x (t) ,x (t read more..

##### Page - 258

Introduction to Modern Economic GrowthNow, heuristically we can derive the transversality condition as an extension of condition(6.32) to T →∞. Take this limit, which implieslimT →∞βT ∂U (x∗ (T ) ,x∗ (T +1))∂x (T +1)x∗ (T +1) = 0.Moreover, as T →∞,wehavethe Eulerequation∂U (x∗ (T ) ,x∗ (T +1))∂x (T +1)+ β∂U (x∗ (T +1) ,x∗ (T +2))∂x (T +1)=0.Substituting this relationship into the previous equation, we obtain− limT →∞βT +1 ∂U (x∗ (T +1) ,x∗ (T read more..

##### Page - 259

Introduction to Modern Economic GrowthWe continue to make the standard assumptions on the production function as in Assump-tions 1 and 2. In addition, we assume that:Assumption 30. u :[c, ∞)→ R is continuously diﬀerentiable and strictly concave forc ∈ [0, ∞).This is considerably stronger than what we need. In fact, concavity or even continuityis enough for most of the results. But this assumption helps us avoid inessential technicaldetails. The lower bound on consumption is imposed to read more..

##### Page - 260

Introduction to Modern Economic Growthfunction c (k). The capital stock of the next period is given by s (k)= f (k)+ (1 − δ) k −c(k).Moreover, V (k) is strictly increasing and concave and s (k) is nondecreasing in k.Proof. Optimality of the solution to the value function (6.35) for the problem (6.33)and (6.34) follows from Theorems 6.1 and 6.2. That V (k) exists follows from Theorem 6.3,and the fact that it is increasing and strictly concave, with the policy correspondence beinga policy read more..

##### Page - 261

Introduction to Modern Economic GrowthConsequently, from Theorem 6.10, we can look at the Euler equations. The Euler equationfrom (6.35) takes the simple form:u0 (c)= βV0 (s)where s denotes the next date’s capital stock. Applying the envelope condition, we haveV0 (k)=£f0(k)+(1−δ)¤u0(c).Consequently, we have the familiar condition(6.37)u0 (c (t)) = β£f0(k(t+1))+(1−δ)¤u0(c(t+1)).As before, a steady state is as an allocation in which the capital-labor ratio and consumptiondo not read more..

##### Page - 262

Introduction to Modern Economic GrowthNext consider the case in which k (1) = s (k (0)) <k (0). The same argument as aboveapplied in reverse now establishes that {k (t)}∞t=0 is a nonincreasing sequence in the compactseth0,ki, thus it converges to a uniquely limit point k (∞). In this case, there are twocandidate values for k (∞), k (∞)= 0 or k (∞)= k∗. The former is not possible, since,as Exercise 6.18 shows, Assumption 2 implies that s (ε) >ε for ε suﬃciently small. Thusk read more..

##### Page - 263

Introduction to Modern Economic Growthk*0Tk(0)Figure 6.1. Turnpike dynamics in a ﬁnite-horizon (T -periods) neoclassicalgrowth model starting with initial capital-labor ratio k (0).The path of the capital-labor ratio thus resembles a turnpike approaching a highway as shownin Figure 6.1 (see Exercise 6.17).6.7. Competitive Equilibrium GrowthOur main interest is not optimal growth, but equilibrium growth. Nevertheless, the Sec-ond Welfare Theorem, Theorem 5.7 of the previous chapter, implies read more..

##### Page - 264

Introduction to Modern Economic GrowthDefinition 6.3. A competitive equilibrium consists of paths of consumption, capital stock,wage rates and rental rates of capital, {C (t) ,K (t) ,w (t) ,R (t)}∞t=0, such that the representa-tive household maximizes its utility given initial capital stock K0 and the time path of prices{w (t) ,R (t)}∞t=0, and thetimepath ofprices {w (t) ,R (t)}∞t=0 is such that given the time pathof capital stock and labor {K (t) ,L (t)}∞t=0 all markets clear.Households read more..

##### Page - 265

Introduction to Modern Economic GrowthNext, market clearing immediately implies that 1+ r (t +1) is given by (6.39), so the capital-labor ratio of the competitive equilibrium is given byβ£f0(k(t+1))+(1−δ)¤=1,Thesteady stateisgiven byβ£f0(k∗)+(1−δ)¤=1.These two equations are identical to equations (6.38) and (6.39), which characterize thesolution to the optimum growth problem. In fact, a similar argument establishes that theentire competitive equilibrium path is identical to the read more..

##### Page - 266

Introduction to Modern Economic GrowthIt is important to emphasize that the treatment in this chapter has assumed away anumber of diﬃcult technical issues. First, the focus has been on discounted problems, whichare simpler than undiscounted problems. In economics, very few situations call for modelingusing undiscounted objective functions (i.e., β =1 rather than β ∈ (0, 1)). More important,throughout we have assumed that payoﬀs are bounded and the state vector x belongs to acompact read more..

##### Page - 267

Introduction to Modern Economic Growthand we have limited the analysis to the case with compact sets and bounded payoﬀ functions.The reader can ﬁnd generalizations of Theorems 6.1-6.6 to certain problems with unboundedreturns and choice sets in Stokey, Lucas and Prescott (1989), Chapter 4 for the deterministiccase, and the equivalent theorems for stochastic dynamic programming problems in theirChapter 9.A much simpler but insightful exposition of dynamic programming is in Sundaram read more..

##### Page - 268

Introduction to Modern Economic Growth(4) Using the previous inequality, the fact that for any B< ∞, Bn/n! → 0 as n → 0and the result in Exercise 6.2, prove that the diﬀerential equation has a uniquecontinuous solution on the compact interval [0,s] for any s ∈ R+.Exercise 6.5. * Recall the Implicit Function Theorem, Theorem A.23 in Appendix ChapterA. Here is a slightly simpliﬁed version of it: consider the function φ (y, x) such that thatφ : R×[a,b] → R is continuously read more..

##### Page - 269

Introduction to Modern Economic Growthsubject tok (t +1) = Ak (t) − c (t)and k (0) = k0. Assume that k (t) ∈£0,¯k¤anda<¯k−1,sothattheutilityfunctionisalwaysincreasing in consumption.(1) Formulate this maximization problem as a dynamic programming problem.(2) Argue without solving this problem that there will exist a unique value functionV (k) and a unique policy rule c = π (k) determining the level of consumption as afunction of the level of capital stock.(3) Solve explicitly for V read more..

##### Page - 270

Introduction to Modern Economic Growth(3) Consider the special case where u (c)=ln c. Provide a closed-form solution for c (0).(4) Next, returning to the general utility function u (·), consider a change in the earningsproﬁle to a new sequence { ˜w (t)}∞t=0 such that for some T< ∞, w (t) < ˜w (t) for allt< T , w (t) ≥ ˜w (t) for all t ≥ T ,andP∞t=0(1+r)−tw(t)=P∞t=0(1+r)−t ˜w (t).What is the eﬀect of this on the initial consumption level and the consumption read more..

##### Page - 271

Introduction to Modern Economic Growth(4) Assuming that V (k) and all the other functions are diﬀerentiable, characterize theEuler equation that determines the optimal path of consumption and capital accu-mulation.(5) Is this Euler equation enough to determine the path of k and c? If not, what othercondition do we need to impose? Write down this condition and explain intuitivelywhy it makes sense.Exercise 6.16. Prove that, as claimed in Proposition 6.4, in the basic discrete-time optimalgrowth read more..

##### Page - 272

CHAPTER 7Review of the Theory of Optimal ControlThe previous chapter introduced the basic tools of dynamic optimization in discrete time.I will now review a number of basic results in dynamic optimization in continuous time–particularly the so-called optimal control approach. Both dynamic optimization in discretetime and in continuous time are useful tools for macroeconomics and other areas of dynamiceconomic analysis. One approach is not superior to another; instead, certain problems read more..

##### Page - 273

Introduction to Modern Economic Growthcontrols are given by y(t) and the resulting behavior of the state variable is summarized byx(t). We also refer to f as the objective function (or the payoﬀ function) and to g as theconstraint function.This problem formulation is general enough to incorporate discounting, since both theinstantaneous payoﬀ function f and the constraint function g depend directly on time inan arbitrary fashion. We will start with the ﬁnite-horizon case and then treat the read more..

##### Page - 274

Introduction to Modern Economic GrowthThese features make it diﬃcult for us to know what type of optimal policy to look for.For example, y may be a highly discontinuous function. It may also hit the boundary ofthe feasible set–thus corresponding to a “corner solution”. Fortunately, in most economicproblems there will be enough structure to make optimal solutions continuous functions.Moreover, in most macroeconomic and growth applications, the Inada conditions make surethat the optimal read more..

##### Page - 275

Introduction to Modern Economic GrowthWe now exploit these features to derive necessary conditions for an optimal path of thisform. To do this, consider the following variationy (t, ε) ≡ ˆy (t)+ εη (t) ,where η (t) is an arbitrary ﬁxed continuous function and ε ∈ R is a scalar. We refer to thisas a variation, because given η (t),by varying ε,we obtain diﬀerent sequences of controls.The problem, of course, is that some of these may be infeasible, i.e., y (t, ε) /∈ Y (t) for read more..

##### Page - 276

Introduction to Modern Economic Growthany λ (·) function, but only for a λ (·) that is chosen appropriately to play the role of thecostate variable.Adding (7.6) to (7.5), we obtain(7.7)W (ε) ≡Z t10{f (t, x (t, ε) ,y (t, ε)) + λ (t)[g (t, x (t, ε) ,y (t, ε)) − ˙x (t, ε)]}dt.To evaluate (7.7), let us ﬁrst consider the integralRt10λ(t)˙x(t,ε)dt. Integrating this expres-sion by parts (see Appendix Chapter B), we obtainZ t10λ (t) ˙x (t, ε) dt = λ (t1) x (t1,ε) − λ (0) x0 read more..

##### Page - 277

Introduction to Modern Economic Growththe case with Lagrange multipliers, the function λ (·) has to be chosen appropriately, and inthis case, it must satisfy(7.9)fy (t, ˆx (t) , ˆy (t)) + λ (t) gy (t, ˆx (t) , ˆy (t)) ≡ 0 for all t ∈ [0,t1] .This immediately implies thatZ t10[fy (t, ˆx (t) , ˆy (t)) + λ (t) gy (t, ˆx (t) , ˆy (t))] η (t) dt =0 for all η (t) .Since η (t) is arbitrary, this implies that xε (t, 0) is also arbitrary. Thus the condition in (7.8)canholdonlyifthe read more..

##### Page - 278

Introduction to Modern Economic Growthsubject to (7.2) and (7.3). The only diﬀerence is that there is no longer a choice over theterminal value of the state variable, x1. In this case, we have:Theorem 7.2. (Necessary Conditions II) Consider the problem of maximizing (7.11)subject to (7.2) and (7.3), with f and g continuously diﬀerentiable. Suppose that this problemhas an interior continuous solution ˆy (t) ∈IntY (t) with corresponding path of state variableˆx (t). Then there exists a read more..

##### Page - 279

Introduction to Modern Economic Growthand a (t) ≥ 0,withaninitial valueof a (0) > 0. In this problem, consumption is the controlvariable, while the asset holdings of the individual are the state variable.To be able to apply Theorem 7.2, we need a terminal condition for a (t), i.e., some valuea1 such that a (1) = a1. The economics of the problem makes it clear that the individualwould not like to have any positive level of assets at the end of his planning horizon (sincehe could consume all read more..

##### Page - 280

Introduction to Modern Economic Growthfront-loaded consumption proﬁle. In contrast, when ρ<r, the opposite reasoning applies andthe individual chooses a back-loaded consumption proﬁle. These are of course identical to theconclusions we reached in the discrete time intertemporal consumer optimization problem inExample 6.5, in particular, equation (6.31).The only variable to determine in order to completely characterize the consumption proﬁleis the initial value of the costate variable. read more..

##### Page - 281

Introduction to Modern Economic Growth7.2. The Maximum Principle: A First Look7.2.1. The Hamiltonian and the Maximum Principle.By analogy with the La-grangian, a much more economical way of expressing Theorem 7.2 is to construct the Hamil-tonian:3(7.12)H (t, x, y, λ) ≡ f (t, x (t) ,y (t)) + λ (t) g (t, x (t) ,y (t)) .Since f and g are continuously diﬀerentiable, so is H. Denote the partial derivatives of theHamiltonian with respect to x (t), y (t) and λ (t),by Hx, Hy and Hλ. Theorem 7.2 read more..

##### Page - 282

Introduction to Modern Economic Growth(1) As in the usual constrained maximization problems, we ﬁnd the optimal solution bylooking jointly for a set of “multipliers” λ (t) and the optimal path of the controland state variables, ˆy (t) and ˆx (t). Here the multipliers are referred to as the costatevariables.(2) Again as with the Lagrange multipliers in the usual constrained maximization prob-lems, the costate variable λ (t) is informative about the value of relaxing the con-straint (at read more..

##### Page - 283

Introduction to Modern Economic GrowthTheorem 7.5. (Arrow’s Suﬃcient Conditions) Consider the problem of maximizing(7.1) subject to (7.2) and (7.3), with f and g continuously diﬀerentiable. Deﬁne H (t, x, y, λ)as in (7.12), and suppose that an interior continuous solution ˆy (t) ∈IntY (t) and the corre-sponding path of state variable ˆx (t) satisfy (7.13)-(7.15). Given the resulting costate variableλ (t),deﬁne M (t, ˆx, λ) as the maximized Hamiltonian as in (7.16). If M (t, read more..

##### Page - 284

Introduction to Modern Economic GrowthIntegrating the last term by parts and using the fact that by feasibility x (0) = ˆx (0) = x0and by the transversality condition λ (t1)= 0,we obtainZ t10˙λ (t)(x (t) − ˆx (t)) dt = −Z t10λ (t)µ˙x(t)− ·ˆx (t)¶dt.Substituting this into (7.19), we obtainW (x (t) ,y (t)) ≤ W (ˆx (t) , ˆy (t))(7.20)+Z t10λ (t)[g (t, ˆx (t) , ˆy (t)) − g (t, x (t) ,y (t))] dt+Z t10λ (t)∙˙x(t)− ·ˆx (t)¸dt.Since by deﬁnition of the admissible read more..

##### Page - 285

Introduction to Modern Economic GrowthSimon and Blume, 1994). These are slightly more messy to express, and since we will make nouse of the constrained maximization problems in this book, we will not state these theorems.The vector-valued theorems are direct generalizations of the ones presented above andare useful in growth models with multiple capital goods. In particular, let(7.21)maxx(t),y(t)W (x (t) , y(t)) ≡Z t10f (t, x(t) , y(t)) dtsubject to(7.22)˙x(t)= g (t, x(t) , y(t)) read more..

##### Page - 286

Introduction to Modern Economic Growththen ˆy(t) and the corresponding ˆx(t) achieves a global maximum of (7.21). Moreover, ifH (t, x, y, λ) is strictly jointly concave, then the pair (ˆx(t) , ˆy(t)) achieves the unique globalmaximum of (7.21).Theorem 7.8. (Arrow’s Suﬃcient Conditions) Consider the problem of maximiz-ing (7.21) subject to (7.22) and (7.23), with f and g continuously diﬀerentiable. DeﬁneH (t, x, y, λ) as in (7.24), and suppose that an interior continuous solution read more..

##### Page - 287

Introduction to Modern Economic Growthsubject to(7.29)˙x (t)= g (t, x (t) ,y (t)) ,and(7.30)y (t) ∈ R for all t, x (0) = x0 and limt→∞x (t) ≥ x1.The main diﬀerence is that now time runs to inﬁnity. Note also that this problem allowsfor an implicit choice over the endpoint x1, since there is no terminal date. The last part of(7.30) imposes a lower bound on this endpoint. In addition, we have further simpliﬁed theproblem by removing the feasibility requirement that the control y (t) read more..

##### Page - 288

Introduction to Modern Economic Growthassumed that all admissible pairs give ﬁnite value, so that V (t0,x0) < ∞, and our focusthroughout will be on admissible pairs (ˆx (t) , ˆy (t)) that are optimal solutions to (7.28) subjectto (7.29) and (7.30), and thus reach the value V (t0,x0).Our ﬁrst result is a weaker version of the Principle of Optimality, which we encounteredin the context of discrete time dynamic programming in the previous chapter:Lemma 7.1. (Principle of Optimality) read more..

##### Page - 289

Introduction to Modern Economic Growthdiscounting in (7.33). This is not the case, since the discounting is embedded in the instanta-neous payoﬀ function f , and is thus implicit in V (t1, ˆx (t1)). Second, this lemma may appearto contradict our discussion of “time consistency” in the previous chapter, since the lemmais stated without additional assumptions that ensure time consistency. The important pointhere is that in the time consistency discussion, the decision-maker considered read more..

##### Page - 290

Introduction to Modern Economic GrowthTheorem 7.9 states that if the optimal control, ˆy (t), is a continuous function of time, theconditions (7.34)-(7.36) are also satisﬁed. This qualiﬁcation is necessary, since we now allowˆy (t) to be a piecewise continuous function of time. The fact that ˆy (t) is a piecewise contin-uous function implies that the optimal control may include discontinuities, but these will berelatively “rare”–in particular, it will be continuous “most of the read more..

##### Page - 291

Introduction to Modern Economic GrowthV with respect to both time and the state variable x. Third, this partial diﬀerential equationalso has a similarity to the Euler equation derived in the context of discrete time dynamicprogramming. In particular, the simplest Euler equation (6.22) required the current gainfrom increasing the control variable to be equal to the discounted loss of value. The currentequation has a similar interpretation, with the ﬁrst term corresponding to the current read more..

##### Page - 292

Introduction to Modern Economic Growthonly concavity (not strict concavity), Theorems 7.11 and 7.12 can be applied to models withconstant returns and endogenous growth, thus will be particularly useful in later chapters.Notice that both of these suﬃciency theorems involve the diﬃcult to check condition thatlimt→∞ λ (t)(x (t) − ˜x (t)) ≤ 0 for all ˜x (t) implied by an admissible control path ˜y (t).Thiscondition will disappear when we can impose a proper transversality read more..

##### Page - 293

Introduction to Modern Economic GrowthTherefore, maximizing (7.40) is equivalent to maximizing instantaneous returns as givenby the function f (t, ˆx (t) ,y (t)),plusthe value ofstock of x (t),asgiven by λ (t),times theincrease in the stock, ˙x (t). This implies that the essence of the Maximum Principle is tomaximize the ﬂow return plus the value of the current stock of the state variable. Thisstock-ﬂow type maximization has a clear economic logic.Let us next turn to the interpreting the read more..

##### Page - 294

Introduction to Modern Economic Growthderivation is provided in Exercise 7.16. Intuitively, we can think of V as the value of anassettradedinthe stockmarketand ρ as the required rate of return for (a large numberof) investors. When will investors be happy to hold this asset? Loosely speaking, they willdo so when the asset pays out at least the required rate of return. In contrast, if the assetpays out more than the required rate of return, there would be excess demand for it fromthe investors read more..

##### Page - 295

Introduction to Modern Economic Growththe state variable resulting from the perturbed control yδ,with xδ (t0 + ∆t) being the valueat time t0 + ∆t. Note by construction xδ (t0)= ˆx (t0) (since yδ (t)= ˆy (t) for all t ∈ [0,t0]).Since the pair (ˆx (t) , ˆy (t)) is optimal, we have thatV (t0, ˆx (t0)) =Z ∞t0f (t, ˆx (t) , ˆy (t)) dt≥Z ∞t0f (t, xδ (t) ,yδ (t)) dt=Z t0+∆tt0f (t, xδ (t) ,yδ (t)) dt + V (t0 + ∆t, xδ (t0 + ∆t)) ,where the last equality uses the fact read more..

##### Page - 296

Introduction to Modern Economic Growthinequality (7.43) can be written asf (t0, ˆx (t0) , ˆy (t0)) + λ (t0) g (t0, ˆx (t0) , ˆy (t0)) ≥ f (t0,xδ (t0) ,yδ (t0))+λ (t0) g (t0,xδ (t0) ,yδ (t0)) ,or equivalently,H (t0, ˆx (t0) , ˆy (t0)) ≥ H (t0,xδ (t0) ,yδ (t0))for all admissible (xδ (t0) ,yδ (t0)) .Therefore,H (t, ˆx (t) , ˆy (t)) ≥ maxyH (t, ˆx (t) ,y) .This establishes the Maximum Principle.The necessary condition (7.34) directly follows from the Maximum Principle read more..

##### Page - 297

Introduction to Modern Economic Growthwhere c∗ ≡ [k∗]α − δk∗ and k∗ ≡ (α/δ)1/(1−α).In other words, c∗ is the maximum level ofconsumption that can be achieved in the steady state of this model and k∗ is the correspondingsteady-state level of capital. This way of writing the objective function makes sure that theintegral converges and takes a ﬁnite value (since c (t) cannot exceed c∗ forever).The Hamiltonian is straightforward to construct; it does not explicitly read more..

##### Page - 298

Introduction to Modern Economic GrowthNowtakethe limitas t →∞.Since limt→∞ V (t, ˆx (t)) exists, wehavethateitherlimt→∞ ∂V (t, ˆx (t)) /∂t > 0,so that limt→∞ V (t, ˆx (t)) = +∞,or limt→∞ ∂V (t, ˆx (t)) /∂t < 0everywhere, so that limt→∞ V (t, ˆx (t)) = −∞ or limt→∞ ∂V (t, ˆx (t)) /∂t =0.The ﬁrst twopossibilities are ruled out by the hypothesis that an optimal solution that reaches the maxi-mum exists. Thus we must have limt→∞ ∂V read more..

##### Page - 299

Introduction to Modern Economic GrowthThe following result establishes the necessity of a stronger transversality condition undersome additional assumptions, which are typically met in economic applications. In prepa-ration for this result, let us refer to the functions f (x, y) and g (x, y) as weakly monotone,if each one is monotone in each of its arguments (for example, nondecreasing in x and non-increasing in y). Furthermore, let us simplify the statement of this theorem by assumingthat the read more..

##### Page - 300

Introduction to Modern Economic Growthfail to exist). The latter fact also implies that limt→∞ ˙x (t) exists (though it may also be inﬁ-nite). Moreover, limt→∞ ˙x (t) is nonnegative, since otherwise the condition limt→∞ x (t) ≥ x1would be violated. From (7.53), (7.55) implies that as t →∞, λ (t) ≡ exp(−ρt)μ(t) → κ forsome κ ∈ R+.Suppose ﬁrst that limt→∞ ˙x (t)=0.Then limt→∞ ˆx (t)= ˆx∗ ∈ R (i.e., a ﬁnite value). Thisalso implies that f (ˆx read more..

##### Page - 301

Introduction to Modern Economic Growthtransversality condition (7.56) will be necessary. Notice that compared to the transversalitycondition in the ﬁnite-horizon case (e.g., Theorem 7.1), there is the additional term exp (−ρt).This is because the transversality condition applies to the original costate variable λ (t),i.e.,limt→∞ [x (t) λ (t)] = 0, and as shown above, the current-value costate variable μ (t) is givenby μ (t)=exp (ρt) λ (t). Note also that the stronger read more..

##### Page - 302

Introduction to Modern Economic Growthso that his objective function at time t =0 is to maximizeZ ∞0exp (−ρt)u(y (t)) dt.The constraint is that the remaining size of the resource at time t, x (t) evolves according to˙x (t)= −y (t) ,which captures the fact that the resource is not renewable and becomes depleted as more ofit is consumed. Naturally, we also need that x (t) ≥ 0.The current-value Hamiltonian takes the formˆH (x (t) ,y (t) ,μ (t)) = u (y (t)) − μ (t) y (t) .Theorem 7.14 read more..

##### Page - 303

Introduction to Modern Economic GrowthSince along any optimal path we must have limt→∞ ˆx (t)= 0, wehavethatZ ∞0u0−1 [μ (0) exp (ρs)] ds =1.Therefore, the initial value of the costate variable μ (0) must be chosen so as to satisfy thisequation.Notice also that in this problem both the objective function, u (y (t)),and the constraintfunction, −y (t), are weakly monotone in the state and the control variables, so the strongerform of the transversality condition, (7.56), holds. You read more..

##### Page - 304

Introduction to Modern Economic Growthstructure to ensure this. For example, in most of the problems we will encounter Inada-type conditions ensure that optimal controls remain within the interior of the feasible set andconsumption-smoothing and no-arbitrage arguments rule out discontinuous controls. In thesecases, continuous solutions can be shown to exist. Moreover, using Theorem 7.15, we canoften establish that these solutions are unique. Throughout the rest of the book, I will followthe read more..

##### Page - 305

Introduction to Modern Economic GrowthMoreover, the ﬁrst necessary condition immediately implies that μ (t) > 0 (since u0 > 0everywhere). Consequently, the current-value Hamiltonian given in (7.60) consists of thesum of two strictly concave functions and is itself strictly concave and thus satisﬁes theconditions of Theorem 7.15. Therefore, a solution that satisﬁes these necessary conditions infact gives a global maximum. Characterizing the solution of these necessary conditions read more..

##### Page - 306

Introduction to Modern Economic GrowthNotice that φ (i) does not contribute to capital accumulation; it is simply a cost. More-over, since φ is strictly convex, it implies that it is not optimal for the ﬁrm to make “large”adjustments. Therefore it will act as a force towards a smoother time path of investment.To characterize the optimal investment plan of the ﬁrm, let us write the current-valueHamiltonian:ˆH (k, i, q) ≡ [f (k (t)) − i (t) − φ (i (t))] + q (t)[i (t) − δk (t)] read more..

##### Page - 307

Introduction to Modern Economic Growthof capital satisﬁes the ﬁrst-order condition (corresponding to the right-hand side of (7.64)being equal to zero):f0 (k∗)=(r + δ)¡1+φ0(δk∗)¢.This ﬁrst-order condition diﬀers from the standard “modiﬁed golden rule” condition, whichrequires the marginal product of capital to be equal to the interest rate plus the depreciationrate, because an additional cost of having a higher capital stock is that there will have tobe more investment to read more..

##### Page - 308

Introduction to Modern Economic Growthi(t)k(t)0k(t)=0i(t)=0k*k(0)i’(0)i’’(0)Stable armδk*Figure 7.1. Dynamics of capital and investment in the q-theory.Theorem 7.19. Consider the following nonlinear autonomous diﬀerential equation(7.66)˙x(t)= G[x (t)]where G: Rn → Rn and suppose that Gis continuously diﬀerentiable, with initial value x(0).Let x∗ be a steady-state of this system, given by F(x∗)=0.DeﬁneA=DG (x∗) ,and suppose that m ≤ n of the eigenvalues of Ahave negative read more..

##### Page - 309

Introduction to Modern Economic GrowthThe curve corresponding to ˙k =0, (7.61), is upward sloping, since a greater level of capitalstock requires more investment to replenish the depreciated capital. When we are above thiscurve, there is more investment than necessary for replenishment, so that ˙k> 0.When weare below this curve, then ˙k< 0. On the other hand, the curve corresponding to ˙i =0, (7.64),can be nonmonotonic. Nevertheless, it is straightforward to verify that in the read more..

##### Page - 310

Introduction to Modern Economic Growthviolating the transversality condition. In contrast, if we start with i00 (0) <i (0) as the initiallevel, i (t) wouldtendto0in ﬁnite time (as shown by the fact that the trajectories hitthe horizontal axis) and k (t) would also tend towards zero (though not reaching it in ﬁnitetime). After the time where i (t)=0,wealsohave q (t)=1 and thus ˙q (t)=0 (from (7.62)).Moreover, by the Inada conditions, as k (t) → 0, f0 (k (t)) →∞. Consequently, after read more..

##### Page - 311

Introduction to Modern Economic Growthmaterial covered here may have been less familiar than the discrete time optimization methodspresented in the previous chapter. Part of the diﬃculty arises from the fact that optimizationhere is with respect to functions, even when the horizon is ﬁnite (rather than with respectto vectors or inﬁnite sequences as in the discrete time case). This introduces a range ofcomplications and some technical diﬃculties, which are not of great interest in the read more..

##### Page - 312

Introduction to Modern Economic Growth7.10. References and LiteratureThe main material covered in this chapter is the topic of many excellent applied mathe-matics and engineering books. The purpose here has been to provide a review of the resultsthat are most relevant for economists, together with simpliﬁed versions of themostimportantproofs. The ﬁrst part of the chapter is closer to the calculus of variations theory, because itmakes use of variational arguments combined with continuity read more..

##### Page - 313

Introduction to Modern Economic GrowthKamien and Schwartz (1991) and Leonard and Van Long (1992). Another classic is Arrowand Kurz’s (1970) book, which covers the same material and also presents rich economicinsightongrowth theoryand relatedproblems. This book also states and provides a proofof Arrow’s suﬃciency theorem, which also appears in Arrow (1968).Two recent books on applications of optimal control in economics, Caputo (2005) andWeitzman (2003), might be more readable. My treatment read more..

##### Page - 314

Introduction to Modern Economic Growthfound in any graduate-level economics textbook, for example, Blanchard and Fisher (1989) orRomer (1996), as well as in Dixit and Pindyck’s (1994) book on investment under uncertaintyand Caballero’s (1999) survey. Caballero (1999) also includes a critique of the q-theory.7.11. ExercisesExercise 7.1. Consider the problem of maximizing (7.1) subject to (7.2) and (7.3) as inSection 7.1. Suppose that for the pair (ˆx (t) , ˆy (t)) there exists a time read more..

##### Page - 315

Introduction to Modern Economic GrowthNow, without loss of any generality let us take (z0,u0)=(0, 0) and let t = z to transformthe problem into a more familiar form, which becomes that of maximizing−Z t10q1+[x0(t)]2dt.Prove that the solution to this problem requiresdhx0(t)³1+(x0(t))2´idt=0.Show that this is only possible if x00 (t)=0, so that the shortest path between two points isa straight-line.Exercise 7.5. Prove Theorem 7.2, in particular, paying attention to constructing read more..

##### Page - 316

Introduction to Modern Economic Growth(1) Show that W (x (t) ,y (t)) can be written asW (x (t) ,y (t)) =Z t10hH(t,x(t),y(t))+˙λ(t)x(t)idt−λ(t1)x(t1)+λ(0)x0,where H (t, x, y) ≡ f (t, x (t) ,y (t)) + λ (t) g (t, x (t) ,y (t)) is the Hamiltonian andλ (t) is the costate variable.(2) Now suppose that the pair (ˆx (t) , ˆy (t)) together with terminal date ˆt1 constitutesan optimal solution for this problem. Consider the following class of variations:y (t, ε)=ˆy (t)+ εη (t) for t read more..

##### Page - 317

Introduction to Modern Economic Growthwhich is well deﬁnedinviewofExercise7.15. NowwriteV (x (0)) = f ((ˆx (t) , ˆy (t))) ∆t + o (∆t)+Z ∞∆texp (−ρt)f ((ˆx (t) , ˆy (t))) dt,where o (∆t) denotes second-order terms that satisfy lim∆t→0 o (∆t) /∆t =0.Explain whythis equation can be written asV (x (0)) = f ((ˆx (t) , ˆy (t))) ∆t + o (∆t)+exp (−ρ∆t)V (x (∆t))[Hint: again use Exercise 7.15]. Now subtract V (x (∆t)) from both sides and divide bothsides by ∆t read more..

##### Page - 318

Introduction to Modern Economic Growth(4) Show that this optimal solution violates the condition that limt→∞ exp (−ρt)μ(t),but satisﬁes (7.56).Exercise 7.20. Consider the following optimal growth model without discounting:maxZ ∞0[u (c (t)) − u (c∗)] dtsubject to˙k (t)= f (k (t)) − c (t) − δk (t)with initial condition k (0) > 0,and c∗ deﬁned as the golden rule consumption levelc∗ = f (k∗) − δk∗where k∗ is the golden rule capital-labor ratio given by f0 read more..

##### Page - 319

Introduction to Modern Economic GrowthSuppose that u (·) is strictly increasing and strictly concave, with limc→∞ u0 (c)= 0 andlimc→0 u0 (c)= ∞,and g (·) is increasing and strictly concave with limx→∞ g0 (x)= 0 andlimx→0 g0 (x)= ∞.(1) Set up the current value Hamiltonian and derive the Euler equations for an optimalpath.(2) Show that the standard transversality condition and the Euler equations are neces-sary and suﬃcient for a solution.(3) Characterize the optimal path of read more..

##### Page - 320

Part 3Neoclassical Growth read more..

##### Page - 321

This part of the book covers the basic workhorse models of the theory of economic growth.We start with the inﬁnite-horizon neoclassical growth model, which we have already encoun-tered in the previous chapters. A closely related model is the baseline overlapping-generationsmodel of Samuelson and Diamond, and this is the topic of Chapter 9. Despite the similaritiesbetween the two models, they have quite diﬀerent normative and positive implications, andeach model may be appropriate for read more..

##### Page - 322

CHAPTER 8The Neoclassical Growth ModelWe are now ready to start our analysis of the standard neoclassical growth model (alsoknown as the Ramsey or Cass-Koopmans model). This model diﬀers from the Solow modelonly in one crucial respect: it explicitly models the consumer side and endogenizes savings.In other words, it allows consumer optimization. Beyond its use as a basic growth model,this model has become a workhorse for many areas of macroeconomics, including the analysisof ﬁscal policy, read more..

##### Page - 323

Introduction to Modern Economic Growthcooperatively. This implies that the objective function of each household at time t =0,U (0), can be written as(8.3)U (0) ≡Z ∞0exp (−(ρ − n) t) u (c (t)) dt,where c (t) is consumption per capita at time t, ρ is the subjective discount rate, and theeﬀective discount rate is ρ −n, since it is assumed that the household derives utility from theconsumption per capita of its additional members in the future as well (see Exercise 8.1).It is useful read more..

##### Page - 324

Introduction to Modern Economic Growththat, output per capita is given byy (t) ≡Y (t)L (t)= F∙K(t)L (t), 1¸≡ f (k (t)) ,where, as before,(8.4)k (t) ≡K (t)L (t).Competitive factor markets then imply that, at all points in time, the rental rate of capitaland the wage rate are given by:(8.5)R (t)= FK[K(t),L(t)] = f0 (k(t)).and(8.6)w (t)= FL[K(t),L(t)] = f (k (t)) − k (t) f0 (k(t)).The household optimization side is more complicated, since each household will solve acontinuous time read more..

##### Page - 325

Introduction to Modern Economic Growthper capita will be equal to the capital stock per capita (or the capital-labor ratio in theeconomy), that is,a (t)= k (t) .Moreover, since there is no uncertainty here and a depreciation rate of δ, the market rate ofreturn on assets will be given by(8.8)r (t)= R (t) − δ.The equation (8.7) is only a ﬂow constraint. As already noted above, it is not suﬃcientas a proper budget constraint on the individual (unless we impose a lower bound on assets,such read more..

##### Page - 326

Introduction to Modern Economic Growththe chain-letter scheme) and pay his or her previous debts using current borrowings. Theconsequence of this scheme would be that the asset holding of the individual would tend to−∞ as time goes by, clearly violating feasibility at the economy level.To understand where this form of the no-Ponzi-game condition comes from, multiply bothsides of (8.9) by exp³−RT0r(s)ds´toobtainZ T0c (t) L(t)expµ−Z t0r (s) ds¶dt+expµ−Z T0r (s) ds¶A(T)=Z T0w (t) L read more..

##### Page - 327

Introduction to Modern Economic GrowthDefinition 8.2. A competitive equilibrium of the Ramsey economy consists of pathsof per capita consumption, capital-labor ratio, wage rates and rental rates of capital,[c (t) ,k (t) ,w (t) ,R (t)]∞t=0, such that the representative household maximizes (8.3) subjectto (8.7) and (8.10) given initial capital-labor ratio k (0),factor prices [w (t) ,R (t)]∞t=0 as in(8.5) and (8.6), and the rate of return on assets r (t) given by (8.8).8.2.2. Household read more..

##### Page - 328

Introduction to Modern Economic GrowthTo make more progress, let us diﬀerentiate this with respect to time and divide by μ (t),which yieldsu00 (c (t)) c (t)u0 (c (t))˙c (t)c (t)=˙μ (t)μ (t).Substituting this into (8.12), we obtain another form of the famous consumer Euler equation:(8.14)˙c (t)c (t)=1εu (c(t))(r (t) − ρ)where(8.15)εu (c (t)) ≡−u00 (c (t)) c (t)u0 (c (t))is the elasticity of the marginal utility u0 (c(t)). This equation is closely related to the con-sumer Euler read more..

##### Page - 329

Introduction to Modern Economic Growthwhere the second line uses the ﬁrst optimality condition of the current-value Hamiltonian attime t =0. Now substituting into the transversality condition, we havelimt→∞∙exp(−(ρ−n)t)a(t)u0(c(0))expµ−Z t0(r (s) − ρ) ds¶¸ =0,limt→∞∙a(t)expµ−Z t0(r (s) − n) ds¶¸ =0,which implies that the strict no-Ponzi condition, (8.11) has to hold. Also, for future reference,notes that, since a (t)= k (t), the transversality condition is also read more..

##### Page - 330

Introduction to Modern Economic Growthfor example, εu (c (s)) = θ, this equation simpliﬁes toc (t)= c (0) expµµ¯r(t)−ρθ¶t¶,and moreover, the lifetime budget constraint simpliﬁes toZ ∞0c (t)exp (−(¯r(t) − n) t) dt = a (0) +Z ∞0w (t)exp (−(¯r(t) − n) t) dt,and substituting for c (t) into this lifetime budget constraint in this iso-elastic case, we obtain(8.19)c (0) =Z ∞0expµ−µ(1−θ)¯r(t)θ−ρθ+ n¶t¶dt∙a(0)+Z ∞0w (t)exp (−(¯r(t) − n) t) dt¸as the read more..

##### Page - 331

Introduction to Modern Economic Growthdo not need to appeal to these theorems since in this together case it is straightforward toshow the equivalence of the two problems.To do this, let us once again set up the current-value Hamiltonian, which in this casetakes the formˆH (k, c, μ)= u (c (t)) + μ (t)[f (k (t)) − (n + δ)k (t) − c (t)] ,with state variable k, control variable c and current-value costate variable μ. As noted inthe previous chapter, in the relevant range for the capital read more..

##### Page - 332

Introduction to Modern Economic Growthwhich is the equivalent of the steady-state relationship in the discrete-time optimal growthmodel.2 This equation pins down the steady-state capital-labor ratio only as a function ofthe production function, the discount rate and the depreciation rate. This corresponds tothe modiﬁed golden rule, rather than the golden rule we saw in the Solow model (see Exercise8.8). The modiﬁed golden rule involves a level of the capital stock that does not read more..

##### Page - 333

Introduction to Modern Economic Growthc∗ (a, ρ, n, δ) when the underlying parameters are a, ρ, n and δ. Then we have∂k∗ (a, ρ, n, δ)∂a> 0,∂k∗ (a, ρ, n, δ)∂ρ< 0,∂k∗ (a, ρ, n, δ)∂n=0 and∂k∗ (a, ρ, n, δ)∂δ< 0∂c∗ (a, ρ, n, δ)∂a> 0,∂c∗ (a, ρ, n, δ)∂ρ< 0,∂c∗ (a, ρ, n, δ)∂n< 0 and∂c∗ (a, ρ, n, δ)∂δ< 0.Proof. See Exercise 8.5.¤The new results here relative to the basic Solow model concern the comparative read more..

##### Page - 334

Introduction to Modern Economic GrowthAs we already discussed in the context of the q-theory of investment, this combination of aninitial condition and a transversality condition is quite typical for economic optimal controlproblems where we are trying to pin down the behavior of both state and control variables.This means that we will again use the notion of saddle-path stability introduced in Theorems7.18 and 7.19 instead of those in Theorems 2.4, 2.5 and 2.6. In particular, the read more..

##### Page - 335

Introduction to Modern Economic Growthc(t)kgold0k(t)k(0)c’(0)c’’(0)c(t)=0k(t)=0k*c(0)c*kFigure 8.1. Transitional dynamics in the baseline neoclassical growth model.accumulate continuously until the maximum level of capital (reached with zero consumption)¯k> kgold. Continuous capital accumulation towards ¯k with no consumption would violate thetransversality condition. This establishes that the transitional dynamics in the neoclassicalgrowth model will take the following simple form: c read more..

##### Page - 336

Introduction to Modern Economic Growthand˙c (t)c (t)=1εu (c (t))¡f0(k(t))−δ−ρ¢.Linearizing these equations around the steady state (k∗,c∗), we have (suppressing time de-pendence)˙k = constant +¡f0(k∗)−n−δ¢(k−k∗)−c˙c =constant +c∗f00 (k∗)εu (c∗)(k − k∗) .Moreover, from (8.21), f0 (k∗) − δ = ρ, so the eigenvalues of this two-equation system aregiven by the values of ξ that solve the following quadratic form:detÃ ρ − n − read more..

##### Page - 337

Introduction to Modern Economic GrowthThe constant returns to scale feature again enables us to work with normalized variables.Now let us deﬁneˆy (t) ≡Y (t)A (t) L (t)= F∙ K (t)A (t) L (t), 1¸≡ f (k (t)) ,where(8.27)k (t) ≡K (t)A (t) L (t).is the capital to eﬀective capital-labor ratio, which is deﬁned taking into account that eﬀectivelabor is increasing because of labor-augmenting technological change. Naturally, this is similarto the way that the eﬀective capital-labor read more..

##### Page - 338

Introduction to Modern Economic GrowthConstant relative risk aversion (CRRA) utility function satisﬁes the property that R is con-stant. Now integrating both sides of the previous equation, setting R to a constant, impliesthat the family of CRRA utility functions is given byU (c)=½ c1−θ−11−θif θ 6=1 and θ ≥ 0ln cif θ =1,with the coeﬃcient of relative risk aversion given by θ. In writing this expression, we separatedthecasewhere θ =1,since¡c1−θ−1¢/(1−θ) is undeﬁned read more..

##### Page - 339

Introduction to Modern Economic Growthis the case because most of the models we focus on in this book do not feature uncertainty,so that attitudes towards risk are not important. However, as noted before and illustratedin Exercise 5.2 in Chapter 5, with time-separable utility functions the coeﬃcient of relativerisk aversion in the inverse of the intertemporal elasticity of substitution are identical. Theintertemporal elasticity of substitution is particularly important in growth models because read more..

##### Page - 340

Introduction to Modern Economic GrowthWe will see that this normalized consumption level will remain constant along the BGP. Inparticular, we have˙c (t)c (t)≡·˜c (t)˜c (t)− g=1θ(r (t) − ρ − θg) .Moreover, for the accumulation of capital stock, we have˙k (t)= f (k (t)) − c (t) − (n + g + δ) k (t) ,where recall that k (t) ≡ K (t) /A (t) L (t).The transversality condition, in turn, can be expressed as(8.32)limt→∞½k(t)expµ−Z t0£f0(k(s))−g−δ−n¤ds¶¾=0.In read more..

##### Page - 341

Introduction to Modern Economic GrowthNote that this assumption strengthens Assumption 40 when θ< 1. Alternatively, recallthat in steady state we have r = ρ + θg and the growth rate of output is g + n. Therefore,Assumption 4 is equivalent to requiring that r> g + n. We will encounter conditions likethis all throughout, and they will also be related to issues of “dynamic eﬃciency” as we willsee below.The following is an immediate generalization of Proposition 8.2:Proposition 8.6. read more..

##### Page - 342

Introduction to Modern Economic GrowthProof. See Exercise 8.9.¤It is also useful to brieﬂy look at an example with Cobb-Douglas technology.Example 8.2. Consider the model with CRRA utility and labor-augmenting technologi-cal progress at the rate g. Assume that the production function is given by F (K, AL)=Kα(AL)1−α,sothatf (k)= kα,and thus r = αkα−1 − δ. In this case, suppressing time dependence to simplify notation, theEuler equation read more..

##### Page - 343

Introduction to Modern Economic Growthrepresented by (k∗,c∗). Now imagine that the discount rate declines to ρ0 <ρ.How does theequilibrium path change?We know from Propositions 8.6 and 8.7 that at the new discount rate ρ0 > 0,there existsa unique steady state equilibrium that is saddle path stable. Let this steady state be de-noted by (k∗∗,c∗∗). Therefore, the equilibrium will ultimately tend to this new steady-stateequilibrium. Moreover, since ρ0 <ρ, we know that the read more..

##### Page - 344

Introduction to Modern Economic Growthc(t)kgold0k(t)k*c(t)=0k(t)=0k**c**c*kFigure 8.2. The dynamic response of capital and consumption to a declinein the discount rate from ρ to ρ0 <ρ.If we were to go back to the proximate causes of diﬀerences in income per capita ofeconomic growth across countries, this model would give us a way of understanding thosediﬀerences only in terms of preference and technology parameters. However, as alreadydiscussed in Chapter 4, and we would also like to read more..

##### Page - 345

Introduction to Modern Economic Growthlinear tax policy. Suppose that returns on capital net of depreciation are taxed at the rateτ and the proceeds of this are redistributed back to the consumers. In that case, the capitalaccumulation equation, in terms of normalized capital, still remains as above:˙k (t)= f (k (t)) − c (t) − (n + g + δ) k (t) ,but the net interest rate faced by households now changes to:r (t)=(1 − τ )¡f0(k(t))−δ¢,because of the taxation of capital returns. The read more..

##### Page - 346

Introduction to Modern Economic GrowthLet us assume that there is no population growth, so that cj is both total or per capitaconsumption. Equation (8.38) imposes that all countries have the same discount rate ρ (seeExercise 8.16).All countries also have access to the same production technology given by the Cobb-Douglas production function(8.39)Yj = K1−αj(AHj)α ,with Hj representing the exogenously given stock of eﬀective labor (human capital). Theaccumulation equation is˙Kj = Ij − read more..

##### Page - 347

Introduction to Modern Economic GrowthNow substituting this into (8.39), and comparing two countries with diﬀerent taxes (butthe same human capital), we obtain the relative incomes as(8.41)Y (τ )Y (τ0)=µ1+τ01+ τ¶1−ααSo countries that tax investment, either directly or indirectly, at a higher rate will be poorer.The advantage of using the neoclassical growth model for quantitative evaluation relativeto the Solow growth model is that the extent to which diﬀerent types of distortions read more..

##### Page - 348

Introduction to Modern Economic Growthincome per capita across countries are unlikely to be accounted for by diﬀerences in capital perworker alone. Instead, to explain such large diﬀerences in income per capita across countries,we need sizable diﬀerences in the eﬃciency with which these factors are used. Such diﬀerencesdo not feature in this model. Therefore, the simplest neoclassical model does not generatesuﬃcient diﬀerences in capital-labor ratios to explain cross-country income read more..

##### Page - 349

Introduction to Modern Economic Growth8.11. Taking StockThis chapter presented arguably the most important model in macroeconomics; the one-sector neoclassical growth model. Recall that our study of the basic models of economicgrowth started in Chapter 2, with the Solow growth model. We saw that while this modelgives a number of important insights, it treats much of the mechanics of economic growthas a “black box”. Growth can only be generated by technological progress (unless we are inthe read more..

##### Page - 350

Introduction to Modern Economic Growth8.12. References and LiteratureThe neoclassical growth model goes back to Frank Ramsey’s (1928) classic treatment andfor that reason is often referred to as the “Ramsey model”. Ramsey’s model is very similar tostandard neoclassical growth model, except that it did not feature discounting. Another earlyoptimal growth model was presented by John von Neumann (1935), focusing more on thelimiting behavior of the dynamics in a linear model. The current read more..

##### Page - 351

Introduction to Modern Economic Growthhas total consumption C (t) to allocate at time t. The household has “utilitarian” preferenceswith instantaneous utility function u (c) and discount the future at the rate ρ> 0.(1) Show that the problem of the household can be written asmaxZ ∞0exp (−ρt)"Z L(t)0u (ci (t)) di#dt,subject toZ L(t)0ci (t) di ≤ C (t) ,and subject to the budget constraint of the household,˙A(t)= r (t) A(t)+ W (t) − C (t) ,where i denotes a generic member of read more..

##### Page - 352

Introduction to Modern Economic Growth(2) Show that in contrast to the Solow model, the saving rate s∗ can never be so highthat a decline in savings (or an increase in ρ)can raisethe steady-state levelofconsumption per capita.Exercise 8.8. In the dynamics of the basic neoclassical growth model, depicted in Figure8.1, prove that the ˙c =0 locus intersects the ˙k =0 locus always to the left of kgold.Basedon this analysis, explain why the modiﬁed golden rule capital-labor ratio, k∗, given read more..

##### Page - 353

Introduction to Modern Economic Growth(5) What determines the speed of adjustment of k (t) towards its steady-state value k∗?Exercise 8.12. Derive closed-form equations for the solution to the diﬀerential equations oftransitional dynamics presented in Example 8.2 with log preferences.Exercise 8.13.(1) Analyze the comparative dynamics of the basic model in responseto unanticipated increase in the rate of labor-augmenting technological progress willincrease to g0 >g. Does consumption read more..

##### Page - 354

Introduction to Modern Economic Growthwhere l (t) ∈ (0, 1) is labor supply. In a symmetric equilibrium, employment L (t) is equalto l (t). Assume that the production function is given by Y (t)= F [K(t),A (t) L (t)],whichsatisﬁes all the standard assumptions and A (t)=exp (gt) A (0).(1) Deﬁne a competitive equilibrium.(2) Set up the current value Hamiltonian that each household solves taking wages andinterest rates as given, and determine ﬁrst-order conditions for the allocation read more..

##### Page - 355

Introduction to Modern Economic Growth(4) What value of τ maximizes the level of utility at the steady state. Starting awayfrom the state state, is this also the tax rate that would maximize the initial utilitylevel? Why or why not?Exercise 8.19. Consider the neoclassical growth model with a government that needs toﬁnance a ﬂow expenditure of G. Suppose that government spending does not aﬀect utilityand that the government can ﬁnance this expenditure by using lump-sum taxes (that read more..

##### Page - 356

Introduction to Modern Economic Growthwhere Km denotes the mth type of capital and L is labor. F is homogeneous of degree 1 inall of its variables. Capital in each sector accumulates in the standard fashion, with˙Km (t)= Im (t) − δmKm (t) ,for m =1, ..., M . The resource constraint of the economy isC (t)+MXm=1Im (t) ≤ Y (t)for all t.(1) Write budget constraint of the representative household in this economy. Show thatthis can be done in two alternative and equivalent ways; ﬁrst, with M read more..

##### Page - 357

Introduction to Modern Economic GrowthExercise 8.25. Again in the discrete time version of the neoclassical growth model, supposethat there is labor-augmenting technological progress at the rate g, i.e.,A (t +1) = (1 + g) A (t) .For simplicity, suppose that there is no population growth.(1) Prove that balanced growth requires preferences to take the CRRA formU (0) =( P∞t=0βt[c(t)]1−θ−11−θif θ 6=1 and θ ≥ 0P∞t=0βtlnc(t)if θ =1.(2) Assuming this form of preferences, prove that read more..

##### Page - 358

CHAPTER 9Growth with Overlapping GenerationsA key feature of the neoclassical growth model analyzed in the previous chapter is thatit admits a representative household. This model is useful as it provides us with a tractableframework for the analysis of capital accumulation. Moreover, it enables us to appeal to theFirst and Second Welfare Theorems to establish the equivalence between equilibrium andoptimum growth problems. In many situations, however, the assumption of a representativehousehold read more..

##### Page - 359

Introduction to Modern Economic Growth9.1. Problems of InﬁnityLet us discuss the following abstract general equilibrium economy introduced by KarlShell. We will see that the baseline overlapping generations model of Samuelson and Diamondis very closely related to this abstract economy.Consider the following static economy with a countably inﬁnite number of households,each denoted by i ∈ N, and a countably inﬁnite number of commodities, denoted by j ∈ N.Assume that all households behave read more..

##### Page - 360

Introduction to Modern Economic GrowthProposition 9.2. In the above-described economy, the competitive equilibrium at (¯p, ¯x)is not Pareto optimal.So why does the First Welfare Theorem not apply in this economy? Recall that the ﬁrstversion of this theorem, Theorem 5.5, was stated under the assumption of a ﬁnite number ofcommodities, whereas we have an inﬁnite number of commodities here. Clearly, the sourceof the problem must be related to the inﬁnite number of commodities. The read more..

##### Page - 361

Introduction to Modern Economic Growthhouseholds i have one unit of good i +1. At the price vector ¯p,household 0 has a budget setc00 + c11≤ 2,thus chooses c00 = c01 =1. All other households have budget sets given bycii + cii+1≤ 1,thus it is optimal for each household i> 0 to consume one unit of the good cii+1,which iswithin its budget set and gives as high utility as any other allocation within his budget set,establishing that ˜xis a competitive equilibrium.¤9.2. The Baseline read more..

##### Page - 362

Introduction to Modern Economic Growthwhere f (k) ≡ F (k, 1) is the standard per capita production function. As usual, the wagerate is(9.4)w (t)= f (k (t)) − k (t) f0 (k (t)) .9.2.2. Consumption Decisions.Let us start with the individual consumption deci-sions. Savings by an individual of generation t, s (t), is determined as a solution to thefollowing maximization problemmaxc1(t),c2(t+1),s(t)u (c1 (t)) + βu (c2 (t +1))subject toc1 (t)+ s (t) ≤ w (t)andc2 (t +1) ≤ R (t +1) s (t) ,where read more..

##### Page - 363

Introduction to Modern Economic Growthwhere L (t) denotes the size of generation t, who are saving for time t +1. Since capitaldepreciates fully after use and all new savings are invested in the only productive asset of theeconomy,capital,the lawofmotionofthe capitalstock is givenby(9.7)K (t +1) = L (t) s (w (t) ,R (t +1)) .9.2.3. Equilibrium.A competitive equilibrium in the overlapping generations economycan be deﬁned as follows:Definition 9.1. A competitive equilibrium can be represented by read more..

##### Page - 364

Introduction to Modern Economic Growth45°0k(t+1)k(t)k3*k2*k4*k1*Figure 9.1. Various types of steady-state equilibria in the baseline overlap-ping generations model.where θ> 0 and β ∈ (0, 1). Furthermore, assume that technology is Cobb-Douglas, so thatf (k)= kαThe rest of the environment is as described above. The CRRA utility simpliﬁes the ﬁrst-ordercondition for consumer optimization and impliesc2 (t +1)c1 (t)=(βR (t +1))1/θ .Once again, this expression is the discrete-time read more..

##### Page - 365

Introduction to Modern Economic Growthwhich ensures that savings are always less than earnings. The impact of factor prices onsavings is summarized by the following and derivatives:sw≡∂s (t)∂w (t)=1ψ (t +1)∈ (0, 1) ,sr ≡∂s (t)∂R (t +1)=µ1−θθ¶(βR(t+1))−1/θ s (t)ψ (t +1).Since ψ (t +1) > 1, wealsohavethat 0 <sw < 1. Moreover, in this case sr > 0 if θ< 1,sr < 0 if θ> 1,and sr =0 if θ =1. The relationship between the rate of return on savingsandthe read more..

##### Page - 366

Introduction to Modern Economic GrowthProposition 9.4. In the overlapping-generations model with two-period lived households,Cobb-Douglas technology and CRRA preferences, there exists a unique steady-state equilib-rium with the capital-labor ratio k∗ given by (9.15) and as long as θ ≥ 1, this steady-stateequilibrium is globally stable for all k (0) > 0.Proof. See Exercise 9.5¤In this particular (well-behaved) case, equilibrium dynamics are very similar to the basicSolow model, and are read more..

##### Page - 367

Introduction to Modern Economic Growth45°k*k*k(0)k’(0)0k(t+1)k(t)Figure 9.2. Equilibrium dynamics in the canonical overlapping generations model.which corresponds to a constant saving rate, equal to β/ (1 + β), out of labor income for eachindividual. This constant saving rate makes this model very similar to the baseline Solowgrowth model of Chapter 2.Now combining this with the capital accumulation equation (9.8), we obtaink (t +1) =s(t)(1 + n)=βw(t)(1 + n)(1 + β)=β (1 − α)[k(t)]α(1 read more..

##### Page - 368

Introduction to Modern Economic GrowthProposition 9.5. In the canonical overlapping generations model with log preferencesand Cobb-Douglas technology, there exists a unique steady state, with capital-labor ratio k∗given by (9.20). Starting with any k (0) ∈ (0,k∗), equilibrium dynamics are such that k (t) ↑k∗, and starting with any k0 (0) >k∗, equilibrium dynamics involve k (t) ↓ k∗.Exercise 9.6 asks you to introduce technological progress into this canonical model andto read more..

##### Page - 369

Introduction to Modern Economic Growthas captured by the parameter βS. In particular, it can be shown that the socially plannedeconomy will converge to a steady state with capital-labor ratio kS such thatβSf0¡kS¢=1+n,which is similar to the modiﬁed golden rule we saw in the context of the Ramsey growth modelin discrete time (cf., Chapter 6). In particular, the steady-state level of capital-labor ratiokSchosen by the social planner does not depend on preferences (i.e., on the utility read more..

##### Page - 370

Introduction to Modern Economic Growththe heterogeneity inherent in the overlapping generations model, which removes the transver-sality condition.In particular, suppose we start from steady state at time T with k∗ >kgold.Considerthe following variation where the capital stock for next period is reduced by a small amount.In particular, change next period’s capital stock by −∆k,where ∆k> 0, and from then on,imagine that we immediately move to a new steady state (which is clearly read more..

##### Page - 371

Introduction to Modern Economic Growthin turn, is a result of the fact that the current young generation needs to save for old age.However, the more they save, the lower is the rate of return to capital and this may encouragethem to save even more. Once again, the eﬀect of the savings by the current generation onthe future rate of return to capital is a pecuniary externality on the next generation. Wemay reason that this pecuniary externality should not lead to Pareto suboptimal allocations,as read more..

##### Page - 372

Introduction to Modern Economic GrowthIt is also no longer the case that individuals will always choose s (t) > 0, since theyhave the income from Social Security. Therefore this economy can be analyzed under twoalternative assumptions, with the constraint that s (t) ≥ 0 and without.It is clear that as long as s (t) is free, whatever the sequence of feasible Social Securitypayments {d(t)}∞t=0, the competitive equilibrium applies. When s (t) ≥ 0 is imposed asa constraint, then the read more..

##### Page - 373

Introduction to Modern Economic GrowthWhat this implies is that the rate of return on Social Security payments is n rather thanr (t +1) = R (t +1) − 1, because unfunded Social Security is a pure transfer system. Onlys (t)–rather than s (t) plus d (t) as in the funded scheme–goes into capital accumulation.This is the basis of the claim that unfunded Social Security systems discourage aggregatesavings. Of course, it is possible that s (t) will change in order to compensate this eﬀect,but read more..

##### Page - 374

Introduction to Modern Economic Growthrelevant form of altruism is one in which parents care about certain dimensions of the con-sumption vector of their oﬀspring instead of their total utility. These types of preferences areoften referred to as “impure altruism” to distinguish it from the pure altruism discussed inSection 5.3. A particular type of impure altruism, commonly referred to as “warm glow pref-erences”, plays an important role in many growth models because of its read more..

##### Page - 375

Introduction to Modern Economic Growthwhere yi (t) denotes the income of this individual,(9.24)w (t)= f (k (t)) − k (t) f0 (k (t))is the equilibrium wage rate,(9.25)R (t)= f0 (k (t))is the rate of return on capital and bi (t − 1) is the bequest received by this individual fromhis own parent.The total capital-labor ratio at time t +1 is given by aggregating the bequests of alladults at time t:(9.26)k (t +1) =Z 10bi (t) di,which exploits the fact that the total measure of workers is 1, so that read more..

##### Page - 376

Introduction to Modern Economic Growthratio of the economy via equation (9.26). Integrating (9.27) across all individuals, we obtaink (t +1) =Z 10bi (t) di=β1+ βZ 10[w (t)+ R (t) bi (t − 1)] di=β1+ βf (k (t)) .(9.28)The last equality follows from the fact thatR10bi(t−1)di = k (t) and because by Euler’sTheorem, Theorem 2.1, w (t)+ R (t) k (t)= f (k (t)).Consequently, aggregate equilibrium dynamics in this economy are straightforward andagain closely resemble those in the baseline Solow read more..

##### Page - 377

Introduction to Modern Economic GrowthMoreover, it can be veriﬁed that the steady-state equilibrium must involve R∗ < (1 + β) /β.This follows from the fact that in steady stateR∗ = f0 (k∗)<f (k∗)k∗=1+ ββ,where the second line exploits the strict concavity of f (·) and the last line uses the deﬁnitionof the steady-state capital-labor ratio, k∗, from (9.29).The following proposition summarizes this analysis:Proposition 9.9. Consider the overlapping generations economy read more..

##### Page - 378

Introduction to Modern Economic Growtha ﬁnite expected life, all individuals who have survived up to a certain date have exactlythe same expectation of further life. Therefore, individuals who survive in this economy are“perpetually young”; their age has no eﬀect on their future longevity and has no predictivepower on how many more years they will live for.Individual i’s ﬂow budget constraint can be written as(9.32)ai (t +1) = (1 + r (t +1)) ai (t) − ci (t)+ w (t)+ zi (t) ,which is read more..

##### Page - 379

Introduction to Modern Economic GrowthWith free entry, insurance companies should make zero expected proﬁts (in terms of netpresent discounted value), which requires that π (a (t) ,t)=0 for all t and a,thus(9.33)z (a (t)) =ν1 − νa (t) .The other important element of the model is the evolution of the demographics. Sinceeach agent faces a probability of death equal to ν at every date, there is a natural forcetowards decreasing population. We assume, however, that there are also new agents read more..

##### Page - 380

Introduction to Modern Economic Growthan individual of generation τ canbewrittenas:(9.35)a (t +1 | τ )=µ1+r(t+1)+ ν1 − ν¶a(t|τ)−c(t|τ)+w(t).A competitive equilibrium in this economy can then be deﬁned as follows:Definition 9.3. A competitive equilibrium consists of paths of capital stock, wage ratesand rental rates of capital, {K (t) ,w (t) ,R (t)}∞t=0, and paths of consumption for each gen-eration, {c(t | τ )}∞t=0,τ≤t, such that each individual maximizes utility and the read more..

##### Page - 381

Introduction to Modern Economic GrowthIn that case, (9.36) simpliﬁes to(9.37)c (t +1 | τ )c (t | τ )= β [(1 + r (t + 1)) (1 − ν)+ ν] ,and implies that the growth rate of consumption must be equal for all generations. Usingthis observation, it is possible to characterize the behavior of the aggregate capital stock,though this turns out to be much simpler in continuous time. For this reason, we now turnto the continuous time version of this model (details on the discrete time model are read more..

##### Page - 382

Introduction to Modern Economic Growthtime t is˙a (t | τ )= r (t) a (t | τ ) − c (t | τ )+ w (t)+ z (a (t | τ ) | t, τ ) ,where again z (a (t | τ ) | t, τ ) is the transfer payment or annuity payment at time t to anindividual born at time τ holding assets a (t | τ ). We follow Yaari and Blanchard and againassume complete annuity markets, with free entry. Now the instantaneous proﬁts of a lifeinsurance company providing such annuities at time t for an individual born at time τ read more..

##### Page - 383

Introduction to Modern Economic GrowthLet us start with consumer optimization. The maximization of (9.38) subject to (9.41)gives the usual Euler equation(9.44)˙c (t | τ )c (t | τ )= r (t) − ρ,where ˙c (t | τ ) ≡ ∂c (t | τ ) /∂t. Notice that, in contrast to the discrete time version of thisequation, (9.37), the probability (ﬂow rate) of death, ν, does not feature here, since it exactlycancels out (the rate of return on assets is r (t)+ ν and the eﬀective discount factor is ρ read more..

##### Page - 384

Introduction to Modern Economic Growthwhere a (t) is average assets per capita. Since the only productive assets in this economy iscapital, we also have that a (t)= k (t). Finally, diﬀerentiating (9.47), we obtain(9.48)˙c (t)=(ρ + ν)( ˙a (t)+ ˙ω (t)) .The law of motion of assets per capita can be written as˙a (t)=(r (t) − (n − ν)) a (t)+ w (t) − c (t) .This equation is intuitive. Aggregate wealth (a (t) L (t)) increases because of the returns tocapital at the rate r (t) and also read more..

##### Page - 385

Introduction to Modern Economic Growthtransversality condition (9.45) applied to average assets, thus to the capital-labor ratio, k (t).First, a steady-state equilibrium is obtained when both ˙k (t) /k (t) and ˙c (t) /c (t) are equalto zero, and thus satisﬁes the following two equations:(9.50)c∗k∗=(ρ + ν) nf0 (k∗) − δ − ρ(9.51)f (k∗)k∗ − (n − ν + δ) −(ρ + ν) nf0 (k∗) − δ − ρ=0.The second equation pins down a unique positive level of steady-state read more..

##### Page - 386

Introduction to Modern Economic Growthc(t)kgold0k(t)k(0)c’(0)c’’(0)k(t)=0k*c(0)c*c(t)=0kmgrkFigure 9.3. Steady state and transitional dynamics in the overlapping gen-erations model in continuous time.the level of capital-labor ratio that satisﬁes the modiﬁed golden rule, kmgr. Starting with anyk (0) > 0, equilibrium dynamics monotonically converge to this unique steady state.Perhaps the most interesting feature of this equilibrium is that, despite ﬁnite lives andoverlapping read more..

##### Page - 387

Introduction to Modern Economic Growthwhere now ω (t | τ ) is the human wealth of an individual of generation τ at time t,given by(see Exercise 9.28):(9.52)ω (t | τ )=Z ∞texp (−(¯r(t − s)+ ν)) exp (−ζ (s − τ )) w (s) ds,where exp (−ζ (s − τ )) is the correction factor taking into account the decline in eﬀectivelabor units. Repeating the same steps as before with this new expression for human wealth,we obtain(9.53)˙c (t)c (t)= f0 (k (t)) − δ − ρ − ζ − (ρ + read more..

##### Page - 388

Introduction to Modern Economic Growththe extensive study of the baseline overlapping generations models were partly motivated bythe possibility of Pareto suboptimal allocations in such models. We have seen that theseequilibria may be “dynamically ineﬃcient” and feature overaccumulation–a steady-statecapital-labor ratio greater than the golden rule capital-labor ratio. We have also seen howan unfunded Social Security system can reduce aggregate savings and thus ameliorate read more..

##### Page - 389

Introduction to Modern Economic Growthprogress) and why some countries are much poorer than others (related to the fundamentalcause of income diﬀerences). Of course, its purpose was not to provide such answers in theﬁrst place.9.10. References and LiteratureThe baseline overlapping generations model with two-period lived agents is due to Samuel-son (1958) and Diamond (1965). A related model appears in French in Maurice Allais’ work.Blanchard and Fischer (1989, Chapter 3) provide an read more..

##### Page - 390

Introduction to Modern Economic Growthwhile Barro (1974) is the reference for the original statement of the Ricardian Equivalence hy-pothesis. Another important application of overlapping generations models is to generationalaccounting, for example, as in the work by Auerbach and Kotlikoﬀ (1987).9.11. ExercisesExercise 9.1. Prove that the allocation characterized in Proposition 9.1 is the unique com-petitive equilibrium. [Hint: ﬁrst, show that there cannot be any equilibrium with pj read more..

##### Page - 391

Introduction to Modern Economic Growth(4) What is the eﬀect of an increase in β on the equilibrium? Provide an intuition forthis result.Exercise 9.7. Consider the canonical model with log preferences, log (c1 (t)) + β log (c2 (t)),and the general neoclassical technology F (K, L) satisfying Assumptions 1 and 2. Show thatmultiple steady-state equilibria are possible in this economy.Exercise 9.8. Consider again the canonical overlapping generations model with log prefer-ences and Cobb-Douglas read more..

##### Page - 392

Introduction to Modern Economic Growthsystem will increase the welfare of the current old generation and reduce the welfare of somefuture generation.Exercise 9.15. Derive equation (9.31).Exercise 9.16. Consider the overlapping generations model with warm glow preferences inSection 9.6, and suppose that preferences are given byc (t)η b (t +1)1−η ,with η ∈ (0, 1), instead of equation (9.21). The production side is the same as in Section 9.6.Characterize the dynamic equilibrium of this read more..

##### Page - 393

Introduction to Modern Economic Growth(2) Show that the probability that an individual born at the time τ is alive at time t ≥ τis exp (−ν (t − τ )).(3) Derive equation (9.40).(4) Show that this equation would not apply at any ﬁnite time if the economy starts att =0 with an arbitrary age distribution.Exercise 9.24. Derive equation (9.46). [Hint: ﬁrst integrate the ﬂow budget constraintof the individual, (9.41) using the transversality condition (9.45) and then use the read more..

##### Page - 394

Introduction to Modern Economic Growth• All agents are inﬁnitely lived, with preferences∞Xt=0βt ln c (t)• An overlapping generations model where agents work in the ﬁrst period, and con-sume the capital income from their savings in the second period. The preferencesof a generation born at time t,deﬁned over consumption when young and old, aregiven byln cy (t)+ β ln c0 (t)(1) Characterize the equilibria in these two economies, and show that in the ﬁrst econ-omy, taxation reduces read more..

##### Page - 395

##### Page - 396

CHAPTER 10Human Capital and Economic GrowthIn this chapter, we will discuss human capital investments and the role of human capital ineconomic growth and in cross-country income diﬀerences. As already discussed in Chapter 3,human capital can play a major role in economic growth and cross-country income diﬀerences,and our main purpose is to understand which factors aﬀect human capital investments andhow these inﬂuence the process of economic growth and economic development. Humancapital read more..

##### Page - 397

Introduction to Modern Economic Growthis allowed), discounts the future at the rate ρ> 0 and faces a constant ﬂow rate of deathequal to ν ≥ 0 (as in the perpetual youth model studied in the previous chapter). Standardarguments imply that the objective function of this individual at time t =0 is(10.1)maxZ T0exp (−(ρ + ν) t) u (c (t)) dt.Now suppose that this individual is born with some human capital h (0) ≥ 0. Supposethat his human capital evolves over time according to the read more..

##### Page - 398

Introduction to Modern Economic Growthmaximizes(10.5)Z T0exp (−rt)w (t)[1 − s (t)] [h (t)+ ω (t)] dtsubject to (10.2) and (10.3), and [ˆc (t)]Tt=0maximizes (10.1) subject to (10.4) givenhˆs(t),ˆh(t)iTt=0. That is, human capital accumulation and supply decisions can be separatedfrom consumption decisions.Proof. To prove the “only if” part, suppose thathˆs(t),ˆh(t)iTt=0does not maximize(10.5), but there exists ˆc (t) such thathˆc(t),ˆs(t),ˆh(t)iTt=0is a solution to (10.1). Let read more..

##### Page - 399

Introduction to Modern Economic Growth10.2. Schooling Investments and Returns to EducationWe now turn to the simplest model of schooling decisions in partial equilibrium, whichwill illustrate the main tradeoﬀs in human capital investments. The model presented here isa version of Mincer’s (1974) seminal contribution. This model also enables a simple mappingfrom the theory of human capital investments to the large empirical literature on returns toschooling.Let us ﬁrstassumethat T = ∞, read more..

##### Page - 400

Introduction to Modern Economic GrowthIntegrating both sides of this equation with respect to S,we obtain(10.11)ln η (S∗)= constant +(r + ν − gw) S∗.Now note that the wage earnings of the worker of age τ ≥ S∗ in the labor market at time twill be given byW (S, t)= exp (gwt)exp (gh (t − S)) η (S) .Taking logs and using equation (10.11) implies that the earnings of the worker will be givenbyln W (S∗,t)= constant +(r + ν − gw) S∗ + gwt + gh (t − S∗) ,where t − S can be read more..

##### Page - 401

Introduction to Modern Economic Growthlifetime of the individual. In particular, we now let s (t) ∈ [0, 1] for all t ≥ 0. Togetherwith the Mincer equation (10.12) (and the model underlying this equation presented in theprevious section), the Ben-Porath model is the basis of much of labor economics. Here it issuﬃcient to consider a simple version of this model where the human capital accumulationequation, (10.2), takes the form(10.13)˙h (t)= φ (s (t) h (t)) − δhh (t) ,where δh > 0 read more..

##### Page - 402

Introduction to Modern Economic GrowthTo solve for the optimal path of human capital investments, let us adopt the followingtransformation of variables:x (t) ≡ s (t) h (t) .Instead of s (t) (or μ (t))and h (t), wewillstudy thedynamicsofthe optimalpathin x (t)and h (t).The ﬁrst necessary condition then implies that(10.14)1= μ (t) φ0 (x (t)) ,while the second necessary condition can be expressed as˙μ (t)μ (t)= r + ν + δh − s (t) φ0 (x (t)) −1 − s (t)μ (t).Substituting for μ read more..

##### Page - 403

Introduction to Modern Economic Growthh(t)0h(t)=0h*x*x(t)x(t)=0h(0)x’’(0)x’(0)Figure 10.1. Steady state and equilibrium dynamics in the simpliﬁed BenPorath model.is the elasticity of the function φ0 (·) and is positive since φ0 (·) is strictly decreasing (thusφ00 (·) < 0). Combining this equation with (10.15), we obtain(10.18)˙x (t)x (t)=1εφ0 (x (t))¡r+ν+δh−φ0(x(t))¢.Figure 10.1 plots (10.13) and (10.18) in the h-x space. The upward-sloping curve cor-responds to the read more..

##### Page - 404

Introduction to Modern Economic Growthh(t)t0h*h(0)Figure 10.2. Time path of human capital investments in the simpliﬁed BenPorath model.schooling. After full-time schooling, the individual starts working (i.e., s (t) < 1). But evenon-the-job, the individual continues to accumulate human capital (i.e., s (t) > 0), which canbe interpreted as spending time in training programs or allocating some of his time on thejob to learning rather than production. Moreover, because the horizon is read more..

##### Page - 405

Introduction to Modern Economic Growth10.4. Neoclassical Growth with Physical and Human CapitalOur next task is to incorporate human capital investments into the baseline neoclassicalgrowth model. This is useful both to investigate the interactions between physical and hu-man capital, and also to generate a better sense of the impact of diﬀerential human capitalinvestments on economic growth. Physical-human capital interactions could potentially beimportant, since a variety of evidence read more..

##### Page - 406

Introduction to Modern Economic GrowthL for all t. This production function is assumed to satisfy Assumptions 10 and 20 in Chapter3, which generalize Assumptions 1 and 2 to this production function with three inputs. Asalready discussed in that chapter, a production function in which “raw” labor and humancapital are separate factors of production may be less natural than one in which humancapital increases the eﬀective units of labor of workers (as in the analysis of the previoustwo read more..

##### Page - 407

Introduction to Modern Economic Growthconstraint and write the current-value Hamiltonian asH (k (t) ,h (t) ,ik (t) ,ih (t) ,μk (t) ,μh (t)) = u (f (k (t) ,h (t)) − ih (t) − ik (t))+μh (t)(ih (t) − δhh (t)) + μk (t)(ik (t) − δkk (t)) ,(10.23)where we now have two control variables, ik (t) and ih (t) and two state variables, k (t) andh (t), as well as two costate variables, μk (t) and μh (t), corresponding to the two constraints,(10.20) and (10.21). The necessary conditions for an read more..

##### Page - 408

Introduction to Modern Economic GrowthProposition 10.1. In the neoclassical growth model with physical and human capitalinvestments described above, the optimal path of physical capital and consumption are given asin the one-sector neoclassical growth model, and satisfy the following two diﬀerential equations˙c (t)c (t)=1εu (c (t))[fk (k (t) ,ξ (k (t))) − δk − ρ] ,˙k (t)=11+ ξ0 (k)[f (k (t) ,ξ (k (t))) − δhξ (k (t)) − δkk (t) − c (t)] ,where εu (c (t)) = −u00 (c (t)) c read more..

##### Page - 409

Introduction to Modern Economic Growthwhere τ ≥ 0 is a tax aﬀecting both types of investments. Using an analysis parallel to that inSection 8.9, we can characterize the steady-state income ratio of two countries with diﬀerentpolicies represented by τ and τ0. In particular, let us suppose that the aggregate productionfunction takes the Cobb-Douglas formY= F (K, H, L)= Kαk Hαh L1−αk−αh.In this case, the ratio of income in the two economies with taxes/distortions of τ and τ0 read more..

##### Page - 410

Introduction to Modern Economic Growthpotentially increase the role of human capital; large human capital externalities and signiﬁcantdiﬀerences in the quality of schooling across countries. These issues will be discussed below.10.5. Capital-Skill Complementarity in an Overlapping Generations ModelOur analysis in the previous section suggests that the neoclassical growth model withphysical and human capital does not generate signiﬁcant imbalances between these two dif-ferent types of read more..

##### Page - 411

Introduction to Modern Economic Growthindividual i of generation t can be written as(10.27)η−η (1 − η)−(1−η) ci (t)η bi (t)1−η − γµhi(t)a¶.The budget constraint of the individual is(10.28)ci (t)+ bi (t) ≤ mi (t)= w (t) hi (t)+ R (t) bi (t − 1) ,which deﬁnes mi (t) as the current income of individual i at time t consisting of labor earnings,w (t) hi (t), and asset income, R (t) bi (t − 1) (we use m rather than y,since y will have adiﬀerent meaning below).The read more..

##### Page - 412

Introduction to Modern Economic Growthk for capital per worker in the next section. From the deﬁnition of κ,the law of motion ofeﬀective capital-labor ratios can be written as(10.30)κ (t) ≡K (t)H (t)=R10bi(t−1)diR10hi(t)di.Factor prices are then given by the usual competitive pricing formulae:(10.31)R (t)= f0 (κ (t)) and w (t)= f (κ (t)) − κ (t) f0 (κ (t)) ,with the only noteworthy feature that w (t) is now wage per unit of human capital, in a wayconsistent with (10.28).An read more..

##### Page - 413

Introduction to Modern Economic GrowthNext, note that since bequest decisions are linear as shown (10.32), we haveK (t +1) =Z 10bi (t) di=(1 − η)Z 10mi (t) di=(1 − η) f (κ (t)) h (t) ,where the last line uses the fact that, since all individuals choose the same human capitallevel given by (10.35), H (t)= h (t), and thus Y (t)= f (κ (t)) h (t).Now combining this with (10.30), we obtainκ (t +1) =(1 − η) f (κ (t)) h (t)h (t +1).Using (10.35), this becomesκ (t +1) read more..

##### Page - 414

Introduction to Modern Economic Growth10.6. Physical and Human Capital with Imperfect Labor MarketsIn this section, we analyze the implications of labor market frictions that lead to factorprices diﬀerent from the ones we have used so far (in particular, in terms of the model of thelast section, deviating from the competitive pricing formulea (10.31)). The literature on labormarket imperfections is vast and our purpose here is not to provide an overview. For thisreason, we will adopt the read more..

##### Page - 415

Introduction to Modern Economic GrowthAn equilibrium is deﬁned similarly except that factor prices are no longer determined by(10.31). Let us start the analysis with the physical capital choices of ﬁrms. At the time eachﬁrm chooses its physical capital it is unsure about the human capital of the worker he willbe facing. Therefore, the expected return of ﬁrm j can be written as(10.38)(1 − λ)Z 10F (kj (t) ,hi (t)) di − R (t) kj (t) .This expression takes into account that the ﬁrm read more..

##### Page - 416

Introduction to Modern Economic GrowthFinally, to satisfy market clearing in the capital market, the rate of return to capital, R (t),has to adjust, such thatˆk (t)=Z 10bi (t − 1) di,which follows from the facts that all ﬁrms choose the same level of capital investment and thatthe measure of ﬁrms is normalized to 1. This equation implies that in the closed economyversion of the current model, capital per ﬁrm is ﬁxed by bequest decisions from the previousperiod. The main economic read more..

##### Page - 417

Introduction to Modern Economic Growthto observe that there is underinvestment both in human capital and physical capital (thisrefers to the positive activity equilibrium; clearly, there is even a more severe underinvestmentin the no-activity equilibrium). Consider a social planner wishing to maximize output (orone who could transfer resources across individuals in a lump-sum fashion). Suppose thatthe social planner is restricted by the same random matching technology, so that she cannotallocate read more..

##### Page - 418

Introduction to Modern Economic Growthand this encourages their human capital investments. But symmetrically, it discourages thephysical capital investments of ﬁrms, since they will only receive a small fraction of theoutput. Therefore, high level of λ (as long as we have λ< 1) creates an imbalance with toohigh a level of human capital relative to physical capital. This imbalance eﬀect becomes moreextreme as λ → 1. In this limit, workers’ investment behavior is converging to the read more..

##### Page - 419

Introduction to Modern Economic GrowthUsing this speciﬁc structure, the ﬁrst-order condition of ﬁrms, (10.39), can be written as(10.41)(1 − λ)⎡⎣χ∂F³ˆk,ˆh1³ˆk´´∂k+(1 − χ)∂F³ˆk,ˆh2³ˆk´´∂k⎤⎦=R∗,while the ﬁrst-order conditions for human capital investments for the two types of workerstake the form(10.42)λaj∂F³ˆk,ˆhj³ˆk´´∂h= γ0⎛⎝ˆhj³ˆk´aj⎞⎠forj=1,2.Clearly, ˆh1 (k) > ˆh2 (k) since a1 >a2. Now imagine an increase in χ, read more..

##### Page - 420

Introduction to Modern Economic GrowthMany economists believe that the human capital stock of the workforce creates a direct non-pecuniary (technological) spillover on the productivity of each worker. In The Economy ofCities, Jane Jacobs, for example, argued for the importance of human capital externalities,and suggested that the concentration of economic activity in cities is partly a result of theseexternalities and also acts as an engine of economic growth because it facilitates the read more..

##### Page - 421

Introduction to Modern Economic Growth3. However, Rauch’s regressions exploited diﬀerences in average schooling levels across cities,which could reﬂect many factors that also directly aﬀect wages. For example, wages aremuch higher in NewYorkCitythanAmes, Iowa, but this is not only the result of thehigher average education of New Yorkers. A more convincing estimate of external returnsnecessitates a source of exogenous variation in average schooling.Acemoglu and Angrist (2000) exploited read more..

##### Page - 422

Introduction to Modern Economic Growthview of human capital has played an important role in a variety of diﬀerent literatures andfeatures in a number of growth models. Here we will provide a simple presentation of themain ideas along the lines of Nelson and Phelps’ original model and a discussion of how thisnew dimension of human capital will change our views of its role in economic growth anddevelopment. This model will also act as a steppingstone towards our study of technologyadoption read more..

##### Page - 423

Introduction to Modern Economic Growthwhere ¯h> 0.This speciﬁcation implies that the human capital of the workforce regulates theability of the economy to cope with new developments embedded in the frontier technologies;if the workforce has no human capital, there will be no adoption or implementation of frontiertechnologies and A (t) will grow at the rate g.If, in contrast, h ≥¯h, there will be very quickadaptation to the frontier technologies.Since AF (t)=exp (gF t) AF (0), the read more..

##### Page - 424

Introduction to Modern Economic Growthhuman capital externalities and it should have shown up in the estimates on local externaleﬀects of human capital. It therefore would appear that, unless this particular role of humancapital is also external and these external eﬀects work at a global level, the calibration-typeexercises in Chapter 3 should not be seriously underestimating the contribution of humancapital to cross-country diﬀerences in income per capita.10.9. Taking StockHuman capital read more..

##### Page - 425

Introduction to Modern Economic Growthlarger diﬀerences across countries. In addition, most available empirical approaches measurehuman capital diﬀerences across countries by using diﬀerences in formal schooling. However,the Ben-Porath model, analyzed in Section 10.3, suggests that human capital continues tobe accumulated even after individuals complete their formal schooling. When human cap-ital is highly rewarded, we expect both higher levels of formal schooling and greater levelsof read more..

##### Page - 426

Introduction to Modern Economic Growth10.10. References and LiteratureThe concept of human capital is due to Ted Shultz (1965), Gary Becker (1965), andJacob Mincer (1974). The standard models of human capital, used extensively in labor eco-nomics and in other areas economics, have been developed by Becker (1965), Mincer (1974)and Yoram Ben-Porath (1967). These models have been the basis of the ﬁrst three sec-tions of this chapter. Recently there has been a renewed interest in the Ben-Porath read more..

##### Page - 427

Introduction to Modern Economic Growthsituations). Nelson and Phelps (1966) formulated the same ideas and presented a simplemodel, essentially identical to that presented in Section 10.8 above. Foster and Rosenzweig(1995) provide evidence consistent with this role of human capital. Benhabib and Spiegel(1994) and Aghion and Howitt (1999) also include extensive discussions of the Nelson-Phelpsview of human capital. Recent macroeconomic models that feature this role of human capitalinclude Galor read more..

##### Page - 428

Introduction to Modern Economic Growthof ν> 0. Show that the optimal path of human capital investments involves s (t)=1 forsome interval [0,T ] and then s (t)= s∗ for t ≥ T .Exercise 10.6. Modify the Ben-Porath model studied in Section 10.3 as follows. First,assume that the horizon is ﬁnite. Second, suppose that φ0 (0) < ∞. Finally, suppose thatlimx→h(0) φ0 (x) > 0. Show that under these conditions the optimal path of human capitalaccumulation will involve an interval of read more..

##### Page - 429

Introduction to Modern Economic GrowthExercise 10.14. * Characterize the optimal growth path of the economy in Section 10.4subject to the additional constraints that ik (t) ≥ 0 and ih (t) ≥ 0.Exercise 10.15. Derive equation (10.25).Exercise 10.16. Derive equations (10.32) and (10.33).Exercise 10.17. Provide conditions on f (·) and γ (·) such that the unique steady-stateequilibrium in the model of Section 10.5 is locally stable.Exercise 10.18. Analyze the economy in Section 10.6 under the read more..

##### Page - 430

CHAPTER 11First-Generation Models of Endogenous GrowthThe models presented so far focused on physical and human capital accumulation. Eco-nomic growth is generated by exogenous technological progress. While such models are usefulin thinking about sources of income diﬀerences among countries that have (free) access tothe same set of technologies, they do not generate sustained long-run growth (of the countryor of the world economy) and have relatively little to say about sources of technology read more..

##### Page - 431

Introduction to Modern Economic Growththe interest in economic growth among economists. While Romer’s objective was to model“technological change,” he achieved this by introducing technological spillovers–similar tothose we encountered in Chapter 10. Consequently, while the competitive equilibrium ofRomer’s model is not Pareto optimal and the engine of economic growth can be interpretedas a form “knowledge accumulation,” in many ways the model is still neoclassical in nature.In read more..

##### Page - 432

Introduction to Modern Economic GrowthThe Euler equation for the representative household is the same as before and impliesthe following rate of consumption growth per capita:(11.4)˙c (t)c (t)=1θ(r (t) − ρ).The other necessary condition for optimality of the consumer’s plans is the transversalitycondition,(11.5)limt→∞½a(t)exp∙−Z t0[r(s) − n] ds¸¾=0.As before, the problem of the consumer is concave, thus any solution to these necessaryconditions is in fact an optimal plan.The read more..

##### Page - 433

Introduction to Modern Economic Growth11.1.2. Equilibrium.A competitive equilibrium of this economy consists of pathsof per capita consumption, capital-labor ratio, wage rates and rental rates of capital,[c (t) ,k (t) ,w (t) ,R (t)]∞t=0, such that the representative household maximizes (11.1) subjectto (11.2) and (11.3) given initial capital-labor ratio k (0) and factor prices [w (t) ,r (t)]∞t=0such that w (t)=0 for all t,and r (t) is givenby(11.7).To characterize the equilibrium, we again read more..

##### Page - 434

Introduction to Modern Economic GrowthTo do this, let us substitute for c(t) from equation (11.11) into equation (11.8), whichyields(11.13)˙k (t)= (A − δ − n)k (t) − c(0) expµ1θ(A − δ − ρ)t¶,which is a ﬁrst-order, non-autonomous linear diﬀerential equation in k (t).This type ofequation can be solved easily. In particular recall that if˙z (t)= az (t)+ b (t) ,then, the solution isz (t)= z0 exp (at)+exp (at)Z t0exp (−as)b(s)ds,for some constant z0 chosen to satisfy the read more..

##### Page - 435

Introduction to Modern Economic Growthlevel of income per capita–it could have no eﬀect on the growth rate, which was determinedby the exogenous labor-augmenting rate of technological progress. Here, is straightforwardto verify that an increase in the discount rate, ρ, will reduce the growth rate, because it willmake consumers less patient and will therefore reduce the rate of capital accumulation. Sincecapital accumulation is the engine of growth, the equilibrium rate of growth will read more..

##### Page - 436

Introduction to Modern Economic Growth11.1.4. The Role of Policy.It is straightforward to incorporate policy diﬀerences into this framework and investigate their implications on the equilibrium growth rate. Thesimplest and arguably one of the most relevant classes of policies are, as also discussed above,those aﬀecting the rate of return to accumulation. In particular, suppose that there is aneﬀective tax rate of τ on the rate of return from capital income, so that the ﬂow read more..

##### Page - 437

Introduction to Modern Economic Growth11.2. The AK Model with Physical and Human CapitalAs pointed out in the previous section, a major shortcoming of the baseline AK modelis that theshare of capitalaccruingtonationalincomeisequal to1(or limits to1asinthe variant of the AK model studied in Exercises 11.3 and 11.4). One way of enriching theAK model and avoiding these problems is to include both physical and human capital. Wenow brieﬂy discuss this extension. Suppose the economy admits a read more..

##### Page - 438

Introduction to Modern Economic GrowthTo characterize the competitive equilibrium, let us ﬁrst set up at the current-value Hamil-tonian for the representative household with costate variables μa and μh:H (a, h, c, ih,μa,μk)=c (t)1−θ − 11 − θ+ μa (t)[(r (t) − n)a (t)+ w (t) h (t) − c (t) − ih (t)]+μh (t)[ih (t) − δhh (t)] .Now the necessary conditions of this optimization problem imply the following (see Exercise11.8):μa (t)= μh (t)= μ (t) for all t(11.25)w (t) − δh read more..

##### Page - 439

Introduction to Modern Economic Growthsuch, the empirical mechanism through which these large cross-country income diﬀerencesare generated may again not ﬁt with the empirical patterns discussed in Chapter 3. More-over, given substantial diﬀerences in policies across economies in the postwar period, likethe baseline AK economy, the current model would suggest signiﬁcant changes in the worldincome distribution, whereas the evidence in Chapter 1 points to a relatively stable postwarworld read more..

##### Page - 440

Introduction to Modern Economic Growthwhere the subscript “C” denotes that these are capital and labor used in the consumptionsector, which has a Cobb-Douglas technology. In fact, the Cobb-Douglas assumption hereis quite important in ensuring that the share of capital in national income is constant (seeExercise 11.12). The capital accumulation equation is given by:˙K (t)= I (t) − δK (t) ,where I (t) denotes investment. Investment goods are produced with a diﬀerent technologythan read more..

##### Page - 441

Introduction to Modern Economic Growthnumeraire, so that pC (t)=1 for all t.Then diﬀerentiating (11.29) implies that at the steadystate:(11.30)˙pI (t)pI (t)= −(1 − α) gK,where gK is the steady-state (BGP) growth rate of capital.As noted above, the Euler equation for consumers, (11.4), still holds, but the relevantinterest rate has to be for consumption-denominated loans,denoted by rC (t). In other words,it is the interest rate that measures how many units of consumption good an read more..

##### Page - 442

Introduction to Modern Economic GrowthSubstituting this into (11.32), we have(11.33)g∗K =A − δ − ρ1 − α (1 − θ)and(11.34)g∗C = αA − δ − ρ1 − α (1 − θ).What about wages? Because labor is being used in the consumption good sector, there willbe positive wages. Since labor markets are competitive, the wage rate at time t is given byw (t)= (1 − α) pC (t) Bµ(1−κ(t))K(t)L¶α.Therefore, in the balanced growth path, we obtain˙w (t)w (t)=˙pC (t)pC (t)+ α˙K (t)K read more..

##### Page - 443

Introduction to Modern Economic Growth11.4. Growth with ExternalitiesThe model that started much of endogenous growth theory and revived economists’ in-terest in economic growth was Paul Romer’s (1986) paper. Romer’s objective was to modelthe process of “knowledge accumulation”. He realized that this would be diﬃcult in the con-text of a competitive economy. His initial solution (later updated and improved in his andothers’ work during the 1990s) was to consider knowledge read more..

##### Page - 444

Introduction to Modern Economic GrowthandZ 10Li (t) di = L,where L is the constant level of labor (supplied inelastically) in this economy. Firms arecompetitive in all markets, which implies that they will all hire the same capital to eﬀectivelabor ratio, and moreover, factor prices will be given by their marginal products, thusw (t)=∂F (K (t) ,A (t) L)∂LR (t)=∂F (K (t) ,A (t) L)∂K (t).The key assumption of Romer (1986) is that although ﬁrms take A (t) as given, this stockof read more..

##### Page - 445

Introduction to Modern Economic GrowthOutput per capita can therefore be written as:y (t) ≡Y (t)L=Y (t)K (t)K (t)L= k (t) ˜f (L) ,where again k (t) ≡ K (t) /L is the capital-labor ratio in the economy.As in the standard growth model, marginal products and factor prices can be expressedin terms of the normalized production function, now ˜f (L).In particular, we have(11.37)w (t)= K (t) ˜f0 (L)and(11.38)R (t)= R = ˜f (L) − L ˜f0 (L) ,which is constant.11.4.2. Equilibrium.An equilibrium read more..

##### Page - 446

Introduction to Modern Economic Growtha unique equilibrium path where starting with any level of capital stock K (0) > 0, capital,output and consumption grow at the constant rate (11.39).Proof. Much of this proposition is proved in the preceding discussion. You are askedto verify the transversality conditions and show that there are no transitional dynamics inExercise 11.16.¤Population must be constant in this model because of the scale eﬀect.Since ˜f (L)−L˜f0 (L)is always increasing read more..

##### Page - 447

Introduction to Modern Economic GrowthProposition 11.6. In the above-described Romer model with physical capital externali-ties, the decentralized equilibrium is Pareto suboptimal and grows at a slower rate than theallocation that would maximize the utility of the representative household.Exercise 11.18 asks you to characterize various diﬀerent types of policies that can closethe gap between the equilibrium and Pareto optimal allocations.11.5. Taking StockThis chapter ends our investigation of read more..

##### Page - 448

Introduction to Modern Economic Growthwe observe in the data. Even if we choose quite large diﬀerences in cross-country distortions(for example, eightfold diﬀerences in eﬀective tax rates), the implied steady-state diﬀerencesin income per capita are relatively modest. We have seen that this has generated a largeliterature that seeks reasonable extensions of the neoclassical growth model in order to derivemore elastic responses to policy distortions or other diﬀerences across countries. read more..

##### Page - 449

Introduction to Modern Economic Growthwith exogenous savings), but dismissed it as uninteresting. A more complete treatment ofsustained neoclassical economic growth is provided in Jones and Manuelli (1990), who showthat even convex models (with production function is that satisfy Assumption 1, but naturallynot Assumption 2) are consistent with sustained long-run growth. Exercise 11.4 is a versionof the convex neoclassical endogenous growth model of Jones and Manuelli.Barro and Sala-i-Martin read more..

##### Page - 450

Introduction to Modern Economic Growthwhere A, B > 0.(1) Deﬁne a competitive equilibrium for this economy.(2) Set up the current-value Hamiltonian for an individual and characterize the neces-sary conditions for consumer maximization. Combine these with equilibrium factormarket prices and derive the equilibrium path. Show that the equilibrium pathdisplays non-trivial transitional dynamics.(3) Determine the evolution of the labor share of national income over time.(4) Analyze the impact of read more..

##### Page - 451

Introduction to Modern Economic GrowthSuppose that the ﬁrst country has a capital income tax rate of τ =0.2,while the secondcountry has a tax rate of τ0 =0.4. Suppose that the two countries start with the same levelof income in 1900 and experience no change in technology or policies for the next 100 years.What will be the relative income gap between the two countries in the year 2000? Discussthis result and explain why you do (or do not) ﬁnd the implications plausible.Exercise 11.8. Prove read more..

##### Page - 452

Introduction to Modern Economic GrowthExercise 11.15. In the Romer model presented in Section 11.4, let g∗C be thegrowthrateofconsumption and g∗ the growth rate of aggregate output. Show that g∗C >g∗ is not feasible,while g∗C <g∗ would violate the transversality condition.Exercise 11.16. Consider the Romer model presented in Section 11.4. Prove that the allo-cation in Proposition 11.5 satisﬁes the transversality condition. Prove also that there are notransitional dynamics in read more..

##### Page - 453

Introduction to Modern Economic Growthcondition on F , β and θ such that this growth rate is positive, but the transversalitycondition is still satisﬁed.(4) Argue (without providing the math) why any equilibrium must be along the balancedgrowth path characterized in part 3 at all points.(5) Is this a good model of endogenous growth? If yes, explain why. If not, contrast itwith what you consider to be better models.Exercise 11.20. * Consider the following endogenous growth model due to Uzawa read more..

##### Page - 454

Introduction to Modern Economic Growth(5) Now analyze the transitional dynamics of the economy starting with K/H diﬀerentfrom k∗ [Hint: look at dynamics in three variables, k ≡ K/H, χ ≡ C/K and φ,andconsider the cases α<θ and α ≥ θ separately].441 read more..

##### Page - 455

##### Page - 456

Part 4Endogenous Technological Change read more..

##### Page - 457

This part of the book focuses on models of endogenous technological change. Chapter 12discusses various diﬀerent approaches to technological change and provides a brief overview ofsome models of technological progress from the industrial organization literature. Chapters13 and 14 present the baseline endogenous technological progress models developed by Romer,Grossman and Helpman and Aghion and Howitt. Chapter 15 considers a richer class of modelsin which the direction of technological change, read more..

##### Page - 458

CHAPTER 12Modeling Technological ChangeWe have so far investigated models of economic growth of exogenous or endogenousvariety. But economic growth has not resulted from technological change. Either it has beenexogenous, or it has been sustained because of a linear neoclassical technology, or it hastaken place as a byproduct of knowledge spillovers. Since our purpose is to understand theprocess of economic growth, models in which growth results from technological progress andtechnological change read more..

##### Page - 459

Introduction to Modern Economic Growthvintages of the same good or machine and also to potential competition between existingproducers and the innovator. In addition, in the context of this type of innovation, onemight want to distinguish between the introduction of a higher-quality DVD player and theproduction of a cheaper DVD player because heterogeneous consumers may have diﬀerentialwillingness to pay for quality than for quantity. Issues of diﬀerential willingness to pay forquality are read more..

##### Page - 460

Introduction to Modern Economic GrowthAnother important distinction in the technological change literature is between “macro”and “micro” innovations (see Mokyr, 1990). The ﬁrst refers to radical innovations, perhapsthe introduction of general-purpose technologies, such as electricity or the computer, whichpotentially changed the organization of production in many diﬀerent product lines. In con-trast, micro innovations refer to the more common innovations that introduce newer models read more..

##### Page - 461

Introduction to Modern Economic Growthand we are willing to assume that individuals can make calculations about the eﬀect of theiractions on the probability of success and quality of the research project. Naturally, some mayargue that such calculations are not possible. But, without such calculations we would havelittle hope of modeling the process of technological change (or technology adoption). Sinceour objective is to model purposeful innovations, to assume that individuals and ﬁrms read more..

##### Page - 462

Introduction to Modern Economic Growthit is already out there available for all ﬁrms to use. In that case, F (K, L, A) will exhibitsconstant returns in K and L,and increasing returns to scale in K, L and A.Thus the non-rivalry of ideas and increasing returns to scale to all factors of production,including technology, are intimately linked. This has motivated Romer to develop diﬀerenttypes of endogenous growth models, exhibiting diﬀerent sources of increasing returns to scale,but the read more..

##### Page - 463

Introduction to Modern Economic Growthprogress of science, and how important breakthroughs–perhaps macro innovations discussedabove–have taken place as scientists build on each other’s work, with little emphasis onproﬁt opportunities. For example, in his History of Modern Computing, Ceruzzi emphasizesthe importance of a number of notable scientiﬁc discoveries and the role played by certaintalented individuals, such as John von Neumann, J. Presper Eckert, John Maucly, JohnBackus, read more..

##### Page - 464

Introduction to Modern Economic Growthpotentially proﬁtable opportunity to be seized...” (1966, p. 199). Other studies of innovationin particular industries also reach similar conclusions, see, for example, Myers and Marquis(1969) or Langrish et al. (1974).If potential proﬁts are a main driver of technological change, then the market size thatwill be commanded by new technologies or products will be a key determinant of innovations.A greater market size increases proﬁts and makes read more..

##### Page - 465

Introduction to Modern Economic Growthof diﬀerent types and ﬁnd a signiﬁcant response in the rate of innovation to these changes inmarket sizes.Overall, the evidence suggests that the market size is a major determinant of innovationincentives and the amount and type of technological change. This evidence motivates thetypes of models we will study, where technological change will be an economic activity and willrespond to proﬁt incentives rather than simply being driven by exogenous read more..

##### Page - 466

Introduction to Modern Economic Growthuncertainty in research, and if the ﬁrm incurs a cost μ> 0, it can innovate and reduce themarginal cost of production to ψ/λ,where λ> 1. Let us suppose that this innovation isnon-rival and is also non-excludable, either because it is not patentable or because the patentsystem does not exist.Let us now analyze the incentives of this ﬁrm in undertaking this innovation. We ﬁrstlook at the equilibrium without the innovation. Clearly, the presence read more..

##### Page - 467

Introduction to Modern Economic GrowthThe ﬁrst term in the second line is the increase in consumer surplus because of the expansionof output as the price falls from ψ to λ−1ψ (recall that price is equaltomarginalcostinthissocial planner’s allocation). The second term is the savings in costs for already producedunits; in particular, there is a saving of λ−1 (λ − 1) ψ on D (ψ) units. Finally, the last termis the cost of innovation. Depending on the shape of the function D (p),the read more..

##### Page - 468

Introduction to Modern Economic GrowthLet us now analyze this situation in a little more detail. It is useful to separate two cases:(1) Drastic innovation: a drastic innovation corresponds to a suﬃciently high value of λsuch that ﬁrm 1 becomes an eﬀective monopolist after the innovation. To determinewhich values of λ will lead to a situation of this sort, let us ﬁrst suppose that ﬁrm1 does indeed act like a monopolist. This implies that it will choose its price tomaximizeπI1 = D read more..

##### Page - 469

Introduction to Modern Economic Growthλ ≥ λ∗), then ﬁrm 1 sets the unconstrained monopoly price p1 = pM and makes proﬁts(12.3)ˆπI1 = D¡pM¢¡pM−λ−1ψ¢−μ.If pM >ψ (if λ<λ∗), then ﬁrm 1 sets the limit price p1 = ψ and makes proﬁts(12.4)πI1 = D (ψ) λ−1 (λ − 1) ψ − μ< ˆπI1.Proof. The proof of this proposition involves solving for the equilibrium of an asym-metric cost Bertrand competition game. While this is standard, it is useful to repeat read more..

##### Page - 470

Introduction to Modern Economic GrowthProposition 12.2. We have thatπI1<ˆπI1 < SI .SI1<bSI1 < SI .Proof. See Exercise 12.3.¤This proposition states that the social value of innovation is always greater than theprivate value in two senses. First, the ﬁrst line states that a social planner interested in max-imizing consumer and producer surplus will always be more willing to adopt an innovation,because of an appropriability eﬀect ;the ﬁrm, even if it has ex post monopoly read more..

##### Page - 471

Introduction to Modern Economic GrowthProposition 12.3. We have that∆ˆπI1 <πI1 < ˆπI1,so that a monopolist always has lower incentives to undertake innovation than a competitiveﬁrm.Proof. See Exercise 12.6.¤This result, which was ﬁrst pointed out in Arrow’s (1962) seminal paper, is referred to asthe replacement eﬀect. The terminology reﬂects the intuition for the result; the monopolisthas lower incentives to undertake innovation than the ﬁrm in a competitive industry read more..

##### Page - 472

Introduction to Modern Economic Growthexpect incumbents to be politically powerful, this implies that many economic systems willcreate powerful barriers against the process of economic growth. Political economy of growthis partly about understanding the opposition of certain ﬁrms and individuals to technologicalprogress and studying whether this opposition will be successful.There is another, perhaps more surprising, implication of the analysis in this subsection.This relates to the business read more..

##### Page - 473

Introduction to Modern Economic Growthstructure of many industries is one in which each ﬁrm faces a downward sloping demand curve(thus has some degree of monopoly power), but there is also free entry into the industry, sothat each ﬁrm (or at the very least, the marginal ﬁrm) makes zero proﬁts.The distinguishing feature of the Dixit-Stiglitz-Spence model (or Dixit-Stiglitz model forshort) is that it allows us to specify a structure of preferences that leads to constant monopolymarkups. read more..

##### Page - 474

Introduction to Modern Economic Growthfeature, but also the fact that individual demands take a very simple iso-elastic form. Toderive the demand for individual varieties, let us normalize the price of the y good to 1 anddenote the price of variety i by pi and the total money income of the individual by m.Thenthe budget constraint of the individual takes the form(12.9)NXi=1pici + y ≤ m.The maximization of (12.7) subject to (12.9) implies the following ﬁrst-order condition be-tween read more..

##### Page - 475

Introduction to Modern Economic Growthconstraint, implies that this ﬁrst-order condition can be expressed asy= g (P, m)(12.12)C=m − g (P, m)P,for some function g (·,·).Next, let us consider the production of the varieties. Suppose that each variety canonly be produced by a single ﬁrm, who is thus an eﬀective monopolist for this particularcommodity. Also assume that all monopolists are owned by the representative household andmaximize proﬁts.Recall that the marginal cost of producing read more..

##### Page - 476

Introduction to Modern Economic Growthandπ =1εNµm−gµN− 1ε−1εε − 1ψ, m¶¶.It can be veriﬁed that depending on the form of the g (·) function, which in turn depends onthe shape of the utility function u in (12.7), proﬁts can be increasing in the number of varieties(see Exercise 12.12). This may at ﬁrst appear somewhat surprising: typically, we expect agreater number of competitors to reduce proﬁts. But the love-for-variety eﬀect embeddedin the Dixit-Stiglitz preferences read more..

##### Page - 477

Introduction to Modern Economic GrowthUsing the deﬁnition of the ideal price index and (12.10), we obtain the budget constraint asPC + y ≤ m.Equation (12.12) then determines y and C. Since the supplier of each variety is inﬁnitesimal,their prices have no eﬀect on P and C. Consequently, the proﬁt-maximizing pricing decisionin (12.14) obtains exactly, and each ﬁrm has proﬁts given byπ =1εNψµm−gµN− 1ε−1εε − 1ψ, m¶¶,where g (·) is deﬁned as (12.12) in the previous read more..

##### Page - 478

Introduction to Modern Economic Growththeir portfolios and there would be entry by other proﬁt-maximizing ﬁrms instead. Thus, aslong as the representative household or the set of households on the consumer side act asprice-takers (as we have assumed to be the case throughout), proﬁt maximization is the onlyconsistent strategy for the monopolistically competitive ﬁrms.The only caveat to this arises from a diﬀerent type of deviation on the production side.In particular, a single ﬁrm read more..

##### Page - 479

Introduction to Modern Economic Growthestablish that this limit price must take the formp = γψ <εε − 1ψ.It is then straightforward to see that the entry condition that determines the number ofvarieties in the market will change toN−εε−1 γψµm−gµN− 1ε−1εε − 1ψ¶¶=μ.12.4.5. Limitations.The most important limitation of the Dixit-Stiglitz model is thefeature that makes it tractable: the constancy of markups as in equation (12.14). In particu-lar, the model implies read more..

##### Page - 480

Introduction to Modern Economic GrowthExample 12.2. Suppose that the representative household has a utility function over con-sumption given by u (c),where u (·) is strictly increasing, continuously diﬀerentiable andstrictly concave, so that individual is risk averse. Moreover, let us assume that limc→0 u0 (c)=∞, so that the marginal utility of consumption at zero is very high. The individual startswith an endowment equal to y> 0. This endowment can be consumed or it can be invested read more..

##### Page - 481

Introduction to Modern Economic Growth12.6. Taking StockThis chapter has reviewed a number of conceptual and modeling issues related to theeconomics of research and development. We have introduced the distinction between processand product innovations, macro and micro innovations, and also discussed the concept of theinnovation possibilities set and the importance of the non—rivalry of ideas.We have also seen why ex post monopoly power is important to create incentives forresearch spending, read more..

##### Page - 482

Introduction to Modern Economic Growthwhich we will encounter in Section 14.4 in Chapter 14. A more up-to-date reference thatsurveys the recent developments in the economics of innovation is Scotchmer (2005).The classic reference on the private and social values of innovation is Arrow (1962).Schumpeter (1943) was the ﬁrst to emphasize the role of monopoly in R&D and innovation.The importance of monopoly power for innovation and the indications of the non-rival natureof ideas are discussed read more..

##### Page - 483

Introduction to Modern Economic Growth12.8. ExercisesExercise 12.1. Derive equation (12.2).Exercise 12.2. Prove Proposition 12.1. In particular:(1) Show that even if pM = ψ, the unique (Nash) equilibrium involves q1 = D¡pM¢andqj =0 for all j> 1. Why is this?(2) Show that when pM >ψ,any price p1 >ψ or p1 <ψ cannot be proﬁt-maximizing.Show that there cannot be an equilibrium in which p1 = ψ and qj > 0 for somej> 1 [Hint: ﬁnd a proﬁtable deviation for ﬁrm 1].(3) read more..

##### Page - 484

Introduction to Modern Economic Growthpolicy is licensing, where ﬁrms that have made an innovation can license the rights to usethis innovation to others. This exercise asks you to work through the implications of thistype of licensing. Throughout, we think of the licensing stage as follows: the innovator canmake a take-it-or-leave-it-oﬀer to one or many ﬁrms so that they can buy the rights to usethe innovation (and produce as many units of the output as they like) in return for read more..

##### Page - 485

Introduction to Modern Economic Growthdeclining in the number of ﬁrms. Imagine that consumers are located uniformly around acircle with perimeter equal to 1. The circle indexes both the preferences of heterogeneousconsumers and the types of goods. The point where the consumer is located along the circlecorresponds to thetypeofproductthathemost prefers. When a consumer at point x aroundthecircleconsumesagood of type z, his utility isR − t |z − x| − p,whileifhechooses nottoconsume, his read more..

##### Page - 486

CHAPTER 13Expanding Variety ModelsAs emphasized in the previous chapter, the key to understanding endogenous technolog-ical progress is that R&D is a purposeful activity, undertaken for proﬁts, and the knowledge(machines, blueprints, or new technologies) that it generates increases the productivity ofexisting factors of production. The ﬁrst endogenous technological change models were for-mulated by Romer (1987 and 1990). Diﬀerent versions have been analyzed by Segerstrom,Anant and read more..

##### Page - 487

Introduction to Modern Economic Growth13.1.1. Demographics, Preferences and Technology.Imagine an inﬁnite-horizoneconomy in continuous time admitting a representative household with preferences(13.1)Z ∞0exp (−ρt)C (t)1−θ − 11 − θdt.There is no population growth, and the total population of workers, L, supplies labor inelas-tically throughout. We also assume, as discussed in the previous chapter, that the represen-tative household owns a balanced portfolio of all the ﬁrms in the read more..

##### Page - 488

Introduction to Modern Economic Growthwith εβ ≡ 1/β as the elasticity of substitution between inputs. This form emphasizes boththe constant returns to scale properties of the production function and the continuity betweenthe model here and the Dixit-Stiglitz model of the previous chapter.The resource constraint of the economy at time t is(13.3)C (t)+ X (t)+ Z (t) ≤ Y (t) ,where X (t) is investment or spending on inputs at time t and Z (t) is expenditure on R&Dat time t, which comes read more..

##### Page - 489

Introduction to Modern Economic GrowthThe ﬁrst-order condition of this maximization problem with respect to x (ν, t) for anyν ∈ [0,N (t)] yields the demand for machines from the ﬁnal good sector. These demands takethe convenient isoelastic form:(13.6)x(ν, t)= px(ν, t)−1/βL,which is intuitive in view of the fact that elasticity of demand for diﬀerent machine varietiesis εβ =1/β (so that x(ν, t)= px(ν, t)−εβL). This equation implies that the demand formachines only depends read more..

##### Page - 490

Introduction to Modern Economic Growthmarket clearing, and the time paths of [C (t) ,X (t) ,Z (t)]∞t=0 are consistent with consumeroptimization. We now characterize the unique equilibrium of this economy.Let us start with the ﬁrm side. Since (13.6) deﬁnes isoelastic demands, the solution tothe maximization problem of any monopolist ν ∈ [0,N (t)] involves setting the same price inevery period (see Exercise 13.2):(13.9)px(ν, t)=ψ1 − βfor all ν and t.That is, all monopolists charge a read more..

##### Page - 491

Introduction to Modern Economic Growthwhere V (ν, t) is given by (13.7). To understand (13.14), recall that one unit of ﬁnal good spenton R&D leads to the invention of η units of new inputs, each with a net present discountedvalue of proﬁts given by (13.7). This free entry condition is written in the complementaryslackness form, since research may be very unproﬁtable and there may be zero R&D eﬀort,in which case ηV (ν, t) could be strictly less than 1. Nevertheless, for the read more..

##### Page - 492

Introduction to Modern Economic Growthrate is constant. Let us therefore look for an equilibrium allocation in whichr (t)= r∗ for all t,where “*” refers to BGP values. Since proﬁts at each date are given by (13.11) and since theinterest rate is constant, (13.8) implies that ˙V (t)=0. Substituting this in either (13.7) or(13.8), we obtain(13.18)V∗ =βLr∗.This equation is intuitive: a monopolist makes a ﬂow proﬁtof βL, and along the BGP, thisis discounted at the constant interest read more..

##### Page - 493

Introduction to Modern Economic GrowthProof. The preceding discussion establishes all the claims in the proposition except thatthe transversality condition holds. You are asked to check this in Exercise 13.7.¤An important feature of this class of endogenous technological progress models is thepresence of the scale eﬀect, which we encountered in Section 11.4 in Chapter 11: the largeris L, the greater is the growth rate. The scale eﬀect comes from a very strong form of themarket size eﬀect read more..

##### Page - 494

Introduction to Modern Economic Growththat drive R&D, and R&D drives economic growth. We have therefore arrived to our ﬁrstmodel in which market-shaped incentives determine the rate at which the technology of theeconomy evolves over time.13.1.5. Pareto Optimal Allocations.The presence of monopolistic competition im-plies that the competitive equilibrium is not necessarily Pareto optimal. In particular, thecurrent model exhibits a version of the aggregate demand externalities discussed read more..

##### Page - 495

Introduction to Modern Economic GrowthThe aggregate resource constraint is still given by (13.3). Let us deﬁne net output, whichsubtracts the cost of machines from total output, as˜YS (t) ≡ YS (t) − XS (t) .This is relevant, since it is net output that will be distributed between R&D expenditure andconsumption. We obtain˜YS (t)=(1 − β)−1/β NS (t) L −Z N S (t)0ψxS (ν, t) dν=(1 − β)−1/β NS (t) L − (1 − β)−(1−β)/β NS (t) L=(1 − β)−1/β βNS (t) L.Given read more..

##### Page - 496

Introduction to Modern Economic Growthand it is straightforward to see that the former is always greater since (1 − β)−1/β > 1 byvirtue of the fact that β ∈ (0, 1). This implies that the socially-planned economy will alwaysgrow faster than the decentralized economyProposition 13.3. In the above-described expanding input variety model, the decentral-ized equilibrium is always Pareto suboptimal. Starting with any N (0) > 0, the Pareto optimalallocation involves a constant growth read more..

##### Page - 497

Introduction to Modern Economic GrowthMoreover, it is noteworthy that as in the ﬁrst-generation endogenous growth models, avariety of diﬀerent policy interventions, including taxes on investment income and subsidiesof various forms, will have growth eﬀects not just level eﬀects in this framework (see, forexample, Exercise 13.13).Naturally, once we start thinking of policy in order to close the gap between the decen-tralized equilibrium and the Pareto optimal allocation, we also have to read more..

##### Page - 498

Introduction to Modern Economic Growthwhich is less than g∗ given in (13.20). Therefore, in this model, greater competition, whichreduces markups (and thus static distortions), also reduces long-run growth. This might atﬁrst appear counter-intuitive, since the monopoly markup may be thought to be the keysource of ineﬃciency and greater competition (lower γ) reduces this markup. Nevertheless,as mentioned above, ineﬃciency results both because of monopoly markups and because theset of read more..

##### Page - 499

Introduction to Modern Economic GrowthAn alternative is to have “scarce factors” used in R&D. In other words, instead of thelab equipment speciﬁcation, we now have scientists as the key creators of R&D. The labequipment model generated sustained economic growth by investing more and more resourcesin the R&D sector. This is impossible with scarce factors, since, by deﬁnition, a sustainedincrease in the use of these factors in the R&D sector is not possible. Consequently, read more..

##### Page - 500

Introduction to Modern Economic Growththe workers are working in the R&D sector. Labor market clearing then requires thatLR (t)+ LE (t) ≤ L.The fact that not all workers are in the ﬁnal good sector implies that the aggregate outputof the economy (by an argument similar to before) is given by(13.25)Y (t)=11 − βN (t) LE (t) ,and proﬁts of monopolists from selling their machines is(13.26)π (t)= βLE (t) .The net present discounted value of a monopolist (for a blueprint ν)isstill read more..

##### Page - 501

Introduction to Modern Economic Growththe rate of technological progress, thus g∗ = ˙N (t) /N (t). This implies that the BGP level ofemployment is uniquely pinned down as(13.30)L∗E =θηL + ρ(1 − β) η + θη.The rest of the analysis is unchanged. It can also be veriﬁed that there are no transitionaldynamics in the decentralized equilibrium (see Exercise 13.17). It is also useful to note thatthere is again a scale eﬀect here–greater L increases the interest rate and the growth read more..

##### Page - 502

Introduction to Modern Economic Growth(3) In the data, the total amount of resources devoted to R&D appears to increasesteadily, but there is no associated increase in the aggregate growth rate.Each one of these arguments against scale eﬀects can be debated (for example, by arguingthat countries do not provide the right level of analysis because of international trade linkagesor that the growth rate of the world economy has indeed increased when we look at the past2000 years rather than read more..

##### Page - 503

Introduction to Modern Economic GrowthSuppose that this BGP involves positive growth, so that the free entry condition holds asequality. Then, the BGP free entry condition can be written as (see Exercise 13.18)(13.35)ηN (t)φ βLE (t)r∗ − n= w (t) .As before, the equilibrium wage is determined by the production side, (13.13), as w (t)=βN (t) / (1 − β). Combining this with the previous equation gives the following free entryconditionηN (t)φ−1(1 − β) LE (t)r∗=1.Now read more..

##### Page - 504

Introduction to Modern Economic Growthof spillovers. Without population growth, these spillovers would aﬀect the level of output,but would not be suﬃcient to sustain long-run growth. Continuous population growth, on theother hand, steadily increases the market size for new technologies and generates growth fromthese limited spillovers. While this pattern is referred to as “growth without scale eﬀects,”it is useful to note that there are two senses in which there are limited scale read more..

##### Page - 505

Introduction to Modern Economic Growthpatent on this product. Each product can be produced with the technology(13.40)y (ν, t)= l (ν, t) ,where l (ν, t) is labor allocated to the production of this variety. Since the economy is closed,y (ν, t)= c (ν, t).As in model with knowledge spillovers of Section 13.2, we assume that new products canbe produced with the production function(13.41)˙N (t)= ηN (t) LR (t) .The reader will notice that there is a very close connection between the model here read more..

##### Page - 506

Introduction to Modern Economic GrowthWith this choice of numeraire, we obtain the consumer Euler equation as (see Exercise 13.25):(13.45)˙C (t)C (t)= r (t) − ρ.With similar arguments to before, the net present discounted value of the monopolistowning the patent for product ν can be written asV (ν, t)=Z ∞texp∙−Z str¡s0¢ds0¸[px(ν,s)c(ν,s)−w(s)c(ν,s)]ds,where w (t) c(ν, t) is the total expenditure of the ﬁrm to produce a total quantity of c(ν, t)(given the production read more..

##### Page - 507

Introduction to Modern Economic GrowthCombining this equation with (13.47), we see thatπ (t)=1ε − 1(L − LR (t)) ηV (t) ,whereweuse π (t) to denote the proﬁts of all monopolists at time t, which are equal. InBGP, where the fraction of the workforce working in research is constant, this implies thatproﬁts and the net present discounted value of monopolists are also constant. Moreover, inthis case we must haveV (t)=π (t)r∗,where r∗ denotes the BGP interest rate. The previous two read more..

##### Page - 508

Introduction to Modern Economic Growthdepends on what we choose as the numeraire. The natural numeraire is the one setting theideal price index, (13.44), equal to 1, which amounts to measuring incomes in similar units atdiﬀerent dates. With this choice of numeraire, real incomes grow at the same rate as C (t),atthe rate g∗. Second, even though the equilibrium was characterized in a somewhat diﬀerentmanner than our baseline expanding input variety model, there is a close parallel read more..

##### Page - 509

Introduction to Modern Economic Growthunderstand the mechanics of economic growth. In this respect, the models presented in thischapterconstituteamajorimprovement over thosewehaveencountered so far.The models studied in this chapter, like those of the previous chapter, emphasize theimportance of proﬁts in shaping technology choices. We have also seen the role of monopolypower and patent length on the equilibrium growth rate. In addition, the same factors thatinﬂuenced the equilibrium growth read more..

##### Page - 510

Introduction to Modern Economic Growthpresented in Section 13.1 appears in Rivera-Batiz and Romer (1991). The model in Romer(1990) is similar to that presented in Section 13.2, but with skilled workers working in R&D.Gancia and Zilibotti (2005) provide an excellent survey of many of the models discussed inthis chapter. Matsuyama (1995) gives a very lucid and informative discussion of the sourcesof ineﬃciency in Dixit-Stiglitz type models, which is related to the sources of ineﬃciency read more..

##### Page - 511

Introduction to Modern Economic Growth(1) Rewrite (13.7) at time t as:V (ν, t)=Z t+∆ttexp∙−Z str (τ ) dτ¸(px(ν,s)−ψ)x(ν,s)ds+Z ∞t+∆texp∙−Z st+∆tr (τ ) dτ¸[px(ν,s)x(ν,s)−ψx(ν,s)]dswhich is just an identity for any ∆t. Interpret this equation and relate this to thePrinciple of Optimality.(2) Show that for small ∆t, this can be written asV (ν, t)= ∆t (px(ν, t) − ψ) x(ν, t)+exp (r (t) ∆t) V (ν, t + ∆t)+ o (∆t) ,and thus derive the equation∆t read more..

##### Page - 512

Introduction to Modern Economic GrowthExercise 13.6. This exercise asks you to construct and analyze the equivalent of the lab-equipment expanding variety model of Section 13.1 in discrete time. Suppose that the econ-omy admits a representative household with preferences at time 0 given by∞Xt=0βt C (t)1−θ − 11 − θ,with β ∈ (0, 1) and θ ≥ 0. Production technology is the same as in the text and the innovationpossibilities frontier of the economy is given byN (t +1) − N (t)= ηZ read more..

##### Page - 513

Introduction to Modern Economic Growthmodel can be mapped to reality. If not, explain which features of the model appearunsatisfactory to you and how you would want to change them.Exercise 13.9. Derive the consumption growth rate in the socially-planned economy,(13.22).Exercise 13.10. Consider the expanding input variety model of Section 13.1. Show thatit is possible for the equilibrium allocation to satisfy the transversality condition, while thesocial planner’s solution may violate it. read more..

##### Page - 514

Introduction to Modern Economic Growth(1) Characterize the equilibrium in this case and show how the equilibrium growth ratedepends on ι. [Hint: notice that there will be two diﬀerent types of machines,supplied at diﬀerent prices].(2) What is the value of ι that maximizes the equilibrium rate of economic growth?(3) Show that a policy of ι =0 does not necessarily maximize social welfare at timet =0.Exercise 13.15. Consider the formulation of competition policy in subsection 13.1.6.(1) read more..

##### Page - 515

Introduction to Modern Economic GrowthExercise 13.21. Consider the lab equipment model of Section 13.1, but modify the innova-tion possibilities frontier to˙N (t)= ηN (t)−φ Z (t) ,where φ> 0.(1) Deﬁne an equilibrium.(2) Characterize the market clearing factor prices and determine the free entry condition.(3) Show that without population growth, there will be no sustained growth in thiseconomy.(4) Now consider population growth at the exponential rate n, and show that this read more..

##### Page - 516

Introduction to Modern Economic Growthwho discovers a new good becomes the monopoly supplier, with a perfectly and indeﬁnitelyenforced patent.(1) Characterize the balanced growth path in the case where φ =1 and n =0,andshow that there are no transitional dynamics. Why is this? Why does the long-rungrowth rate depend on θ? Why does thegrowth ratedependon L?Do you ﬁndthis plausible?(2) Now suppose that φ =1 and n> 0. What happens? Interpret.(3) Now characterize the balanced growth path read more..

##### Page - 517

##### Page - 518

CHAPTER 14Models of Schumpeterian GrowthThe previous chapter presented the basic endogenous technological change models basedon expanding input or product varieties. The advantage of these models is their relativetractability. While the expansion of the variety of machines used in production capturescertain aspects of process innovation, most process innovations we observe in practice eitherincrease the quality of an existing product or reduce the costs of production. Therefore,typical process read more..

##### Page - 519

Introduction to Modern Economic Growthcase, however. In this chapter, we will present the basic models of competitive innovations,ﬁrst proposed by Aghion and Howitt (1992) and then further developed by Grossman andHelpman (1991a,b) and Aghion and Howitt (1998). The literature on models of Schum-peterian economic growth is now large and an excellent survey is presented in Aghion andHowitt (1998). Our purpose here is not to provide a detailed survey, but to emphasize themost important read more..

##### Page - 520

Introduction to Modern Economic GrowthThe production function of the ﬁnal good is similar to that in the previous chapter,except that now the quality of the machines matters for productivity. We write the aggregateproduction function of the economy as follows:(14.3)Y (t)=11 − β∙Z 10q(ν, t)x(ν, t | q)1−βdν¸Lβ,where x(ν, t | q) is the quantity of the machine of type ν of quality q used in the productionprocess. As in the previous chapter, this production function can be written read more..

##### Page - 521

Introduction to Modern Economic Growththe patent system does not preclude other ﬁrms undertaking research based on the productinvented by this ﬁrm. We will discuss below how diﬀerent patenting arrangements mightaﬀect incentives in this model.Once a particular machine of quality q (ν, t) has been invented, any quantity of thismachine can be produced at the marginal cost ψq (ν, t). Once again, the fact that the marginalcost is proportional to the quality of the machine is natural, since read more..

##### Page - 522

Introduction to Modern Economic Growthwhere px (ν, t | q) refers to the price of machine type ν of quality q (ν, t) at time t.Thisexpression stands for px (ν, t | q (ν, t)), but there should be no confusion in this notation sinceit is clear that q here refers to q (ν, t), and we will use this notation for other variables as well.The price px (ν, t | q) will be determined by the proﬁt-maximization of the monopolist holdingthepatentfor machineoftype ν of quality q (ν, t). Note that the read more..

##### Page - 523

Introduction to Modern Economic Growthis the average total quality of machines. This expression closely parallels the derived produc-tion function (13.12) in the previous chapter, except that instead of the number of machinevarieties, N (t), labor productivity is determined by the average quality of the machines,Q (t). This expression also clariﬁes the reasoning for the particular functional form assump-tions above. In particular, the reader can verify that it is the linearity of the read more..

##### Page - 524

Introduction to Modern Economic Growthq, which will have a net present value gain of V (ν, t | q). If there is positive R&D, i.e.,Z (ν, t | q) > 0, then the free entry condition must hold as equality.Note also that even though the quality of individual machines, the q (ν, t)’s, are stochastic(and depend on success in R&D), as long as R&D expenditures, the Z (ν, t | q)’s, are nonsto-chastic, average quality Q (t), and thus total output, Y (t), and total spending on read more..

##### Page - 525

Introduction to Modern Economic Growthfor all of them. This, in turn, implies that(14.16)V (ν, t | q)=q (ν, t)λη.Moreover, if it holds between t and t + ∆t, ˙V (ν, t | q)=0, because the right-hand side ofequation (14.16) is constant over time–q (ν, t) refers to the quality of the machine suppliedby the incumbent, which does not change. Since R&D for each machine type has the sameproductivity, this implies that z (ν, t) must also be the same for all machine types, thus equalto read more..

##### Page - 526

Introduction to Modern Economic GrowthNow combining (14.18)-(14.20), we obtain the BGP growth rate of output and consumptionas:(14.21)g∗ =ληβL − ρθ +(λ − 1)−1.This establishes the following propositionProposition 14.1. Consider the model of Schumpeterian growth described above. Sup-pose that(14.22)ληβL > ρ > (1 − θ) ληβL .Then, there exists a unique balanced growth path in which average quality of machines, outputand consumption grow at rate g∗ given by (14.21). read more..

##### Page - 527

Introduction to Modern Economic Growth14.1.4. Pareto Optimality.This equilibrium, like that of the endogenous technologymodel with expanding varieties, is typically Pareto suboptimal. The ﬁrst reason for thisis the appropriability eﬀect, which results because monopolists are not able to capture theentire social gain created by an innovation. However, Schumpeterian growth also introducesthe business stealing eﬀect discussed in Chapter 12. Consequently, the equilibrium rate ofinnovation and read more..

##### Page - 528

Introduction to Modern Economic Growthsatisﬁes the necessary conditions in Theorem 7.9 will give the unique optimal growth path.To characterize this solution, let us set up the current-value Hamiltonian asˆH¡QS,CS,μS¢= CS(t)1−θ − 11 − θ+ μS (t)hη(λ−1)(1−β)−1/ββQS(t)L−η(λ−1)CS(t)i.The necessary conditions for a maximum areˆHC¡QS,CS,μS¢ = CS (t)−θ − μS (t) η (λ − 1) = 0ˆHQ¡QS,CS,μS¢ = μS (t) η (λ − 1) (1 − β)−1/β βL = ρμS (t) − read more..

##### Page - 529

Introduction to Modern Economic Growthof the previous few chapters, anti-trust policy, patent policy and taxation will aﬀect the equi-librium growth rate. For example, two economies that tax corporate incomes at diﬀerentrates, say τ and τ0,willgrow atdiﬀerent rates.There is a sense in which the current model is much more appropriate for conductingpolicy analysis than the expanding varieties models, however. In those models, there wasno reason for any agent in the economy to support read more..

##### Page - 530

Introduction to Modern Economic Growthmodel, slowing down the process of creative destruction is beneﬁcial for incumbents, creatinga rationale for growth-retarding policies to emerge in equilibrium.Therefore, an important advantage of models of Schumpeterian growth is that they startproviding us clues about why some societies may adopt policies that reduce the growth rate.Since taxing R&D by new entrants beneﬁts incumbent monopolists, when incumbents aresuﬃciently powerful politically, read more..

##### Page - 531

Introduction to Modern Economic Growthgenerates a ﬂow rate η of a new machine. When the current machine used in production hasquality q (t), the new machine has quality λq (t).Letusonceagainassumethat(14.5)above is satisﬁed, so that the monopolist can chargethe unconstrained monopoly price. Then, an analysis similar to that in the previous sectionimmediately implies that the demand for the leading-edge machine of quality q is given byx (t | q)= px (t)−1/β q (t) LE (t) ,where again px read more..

##### Page - 532

Introduction to Modern Economic Growthwhere we have used the fact that in steady state total employment in the ﬁnal good sectoris equal to L∗E = L − L∗R.To simplify the notation, we also wrote V as a function of qonly rather than a function of both q and time. Free entry requires that when the currentmachine quality is q, the wage paid to one more R&D worker, w (q),must be equal to theﬂow beneﬁts, ηV (λq),thusw (q)= ηV (λq) .Flow beneﬁts from R&D are equal to ηV (λq), read more..

##### Page - 533

Introduction to Modern Economic GrowthProof. Much of the proof is provided by the preceding analysis. Exercise 14.16 asks youto verify that the average growth is given by g∗ = ηL∗R ln λ and that (14.27) is necessary forthe above described equilibrium to exist and to satisfy the transversality condition.¤Therefore, this analysis shows that the basic insights of the one-sector Schumpeterianmodel, as originally developed by Aghion and Howitt (1992), are very similar to the baselinemodel of read more..

##### Page - 534

Introduction to Modern Economic Growtheven-numbered innovation (say with the number of innovations counted starting from somearbitrary date t =0). This type of equilibrium is possible when all agents in the economyexpect there to be such an equilibrium (i.e., it is a “self-fulﬁlling” equilibrium). Denote thenumber of workers in R&D for odd and even-numbered innovations by L1R and L2R. Then,following the analysis in the previous subsection, in any equilibrium with a cyclical patternthe read more..

##### Page - 535

Introduction to Modern Economic Growth14.3. Innovation by Incumbents and Entrants and Sources of ProductivityGrowthA key aspect of the growth process is the interplay between innovations and productivityimprovements by existing ﬁrmsonthe onehandand entrybymoreproductive, new ﬁrmson the other. The evidence from industry studies, which will be discussed in greater detailin Section 18.1, suggests that a large part of productivity growth at the industry level (andthus in the aggregate) comes read more..

##### Page - 536

Introduction to Modern Economic Growthwith diﬀerent productivity levels will have diﬀerent levels of sales (see Exercise 14.26). It willtherefore enable us to make predictions about the size distribution of ﬁrmsaswell.The engine of economic growth is again quality improvements, but these will be drivenby two types of innovations:(1) Innovation by incumbents(2) Creative destruction by entrants.Let q (ν, t) be the quality of machine line ν at time t. We assume the following read more..

##### Page - 537

Introduction to Modern Economic Growthstill ﬁnd it proﬁtable to use the technology for incremental innovations, which is not availableto entrants.The presence of the strictly decreasing function η, which was also used in Section 14.2,captures the fact that when many ﬁrms are undertaking R&D to replace the same machineline, they are likely to try similar ideas, thus there will be some amount of “external” di-minishing returns (new entrants will be “ﬁshing out of the same read more..

##### Page - 538

Introduction to Modern Economic Growthby incumbents maximize their net present discounted value, consumers choose the path ofconsumption optimally, and the labor market clears.Let us start with the aggregate production function for the ﬁnal good producers. Proﬁt-maximization by the ﬁnal good sector implies the demand for machines of highest-quality isgiven by a slight variant of equation (14.4) in Section 14.1:(14.36)x(ν, t | q)= px (ν, t | q)−1/β q (ν, t) Lfor all ν ∈ [0, 1] and read more..

##### Page - 539

Introduction to Modern Economic Growthhighest quality of machine q at time t in machine line ν. This value satisﬁes the standardHamilton-Jacobi-Bellman equation:r (t) V (ν, t | q) −˙V (ν, t | q)=maxz(ν,t|q)≥o{π(ν, t | q) − z (ν, t | q) q (ν, t)(14.43)+φz (ν, t | q)(V (ν, t | λq) − V (ν, t | q)) − ˆz(ν, t | q)η (ˆz(ν, t | q)) V (ν, t | q)},where ˆz(ν, t | q)η (ˆz(ν, t | q)) is the rate at which radical innovations by entrants occur insector ν at time t and z read more..

##### Page - 540

Introduction to Modern Economic GrowthAs usual, I deﬁne a BGP (balanced growth path) as an equilibrium path in which innova-tion, output and consumption growth a constant rate. Notice that in BGP, aggregates growat the constant rate, but there will be ﬁrm deaths and births, and the ﬁrm size distributionmay also change.The requirement that consumption grows at a constant rate in the BGP implies thatr (t)= r∗ (from (14.14)). Moreover, in BGP, we must also have z (ν, t | q)= z (q) andˆz read more..

##### Page - 541

Introduction to Modern Economic GrowthFrom (14.14), the growth rate of consumption and output is therefore given by(14.51)g∗ =1θ(φ (λ − 1) βL − ˆz∗η (ˆz∗) − ρ) .Equation (14.51) already has some interesting implications. In particular, it determines therelationship between the rate of innovation by entrants ˆz∗ and the BGP growth rate g∗.Instandard Schumpeterian models, this relationship is positive. In contrast, here we have thefollowing immediate result:Proposition read more..

##### Page - 542

Introduction to Modern Economic Growthwe need to verify that R&D is proﬁtable both for entrants and incumbents. The conditionthat the BGP interest rate, r∗, given by (14.50), should be greater than the discount rate ρ issuﬃcient for there to be positive aggregate growth. In addition, this interest rate should notbe so high that the transversality condition of the consumers is violated. Finally, we need toensure that there is also innovation by incumbents. The following condition read more..

##### Page - 543

Introduction to Modern Economic GrowthProposition 14.6. Consider the above-described economy starting with an initial condi-tion Q (0) > 0. Suppose that (14.33) and (14.56) are satisﬁed and focus on equilibrium inwhich all incumbents exert the same level of R&D eﬀort. Then there exists a unique equilib-rium. In this equilibrium growth is always balanced, and technology, Q (t),aggregate output,Y (t), and aggregate consumption, C (t),grow at the rate g∗ as in (14.51) with ˆz∗ read more..

##### Page - 544

Introduction to Modern Economic Growthwhereεη (ˆz) ≡−η0 (ˆz)ˆzη (ˆz)> 0is the elasticity of the η function.[B] The value of a ﬁrm with a machine of quality q at time t can be written as(14.61)V (ν, t | q)=Z ∞texp∙−Z st¡r¡s0¢+ˆz(ν,s0|q)η¡ˆz(ν,s0|q)¢¢ds0¸π(ν,s|q)ds.Foranyﬁniteq,thisvalueisﬁnitesince,fromobservation[A],(r (t)+ ˆz(ν, t | q)η (ˆz(ν, t | q)))>0forallν,qandt,andbecauseπ (ν, s | q)= βqL ∈ (0, ∞). In addition, in view of (14.33), read more..

##### Page - 545

Introduction to Modern Economic GrowthSimilarly,∂ ˆz(ν, t | q)/∂tˆz(ν, t | q)=1εη (ˆz(ν, t | q))[r (t)+ ˆz (ν, t | κq) η (ˆz (ν, t | κq)) − βLη (ˆz(ν, t | q))] ,and so on. It is then straightforward to see that the system of diﬀerential equations forˆz (ν, t | q) for all q is unstable, in the sense that if ∂ ˆz(ν, t | q)/∂t > 0,then it will growcontinuously and if ∂ ˆz(ν, t | q)/∂t < 0, then it will shrink continuously. This implies thatwe must have read more..

##### Page - 546

Introduction to Modern Economic Growththirds of national income accrues to labor and one third to proﬁts. The requirement in(14.37) then implies that κ> 1.7. I also take the benchmark value of κ =3,so that entryby new ﬁrmsissuﬃciently “radical” as suggested by some of the qualitative accounts of theinnovation process (e.g., Freeman, 1982, Scherer, 1984). Innovation by incumbents is takento be correspondingly smaller λ =1.2. This implies that productivity gains from a read more..

##### Page - 547

Introduction to Modern Economic Growthinvestigate whether this is the case, let us suppose that there is a tax τ e on R&D expenditureby entrants and a tax τ i on R&D expenditure by incumbents (naturally, these can be takento be negative and interpreted as subsidies as well). Note also that the tax on entrants, τ e,can be interpreted as a more strict patent policy than the one in the baseline model, wherethe entrant did not have to pay the incumbent for partially beneﬁting from its read more..

##### Page - 548

Introduction to Modern Economic Growththe decentralized equilibrium of this economy there tends to be too much entry, so a taxon entry also tends to improve welfare. The intuition for this result is related to the maindeparture of this model from the standard Schumpeterian models. In contrast to the baselineSchumpeterian models, the engine of growth is still quality improvements, but these areundertaken both by incumbents and entrants. Entry barriers, by protecting incumbents,increase their read more..

##### Page - 549

Introduction to Modern Economic Growth1−Γy−χ). The Pareto distribution is attractive both because of its simplicity and tractability(see, for example, Section 15.8 in the next chapter), but also because the actual distributionof ﬁrm sizes in the US appears to fairly well approximated by a Pareto distribution with anexponent of one (e.g., Axtell, 2001). It is therefore a somewhat surprising and remarkableresult that the simple model developed here, which was not designed to match the read more..

##### Page - 550

Introduction to Modern Economic GrowthSubstituting this conjecture into the previous expression, we obtainΓy−χ = φz∗∆tΓµ(1+g∗∆t)yλ¶−χ+ˆz∗η(ˆz∗)∆tΓµ(1+g∗∆t)yκ¶−χ+(1 − φz∗∆t − ˆz∗η (ˆz∗) ∆t) Γ ((1 + g∗∆t) y)−χ + o (∆t) .Rewriting this,φz∗∆tµ(1+g∗∆t)λ¶−χ+ˆz∗η(ˆz∗)∆tµ(1+g∗∆t)κ¶−χ(14.69)+(1 − φz∗∆t − ˆz∗η (ˆz∗) ∆t)(1 + g∗∆t)−χ + o (∆t) Γ−1y−χ =1.Now subtracting 1 from read more..

##### Page - 551

Introduction to Modern Economic Growththat a stationary ﬁrm-size distribution exists. Unfortunately, the next corollary shows thatthis will not be the case.Corollary 14.1. Let us focus on the equilibrium in which all incumbents choose R&Deﬀort z∗. Then a stationary ﬁrm-size distribution does not exist.Proof. We know from Proposition 14.8 that if a stationary distribution exists, it musttake the form Pr [˜x ≤ y]=1 − Γ/y with Γ > 0. But the Pareto distribution is deﬁned for read more..

##### Page - 552

Introduction to Modern Economic GrowthProposition 14.9. Consider the BGP equilibrium in which incumbent R&D eﬀort levelsare given by (14.71) for ε> 0 and ¯z+ > ¯z∗ > 0. Consider the limiting case where ¯z+ →∞.Then there exists a unique stationary ﬁrm-size distribution given by the Pareto distributionwith an exponent of one.Proof. As ¯z+ →∞,no ﬁrm wouldhaverelativequality ˜q< ε (since there will beimmediate innovation at ˜q = ε). This implies that relative read more..

##### Page - 553

Introduction to Modern Economic Growthindustries by stating that: “each of the major companies seems to have made more frequentcontributions in a particular area,” and argues that this is because previous innovations ina ﬁeld facilitate future innovations. This aspect is entirely missing from the baseline modelof Schumpeterian growth, where any ﬁrm can engage in research to develop the next higher-quality machine (and in addition Arrow’s replacement eﬀect implies that incumbents do read more..

##### Page - 554

Introduction to Modern Economic GrowthLet Y (t) be the total production of the ﬁnal good at time t. We assume that the economyis closed and the ﬁnal good is used only for consumption (i.e., there is no investment orspending on machines), so that C (t)= Y (t). The standard Euler equation from (14.72)then implies that(14.73)g (t) ≡˙C (t)C (t)=˙Y (t)Y (t)= r (t) − ρ,where this equation deﬁnes g (t) as the growth rate of consumption and thus output, andr (t) is the interest rate at date read more..

##### Page - 555

Introduction to Modern Economic GrowthBertrand competition between the two ﬁrms implies that all intermediates will be suppliedby the leader at the limit price (see Exercise 14.27):(14.78)pyi (ν, t)=w (t)q−i (ν, t).Equation (14.75) then implies the following demand for intermediates:(14.79)y (ν, t)=q−i (ν, t)w (t)Y (t) .R&D by the leader or the follower stochastically leads to innovation. We assume thatwhen the leader innovates, its technology improves by a factor λ> 1. The read more..

##### Page - 556

Introduction to Modern Economic Growthand that the follower −i’s technology at time t is(14.83)q−i (ν, t)= λn−i(ν,t),where, naturally, ni (ν, t) ≥ n−i (ν, t). Let us denote the technology gap in industry ν attime t by n (ν, t) ≡ ni (ν, t) − n−i (ν, t). If the leader undertakes an innovation within a timeinterval of ∆t, then the technology gap rises to n (ν, t + ∆t)= n (ν, t)+ 1 (the probability oftwo or more innovations within the interval ∆t is again o read more..

##### Page - 557

Introduction to Modern Economic Growthan industry that is neck-and-neck, i.e., in industries with n (j, t)=0. Followers also makezero proﬁts, since they have no sales.The Cobb-Douglas aggregate production function in (14.74) is responsible for the simpleform of the proﬁts (14.85), since it implies that proﬁts only depend on the technology gapof the industry and aggregate output. This will simplify the analysis below by making thetechnology gap in each industry the only read more..

##### Page - 558

Introduction to Modern Economic GrowthWe can deﬁne an equilibrium as follows. A Markov Perfect Equilibrium is represented bytime paths [ξ∗ (t) ,w∗ (t) ,r∗ (t) ,Y∗ (t)]∞t=0 such that (i)£py∗i(ν,t)¤∞t=0and[y∗i(ν,t)]∞t=0impliedby [ξ∗ (t)]∞t=0 satisfy (14.78) and (14.79); (ii) R&D policies [z∗ (t)]∞t=0 are best responses tothemselves, i.e., [z∗ (t)]∞t=0 maximizes the expected proﬁts of ﬁrms taking aggregate output[Y∗ (t)]∞t=0, factor prices [w∗ (t) read more..

##### Page - 559

Introduction to Modern Economic GrowthGiven this, the equilibrium wage can be written as (see Exercise 14.28):(14.91)w (t)= Q (t) λ− ∞n=0 nμn(t).14.4.3. Steady-State Equilibrium.Let us now focus on steady-state (Markov Per-fect) equilibria, where the distribution of industries μ(t) ≡ {μn (t)}∞n=0 is stationary, ω (t)deﬁned in (14.89) and g∗, the growth rate of the economy, is constant over time (we referto this as a steady-state Markov perfect equilibrium, since the potentially read more..

##### Page - 560

Introduction to Modern Economic Growthwhile the values for followers are given byρv−n =maxz−n {−ω∗G(z−n)+ [z−n + κ][v0 − v−n]}.It is clear that these value functions and proﬁt-maximizing R&D decision for followers shouldnot depend on how many steps behind the leader they are, since a single innovation is suﬃcientto catch-up with the leader. Therefore, we can write(14.96)ρv−1 =maxz−1 {−ω∗G(z−1)+ [z−1 + κ][v0 − v−1]},where v−1 represents the value of read more..

##### Page - 561

Introduction to Modern Economic GrowthThe ﬁrst expression equates exit from state n+1 (which takes the form of the leader going onemore step ahead or the follower catching-up the leader) to entry into this state (which takesthe form of a leader from the state n making one more innovation). The second equation,(14.101), performs the same accounting for state 1, taking into account that entry into thisstate comes from innovation by either of the two ﬁrms that are competing read more..

##### Page - 562

Introduction to Modern Economic Growthfar, because such policies will change the equilibrium market structure (i.e., the compositionof industries).Definition 14.1. A steady-state equilibrium is given by hμ∗, v, z∗,ω∗,g∗i suchthat the distribution of industries μ∗ satisfy (14.100), (14.101) and (14.102), the valuesv≡{vn}∞n=−∞ satisfy (14.94), (14.95) and (14.96), the R&D decisions z∗ are given by,(14.97), (14.98) and (14.99), the steady-state labor share ω∗ read more..

##### Page - 563

Introduction to Modern Economic GrowthSince λ−n−1 <λ−n,this implies vn <vn+1, contradicting the hypothesis that vn ≥ vn+1,andestablishing the desired result, vn <vn+1.Consequently, {vn}∞n=−1 is nondecreasing and {vn}∞n=0 is (strictly) increasing. Since anondecreasing sequence in a compact set must converge, {vn}∞n=−1 converges to its limitpoint, v∞, which must be strictly positive, since {vn}∞n=0 is strictly increasing and has anonnegative initial value. This read more..

##### Page - 564

Introduction to Modern Economic Growthwhere ¯ρ ≡ ρ + z∗−1+ κ.Since z∗n+1, z∗n and z∗n−1are maximizers of the value functions vn+1,vn and vn−1, (14.108) implies:¯ρvn+1 =1 − λ−n−1 − ω∗G¡z∗n+1¢+z∗n+1[vn+2−vn+1]+¡z∗−1+κ¢v0,(14.109)¯ρvn ≥ 1 − λ−n − ω∗G¡z∗n+1¢+z∗n+1[vn+1−vn]+¡z∗−1+κ¢v0,,¯ρvn ≥ 1 − λ−n − ω∗G¡z∗n−1¢+z∗n−1[vn+1−vn]+¡z∗−1+κ¢v0,,¯ρvn−1 =1 − λ−n+1 − read more..

##### Page - 565

Introduction to Modern Economic GrowthExercise 14.31 shows how a relaxation of intellectual property rights protection can increasethe growth rate in the economy.So far, we have not provided a closed-form solution for the growth rate in this economy.It turns out that this is generally not possible, because of the endogenous market structure inthese types of models. Nevertheless, it can be proved that a steady state equilibrium exists inthis economy, though the proof is somewhat more involved and read more..

##### Page - 566

Introduction to Modern Economic GrowthAn important insight of Schumpeterian models is that growth comes with potential con-ﬂict of interest. The process of creative destruction destroys the monopoly rents of previousincumbents. This raises the possibility that distortionary policies may arise endogenouslyas a way of protecting the rents of politically powerful incumbents. Models of creative de-struction therefore raise the political economy issues that are central for understanding read more..

##### Page - 567

Introduction to Modern Economic Growthimprovements and costs of production and R&D increasing proportionally with quality. Theoriginal Aghion and Howitt (1992) approach is very similar to that used in Section 14.2.Aghion and Howitt (1992) also discuss uneven growth and potential growth cycles, whichwere presented in Section 14.2. Uneven growth and cycles are also possible in expandingproduct or input variety models as shown by Matsuyama (1999, 2001). I only discussed thepossibility of such read more..

##### Page - 568

Introduction to Modern Economic Growthat the same time generating the process of productivity growth of continuing plans and newentrants endogenously.Step-by-step or cumulative innovations havebeenanalyzedinAghion, Harrisand Vick-ers (1999) and Aghion, Harris, Howitt and Vickers (2001). The model presented here is asimpliﬁed version of Acemoglu and Akcigit (2006), which includes a detailed analysis of theimplications of intellectual property rights policy and licensing in this class of models. read more..

##### Page - 569

Introduction to Modern Economic GrowthExercise 14.4. Prove Proposition 14.2.Exercise 14.5. * In the baseline Schumpeterian growth model, instead of (14.3), supposethat the production function of the ﬁnal good sector is given byY (t)=11 − β∙Z 10q(ν, t)ζ1 x(ν, t | q)1−βdν¸Lβ.Suppose also that producing one unit of an intermediate could of quality q costs ψqζ2 andthat one unit of ﬁnalgood devotedtoresearchonthe machinelinewithquality q generatesa ﬂow rate of innovation equal read more..

##### Page - 570

Introduction to Modern Economic Growth(1) Show that with this production function, a BGP does not exist. Explain why thisis.(2) What would you change in the model to ensure the existence of a BGP.Exercise 14.10. Suppose that there is constant exponential population growth at the raten. Modify the baseline model of Section 14.1 so that there is no scale eﬀect and the economygrows at the constant rate (with positive growth of income per capita). [Hint: suppose thatone unit of ﬁnal good spent read more..

##### Page - 571

Introduction to Modern Economic GrowthExercise 14.13. Consider the model in Section 14.2, with R&D performed by workers.Suppose instead that the aggregate production function for the ﬁnal good is given byY (t)=11 − β∙Z 10q(ν, t)x(ν, t | q)1−βdν¸LE(t)β,where LE (t) denotes the number of workers employed in ﬁnal good production at time t.(1) Show that in this case, there will only be R&D for the machine with the highestq(ν, t).(2) How would you modify the model so that read more..

##### Page - 572

Introduction to Modern Economic Growth(1) Deﬁne an equilibrium in this economy.(2) Characterize the BGP and specify restrictions on parameters so that the transver-sality condition is satisﬁed.(3) Compare the BGP growth rate to the Pareto optimal growth rate of the economy.Show that the size of innovations is always too small relative to the size of innovationsin the Pareto optimal allocation.Exercise 14.18. * Consider the one-sector Schumpeterian model in discrete time analyzedin the read more..

##### Page - 573

Introduction to Modern Economic GrowthExercise 14.23. Set up the social planner’s problem (of maximizing the utility of the rep-resentative household) in Section 14.3.(1) Show that this maximization problem corresponds to a concave current-value Hamil-tonian and derive the unique solution to this problem. Show that this solution in-volves the consumption of the representative household growing at a constant rateat all points.(2) Show that the social planner will tend to increase growth because read more..

##### Page - 574

Introduction to Modern Economic Growth(2) Show that the BGP characterized in Proposition 14.6 also applies when the distri-bution of R&D eﬀorts across incumbents is given by (14.71).Exercise 14.26. Consider the model of Section 14.3, but modify the production functionto Y (t)=hR10q(ν,t)x(ν,t|q)1−βdνiLβ/(1−β)andassumethatproductionofaninputofquality q requires ψq units of the ﬁnal good as in the baseline model of Section 14.1. Showthat the equilibrium growth rate and the read more..

##### Page - 575

Introduction to Modern Economic GrowthExercise 14.33.(1)Whatisthe eﬀect of competition on the rate of growth of theeconomy in a standard product variety model of endogenous growth? What aboutthe quality-ladder model? Explain the intuition.(2) Now consider the following one-period model. There are two Bertrand duopolists,producing a homogeneous good. At the beginning of each period, duopolist 1’s mar-ginal cost of production is determined as a draw from the uniform distribution [0, ¯c1]and read more..

##### Page - 576

CHAPTER 15Directed Technological ChangeThe previous two chapters introduced the basic models of endogenous technologicalchange. These models provide us with a tractable framework for the analysis of aggregatetechnological change, but focus on a single type of technological change. Even when there aremultiple types of machines, these all play the same role in increasing aggregate productivity.Consequently, technological change in these models is always “neutral” (that is, Hicks-neutralas read more..

##### Page - 577

Introduction to Modern Economic Growth15.1. Importance of Biased Technological ChangeTo see the potential importance of the biased technological change, let us ﬁrst review anumber of examples:(1) Perhaps the most important example of biased technological change is the so-calledskill-biased technological change, which has played an important role in the analysisof recent labor market developments and changes in the wage structure. Figure 15.1plots a measure of the relative supply of skills read more..

##### Page - 578

Introduction to Modern Economic GrowthCollege wage premiumRelative Supply of College Skills and College PremiumyearRel. supply of college skills College wage premiumRel. supply of college skills39495969798996.3.4.5.60.2.4.6.8Figure 15.1. Relative supply of college graduates and the college premiumin the U.S. labor market.parts and the assembly line. Products previously manufactured by skilled artisansstarted to be produced in factories by workers with relatively few skills, and manypreviously read more..

##### Page - 579

Introduction to Modern Economic Growth(3) Beginning in the late 1960s and the early 1970s, both unemployment and the share oflabor in national income increased rapidly in a number of continental European coun-tries. During the 1980s, unemployment continued to increase, but the labor sharestarted a steep decline, and in many countries it fell below its initial level. Blanchard(1997) interprets the ﬁrst phase as the response of these economies to a wage-pushby workers, and the second phase as a read more..

##### Page - 580

Introduction to Modern Economic GrowthWe will seethatthis marketsize eﬀect will be powerful enough to outweigh the priceeﬀect. In fact, our analysis will show that under fairly general conditions the following tworesults will hold:• Weak Equilibrium (Relative) Bias: an increase in the relative supply of a factoralways induces technological change thatisbiasedinfavor of this factor.• Strong Equilibrium (Relative) Bias: if the elasticity of substitution between factorsis suﬃciently read more..

##### Page - 581

Introduction to Modern Economic GrowthSkill premiumRelative supply of skillsH/LSkill-biased tech. changeωω’Relative demand for skillsFigure 15.2. The eﬀect of H-biased technological change on relative demandand relative factor prices.where L (t) is labor, and H (t) denotes another factor of production, which could be skilledlabor, capital, land or some intermediate goods, and A (t) represents technology. Withoutloss of generality imagine that ∂F/∂A > 0, so a greater level of A read more..

##### Page - 582

Introduction to Modern Economic GrowthThese concepts can be further clariﬁed using the constant elasticity of substitution (CES)production function, which we will use for the rest of this chapter. The CES productionfunction takes the formY (t)=hγL(AL(t)L(t))σ−1σ + γH (AH (t) H (t))σ−1σi σσ−1 ,where AL (t) and AH (t) are two separate technology terms, the γis determine the importanceof the two factors in the production function, and γL + γH =1. Finally, σ ∈ (0, ∞) isthe read more..

##### Page - 583

Introduction to Modern Economic Growthincreases the demand for labor, L, by more than the demand for H. As a result, the marginalproduct of labor increases by more than the marginal product of H. This can be seen mostclearly in the extreme case where σ → 0, so that the two factors become Leontieﬀ.In thiscase, starting from a situation in which γLAL (t) L (t)= γHAH (t) H (t), a small increase inAH (t) will create an “excess of the services” of the H factor (and thus “excess demand” read more..

##### Page - 584

Introduction to Modern Economic Growthexternalities. Section 15.4 will consider a model of directed technological with change knowl-edge spillovers. Exercise 15.19 shows that all of the results presented here generalize to amodel of Schumpeterian growth, thus the assumption of expanding varieties is only adoptedfor convenience.The baseline economy has a constant supply of two factors, L and H,and admits arepresentative household with the standard CRRA preferences given by(15.2)Z ∞0exp read more..

##### Page - 585

Introduction to Modern Economic Growthtwo production functions (15.5) and (15.6) use diﬀerent types of machines. The range ofmachines complementing labor, L,is [0,NL (t)], while the range of machines complementingfactor H is [0,NH (t)].Again as in Chapter 13, we assume that all machines in both sectors are supplied bymonopolists that have a fully-enforced perpetual patent on the machines. We denote theprices charged by these monopolists at time t by pxL (ν, t) for ν ∈ [0,NL (t)] and pxH read more..

##### Page - 586

Introduction to Modern Economic Growth15.3.1. Characterization of Equilibrium.An allocation in this economy is de-ﬁned by the following objects:time paths of consumption levels, aggregate spend-ing on machines, and aggregate R&D expenditure [C (t) ,X (t) ,Z (t)]∞t=0,timepathsof available machine types, [NL (t) ,NH (t)]∞t=0,time paths of prices and quantitiesof each machine and the net present discounted value of proﬁts from that machine,[pxL (ν, t) ,xL (ν, t) ,VL (ν, read more..

##### Page - 587

Introduction to Modern Economic GrowthSubstituting these prices into (15.13) and (15.14), we obtainxL (ν, t)= pL (t)1/β Lfor all ν and all t,andxH (ν, t)= pH (t)1/β Hfor all ν and all t.Since these quantities do not depend on the identity of the machine, only on the sector thatis being served, proﬁts are also independent of the machine type. In particular, we have(15.15)πL (t)= βpL (t)1/β L and πH (t)= βpH (t)1/β H.This implies that the net present discounted values of monopolists read more..

##### Page - 588

Introduction to Modern Economic GrowthUsing the latter equation, we can also calculate the relative factor prices in this economy as:ω (t) ≡wH (t)wL (t)= p (t)1/β NH (t)NL (t)= γεσµNH(t)NL (t)¶σ−1σµHL¶−1σ.(15.19)The ﬁrst line of (15.19) deﬁnes ω (t) as the relative wage of factor H compared to factor L.The second line uses the deﬁnition of marginal product combined with (15.16) and (15.17),and the third line uses (15.18). We refer to σ as the (derived) elasticity of read more..

##### Page - 589

Introduction to Modern Economic Growthabove, the greater is VH relative to VL, the greater are the incentives to develop H-augmentingmachines, NH, rather than NL. Taking the ratio of these two expressions, we obtainVHVL=µpHpL¶1β HL.This expression highlights the two eﬀects on the direction of technological change dis-cussed in Section 15.1.(1) The price eﬀect manifests itself because VH/VL is increasing in pH/pL. The greater isthis relative price, the greater are the incentives to invent read more..

##### Page - 590

Introduction to Modern Economic Growthwhere η ≡ ηH/ηL and the *’s denote that this expression refers to the BGP value. The notablefeature here is that relative productivities are determined by the innovation possibilitiesfrontier and the relative supply of the two factors. In this sense, this model totally endogenizestechnology. Equation (15.27) contains most of the economics of directed technology. However,before discussing this, it is useful to characterize the BGP growth rate of the read more..

##### Page - 591

Introduction to Modern Economic Growth15.3.2. Directed Technological Change and Factor Prices.Let us start by study-ing (15.27). This equation implies that, in BGP, there is a positive relationship between therelative supply of the H factor, H/L, and the relative factor-augmenting technologies, N∗H /N∗Lonly when σ> 1. In contrast, if the derived elasticity of substitution, σ, islessthan1, therelationship is reversed. This might suggest that, depending on the elasticity of read more..

##### Page - 592

Introduction to Modern Economic Growthseemingly paradoxical result that relative demand curves can be upward-sloping once theendogeneity of technology is taken into account. To obtain this result, let us substitute for(NH/NL)∗ from (15.27) into the expression for the relative wage given technologies, (15.19),and obtain the following BGP relative factor price ratio (see Exercise 15.4):(15.30)ω∗ ≡µwHwL¶∗=ησ−1γεµHL¶σ−2.Inspection of this equation immediately establishes read more..

##### Page - 593

Introduction to Modern Economic Growthcurve is downward-sloping follows from basic producer theory. The curve marked as ET1applies when technology is endogenous, but the condition in Proposition 15.4, that σ> 2,is not satisﬁed. We know from Proposition 15.3 that even in this case an increase in H/Lwill induce skill-biased (H-biased) technological change. This implies that when H/L ishigher than its initial level, the induced-technology eﬀect will shift the constant-technologydemand curve read more..

##### Page - 594

Introduction to Modern Economic Growthduring the 1970s in the face of the very large increase in the supply of college-educated work-ers, the skill (college) premium has increased very sharply throughout the 1980s and the1990s, to reach a level not experienced in the postwar era. Figure 15.1 above showed thesegeneral patterns.In the labor economics and parts of the macroeconomics literature, the most popularexplanation for these patterns is skill-biased technological change. For example, read more..

##### Page - 595

Introduction to Modern Economic Growthunskilled workers available to be employed in the factories during this time periods.Bairoch (1988, p. 245), for example, describes this rapid expansion of the supply ofunskilled labor as follows:“ ... between 1740 and 1840 the population of England ... went up from 6 millionto 15.7 million. ... while the agricultural labor force represented 60-70% of the totalwork force in 1740, by 1840 it represented only 22%.”Habakkuk’s well-known account of read more..

##### Page - 596

Introduction to Modern Economic GrowthSkill premiumLong-run relative demand for skillsExogenous Shift in Relative SupplyInitial premiumShort-runResponseLong-run premiumFigure 15.4. Dynamics of the skill premium in response to an exogenousincrease in the relative supply of skills, with an upward-sloping endogenous-technology relative demand curve.up below its initial level. To explain the larger increase in the college premium in the 1980s,in this case we would need some exogenous skill-biased read more..

##### Page - 597

Introduction to Modern Economic GrowthSkill premiumLong-run relative demand for skillsExogenous Shift in Relative SupplyInitial premiumShort-runResponseLong-run premiumFigure 15.5. Dynamics of the skill premium in response to an increase inthe relative supply of skills, with a downward-sloping endogenous-technologyrelative demand curve.when scale eﬀects are removed. Third, we would like to apply these ideas to investigatewhether there are reasons for technological change to be endogenously read more..

##### Page - 598

Introduction to Modern Economic Growthsubject toCS (t)=(1 − β)−1/β∙γεL¡NSL(t)L¢σ−1σ+γεH¡NSH(t)H¢σ−1σ¸σσ−1− ZSL (t) − ZSH (t) .The necessary conditions for this problem give the following characterization of the Paretooptimal allocation in this economy.Proposition 15.5. The stationary solution of the Pareto optimal allocation involvesrelative technologies given by (15.27) as in the decentralized equilibrium. The stationarygrowth rate is higher than the equilibrium read more..

##### Page - 599

Introduction to Modern Economic Growthstate dependence, because productivity in each sector can depend on the state of knowledgein both sectors. A ﬂexible formulation is the following:(15.32)˙NL (t)= ηLNL (t)(1+δ)/2 NH (t)(1−δ)/2 SL (t) and ˙NH (t)= ηHNL (t)(1−δ)/2 NH (t)(1+δ)/2 SH (t) ,where δ ≤ 1,and SL (t) is the number of scientists working to produce L-augmenting ma-chines, while SH (t) denotes the number of scientists working on H-augmenting machines.Clearly, market read more..

##### Page - 600

Introduction to Modern Economic Growththe direction of technological change would be identical to those from the lab equipmentspeciﬁcation.This is no longer true when δ> 0.To characterize the results in this case, let uscombine condition (15.36) with equations (15.15) and (15.18), we obtain the equilibriumrelative technology as (see Exercise 15.9):(15.37)µNHNL¶∗=η σ1−δσ γε1−δσµHL¶σ−11−δσ,where recall that γ ≡ γH/γL and η ≡ ηH/ηL. This expression shows read more..

##### Page - 601

Introduction to Modern Economic GrowthProof. See Exercise 15.10.¤In contrast to the model with the lab equipments technology, transitional dynamicsdo not always take the economy to the BGP equilibrium, however.This is because ofthe additional increasing returns to scale mentioned above. With a high degree of statedependence, when NH (0) is very high relative to NL (0), it may no longer be proﬁtable forﬁrms to undertake further R&D directed at labor-augmenting (L-augmenting) read more..

##### Page - 602

Introduction to Modern Economic GrowthProposition 15.9. Consider the directed technological change model with knowledgespillovers and state dependence in the innovation possibilities frontier. Then ifσ> 2 − δ,there is strong equilibrium (relative) bias in the sense that an increase in H/L raises therelative marginal product and the relative wage of the H factor compared to the L factor.Intuitively, the additional increasing returns to scale coming from state dependence makesstrong bias read more..

##### Page - 603

Introduction to Modern Economic Growthfrom past research are limited, and this economy will not have steady growth in the absenceof population growth.Let us also modify the baseline environment by assuming that total population, in partic-ular, the population of scientists, grows at the exponential rate n. With a similar argumentsto that in Section 13.3 in Chapter 13, it can be veriﬁed that aggregate output in this economywill grow at the rate (see Exercise 15.15):(15.42)g∗ =n1 − read more..

##### Page - 604

Introduction to Modern Economic Growth15.6. Endogenous Labor-Augmenting Technological ChangeOne of the advantages of the models of directed technical change is that they allow usto investigate why technological change might be purely labor-augmenting as required forbalanced growth. We will see that models of directed technological change create a naturalreason for technology to be more labor-augmenting than capital-augmenting. However, undermost circumstances, the resulting equilibrium is not read more..

##### Page - 605

Introduction to Modern Economic GrowthThe next proposition summarizes the main idea of the previous paragraph. For simplic-ity, this proposition treats the increase in K (t) /L as a sequence of one-time increases (fullequilibrium dynamics are investigated in the next two propositions).Proposition 15.12. In the baseline model of directed technological change with H (t)=K (t) as capital, if K (t) /L is increasing over time and σ< 1,then NL (t) /NK (t) will alsoincrease over time, i.e., read more..

##### Page - 606

Introduction to Modern Economic Growthfollowing relationship in BGP (see Exercise 15.23):(15.47)r (t) K (t)wL (t) L= η−1.Thus, directed technological change implies that in the long-run the share of capital is con-stant in national income. Long-run constant factor shares (combined with capital deepening)means that asymptotically all technological change must be purely-labor-augmenting. Morespeciﬁcally, recall from (15.19) thatr (t)wL (t)= γεσµNK(t)NL (t)¶σ−1σµK(t)L¶−1σ,where read more..

##### Page - 607

Introduction to Modern Economic GrowthNotice that Proposition 15.14 does not imply that all technological change must be Harrodneutral (purely labor-augmenting). Along the transition path, there can be (and in factthere will be) capital-augmenting technological change. However, in the long run (that is,asymptotically or as t →∞), all technological change will be labor-augmenting.It can also be veriﬁed that the constant growth path allocation with purely labor-augmenting technological read more..

##### Page - 608

Introduction to Modern Economic Growthsuﬃciently high. However, once we allow for a richer menu of technological changes, theseresults do not necessarily hold. Nevertheless, the essence of the results appears to be muchmore general. Acemoglu (2007) deﬁnes the complementary notions of weak and strong ab-solute equilibrium bias, which refer to whether the equilibrium price of a factor change as thesupply of that factor changes (rather than the price of a factor relative to the price of read more..

##### Page - 609

Introduction to Modern Economic Growththis approach can be used to provide conditions under which technological change will beendogenously labor-augmenting (recall that this type of technological progress is necessaryfor balanced growth). An alternative approach to this problem is suggested in the recentpaper by Jones (2005). I now brieﬂy discussed this alternative approach.The models developed so far treat the diﬀerent types of technologies (e.g., NL and NH inthe previous sections) as state read more..

##### Page - 610

Introduction to Modern Economic Growththe economy from the set of available ideas. To do this, we ﬁrst need to specify how a givenidea is used for production. Let us suppose that there is a single ﬁnal good, Y ,which canbeproduced using any idea i ∈ I with a Leontief production function given by(15.50)Y (t)=min{biK (t) ,aiL (t)},where K (t) and L (t) are the amounts of capital and labor in the economy. In general, theeconomy may use multiple ideas, thus K (t) and L (t) should be indexed by read more..

##### Page - 611

Introduction to Modern Economic GrowthNow, given this structure, let us deﬁne the function(15.51)G(b, a) ≡ Pr [ai≥ a and bi ≥ b ]=µbγb¶−βµaγa¶−αas the joint probability ai ≥ a and bi ≥ b. Denote the level of aggregate output that canbe produced using technique i with capital K and labor L by ˜Yi(K, L). Beforeweknowthe realizations of ai and bi for idea i, this level of output is a random variable. Since theproduction function is Leontief, the distribution of ˜Yi can be read more..

##### Page - 612

Introduction to Modern Economic Growthof output as N (t) →∞. Instead, we have to look at aggregate output normalized by anappropriate variable, such as its “expected value” (and apply a type of reasoning similar tothe Central Limit Theorem). Given the Pareto distribution, the normalizing factor turns outto be n (t) ≡³γN(t)K(t)βL(t)α´ 1α+β ,sothatwecan writePr∙˜Y(t;N(t))≤³γN(t)K(t)βL(t)α´ 1α+β y¸ =³1−γK(t)βL(t)α(n(t)y)−(α+β)´N(t)=Ã1−y−(α+β)N read more..

##### Page - 613

Introduction to Modern Economic Growthderivation of a static production function also imply that the short-run production functionwill evolve endogenously on average with labor-augmenting technological change dominatingthe limiting behavior and making sure that the economy, in the long run, acts as if it has aCobb-Douglas production function.Although this is an interesting idea, and as we have already seen in Section 14.3, thePareto distribution appears in many important contexts and has various read more..

##### Page - 614

Introduction to Modern Economic Growththe past 100 years, the causes of acceleration in skill-biased technological change duringmore recent decades, the causes of unskilled-biased technological developments during the19th century, the impact of international trade on the direction of technological change,the relationship between labor market institutions and the types of technologies that aredeveloped and adopted, and last but not least, an investigation of why technological changein read more..

##### Page - 615

Introduction to Modern Economic Growth15.10. References and LiteratureModels of directed technological change were developed in Acemoglu (1998, 2002a,2003a,b, 2007a), Kiley (1999), and Acemoglu and Zilibotti (2001). These papers use the termdirected technical change, but here we used the related term directed technological change,to emphasize continuity with the models of endogenous technological change studied in theprevious chapters. The framework presented here builds on Acemoglu (2002a). A read more..

##### Page - 616

Introduction to Modern Economic Growthin the analysis. Similar problems are present in the other earlier works as well. It was alsonot clear who undertook the R&D activities and how they were ﬁnanced and priced. Theseshortcomings reduced the interest in this literature, and there was little research for almost30 years, with the exception of some empirical work, such as that by Hayami and Ruttan(1970) on technical change in American and Japanese agriculture.The analysis in Acemoglu (1998) read more..

##### Page - 617

Introduction to Modern Economic Growthfor high-cost labor. This is the basis of Exercise 15.27 below. Caballero and Hammour (1999)provide an alternative and complementary explanation to that suggested here.Acemoglu and Zilibotti (2001) discuss implications of directed technological change forcross-country income diﬀerences. We have not dwelled on this topic here, since this will bediscussed in greater detail in Chapter 18.4 in the context of appropriate technologies.Acemoglu (2003b) suggested read more..

##### Page - 618

Introduction to Modern Economic GrowthExercise 15.5. Explain why in Proposition 15.1 the eﬀect of γ on the BGP growth rate,(15.29), is ambiguous. When is this eﬀect positive? Provide an intuition.Exercise 15.6. Derive equation 15.31.Exercise 15.7. Prove Proposition 15.5. [Hint: ﬁrst substitute for C (t) from the constraint.Then show that μH (t) /μL (t)=(ηH (t) /ηL (t))−1. Then use the necessary conditions with˙μH (t)= ˙μL (t)].Exercise 15.8. Derive the free entry conditions read more..

##### Page - 619

Introduction to Modern Economic Growth(2) Specify the free entry conditions for each machine variety.(3) Characterize the BGP equilibrium, show that it is uniquely deﬁned and determineconditions such that the growth rate is positive and the transversality condition issatisﬁed.(4) Show that the equivalents of Propositions 15.3 and 15.4 hold in this environment.(5) Characterize the transitional dynamics and show that they are similar to those inProposition 15.2.(6) Characterize the Pareto read more..

##### Page - 620

Introduction to Modern Economic Growtheconomy during the 19th century was faster than in Britain because of relative labor scarcityin the former (which increased wages and encouraged technology adoption).(1) First, consider a neoclassical-type model with two factors, labor and technology,F (A, L),where F exhibits constant returns to scale. Show that an increase inwages, either caused by a decline in labor supply or an exogenous increase in wagesbecause of the minimum wage, cannot increase A.(2) read more..

##### Page - 621

Introduction to Modern Economic Growthmarket clearing level will ﬁrst cause unemployment and then if σ< 1, it will cause capital-biased technological change. Can this model shed light on the persistent unemploymentdynamics in continental Europe? [Hint: distinguish two cases: (i) the minimum wage ﬂooris constant; (ii) the minimum wage ﬂoor increases at the same rate as the growth of theeconomy].Exercise 15.28. * The analysis in the text has treated the supply of the two factors read more..

##### Page - 622

Introduction to Modern Economic Growthproduced asY (t)=∙Z n0y (ν, t)ε−1εdν¸ εε−1where ε> 1 and intermediate y (ν, t) can be produced using either skilled or unskilled labor.In particular, when a new intermediate is invented, it is ﬁrst produced using skilled laboronly, with the production function y (ν, t)= h (ν, t), and eventually, another ﬁrm may ﬁnd away to produce this good using unskilled labor with the production function y (ν, t)= l (ν, t).Assume that when read more..

##### Page - 623

Introduction to Modern Economic GrowthExercise 15.30. Consider the model presented in Section 15.8.(1) Show that if capital and labor are allocated in competitive markets, in general morethan one technique will be used in equilibrium. [Hint: construct an example inwhich there are three ideas i =1, 2 and 3, such that when only one can be used, itwill be i =1, but output can be increased by allocating some of labor and capitalto ideas 2 and 3].(2) * Show that in this case the exact aggregation read more..

##### Page - 624

Introduction to ModernEconomic Growth: Parts 6-9Daron AcemogluDepartment of Economics,Massachusetts Institute of Technology read more..

##### Page - 625

ContentsPrefacexiPart 1.Introduction1Chapter 1.Economic Growth and Economic Development:The Questions31.1.Cross-Country Income Diﬀerences31.2.Income and Welfare61.3.Economic Growth and Income Diﬀerences101.4.Origins of Today’s Income Diﬀerences and World Economic Growth121.5.Conditional Convergence161.6.Correlates of Economic Growth201.7.From Correlates to Fundamental Causes221.8.The Agenda251.9.References and Literature27Chapter 2.The Solow Growth Model312.1.The Economic Environment of read more..

##### Page - 626

Introduction to Modern Economic Growth4.2.Economies of Scale, Population, Technology and World Growth1334.3.The Four Fundamental Causes1364.4.The Eﬀect of Institutions on Economic Growth1474.5.What Types of Institutions?1644.6.Disease and Development1674.7.Political Economy of Institutions: First Thoughts1704.8.Taking Stock1714.9.References and Literature1714.10.Exercises174Part 2.Towards Neoclassical Growth177Chapter 5.Foundations of Neoclassical Growth1795.1.Preliminaries1795.2.The read more..

##### Page - 627

Introduction to Modern Economic Growth7.10.References and Literature2997.11.Exercises301Part 3.Neoclassical Growth307Chapter 8.The Neoclassical Growth Model3098.1.Preferences, Technology and Demographics3098.2.Characterization of Equilibrium3138.3.Optimal Growth3178.4.Steady-State Equilibrium3188.5.Transitional Dynamics3208.6.Technological Change and the Canonical Neoclassical Model3238.7.Comparative Dynamics3298.8.The Role of Policy3308.9.A Quantitative read more..

##### Page - 628

Introduction to Modern Economic Growth11.3.The Two-Sector AK Model42611.4.Growth with Externalities43011.5.Taking Stock43411.6.References and Literature43511.7.Exercises436Part 4.Endogenous Technological Change443Chapter 12.Modeling Technological Change44512.1.Diﬀerent Conceptions of Technology44512.2.Science and Proﬁts44912.3.The Value of Innovation in Partial Equilibrium45212.4.The Dixit-Stiglitz Model and “Aggregate Demand Externalities”45912.5.Individual R&D Uncertainty and the read more..

##### Page - 629

Introduction to Modern Economic GrowthChapter 16.Stochastic Dynamic Programming61316.1.Dynamic Programming with Expectations61316.2.Proofs of the Stochastic Dynamic Programming Theorems*62116.3.Stochastic Euler Equations62616.4.Generalization to Markov Processes*62916.5.Applications of Stochastic Dynamic Programming63116.6.Taking Stock63916.7.References and Literature64016.8.Exercises641Chapter 17.Stochastic Growth Models64517.1.The Brock-Mirman Model64617.2.Equilibrium Growth under read more..

##### Page - 630

Introduction to Modern Economic Growth20.3.Agricultural Productivity and Industrialization83520.4.Taking Stock84120.5.References and Literature84320.6.Exercises844Chapter 21.Structural Transformations and Market Failures in Development84921.1.Financial Development85121.2.Fertility, Mortality and the Demographic Transition85721.3.Migration, Urbanization and The Dual Economy86521.4.Distance to the Frontier and Changes in the Organization of Production87621.5.Multiple Equilibria From Aggregate read more..

##### Page - 631

Introduction to Modern Economic GrowthChapter A.Odds and Ends in Real Analysis and Applications to Optimization1117A.1.Distances and Metric Spaces1118A.2.Mappings, Functions, Sequences, and Continuity1121A.3.A Minimal Amount of Topology: Continuity and Compactness1126A.4.The Product Topology1131A.5.Correspondences and Berge’s Maximum Theorem1134A.6.Convexity, Concavity, Quasi-Concavity and Fixed Points1138A.7.Diﬀerentiation, Taylor Series and the Mean Value Theorem1142A.8.Functions of read more..

##### Page - 632

Part 5Stochastic Growth read more..

##### Page - 633

This part of the book focuses on stochastic growth models and provides a brief intro-duction to basic tools of stochastic dynamic optimization. Stochastic growth models areuseful for two related reasons. First, a range of interesting growth problems involve eitheraggregate uncertainty or nontrivial individual level uncertainty interacting with investmentdecisions and the growth process. Some of these models will be discussed in Chapter 17.Second, the stochastic neoclassical growth model has a read more..

##### Page - 634

CHAPTER 16Stochastic Dynamic ProgrammingThis chapter provides an introduction to basic stochastic dynamic programming. To avoidthe use of measure theory in the main body of the text, I will ﬁrst focus on economies inwhich stochastic variables take ﬁnitely many values. This will enable us to use Markov chains,instead of general Markov processes, to represent uncertainty. Since many commonly-usedstochastic processes, such as those based on normal or uniform distributions, fall outsidethis read more..

##### Page - 635

Introduction to Modern Economic GrowthWe assume that the stochastic variable z (t) follows a (ﬁrst-order) Markov chain.1 Theimportant property implied by the Markov chain assumption is that the current value of z (t)only depends on its last period value, z (t − 1). Mathematically, this can be expressed asPr [z (t)= zj | z (0) ,...,z (t − 1)] ≡ Pr [z (t)= zj | z (t − 1)] .The simplest example of an economic model with uncertainty represented by a Markov chainwould be one in which the read more..

##### Page - 636

Introduction to Modern Economic Growthof consumption per capita is stochastic (as they will depend on the realization of future z’s).In particular, suppose that the production function (per capita) takes the formy (t)= f (k (t) ,z (t)) ,where k (t) again denotes the capital-labor ratio and z (t) ∈ Z ≡ {z1,...,zN} representsa stochastic variable that aﬀects how much output will be produced with a given amountof inputs. We continue to assume that z (t) follows a Markov chain. The most read more..

##### Page - 637

Introduction to Modern Economic Growthvalue of the stochastic variable). Let x (t)= k (t),sothatx (t +1) = k (t +1)= f (k (t) ,z (t)) + (1 − δ) k (t) − ˜c£zt¤≡˜k£zt¤,where the second line simply uses the resource constraint with equality and the third linedeﬁnes the function ˜k£zt¤. With this notation, feasibility is easier to express, sincek (t +1) ≡˜k£zt¤by deﬁnition depends only on the history of the stochastic shocks up to time t and not onz (t +1). In addition, read more..

##### Page - 638

Introduction to Modern Economic GrowthWith the same reasoning, the recursive characterization would naturally take the form(16.3)V (k, z)=supy∈[0,f(k,z)+(1−δ)k]©u(f(k,z)+(1−δ)k−y)+βE£V ¡y,z0¢|z¤ª,where E [·| z] denotes the expectation conditional on the current value of z and incorporatesthe fact that the random variable z is a Markov chain. Let us suppose that this program hasa solution, meaning that there exists a feasible plan that achieves the value V (k, z) startingwith read more..

##### Page - 639

Introduction to Modern Economic GrowthSimilar to (16.3) in Example 16.1, the functional equation corresponding to the recursiveformulation of this problem can be written as:Problem B2:V (x, z)=supy∈G(x,z)©U(x,y,z)+βE£V(y,z0)|z¤ª, for all x ∈ X and z ∈ Z(16.4)where V : X ×Z → R is a real-valued function and y ∈ G(x, z) represents the constrainton next period’s state vector as a function of the realization of the stochastic variable z.Problem B2 is a direct generalization of the read more..

##### Page - 640

Introduction to Modern Economic GrowthWe now introduce analogs of Assumption 6.1-6.5 from Chapter 6 and the appropriategeneralizations of Theorems 6.1-6.6.Assumption16.1.G (x, z)isnonemptyforallx∈ Xandz∈Z.Moreover, forallx (0)∈ X,z (0)∈ Z,and x∈Φ(x(0),z (0)),limn→∞ E£Pnt=0βtU(˜x£zt−1¤,˜x£zt¤,z(t))|z(0)¤existsandisﬁnite.Assumption 16.2. X is a compact subset of RK , G is nonempty, compact-valued andcontinuous. Moreover, let XG = {(x,y,z) ∈ X × X ×Z : y ∈ G(x, z)} read more..

##### Page - 641

Introduction to Modern Economic GrowthThe next result establishes the uniqueness of the value function and existence of solutions.Theorem 16.3. (Existence of Solutions) Suppose that Assumptions 16.1 and 16.2hold. Then the unique function V : X ×Z → R that satisﬁes (16.4) is continuous andbounded in x for each z ∈ Z. Moreover, an optimal plan x∗ ∈Φ(x(0),z (0)) exists for anyx (0) ∈ X and any z (0) ∈ Z.The remaining results, as their analogs in Chapter 6 use further assumptions to read more..

##### Page - 642

Introduction to Modern Economic GrowthThese theorems have exact analogs in Chapter 6. Since the value function now alsodepends on the stochastic variable z, an additional monotonicity result can also be obtained.For this, let us introduce the following additional assumption:Assumption 16.6. (i) G is monotone in z in the sense that z ≤ z0 implies G(x, z) ⊂G(x, z0) for each any x ∈ X and z, z0 ∈ Z such that z ≤ z0.(ii) For each (x, y, z) ∈ XG, U (x, y, z) is strictly increasing in read more..

##### Page - 643

Introduction to Modern Economic Growthandfor any ε> 0,there exists y0 ∈ G (x (0) ,z (0))(16.11)s.t.V (x (0) ,z (0)) ≤ U (x (0) ,y0,z (0)) + βE [V (y, z (1)) | z (0)] + ε.The following lemma is as straightforward generalization of Lemma 6.1 in 6Lemma 16.1. Suppose that Assumption 16.1 holds. Then for any x (0) ∈ X,any z (0) ∈Z,any x≡©˜x£zt¤ª∞t=−1∈Φ(x(0),z(0)),wehavethat¯U(x,z (0)) = U (x (0) , ˜x£z0¤,z(0))+βE£¯U(©˜x£zt¤ª∞t=0,z(1))|z(0)¤.Proof. See read more..

##### Page - 644

Introduction to Modern Economic GrowthBy the feasibility of x0ε,we have x00ε=¡˜x0ε£z0¤,˜x0ε£z1¤,...¢∈Φ¡˜x0ε£z0¤,z(1)¢foranyz(1)∈Z.Moreover, also by deﬁnition V∗¡˜x0ε£z0¤,z(1)¢isthesupremuminProblemB1startingwiththe initial conditions ˜x0ε£z0¤andz(1). Then Lemma 16.1 implies that for any ε> 0,V∗ (x (0) ,z (0)) − ε ≤ U¡x(0),˜x0ε£z0¤,z(0)¢+βE£¯U¡©˜x£zt¤ª∞t=0,z(1)¢|z(0)¤= read more..

##### Page - 645

Introduction to Modern Economic GrowthWe ﬁrst show that for any t ≥ 0, x∗t attains the supremum starting from ˜x∗£zt−1¤andany z (t) ∈ Z,thatis,(16.12)¯U(x∗t ,z (t)) = V∗¡˜x∗£zt−1¤,z(t)¢.The proof is by induction. The hypothesis is trivially satisﬁed for t =0 since, by deﬁnition,x∗0 = x∗ attains V∗ (x (0) ,z (0)).Next suppose that the statement is true for t,sothat x∗t attains the supremum startingfrom read more..

##### Page - 646

Introduction to Modern Economic Growthloss of generality taking ˆz = z1,E£¯U(ˆx∗t+1,z(t+1))|z(t)¤ =NXj=1qjj0 ¯U(ˆx∗t+1,zj )= q1j0 ¯U(ˆxt+1,zj)+NXj=2qjj0 ¯U(x∗t+1,zj )>q1j0 ¯U(x∗t+1,zj )+NXj=2qjj0 ¯U(x∗t+1,zj )= E£¯U(x∗t+1,z(t+1))|z(t)¤,contradicting (16.14) and completing the induction step, which establishes that x∗t+1 attainsthe supremum starting from ˜x∗£zt¤andanyz(t+1)∈Z.Equation (16.12) then implies thatV∗¡˜x∗£zt−1¤,z(t)¢ =¯U(x∗t ,z (t))= read more..

##### Page - 647

Introduction to Modern Economic GrowthNow deﬁne the operator T(16.15)TV (x, z)=maxy∈G(x,z)©U(x,y,z)+βE£V ¡y,z0¢|z¤ª.Suppose that V (x, z) is continuous and bounded. Then E [V (y, z0) | z] is also continuous andbounded, since it is simply given byE£V ¡y,z0¢|z¤≡NXj=1qjj0V (y, zj) ,with j0 deﬁned such that z = zj0.Moreover, U (x, y, z) is also continuous and boundedover XG.A ﬁxed point of the operator T , V (x, z)= TV (x, z),will then be a solutionto Problem B2 for given z ∈ read more..

##### Page - 648

Introduction to Modern Economic GrowthLet us follow the treatment in Chapter 6 and also build on the results from Section 16.1.Let us use ∗’s to denote optimal values and D for gradients. Then using Assumption 16.5and Theorem 16.6, we can write the necessary conditions for an interior optimal plan as(16.16)DyU (x, y∗,z)+ βE£DxV ¡y∗,z0¢|z¤=0,where x ∈ RK is the current value of the state vector, z ∈ Z is thecurrent valueofthestochastic variable, and DxV (y∗,z0) denotes the read more..

##### Page - 649

Introduction to Modern Economic GrowthTheorem 16.8. (Euler Equations and the Transversality Condition) Let X ⊂RK+and suppose that Assumptions 16.1-16.5 hold.Then the sequence of feasible plans©˜x∗£zt¤ª∞t=−1,with ˜x∗£zt¤∈IntG(˜x∗£zt−1¤,z(t))foreachz(t)∈Z and each t =0, 1,... , isoptimal for Problem B1 given x (0) and z (0) ∈ Z if it satisﬁes (16.18) and (16.19).Proof. Consider an arbitrary x (0)∈ X and z (0)∈ Z,and let x∗ ≡©˜x∗£zt¤ª∞t=−1 read more..

##### Page - 650

Introduction to Modern Economic Growthzero. Finally, since U is increasing in x, DxU ≥ 0,and x ≥ 0, the fourth line is nonnega-tive, establishing that E [∆x (z∞) | z (s)] ≥ 0 for any x∈Φ(x(0),z (0)) and any z (s) ∈ Z.Consequently, x∗ yields higher value than any feasible x∈Φ(x(0),z (0)), and is thereforeoptimal.¤16.4. Generalization to Markov Processes*What happens if z does not take on ﬁnitelymanyvalues?Forexample, z may berepresented by a general Markov process, taking read more..

##### Page - 651

Introduction to Modern Economic Growthselection for all x (t) ∈ X and zt ∈ Zt. For this reason, I will refer to these assumptions witha * (i.e., instead of Assumption 16.2, I will refer to Assumption 16.2*).Theorem 16.9. (Existence of Solutions) Suppose that Φ¡x(0),z0¢ is nonemptyfor all z0∈ Z and all x (0) ∈ X.Suppose also that for any x∈Φ¡x(0),z0¢,E£P∞t=0βtU¡˜x£zt−1¤,˜x£zt¤,z(t)¢|z(0)¤ is well-deﬁned and ﬁnite-valued. Then any so-lution V (x, z) to Problem B2 read more..

##### Page - 652

Introduction to Modern Economic Growth16.5. Applications of Stochastic Dynamic ProgrammingWe now present a number of applications of the methods of stochastic dynamic program-ming. Some of the most important applications, related to stochastic growth and growthwith incomplete markets, are left for next chapter. In each application, I try to point outhow formulating the problem recursively and using stochastic dynamic programming methodssimplify the analysis.16.5.1. The Permanent Income read more..

##### Page - 653

Introduction to Modern Economic Growthdistribution. Nevertheless, for our purposes this is also a technicality and not much morethan the requirement that the lifetime budget constraint (16.21) should hold almost surelyis necessary for our analysis.Leaving technicalities aside, the fact that the lifetime budget constraint is stochastic hasimportant economic implications. In particular, although we have not introduced an explicitborrowing constraint, the fact that the lifetime budget constraint read more..

##### Page - 654

Introduction to Modern Economic Growthprices for all possible claims to consumption contingent on any realization of history areintroduced, is much more tractable and gives similar results to the recursive approach be-low. I will introduce this contingent-claims formulation in the analysis of the competitiveequilibrium of the neoclassical growth model under uncertainty in the next chapter.Instead, if we formulate the same problem recursively, sharper results can be easily ob-tained. Using the read more..

##### Page - 655

Introduction to Modern Economic Growthwith φ suﬃciently large that in the relevant range u (·) is increasing in c. Using this quadraticform with (16.24), we obtain Hall’s famous stochastic equation that(16.25)c (t)= (1 − κ) φ + κEtc (t +1) ,where κ ≡ β (1 + r). A striking prediction of this equation is that variables, such as currentor past income, should not predict future consumption growth. A large empirical literatureinvestigates whether or not this is the case in aggregate or read more..

##### Page - 656

Introduction to Modern Economic Growthengage in production using one of the techniques he has already discovered, or spend thatperiod searching for a new technique. Let us assume that each period in which he engages insuch a search, he gets an independent draw from a time-invariant distribution function H (a)deﬁned over a bounded interval [0, ¯a].Therefore, the decision of the entrepreneur at each date is whether to search for a newtechnique or to produce with one of the techniques he has read more..

##### Page - 657

Introduction to Modern Economic Growthprogramming formulation will be quite tractable even when the sequence problem may lookquite complicated.To demonstrate this, we now write this optimization problem recursively using dynamicprogramming techniques. Let us simplify the formulation of the recursive form of this problemby making two observations (which will both beprovedinExercise16.11). First, becausetheproblem is stationary we can discard all of the techniques that the individual has read more..

##### Page - 658

Introduction to Modern Economic Growthequations and writeV¡a0¢ =max½ a01 − β,βZ ¯a0V (a) dH (a)¾,(16.28)= TV¡a0¢,where the second line deﬁnes the mapping T . Now (16.28) is in a form to which we canapply the above theorems. Blackwell’s suﬃciency theorem (Theorem 6.9) applies directlyand implies that T is a contraction since it is monotonic and satisﬁes discounting.Next, let V ∈ C([0, ¯a]), i.e., the set of real-valued continuous (hence bounded) functionsdeﬁned over the read more..

##### Page - 659

Introduction to Modern Economic Growthso that the individual is just indiﬀerent between accepting the technology a = R and waitingfor one more period. Next we also have that since a<R are turned down, for all a<RV (a)= βZ ¯a0V (a) dH (a)=R1 − β,and for all a ≥ R,wehaveV (a)=a1 − β.Using these observations, we obtainZ ¯a0V (a) dH (a)=RH (R)1 − β+Za≥Ra1 − βdH (a) .Combining this equation with (16.29), we have(16.30)R1 − β= β∙RH(R)1 − β+Za≥Ra1 − βdH read more..

##### Page - 660

Introduction to Modern Economic GrowthThis implies that equation (16.31) has a unique solution. It can be easily veriﬁed that ahigher β, by making the entrepreneur more patient, increases the cutoﬀ threshold R.16.5.3. Other Applications.There are numerous other applications of stochastic dy-namic programming. In addition to the three growth models we will study in the nextchapter, the following are noteworthy.(1) Asset Pricing: following Lucas (1978), we can consider an economy in which read more..

##### Page - 661

Introduction to Modern Economic Growthof individual consumption. The other substantial model introduced in this chapter is thesearch for ideas model in subsection 16.5.2, which is adapted from McCall’s (1978) labormarket search model. McCall’s model is the basis of much of the modern equilibrium theoryof unemployment. While the model here has been cast in terms of searching for ideas, thereader can easily adapt it to unemployment and use it as an introduction to equilibriumunemployment read more..

##### Page - 662

Introduction to Modern Economic Growth16.8. ExercisesExercise 16.1. Show that Assumption 16.6 (iii) is satisﬁed if and only if for any j00 >j0and any ¯j =1,...,N , wehavethatPNj=¯jqjj00 ≥PNj=¯jqjj0. What does this imply about therelationship between the conditional distribution of z given zj00 and given zj0?Exercise 16.2. * Prove Lemma 16.1.Exercise 16.3. * Prove Theorem 16.4.Exercise 16.4. * Prove Theorem 16.5.Exercise 16.5. * Prove Theorem 16.6.Exercise 16.6. * Prove Theorem read more..

##### Page - 663

Introduction to Modern Economic Growth(2) * Provethatif u (·) takes the CRRA form and β< (1 + r)−1, then there exists some¯a< ∞ such that a (t) ∈ (0, ¯a) for all t.(3) * Prove that when β ≤ (1 + r)−1, there exists no ¯a< ∞ such that a (t) ∈ (0, ¯a)for all t. [Hint: consider the case where β =(1 + r)−1 and take the stochasticsequence where w (t)= wN for an arbitrarily large number of periods, which isa positive probability sequence. Then generalize this argument read more..

##### Page - 664

Introduction to Modern Economic Growth(5) Show that if the wages in the wage distribution H (w) are oﬀered by ﬁrms and allworkers are identical, the wage oﬀers of all ﬁrms other than those oﬀering w = R arenot proﬁt-maximizing. What does this observation imply about the McCall searchmodel?Exercise 16.13. Consider an economy populated by identical households each with pref-erences given by E£P∞t=0βtu(c(t))¤,where u (·) is strictly increasing, strictly concave andtwice read more..

##### Page - 665

Introduction to Modern Economic GrowthHere f (k (t) ,z (t)) is the revenue or proﬁtof the ﬁrm as a function of its capital stock, k (t),and a stochastic variable, representing productivity or demand, z (t). As in Section 7.8, i (t)is investment and φ (i (t)) represents adjustment costs.(1) Assume that z (t) has a distribution represented by a Markov chain. Formulate thesequence version of the maximization problem of the ﬁrm.(2) Formulate the recursive version of the maximization problem read more..

##### Page - 666

CHAPTER 17Stochastic Growth ModelsIn this chapter, I present four models of stochastic growth emphasizing diﬀerent aspectsof the interaction between growth and uncertainty. The ﬁrst is the baseline neoclassicalgrowth model (with complete markets) augmented with stochastic productivity shocks, ﬁrststudied by Brock and Mirman (1972). This model is not only an important generalizationof the baseline neoclassical growth of Chapter 8 but also provides the starting point of theinﬂuential Real read more..

##### Page - 667

Introduction to Modern Economic Growthrisk, income ﬂuctuations and policy. Consequently, over the past decade or so, it has becomea workhorse model for macroeconomic analysis.The last two sections, Sections 17.5 and 17.6 turn to stochastic overlapping generationsmodels. The ﬁrst presents a simple extension of the canonical overlapping generations modelthat includes stochastic elements.Section 17.6 shows how stochastic growth models can be useful in understanding theprocess of takeoﬀ from read more..

##### Page - 668

Introduction to Modern Economic Growthproblem and solved for the social planner’s maximization problem in a dynamic neoclassicalenvironment with uncertainty. Since, with competitive and complete markets, the First andSecond Welfare Theorems still hold, the equilibrium growth path is identical to the optimalgrowth path. Nevertheless, the analysis of equilibrium growth is more involved and alsointroduces a number of new concepts. I start with the Brock-Mirman model approach hereand then discuss read more..

##### Page - 669

Introduction to Modern Economic Growthsubject to(17.3)k (t +1) = f (k (t) ,z (t)) + (1 − δ) k (t) − c (t) and k (t) ≥ 0,with given k (0) > 0. To characterize the optimal growth path using the sequence problem wewould need to deﬁne feasible plans, in particular, the mappings ˜k£zt¤and˜c£zt¤introducedinthe previous chapter, with zt ≡ (z (0) ,..., z (t)) again standing for the history of (aggregate)shocks up to date t. Rather than going through these steps again, let us directly read more..

##### Page - 670

Introduction to Modern Economic GrowthIt is also straightforward to derive the stochastic Euler equations corresponding to theneoclassical growth model with uncertainty. For this purpose, let us ﬁrst deﬁne the policyfunction for consumption asπc (k, z) ≡ f (k, z)+(1 − δ) k − π (k, z) ,where π (k, z) is the optimal policy function for next date’s capital stock determined inProposition 17.1. Using this notation, the stochastic Euler equation can be written as(17.5)u0 (πc (k, z)) = read more..

##### Page - 671

Introduction to Modern Economic Growthhorizons, the distribution of k should be independent of k (0). Moreover, the average value ofk (t) in this invariant limiting distribution will be the same as the time average of {k (t)}Tt=0as T →∞ (so that the stochastic process for the capital stock is “ergodic”). Consequently,a “steady-state” equilibrium now corresponds not to speciﬁc values of the capital-labor ratioand output per capita but to invariant limiting distributions. If the read more..

##### Page - 672

Introduction to Modern Economic GrowthNow taking zj0 and k out of the summation and using the fact that, by deﬁnition,PNj=1qjj0=1, we can cancel the remaining terms and obtainB1 = αβ,so that irrespective of the exact Markov chain for z,the optimalpolicyruleisπ (k, z)= αβzkα.The reader can verify that this is identical to the result in Example 6.4 in Chapter 6, withz there corresponding to a non-stochastic productivity term. Consequently, in this case thestochastic elements have not read more..

##### Page - 673

Introduction to Modern Economic Growthideas related to the pricing of various contingent claims in competitive equilibrium underuncertainty. The reference to complete markets in this context implies that, in principle, anycommodity, including any contingent claim, can be traded competitively. Nevertheless, asshown by our analysis in Section 5.8 in Chapter 5, in practice there is no need to specifyor trade all of these commodities and a subset of the available commodities is suﬃcient toprovide read more..

##### Page - 674

Introduction to Modern Economic Growthin the standard theory of general equilibrium. More explicitly, the household buys claims todiﬀerent “contingent” consumption bundles. These bundles are contingent in the sense thatthey are conditioned on the history of the aggregate state variable (stochastic shock) zt andthus whether they are realized and delivered depends on the realization of the sequence ofthe stochastic shock. For example, c£zt¤denotesunitsofﬁnalgoodallocatedtoconsumptionat read more..

##### Page - 675

Introduction to Modern Economic Growththe ﬁrst-order conditions of this problem is(17.12)βtq£zt|z0¤u0¡c£zt¤¢=λp0£zt¤for all t and all zt,where λ is the Lagrange multiplier on (17.10) and now does in factcorrespond to the marginal utility of income at date t =0 (see Exercise 17.11 on why asingle multiplier for the lifetime budget constraint is suﬃcient in this case). Combining thisﬁrst-order condition for two diﬀerent date t histories zt and ˆzt,weobtainu0¡c£ˆzt¤¢u0 (c read more..

##### Page - 676

Introduction to Modern Economic GrowthUsing constant returns to scale and expressing everything in per capita terms, these ﬁrst-orderconditions can be written asp0£zt¤¡f0¡ke£zt¤,z(t)¢+(1−δ)¢ = R0£zt¤(17.13)p0£zt¤¡f¡ke£zt¤,z(t)¢−ke£zt¤f0¡ke£zt¤,z(t)¢¢ = w0£zt¤,where f0 denotes the derivative of the per capita production function with respect to thecapital-labor ratio ke ≡ Ke/L.The ﬁrst equation relates the price of the ﬁnal good to theprice of capital goods read more..

##### Page - 677

Introduction to Modern Economic Growthfor any zt+1 =¡zt,z(t+1)¢.The capital market clearing condition also implies a particular no arbitrage conditionlinking the price of capital conditional on zt+1 (R0£zt+1¤) to the price of the ﬁnal good attime t (p0£zt¤). In particular, consider the following riskless arbitrage; buy one unit of theﬁnal good after history ztto be used as capital at time t +1 and simultaneously sell claimson capital goods for each zt+1 =¡zt,z(t+1)¢. These combined read more..

##### Page - 678

Introduction to Modern Economic GrowthProof. See Exercise 17.13.¤Complementary insights can be obtained by considering the equilibrium problem in itsequivalent form with sequential trading rather than all trades taking place at the initialdate t =0. To do this, we will write the budget constraint of the representative householdsomewhat diﬀerently. First, normalize the price of the ﬁnal good at each date to 1 (recall thediscussion in Section 5.8 in Chapter 5). The read more..

##### Page - 679

Introduction to Modern Economic Growthof equilibrium prices ¯p as given, the value function of the representative household can bewritten as(17.19)V (a, z)=sup{a0[z0|z]}z0∈Z (uÃa+w− Xz0∈Z¯p£z0|z¤a0£z0|z¤!+β Xz0∈Zq£z0|z¤V ¡a0£z0|z¤,z0¢).Theorems 16.1-16.7 from the previous chapter can again be applied to this value function(see Exercise 17.15). The ﬁrst-order condition for current consumption can now be writtenas¯p£z0|z¤u0(c[a,z])=βq£z0|z¤∂V (a0 [z0 | z] read more..

##### Page - 680

Introduction to Modern Economic GrowthNext the market clearing condition for capital, combined with the fact that the only asset inthe economy is capital, implies thata = k.Therefore, this ﬁrst-order condition can be written asu0 (c [k, z]) = βE£R£z0|z¤u0¡c£k0,z0¤¢|z¤which is identical to (17.6). This again shows the equivalence between the social planner’sproblem and the competitive equilibrium path.Given the equivalence between the social planner’s problem (the optimal growth read more..

##### Page - 681

Introduction to Modern Economic Growthchoices into the neoclassical growth model under uncertainty generates new insights. So farI have assumed, except in Exercise 8.17 in Chapter 8, that labor is supplied inelasticallyand this choice has enabled us to focus on the ﬁrst-order issues related to economic growth.Because the issue of labor supply is central to a number of questions in macroeconomics, abrief analysis of the neoclassical growth model with labor supply is also useful.The economic read more..

##### Page - 682

Introduction to Modern Economic Growththe equilibrium. In particular, the two key ﬁrst-order conditions determine the evolution ofconsumption over time and the equilibrium level of labor supply. Denoting the derivatives ofthe u function with respect to its ﬁrst and second arguments by uc and ul, the derivatives ofthe F function by Fk and Fl, and deﬁning the policy function for consumption asπc (K, z) ≡ F³K,πl(K,z),z´+(1−δ)K−πk(K,z),these take the read more..

##### Page - 683

Introduction to Modern Economic Growthsavings and thus future levels of capital stock (though this does also depend on the form ofthe utility function, which regulates the desire for consumption smoothing and the balancebetween income and substitution eﬀects).This brief discussion suggests that the neoclassical growth model under uncertainty withlabor supply choices and with aggregate productivity shocks may generate some of the majorqualitative features of macroeconomic ﬂuctuations. The RBC read more..

##### Page - 684

Introduction to Modern Economic Growthwhich yieldsB = αβ.The resulting policy function for the capital stock is thereforeπk (K, z)= αβzKαL1−α,which is identical to that in Example 17.1. Next, considering the ﬁrst-order condition forlabor, we obtain(1 − α) zKαL−α(1 − B) zKαL1−α= γ.The resulting policy function for labor asπl (K, z)=(1 − α)γ (1 − αβ),which implies that labor supply is constant. This is because with the preferences as speciﬁedhere, the income and read more..

##### Page - 685

Introduction to Modern Economic Growthvary over time. In particular, each household h ∈ H has a labor endowment of zh (t) at timet,where zh (t) is an independent draw from the set Z ≡ [zmin,zmax],where 0 <zmin <zmax <∞, so that the minimum labor endowment is zmin. We assume that the labor endowment ofeach household is identically and independently distributed with distribution function G (z)deﬁned over [zmin,zmax].The production side of the economy is the same as in the read more..

##### Page - 686

Introduction to Modern Economic Growthfor all t. We can then write the maximization problem of household h ∈ H recursively as(17.24)Vh (a, z)=supa0∈[−b,Ra+wz]nu¡Ra+wz−a0¢+βEhVh¡a0,z0¢|zio.Standard dynamic programming arguments then establish the following proposition:Proposition 17.5. The value function V h (a, z) deﬁned in (17.24) is uniquely deﬁned,continuous and strictly concave in a, strictly increasing in a and z,and diﬀerentiable ina ∈ (−b,Ra + wz). Moreover, the read more..

##### Page - 687

Introduction to Modern Economic GrowthNext turning to the production side, we have the same factor prices as the neoclassicalgrowth model under certainty, i.e.,R = f0 (k∗)+(1 − δ)w= f (k∗) − k∗f0 (k∗) .Recall from Chapters 6 and 8 that the neoclassical growth model with complete markets andno uncertainty implies that there exists a unique steady state in which βR =1,i.e.,(17.25)f0 (k∗∗)= β−1 − (1 − δ) ,where k∗∗ refers to the capital-labor ratio of the neoclassical read more..

##### Page - 688

Introduction to Modern Economic Growthineﬃciency coming from overaccumulation of capital is unlikely to be important for explainingincome per capita diﬀerences across countries. Thus the Bewley model is not interestingbecause of the greater capital-labor ratio that it generates. Instead, it is important as anillustration of how an economy might exhibit a stationary equilibrium in which aggregatesare constant while individual households have uncertain and ﬂuctuating consumption andincome read more..

##### Page - 689

Introduction to Modern Economic Growthbe expressed asR (k, z)= αzkα−1(17.30)w (k, z)= (1 − α) zkα.The consumption Euler equation for an individual of generation t, then can be expressed asc2 (t +1)c1 (t)= βR (t +1)= βR (k, z) ,with R (k, z) given by (17.30). The total amount of savings at time t is then given bys (t)= s (k (t) ,z (t)) such that(17.31)s (k, z)=β1+ βw (k, z) ,which, as in the canonical overlapping generations model of Section 9.3 in Chapter 9 and alsoas in the baseline read more..

##### Page - 690

Introduction to Modern Economic Growthis arbitrarily large when the capital stock is close to zero. The stochastic correspondenceenables a simple analysis of the dynamics of stochastic models as in this case. For exam-ple, Figure 17.1 plots a particular sample path of capital-labor ratio in this economy, wherestarting with k (0), the economy ﬁrst receives a fairly favorable productivity shock moving tok (1). Following this, there is another moderately favorable productivity realization and read more..

##### Page - 691

Introduction to Modern Economic Growth17.6. Risk, Diversiﬁcation and GrowthIn this section, I present a stochastic model of long-run growth based on Acemoglu andZilibotti (1997). This model is useful for two distinct purposes. First, because it is simplerthan the baseline neoclassical growth model under uncertainty, it will provide a completecharacterization of the stochastic dynamics of growth and show how simple ideas from thetheory of Markov processes can be used in the context of the study read more..

##### Page - 692

Introduction to Modern Economic Growthof endogenously incomplete markets, it also enables us to show that price-taking behaviorby itself is not suﬃcient to guarantee Pareto optimality, and the form of ineﬃciency ofthe equilibrium in this economy will be interesting both on substantive and methodologicalgrounds.17.6.1. The Environment.We consider an overlapping generations economy. Eachgeneration lives for two periods. There is no population growth and the size of each generationis normalized read more..

##### Page - 693

Introduction to Modern Economic Growthso that the safe option is also less productive.The requirement that I (j, t) ≥ M (j) combined with the fact that the amount of capitalobtained from savings I (j, t) in state j is equal to QI (j, t) implies that all intermediatesectors have linear technologies, but only after the minimum size requirement, M (j),ismet. For any I (j, t) <M (j), the output is equal to 0. In order to simplify the expositionand the computations, let us adopt a simple read more..

##### Page - 694

Introduction to Modern Economic Growth(3) The mathematical formulation here implies a simple relationship between invest-ments and returns. As already hinted above, if an individual holds a portfolioconsisting of an equi-proportional investment I in all sectors j ∈_J ⊆ [0, 1],and the(Lebesgue) measure of the set_J is p, then the portfolio pays the return QI withprobability p, and nothing with probability 1 − p.The ﬁrst two features imply that if the aggregate production set of this read more..

##### Page - 695

Introduction to Modern Economic GrowthOLDYOUNGConsumption (c1(t))Wage (w(t))Savings (s(t))Riskless asset (X(t))Risky assets {I(j,t)}Capital (K(j,t+1))Cap.inc. (R(j,t+1) K(j,t+1))Consumption (c2(j,t+1))Realiz. ofuncertaintyFigure 17.3. Life cycle of a typical household.Since the capital stock is potentially random, so will be output and factor prices. Inparticular, both labor and capital are assumed to be traded in competitive markets, so theequilibrium factor prices will be given by their read more..

##### Page - 696

Introduction to Modern Economic Growththat each ﬁnancial intermediary can operate only a single sector.4 Thus we are ruling outthe formation of a grand ﬁnancial intermediary managing all investments. We return tothis issue in subsection 17.6.5. We denote the price charged for a security associated withintermediate sector j at time t by P (j, t). Although the decentralized equilibrium in thiseconomy will be deﬁned and analyzed in the next section, we can already make a few read more..

##### Page - 697

Introduction to Modern Economic Growth(17.44)c (t)+ s (t) ≤ w (t) ,where I have suppressed the superscript h to simplify the notation. Here equation (17.40) isthe expected utility (objective function) of the representative household. Equations (17.41)-(17.44) are the constraints on this maximization problem. The ﬁrst one, (17.41), requires thatthe investment in the safe sector and the sum of the investments in all other securities areequal to the total savings of the individual, s (t). read more..

##### Page - 698

Introduction to Modern Economic GrowthBecause preferences in (17.40) are logarithmic, the saving rate of all households will beconstant as in the canonical overlapping generations model. Consequently, we obtain thefollowing saving rule regardless of the risk-return tradeoﬀ:(17.45)s∗ (t) ≡ s∗(w (t)) =β1+ βw (t) .Given this result, a household’s optimization problem can be broken into two parts: ﬁrst, theamount of savings is determined, and then an optimal portfolio is chosen. This read more..

##### Page - 699

Introduction to Modern Economic Growthis the marginal product of capital in the “bad” state, when the realized state is j> n∗ (t)and no risky investment pays oﬀ,andRG (t)= α (qX (t)+ QI (t))α−1applies in the “good state”, i.e. when the realized state is j [0,n∗ (t)].Straightforward maximization of (17.46) subject to (17.47) yields the unique solution tothe household’s problem as:(17.48)X∗ (t)=(1 − n∗ (t))QQ − qn∗ (t)s∗ (t) ,and(17.49)I∗ (j, t)=( I∗ (n∗ read more..

##### Page - 700

Introduction to Modern Economic Growththere might be multiple solutions, corresponding to multiple equilibria. These equilibriawould involve diﬀerent number of active sectors. When there are only a few active sectors,households invest a large fraction of their resources in the safe asset, and in equilibrium onlya few risky sectors can be operated. In contrast, when there is a signiﬁcant number of activerisky sectors, each household invests a large fraction of its resources in risky assets. read more..

##### Page - 701

Introduction to Modern Economic Growthlaw of motion for the capital stock:(17.51)K (t +1) =( q(1−n∗[K(t)])Q−qn∗[K(t)]QΓK (t)αwith probability 1 − n∗ [K (t)]QΓK (t)αwith probability n∗ [K (t)]where n∗ [K (t)] is given by equation (17.50) and recall that Γ ≡ (1 − α)β (1 + β)−1.Noticethat the ﬁrst line of (17.51) is always less than the second line, which reﬂects the fact thatthe second line refers to the case in which the investments in the intermediate sectors read more..

##### Page - 702

Introduction to Modern Economic Growthso that their investments always had positive return. The second, inverse U-shaped curvecorresponds to q(1 − n∗ [K (t)])QΓK (t)α / (Q − qn∗ [K (t)]) and thus applies if the economyis unlucky at each date. Both curves start above the 450 line near zero for the same reasonas that given for the similar pattern in Figure 17.1 (i.e., because the aggregate productionfunction (17.34) satisﬁes the Inada conditions). The economy will be on the upper curve read more..

##### Page - 703

Introduction to Modern Economic Growthit. This is because once the economy accumulates suﬃcient capital to open all intermediatesectors, it would eliminate all risk and would always be on the upper curve in Figure 17.4.Equations (17.50) and (17.53) show that the condition for this good steady state to exist,i.e., for n∗£KQSSG¤ =1, is that the saving level corresponding to KQSSG be suﬃcient toensure a balanced portfolio of investments, of at least D, in all the intermediate sectors. Itis read more..

##### Page - 704

Introduction to Modern Economic GrowthThis discussion, combined with Figure 17.4, gives a fairly complete characterization of thestochastic equilibrium growth path. In particular, an economy that starts with a low enoughcapital stock will ﬁrst experience some growth, but then spend a long time ﬂuctuating betweensuccessful periods and periods of severe crises. Eventually, a string of good news will take theeconomy to a level of capital stock such that much (here all) of the risks can be read more..

##### Page - 705

Introduction to Modern Economic GrowthIt is clear from this equation that capital (and output) growth volatility, after removing thedeterministic “convergence eﬀects” due to the standard neoclassical eﬀects, are determinedby the stochastic component σ. Denoting the (conditional) variance of σ(n∗ [K (t)] givenK (t) by Vn, we can state the following proposition.Proposition 17.11. Let Vn ≡Var(σ(n∗) | n∗)= n∗(1−n∗)[Q(Q − q)/ (Q − qn∗)]2.Thenwe have that• If γ ≥ Q/ read more..

##### Page - 706

Introduction to Modern Economic Growthconsidering the social planner’s problem. This problem can be written as:(17.56)maxn(t),X(t),[I(j,t)]0≤j≤n(t)Z n(t)0log(qX (t)+ QI (j, t))dj +(1 − n (t)) log(qX (t))subject toX (t)+Z n(t)0I (j, t) dj ≤ s (t) .More speciﬁcally, the social planner chooses the set of sectors that are active, which is denotedby [0,n (t)], the amount that will be invested in the safe sector X (t) and the allocation offunds among the other sectors denoted by [I (j, read more..

##### Page - 707

Introduction to Modern Economic Growthj*1γns[K(t)]M(j)M(j*)M(ns[K(t)])Dj, nFigure 17.5. The eﬃcient portfolio allocation.The deviation from the balance portfolio implies that the social planner is implicitly cross-subsidizing the sectors with high a minimum size requirements at the expense of sectors withlow or no minimum size requirements. This is because, starting with a balanced portfolio,opening a few more sectors always beneﬁts consumers, who will be able to achieve better read more..

##### Page - 708

Introduction to Modern Economic Growthin a non-balanced portfolio. Market prices do not induce the households to hold the rightportfolio.At this point, the reader may wonder why the First Welfare Theorem does not apply.In particular, all households are price takers. The reason why the First Welfare Theoremdoes not apply that the decentralized equilibrium here does not correspond to an Arrow-Debreu equilibrium. In particular, this is an equilibrium for an economy with endogenouslyincomplete read more..

##### Page - 709

Introduction to Modern Economic Growtha speciﬁc household, and if it is proﬁtable for other households to join the coalition, theorganizer of the coalition can charge a premium (or a joining fee) thus making proﬁts. Weassume that there is free entry into ﬁnancial intermediation or coalition-building, so that anyhousehold can attempt to exploit proﬁt opportunities if there are any. Let us also imposesome structure on how the timing of ﬁnancial intermediation works and also how read more..

##### Page - 710

Introduction to Modern Economic Growth17.7. Taking StockThis section presented a number of diﬀerent models of stochastic growth. My selection oftopics was geared towards achieving two objectives. First, I introduced a number of workhorsemodels of macroeconomics, such as the neoclassical growth model under uncertainty and thebasic Bewley model. These models not only useful for the analysis of economic growth butalso have a wide range of applications in the macroeconomics literature.Second, the read more..

##### Page - 711

Introduction to Modern Economic Growthparallels between the fact that an insuﬃcient number of markets are open in this model andtoo few intermediate goods being produced in the baseline endogenous technological changemodel of Chapter 13.17.8. References and LiteratureThe neoclassical growth model under uncertainty, presented in Section 17.1, was ﬁrstanalyzed by Brock and Mirman (1972). Because the analysis of the social planner’s problemis considerably easier than the study of equilibrium read more..

##### Page - 712

Introduction to Modern Economic Growthoptimal ﬁscal policy, monetary policy, and asset pricing. A more modern treatment is pro-vided in Aiyagari (1994), though the published version of the paper does not contain any ofthe mathematical analysis. The reader is referred to Bewley (1977, 1980) and to the workingpaper version, Aiyagari (1993), for more details on some of the propositions stated in Section17.4 as well as a proof of existence of a stationary equilibrium, which I did not provide inthe read more..

##### Page - 713

Introduction to Modern Economic GrowthExercise 17.3. Consider the same production structure as in Sections 17.1 and 17.2 butassume that irrespective of the level of the capital stock and the realization of the stochasticvariable, each household saves a constant fraction s of its income. Characterize the stochasticlaws of motion of this economy. How does behavior in this economy diﬀer from that in thecanonical neoclassical growth model under uncertainty.Exercise 17.4. Consider the neoclassical read more..

##### Page - 714

Introduction to Modern Economic Growth(1) Setup and analyze the optimal growth problem in this economy. Show that theoptimal consumption sequence satisﬁes a modiﬁed stochastic Euler equation.(2) Prove that Theorem 5.7 can be applied to this economy and the optimum growthpath can be decentralized as a competitive growth path.Exercise 17.10. Write the maximization problem of the social planner explicitly as a se-quence problem, with output, capital and labor following diﬀerent histories read more..

##### Page - 715

Introduction to Modern Economic GrowthU (C, L) such that the optimal growth path corresponds to a “balanced growth path,” wherelabor supply does not (with probability 1) go to zero or inﬁnity?Exercise 17.19. In Example 17.2, suppose that the utility function of the representativehousehold is u (C, L)=log C + h (L),where h (·) is a continuous, decreasing and concavefunction. Show that the equilibrium level of labor supply is constant and independent of thelevel of capital stock and the read more..

##### Page - 716

Introduction to Modern Economic Growthrequires a minimum size investment M (j) and without loss of generality rank the projects inascending order of minimum size.There is continuum of consumers with measure normalized to 1, each with the utilityfunction u(c)+ Ev(c0),where c is consumption today, c0 is consumption tomorrow, so thatEv(c0) denotes expected utility from tomorrow’s consumption. Each consumer has totalresources equal to w and decides how much to consume and how much to save and then read more..

##### Page - 717

##### Page - 718

Part 6Technology Diﬀusion, Trade andInterdependences read more..

##### Page - 719

One of the most major shortcomings of the models presented so far is that each countryis treated as an “island” on to itself, not interacting with the rest of the countries in theworld. This is problematic for at least two reasons. The ﬁrst is related to the technologicalinterdependences across countries and the second to international trade (in commodities andin assets). In this part of the book, we will investigate the implications of technological andtrade interdependences on the read more..

##### Page - 720

Introduction to Modern Economic Growthworld level together with growth in each speciﬁc country that depends on technological andother developments at the world level.In Chapter 18, we will start with models of technology adoption and investigate thefactors aﬀecting the speed and nature of technology adoption. We will also place specialemphasis on whether or not technologies that are available from the world technology fron-tier are appropriate for the needs of less-developed countries. read more..

##### Page - 721

##### Page - 722

CHAPTER 18Diﬀusion of TechnologyIn many ways, the problem of innovation ought to be harder to model than the problem oftechnology adoption. Nevertheless, the literature on economic growth and development hasmade more progress on models of innovation, such as those we discussed in Chapters 13-15,than on models of technology diﬀusion. This is in part because the process of technologyadoption involves many challenging features. First, even within a single country, we observeconsiderable read more..

##### Page - 723

Introduction to Modern Economic Growththat productivity and technology diﬀerences are ubiquitous even across ﬁrms within narrowsectors in the same country.18.1.1. Productivity and Technology Diﬀerences within Narrow Sectors.Alarge literature uses longitudinal micro-data (often for the manufacturing sector) to studylabor and total factor productivity diﬀerences across plants within narrow sectors (for examplethree-digit or four-digit manufacturing sectors). For our focus, the most read more..

##### Page - 724

Introduction to Modern Economic Growth14, Bartelsman and Doms (2000) and Foster, Haltiwanger and Krizan (2000) document thatentry of new plants plays has an important contribution to industry productivity growth.Nevertheless, entry and exit appear to account for only about 25% of average TFP growth,with the remaining productivity improvements are accounted for by continuing plants. Thissuggests that models in which ﬁrms continually invest in technology and productivity (forexample such as the read more..

##### Page - 725

Introduction to Modern Economic Growthentirely surprising. Nevertheless, the causes of within-country and cross-country productiv-ity and technology diﬀerences might be diﬀerent, and despite the presence of within-countrydiﬀerences, the signiﬁcant cross-country diﬀerences do pose a puzzle that requires investi-gation. For example, within-country productivity diﬀerences might be due to diﬀerences inmanagerial (entrepreneurial) ability or related to the success of the match between read more..

##### Page - 726

Introduction to Modern Economic GrowthUsing our usual approach, we can write income per capita asyj (t) ≡Yj (t)Lj (t)= Aj (t) Fµ Kj (t)Aj (t) Lj (t), 1¶≡ Aj (t) f (kj (t)) ,wherekj (t) ≡Kj (t)Aj (t) Lj (t)is the eﬀective capital-labor ratio of country j at time t.We assume that time is continuous, that there is population growth at the constant ratenj≥ 0 in country j, and that there is an exogenous saving rate equal to sj ∈ (0, 1) in countryj and a depreciation rate of δ ≥ 0 for read more..

##### Page - 727

Introduction to Modern Economic Growthinstitutional or policy barriers aﬀecting technology adoption. This parameter multiplies thediﬀerence A (t) −Aj (t), since it is this diﬀerence that remains to be absorbed by the countryin question–if A (t)= Aj (t), there is nothing to absorb from the world technology frontier.Though natural, this formulation has important economic consequences. In particular, itimplies that countries that are relatively “backward” in the sense of having a low read more..

##### Page - 728

Introduction to Modern Economic GrowthClearly, the initial conditions A (0) > 0 and Aj (0) > 0 give a unique initial condition for thediﬀerential equation for aj, aj (0) ≡ Aj (0) /A (0) > 0.Given the description of the environment above, the dynamics of the world income percapita levels and technology are determined by 2J diﬀerential equations. For each j,we haveone of (18.1) and one of (18.4). These equations characterize the state-state distributionof technology and income per read more..

##### Page - 729

Introduction to Modern Economic GrowthThe steady-state world equilibriumnk∗j,a∗joJj=1is globally stable in the sense that starting withany strictly positive initial values {kj (0) ,aj (0)}Jj=1, the equilibrium path {kj (t) ,aj (t)}Jj=1converges tonk∗j,a∗joJj=1.Proof. (Sketch) First solve (18.1) and (18.4) for each j =1,...,J imposing the steady-state condition that ˙kj (t)= ˙aj (t)= 0. This yields a unique solution, establishing theuniqueness of the steady-state equilibrium. Then read more..

##### Page - 730

Introduction to Modern Economic GrowthProof. See Exercise 18.7.¤A particularly convenient–but also restrictive–feature of the equilibrium studied here isthat even though there is technology diﬀusion and interdependence in this world equilibrium,there is no interaction among countries. Each country’s steady-state income per capita (andin fact path of income per capita) only depends on the behavior of the world technologyfrontier and its own parameters. Later in this chapter, we will see read more..

##### Page - 731

Introduction to Modern Economic GrowthMoreover, the steady-state world equilibrium is globally saddle-path stable in the sensethat starting with any strictly positive initial values {kj (0) ,aj (0)}Jj=1,initial consumptionto eﬀective labor ratios are {cj (0)}Jj=1 and the equilibrium path {kj (t) ,aj (t) ,cj (t)}Jj=1 con-verges tonk∗j,a∗j,c∗joJj=1,where c∗j is the steady-state consumption to eﬀective labor ratio ineconomy j.Proof. (Sketch) We can ﬁrst show that a∗j can be read more..

##### Page - 732

Introduction to Modern Economic Growthmore generally) to be in perpetual “disequilibrium” and argued that the main role of humancapital was to enable individuals to deal with and adapt to situations of disequilibrium.Today, we would recognize what Schultz dubbed “disequilibrium” as an equilibrium of adynamical system that is far from steady state (see Chapter 20 for a similar perspective).Thus his argument can be viewed as emphasizing the role of human capital in environmentswhere read more..

##### Page - 733

Introduction to Modern Economic Growthon whether the beneﬁts created by human capital are internalized by ﬁrms and workers orwhether they take the form of externalities. Consequently, as in the Becker-Mincer approach,if there are signiﬁcant external eﬀects, many of the empirical strategies discussed in Chapter 3will understate the role of human capital. This discussion, however, emphasizes that were thisto happen, it would be a consequence of human capital externalities, not a direct read more..

##### Page - 734

Introduction to Modern Economic Growththe endogenous technological change models we have seen above are more widely used andoﬀer richer insights about the nature of technology, I will introduce endogenous technologyadoption decisions in the context of these models. The question of why some societies blocktechnology adoption will be the topic of Part 8 below.18.3. Technology Diﬀusion and Endogenous GrowthIn the previous section, technology diﬀusion took place “exogenously,” in the sense read more..

##### Page - 735

Introduction to Modern Economic Growthtime is(18.8)Cj (t)+ Xj (t)+ ζjZj (t) ≤ Yj (t) ,where Xj (t) is investment or spending on inputs at time t and Zj (t) is expenditure ontechnology adoption at time t, which may take the form of R&D or other expenditures,such as the purchase or rental of machines embodying new technologies. The parameter ζjis introduced as a potential source of diﬀerences in the cost of technology adoption acrosscountries, which may result from institutional barriers read more..

##### Page - 736

Introduction to Modern Economic Growthcounterbalanced against the cost of ζj. Free-entry (with positive activity) then requires(18.11)ν∗j =µηjβLjζjr∗¶1/φ,where I have also used the fact that given the preferences (18.6), equal growth rate acrosscountries implies that the interest rate will be the same in all countries (and in fact it willbe equal to r∗ = ρ + θg).Since a higher νj implies that country j is technologically more advanced and thus richerthan others, equation (18.11) read more..

##### Page - 737

Introduction to Modern Economic Growth18.3.2. Endogenous Growth in the World.The model in the previous subsectionwas simpliﬁed by the fact that the world growth rate was exogenous. A more satisfactorymodel would derive the world growth rate from the technology adoption and R&D activitiesof each country. Such models are typically more involved, because the degree of interactionamong countries in the world equilibrium is now considerably greater. In addition, a certainamount of care needs to read more..

##### Page - 738

Introduction to Modern Economic Growthdepend on the speciﬁcation in (18.12). Besides equation (18.12), we assume that all the otherequations from the previous subsection continue to hold.The main result of this section is that the pattern of cross-country growth will be similarto that in the previous subsection, but now the growth rate of the world economy, g,will beendogenous, resulting from the investments in technologies made by ﬁrms in each country. Inparticular, suppose that there read more..

##### Page - 739

Introduction to Modern Economic GrowthA number of features about this equilibrium are noteworthy. First, taking the worldgrowth rate as given, the structure of the equilibrium is very similar to that in Proposition18.4. Thus the fact that all countries grow at the same rate and that diﬀerences in theinnovation possibilities frontier, ηj, the size the labor force, Lj, and the extent of potentialdistortions in technology investments, ζj, translate into level diﬀerences across countries read more..

##### Page - 740

Introduction to Modern Economic Growthtechnology diﬀusion to later, in this section we focus on how “technology” diﬀerences andincome gaps can remain substantial even with free ﬂow of ideas.A ﬁrst important idea is that productivity diﬀerences may remain even if all diﬀerencesin “techniques” disappear, because production is organized diﬀerently and the extent ofineﬃciency in production may vary across countries. A model along these lines will bediscussed later in this read more..

##### Page - 741

Introduction to Modern Economic Growth(in this case, the J0 countries aﬀected by malaria). In fact, in this extreme case, a technologicaladvance that is freely available to all countries in the world increases productivity in a subsetof the countries and creates cross-country income diﬀerences.Is there any reason to expect that issues of the sort might be important? The answeris both yes and no. Over 90% of the world R&D is carried out in OECD economies. Thereis therefore natural reasons read more..

##### Page - 742

Introduction to Modern Economic Growthwhere k = K/L is the capital-labor ratio, and A (k | k0) is the (total factor) productivity oftechnology designed to be used with capital-labor ratio k0 when used instead with capital-labor ratio k. I have suppressed the time and country indices to simplify notation.For example, suppose thatA¡k|k0¢=Amin½1,µkk0¶γ¾for some γ ∈ (0, 1). That is, when a technology designed for the capital labor ratio k0 is usedwith a lower capital-labor ratio, there is read more..

##### Page - 743

Introduction to Modern Economic Growthwe may expect a mismatch between the skill requirements of frontier technologies and theavailable skills of the workers in less-developed countries to be potentially more importantthan diﬀerences in capital intensity. In this subsection, I outline the model introduced inAcemoglu and Zilibotti (2001), which emphasizes the implications of the mismatch betweentechnologies developed in advanced economies and the skills of the work force of the less-developed read more..

##### Page - 744

Introduction to Modern Economic GrowthLet us assume that the technology for producing intermediate i in country j at time t isgiven as follows:yj(i, t)=11 − β"Z NL(t)0xL,j(i, ν, t)1−βdν#[(1−i)lj (i, t)]β(18.17)+11 − β"Z NH (t)0xH,j(i, ν, t)1−βdν#[iωhj (i, t)]β .A number of features about this intermediate production function is worth noting. First eachintermediate can be produced using two alternative technologies, one using skilled workers,the other one using read more..

##### Page - 745

Introduction to Modern Economic GrowthNorth will be the ﬁrm that invents the new type of machine, so here the analysis is identicalto that in Chapters 13 and 15.What about in the South? To keep the treatment of Northern and Southern economiessymmetric, we assume that in each Southern economy a “technology” ﬁrm adopts the newtechnology invented in the North (at no cost) and acts as the monopolist supplier of thatmachine for the producers in its own country. Moreover, we assume that the read more..

##### Page - 746

Introduction to Modern Economic Growthand the equilibrium threshold Ij(t) is uniquely pinned down by(18.21)Ij (t)1 − Ij (t)=µNH(t)NL (t)ωHjLj¶−1/2.Combining these two equations, we can also derive the level of total output in economy j as(18.22)Yj (t)=exp(−β)h(NL(t)Lj)1/2+(NH(t)ωHj)1/2i2,and the skill premium as(18.23)wH,j (t)wL,j (t)= ωµNH(t)NL (t)¶1/2µωHjLj¶−1/2(see Exercise 18.22). An interesting feature of this characterization, apparent from equation(18.22) is that the read more..

##### Page - 747

Introduction to Modern Economic Growththefactthatthe relevant market sizes aregiven by Hn and Ln (because research ﬁrms canonly sell their technologies to Northern ﬁrms) implies that the steady-state (balanced growth)equilibrium must take the following form:Proposition 18.7. With the lab equipment speciﬁcation of directed technical change asin (18.24) and no intellectual property rights in the South, the unique steady-state equilibriuminvolves Northern relative pricesP nHP read more..

##### Page - 748

Introduction to Modern Economic GrowthThe next result shows that the steady-state technologies N∗L and N∗H are indeed “appro-priate” for the conditions (factor proportions) in the North, and that this creates endogenousincome diﬀerences between the North and the South.Proposition 18.8. Consider the above-described model. Then:(1) The steady-state equilibrium technology ratio N∗H /N∗L is such that, given a constantlevel of for given NH + NL, it achieves the unique maximum of net read more..

##### Page - 749

Introduction to Modern Economic Growthin providing a perspective in discussions. To make further progress, we need more micro-founded models of why there are barriers to technology adoptions and how these barriersaﬀect technology choices. The reasons why certain groups may want to erect barriers againstthe introduction of new technologies will be discussed in detail in Part 8 below. In Part 7,we will discuss other factors aﬀecting the eﬃciency of the organization of production, whichcan read more..

##### Page - 750

Introduction to Modern Economic Growththe decisions of a single ﬁrm. The resulting revenue function for the ﬁrm can therefore bewritten as(18.27)R = A1−βqβ.Production depends on the technology choice of the ﬁrm, which is denoted by N ∈R+. More advanced technologies involve a greater range of intermediate goods (inputs),supplied by diﬀerent suppliers. The transactions between the producer and the supplierswill necessitate contracting relationships. For each j ∈ [0,N ],let X (j) be read more..

##### Page - 751

Introduction to Modern Economic Growth(ii) For all N> 0, N Γ00 (N ) / [Γ0 (N )+ w0] > [β (κ +1) − 1] / (1 − β).These restrictions are standard. In particular, they introduce enough convexity to ensureinterior solutions.The relationship between the producer and its suppliers requires contracts to ensure thatthe suppliers deliver the required inputs. Let the payment to supplier j consist of two parts:an ex ante payment τ (j) ∈ R before the investment levels x (i, j) take place, read more..

##### Page - 752

Introduction to Modern Economic Growthsubject to (18.32) and the suppliers’ participation constraint,(18.34)s (j)+ τ (j) − ψZ 10x (i, j) di ≥ w0 for all j ∈ [0,N ] .Since the ﬁrm has no reason to provide rents to the suppliers, it chooses payments s (j) andτ (j) that satisfy (18.34) with equality. Moreover, with complete contracts, τ (j) and s (j) areperfect substitutes, so only the sum s (j)+ τ (j) matters and is determined in equilibrium–this will not be the case when read more..

##### Page - 753

Introduction to Modern Economic Growthboth suppliers’ and the producer’s investments more productive.The other noteworthyimplication of this proposition is that under complete contracts, the level of technology andthus productivity do not depend on the elasticity of substitution between intermediate inputs,1/ (1 − α).18.5.3. Equilibrium under Incomplete Contracts.We now consider the same en-vironment under incomplete contracts. We model the imperfection of the contracting insti-tutions by read more..

##### Page - 754

Introduction to Modern Economic GrowthBehavior along the SSPE can be described by a tuplen˜N,˜xc,˜xn,˜τoinwhich ˜N representsthe level of technology, ˜xc the investment in contractible activities, ˜xn the investment innoncontractible activities, and ˜τ the upfront payment to every supplier. That is, for everyj ∈h0, ˜Nitheupfront payment is τ (j)= ˜τ , and the investment levels are x (i, j)= ˜xc fori ∈ [0,μ] and x (i, j)= ˜xn for i ∈ (μ, 1]. With a slight abuse of read more..

##### Page - 755

Introduction to Modern Economic GrowthIn other words, given N and (xc,τ ), each supplier j ∈ [0,N ] should expect her Shapley valueplus the upfront payment to cover the cost of investment in contractible and noncontractibleactivities and the value of her outside option.The maximization problem of the ﬁrm can then be written as:maxN,xc,xn,τsq (N, xc,xn) − Nτ − Γ (N )subject to (18.39) and (18.40).With no restrictions on τ , the participation constraint (18.40) will be satisﬁed with read more..

##### Page - 756

Introduction to Modern Economic GrowthA number of features of (18.43) are worth noting. First, the derived parameter γ ≡α/ (α + β) represents the bargaining power of the ﬁrm; it is increasing in α and decreasingin β. A higher elasticity of substitution between intermediate inputs, i.e., a higher α,raisesthe ﬁrm’s bargaining power, because it makes every supplier less essential in production andtherefore raises the share of revenue appropriated by the ﬁrm. In contrast, a higher read more..

##### Page - 757

Introduction to Modern Economic Growthand solving for the ﬁxed point by substituting xn (j)= xn yields a unique xn:(18.47)xn =¯xn (N, xc) ≡hα(1−γ)ψ−1xβμcA1−βNβ(κ+1)−1i1/[1−β(1−μ)].This equation implies that investments in noncontractible activities are increasing in α.Math-ematically, this follows from the fact that α (1 − γ)= αβ/ (α + β) is increasing in α.Theeconomics of this relationship is the outcome of two opposing forces.The share of thesuppliers in read more..

##### Page - 758

Introduction to Modern Economic Growth(18.49), and (18.50) gives the level of investment in noncontractible activities as(18.51)˜xn =α (1 − γ)[1 − β (1 − μ)]β [1 − α (1 − γ)(1 − μ)]⎛⎝Γ0³˜N´+w0κψ⎞⎠.Comparing (18.37) to (18.50), we see that for a given N the implied level of investment incontractible activities under incomplete contracts, ˜xc, is identical to the investment level incontractible activities under complete contracts, x∗. This highlights the fact read more..

##### Page - 759

Introduction to Modern Economic GrowthThis proposition states that suppliers invest less in noncontractible activities than incontractible activities. In particular, we have that(18.52)˜xn˜xc=α (1 − γ)[1 − β (1 − μ)]β [1 − α (1 − γ)(1 − μ)]< 1,which follows from equations (18.50) and (18.51) and from the fact that α (1 − γ)=αβ/ (α + β) <β (recall (18.44)). This is intuitive: the producer ﬁrm is the full residualclaimant of the return to investments in read more..

##### Page - 760

Introduction to Modern Economic GrowthThis proposition implies that since incomplete contracts lead to the choice of less ad-vanced (lower N ) technologies, they also reduce productivity and investments in contractibleand noncontractible activities. Acemoglu, Antras and Helpman (2007) also show that thetechnology and income diﬀerences resulting from relatively modest diﬀerences in contractinginstitutions can be quite large. Therefore, the link between contracting institutions and tech-nology read more..

##### Page - 761

Introduction to Modern Economic Growthwhen supplier j is not in the coalition. Notice that even when n<N ,the term N β(κ+1−1/α)remains in front, because it represents a feature of the technology aﬀecting output indepen-dent of the amount and quality of the inputs provided by the suppliers. On the other hand,productivity suﬀers because the term in square brackets is lower.TheShapley valueofplayer j is then(18.53)sj =1(T +1)!Xg∈G£v¡zjg∪j¢−v¡zjg¢¤.The fraction of read more..

##### Page - 762

Introduction to Modern Economic GrowthNow taking the limit as T →∞, which is also equivalent to the limit ξ = N/T → 0,weobtain limξ→0 o (ξ) /ξ =0, so that we are left with the Riemann integral:limT →∞µsjξ¶= A1−βNβ(κ+1−1/α) (β/α)h xn(j)xn(−j)i(1−μ)αxβμcxn(−j)β(1−μ)N 2Z N0zβ/αdz.Solving this integral deliverslimT →∞(sj/ξ)= (1 − γ) A1−β∙ xn (j)xn (−j)¸(1−μ)αxβμc xn (−j)β(1−μ) Nβ(κ+1)−1,with γ ≡ α/ (α + β). This read more..

##### Page - 763

Introduction to Modern Economic Growth(1) We can make considerable progress in understanding technology and productivitydiﬀerences across nations by positing a slow process of technology transfer acrosscountries. Namely, in light of the within-country evidence, which suggests thateven within narrowly-deﬁned sectors in the same country diﬀerent technologies cansurvive side-by-side for long periods of time, it seems reasonable to assume thattechnologically backward economies will only slowly read more..

##### Page - 764

Introduction to Modern Economic Growthargued that the answer to this question is also yes and is related to the “appropri-ateness” of technologies. A given technology will not have the same impact on theproductivity of all economies, because it may be a better match to the conditionsor to the factor proportions of some countries than of others. Part of this chapterwas devoted to explaining how the issue of appropriate technologies can play a rolein diﬀerent contexts. In our current age of read more..

##### Page - 765

Introduction to Modern Economic Growth18.7. References and LiteratureThe large literature documenting productivity and technology diﬀerences across ﬁrms andthe patterns of technology diﬀusion were discussed in Section 18.1 and the relevant referencescan be found there. The simple model of technology diﬀusion presented in Section 18.2 isinspired by Gerschenkron (1962) essay and Nelson and Phelps’s (1966) classic paper, though Iam not aware of a paper that presents a simple general read more..

##### Page - 766

Introduction to Modern Economic Growthchange to show how there will be a bias towards technologies inappropriate to the needsof poorer nations. Acemoglu and Zilibotti also provide evidence that these eﬀects could bequantitatively large and patterns of sectoral diﬀerences are consistent with the importanceof this type of technology-skill mismatch. Acemoglu (2002b) shows that technology-skillmismatch applies in a more general model of directed technical change than the one inAcemoglu and read more..

##### Page - 767

Introduction to Modern Economic GrowthExercise 18.9. In the model of Section 18.2 with consumer optimization, suppose thatpreferences in country j are given byUj =Z ∞0exp¡−¡ρj−nj¢t¢h³cj(t)1−θ−1´/(1−θ)idt,where ρj diﬀers across countries.(1) Show that a unique steady-state world equilibrium still exists and all countries growat the rate g.(2) Provide an intuition for why countries grow at the same rate despite diﬀerent ratesof discounting.(3) Show that this steady-state read more..

##### Page - 768

Introduction to Modern Economic Growthwe have seen so far. If not, suggest what important features are missing and howthey might be introduced.Exercise 18.13. * Consider the model in subsection 18.3.1. Suppose that preferences aregiven by Uj =R∞0exp¡−ρjt¢h³cj(t)1−θ−1´/(1−θ)idt,whereρj diﬀersacrosscountries.Show that an equivalent of Proposition 18.4, with a unique globally saddle-path stable worldequilibrium where all countries grow at the same rate, applies.Exercise 18.14. read more..

##### Page - 769

Introduction to Modern Economic Growth(1) Characterize the steady-state world equilibrium (that is, the steady-state capital-labor ratios in both countries).(2) Characterize the output per capita dynamics in the two economies. How does anincrease in γ aﬀect these dynamics?(3) Show that the implied income per capita diﬀerences (in steady state) between thetwo countries are increasing in γ. Interpret this result.(4) Do you think this model provides a good/plausible mechanism for generating read more..

##### Page - 770

Introduction to Modern Economic Growth(3) Derive the equivalents of Proposition 18.8.(4) Do the implications of inappropriate technologies become more or less importantwhen σ increases?Exercise 18.27. Prove Proposition 18.9.Exercise 18.28. Prove Proposition 18.11.Exercise 18.29. Prove Proposition 18.12.Exercise 18.30. * Consider the model of Section 18.5. Suppose that there is a total popula-tion of L. Assume that each individual can work as a supplier for one of the M products, orhe can work read more..

##### Page - 771

##### Page - 772

CHAPTER 19Trade and GrowthThe previous chapter discussed how technological linkages across countries and technol-ogy adoption decisions lead to a pattern of interdependent growth across countries. In thischapter, we study world equilibria when countries can trade ﬁnancial assets or commodities.We start with growth in economies that can borrow and lend internationally, and discusshow this aﬀects cross-country income diﬀerences and growth dynamics. We then turn to thegrowth implications of read more..

##### Page - 773

Introduction to Modern Economic Growthto illustrate the implications of international capital ﬂows for economic growth and show howthey signiﬁcantly change transitional dynamics in the basic neoclassical growth model. Oursecond task is to highlight what new lessons can be derived from the analysis of economicgrowth in the presence of international capital ﬂows. In particular, we will see that thepresence of international capital ﬂows raises a number of puzzles, most notably, the read more..

##### Page - 774

Introduction to Modern Economic Growthwithout loss of any generality, is normalized to 1, i.e., Lj (0) = 1 for all j =1,...,J ,sothatLj (t)= L (t)= exp (nt) ,for all j. In addition, we assume that Assumption 4 from Chapter 8 is satisﬁed, i.e., ρ − n>(1 − θ) g.The key feature of this economy is the presence of international borrowing and lending.Consistent with the permanent income hypothesis for individual consumption decisions, bor-rowing and lending will allow a smoother consumption read more..

##### Page - 775

Introduction to Modern Economic Growthis positive, the country is a net lender and has positive claims on output produced in othercountries, while if it is negative, the country is a net borrower. The ﬂow international budgetconstraint for country j at time t can then be written as:(19.3)˙Aj (t)= r (t) Aj (t) − Bj (t) ,which simply states that the country earns the world interest rate, r (t), on its existing assetposition A(t) (or accumulates further debt if the latter is negative) and in read more..

##### Page - 776

Introduction to Modern Economic GrowthWith access to international capital markets, the problem of the representative householdin each country can be written as maximizing (19.1) subject to (19.2), (19.4) and (19.5).A world equilibrium is now deﬁned as a sequence of normalized consumption levels, capitalstocks and asset positions for each country, that is,n[kj(t),cj(t),aj(t)]t≥0oJj=1and a timepath of world interest rates, [r (t)]t≥0, such that each country’s allocation maximizes read more..

##### Page - 777

Introduction to Modern Economic GrowthAt some level this result is very intuitive: with free capital ﬂows, we have an integratedworld economy. This integrated world economy has a unique steady-state equilibrium similarto that in the standard neoclassical growth model. This steady-state equilibrium not onlydetermines the eﬀective capital-labor ratio and its growth rate, but also the distribution of theavailable capital across diﬀerent countries in the world economy. Even though this read more..

##### Page - 778

Introduction to Modern Economic GrowthThis result implies that in the world economy with free ﬂows of capital, there are onlytransitional dynamics for the aggregate world economy, but no transitional dynamics sepa-rately for each country (in particular, kj (t) /kj0 (t)=1 for all t and any j and j0). This isintuitive, since international capital ﬂows will ensure that each country has the same eﬀec-tive capital-labor ratio, thus dynamics resulting from slow capital accumulation are read more..

##### Page - 779

Introduction to Modern Economic Growthdiﬀerences across countries as given. Nevertheless, it explains how, given these productivitydiﬀerences, there is no compelling reason to expect capital to ﬂow from rich to poor countries.Proposition 19.4. Consider a world economy with identical neoclassical preferencesacross countries and free ﬂows of capital. Suppose that countries diﬀer according to theirproductivities, the Aj’s. Then there exists a unique steady state equilibrium in which read more..

##### Page - 780

Introduction to Modern Economic Growthfrom rich to poor countries, we may expect large diﬀerences in the return to capital acrosscountries.Existing evidence on this topic is mixed. Three diﬀerent types of evidence are relevant.First, a number of studies, including Treﬂer’s (1993) important work discussed in Chapter3 and recent work by Caselli and Feyrer (2007), suggest that diﬀerences in the return tocapital across countries are relatively limited. These estimates are directly relevant read more..

##### Page - 781

Introduction to Modern Economic Growthfor OECD economies. Similar results have been found for other samples of countries, thoughother studies, most notably Taylor (1994), argue that including additional controls removesthe puzzle. Feldstein and Horioka and much of the literature that has followed them hasinterpreted the positive correlation between investment and savings as evidence against freecapital ﬂows. Naturally, in practice there are a number of econometric issues one needs toworry read more..

##### Page - 782

Introduction to Modern Economic Growthrole in the analysis below. These intermediate inputs, XLj (t) and XKj(t) are respectivelylabor and capital intensive. We use the letter X to denote these inputs, since they refer tothe amounts of these inputs used in production rather than the amount of inputs producedin country j. In the presence of international trade these two quantities will typically diﬀer.We assume throughout that the production of the ﬁnal good is competitive.The theory of read more..

##### Page - 783

Introduction to Modern Economic Growthequal to their “world prices”. Then the world supply and demand for these commoditieswill determine these prices. In particular, we denote the world price of the labor-intensiveintermediateattime t by pL (t) and the price of the capital-intensive intermediate by pK (t).Both of these prices are in terms of the ﬁnal good in the world market, which is taken asnumeraire, with price normalized to 1.1Given the production technologies in (19.8) and (19.9), read more..

##### Page - 784

Introduction to Modern Economic Growthabundant in one factor will sell enough of the goods embedding that factor to equalize fac-tor prices across countries. In the jargon of international trade theory, with free trade ofcommodities, there will exist a cone of diversiﬁcation, such that when factor proportions ofdiﬀerent countries are within this cone, there will be (conditional) factor price equalization.Our extreme assumption that labor is used in the production of the labor-intensive read more..

##### Page - 785

Introduction to Modern Economic GrowthIn addition to this trade balance equation, there is the usual resource constraint aﬀectingeach country at each time, which we can write as(19.12)˙Kj (t)= F¡XKj(t),XLj(t)¢−Cj(t)−δKj(t),for all j and t. In addition, world market clearing requires(19.13)JXj=1XLj (t)=JXj=1YLj (t) andJXj=1XKj(t)=JXj=1YKj(t) for all t.The important feature in this equation is that both the consumption good and the capitalgood are produced with the same technology–one read more..

##### Page - 786

Introduction to Modern Economic Growthcountry j.We also deﬁne kj (t) ≡ Kj (t) /Lj (t) as the capital labor ratio in country j at timet.A world equilibrium can be expressed as a sequence of consumption, capital accumu-lation and capital intermediate intensity decision for each country and world prices, i.e.,h{cj(t),kj(t),xj(t)}Jj=1,pK(t),pL(t)it≥0such that [cj (t) ,kj (t) ,xj (t)]t≥0maximizes theutility of the representative household in country j subject to (19.11) and (19.12) given read more..

##### Page - 787

Introduction to Modern Economic Growthof diﬀerent countries to obtain the behavior of world aggregates. In particular, let c (t) bethe average consumption per capita in the world and k (t) be the average capital-labor ratioin the world, given byc (t) ≡1JJXj=1cj (t) and k (t) ≡1JJXj=1kj (t) .The next proposition shows that world aggregates follow laws of motion very similar to thatof the standard neoclassical closed economy.Proposition 19.6. Consider the above-described model. Then in any read more..

##### Page - 788

Introduction to Modern Economic Growthdiﬀerential equations of the neoclassical model. Now using the previous two propositions, wecan characterize the form of the steady-state world equilibrium.Proposition 19.7. Consider the above-described model. There exists a unique steady-state equilibrium whereby(19.16)f0¡x∗j¢=f0µk∗A¶=ρ+δ for all j,where(19.17)x∗j = x∗ =PJj=1Kj(t)L (t)PJj=1Ajand k∗ =PJj=1Kj(t)JL (t).Moreover(19.18)pK∗ = ρ + δ.Proof. The proof follows from Proposition read more..

##### Page - 789

Introduction to Modern Economic Growtheach country j at any time t satisﬁes the diﬀerential equations:˙cj (t)cj (t)=1θ¡pK(t)−δ−ρ¢˙kj (t)=£pK(t)−(n+δ)¤kj(t)+pL(t)Aj−cj(t).Standard arguments from Chapter 8 applied to world averages in Proposition 19.6 implythat world averages converge to the unique world steady state equilibrium and£pK(t)¤t≥0converges to ρ + δ. This immediately implies that the law of motion for the consumptionand capital-labor ratio of each country read more..

##### Page - 790

Introduction to Modern Economic Growthtype of short run (or “medium run”) dynamics, especially for countries that have diﬀerentsaving rates than others.To illustrate this possibility in the simplest possible way, consider the following thoughtexperiment. Let us start with the world economy in steady state and suppose that one ofthe countries experiences a decline in its discount rate from ρ to ρ0 <ρ. What will happen?The answer is provided in the next proposition.Proposition 19.9. read more..

##### Page - 791

Introduction to Modern Economic Growthpoint, country 1 will become so large relative to the rest of the world that it will essentiallyown almost all of the capital of the world. At that point or in fact even before this point isreached, country 1 can no longer be considered a “small” country; it will have a major impact,and it will recognize this impact, on the relative price of the capital-intensive intermediate.Consequently, the rate of return on capital will eventually fall so that read more..

##### Page - 792

Introduction to Modern Economic GrowthThe model economy presented here builds on Acemoglu and Ventura (2002). I will startwith a simpliﬁed version of this model, which features physical capital as the only factor ofproduction. I will then present the full model in which both physical capital and labor areused to produce consumption and investment goods.In addition to the nature of trade (Heckscher-Ohlin versus Ricardian), another majordiﬀerence between the model in this section and the read more..

##### Page - 793

Introduction to Modern Economic GrowthThe budget constraint of the representative household in country j at time t ispIj (t) ˙Kj (t)+ pCj Cj (t)= Yj (t)(19.20)= rj (t) Kj (t)+ wj (t) ,where pIj and pCjare the prices of the investment and consumption goods in country j (interms of the numeraire, which will be the ideal price index of traded intermediates; see below).Despite international trade in intermediates, because consumption and investment goods arenot traded, their prices might diﬀer read more..

##### Page - 794

Introduction to Modern Economic Growthtime t are given by(19.22)pj (t)= rj (t) ,where recall that rj (t) is the rental rate of return in country j at time t.19.4.2. TheAK Model.Before presenting the full model, it is convenient to start withasimpliﬁed version, where capital is the only factor of production. Consequently, in termsof equation (19.20), we have wj (t)= 0,andYj (t)= rj (t) Kj (t) .We assume that both consumption and investment goods are produced using domestic capitalas well as a read more..

##### Page - 795

Introduction to Modern Economic Growthtrade in GDP for all countries in this world economy (see Exercise 19.13). Finally, χ is aconstant introduced for normalization (see Exercise 19.11).The production function for investment goods in country j is:(19.24)Ij (t)= ζ−1jχKIj (t)1−τµZ N0xIj (t, ν)ε−1εdν¶τεε−1,which is identical to that for consumption goods, except for the presence of the term ζj.This allows diﬀerential levels of productivity, due to technology or policy, in read more..

##### Page - 796

Introduction to Modern Economic Growth(19.27)BIj³rj(t),[p(t,ν)]ν∈[0,N]´=ζjrj(t)1−τ"µZ N0p(t, ν)1−εdν¶ τ1−ε#,where p(t, ν) is the price of the intermediate ν at time t and the constant χ in (19.23)and (19.24) is chosen appropriately (see Exercise 19.11). Notice that these prices are notindexed by j, since there is free trade in intermediates and thus all countries face the sameintermediate prices. The speciﬁcation using the unit cost functions simpliﬁes the read more..

##### Page - 797

Introduction to Modern Economic GrowthEquation (19.29) is the transversality condition. Integrating the budget constraint andusing the Euler and transversality conditions, we obtain a particularly simple consumptionfunction in this case:(19.30)pCj (t) Cj (t)= ρjpIj (t) Kj (t) ,which can be interpreted as individuals spending a fraction ρj of their wealth on consumptionat every instant (recall that in this simpliﬁed model, there is no labor income and pIj (t) Kj (t)is consumer wealth at read more..

##### Page - 798

Introduction to Modern Economic Growthwhere Y (t) ≡PJj=1Yj(t)istotalworldincomeattimet. To see why this equation ensuresbalanced trade, note that each country spends a fraction τ of its income on intermediates, andsince each country is small, this implies a fraction τ of its income being spent on imports.At the same time, the rest of the world spends a fraction τμjpj (t)1−ε of its total incomeon intermediates produced by country j (this follows because of the CES aggregator read more..

##### Page - 799

Introduction to Modern Economic Growthfor j =1,...,J , and the world steady-state growth rate g∗ is the unique solution to equation(19.37)JXj=1μj£ζj¡ρj+g∗¢¤(1−ε)/τ =1.The steady-state rental rate of capital and the terms of trade in country j are given by(19.38)r∗j = p∗j =£ζj¡ρj+g∗¢¤1/τ.This unique steady-state equilibrium is globally saddle-path stable.Proof. (Sketch) By deﬁnition, a steady-state equilibrium must have constant prices,thus a constant r∗j . This read more..

##### Page - 800

Introduction to Modern Economic Growth(19.34) implies that this country will tend to accumulate more capital than others. But(19.35) makes it clear that this cannot go on forever and country j,by virtue ofbeing richerthan the world average, will also have a lower rate of return on capital. This lower rate ofreturn will ultimately compensate the greater incentive to accumulate in country j,so thatcapital accumulation in this country converges to the same rate as in the rest of the read more..

##### Page - 801

Introduction to Modern Economic Growthdistribution to diﬀerences in savings rates and technology. In particular, recall that in aworld with a Cobb-Douglas aggregate production function and no human capital diﬀerences,the Solow model implies that(19.40)y∗j = Ajµsjg∗¶α/(1−α),where Aj is the relative labor-augmenting productivity of country j, sj is its savings rate, g∗ isagain the world growth rate and α is the exponent of capital in the Cobb-Douglas productionfunction, which is read more..

##### Page - 802

Introduction to Modern Economic Growthfor some γ ∈ (0, 1).Here χ is again a normalizing constant and where Lj (t) is the totallabor supply of country j at time t. All of this labor supply is used in the production of theconsumption good, since neither the production of intermediates nor the production of theinvestment good use labor. The labor endowment in the economy is supplied inelastically bythe representative household in the economy, and without loss of any generality, we normalizeLj read more..

##### Page - 803

Introduction to Modern Economic Growthconvenient equation:(19.44)pCj (t) Cj (t)=ρj1 − (1 − γ)(1 − τ )pIj (t) Kj (t) .In other words, households again consume a constant fraction of the value of the capitalstock, but this fraction now depends not only on their discount rate, ρj, but also on thetechnology parameters, τ and γ. In light of this derivation, the following two propositionsare straightforward:Proposition 19.11. In the general model with labor, the world equilibrium is read more..

##### Page - 804

Introduction to Modern Economic Growtharguethatitmustbe due to frictionsaﬀecting the investment sector in poor economies.However, only models that allow for trade and diﬀerent production function for consumptionand investment goods can be truly useful for understanding the sources of diﬀerences in theserelative prices. The current model, which incorporates these features, naturally generates thispattern of relative prices. The equilibrium derivation above immediately implies the read more..

##### Page - 805

Introduction to Modern Economic Growth(e.g., Vogel, 2006). However, as countries become richer they also produce and consumemore specialized goods. These goods often come in diﬀerentiated varieties and thus a greatersupply of any one of these goods will create terms of trade eﬀects. Consequently, if a countryis in the stage of development where it produces more of the specialized goods, furthercapital accumulation will run into diminishing returns because of terms of trade eﬀects. read more..

##### Page - 806

Introduction to Modern Economic GrowthEach country admits a representative household with dynamic preferences deﬁned overstreams of consumption Cj (t). For our purposes here, we do not need to specify what thesedynamic preferences are, but for concreteness, the reader may want to assume that these aregiven by the standard CRRA preferences as in (19.1).The key assumption of the model is that goods fall into two categories: new goods arejust invented in the North and can only be produced there; read more..

##### Page - 807

Introduction to Modern Economic GrowthSouth ws (t). By its very nature, a specialization equilibrium implies thatpn (t)= wn (t)(19.48)po (t)= ws (t) .It must be thecasethat wn (t) ≥ ws (t), since otherwise Northern workers would prefer toproduce the old goods. Thus a specialization equilibrium can exist only if when all old goodsare produced in the South, the implied equilibrium wage rate in the South is lower thanthat in the North. To ﬁnd out when this will be so is straightforward. The CES read more..

##### Page - 808

Introduction to Modern Economic Growthcase the relative supply of labor in the North is suﬃciently large that we entered region ofequalization equilibrium.Nn /NoLn /Lswn /ws1OEFigure 19.1. Determination on the relative wages in the North and the Southin the basic product cycle model.An interesting implication of this equilibrium is that even when there is a technologygap between the North and the South, Northern and Southern incomes may be equalized.There will only be an income gap between the read more..

##### Page - 809

Introduction to Modern Economic Growthtypically the case, but perhaps surprisingly, not always so. Exercise 19.26 goes through theimplications of trade on cross-country income diﬀerences and shows that even in the contextof the current model, it can sometimes lead to a larger gap of income between rich and poorcountries.19.5.2. Product Cycles and Technology Transfer.The characterization of the equi-librium in the previous subsection was for a given number of new and old goods. Our interestin read more..

##### Page - 810

Introduction to Modern Economic Growthnot absolutely so), while a higher rate of imitation by the South makes the South relativelyricher andthe Northrelativelypoorer(seeExercise 19.24). In view of the results from theprevious chapter, these results are not surprising.An important and interesting feature of this steady-state equilibrium is the product cycle.Let us focus on the specialization equilibrium. Then new goods are invented in the Northand produced there by workers that receive relatively read more..

##### Page - 811

Introduction to Modern Economic Growththough at least some part of this correlation is likely to be due to selection (Melitz, 2003).Similarly, ﬁrms in developing countries that import machinery from more advanced economiesappear to be more productive (e.g., Goldberg and Pavnik, 2007). Nevertheless, a numberof economists are skeptical of the growth eﬀects of trade. Rodrik (1997) and Rodriguez andRodrik (1999) argue that the empirical evidence that trade promotes growth is not read more..

##### Page - 812

Introduction to Modern Economic Growthholds. Then in autarky there exists a unique equilibrium in which starting from any level oftechnology, both countries innovate and grow at the same rate(19.55)gA =1θ(ηβ − ρ) .Proof. See Exercise 19.27.¤Next, we will analyze what happens when these two economies start trading. The exactimplications of trade will depend on whether, before trade opening, the two countries wereproducing some of the same inputs or not (recall that there is a continuum of read more..

##### Page - 813

Introduction to Modern Economic Growthfocus on an economy with semi-endogenous growth (as the model studied in Section 13.3 inChapter 13), trade opening will increase innovation temporarily, but not in the long run.19.7. Learning-by-Doing, Trade and GrowthThe previous section showed how international trade can increase economic growth inall countries in the world by encouraging faster technological progress. In addition to thiseﬀect of trade on growth working via technological change, the read more..

##### Page - 814

Introduction to Modern Economic Growthwe write this asYj (t)=hγX1j(t)ε−1ε +(1 − γ) X2j (t)ε−1εi εε−1 ,where ε is the elasticity of substitution between the two intermediates. Throughout thissection we assume that these two intermediates are gross substitutes, so that ε> 1.How-ever, the case of ε =1 (where the production function becomes Cobb-Douglas) is also ofspecial-interest, so I will treat this case separately. Moreover, to simplify the algebra and theexposition below, read more..

##### Page - 815

Introduction to Modern Economic Growthﬁnal good producers immediately implies thatp1j (t)p2j (t)=ÃX1j(t)X2j (t)!−1ε=ÃAj(t)L1j(t)1 − L1j (t)!−1ε,where L1j (t) denotes the amount of labor allocated the sector 1 in country j at time t,andnaturally, the amount of labor allocated to sector 2 is L2j (t)=1 − L1j (t). Combining thiswith (19.58), we obtain(19.59)L1j (t)=Aj (t)ε−11+ Aj (t)ε−1.The evolution of the productivity of sector 1 is then given by (19.56).Proposition 19.15. read more..

##### Page - 816

Introduction to Modern Economic GrowthProposition 19.16. Consider the above-described model. Then with free internationaltrade, the equilibrium is as follows: L1n (t)=1 and L1s (t)=0 for all t. In this equilibrium,we have that˙An (t)An (t)= η and˙As (t)As (t)=0.The world economy converges to a growth rate of g∗ = η in the long run. Throughout, theratio of income in the North and the South is given byYn (t)Ys (t)= An (t)ε−1ε.Consequently, if ε> 1, the North becomes progressively read more..

##### Page - 817

Introduction to Modern Economic Growthgrowth, the debate can be resolved only by empirical work. Having said that, the theoreticalperspectives are still useful. A couple of issues are particularly worth noting. First, the eﬀectof trade integration on the rate of endogenous technological progress may be limited becauseof the factors already discussed at the end of the previous section. For example, signiﬁcanteﬀects are possible only when trade opening does not increase wages in the ﬁnal read more..

##### Page - 818

Introduction to Modern Economic Growthinternational trade in commodities change both the dynamics and potentially long-run impli-cations of the closed-economy neoclassical growth models. For example, international capitalﬂows remove transitional dynamics, because economies that are short of capital do not need toaccumulate it slowly, but can borrow in international markets. Naturally, there are limits tohow much international borrowing can take place. Countries are sovereign entities, thus it read more..

##### Page - 819

Introduction to Modern Economic Growthcountries. In contrast, the model in Section 19.4 emphasized how Ricardian trade, based ontechnological comparative advantage, creates a new source of diminishing returns to accu-mulation for each country based on terms of trade eﬀects. As a country accumulates morecapital, it starts exporting more of the goods in which it specializes. The result is a worseningof its terms of trade, eﬀectively reducing the rate of return to further capital read more..

##### Page - 820

Introduction to Modern Economic Growth19.9. References and LiteratureThis chapter covered a variety of models. Section 19.1 focused on the implications ofinternational ﬁnancial ﬂows on economic growth. This topic is discussed in detail in Chapter3 of Barro and Sala-i-Martin (2004), both with and without limits to ﬁnancial ﬂows. Ob-stfeld and Rogoﬀ (1996) Chapters 1 and 2 provide a more detailed analysis of internationalborrowing and lending. Chapter 6 of Obstfeld and Rogoﬀ provides read more..

##### Page - 821

Introduction to Modern Economic GrowthDixit and Norman, 1990), but their growth implications had not previously been recognized.The model presented in Acemoglu and Ventura (2002) is a much simpliﬁed Ricardian model,exploiting the structure of preferences ﬁrst introduced by Armington (1969), but in theproduction of the ﬁnal good rather than in preferences. Richer Ricardian models typicallybuild on the seminal article by Dornbusch, Fischer and Samuelson (1977), though this richersetup has read more..

##### Page - 822

Introduction to Modern Economic Growthbe costly rely on diﬀerences in the amount of rents generated by diﬀerent sectors becauseof imperfections in the labor market or institutional problems. Levchenko (2008) and Nunn(2007) present models in which trade leads to the transfer of rent-creating jobs from countrieswith weak institutions to those with better institutions and may be harmful to countries withweak institutions. Davis and Harrigan (2007) present a model in which trade leads to read more..

##### Page - 823

Introduction to Modern Economic GrowthExercise 19.4. * Consider a world economy with international capital ﬂows, but supposethat because of sovereign risk a country cannot borrow more than a fraction φ> 0 of itscapital stock. Consequently, in terms of the model in Section 19.1, we have the restrictionthatbj (t) ≤ φkj (t) .(1) Show that the steady-state equilibrium of the world economy is not aﬀected by thisconstraint. Explain the intuition for this result carefully.(2) Characterize read more..

##### Page - 824

Introduction to Modern Economic Growth(2) Now let us go back to the preferences as in (19.14), but suppose that productivityof labor in each country is given byAj (t)= Aj exp (gt) .Show that all of the results from the text continue to apply, and in particular, derivethe equivalent of Proposition 19.4.(3) Finally, let us suppose that F in (19.7) does not satisfy Assumption 2. How doesthis aﬀect the analysis and the results?Exercise 19.11. Derive the unit cost functions (19.26) and (19.27) from read more..

##### Page - 825

Introduction to Modern Economic Growthof its income. Show that terms of trade eﬀects will be present in equilibrium, but the steadystate will be “degenerate,” with the relative prices of goods supplied by the highest savingcountry going to zero. Explain why exogenous savings versus dynamic utility maximizationgive diﬀerent answers in this case.Exercise 19.21. * Consider the model of Section 19.4, but assume that ε< 1. Characterizethe equilibrium. Show that in this case countries that read more..

##### Page - 826

Introduction to Modern Economic GrowthExercise 19.25. This exercise asks you to endogenize innovation decisions in the model ofSection 19.5. Assume that new goods are created by technology ﬁrmsinthe Northasinthe model in Section 13.4 in Chapter 13, and these ﬁrms are monopolist suppliers until thegood they have invented is copied by the South. The technology of production is the same asbefore, and assume that new goods can be produced by using ﬁnal goods, with the technologyN (t)= ηZ read more..

##### Page - 827

Introduction to Modern Economic Growthfor country j,where N (t)= N 1 (t)+ N 2 (t) and LjR (t) is the number workers working inR&D in country j. Consequently, trade opening does not change the structure of knowledgespillovers.(1) Show that in this model, trade opening has no eﬀect on the equilibrium growth rate.Provide a precise intuition for this result.(2) Next assume that before trade opening the innovation possibly the frontier takesthe form ˙N j (t)= ηN j (t) LjR (t). Show that in read more..

##### Page - 828

Part 7Economic Development and EconomicGrowth read more..

##### Page - 829

In this part of the book, I discuss the relationship between economic development andeconomic growth. The ﬁrst question that the reader will rightly ask is why there is (orthere should be) a distinction between economic development and economic growth. Thisquestion is particularly apt because I have argued in Chapter 1 that societies that are rich–developed –today are those that have grown steadily over the past 200 years and those thatare poor or less-developed are those that have not read more..

##### Page - 830

Introduction to Modern Economic GrowthAlthough one might debate whether this is the most functional deﬁnition of economicgrowth, it does capture a range of important changes that accompany economic growth inmost societies. And yet, the models of economic growth we have studied so far do not dojustice to the complex process described by Kuznets. They provide a framework for explainingthe sustained increase in income per capita or output per worker. But our models do notfeature Kuznets’s read more..

##### Page - 831

Introduction to Modern Economic Growtheconomies. Economic development, on the other hand, becomes the study of structuraltransformations, and the eﬃciency implications of these transformations, at the early stagesof development. Models of economic development would then focus on structural changes inthe production and consumption, on urbanization, on the size and the composition of thepopulation, on the occupational structure, and on changes in living and social arrangements.The study of read more..

##### Page - 832

Introduction to Modern Economic GrowthThese two reasons motivate my acceptance of the standard distinction between economicdevelopment and economic growth. Although I go along with this standard distinction,throughout I emphasize how it is exactly the same tools that are useful for understanding theprocess of economic development–the structural transformations emphasized by Kuznets,Hirschman, Nurske and Rosenstein-Rodan–as well as the more orderly process of economicgrowth.My hope is that read more..

##### Page - 833

##### Page - 834

CHAPTER 20Structural Change and Economic GrowthIn this chapter, I discuss various diﬀerent approaches to the analysis of structural change.The next two sections focus on the shift of employment and production from agriculture tomanufacturing, and then from manufacturing to services. This is a useful starting pointboth because changes in the composition of employment and production are an importantpart of the process of economic development and also because, as emphasized by Kuznetsand others, read more..

##### Page - 835

Introduction to Modern Economic Growthagricultural goods). The changes in the composition of employment in the British economytowards the end of the 18th century are also consistent with the US patterns shown in Figure20.1 (see, for example, Mokyr, 1989). Consequently, similar patterns are present in all OECDeconomies. Some of the less-developed economies are still largely agricultural but the timetrend is inexorably towards a smaller share of agriculture. Because of Kuznets’s emphasis read more..

##### Page - 836

Introduction to Modern Economic GrowthKaldor facts of aggregate balanced growth. Even though it is designed to match the Kaldorfacts regardless of the stage of development, its tractability makes this model a useful start-ing point for our analysis. Moreover, as emphasized by Kuznets, once we look beneath theaggregate facts of balanced growth structural changes in the composition of employment andproduction are present even in relatively advanced economies. A model consistent with theKaldor read more..

##### Page - 837

Introduction to Modern Economic Growthhas been achieved, the household starts to demand other items, in particular, manufacturedgoods (e.g., textiles and durables) and services (entertainment, retail, etc.). However, as wewill see shortly, the presence of the γS term in the aggregator implies that the household willspend a positive amount on services only after certain levels of agricultural and manufacturingconsumption have been reached.We assume that the economy is closed, thus agricultural, read more..

##### Page - 838

Introduction to Modern Economic Growthand(20.7)LA (t)+ LM (t)+ LS (t)= L (t) ,where K (t) and L (t) are the total supplies of capital and labor at time t.Another key assumption of the Kongsamut, Rebelo and Xie model builds on Rebelo(1991) and imposes that it is the manufacturing good that is used in the production of theinvestment good. Consequently, market clearing for the manufacturing good takes the form(20.8)˙K (t)+ cM (t) L (t)= YM (t) ,where, for simplicity, we have ignored capital read more..

##### Page - 839

Introduction to Modern Economic GrowthA competitive equilibrium is deﬁned in the usual manner as sequences of sectoral fac-tor demands£KA(t),KM(t),KS(t),LA(t),LM(t),LS(t)¤∞t=0 that maximize proﬁts giventhe sequence of the total supplies of capital and labor [K (t) ,L (t)]∞t=0 and the sequence ofprices£pA(t),pM(t),w(t),r(t)¤∞t=0;pricesequences£pA(t),pM(t),w(t),r(t)¤∞t=0thatsat-isfy (20.10)-(20.13) given£KA(t),KM(t),KS(t),LA(t),LM(t),LS(t)¤∞t=0; and sequencesof consumption read more..

##### Page - 840

Introduction to Modern Economic GrowthProposition 20.2. Suppose (20.14) holds. Then, in any equilibrium, we have that(20.18)˙cM (t)cM (t)=1θ(r (t) − ρ)for all t and moreover, provided that Assumption 4 holds, the transversality condition of therepresentative household is satisﬁed. In addition, we have that for all t(20.19)pA (t)¡cA(t)−γA¢ηA=cM(t)ηMand(20.20)ps(t)¡cS(t)+γS¢ηS=cM(t)ηM.Proof. See Exercise 20.3.¤In analogy to previous models we have seen so far, we may want to read more..

##### Page - 841

Introduction to Modern Economic GrowthIn a CGP k (t)= k∗ for all t, and moreover we have the following evolution of consumptionand employment in the three sectors(20.22)˙cA (t)cA (t)= gcA (t) − γAcA (t),˙cM (t)cM (t)= g,and˙cS (t)cS (t)= gcS (t)+ γScS (t),and˙LA (t)LA (t)= n − gγA/LA (t)BAX (t) F (k∗, 1),˙LM (t)LM (t)= n,and˙LS (t)LS (t)= n + gγS/LS (t)BSX (t) F (k∗, 1)for all t.Moreover, in the CGP the share of national income accruing to capital is constant.Proof. See read more..

##### Page - 842

Introduction to Modern Economic GrowthSecond, the assumption that all three sectors have the same production function appearsrestrictive. Nevertheless, this assumption can be relaxed to some degree. Exercise 20.7discusses how this can be done. Perhaps more important is the assumption that investmentsfor all three sectors use only the manufacturing good. This assumption is similar in nature tothe assumption that only capital is used to be produce capital (investment) goods in Rebelo’s(1991) read more..

##### Page - 843

Introduction to Modern Economic Growthfactors. Nevertheless, isolating these factors in separate models is both more tractable andalso conceptually more transparent. For this reason, in this section I focus on the supplyside, abstracting from Engel’s law throughout, and will only return to the combination of thesupply-side and the demand-side factors in Exercise 20.17.20.2.1. General Insights.At some level, Baumol’s theory of non-balanced growth canbe viewed as self-evident–if some sectors read more..

##### Page - 844

Introduction to Modern Economic Growthwhere L1 (t), L2 (t), K1 (t) and K2 (t) denote the amount of labor and capital employed inthe two sectors, and the functions G1 and G2 are also assumed to satisfy the equivalents ofAssumptions 1 and 2. The terms A1 (t) and A2 (t) are Hicks-neutral technology terms.Market clearing for capital and labor implies thatK1 (t)+ K2 (t)= K (t) ,(20.25)L1 (t)+ L2 (t)= L (t) ,at each t. Without loss of any generality, I ignore capital depreciation.Let us take the read more..

##### Page - 845

Introduction to Modern Economic Growthand the “per capita production functions” (without the Hicks-neutral technology term) as(20.29)g1 (k1 (t)) ≡G1 (K1 (t) ,L1 (t))L1 (t)and g2 (k2 (t)) ≡G2 (K2 (t) ,L2 (t))L2 (t).Since G1 and G2 are twice continuously diﬀerentiable by assumption, so are g1 and g2 anddenote their ﬁrst and second derivatives by g01, g02, g001 and g002 .Now, diﬀerentiating the production functions for the two sectors,˙Y1 (t)Y1 (t)=˙A1 (t)A1 (t)+ σ1 (t)˙K1 (t)K1 read more..

##### Page - 846

Introduction to Modern Economic GrowthThe intuition for this result is straightforward. Suppose that there is capital deepeningand that, for concreteness, sector 2 is more capital-intensive (i.e., σ1 <σ2). Now, if bothcapital and labor were allocated to the two sectors at constant proportions over time, themore capital-intensive sector, sector 2, would grow faster than sector 1. In equilibrium, thefaster growth in sector 2 would change equilibrium prices, and the decline in the relative read more..

##### Page - 847

Introduction to Modern Economic GrowthFor this purpose, I now specialize the environment of the previous subsection by incorpo-rating speciﬁc preferences and production functions and then provide a full characterizationof a simpler economy. The economy is again in inﬁnite horizon and population grows at theexogenous rate n> 0 according to (20.1). Let us also assume that the economy admits arepresentative consumer, with standard preferences given by (20.2), who also supplies read more..

##### Page - 848

Introduction to Modern Economic GrowthLet us also denote the wage and the interest rate (the rental rate of capital) by w (t) and r (t)and the prices of the two intermediate goods by p1 (t) and p2 (t). We again normalize the priceof the ﬁnal good to 1 at each instant. An equilibrium is deﬁned in the usual manner, as se-quences of labor and capital allocations and prices, such that [K1 (t) ,K2 (t) ,L1 (t) ,L2 (t)]∞t=0maximize intermediate sector proﬁts given the prices [w (t) ,r (t) ,p1 read more..

##### Page - 849

Introduction to Modern Economic GrowthThe key to the characterization of the static equilibrium is to determine the fractionof capital and labor employed in the two sectors. Let us deﬁne κ (t) ≡ K1 (t) /K (t) andλ (t) ≡ L1 (t) /L (t). Combining equations (20.39), (20.40), (20.42), and (20.43), we obtain:(20.46)κ (t)="1+α2α1µ1−γγ¶µY1(t)Y2 (t)¶1−εε#−1,and(20.47)λ (t)=∙1+α1α2µ1−α21 − α1¶µ1−κ(t)κ (t)¶¸−1.Equation (20.47) makes it clear that the read more..

##### Page - 850

Introduction to Modern Economic GrowthCombining (20.44) and (20.45), we also obtain relative factor prices as(20.50)w (t)r (t)=1 − α1α1µκ(t)K(t)λ (t) L (t)¶,and the capital share in the economy as:(20.51)σK (t) ≡r (t) K (t)Y (t)= γα1µY1(t)Y (t)¶ε−1εκ(t)−1.Proposition 20.8. In equilibrium,(20.52)d ln (w (t) /r (t))d ln K (t)= −d ln (w (t) /r (t))d ln L (t)=11+ (1 − ε)(α2 − α1)(κ (t) − λ (t))> 0.d ln (w (t) /r (t))d ln A2 (t)= −d ln (w (t) /r (t))d ln A1 read more..

##### Page - 851

Introduction to Modern Economic GrowthThe most important result in this proposition is (20.54), which links the equilibriumrelationship between the capital share in national income and the capital stock to the elasticityof substitution. Since a negative relationship between the share of capital in national incomeand the capital stock is equivalent to capital and labor being gross complements in theaggregate, this result also implies that the elasticity of substitution between capital and laboris read more..

##### Page - 852

Introduction to Modern Economic GrowthA dynamic equilibrium is given by paths of wages, interest rates, labor and capital alloca-tion decisions, w, r, λ and κ, satisfying (20.44), (20.42), (20.45), (20.43), (20.46) and (20.47),and of consumption per capita, c, capital stock, K,employment, L, and technology, A1 andA2, satisfying (20.1), (20.35), (20.38), (20.58), and (20.59).Let us also introduce the following notation for growth rates of the key objects in thiseconomy:˙Ls (t)Ls (t)≡ ns (t) read more..

##### Page - 853

Introduction to Modern Economic GrowthProof. Diﬀerentiating the production function for the ﬁnal good (20.34) we obtain:(20.62)g (t)=γY1 (t)ε−1εg1 (t)+(1 − γ) Y2 (t)ε−1εg2 (t)γY1 (t)ε−1ε+(1 − γ) Y2 (t)ε−1ε.This equation, combined with ε< 1, implies that as t →∞, g∗ =min {g∗1,g∗2}. Similarly,combined with ε> 1, it implies that as t →∞, g∗ =max {g∗1,g∗2}.¤Consequently, when the elasticity of substitution is less than 1, the asymptotic read more..

##### Page - 854

Introduction to Modern Economic Growthwhich will make it easier to state this result. In particular, this condition ensures that sector1isthe asymptotically dominant sector, either because it has a slower rate of technologicalprogress and ε< 1, or it has more rapid technological progress and ε> 1.Notice also that, forthe reasons noted above, the appropriate comparison is not between a1 and a2, but betweena1/ (1 − α1) and a2/ (1 − α2). Exercise 20.14 generalizes the results in this read more..

##### Page - 855

Introduction to Modern Economic GrowthTo complete the proof, we need to establish that in all CGPs g∗2≥ g∗1 > 0 when ε< 1(g∗1≥ g∗2 > 0 when ε> 1 is again left to Exercise 20.14). We now separately derive acontradiction for two conﬁgurations, (1) g∗1≥ g∗2 ,or(2) g∗2≥ g∗1 but g∗1≤ 0.(1) Suppose g∗1≥ g∗2 and ε< 1. Then, following the same reasoning as above, the uniquesolution to the equilibrium conditions (20.36), (20.60) and (20.61), when read more..

##### Page - 856

Introduction to Modern Economic Growthsector (cf., condition (20.64)). If, in addition, we also have ε< 1, the model leads to therichest set of dynamics, whereby the more slowly growing sector determines the long-rungrowth rate of the economy, while the more rapidly growing sector continually sheds capitaland labor, but does so at exactly the right rate to ensure that it still grows faster than therest of the economy.Third, in the limiting equilibrium the share of capital and labor allocated read more..

##### Page - 857

Introduction to Modern Economic Growththe picture presented in Chapter 1, this question might hold important clues about the cross-country diﬀerences in income per capita today.In this light, it would be useful to have a number of diﬀerent approaches to this questionand evaluate their pros and cons. Although this is part of my objective, I will not present thesemodels all in one place. The ﬁrst approach, based on the model of Acemoglu and Zilibotti(1997), was already presented as an read more..

##### Page - 858

Introduction to Modern Economic Growthwe have already studied in this chapter, because it is, at some level, a model of structuralchange. It combines Engel’s law and learning-by-doing externalities in the industrial sector.The model is not only a tractable framework for the analysis of the relationship betweenagricultural productivity and industrialization, but it also enables an insightful analysis ofthe impact of international trade on industrialization.Consider the following read more..

##### Page - 859

Introduction to Modern Economic Growthcountries, reﬂecting either previous technological progress in terms of new agricultural meth-ods or diﬀerences in land quality (even though here, for simplicity, we are focusing on a singlecountry). Existing evidence shows that there are very large (perhaps too large) diﬀerences inlabor productivity and TFP of agricultural activities among countries even today, thus allow-ing for potential productivity diﬀerences in agriculture is reasonable. read more..

##### Page - 860

Introduction to Modern Economic Growth(20.75)BAG0(1 − n (t)) = p (t) X (t) F0(n (t)).Thepresenceofthe term γA > 0 implies that as in Section 20.1, preferences are non-homothetic and that the income elasticity of demand for agricultural goods will be less thanunity (while that for manufacturing goods will be greater than unity). As we have alreadyseen, this is the simplest way of introducing Engel’s law.Let us also assume that aggregate productivity is high enough to meet the read more..

##### Page - 861

Introduction to Modern Economic Growthis a strictly decreasing function. Moreover, we have that φ(0) = G(1) and φ(1) < 0.The φfunction can be interpreted as the “excess demand” function for manufacturing over agricul-ture. An equilibrium has to satisfy (20.78). From Assumption (20.76) and the properties ofthe φ function, we can conclude that the equilibrium condition (20.78) has a unique interiorsolution in whichn (t)= n∗ ∈ (0, 1) .Notice an important implication of this equation. read more..

##### Page - 862

Introduction to Modern Economic Growthgrowth of manufacturing production. However, since manufacturing and agricultural goodsare imperfect substitutes, the relative prices change, so expenditure on agricultural goodsincreases (see Exercise 20.18).We can summarize these results as follows:Proposition 20.12. In the above-described model, the combination of learning-by-doingand Engel’s law generates a unique equilibrium in which the share of employment of manu-facturing is constant at n∗ ≡ read more..

##### Page - 863

Introduction to Modern Economic Growthfrom agriculture to manufacturing and to services and with the changes between sectors ofdiﬀerent capital intensities. Section 20.1 focused on demand-side reasons for why growth canbe non-balanced. In particular, it incorporated Engel’s law into the basic neoclassical growthmodel so that households spend a smaller fraction of their budget on agricultural goods asthey become richer. This framework is ideally suited for the analysis of the structural read more..

##### Page - 864

Introduction to Modern Economic Growthof industrialization, sheds light on both the process of economic growth and the process ofeconomic development. In this sense, the models in this section enrich our understanding ofeconomic growth considerably. And yet, this is only a modest step towards the investigationof the sweeping structural changes emphasized by Kuznets because we have not departedfrom the neoclassical approach to economic growth. In particular, Sections 20.1 and 20.2used generalized read more..

##### Page - 865

Introduction to Modern Economic Growthon sources of agricultural productivity and emphasizes that diﬀerences in agricultural pro-ductivity across countries are often as large as or even larger than productivity diﬀerences inother sectors. Gollin, Parente and Rogerson (2002) is one of the ﬁrstpapersinthisvein.The works mentioned in the previous paragraph, like the model I presented in Section20.1, appeal to Engel’s law and model the resulting non-homothetic preferences by read more..

##### Page - 866

Introduction to Modern Economic Growth(2) Show that even though a balanced growth path does not exist, an equilibrium pathalways exists.Exercise 20.5.(1) Prove Proposition 20.4. In particular, show that if (20.21) is notsatisﬁed, a CGP cannot exist, and that this condition is suﬃcient for a CGP toexist.(2) Characterize the CGP eﬀective capital-labor ratio, k∗.Exercise 20.6. In the model of Section 20.1, show that as long as condition (20.21) is satisﬁedwhen the economy starts with an read more..

##### Page - 867

Introduction to Modern Economic GrowthExercise 20.10. Consider the model of Section 20.1 but assume that there exist a ﬁnal goodproduced according to the technology Y (t)=¡YA(t)−γA¢ηAYM(t)ηM¡YS(t)+γS¢ηS.(1) Show that all the results in Section 20.1 hold without any change as long as capitalgoods are produced out of intermediate Y M as implied by equation (20.8).(2) Next assume that capital goods are produced out of the ﬁnal good, so that theresource constraint becomes ˙K (t)+ c read more..

##### Page - 868

Introduction to Modern Economic Growthχ (t) in terms of κ (t), λ (t) and η (t) as deﬁned in (20.81), and then diﬀerentiate(20.46).](2) State the appropriate transversality condition.(3) Show that if an allocation satisﬁes the three diﬀerential equations in (20.80) and theappropriate transversality condition, then it corresponds to an equilibrium path.(4) Show that in a CGP equilibrium ϕ (t) must be constant. Using this, show that theCGP requires that κ (t) → 1 and that χ (t) be read more..

##### Page - 869

Introduction to Modern Economic Growthtype ν used by the manufacturing sector. Assume as in Part 4 that these machines aresupplied by technology monopolists with perpetual patents and can be produced by usingthe manufacturing good only at constant marginal cost of (1 − β) units of the manufacturinggood. Also assume the lab-equipment speciﬁcation for creating new machines as in Section15.7. Characterize the equilibrium of this economy and show that the qualitative featuresare the same as read more..

##### Page - 870

CHAPTER 21Structural Transformations and Market Failures inDevelopmentTogether with the process of economic development and the changes in the structure ofproduction, there is also a transformation of the economy, which both involves major so-cial changes and induces greater (and perhaps more “complex”) coordination of economicactivities. Loosely speaking, we can think of a society that is relatively developed as func-tioning along (or at any rate, nearer) the frontier of its production read more..

##### Page - 871

Introduction to Modern Economic Growthallocating workers to activities in which their marginal product is higher and also how a dualeconomy structure can emerge in equilibrium and slow down this reallocation process. Allthree of these sections are meant to give a ﬂavor of vast literatures dealing with issues ofﬁnancial development, the demographic transition, population growth, fertility, migration,urbanization, and other social changes taking place in the course of the development read more..

##### Page - 872

Introduction to Modern Economic Growthdemand externalities. In the model presented in this section, one of the multiple equilibriaapproximates a situation without industrialization and growth. Section 21.6 investigates theimportance of income inequality for economic development and shows how the interaction ofimperfect capital markets with income inequality can lead to multiple steady states, againwith diﬀerent levels of eﬃciency and productivity. I also use the models in this section read more..

##### Page - 873

Introduction to Modern Economic Growthand relaxation of credit market constraints might lead to better income and risk sharing. Onthe other hand, as the possibility of such risk-sharing arrangements reduce consumption risk,individuals might also take riskier actions, potentially aﬀecting the distribution of income. Acomplete analysis of the issues surrounding ﬁnancial development and its interactions witheconomic growth are beyond the scope of this chapter. As already hinted, existing read more..

##### Page - 874

Introduction to Modern Economic GrowthMoreover, we assume that agents can either save all of their labor earnings from the ﬁrstperiod of their lives using a safe technology with rate of return q (in terms of capital at thenext date) or invest all of their labor income in the risky technology with return Q + ε,whereε is a mean zero independently and identically distributed stochastic shock andQ>q.This implies that the risky technology is more productive. The assumption that individualshave read more..

##### Page - 875

Introduction to Modern Economic Growthsigniﬁcantly simpliﬁes the analysis of the model. Now suppose that the economy starts withsome initial capital stock of K (0). This implies that an individual with labor endowment liwill have labor earnings ofWi (0) = w (0) li,where(21.2)w (t)=(1 − α) K (t)αis the competitive wage rate at time t. After labor incomes are realized, individuals ﬁrstmake their savings decisions and then choose which assets to invest in. The preferences in(21.1) imply read more..

##### Page - 876

Introduction to Modern Economic Growthindependent of the rate of return on capital in the second period of the lives of the individuals,R. This is an implication of log preferences in (21.1).Now that we have determined the behavior of individuals concerning whether they willjoin the ﬁnancial market, we can determine the evolution of the economy by studying theevolution of individual earnings. Individual earnings are determined by two factors: individ-ual labor endowments and the capital stock read more..

##### Page - 877

Introduction to Modern Economic Growthsocieties diﬀer accordingtotheir ξs, which can be interpreted as a measure of theinstitutionally- or technologically-determined costs of monitoring or some cost of ﬁ-nancial transactions that depend on the degree of investor protection. Societies withlower ξs will have a greater participation in ﬁnancial markets and this will endoge-nously increase their productivity. Thus while the equilibrium behavior of ﬁnancialand economic development are read more..

##### Page - 878

Introduction to Modern Economic Growthis indeed a Kuznets curve in general and if so, whether the mechanism highlighted here playsan important role in generating this pattern are areas for future theoretical and empiricalwork.21.2. Fertility, Mortality and the Demographic TransitionChapter 1 highlighted the big questions related to growth of income per capita over timeand its dispersion across countries today. Our focus so far has been on these per capita incomediﬀerences. Equally striking read more..

##### Page - 879

Introduction to Modern Economic Growth1000000200000030000004000000Population0500100015002000YearEuopeLatin AmericaAfricaWestern OffshootsAsiaFigure 21.1. Totalpopulationindiﬀerent parts of the world over the past2000 years.of its dire prediction that population will adjust up or down (by births or deaths) until allindividuals are at the subsistence level of consumption. Nevertheless, this dire prediction isnot the most important part of the Malthusian model. Instead, at the heart of this model read more..

##### Page - 880

Introduction to Modern Economic Growth21.2.1. A Simple Malthusian Model.Consider the following non-overlapping gener-ations model. We start with a population of L (0) > 0 at time t =0. Each individual livingat time t supplies one unit of labor inelastically and has preferences given by(21.6)c (t)β∙y(t+1)(n(t+1)−1)−12η0n (t +1)2¸,where c (t) denotes the consumption of the unique ﬁnal good of the economy by the individualhimself, n (t +1) denotes the number of oﬀsprings the read more..

##### Page - 881

Introduction to Modern Economic Growthto their oﬀsprings. This, however, introduces another layer of complication, and since mypurpose here is to illustrate the basic ideas, I will follow the unsatisfactory assumption oftenmade in the literature, that land is owned by another set of agents, whose behavior will notbe analyzed here. The main important assumption made here is that those receiving landrents do not supply labor and/or their oﬀsprings do not make an important contribution read more..

##### Page - 882

Introduction to Modern Economic Growthto equation (21.8). This maximization problem immediately gives c (t)= w (t) andn (t +1) = (1 − α) L (t +1)−α η−10 .Now substituting for (21.8) and rearranging, we obtain(21.10)L (t +1) = (1 − α)11+αη−11+α0L (t)11+α.This equation implies that L (t +1) is an increasing concave function of L (t).In fact, thelaw of motion for population implied by (21.10) resembles the dynamics of capital-labor ratioin the Solow growth model (or the read more..

##### Page - 883

Introduction to Modern Economic GrowthThe second modiﬁcation is that there are now two production technologies that can be usedfor producing the ﬁnal good. The Malthusian (traditional) technology is still given by (21.7)and any worker can be employed with the Malthusian technology. The modern technology isgiven by(21.12)YM (t)= X (t) S (t) .This equation implies that productivity in the modern technology is potentially time varyingand also states that only skilled workers can be employed read more..

##### Page - 884

Introduction to Modern Economic Growthin the traditional sector and all skilled workers will work in the modern sector, we have wagesof skilled and unskilled workers at time t as(21.15)wU (t)=(1 − α) U (t)−α ,and(21.16)wS (t)= X (t) ,where (21.15) is identical to (21.9) in the previous subsection, except that it features onlythe unskilled workers instead of the entire labor force.Let us next turn to the fertility and quality-quantity decisions of individuals. As before,each individual will read more..

##### Page - 885

Introduction to Modern Economic GrowthThis equilibrium condition implies that there are two possible conﬁgurations. First, X (0)can be so low that (21.19) will hold as a strict inequality. In this case, all oﬀsprings will beunskilled. The condition for this inequality to be strict isX (0) η−11< (1 − α)2 L (1)−2α η−10 ,which uses the fact that when there are no skilled workers there is no production in the modernsector and thus X (1) = X (0). If this inequality satisﬁed, read more..

##### Page - 886

Introduction to Modern Economic Growthin the skills of their children and the rate of population growth declines. Ultimately, the rateof population growth approaches η−11 . Thus this model gives a very stylized representationof the demographic transition.In the literature, there are richer models of the demographic transition. For example,there are many ways of introducing quality-quantity tradeoﬀs in the utility function of theparents, and what spurs a change in the quality-quantity read more..

##### Page - 887

Introduction to Modern Economic Growthdevelopment. I will illustrate the main ideas by focusing on the process of migration fromrural areas and urbanization. Another reason to study migration and urbanization is that thereallocation of labor from rural to urban areas is closely related to the popular concept of thedual economy, which is an important theme of some of the older literature on developmenteconomics. According to this notion, less-developed economies consist of a modern sectorand a read more..

##### Page - 888

Introduction to Modern Economic Growthgainfully and productively employed together with the traditional sector where they are un-deremployed. The general tendency of less-developed economies to have higher levels of un-employment (and lower levels of employment to population ratios) was one of the motivationsfor Lewis’s model. A key feature of Lewis’s model is the presence of some barriers preventing,or slowing down, the allocation of workers away from the traditional sector towards read more..

##### Page - 889

Introduction to Modern Economic GrowthGiven competitive labor markets, the wage rates in the urban and rural areas at date tare given bywU (t)=∂F¡K(t),LU(t)¢∂Land wR (t)= BA.Let us assume that(21.25)∂F (K (0) , 1)∂L>BA,so that even if all workers are employed in the manufacturing sector at the initial capitalstock, they will have a higher marginal product than working in agriculture.Migration dynamics are assumed to take the following simple form:(21.26)˙LR (t)⎧⎨⎩= −μLR read more..

##### Page - 890

Introduction to Modern Economic Growthwhere urban and rural wages are equalized. Once this level is reached, migration will stop,and therefore ν (t) will remain constant. After this level of capital-labor ratio is reached,equilibrium dynamics will again be given by ˙k (t)= sf (k (t)) − δk (t). Therefore, the steadystate must always involve(21.29)sf (ˆk)ˆk= δ.For the analysis of transitional dynamics, which are our primary interest here, there areseveral cases to study. Let us focus on read more..

##### Page - 891

Introduction to Modern Economic GrowthTherefore, this discussion illustrates how a simple model of migration can generate richdynamics of population in rural and urban areas and wage diﬀerences between the modernand the traditional sectors.In dynamics discussed above, especially in the ﬁrst case, the economy has the ﬂavor of adual economy. Wages and the marginal product of labor are higher in the urban area thanin the rural area. If, in addition, μ is low, the allocation of workers from read more..

##### Page - 892

Introduction to Modern Economic Growth21.3.2. Community Enforcement, Migration and Development.Inow presentamodel inspired by Banerjee and Newman (1998) and Acemoglu and Zilibotti (1999). Banerjeeand Newman consider an economy where the traditional sector has low productivity but isless aﬀected by informational asymmetries and thus individuals can engage in borrowing andlending with limited monitoring and incentive costs. In contrast, the modern sector is moreproductive but informational read more..

##### Page - 893

Introduction to Modern Economic Growthand ﬁxed supply (so that there are diminishing returns to labor), and the production functionF satisﬁes our standard assumptions, Assumptions 1 and 2. Moreover, let us assume thatthe technology in the manufacturing sector evolves according to the diﬀerential equation˙X (t)= ηLU (t) X (t)ζ ,where ζ ∈ (0, 1). This equation builds in learning-by-doing externalities along the lines ofRomer’s (1986) paper and is also similar to the industrialization read more..

##### Page - 894

Introduction to Modern Economic Growthclearing implies thatX (t) ˜φ¡LU(t)¢=BA+ξ,orLU (t)=˜φ−1µBA+ξX (t)¶≡ φµ X (t)BA + ξ¶,where again the second line deﬁnes the function φ, which is strictly increasing in view of thefact that ˜φ (and thus ˜φ−1) was strictly decreasing. Therefore, the evolution of this economycan be represented by the diﬀerential equation˙X (t)= ηφµ X (t)BA + ξ¶X(t)ζ.time1Fraction of population living in the cityFigure 21.3. The dynamic read more..

##### Page - 895

Introduction to Modern Economic Growthsector, and associated with that the migration of workers from rural to urban areas will alsofollow an S-shaped pattern.Second and more importantly, the process of technological change in the manufacturingsector and migration to the cities are slowed down by the comparative advantage of the ruralareas in community enforcement. In particular, the greater is ξ, the slower is technologicalchange and migration into urban areas. Since employment in the urban read more..

##### Page - 896

Introduction to Modern Economic Growthexactly equal to h (for example, the skilled workers will be the managers or the supervisorsfor the unskilled workers). Suppose Ah is increasing in h, so that more advanced technologiesare more productive.Now consider a less-developed economy that has access to all technologies Ah for h ∈£0,¯h¤for some ¯h< ∞. Suppose that the population of this economy consists of H skilled and Lunskilled workers, such that H/L < ¯h. This inequality implies read more..

##### Page - 897

Introduction to Modern Economic Growthin situations where less-developed economies import their technologies from more advancednations and these technologies are inappropriate to the needs of less-developed countries.Models of dual economy based on this type of appropriate technology ideas have not beeninvestigated in detail, though the literature on appropriate technology, which was discussedin Chapter 18, suggests that they may be important in practice. While this model focuses onthe dual read more..

##### Page - 898

Introduction to Modern Economic Growthcompetitive fringe forces the monopolist to charge a limit price:(21.33)p (ν, t)= χ> 1.Naturally, this limit price conﬁguration will be an equilibrium when χ is not so high thatthe monopolist prefers to set a lower unconstrained monopoly price. The condition for thisis simplyχ ≤ 1/α,which I impose throughout. Broadly, one can think of the parameter χ as capturing bothtechnological factors and government regulations regarding competitive policy. read more..

##### Page - 899

Introduction to Modern Economic Growthwhere η and ¯γ will be deﬁned below.We assume that the process of imitation and innovation leads to the following law ofmotion of each monopolist’s productivity:(21.37)A (ν, t)= η ¯A (t − 1) + γA (t − 1) + ε (ν, t) ,where η> 0 and γ> 0,and ε (ν, t) is a random variable with zero mean, capturing diﬀerencesin innovation performance across ﬁrms and sectors.In equation (21.37), η ¯A (t − 1) stands for advances in productivity read more..

##### Page - 900

Introduction to Modern Economic Growthlow skill. This growth strategy will involve a high degree of turning (creative destruction)and a large number of young entrepreneurs (as older unsuccessful entrepreneurs are replacedby new young entrepreneurs). The second strategy maintains experienced entrepreneurs inplace even when they are have low skills. This strategy therefore involves an organizationof ﬁrms relying on “longer-term relationships” (here between entrepreneurs in the creditmarket), read more..

##### Page - 901

Introduction to Modern Economic Growththere is greater transfer of technology from the world technology frontier. The ﬁnal part ofthis assumption, that γ < 1+ g simply ensures that imitation-based growth will not leadto faster growth than the world technology frontier. Also in terms of (21.39), we can inter-pret the assumption (21.36) as stating that the world technology frontier advances due toinnovation-based growth strategy, which is natural, since a country at the world read more..

##### Page - 902

Introduction to Modern Economic Growthsheltered from the competition of younger entrepreneurs and this may enable the economyto make better use of the experience of older entrepreneurs or to ﬁnance greater investmentsout of the retained earnings of incumbent entrepreneurs. In contrast, the innovation-basedregime is based on an organizational form relying on greater selection of entrepreneurs andplaces greater emphasis on maximizing innovation at the expense of experience, imitationand read more..

##### Page - 903

Introduction to Modern Economic Growth45ºa(t)1a(t+1)âR = 1R = 0ar(δ)Figure 21.5. Dynamics of the distance to frontier in the underinvestment equilibrium.(2) Underinvestment equilibrium: the second potential equilibrium conﬁguration in-volves the following equilibrium organizational form:R (t)=⎧⎨⎩1 if a (t − 1) <ar (δ)0 if a (t − 1) ≥ ar (δ)where ar (δ) < ˆa. Figure 21.5 depicts this visually, with the thick black lines corre-sponding to the equilibrium law of motion of read more..

##### Page - 904

Introduction to Modern Economic GrowthR (t)= 1, but because the appropriability eﬀect discourages investments, there is aswitch to the innovation-based equilibrium and the associated organizational formsearlier than the growth-maximizing threshold.A notable feature is that although the equilibrium is diﬀerent from the previouscase, it again follows the sequence of R =1 followed by a structural transformationand a switch to greater competition among and selection of entrepreneurs withthe read more..

##### Page - 905

Introduction to Modern Economic Growth45ºa(t)1a(t+1)âR = 1R = 0ar(δ)Figure 21.6. Dynamics of the distance to frontier in the sclerotic equilibrium.The resulting pattern in this case is drawn in Figure 21.6. Now the economyfails to achieve the maximum growth rate for a range of values of a such that a ∈(ˆa, ar (δ)). In this range, the innovation-based regime would be growth-maximizing,but the economy is stuck with the imitation-based regime because of the retainedearnings and the power of read more..

##### Page - 906

Introduction to Modern Economic Growth45ºa(t)1a(t+1)âR = 1R = 0ar(δ)atrapFigure 21.7. Dynamics of the distance to frontier in a non-convergence trap.If the economy starts with a (0) <atrap, it fails to converge to the worldtechnology frontier and instead converges to atrap.never transitions to the innovation-based equilibrium. This not only retards growthfor a temporary interval but pushes the economy into a non-convergence trap. Inparticular, this is the only equilibrium pattern in which read more..

##### Page - 907

Introduction to Modern Economic Growthstrategies will be pursued. Thus, some degree of government intervention might be useful.However, the third and the fourth cases also emphasize that government intervention canhave fairly negative unintended consequences. Such intervention will improve growth perfor-mance during a limited period of time (in the second scenario when a ∈ (ar (δ) , ˆa), but it cancreate much more substantial costs by leading to a non-convergence trap as shown in Figure21.7. read more..

##### Page - 908

Introduction to Modern Economic Growthlead to multiple state states, whereby once an economy ends up in a steady state with loweconomic activity, it may get stuck there (and there is no possibility of a coordination tojump to the other steady state). Models with multiple steady states will be discussed in thenext section, where I will return to a further discussion of the diﬀerence between multipleequilibria and multiple steady states.Murphy, Shleifer and Vishny consider the following read more..

##### Page - 909

Introduction to Modern Economic Growthpresent in our workhorse endogenous growth models. As usual ε is the elasticity of substitu-tion between intermediate goods within a given period and is assumed to be strictly greaterthan one, i.e., ε> 1.The production functions of intermediate goods in the two periods are as follows:y (ν, 1) = l (ν, 1)and(21.42)y (ν, 2) =½ l (ν, 2)with old technologyαl (ν, 2) with new technologywhere α> 1 and l (ν, t) denotes labor devoted to the production read more..

##### Page - 910

Introduction to Modern Economic GrowthConsequently, when the technology is not adopted, we haveY (2) = Land when the technology is adopted by all the ﬁrms, we haveY (2) = αL.We now turn to the pricing decisions. In the ﬁrst date, the designated producers have nomonopoly power because of the competitive fringe, thus they charge price equal to marginalcost, which is w (1), and make zero proﬁts. Since total output is equal to Y (1) = L,this alsoimplies that the equilibrium wage rate is equal read more..

##### Page - 911

Introduction to Modern Economic GrowthThis expression is useful in laying the foundations for the aggregate demand externalities,which we will discuss soon; the demand for intermediate ν depends on the total amount ofproduction, Y (2).3 The familiar feature of the demand curve (21.46) is that it is iso-elastic.To make further progress, ﬁrst imagine the situation in which there is no fringe of competitiveproducers. In that case, each designated producer will act as an unconstrained read more..

##### Page - 912

Introduction to Modern Economic Growththe market, and thus increase its proﬁts. This implies that under (21.47), each monopolistwould make per unit proﬁts equal tow (2) −w (2)α=α − 1αw (2) .The proﬁts of ﬁrms are then obtained from substituting from (21.46) as:(21.48)π (2) =α − 1αw (2)1−ε Y (2) .Thewageratecan be determinedfromincomeaccounting. Total production will be equalto Y (2) = αL, and this has to be distributed between proﬁts and wages, thusα − 1αw (2)1−ε read more..

##### Page - 913

Introduction to Modern Economic Growthfact that w (2) = 1,proﬁts at date 2 areπN (2) =α − 1αL.where the superscript N denotes that no other ﬁrm is undertaking the investment. Therefore,the net discounted proﬁts at date 1 for the ﬁrm in question is∆πN= −F +1ˆRα − 1αL= −F + βα − 1αL.Next consider the case in which all other ﬁrms are undertaking the investment. In thiscase, proﬁts at date 2 areπI (2) = (α − 1) L,where the superscript I designates that all other read more..

##### Page - 914

Introduction to Modern Economic GrowthIt is also straightforward to see that whenever both equilibria exist, the equilibrium withinvestment Pareto dominates the one without investment, since condition (21.51) implies thatall households are better-oﬀ with the upward sloping consumption proﬁle giving them higherconsumption at date 2 (see Exercise 21.8). Therefore, this analysis establishes that whencondition (21.51) is satisﬁed, there will exist two pure strategy SSPE. In one of these, read more..

##### Page - 915

Introduction to Modern Economic Growthhouseholds (and ﬁrms) better oﬀ. Instead, it is much more likely that the ideas related toaggregate demand externalities (or other potential forces leading to multiple equilibria) aremore important as sources of persistence or as mechanisms generating multiple steady states(while still maintaining a unique equilibrium path). In the next section, we will discuss howcertain factors can lead to multiple steady states in dynamic models instead of the read more..

##### Page - 916

Introduction to Modern Economic Growthat the end of adulthood. Preferences are given by(1 − δ)log ci (t)+ δ log ei (t +1)where c is consumption at the end of the individual’s life, and e is the educational spendingon the oﬀspring of this individual. The budget constraint isci (t)+ ei (t +1) ≤ wi (t) ,where w is the wage income of the individual. Notice that preferences here have the “warmglow” type altruism which we encountered in Chapter 9 and in Section 21.2 above. Inparticular, read more..

##### Page - 917

Introduction to Modern Economic GrowthNow, let us look at the dynamics of human capital for a particular dynasty i.If at time0, we have hi (0) < (δA)−1, then (21.53) implies that ei (t) < 1,so the oﬀspring will havehi (1) = ¯h. Given (21.54), we have hi (1) = ¯h< (δA)−1, and repeating this argument, wehave hi (t)= ¯h< (δA)−1 for all t. Therefore, a dynasty that starts with hi (0) < (δA)−1 willnever reach a human capital level greater than read more..

##### Page - 918

Introduction to Modern Economic Growthto human capital or the interest rate) that are being determined in equilibrium here. For thisreason, dynamics in this type of models are sometimes described as “Markovian”–becausethey are summarized by a Markov process without any general equilibrium interactions.Markovian dynamics are much more tractable than dynamics of inequality depending onequilibrium prices. An example of this richer type of model is given in Exercise 21.13.The most important read more..

##### Page - 919

Introduction to Modern Economic Growththe economy. But also, because multiple steady states are possible, the model can be usefulfor thinking about potential development traps.Aside from providing us with a simple example of multiple steady states, this modelshows the importance of the distribution of income in an economy with imperfect creditmarkets (here with no credit markets). In particular, the distribution of income aﬀects whichindividuals will be unable to invest in human capital read more..

##### Page - 920

Introduction to Modern Economic Growthwho will use the monetary bequests to invest in education. Also, the logarithmic formulationwill once again ensure a constant saving rate equal to δ.Education is a binary outcome, and educated (skilled) workers earn wage ws while un-educated workers earn wu. The required education expenditure to become skilled is h,andworkers acquiring education do not earn the unskilled wage, wu, during the ﬁrst period oftheir lives. The fact that education is binary read more..

##### Page - 921

Introduction to Modern Economic GrowthComparing these expressions we obtain that an individual likes to invest in education ifandonlyifx ≥ f ≡(2 + r) wu +(1+ i) h − wsi − rThe dynamics of the system can then be obtained simply by using the bequests of uncon-strained, constrained-investing and constrained-non-investing agents.More speciﬁcally, the equilibrium correspondence describing equilibrium dynamics is(21.56)x (t +1) =⎧⎨⎩bu (x (t)) = δ ((1 + r)(wu + x (t)) + wu)if x (t) read more..

##### Page - 922

Introduction to Modern Economic GrowthAll individuals with x (t) <x∗ converge to the wealth level ¯xU , while all those withx (t) >x∗ converge to the greater wealth level ¯xS. As in the example without credit markets,there is a “poverty trap,” which attracts agents with low initial wealth. The distribution ofincome again has a potentially ﬁrst-order eﬀect on the income level of the economy. If themajority of the individuals start with x (t) <x∗, the economy will have low read more..

##### Page - 923

Introduction to Modern Economic Growththis stationary distribution will exhibit a large amount of persistence.5 In contrast, modelsin which prices determined in general equilibrium aﬀect wealth (income) dynamics generatemore robust multiplicity of steady states. The second potential shortcoming of the currentmodel is that it focuses on human capital investments. Some development economists, suchas Banerjee and Newman (1994), believe that the eﬀect of income inequality on occupationalchoices read more..

##### Page - 924

Introduction to Modern Economic Growthgeneral enough to allow for the case in which greater inequality is productivity-enhancing. Inparticular, even though this aggregator looks like the constant elasticity of substitution pro-duction function we have used many times in this book before, in contrast to that productionfunction, it is deﬁned for σ< 0 as well (whereas recall that the Dixit-Stiglitz aggregator isonly deﬁned for σ ≥ 0, see Exercise 21.14). When σ< 0, greater inequality read more..

##### Page - 925

Introduction to Modern Economic Growththe oﬀsprings. This structure of spillovers may be viewed as quite plausible, for example,because the presence of some low human capital children will slow down learning by those withhigher potential (or because one “bad apple” will spoil the pack). This type of neighborhoodspillovers may then suggest that segregation of high and low human capital parents in diﬀerentneighborhoods might be beneﬁcial for human capital accumulation. Whether or not read more..

##### Page - 926

Introduction to Modern Economic Growthmay reduce long-run inequality leading to a better distribution of income and thus to bettereconomic outcomes as a result.With full segregation, it is straightforward to see that (see Exercise 21.16)mt+1 =ln B −ω22+(α + β + γ) mt + γµσ−1σ¶∆2t2(21.63)∆2t+1=(α + β)2 ∆2t + ω2whereas with full integration, we haveˆmt+1 =ln B −ω22+(α + β + γ)ˆmt +∙γµσ−1σ¶+βµε−1ε¶¸ ˆ∆2t2(21.64)ˆ∆2t+1= α2ˆ∆2t + ω2,where ˆmt read more..

##### Page - 927

Introduction to Modern Economic Growthconﬁrming that there will be greater inequality of human capital and income in this societywith segregation of neighborhoods. The mean of the two distributions will also be diﬀerenthowever. Let us suppose that α + β + γ< 1, so that this steady state distribution existsunder both full segregation and full integration. Then we havem∞ =11 − (α + β + γ)⎡⎣lnB−ω22+γµσ−1σ¶ ω22³1−(α+β)2´⎤⎦,andˆm∞ =11 − (α + β + read more..

##### Page - 928

Introduction to Modern Economic Growth21.7. Towards a Uniﬁed Theory of Development and Growth?There has been a uniﬁed theme to the models discussed in this chapter (and even be-tween those discussed in this and the previous chapter). They have either emphasized thetransformation of the economy and the society over the process of development or potentialreasons for why such a transformation might be halted. This transformation takes the formof the structure of production changing, the process read more..

##### Page - 929

Introduction to Modern Economic Growthdevelopment into a single model will lead to a framework that is complicated and involvedand I believe that relatively abstract representations of reality are more insightful. Second,the economic growth and development literatures have not made great progress towards suchuniﬁed model. So while I believe there is room for thinking and constructing such uniﬁedtheories of economic development, I do not think that one (or at least I) can do justice tothis read more..

##### Page - 930

Introduction to Modern Economic Growthwhere I have suppressed population growth and there is no technological change for simplicity.For a ﬁxed x, capital naturally accumulates in an identical fashion to that in the basic Solowmodel. The structure of this economy is slightly more involved because x (t) also changes.Diﬀerential equations (21.66) and (21.67) provide a simple reduced-form representation ofstructural change driven by economic growth (capital accumulation).To illustrate the types read more..

##### Page - 931

Introduction to Modern Economic Growthx(t)k(t)k(t) = 0k*x*.x(t) = 0.Figure 21.10. Capital accumulation and structural transformation withoutany eﬀect of the “social variable” x on productivity.x(t)k(t)k(t) = 0k*x*.x(t) = 0.k´x´k´´x´´Figure 21.11. Capital accumulation and structural transformation withmultiple steady states.locally stable, so that starting from the neighborhood of one, the economy will converge to thenearest steady state and will tend to stay there. This highlights read more..

##### Page - 932

Introduction to Modern Economic Growthcorresponding to a “development trap”. Interestingly, this development trap is, at least inpart, caused by lack of structural change (i.e., a low value of the social variable x).Figure 21.11 makes it clear that such multiplicity requires the locus for ˙k (t) /k (t)=0 tobe relatively ﬂat, at least over some range. Inspection of equation (21.67) shows that this willbe thecasewhen fx (k, x) is large, at least over some range. Intuitively, multiple read more..

##### Page - 933

Introduction to Modern Economic Growthin the models we have analyzed in Section 17.6 in Chapter 17 and in Sections 21.3 and 21.1of this chapter. This increase in productivity leads to faster capital accumulation and thereis a self-reinforcing (“cumulative”) process of development, with economic growth leading tostructural changes facilitating further growth. However, since the eﬀect of x on productivityis limited, this process ultimately takes us towards a unique steady state.This read more..

##### Page - 934

Introduction to Modern Economic Growthgrowth literature to the forefront of analysis. These include, among other things, the organi-zation of ﬁnancial markets, the distribution of income and wealth, and issues of incentives,such as problems of moral hazard, adverse selection and incomplete contracts both in creditmarkets and in production relationships; unfortunately, space restrictions have precluded mefrom providing a satisfactory discussion of these issues, and instead, I had to read more..

##### Page - 935

Introduction to Modern Economic Growthliterature looking at the eﬀect of ﬁnancial development on economic growth. An excellentsurvey of this literature is provided in Levine (2005). Some of the most well-known empiricalpapers include King and Levine (1993), which documents the cross-country correlation be-tween measures of ﬁnancial development and economic growth, Rajan and Zingales (1998),which shows that lack of ﬁnancial development has particularly pernicious eﬀects on sectorsthat read more..

##### Page - 936

Introduction to Modern Economic Growthsome of the economic literature in this area. The ﬁrst model in Section 21.3 builds on ArthurLewis’s (1954) classic, which argued that early development can be viewed as a situation inwhich there is surplus labor available to the modern technology, thus growth is constrained bycapital and technology but not by labor. A formalization of Lewis’s ideas naturally takes usto the realm of the dual economy, since surplus labor for the modern technology can read more..

##### Page - 937

Introduction to Modern Economic Growthstates. Matsuyama (1996) also provides a very clear discussion of why pecuniary externalitiescan lead to multiple equilibria in the presence of monopolistic competition.The distinction between multiple equilibria and multiple steady states is discussed inKrugman (1994) and Matsuyama (1994). Both of these papers highlight that in modelswith multiple equilibria, expectations determine which equilibrium will be played, while withmultiple steady states, there read more..

##### Page - 938

Introduction to Modern Economic Growthinstead of (21.8), where ε (t) is a random variable that takes one of two values, 1 −ε or 1+ ε,reﬂecting random factors aﬀecting population growth. Characterize the stochastic equilib-rium. In particular, plot the stochastic correspondence representing the dynamic equilibriumbehavior and analyze how shocks aﬀect population growth and income dynamics.Exercise 21.4. Characterize the full dynamics of migration, urban capital-labor ratio andwages in read more..

##### Page - 939

Introduction to Modern Economic Growthproﬁts with the manager (owner) of the ﬁrm to which certain tasks have been outsourced(whereas in a vertically integrated structure, he can keep the entire revenue).(1) Determine the proﬁt-maximizing outsourcing decision for an entrepreneur as a func-tion of the (inverse) distance to frontier a (t). In particular, show that there existsa threshold ¯a such that there will be vertical integration for all a (t) ≤ ¯a and out-sourcing for all a (t) > read more..

##### Page - 940

Introduction to Modern Economic Growthsense that depending on their initial conditions some dynasties become high skilledand others become low skilled.(2) Now suppose that ε is distributed with support [−ψ,∞),where ψ ≤ wu. Show thatin this case there is a unique ergodic distribution of wealth and no poverty trap (inthe sense that every dynasty will become skilled at some point with probabability1). Explain why the results here are diﬀerent from those presented in subsection21.6.2?(3) read more..

##### Page - 941

Introduction to Modern Economic Growthdynamics. The utility of each individual again depends on consumption and bequest, with(1 − δ)−(1−δ) δ−δc1−δbδ − zwhere z denotes whether the individual is exerting eﬀort, with cost of eﬀort normalized to1. Each agent chooses one of four possible occupations. These are (1) subsistence and nowork, which leads to no labor income and has a rate of return on assets equal to ˆr< 1/δ;(2) work for a wage v; (3) self-employment, which read more..

##### Page - 942

Introduction to Modern Economic Growth(8) Show that if Gt (w∗) <μ [1 − Gt (w∗∗)], there will be an excess demand for laborand the equilibrium wage rate will be v =¯v.(9) Now derive the individual wealth (bequest) dynamics (for a worker with wealthw) as follows: (1) subsistence and no work b (t)= δˆrw;(2) worker: b (t)=δ (ˆrw + v);(3) self-employment: b (t)= δ (¯rI +ˆr (w − I)); (4) entrepreneurship:b (t)= δ (¯rμI +ˆr (w − μI) − μv). Explain the intuition for each read more..

##### Page - 943

Introduction to Modern Economic Growthwhere ci (t) is the consumption of the individual and ei (t +1) is educational investment inthe human capital of the oﬀspring. Each individual has some earned income wi (t) and issubject to the budget constraint ci (t)+ei (t +1) ≤ wi (t). The human capital of the oﬀspring,hi (t +1), is given by equation (21.52) in subsection 21.6.1 with ¯h =1 and γ =1.There isalso to continue 1 of ﬁrms, each with the production functionyi,j = A (kj)α (hi)1−α read more..

##### Page - 944

Introduction to Modern Economic Growth(4) Derive a diﬀerence equation for φ (t) using the optimal capital investment level ofﬁrms k∗ [μt, η,β,R] derived in part 2 and the preferences of individuals regardinginvestments in their oﬀspring’s human capital.(5) Prove that there exists some ¯φ ∈ (0, 1), such that if φ (0) > ¯φ, then the dynamicequilibrium involves φ (t) → 1 and the economy achieves a constant growth rate.In contrast, if φ (0) < ¯φ,then h1 (t) →¯h read more..

##### Page - 945

##### Page - 946

Part 8Political Economy of Growth read more..

##### Page - 947

In this part of the book, I turn from the mechanics of economic growth to an investigationof potential causes of economic growth. Almost all of the models we have studied so fartake economic institutions (such as whether property rights are enforced and what types ofcontracts can be written), policies (such as tax rates, distortions, subsidies) and often themarket structure as given. They then derive implications for economic growth and cross-country income diﬀerences. While these models read more..

##### Page - 948

Introduction to Modern Economic Growthpowers. For example, the nature of political conﬂict and the resulting political econ-omy equilibrium is likely to be diﬀerent in a society where much of the land and thecapital stock is concentrated in the hands of a few individuals and families than onein which there is a more equitable distribution of resources. We would also expectpolitics to function diﬀerently in a society where the major assets are in the form ofhuman capital vested in read more..

##### Page - 949

Introduction to Modern Economic Growthhave preferences over political institutions. However, economic institutions and policies alsohave a direct eﬀect on economic outcomes (for example, as illustrated by the eﬀects of taxpolicies, regulation and contracting institutions in previous chapters). Thus we may plausiblyexpect that the major determinant of individual preferences on economic institutions (andpolicies) will be the allocations that result from these arrangements. Based on this read more..

##### Page - 950

Introduction to Modern Economic Growthwords, we presume that individuals are purely consequentialist (and thus ignore any directbeneﬁts they may obtain from institutions). Then their preferences over some economicinstitution R ∈ R is simply given by ui (ρ (R)) ≡ ui ◦ ρ : R → R. This mapping thereforecaptures their induced preferences over economic institutions (as a function of the economicallocations that these institutions will lead to). Preferences on political institutions are read more..

##### Page - 951

##### Page - 952

CHAPTER 22Institutions, Political Economy and GrowthThis chapter will make a ﬁrst attempt towards answering the following fundamentalquestion that has been in the background of much of what we have done so far: why dosimilar societies choose diﬀerent institutions and policies, leading to very diﬀerent economicgrowth outcomes?Our analysis so far has highlighted the role of capital accumulation, human capital andtechnology in economic growth. We have investigated the incentives to accumulate read more..

##### Page - 953

Introduction to Modern Economic Growthcomplementary reasons: (1) the desire of individuals or social groups to transfer resources tothemselves using limited ﬁscal instruments, and (2) the potential conﬂict between diﬀerentsocial groups in the marketplace or in the political arena. I will try to highlight why thesetwo sources of distortionary policies are distinct and also argue that the second source ofdistortions is typically more costly for growth.I will conclude the chapter by pointing read more..

##### Page - 954

Introduction to Modern Economic GrowthWe summarize these variations across societies as “institutional diﬀerences”(or diﬀer-ences in institutions and policies). The term is slightly unfortunate, but is one that is widelyused and accepted in the literature. Institutions mean many diﬀerent things in diﬀerentcontexts, and none of these exactly correspond to the meaning intended here. As alreadyemphasized in Chapter 4, by institutional diﬀerences, we are referring to diﬀerences in read more..

##### Page - 955

Introduction to Modern Economic Growthask the next question: if economic institutions matter so much for economic growth, why dosome societies chooses institutions that do not encourage growth? In fact, based on availablehistorical evidence we can go further: why do some societies choose institutions and policiesthat speciﬁcally block technological and economic progress? The rest of this chapter andmuch of the next chapter will try to provide a framework for answering these questions. Istart read more..

##### Page - 956

Introduction to Modern Economic Growthto retain entry barriers and consumers that wish to dismantle them, it will be the equilib-rium of a political process that decides the outcome. This process may be “orderly” as indemocracies, or disorderly or even chaotic as in other political regimes as illustrated by theall too frequent civil wars throughout human history. Whether it is a democratic or a non-democratic process that will lead to the equilibrium policy, the political power of the read more..

##### Page - 957

Introduction to Modern Economic GrowthPareto ineﬃcient allocation, because there will often exist future policies that can make allparties better-oﬀ, but those policies will not arise as part of the equilibrium.Consider a situation in which political power is in the hands of a speciﬁc group or anindividual–the political elite. To simplify the thought experiment, let us ignore for now anyconstraints on the exercise of this political power (this is essentially where we will begin inthe read more..

##### Page - 958

Introduction to Modern Economic Growththe case, they will use distortionary policies to impoverish their political competitors. I willrefertothisasthe political replacement motive for distortionary policies. The rest of thechapter will illustrate how distortionary policies can be adopted for extracting resources fromdiﬀerent social groups and for factor price manipulation and political replacement motives.An interesting implication of the models I will present will be that factor price read more..

##### Page - 959

Introduction to Modern Economic Growthto block economic growth. The major comparative static exercises will look at the eﬀects ofthe nature of production technology, the distribution of resources within the society, whetherthe politically powerful compete with economically productive agents in factor markets, theextent of holdup problems, the importance of natural resources and whether or not politicalpower is contested. While the ﬁeld of the political economy of growth is still in its read more..

##### Page - 960

Introduction to Modern Economic Growthmade by entrepreneurs will highlight the impact of distributional conﬂict(giventhe setofﬁscal instruments) on equilibrium policies and production in the sharpest possible way. Thepresence of three groups is important for the modeling of the eﬀect of competition between theelite and other producers in the labor market in Sections 22.3 and 22.4. The model will thenbe enriched in various diﬀerent ways in this and the next chapter by introducing read more..

##### Page - 961

Introduction to Modern Economic Growthor group (though when referring to groups, I will use i as superscript, and when referring toindividuals, as subscript). The identity of the agents (their social group membership) doesnot change over time.Each entrepreneur (middle-class agent) i ∈ Sm has access to the following productiontechnology for producing the ﬁnal good:(22.2)Yi (t)= F (Ki (t) ,Li (t)) ,where Yi (t) is ﬁnal output produced by entrepreneur i, Ki (t) and Li (t) are the total read more..

##### Page - 962

Introduction to Modern Economic Growththat there are no transitional dynamics; irrespective of initial conditions, each entrepreneurwill immediately choose the capital-labor ratio k∗ as in (22.4).Another special feature of this economy is that it may fail to achieve full employment.Recall that the total labor force is equal to 1. However, equation (22.5) shows that the levelof employment of each employer may be strictly less than 1/θm because of the maximumsize constraint on ﬁrms. When this read more..

##### Page - 963

Introduction to Modern Economic GrowthAs for policies, we assume that the society has access to four diﬀerent policy instrumentsat each date t:• a linear tax rate on output τ (t) ∈ [0, ¯τ ],where ¯τ ∈ (0, 1] is a maximum tax rate thatmay be imposed constitutionally or technologically (for example, when the tax rateis above this level, all activity ﬂees into the informal sector). In this and the nexttwo sections, we take ¯τ =1, so that any tax rate is allowed. We will later read more..

##### Page - 964

Introduction to Modern Economic Growthprocess announces the tax rate τ (t +1) that will apply at the next date and entrepreneurs,after observing this tax rate, choose their capital stocks for the next date, [Ki (t +1)]i∈Sm.The important feature in this timing of events is that entrepreneurs know exactly what taxrate they will face when choosing their capital stock. The alternative, where the capital stockis chosen before the tax rate, will be discussed in Section 22.3. For now it suﬃces to read more..

##### Page - 965

Introduction to Modern Economic GrowthThis expression makes use of the fact that preferences are linear, thus the value of theentrepreneur can be written simply in terms of the discounted sum of his consumption.His consumption, on the other hand, is simply given by the term in square brackets, sinceoutput is taxed at the rate τ (t) at time t and moreover, a fraction (1 − δ) of last period’scapital stock is left, so an additional investment of (Ki (t +1) − (1 − δ) Ki (t)) is made read more..

##### Page - 966

Introduction to Modern Economic Growthimmediately jumped to its state-state value, this may no longer be the case in an economicequilibrium given the policy sequence pt, since the policy sequence may involve time-varyingtaxes.The most noteworthy feature of the equilibrium capital-labor ratio given in (22.10) isthat, thanks to linear preferences, the choice of the capital-labor ratio by each entrepreneurat time t+1 only depends on the tax rate τ (t +1), and not on future taxes. We can read more..

##### Page - 967

Introduction to Modern Economic Growthexpectations concerning which of these equilibrium would be played conditional on thesepolicy choices would complicate the analysis.122.2.3. Political Economy under Elite Control.As noted at the beginning of thissection, our task of characterizing the political economy equilibrium here is considerablysimpliﬁed by the assumption that political power is entirely in the hands of the elite. Thereis no issue of political power changing hands or the elites read more..

##### Page - 968

Introduction to Modern Economic Growthcould increase their consumption and utility by increasing transfers to themselves), to obtainTe (t)=1θeτ (t)ZSmF (Ki (t) ,Li (t)) di=1θe τ (t) f³ˆk(τ(t))´,(22.14)where the ﬁrst line simply uses the government budget constraint (22.8), while the secondline uses the equilibrium characterization in Proposition 22.1 together with the fact that withfull employment, the total number of workers is equal to 1.The problem of maximizing the utility of the read more..

##### Page - 969

Introduction to Modern Economic Growthany generality, focus on the maximization problem in (22.15). We will see in the next sectionthat this is not always the case and we need to take into account the sequential nature of thedecision-making by the elite and the entrepreneurs.This analysis so far has thus established the following characterization of the politicaleconomy equilibrium:Proposition 22.2. Suppose that (22.6) holds. Then for any initial distribution of capi-tal stocks among read more..

##### Page - 970

Introduction to Modern Economic Growthutility of the middle-class entrepreneurs and the workers without making the elite worse-oﬀ.2This is an important observation, since it implies that when we explicitly incorporate politicaleconomy aspects into the analysis, there are typically no “free lunches”–that is, no way ofmaking all agents better-oﬀ. This is the reason why political economy considerations typi-cally involve tradeoﬀs between losers and winners in the process of various read more..

##### Page - 971

Introduction to Modern Economic Growth22.2.4. The Canonical Cobb-Douglas Model of Distributional Conﬂict.Con-sider a slightly specialized version of the economy analyzed so far, with two diﬀerences. First,the production function of each entrepreneur takes the following Cobb-Douglas form:(22.17)Yi (t)=1α(Ki (t))α(Ai (t) Li (t))1−α,where Ai (t) is a labor-augmenting group-speciﬁc or individual-speciﬁc productivity term,which will be used later in this chapter. For now, we can set Ai read more..

##### Page - 972

Introduction to Modern Economic GrowthBoth the tractability aﬀorded by the Cobb-Douglas production function and the linkbetween the concavity of the production function and equilibrium taxes that it highlightsmake this a very useful framework, which we will next use in a number of applications below.22.3. Distributional Conﬂict and CompetitionIn this and the next section, I will use the canonical framework with Cobb-Douglasproduction functions and full depreciation of capital (δ =1) to read more..

##### Page - 973

Introduction to Modern Economic GrowthSince there are entrepreneurs both from the elite and the middle class, the condition forfull employment is diﬀerent from (22.6). In particular, we assume throughout that θe¯L< 1and θm ¯L< 1, so that neither of the two groups generates enough labor demand by itself toemploy the entire labor force. The following condition then determines whether the elite andthe middle class together will generate enough labor demand for the entire labor read more..

##### Page - 974

Introduction to Modern Economic Growth22.3.1. Competition in the Marketplace: The Factor Price Manipulation Ef-fect.The next proposition is the equivalent of Proposition 22.2, except that it now applieswhen Condition 22.1 fails to hold. The reason for this is that, when this condition holds, therewill also be the competition eﬀect, changing the policy preferences of the elite. Propositions22.4 and 22.5 below will focus on the implications of competition in the factor market.Proposition 22.4. read more..

##### Page - 975

Introduction to Modern Economic Growthmanipulate factor prices. Condition 22.2 is necessary, since otherwise even at the maximaltax rate ¯τ , the middle class entrepreneurs are more productive than the elite and the elitemake zero proﬁts. The noteworthy conclusion of Proposition 22.5 is that the equilibriumtaxrateinthis case, τ FPM , is greater than the tax rate when the only motive for taxationwas revenue extraction (τ RE ). This might at ﬁrst appear paradoxical, but is quite read more..

##### Page - 976

Introduction to Modern Economic Growthsubject to (22.22) and(22.25)θeLe (t)+ θmLm (t)=1, and(22.26)Lm (t)= ¯Lif(1 − τm (t))1/(1−α)Am ≥ Ae,where Lm(t) denotes equilibrium employment by a middle-class entrepreneur and Le (t) isequilibrium employment by an elite entrepreneur. The ﬁrst term in (22.24) is the elite’snet revenues and the second term is the transfer they receive. Equation (22.25) is the labormarket clearing constraint, while (22.26) ensures that middle class producers read more..

##### Page - 977

Introduction to Modern Economic Growthextraction. Therefore, the factor price manipulation motive always increases taxes above thepure revenue maximizing level, and thus beyond the peak of the Laﬀer curve (though neverto as high as 100%). Naturally, if this level of tax is greater than ¯τ , the equilibrium tax willbe ¯τ .Second, since Proposition 22.6 incorporates both the revenue extraction and the factorprice manipulation motives, it contains the main comparative static results of read more..

##### Page - 978

Introduction to Modern Economic Growtheconomy eﬀects arise, however, when the probability that the elite will lose power is endoge-nous. To save space while communicating the main ideas, I use a very reduced-form modeland assume that the probability that the elite will lose power to the middle class is a functionof the net income level of the middle class, in particular,(22.30)η (t)= η (θmCm (t)) ∈ [0, 1] ,where Cm (t) is the net income of a representative middle-class entrepreneur, which read more..

##### Page - 979

Introduction to Modern Economic Growthη0 (·) > 0 and V e (E) − V e (M ) > 0,we have τ m (t)= τ PC >τ RE ≡ min {1 − α, ¯τ } .Theresult that V e (E) − V e (M ) > 0 follows from Exercise 22.7.The important point here is that, as with the factor price manipulation mechanism, theelite tax beyond the peak of the Laﬀer curve. Their objective is not to increase their currentrevenues, but to consolidate their political power (in fact, taxes beyond the peak of the read more..

##### Page - 980

Introduction to Modern Economic Growthis very similar to the replacement eﬀect pointed out by Arrow in the context of innovation(recall Chapter 12).22.3.3. Subgame Perfect Versus Markov Perfect Equilibria.Ihavesofar focusedon Markov Perfect Equilibria (MPE). In general, such a focus can be restrictive. A naturalquestion is whether the set of Subgame Perfect Equilibria (SPE) is larger than the set of theMPE and whether some of the SPE can lead to more eﬃcient allocation of resources read more..

##### Page - 981

Introduction to Modern Economic Growthto a situation without commitment to taxes or policies, so that after entrepreneurs haveundertaken their investments they can be “held up” by higher rates of taxation or by expro-priation. These types of holdup problems are endemic in political economy situations, sincecommitmentstofuturepoliciesisdiﬃcult or impossible. Those who have political power at acertain point in time are likely to make the relevant decisions at that point. Moreover, whenthe read more..

##### Page - 982

Introduction to Modern Economic GrowthIn this model, it is no longer true that the unique MPE is the only SPE, since there isroom for an implicit agreement between diﬀerent groups whereby the elite (credibly) promiseadiﬀerent tax rate than ¯τ . Relatedly, the MPE in this model, provided in Proposition 22.8is Pareto ineﬃcient, and a social planner with access to exactly the same ﬁscal instrumentscan improve the utility of all agents in the economy.To illustrate the diﬀerence between read more..

##### Page - 983

Introduction to Modern Economic Growthof the society that can make all the agents in the society better-oﬀ. This analysis also showsthat whether we use the MPE or the SPE equilibrium concept has important implicationsfor the the structure of the equilibrium and its eﬃciency properties. While the use of theequilibrium concept is a choice for the modeler, diﬀerent equilibrium concepts approximatediﬀerent real-world situations. For example, MPE may be much more appropriate when read more..

##### Page - 984

Introduction to Modern Economic GrowthThe result that the allocation described in the proposition is the unique MPE followsimmediately from the analysis so far. The fact that it is also the unique SPE follows fromProposition 22.7 and implies that the elite would choose exactly this tax rate even if theycould commit to a tax rate sequence at time t =0. The reason is intuitive: in the case of purefactor price manipulation, the only objective of the elite is to reduce the middle class’ read more..

##### Page - 985

Introduction to Modern Economic Growthcould, the elite would commit to a lower tax rate in the future in order to encourage themiddle class producers to undertake technological improvements. Their inability to committo such a tax policy leads to more distortionary policies (and in fact in this case to Paretoineﬃciency). The next proposition states this result and to simplify the statement, I assume¯τ =1.Proposition 22.11. Consider the game with technology adoption, and suppose that read more..

##### Page - 986

Introduction to Modern Economic Growthprovided in Exercise 22.15 below. For now, note that the analysis in Propositions 22.5 and22.6 already provide the beginning of an answer, since they show that the equilibrium taxrate would be strictly above the revenue-maximizing level. Our ﬁrsttaskistoderivesomeimplications from these observations about constitutional limits on taxation by the elite.22.4.1. Emergence of Secure Property Rights.The environment is the same as inthe previous section, with read more..

##### Page - 987

Introduction to Modern Economic GrowthProof. See Exercise 22.12.¤The intuition for this proposition is simple: in the presence of holdup problems, Propo-sition 22.8 shows that the unique MPE involves τ =¯τ H . However, this is (Pareto) ineﬃcientand in fact, if the elite could commit to a tax rate of ¯τ = τ COM , they would increase theirconsumption (and also the middle class and the workers would achieve greater consumptionat each date). If the elite could use economic institutions, for read more..

##### Page - 988

Introduction to Modern Economic GrowthThus, while implicit promises and other informal arrangements might play the role of eco-nomic institutions under some circumstances, there will be limits to how well they can per-form this role and in many environments, constitutional limits on distortionary policies andexpropriation (if feasible) would endogenously emerge in the political equilibrium.22.4.2. Blocking Economic Development.The focus in the previous subsection wason choosing economic read more..

##### Page - 989

Introduction to Modern Economic Growthconsumption. Consequently, in this case, the elite beneﬁt from the increase in the produc-tivity of the middle-class entrepreneurs and thus would like them to be as productive aspossible. Intuitively, there is no competition between the elite and the middle class (either infactor markets or in the political arena), and when the middle class entrepreneurs are moreproductive, they generate greater tax revenues for the elite.However, the situation is read more..

##### Page - 990

Introduction to Modern Economic Growthprovides no beneﬁts to the elite. This is intimately related to the fact that in the absence ofholdup problems and given the menu of ﬁscal instruments, the equilibria characterized abovecorresponded to allocations maximizing a weighted social welfare function (and were thusconstrained Pareto eﬃcient). However, when the elite are unable to commit to future taxes(because of holdup problems), the equilibrium is no longer Pareto eﬃcient and read more..

##### Page - 991

Introduction to Modern Economic Growth(6.38) in Chapter 6. All the other assumptions from Section 22.2, especially those regardingthe production functions and the timing of policies, continue to apply.Using exactly the same notation as in Section 22.2, the dynamic optimization of middle-class entrepreneurs for a given sequence of policies and wages, pt and wt,can be writtenasUi¡{Ki(s),Li(s)}∞s=t|pt,wt¢=∞Xs=tβs−tU[(1 − τ (s)) F (Ki (s) ,Li (s)) − Ki (s +1) − w (s) Li (s)+ Tm read more..

##### Page - 992

Introduction to Modern Economic Growthelite to choose some p0 ∈ P to maximize their discounted utility. This would indeed be thesolution to the political economy problem if the elite could commit to a sequence of policiesat date t =0. But the assumption we have made so far, which is a natural approximation toreality, is that political decisions are made sequentially, and commitment to a future sequenceof policies is not possible. This is exactly where my treatment in the Section 22.2 cut read more..

##### Page - 993

Introduction to Modern Economic GrowthGiven this state variable, the policy choice of the elite can be represented by a policyfunction denoted byP :£0,¯k¤→[0,1],which determines the utility-maximizing tax rate for the elite at the next date, τ (t +1),asa function of the current capital-labor ratio k (t). We could extend this function so that italso determines the amount of transfers. But this is not necessary, since, as in the previoussubsection, the elite will always choose T w (t)= T m read more..

##### Page - 994

Introduction to Modern Economic Growth(capital-labor ratio) is given by ki, those of other entrepreneurs is k,today’s tax rate is τ ,and tomorrow’s tax rate has been announced as τ0. In all of this he takes the policy functionsof the elite, P , and the best-response function of other entrepreneurs, κ, as given. The con-tinuation value is therefore βVi (k0,κ (k, τ , τ0) ,τ0,P (κ (k, τ , τ0)) | P, κ). Here, k0 is his choiceof next period’s capital stock, so it will be the ﬁrst read more..

##### Page - 995

Introduction to Modern Economic Growthtechnology, thus they have to consume their current tax revenue and also normalize θe =1without loss of generality. Then their value function can be written as(22.38)W (k, τ | κ)=maxτ0U (τf (k)) + βW¡κ¡k,τ,τ0¢,τ0|κ¢.Intuitively, in the current period, the elite receive a pre-determined amount of tax revenuegivenbythe taxrate, τ , announced in the previous period times output produced by thethe capital stock on the economy (which is also read more..

##### Page - 996

Introduction to Modern Economic Growthwith τ00 ≡ P (κ (k, τ , τ0)).This ﬁrst-order condition has some similarity to (22.16) fromSection 22.2, but is clearly much more complicated. As with that expression, it trades oﬀ thegain from additional taxation, f (κ (k, τ , τ0)), against the loss that additional taxation willinduce by reducing the equilibrium capital-labor ratio (the second term in the ﬁrst bracketin ﬁrst line). The second line represents the discounted future change in read more..

##### Page - 997

Introduction to Modern Economic Growthheterogeneity among entrepreneurs. The bottom line of the analysis in the next section willbe that the source of distortionary (“ineﬃcient”) policies that arise from the desire of thepolitical system to extract revenues from a subset of the population is quite a bit more generalthan in the simple society investigated in Section 22.2. But before doing this, we will get amodicum of basic social choice theory. Strictly speaking, only a simple form of the read more..

##### Page - 998

Introduction to Modern Economic Growthdeﬁne ui (·) ≡ ui (·| αi)) and is convenient for some of the analysis that will follow. Clearly,the general equilibrium variables, such as prices, represented by Y (x, p) here, need not beuniquely deﬁned for a given set of policies p and vector of individual choices x. Since multipleequilibria are not our focus here, we ignore this complication and assume that Y (x, p) isuniquely deﬁned.We also assume that, given aggregates and policies, read more..

##### Page - 999

Introduction to Modern Economic GrowthLet = be the set of all reﬂexive and complete binary relations on P (but notice notnecessarily transitive). A social ordering is RS ∈ =, i.e., it is a reﬂexive complete binaryrelation over all the policy choices in P. Thus, a social ordering can be represented asφ : <H → =.This mathematical formalism implies that φ (ρ) gives the social ordering for the preferenceproﬁles in ρ.We can alternatively think of φ as a political system mapping read more..

##### Page - 1000

Introduction to Modern Economic GrowthThe main theorem of the ﬁeld of social choice theory is the following:Theorem 22.1. (Arrow’s (Im)Possibility Theorem) If a social ordering, φ,istran-sitive, weakly Paretian and satisﬁes independence from irrelevant alternatives, then it is dic-tatorial.Proof. Suppose to obtain a contradiction that there exists a non-dictatorial and weaklyParetian social ordering, φ, satisfying independence from irrelevant alternatives. We willderive a contradiction read more..

##### Page - 1001

Introduction to Modern Economic Growthminimal strongly decisive set). If D a singleton, then Step 1 applies and implies that φ isdictatorial, completing the proof. Thus suppose that D 6= {i}. Then by unrestricted domain,the following preference proﬁle (restricted to {p1,p2,p3})isfeasiblefor i ∈ Dp1Âi p2 Âi p3for j ∈ D\{i} p3 ºj p1 Âj p2for k/∈ Dp2 Âk p3 Âk p1.By hypothesis, D is strongly decisive between p1 and p2 and therefore p1 ÂS p2.Next ifp3 ÂS p2, then given the preference read more..

##### Page - 1002

Introduction to Modern Economic Growthmake in restricting the policy space to a single policy. In some circumstances, limits on ﬁscalinstruments might be justiﬁed on economic grounds. For example, the assumption that therewas only a linear tax on output in Section 22.2 was justiﬁed with the argument that lump-sumtaxes were not possible. Whether or not this is the case, it is important to recognize thatthe limits on the set of ﬁscal instruments is often responsible for the potential read more..

##### Page - 1003

Introduction to Modern Economic GrowthA3. Open agenda. Citizens vote over pairs of policy alternatives, such that the winningpolicy in one round is posed against a new alternative in the next round and the set ofalternatives includes all feasible policies. Later, we will replace the open agenda assumptionwith parties oﬀering policy alternatives, thus moving away from direct democracy some waytowards indirect/representative democracy. For now it is a good starting point.Now, using the three read more..

##### Page - 1004

Introduction to Modern Economic GrowthWe can easily verify that in the Condorcet paradox, not all agents possessed single-peakedpreferences. For example, taking the ordering to be a, b, c, agent 1 who has preferencesa Â c Â b does not have single-peaked preferences (if we took a diﬀerent ordering of thealternatives, then the preferences of one of the other two agents would violate the single-peakedness assumption, see Exercise 22.21).The next theorem shows that with single-peaked preferences read more..

##### Page - 1005

Introduction to Modern Economic GrowthA20. Strategic voting. Deﬁne a vote function of individual i in a pairwise contest betweenp0 and p00 by vi (p0,p00) ∈ {p0,p00}. Let a voting (counting) rule in a society with H citizensbe V :{p0,p00}H → {p0,p00} for any p0,p00 ∈ P. (For example, the majoritarian voting rule V Mpicks p0 over p00 when this policy receives more votes than p00). Let V (vi (p0,p00) ,v−i (p0,p00))be the policy outcome from voting rule V applied to the pairwise contest read more..

##### Page - 1006

Introduction to Modern Economic GrowthExample 22.2. Consider three individuals with the following preference orderings.1 a Â b Â c2 b Â c Â a3 c Â b Â aThese preferences are clearly single peaked (order them alphabetically to see this). In a oneround vote, b will beat any other policy. But now consider the following dynamic voting setup: ﬁrst, there is a vote between a and b. Then, the winner goes against c, and the winnerof this contest is the social choice. Sincere voting will imply read more..

##### Page - 1007

Introduction to Modern Economic Growthoverview of this situation and derive the Downsian Policy Convergence Theorem, which isthe basis of much applied work in political economy.Suppose that we have a situation in which there is a Condorcet winner, and there aretwo parties, A and B, competing for political oﬃce. Assume that the parties do not havean ideological bias, and would like to come to power (for example, they receive some utilityfrom being in power). In particular, they both maximize read more..

##### Page - 1008

Introduction to Modern Economic Growthand win the election for sure. When pA 6= pm and pB = pm, party A can also announce pmand increase its chance of winning to 1/2.¤Exercise 22.25 asks you to provide a generalization of this theorem without AssumptionA4.This theorem is important because it demonstrates that there will be policy convergencebetween the two parties and that party competition will implement the Condorcet winneramong the voters. Therefore, in situations in which the MVT applies, read more..

##### Page - 1009

Introduction to Modern Economic Growthtoo far aﬁeld from our focus, so will be left to Exercise 22.28. Here we introduce the usefulconcept of single-crossing property, which will enable us to prove a version of Theorem 22.2under somewhat weaker assumptions.Definition 22.3. Consider an ordered policy space P andalsoorder voters accordingtotheir αi’s. Then the preferences of voters satisfy the single-crossing property over the policyspace P when the following statement is true:if p>p0 and read more..

##### Page - 1010

Introduction to Modern Economic GrowthTheorem 22.6. (Extended Downsian Policy Convergence) Suppose that there aretwo parties that ﬁrst announce a policy platform and commit to it and a set of voters that votefor one of the two parties. Assume that A4 holds and that all voters have preferences thatsatisfy the single-crossing property, and denote the median-ranked bliss point by pm.Thenboth parties will choose pm as their policy.Proof. See Exercise 22.27.¤Despite this generalization, which is read more..

##### Page - 1011

Introduction to Modern Economic Growthvote share of party P isπP =GXg=1λg πgP .In our analysis so far, all voters in group g would have cast their votes identically (unlessthey were indiﬀerent between the two parties). The idea of probabilistic voting is to smoothout this behavior by introducing other considerations in the voting behavior of individuals.Put diﬀerently, probabilistic voting models will add “noise” to equilibrium votes, smoothingthe behavior relative to models we read more..

##### Page - 1012

Introduction to Modern Economic GrowthFurthermore, to simplify the exposition here, suppose that parties maximize their ex-pected vote share. In this case, party A sets this policy platform pA to maximize:(22.44)πA =GXg=1λg Hg (Ug (pA) − Ug (pB)).Party B faces a symmetric problem and maximizes πB, which is deﬁned similarly. Inparticular, since πB =1 − πA,party B’s problem is exactly the same as minimizing πA.Equilibrium policies will then be determined as the Nash equilibrium of a read more..

##### Page - 1013

Introduction to Modern Economic GrowthThe important point to note about this result is its seeming generality: as long as a purestrategy symmetric equilibrium in the party competition game exists, it will correspond to amaximum of some weighted social welfare function. This generality is somewhat exaggerated,however, since such a symmetric equilibrium does not always exists. In fact, conditionsto guarantee existence of pure strategy symmetric equilibria are rather restrictive and arediscussed in read more..

##### Page - 1014

Introduction to Modern Economic Growthhe is one of many members in his group, and there is a natural free-rider problem. Hemight let others make the contribution, and simply enjoy the beneﬁts. This will typicallybe an outcome if groups are unorganized (for example, there is no eﬀective organizationcoordinating their lobbying activity and excluding non-contributing members from some ofthe beneﬁts etc.). On the other hand, organized groups might be able to collect contributionsfrom their read more..

##### Page - 1015

Introduction to Modern Economic Growthand satisﬁes ˆγg (pg)=0. That is, the contribution function of each lobby is suchthat there exists a policy that makes no contributions to the politician, and gives herthesameutility.Proof. (Sketch) Conditions 1, 2 and 3 are easy to understand. No group would everoﬀer a contribution schedule that does not satisfy Condition 1. Condition 2 has to hold, sincethe politician chooses the policy. If Condition 3 did not hold, then the lobby could changeits read more..

##### Page - 1016

Introduction to Modern Economic Growthand from the ﬁrst-order condition of each lobby thatλgµ∂ˆγg(p∗)∂pk−∂U g (p∗)∂pk¶+ GXg0=1λg ∂ˆγg0 (p∗)∂pk+aGXg0=1λg ∂U g0 (p∗)∂pk=0 for all k =1, 2,.., K and g =1, 2,.., G0.Combining these two ﬁrst-order conditions, we obtain(22.49)∂ˆγg (p∗)∂pk=∂U g (p∗)∂pkfor all k =1, 2,.., K and g =1, 2,.., G0. Intuitively, at the margin each lobby is willing to payfor a change in policy exactly as much as this policy read more..

##### Page - 1017

Introduction to Modern Economic GrowthInstead, the economy consists of a continuum 1 of yeoman-entrepreneurs, each denoted byi ∈ [0, 1] and with access to a neoclassical production functionYi (t)= F (Ki (t) ,AiLi (t)) ,where Ai is a time-invariant labor-augmenting productivity measure and will be the onlysource of heterogeneity among the entrepreneurs. In particular, F satisﬁes Assumptions 1and 2. We assume that Ai has a distribution given by μ (A) among the entrepreneurs. read more..

##### Page - 1018

Introduction to Modern Economic GrowthLet us deﬁne ki (t) ≡ Ki (t) /Ai as the eﬀective capital-labor ratio (i.e., the ratio of capitalto “eﬀective labor”) of entrepreneur i (bearing in mind that its employment level is equalto 1) and recall that pt determines the sequence of taxes starting from time t.With thisdeﬁnition, we can write the value of each entrepreneur recursively as(22.50)Vi¡ki(t)|pt¢=maxki(t+1)≥0{(1 − τ (t)) Aif (ki (t)) −Aiki (t +1) + T (t)+ read more..

##### Page - 1019

Introduction to Modern Economic GrowthLet us next determine the political bliss point of each entrepreneur, i.e., their most pre-ferred tax rate. To do this, let us write their continuation utility from the end of periodt. Ignoring all the terms that are bygone by this point and substituting for best responses(i.e., for the eﬀective capital-labor ratio from (22.52)), the expected discounted utility ofentrepreneur i from (22.50) can be written as(22.54) read more..

##### Page - 1020

Introduction to Modern Economic GrowthProposition 22.23. Consider the above-described model. Then there exists τ m ∈ [0, 1)such that the unique Markov Perfect Political Economy Equilibrium involves τ (t)= τ m forall t.Moreover,:• if the distribution of productivity among the entrepreneurs, μ (A), is such that Am ≥¯A,then τ m =0;• if Am < ¯A,then τ m > 0;• suppose that Am < ¯A. Then, for given ¯A, τ m is strictly decreasing in Am.Proof. The argument preceding the read more..

##### Page - 1021

Introduction to Modern Economic GrowthSecond, this proposition shows that if the productivity of the median voter is above theaverage, there will be no redistributive taxation. This is intuitive. As the ﬁrst term in (22.55)makes it clear, the beneﬁts of taxation are proportional to the average productivity in theeconomy, while the cost (to the median voter) is related to his productivity. If the medianentrepreneur is more productive than the average, there are two forces making him read more..

##### Page - 1022

Introduction to Modern Economic Growth22.8. The Provision of Public Goods: Weak Versus Strong StatesThe analysis so far has emphasized the distortionary eﬀects of taxation and expropriation.This paints a picture whereby the major determinants of poor economic performance arehigh taxes or some type of expropriation, and political economy does (or should) focus onthe determination of the incentives for redistributive taxation. While the disincentive eﬀectsof taxation cannot be denied, whether read more..

##### Page - 1023

Introduction to Modern Economic Growththe population, then the state is too “strong,” and private investment will be stiﬂed. Thusstates that have intermediate levels of strength are most conducive to economic growth.This model will also enable us to discuss the potential distinction between expropriationand taxes, an issue raised at the beginning of this chapter.22.8.1. The Model.All agents have again linear preferences given in (22.1).Thepopulation consists of a set of read more..

##### Page - 1024

Introduction to Modern Economic Growthwhere G (t) denotes government spending on public goods, and φ> 1, so that there aredecreasing returns in the investment technology of the ruler (a greater φ corresponds to greaterdecreasing returns). The term [αφ/ (1 − α)]1/φ is included as a convenient normalization. Inaddition, (22.58) implies full depreciation of A (t),which simpliﬁes the analysis below. Theconsumption of the elite is given by whatever is left over from tax revenues after read more..

##### Page - 1025

Introduction to Modern Economic GrowthCombining this expression with (22.56) and (22.57), we obtain equilibrium tax revenue as afunction of the level of public goods as:(22.62)T (A (t)) =(β (1 − ¯τ ))α/(1−α) ¯τA (t)α.Finally, the elite will choose public investment, G (t) to maximize his consumption. Tocharacterize this, let us write the discounted net present value of the elite as(22.63)Ve (A (t)) = maxA(t+1)½T(A(t))−1−ααφA (t +1)φ + βVe (A (t +1))¾,which simply follows read more..

##### Page - 1026

Introduction to Modern Economic GrowthProposition 22.24. In the above-described economy, there exists a unique MPE whereτ (A)= ¯τ for all A, A (t) is given by A [¯τ ] as in (22.66) for all t> 0, and, the capital-laborratio of each entrepreneur i ∈ [0, 1] and for all t is given by (22.61). For all t> 0,theequilibrium level of aggregate output is:(22.68)Y (t)= Y [¯τ ] ≡1α(β (1 − ¯τ ))α/(1−α) A [¯τ ] .Proof. See Exercise 22.36.¤Note that because of linear preferences read more..

##### Page - 1027

Introduction to Modern Economic Growthstrength as captured by the parameter ¯τ ; greater power for citizens is beneﬁcial when theirinvestments matter more. When it is the state’s investment that is more important foreconomic development, a higher ¯τ is required (justiﬁed).The above discussion focused on the output-maximizing value of the parameter ¯τ .Equallyrelevant is the level of ¯τ ,say ¯τ e, which maximizes the beginning-of-period payoﬀ to the eliteand ¯τ c, which read more..

##### Page - 1028

Introduction to Modern Economic Growthone can analyze the SPE, where there might be an implicit agreement between the stateand the citizens to allow for some amount of taxation and also correspondingly high levelsof public good provision. In Acemoglu (2005), I referred to this equilibrium conﬁguration asa “consensually-strong state,” since the citizens allow the economic power of the state to behigh (partly because they believe they can politically control the state). The read more..

##### Page - 1029

Introduction to Modern Economic Growthare poor and some are rich, we need to understand why some countries choose growth-enhancing policies while others choose policies that block economic development.This chapter emphasized a number of key themes in developing answers to these questions.First, the sources of diﬀerent institutions (and non-growth-enhancing institutions) must besought in social conﬂict among diﬀerent individuals and groups in the society. Social conﬂictimplies that there read more..

##### Page - 1030

Introduction to Modern Economic Growththe ability) to undertake investments and produce. In contrast, the factor price manipulationand the political replacement eﬀects encourage the elite to pursue policies that harm groupsthat they perceive as their competitors. This typically leads to higher taxes and also toexplicit actions to block technology adoption or other productivity-enhancing investments bycompeting entrepreneurs. The consequences of these types of policies for economic growthcan be read more..

##### Page - 1031

Introduction to Modern Economic Growthproblems provides us with one perspective for thinking about the emergence of secure propertyrights.To move beyond models in which political power rests with a pre-speciﬁed group, herethe political elite, we need systematic ways of aggregating heterogeneous political preferences.After reviewing some basic political economy theory, I used the well-known Median VoterTheorem (MVT), which applies in certain economic environments, to investigate the read more..

##### Page - 1032

Introduction to Modern Economic GrowthThe material in this chapter is no more than an introduction to the exciting and impor-tant ﬁeld of political economy of growth. Many issues have not been addressed. Among thosethe following appear most important: ﬁrst, in addition to taxes, expropriation and publicgoods, whether the society provides a level playing ﬁeld to a broad cross-section of societyis important. For example, broad-based human capital investments, which may be quite im-portant read more..

##### Page - 1033

Introduction to Modern Economic GrowthA detailed analysis of why the political elite may block technological innovations in orderto increase the likelihood of their survival is presented in Acemoglu and Robinson (2007a).That paper also shows how both relatively secure elites and elites that are in competitivepolitical environments will not have incentives to block technological change, but those withintermediate levels of security that might be challenged by new technologies are likely toadopt read more..

##### Page - 1034

Introduction to Modern Economic GrowthRoberts (1977) and Romer (1975). Meltzer and Richard (1981) used the Roberts-Romermodel to relate inequality to taxes and more importantly, to draw implications about theextent of the voting franchise on the size of the government. Meltzer and Richard’s workis a classic as it can be viewed as the beginning of positive political economy, i.e., the useof political economy models in order to explain cross-country and over-time diﬀerences inpublic policies. read more..

##### Page - 1035

Introduction to Modern Economic Growth22.11. ExercisesExercise 22.1. Prove that ˆτ given by (22.16) satisﬁes ˆτ ∈ (0, 1).Exercise 22.2. Consider the model in Section 22.2, with the only diﬀerence that theproduction technology is as in the Romer (1986) model studied in Chapter 11. In par-ticular, recall that each entrepreneur now has access to the production function Yi (t)=F (Ki (t) ,A (t) Li (t)) and A (t)= BR10Ki(t)di=BK(t). Characterize the Markov PerfectPolitical Economy read more..

##### Page - 1036

Introduction to Modern Economic Growth(2) Explain how Proposition 22.9 needs to be modiﬁed if ¯τ< 1 and provide an analysisof the best stationary SPE in this case.Exercise 22.12. Prove Proposition 22.13.Exercise 22.13. Prove Proposition 22.14.Exercise 22.14. Prove Proposition 22.16.Exercise 22.15.(1) Prove Proposition 22.17.(2) Now suppose that in this proposition φ is not equal to 0. Provide an example inwhich in the MPE, the elite would still prefer g =0.(3) Now suppose that the elite read more..

##### Page - 1037

Introduction to Modern Economic Growthφ (cab)= abc (why?); and proceeding this way to show that the social ordering iseither dictatorial or it violates one of the axioms].(2) Now suppose we have the following aggregation rule: individual 1 will (sincerely)rank the three outcomes, his ﬁrst choice will get 6 votes, the second 3 votes, thethird 1 vote. Individual 2 will do the same, his ﬁrstchoicewillget 8votes, thesecond 4 votes, and the third 0 vote. The three choices are ranked according read more..

##### Page - 1038

Introduction to Modern Economic Growthbetween p1 and p2. In the second stage, there is a vote between the winner of theﬁrst stage and p3, until we have a ﬁnal vote against pM . The winner of the ﬁnalvote is the policy choice of the society. Prove that if preferences of all agents aresingle peaked (with a unique bliss point for each), then the unique subgame perfectNash equilibrium implements the bliss point of the median voter.Exercise 22.23. * Prove Theorem 22.2 when H is even.Exercise read more..

##### Page - 1039

Introduction to Modern Economic Growthwhere K(αi) is monotonic (monotonically increasing or monotonically decreasing) in αi,andthe functions J1(p) and J2 (p) are common to all voters. Suppose that A2 holds and votershave intermediate preferences. We deﬁne thebliss point(vector)ofindividual i as in thetext, as p (αi) ∈ P that maximizes individual i’s utility. Prove that when preferences areintermediate a Condorcet winner always exists and coincides with bliss point of the voterwith the read more..

##### Page - 1040

Introduction to Modern Economic Growth(1) Suppose that agents vote over a linear income tax, τ . Because of tax distortions,total tax revenue isTax =(τ − v (τ ))µλrk+(1−λ)wZ hdμ (h)¶where v (τ ) is strictly increasing and convex, with v (0) = v0 (0) = 0 and v0 (1) = ∞(why are these conditions useful?). Tax revenues are redistributed lump sum. Findthe ideal tax rate for each agent. Find conditions under which preferences are singlepeaked, and determine the equilibrium tax rate. How read more..

##### Page - 1041

Introduction to Modern Economic GrowthExercise 22.35. We now consider the model by Alesina and Rodrik, which is similar tothe model studied in Section 22.7. There is a continuum 1 of individuals. All individualshave logarithmic instantaneous utility, so that Ui =P∞t=0βtlnCi(t),where i denotes theindividual and Ci (t) refers to his consumption at time t. Each individual has one unit oflabor, which he supplies inelastically. Final output is produced asYi (t)= AKi (t)1−α G (t)α Li (t)αwhere read more..

##### Page - 1042

Introduction to Modern Economic GrowthExercise 22.36.(1) Prove Proposition 22.24.(2) Derive the output-maximizing tax rate as in (22.69).(3) Characterize the tax rates maximizing the utility of the elite and the citizens andestablish the results in Proposition 22.25.1021 read more..

##### Page - 1043

##### Page - 1044

CHAPTER 23Political Institutions and Economic GrowthThe previous chapter investigated why societies often choose ineﬃcient economic insti-tutions and policies and consequently fail to take advantage of growth opportunities. Itemphasized the importance of social conﬂict between diﬀerent groups and lack of commit-ment to future policies as major sources of non-growth-enhancing policies. Much of thediscussion was in the context of a given set of political institutions, which shaped both read more..

##### Page - 1045

Introduction to Modern Economic Growthemerge and change. This is a major area of current research in political economy and largelyfalls beyond the scope of the current book. Sections 23.4 and 23.5 will therefore give a bird’seye view of some of the main issues involved in the analysis of equilibrium political institutionsand their implications for economic growth. Section 23.4 starts with a general discussion ofwhat types of models we might want to consider for understanding the dynamics of read more..

##### Page - 1046

Introduction to Modern Economic Growthpeople would consider the United States in the 1960s a nondemocracy, though many blackswere disenfranchised. This creates various diﬀerent shades of democracy that one might wishto take into account. For example, Colombia has been a democratic country for over half acentury according to most political scientists, though in many parts of the country electionsare very far from “free and fair” and take place under the threat of explicit violence read more..

##### Page - 1047

Introduction to Modern Economic Growthconstrained by certain constitutional restrictions. In contrast, nondemocracies, rather thanrepresenting the wishes of the population at large, represent the preferences of a subgroupof the population. In the previous chapter, I referred to this subgroup as the “elite,” and Iwill continue to do so here. The identity of the elite diﬀers across nondemocratic societies.In China, it is mainly the wishes of the Communist Party that matters. In Chile read more..

##### Page - 1048

Introduction to Modern Economic Growthtypically envisage. I will further argue that a particularly important reason why democraciesmight be dysfunctional is because they are captured by elites despite the fact that on paperthey are supposed to provide majoritarian decision-making and political equality. Accordingto this deﬁnition, a democracy will be “captured” when its modus operandi–its purposeof creating greater political equality than a typical nondemocratic regime–fails. read more..

##### Page - 1049

Introduction to Modern Economic Growthif present, appear to be much less pronounced than one might have expected on the basisof theory alone. I will argue in Section 23.5 that the same factors that are important inthinking about why democracies do not grow much faster than nondemocracies are likelyto be important in understanding why policies in democracies and nondemocracies do notdiﬀer by much (for example, captured democracies are unlikely to pursue highly redistributivepolicies). However, read more..

##### Page - 1050

Introduction to Modern Economic GrowthLong-run regressions, such as those discussed in Chapter 4, are also consistent with thispattern and show a signiﬁcant eﬀect of a broad cluster of institutions on economic growth.Whilewecannotconﬁdently say that this represents the eﬀect of political institution ongrowth, this cluster of institutions comprises both political and economic elements and it islikely that the growth-enhancing cluster of institutions could not exist without the read more..

##### Page - 1051

Introduction to Modern Economic Growth23.2.1. The Dictatorship of the Middle Class Versus the Dictatorship of theElite.First, let us suppose that the middle class hold political power, so that we have thedictatorship of the middle class instead of the dictatorship of the elites in the previous chapter.The situation is entirely symmetric to that in the previous chapter with the middle class andthe elite having exchanged places. In particular, the analysis leading to Proposition 22.6immediately read more..

##### Page - 1052

Introduction to Modern Economic Growthof the gentry and the previous landowning class had already transitioned into commercialagriculture and other industrial activities. Nevertheless, there were intense political debatessurrounding the Corn Law, with landowners supporting the tariﬀsimposed by thelaw, whichkept the price of their produce high, and industrialists opposing it so that the implicit taxon their inputs, especially labor, would be removed with the import of cheaper corn fromabroad.So read more..

##### Page - 1053

Introduction to Modern Economic Growthand without fully analyzing how their productivity and their economic activities compare tothose of others. Naturally, one can dream of political institutions that will outperform boththe elite dominated politics of the previous chapter and the middle class dominated politics ofthis chapter. For example, we can think of a set of political institutions that constitutionallyforce all taxes to be equal to zero–so that in the context of the simple model we are read more..

##### Page - 1054

Introduction to Modern Economic GrowthThe analysis is again straightforward, though the nature of the political equilibriumdoes depend even more strongly on whether or not Condition 22.1 holds (i.e., whether thereis excess supply or not). The following proposition summarizes the equilibrium choices ofpolicies by the workers when they monopolize political power.Proposition 23.3. Consider the environment of subsection 22.2.4 with Cobb-Douglastechnology and suppose that workers hold political read more..

##### Page - 1055

Introduction to Modern Economic Growthrecognize the impact of taxes on their own wages, democracy will generate more moderatepolitical outcomes.The simple analysis in this section therefore already gives us some clues about why thereare no clear-cut relationships between political regimes and economic growth. If the forceshighlighted here are important, we would expect democracy to generate higher growth undercertain circumstances, for example, when the equivalent of Condition 22.1 holds. In read more..

##### Page - 1056

Introduction to Modern Economic Growth23.3.1. The Baseline Model.The model economy is similar to that analyzed in Sec-tion 22.2 and more speciﬁcally, to the Cobb-Douglas economy in subsection 22.2.4. Theeconomy is populated by a continuum 1 of inﬁnitely-lived agents, each with preferences givenby (22.1) as in Section 22.2. In addition, for reasons that will become clear soon, I assumethat each each individual dies with a small probability ε in every period, and a mass ε ofnew individuals read more..

##### Page - 1057

Introduction to Modern Economic Growthby T (t) ∈ [0, ∞). Notice that I have already imposed an upper bound on taxes ˆτ< 1.This upper bound can be derived from the ability of individuals to hide their output in theinformal sector or because of the standard distortionary eﬀects of taxation. It simpliﬁes theanalysis to take it as given here. This parameter will have important implications on whattype of political regime will lead to greater income per capita. The new policy read more..

##### Page - 1058

Introduction to Modern Economic Growthwith talent ai (t) at time t are:(23.4) π (ki (t) | ai (t) ,w (t) ,τ (t)) = (1 − τ (t)) α−1ki (t)α¡ai(t)¯L¢1−α−w(t)¯L−β−1ki(t),where recall that the tax rate is in the range0 ≤ τ (t) ≤ ˆτ and the cost of capital ismultiplied by β−1 because it is incurred in the previous period. Given this expression, the(instantaneous) gain from entrepreneurship for an agent of talent z ∈ {L,H} at time t as afunction of the tax rate, τ read more..

##### Page - 1059

Introduction to Modern Economic Growthof entrepreneurs. Second, we can think of the inﬁnitely-lived agents as representing dynasties,and the imperfect over-time correlation in ai (t) may represent imperfect correlation betweenthe skills of parents and children. Thrid and perhaps most interestingly, it may be that eachindividual has a ﬁxed competence across diﬀerent activities, and comparative advantage inentrepreneurship changes as the importance of diﬀerent activities evolves over time. read more..

##### Page - 1060

Introduction to Modern Economic Growthknowing their ability level, i.e., ai (t +1) is realized before the decisions on ei (t +1) andki (t +1). Notice also that if an individual does not operate his ﬁrm, he loses “the license”,so next time he wants to set up a ﬁrm, he needs to incur the entry cost (and the assumptionthat li (t)= ¯L rules out the possibility of operating the ﬁrm at a much smaller scale).Finally, we need to specify the initial conditions: I assume that the distribution read more..

##### Page - 1061

Introduction to Modern Economic Growthlevel of the entrepreneur, ai (t), and the level of employment, ¯L, and decreasing in the taxrate, τ (t).Now using (23.9), the net current gain to entrepreneurship for an agent of type z ∈ {L,H}(i.e., of skill level AL or AH ) can be obtained as:(23.10)Πz (τ (t) ,w (t)) = (1 − α) α−1(β (1 − τ (t)))1/(1−α)Az ¯L − w (t) ¯L.Moreover, the labor market clearing condition (23.6) implies that the total mass of entre-preneurs at any time read more..

##### Page - 1062

Introduction to Modern Economic Growthwhere Πz is given by (23.10) and now crucially depends on the skill level of the agent, andCV z¡qt+1¢isthecontinuationvalueforanentrepreneuroftypez:(23.15)CVz¡qt+1¢=σzmax©WH¡qt+1¢;VH¡qt+1¢ª+(1−σz)max©WL¡qt+1¢;VL¡qt+1¢ª.An entrepreneur of ability Az also receives the wage w (t) (working for his own ﬁrm) and thetransfer T (t), and in addition makes proﬁts equal to Πz (τ (t) ,w (t)). The following period,this entrepreneur has high read more..

##### Page - 1063

Introduction to Modern Economic Growth(2) Sclerotic equilibrium where agents with ei (t − 1) = 1 remain entrepreneurs irrespec-tive of their productivity.An entry equilibrium requires the net value of entrepreneurship to be greater for a non-elite high skill agent than for a low-skill elite. Let us deﬁne wH (t) as the threshold wagerate such that high-skill non-elite agents are indiﬀerent between entering and not enteringentrepreneurship. That is, wH (t) has to be such that read more..

##### Page - 1064

Introduction to Modern Economic GrowthLabor Supply/Demandw(t)10LSLDwH(t)wH(t)+b(t)wL(t)-b(t)wL(t)⎯LM⎯LFigure 23.1. Labor market equilibrium when (23.18) holds.equal to(23.19)wE (t)= wH (t) .Note also that when (23.18) holds, naturally NV¡qt|ai(t)=AL,ei(t−1)=1¢≤0,solow-skill incumbents would be worse oﬀ if they remained as entrepreneurs at the equilibrium wagerate wE (t).Figure 23.1 illustrates the entry equilibrium diagrammatically by plotting labor demandand supply in this economy. read more..

##### Page - 1065

Introduction to Modern Economic GrowthLabor Supply/Demandw(t)1LSLD1-εwH(t)+b(t)wL(t)wL(t)-b(t)wH(t)⎯LFigure 23.2. Labor market equilibrium when (23.18) does not hold.In a sclerotic equilibrium, on the other hand, wH (t) <wL (t), and low-skill incumbentsremain in entrepreneurship, i.e., ei (t)= ei (t − 1). If there were no deaths so that ε =0,thetotal number of entrepreneurs would be 1/ ¯L and for any w ∈£wH(t),wL(t)¤,labordemandwould exactly equal labor supply (i.e., 1/ ¯L agents read more..

##### Page - 1066

Introduction to Modern Economic Growth(ei (t − 1) = 1 and ai (t)= AL) who, given the entry barriers, have a higher marginal productof labor than high-skill potential entrants.The equilibrium law of motion of the fraction of high-skill entrepreneurs, μ (t),is:(23.20)μ (t)=½ σH μ (t − 1) + σL(1 − μ (t − 1)) if (23.18) does not hold1if (23.18) holds,starting with some μ (0). The exact value of μ (0) will play an important role below. Recallthat we have assumed above that ei (−1) read more..

##### Page - 1067

Introduction to Modern Economic Growthon strategies in the future except through its impact on payoﬀ-relevant state variables. There-fore, given τ (t)= ˆτ , the utility of agent i with ei (t − 1) = 0 and ai (t)= AL depends on b (t)only through the equilibrium wage, wE (t), and the transfer, T E (t). High-productivity work-ers (those with ei (t − 1) = 0 and ai (t)= AH ) may become entrepreneurs, but as the aboveanalysis shows, in this case, read more..

##### Page - 1068

Introduction to Modern Economic Growththat there is perfect equality because the excess supply of high-skill entrepreneurs ensuresthat they receive no rents.It is useful to note that Y D corresponds to the level of output inclusive of con-sumption and investment.“Net output” and consumption can be obtained by sub-tracting investment costs from Y D, and in this case, they will be given by Y Dn≡¡α−1−β(1−ˆτ)¢(β(1−ˆτ))α/(1−α)AH. All the results stated for output here also read more..

##### Page - 1069

Introduction to Modern Economic GrowthWithout loss of any generality, let us assume that they will set the entry barrier as b (t)=bE (t) in this case.An oligarchic equilibrium then can be deﬁned as a policy sequence ˆpt, wage sequenceˆwt, and economic decisions ˆxt such that ˆwt and ˆxt constitute an economic equilibrium givenˆpt,and ˆpt is such τ (t + n)=0 and b (t + n)= bE (t + n) for all n ≥ 0. In the oligarchicequilibrium, there is no redistributive taxation and entry barriers read more..

##### Page - 1070

Introduction to Modern Economic GrowthNevertheless, it is also possible to imagine societies in which μ (0) <M ,because thereis some other process of selection into the oligarchy in the initial period that is negativelycorrelated with skills in entrepreneurship. In this case, somewhat paradoxically, μ (t) andthus Y E (t) would be increasing over time. While interesting in theory, this case appearsless relevant in practice, where we would expect at least some positive selection in the read more..

##### Page - 1071

Introduction to Modern Economic GrowthIf Condition 23.2 holds, then at some point the democratic society will overtake(“leapfrog") the oligarchic society.As noted above, it is possible to imagine societies in which even in the initial period, thereare “elites” that are not selected into entrepreneurship on the basis of their skills. In thiscase, we will typically have μ (0) < 1. In the extreme case where there is negative selectioninto entrepreneurship in the initial period, we read more..

##### Page - 1072

Introduction to Modern Economic GrowthOutput in democracyOutput in oligarchyOutput in oligarchytt'YDY’E∞YE(0)YE∞Y(t)Figure 23.3. Dynamic comparison of output in oligarchy and democracy.The dashed line represents output in oligarchy when Condition 23.2 holds,and the solid line represents output in oligarchy when this condition does nothold.parameter ˆτ may correspond both to certain institutional impediments limiting re-distribution, or more interestingly, to the speciﬁcity of assets in read more..

##### Page - 1073

Introduction to Modern Economic Growtheconomies during the 19th century. First, the American democracy was not highly redistrib-utive, corresponding to low ˆτ in terms of the model here. More important, the 19th centurywas the age of industry and commerce, where the allocation of high-skill agents to entre-preneurship appears to have been probably quite important, and potentially only a smallfraction of the population were really talented as inventors and entrepreneurs. This can bethought of read more..

##### Page - 1074

Introduction to Modern Economic Growthoutput. A very low level of μ (0) may emerge if the oligarchy is founded by individuals thatare talented in non-economic activities (e.g., by elites specialized in ﬁghting in pre-moderntimes) and these non-economic talents are negatively correlated with entrepreneurial skills.Nevertheless, as noted above, a signiﬁcant amount of positive selection on the basis of skillseven in the initial period seems to be the more reasonable case.On the basis of this read more..

##### Page - 1075

Introduction to Modern Economic Growthequality–the net incomes and consumption of all agents are equalized in democracy becauseof the excess supply of high-skill entrepreneurs.Moreover, non-elites are always better oﬀ in democracy than in oligarchy, where theyreceive zero income. In contrast, and more interestingly, it is possible for low-skill elites tobe better oﬀ in democracy than in oligarchy (though high-skill elites are always better oﬀin oligarchy). This point will play a role in read more..

##### Page - 1076

Introduction to Modern Economic GrowthMoreover, this approach does not provide a framework for the analysis of regime changes inthe context of a dynamic equilibrium.A more satisfactory approach would be to use the induced preferences of the agentsover political regimes and analyze a game over the determination of political institutions,where agents have these induced preferences. Broadly speaking two types of processes canlead to political change in this case. The ﬁrst is some type of read more..

##### Page - 1077

Introduction to Modern Economic Growthentry barriers are kept throughout, low-skill agents will eventually become the majority andsucceed in disbanding the oligarchic regime. One complication is that as μ (t) approaches1/2, high-skill elites may prefer to temporarily reduce the entry barrier and include newentrepreneurs in order to prevent the disbanding of the regime. Nevertheless, this strategywill not be attractive when the future is discounted heavily. Consequently, we can establishthe read more..

##### Page - 1078

Introduction to Modern Economic Growthhowever, the reason for the transition from oligarchy to democracy is not increased ineﬃ-ciency in the oligarchy, but conﬂict between high and low-skill agents within the oligarchy;the transition takes place when the low-skill elites become the majority.The result presented in Proposition 23.8 only leads to equilibrium regime change when(23.32) holds. When it fails to hold, this proposition states that society will remain per-manently as an oligarchy read more..

##### Page - 1079

Introduction to Modern Economic Growthanother, equally important type of political power that features importantly in equilibriumpolitical changes–de facto political power. The political power of protesters that marchedagainst the existing regime before the First Reform Act in Britain in 1832 was not of thede jure kind. The law of the land did not empower them to inﬂuence the political courseof actions–in fact, they were quite explicitly disenfranchised. But they had a diﬀerent kindof read more..

##### Page - 1080

Introduction to Modern Economic Growthleast two reasons. First, most of the issues we are discussing are dynamic in nature–they referto political change. Second, whether the distribution of de facto political power is permanentor changing stochastically over time has major consequences for the structure of politicalequilibrium. When a particular (disenfranchised) group has permanent (and unchanging)amount of de facto political power, it can use this at each date to demand concessions fromthose read more..

##### Page - 1081

Introduction to Modern Economic Growthland, physical or human capital, or technology, and failed to foster economic growth. Theseeconomic institutions also ensured that the monarchs and their allies controlled a large frac-tion of the economic resources in society, solidifying their political power and ensuring thecontinuation of the political regime.The 17th century witnessed major changes in the economic and political institutionsthat paved the way for the development of property rights and read more..

##### Page - 1082

Introduction to Modern Economic Growthwould foster economic growth, they were not appealing to the monarchs who would lose theirrents from predation and expropriation as well as various other privileges associated withtheir monopoly of political power. This is why the institutional changes in England as aresult of the Glorious Revolution were not simply conceded by the Stuart kings. James IIhad to be deposed for the changes to take place.The reason why de facto political power is often used to read more..

##### Page - 1083

Introduction to Modern Economic GrowthImagine a dynamic model in which there are two state variables, political institutionsand the distribution of resources. For example, P (t) ∈ P denotes a speciﬁc set of politicalinstitutions in place at time t. This can be democracy or nondemocracy, parliamentary versuspresidential system, diﬀerent types of oligarchic institutions, etc. The set P denotes the entireset of feasible political institutions relevant for the situation we are studying. read more..

##### Page - 1084

Introduction to Modern Economic Growtheconomic institutions today and also one of the future state variables, the political institu-tions tomorrow, P (t +1) ∈ P. Intuitively speaking, the distribution of de facto and de jurepolitical power regulates what types of economic institutions will be in place today and alsoleaves the door open for potential political change, for example a switch from nondemocracyto democracy if such a change is necessary because of the balance of de facto and de read more..

##### Page - 1085

Introduction to Modern Economic Growthbecause it will not only illustrate the basic ideas highlighted here, but also shed light on theeconomic tradeoﬀs between diﬀerent political regimes.23.4.4. Another Example: The Emergence of Democracy.The framework pre-sented above is largely inspired by the the models of the emergence of democracy developedin Acemoglu and Robinson (2000a, 2006a). Acemoglu and Robinson construct a model ofthe emergence of democracy based on social conﬂict between the read more..

##### Page - 1086

Introduction to Modern Economic GrowthThere is little dissent amongst historians that the motive for the 1832 Reform was toavoid social disturbances. The 1832 Reform Act increased the total electorate from 492,700to 806,000, which represented about 14.5% of the adult male population. Yet, the majorityof British people could not vote, and the elite still had considerable scope for patronage, since123 constituencies still contained less than 1,000 voters. There is also evidence of read more..

##### Page - 1087

Introduction to Modern Economic Growthin Britain fell after the 1870s. At the same time as the ﬁscal reforms were taking place, therewere also major educational reforms changing the distribution of resources and distributionof opportunities in the British society in a major way. The Education Act of 1870 committedthe government to the systematic provision of universal education for the ﬁrst time, and thiswas made free in 1891. The school leaving age was set at 11 in 1893. In 1899, it was read more..

##### Page - 1088

Introduction to Modern Economic Growthelite. Democracy was, in many ways, forced on the elite, because of the threat of revolution.Nevertheless, democratization was not the only potential outcome in the face of pressure fromthe disenfranchised, or even in the face of the threat of revolution. Many other countries facedthe same pressures and political elites decided to repress the disenfranchised rather than makeconcessions to them. This happened with regularity in Europe in the 19th century, read more..

##### Page - 1089

Introduction to Modern Economic Growththe masses are able to solve their collective action problem and can threaten to undertake arevolution. A revolution is very costly for the elite, but generates only limited gains for themasses. These limited gains may nonetheless be better than living under elite control andthe inequitable distribution of resources that this involves. So when they are able to solvetheir collective action problem (with probability q), the revolution constraint of the read more..

##### Page - 1090

Introduction to Modern Economic Growthis constant and does not aﬀect transitions or the distribution of political power). In addi-tion, de jure political power is simply a non-stochastic outcome of political institutions; innondemocracy, the elite make the decisions; and in democracy, there is one person one vote,and the masses, thanks to their majority, become the decisive voters. Finally, there are verylimited economic decisions. The only relevant decision is one of taxation. Thus, in its read more..

##### Page - 1091

Introduction to Modern Economic Growthinstitutions.In the ﬁrst, labor markets are competitive and we index these institutions bythe subscript c (indicating “pro-citizen” or “competitive”). Let I (t) ∈ {e,c} denote theinstitutional choice in period t. Given the production technology each elite will make zeroreturns and each citizen will receive their marginal product of labor, A. When there arecompetitive labor markets, I (t)= c, the wage rate (and the wage earnings of each read more..

##### Page - 1092

Introduction to Modern Economic Growthdeﬁne∆R ≡ Re − Rc=(1 − λ)(1 − δ)ALM> 0,(23.38)and∆w≡ wc − we=(1 − λ (1 − δ)) A> 0(23.39)as the gains to the elite and the citizens from their more preferred economic institutions.Since the citizens are signiﬁcantly more numerous, i.e., L>>M , (23.38) and (23.39) implythat ∆R>> ∆w.There are two possible political regimes, democracy and nondemocracy, denoted respec-tively by D and N . The distribution of de read more..

##### Page - 1093

Introduction to Modern Economic Growthθi,s (t) ≥ 0 is(23.41)PCs (t)= φCsXi∈Cθi,s (t)+ ω (t)+ ηI (s (t)= D) ,where φCs > 0, ω (t) is a random variable drawn independently and identically over time froma given distribution F [·], I (s = D) ∈ {0,1} is an indicator function for s = D,and η is astrictly positive parameter measuring citizens’ de jure power in democracy. Equation (23.41)implies that in democracy the political power of the citizens shifts to the right in the senseof read more..

##### Page - 1094

Introduction to Modern Economic GrowthLet us ﬁrst focus on the symmetric Markov Perfect (Political Economy) Equilibrium(MPE) of this game (the results with non-symmetric equilibria and subgame perfect equilibriaare discussed in Exercise 23.17). As usual, a MPE imposes the restriction that equilibriumstrategies are mappings from payoﬀ-relevant states, which here only include s ∈ {D,N},andsince we formulate the model recursively we drop time subscripts from now on. In a MPEstrategies are not read more..

##### Page - 1095

Introduction to Modern Economic GrowthExpressed diﬀerently, the probability that the elite have political power in state s ∈ {N,D}is(23.43)p¡θi,θEs,θCs|s¢=F£φEs¡(M−1)θEs+θi¢−φCsLθCs−ηI(s=D)¤.As noted above, backward induction within the stage game implies that I (e)= e, I (c)= c,s0 (e)= N and s0 (c)= D. Thus returns to the citizens and the elite will be we and Re asgiven by (23.36) and (23.37) when π = e,and wc and Rc as in (23.34) and (23.35) whenπ = c. Incorporating read more..

##### Page - 1096

Introduction to Modern Economic Growthf0£φEN¡(M−1)θEN+θi¢−φCNLθCN¤<0.6 For future reference, let us also introduce the no-tation that θi ∈ ΓE£θE,θC|N¤ifθi is a solution to (23.45) that satisﬁes the second-ordercondition.Similarly, the value function for a citizen when the initial political state is s = N isVC¡N | θE ,θC¢=maxθi≥0.©−θi+p0¡θi,θEN,θCN | N¢¡we+βVC¡N | θE ,θC¢¢(1 − p0¡θi,θEN,θCN | N¢)¡wc+βVC¡D|θE,θC¢)¢ª,(23.46)which is read more..

##### Page - 1097

Introduction to Modern Economic GrowthhaveVC¡D|θE,θC¢=maxθi≥0©−θi+p0¡θi,θED,θCD|D¢¡we+βVC¡N | θE ,θC¢¢+¡1−p0¡θi,θED,θCD|D¢¢¡wc+βVC¡D|θE,θC¢¢ª,(23.51)where p0¡θi,θED,θCD|D¢isgivenby(23.47). The ﬁrst-order necessary condition is now(23.52)φCDf£φEDMθED−φCD¡(L−1)θCD+θi¢−η¤£∆w+β∆VC¤≤1,and θi≥ 0, with complementary slackness and the second-order conditionf0£φEDMθED−φCD¡(L−1)θCD+θi¢−η¤ > 0. We denote read more..

##### Page - 1098

Introduction to Modern Economic GrowthNext, incorporating symmetry and the fact that θCD = θCN =0 into the ﬁrst-order condi-tions (23.45) and (23.50), and assuming the existence of an interior solution (with θEN > 0and θED > 0), we obtain the following two equations that characterize interior equilibria:(23.54)φEN f£φENMθEN¤£∆R+β∆VE¤=1,and(23.55)φEDf£φEDMθED−η¤£∆R+β∆VE¤=1.The question is whether there exists such an interior solution. The following read more..

##### Page - 1099

Introduction to Modern Economic GrowthThe fact that f is single peaked (which has been assumed above) combined with the second-order conditions implies that φE MθEN = φE MθED− η,orinother words,(23.57)θED = θEN +ηφEM.(23.53) then implies that pD = pN ,which is the invariance result discussed in the Introduc-tion.Intuitively, in democracy the elite invest suﬃciently more to increase their de facto po-litical power so that they entirely oﬀset the democratic (de jure power) advantage read more..

##### Page - 1100

Introduction to Modern Economic GrowthDespite its importance, especially in the context of the debate on the relationship betweenpolitical regimes and growth, Proposition 23.10 may be viewed as a special case, because itdepends on the assumption that the technology for de facto political power for the elite isthe same in democracy and nondemocracy, i.e., φEN = φED. One may reasonably suspect thatthe elite may may be less eﬀective in using its resources to garner de facto political power read more..

##### Page - 1101

Introduction to Modern Economic Growthcan become an absorbing state and changes in political institutions will have more importanteﬀects. This is stated in the next proposition.Proposition 23.12. Suppose that L>>M and that there exists ¯θEN > 0 such that(23.58)φEN fhφENM¯θENi⎡⎣ ∆R − β¯θEN1 − βFhφENM¯θENi⎤⎦=1;and(23.59)η> −ω.Then there exists a symmetric MPE in which pN ∈ (0, 1) and pD =0.Proof. See Exercise 23.18.¤Therefore, if we relax part of read more..

##### Page - 1102

Introduction to Modern Economic GrowthUsing (23.54) and (23.60) and denoting the equilibrium level of θEN by θ∗N , we obtain:(23.61)φE f£φEMθ∗N¤∙∆R+β ηφE M¸=1.Similarly, denoting the equilibrium level of θED by θ∗D,wealsohave(23.62)φE f£φEMθ∗D−η¤∙∆R+β ηφE M¸=1.Finally, let us denote the probability that the elite will have political power by p∗ = pD =pN . This probability corresponds both to the probability that the elite will control politicalpower, read more..

##### Page - 1103

Introduction to Modern Economic GrowthThe factthatahigher β also increases the likelihood of labor repressive institutionsis somewhat more surprising. In many models, a higher discount factor leads to betterallocations. Here, in contrast, a higher discount factor leads to more wasteful activities bythe elite and to labor repressive economic institutions. The reason is that the main pivotalagents in this model are the elite, which, by virtue of their smaller numbers, are the onesinvesting in read more..

##### Page - 1104

Introduction to Modern Economic Growthand the political system. Instead, consistent with the discussion in the previous section, Iwill now assume that political institutions are more durable than economic institutions andpolicies, thus more diﬃcult to change. I will then show how this leads to a more extreme formof captured democracy, where, in equilibrium, democratic political institutions may emergeand survive for extended periods of time, but the elite are still able to impose their read more..

##### Page - 1105

Introduction to Modern Economic GrowthCorrespondingly, the value function for the elite in democracy can be written as:VE¡D|θE,θC¢ =maxθi≥0©−θi+p¡θi,θED,θCD|D¢Re+¡1−p¡θi,θED,θCD|D¢¢Rc+ˆp¡θi,θED,θCD|D¢βVE¡N | θE ,θC¢+¡1−ˆp¡θi,θED,θCD|D¢¢βVE¡D|θE,θC¢ª,(23.64)where I have already imposed that when the citizens have suﬃcient power they will choosedemocracy. With similar arguments to before, the maximization in (23.64) implies the fol-lowing read more..

##### Page - 1106

Introduction to Modern Economic Growthﬁnally, π =(c, e) corresponds to the citizens maintaining de jure power in democracy butlosing control over economic institutions (i.e., P CD (t)+ ξ> P ED (t) ≥ P CD (t)).The interesting result in this case is that once the society becomes democratic, it mayremain so potentially for a long time (i.e., ˆpD can be small), but the elite will still be able tocontrol the economic institutions (i.e., pD could be quite large). This is stated and proved read more..

##### Page - 1107

Introduction to Modern Economic Growthﬂuctuations between democracy and nondemocracy. But in democracy, there is no guaranteethat economic institutions will be those favored by the citizens. While in the baseline modelthe elite were able to impose both their political and economic wishes at the same time,here we have an equilibrium pattern whereby democracy persists, but the elite may be ableto impose their favorite economic institutions. In fact, the proposition shows that the elitemay be read more..

##### Page - 1108

Introduction to Modern Economic GrowthA natural conjecture based on the analysis there is to relate diﬀerences in economic insti-tutions to political institutions. For example, if political power is in the hands of an elitethat is opposed to growth, growth-enhancing policies are less likely to emerge. Our analysisin the previous chapter hinted that such considerations could be important. The empiricalevidence in Chapter 4 also provided support for such a view, whereby the cluster of read more..

##### Page - 1109

Introduction to Modern Economic Growthtradeoﬀs. While democracies may create static distortions because of their greater redis-tributive tendencies, they are likely to outperform oligarchies in the long run because theyavoid political sclerosis, whereby the incumbents are able to dominate the political systemand erect entry barriers to protect their businesses, even when eﬃciency dictates that otherindividuals should enter and form new businesses to replace theirs. Thus democracy may bemore read more..

##### Page - 1110

Introduction to Modern Economic Growththan the tip of the iceberg, and it is meant to entice the reader to think more about theseissues and to introduce the bare minimum that is necessary for a more coherent discussionof the relationship between political institutions and economic growth.23.7. References and LiteratureThis chapter relates to a large literature in political economy and political science. Be-cause of space constraints, I will not provide a comprehensive literature review. Instead, read more..

##### Page - 1111

Introduction to Modern Economic Growthcontrast of industrialization in Britain and France against the experiences of Russia andAustria-Hungary draws on Acemoglu and Robinson (2006b), which includes references tothe original literature. Mosse (1992) and Gross (1973) provide an excellent introduction tothe policies of Russian and Austria-Hungarian monarchies concerning industrialization andeconomic development. The model sketched at the end of Section 23.4 builds on Acemogluand Robinson (2000a, read more..

##### Page - 1112

Introduction to Modern Economic Growth(3) Next consider a model endogenizing q. In particular, imagine that the group out ofpower can choose to take power in any period but to do so must pay a non-pecuniarycost c.This cost c is drawn each period from the cumulative distribution G(c).Firstconsider the case without asset expropriation. Show that there will exist a level ofc∗ such that when c ≤ c∗, the group out of power will take power [Hint: write thevalue functions of the members of the read more..

##### Page - 1113

Introduction to Modern Economic GrowthExercise 23.10. Suppose that Condition 23.1 does not hold. Generalize the results in Propo-sitions 23.5 and 23.6.Exercise 23.11.(1) Prove Proposition 23.7.(2) Show that Condition 23.1 and (23.32) can be jointly satisﬁed.(3) Prove Proposition 23.8.Exercise 23.12. Consider the model in Section 23.3, starting with μ (0) = 1 and an oligarchicregime. Suppose that at some time t0 < ∞ a new technology arises, which is ψ> 1 asproductive as the old read more..

##### Page - 1114

Introduction to Modern Economic Growthwant to become entrepreneurs. Suppose that their talent distribution is also givenby G (a). Characterize the equilibrium in this case and show that it is unique.(4) Now consider the limiting equilibrium in part 3 with ε → 0. Explain why this limitleads to a unique equilibrium while there are multiple equilibria at ε =0.Exercise 23.14. Prove Proposition 23.8.Exercise 23.15. Consider an economy populated by λ rich agents who initially hold power,and 1 − read more..

##### Page - 1115

Introduction to Modern Economic Growth(6) Now suppose that μ (t)= μl with probability 1 − q,and μ (t)= μh with probabilityq,where μh > 1 − θ> μl. Construct a MPE where the rich extend the franchise,and from there on, a poor agent sets that tax rate. Be speciﬁc on the parametervalues that are necessary for such an equilibrium to exist. Explain why extension ofthe franchise is useful for rich agents?(7) Now consider non-Markovian equilibria again. Suppose that the unique MPE read more..

##### Page - 1116

CHAPTER 24Epilogue: Mechanics and Causes of Economic GrowthThis chapter contains concluding remarks. Instead of summarizing the models and ideaspresented so far, I will end with a brief discussion of what we have learned from the modelsand analyses presented in this book and how they oﬀer a useful perspective on world growthand cross-country income diﬀerences. I will then provide a very quick overview of some ofthe many remaining questions, which are important to emphasize both as a measure read more..

##### Page - 1117

Introduction to Modern Economic Growth(3) Endogenous investment decisions: while we can make considerable progress by un-derstanding the role of physical capital and human capital, and using cross-countrydata on diﬀerences in investments in machinery and in education to account for theprocess of economic growth and development, we also need to endogenize these in-vestment decisions. Investments in physical and human capital are forward-lookingand depend on the rewards that individuals expect read more..

##### Page - 1118

Introduction to Modern Economic Growthtechnologies that are developed will be responsive to rewards. Key factors inﬂuenc-ing the types of technologies that societies develop are again reward structures andfactor endowments. For example, changes in relative supplies of diﬀerent factors arelikely to eﬀect which types of technologies will be developed and adopted (Chapter15).(5) Linkages across societies and balanced growth at the world level : while endogenoustechnology and endogenous growth read more..

##### Page - 1119

Introduction to Modern Economic Growthincluded low productivity, high volatility in aggregate and individual outcomes, aMalthusian-type conﬁguration where increases in output were often accompanied byincreases in population does only having a limited eﬀect on per capita income, anda largely rural and agricultural economy. Another major aspect of stagnation hasbeen the failed growth attempts; many societies grew for certain periods of timeand then lapsed back into depressions and stagnation. read more..

##### Page - 1120

Introduction to Modern Economic Growthupon modern economic growth for a variety of interrelated reasons (Chapter 4).First, they directly determine the society’s reward structure, thus shaping whetherinvestments in physical and human capital and technological innovations are prof-itable. Second, they determine whether the infrastructure investments and contract-ing arrangements necessary for modern economic relations are present. For exam-ple, modern economic growth would be impossibleinthe read more..

##### Page - 1121

Introduction to Modern Economic Growthdiﬀerences today, we therefore need to study the following: (1) how political institu-tions aﬀect policies and economic institutions, thus shaping incentives for ﬁrms andworkers; (2) how political institutions themselves change, especially interacting witheconomic outcomes and technology; (3) why political institutions and the associatedeconomic institutions did not lead to sustained economic growth throughout history,why they enabled economic takeoﬀ read more..

##### Page - 1122

Introduction to Modern Economic Growthlink the proximate causes of economic phenomena to fundamental causes, and in particular toinstitutions. And here, I take a shortcut. Although I emphasized in Chapter 23 that there areno perfect political institutions and that each set of diﬀerent political arrangements is likely tofavor some groups at the expense of others, I will simplify the discussion here by making a coredistinction between two sets of institutional arrangements. The ﬁrst, which I read more..

##### Page - 1123

Introduction to Modern Economic Growthdid occur many times in the past. The human history is also full of major technological break-throughs. Even before the Neolithic Revolution, many technological innovations increased theproductivity of hunter gatherers. The transition to farming after about 9000BC is perhaps themost major technological revolution of all times; it led to increased agricultural productivityand to the development of socially and politically more complex societies. read more..

##### Page - 1124

Introduction to Modern Economic Growthadvantages. Thus it is not surprising that the improvements in living standards did not aﬀectthe entire society, but only a minority.Why did these growth episodes not turn into a process of takeoﬀ, ultimately leading tosustained growth? My main answer is related to that oﬀered in Section 23.3 in Chapter23. Growth under authoritarian regimes is possible. Entrepreneurs and workers can becomebetter at what they do, achieve better division of labor, and read more..

##### Page - 1125

Introduction to Modern Economic Growthdid not allow an autonomous middle-class to emerge as an economic and political force (Elvin,1973, Mokyr, 1991, Wong, 1997).The Ancient Greek and Roman civilizations are often viewed as the ﬁrst democraticsocieties. One might therefore be tempted to count them as participatory regimes thatshould have achieved sustained economic growth. But this is not necessarily the case. First,participatory regimes do not guarantee sustained economic growth when other read more..

##### Page - 1126

Introduction to Modern Economic Growththe rest of the society and the economy. Instead of abating, absolutism increased. Trade andindustry remained highly regulated, and groups not directly allied to the crown were viewedsuspiciously and discriminated against. The most extreme example of this, the persecution ofJews that had started under the Inquisition, continued and spilled over to other independentmerchants. Spain enjoyed the transfer of wealth from the colonies, but then experience avery read more..

##### Page - 1127

Introduction to Modern Economic GrowthIf this perspective is correct, there is no reason to expect that a similar takeoﬀ would hap-pen in other societies unless they were subject to similar, and similarly unusual, shocks orsome other process of intervention or change that induced them to undergo similar changes.Alternatively, one might suppose that the impetus for growth is ever present and is kept incheck by certain non-growth-enhancing institutions or market failures (Jones, 1988). Oncethe read more..

##### Page - 1128

Introduction to Modern Economic Growthpower base of the European authoritarian regimes that were largely unchallenged until theend of the Middle Ages (Pirenne, 1937).The second structural transformation was related. With the decline in population, realincomes increased in much of Europe, and many cities created suﬃciently large markets formerchants to seek new imports and for industrialists to seek new products (recall the im-pact of a decline in population on income per capita in the Solow or read more..

##### Page - 1129

Introduction to Modern Economic Growthof the Bank of England and other ﬁnancial reforms, the process of ﬁnancial development gotunderway.These constitutional regimes that emerged, ﬁrst in Britain and the Netherlands, then inFrance and other parts of Western Europe, paved the way for sustained economic growthbased on property rights for a broad cross-section of society, investment in contract enforce-ment, law and order, and infrastructure, and free entry into existing and new business read more..

##### Page - 1130

Introduction to Modern Economic Growthand state action to help them in their economic endeavors. By this time, with the collapseof the feudal order, the foundations of the authoritarian regimes that were in place in thethe Middle Ages were already weak. Nevertheless, the changes leading to the constitutionalregimes did not come easy. The Dutch had to ﬁght the Hapsburg monarchy to gain theirindependence as a republic. Britain had to endure the Civil War and the Glorious Revolu-tion. France had read more..

##### Page - 1131

Introduction to Modern Economic Growtheconomic and political modernization with the Meiji restoration (or perhaps even before)and a central element of this modernization eﬀort was the importation of new technologies.But these attitudes to new technologies were by no means universal. New technologieswere not adopted, they were in fact resisted, in many parts of the world. This includedmost of Eastern Europe, for example Russia and Austria-Hungary, where the existing land-based elites saw the read more..

##### Page - 1132

Introduction to Modern Economic Growthtechnologies and achieved average growth rates in line with the growth of the world technologyfrontier (and often exceeding those during their initial phase of catch-up). In most cases,this meant growth for the new members of the global economy, but not necessarily thedisappearance of the income gap between these new members and the earlier industrializers.In the meantime, many parts of the world continued to suﬀer political instability dis-couraging read more..

##### Page - 1133

Introduction to Modern Economic GrowthThus what I have presented above should be taken for what it is; a suggested answer thatneeds to be investigated more. But this is only one of the many remaining questions. Thepolitical economy of growth is important because it enables us to ask and answer questionsabout the fundamental causes of economic growth. But many other aspects of the process ofgrowth require further investigation. In some sense, the ﬁeld of economic growth is one of themore mature read more..

##### Page - 1134

Introduction to Modern Economic GrowthThe previous section emphasized how many economies started the growth process byimporting technologies and thus integrating into the global economy. Today we live in anincreasingly globalized and globalizing economy. But there is still much to understand abouthow technology is transferred from some ﬁrms to others, and from advanced economies to less-developed ones. The models I presented in Chapter 19 emphasized the importance of humancapital, barriers to read more..

##### Page - 1135

##### Page - 1136

Part 9Mathematical Appendices read more..

##### Page - 1137

##### Page - 1138

CHAPTER AOdds and Ends in Real Analysis and Applications toOptimizationThis chapter is included as a review of some basic material from real analysis. Its mainpurpose is to make the book self-contained and also include explicit statements of some ofthe theorems that are appealed to in the text. The material here is not meant to be acomprehensive treatment of real analysis. Space restrictions preclude me from attempting todo justice to any of the topics here, so my purpose is only a brief review. read more..

##### Page - 1139

Introduction to Modern Economic GrowthA.1. Distances and Metric SpacesThroughout, X denotes a set and x ∈ X is a generic element of the set X.A set Xcan be viewed as a space or as a subset of a larger set (space) Y . I denote a subset Y of Xas Y ⊂ X (which includes the case where Y = X). For any Y ⊂ X, X\Y stands for thecomplement of Y in X,i.e., X\Y = {x : x ∈ X and x/∈ Y }. Whenever I use the expressionX\Y it is implicit that Y is a subset of X.1Of special importance for our purposes read more..

##### Page - 1140

Introduction to Modern Economic Growth1 ≤ p< ∞. An extreme element of this family, which also deﬁnes an equivalentmetric on ﬁnite dimensional Euclidean spaces, is d∞ (x, y)=supi |xi − yi|.(2) For any nonempty set X, one can construct the discrete metric deﬁned as d (x, y)=1if x 6= y and d (x, y)=0 if x = y.In this case (X, d) is a discrete space.(3) Let X ⊂ RK and consider the set of continuous real-valued functions f : X →R denoted by C(X). A natural metric for C(X) is the read more..

##### Page - 1141

Introduction to Modern Economic Growth[0, 1] \(x − ε, x + ε)=[0,x − ε] ∪ [x + ε, 1] is closed in X.Also, Int(x − ε, x + ε)=(x − ε, x + ε),Int([0, 1] \(x − ε, x + ε)) = (0,x − ε)∪(x + ε, 1), (x − ε, x + ε)=[x − ε, x + ε],and [0, 1] \(x − ε, x + ε)=[0,x − ε] ∪ [x + ε, 1].Fact A.1. Let (X, d) be a metric space. X and ∅ are both open and closed sets.The importance of the following theorem will become clear once we turn to the somewhatmore abstract read more..

##### Page - 1142

Introduction to Modern Economic GrowthThe restriction to ﬁnite collections is important in Part 2 of Theorem A.1. Consider, thefollowing example.Example A.4. Xα =¡0,1+α−1¢ and consider the inﬁnite intersectionTα∈NXα.It can beveriﬁed thatTα∈NXα =(0, 1], which is not an open set.Definition A.4. Two metrics d and d0 deﬁned on sX are equivalent if they both generatethe same collection of open sets in X. Alternatively, let Nε and N0ε refer to neighborhoodsdeﬁned by these read more..

##### Page - 1143

Introduction to Modern Economic GrowthBy a function f I typically refer to a real-valued mapping, i.e., f : X → R for somearbitrary set X. I will use lowercase letters to refer to functions. I will use the term corre-spondence to refer to a set-valued mapping, i.e., F : X → P (Z) for some set Z. This meansthat the mapping F assigns a subset of Z to each element of x. I will use uppercase lettersto refer to correspondences. Since these will play an important role below, the followingcommon read more..

##### Page - 1144

Introduction to Modern Economic GrowthFact A.2. If a sequence {xn}∞n=1 or a net {xα}α∈A in X is convergent, then it has aunique limit point x∞ ∈ X.Proof. See Exercise A.7.¤Fact A.3. {xn}∞n=1 in X is convergent if and only if every subsequence of {xn}∞n=1 in Xis convergent.Proof. See Exercise A.8.¤Example A.7. Note, however, that convergence of subsequences (or subnets) does notguarantee convergence of the original sequence. Consider the sequence {xn}∞n=1 such thatxn =(−1)n. read more..

##### Page - 1145

Introduction to Modern Economic Growth(2) {xn}∞n=1 is convergent if and only iflim inf xn =lim sup xn,and in this case, we denote both of these as lim xn = x∞.(3) Let {yn}∞n=1 be another sequence in ¯R with xn ≤ yn for all n.Thenlim inf xn ≤ lim inf yn and lim sup xn ≤ lim sup yn,and moreover, if the limits exist,lim xn≤ lim yn.Proof. See Exercise A.11.¤Definition A.12. Let (X, d) be a metric space. A sequence {xn}∞n=1 in X is a Cauchysequence if for each ε> 0, there exists read more..

##### Page - 1146

Introduction to Modern Economic GrowthFact A.6. Equivalently, φ is continuous at x if for all {xn}∞n=0 → x, {φ(xn)}∞n=0 → φ (x).Fact A.7. Let (X, dX), (Y, dY ) and (Z, dZ) be metric spaces and consider the mappingsφ : X → Y and γ : Y → Z.If φ is continuous at x0 and γ is continuous at φ (x0),thenγ (φ (x)) = γ ◦ φ is continuous at x0.Proof. See Exercise A.12.¤Similarly, sums of continuous functions are continuous and ratios of real-valued continuousfunctions are read more..

##### Page - 1147

Introduction to Modern Economic Growthwe have [a, b] ⊂ f−1 (V ) ∪f−1 (V0), which implies that [a, b] is not connected, which is clearlyincorrect and yields a contradiction. Theorem A.3 then follows immediately since f ([a, b]) isconnected and thus includes any value between f (a) and f (b).¤The Intermediate Value Theorem is, in many ways, the simplest “ﬁxed point theorem”that economists can use in some applications (see Theorems A.16 and A.17 below for moregeneral ﬁxed point read more..

##### Page - 1148

Introduction to Modern Economic Growthby all of the open sets, but in a more economical fashion. Two convenient ways of doing thisare as follows. First, a topological space can be derived from a metric space. In particular,since a topological space (X, τ ) is deﬁned by a collection of open sets and a metric space (X, d)deﬁnes the collection of open sets in the space X, it also immediately deﬁnes a topologicalspace with the topology induced by the metric d. Second, a topological space can read more..

##### Page - 1149

Introduction to Modern Economic GrowthFact A.8. If a topological space (X, τ ) is metrizable with some metric d, then it has allthe topological properties of the metric space (X, d).Proof. This follows immediately from the fact that (X, τ ) and (X, d) have the sameopen sets.¤The preceding deﬁnition and fact are provided, because metric spaces are easier to workwith in practice than topological spaces. Nevertheless, sometimes (as with the product topol-ogy introduced in the next section), it read more..

##### Page - 1150

Introduction to Modern Economic GrowthThe proof of this theorem is identical to that of Theorem A.2 and is thus omitted.Unfortunately, in general topological spaces, convergence in terms of sequences is notsuﬃcient to characterize continuity. However, convergence in terms of nets is.Theorem A.5. (Continuity and Convergence of Nets) Let (X, τ X ) and (Y, τ Y )be topological spaces. The mapping φ : X → Y is continuous at x ∈ X if and only if{φ(xα)}α∈A → φ (x) for any net read more..

##### Page - 1151

Introduction to Modern Economic GrowthA proof of this proposition can be found in any real analysis textbook and I will notrepeat it here. Its main implication for us is that any K-dimensional intervalKQi=1[ai,bi],withai,bi∈ R and ai ≤ bi, is compact. The assumption that X is a Euclidean space is importantfor Theorem A.6, as illustrated by the following example.Example A.10. Consider the topological space ( , τ ) where τ is the topology induced bythe discrete metric and the subset0 read more..

##### Page - 1152

Introduction to Modern Economic GrowthProof. Let {Vα}α∈A0 be an open cover of φ (X0).Since φ is continuous, Theorem A.4implies that φ−1 (Vα) is open for each α ∈ A0.Since X0 is compact, every open cover has aﬁnite subcover and therefore there exists A00 ⊂ A0 such that X0 ⊂Sα∈A00φ−1 (Vα). Since, bydeﬁnition, φ¡φ−1(X00)¢⊂X00 for any X00 ⊂ X, this implies thatφ¡X0¢⊂ Sα∈A00(Vα) ,thus {Vα}α∈A00 is a ﬁnite subcover of {Vα}α∈A0, completing the read more..

##### Page - 1153

Introduction to Modern Economic Growthproperties of such product spaces? The answer is provided by the following deﬁnition andthe famous Tychonoﬀ Theorem, Theorem A.12 below.Definition A.22. Let A ⊂ R and {(Xα,τα)}α∈A be a collection of topological spaces.Let X =Qα∈AXα and for each α ∈ A,deﬁne the projection map Pα : X → Xα such thatP (x)= xα.Then, the product topology τ =Qα∈Aτ α is the topology such that all sets ofthe formSj∈JV j are open, where V j read more..

##### Page - 1154

Introduction to Modern Economic Growthtopology τ belong to τ0.Thus τ0 must be ﬁner than τ and establishes that the producttopology is the weakest topology in which each projection map is continuous.¤An implication of this lemma is that that a mapping φ : Y →Qα∈AXα is continuousaccording to the product topology if Pα◦ φ : Y → Xα is continuous for each α ∈ A.Theproduct topology is particularly useful in dynamic optimization problems because of thecombination of the read more..

##### Page - 1155

Introduction to Modern Economic Growthwhere the ﬁrst line uses the triangle inequality and the fact that {fn}n∈N is uniformlybounded, and the second line uses the deﬁnition of j.This inequality shows that©f¡xj¢ªj∈J → f (x∞) and establishes the continuity of f .¤Discounting is important in the previous result. The following example shows why.Example A.11. Suppose that fn : X → R is continuous and X is a compact metric space,and let f =P∞n=1fn : X∞ → R.It can be veriﬁed read more..

##### Page - 1156

Introduction to Modern Economic Growthreasons. First, even when a mapping into real numbersisawell behavedfunction, f : X → R,its inverse f−1 will typically be set-valued, thus a correspondence. Second, our main interestin most economic problems is with the “arg max”setsdeﬁned above, which are the subsets ofvalues in some set X that maximize a function. These will correspond to utility-maximizingconsumption, investment or price levels in simple economic problems. Finally, and read more..

##### Page - 1157

Introduction to Modern Economic Growth(2) F is lower hemi-continuous at x ∈ X if F (x) is nonempty and for every y ∈F (x) and every sequence {xn}∞n=1 → x, there exists some integer N andasequence{yn}∞n=1 with yn ∈ F (xn) for all n ≥ N ,and {yn}∞n=1 → y.Upper hemicontinuity and lower hemicontinuity according to Deﬁnition A.25 imply thecorresponding concepts in Deﬁnition A.24 for general metric spaces.Fact A.13. Let (X, dX) and (Y, dY ) be metric spaces and consider the read more..

##### Page - 1158

Introduction to Modern Economic GrowthProof. See Exercise A.19.¤For ﬁnite dimensional spaces, correspondences with closed graph are also upper hemi-continuous, provided that they satisfy a simple boundedness hypothesis.Fact A.15. Let X ⊂ RKX and Y ⊂ RKY and consider the correspondence F : X⇒ Y .Suppose that F has closed graph at x ∈ X and that there exists a neighborhood Vx of x suchthat F (Vx) is bounded. Then F that is upper hemi-continuous at x.Proof. Consider sequences {xn}∞n=1 read more..

##### Page - 1159

Introduction to Modern Economic GrowthThe fact that M (x) exists and thus Π (x) is nonempty for all x ∈ X follows from TheoremA.9. Consider a sequence{yn}∞n=1 → y such that yn ∈ Π (xn) for each n.Since G (x) isclosed, we have y ∈ G (x). Moreover, by deﬁnition, f (x, yn)= M (x) for each n.Since fis continuous, f (x, y)= M (x) follows. Therefore, y ∈ Π (x) and thus Π (x) is closed. SinceΠ (x) is a closed subset of the compact set G (x), we can invoke Lemma A.2 to conclude thatΠ read more..

##### Page - 1160

Introduction to Modern Economic Growthresults can be strengthened when we focus on problems with concave objective functions andconvex constraint sets, and then I will provide a brief illustration of how these strengthenedresults can be used. Throughout the rest of this appendix, let X be a vector space (or alinear space) so that if x, y ∈ X and λ is a scalar, then x + y ∈ X and λx ∈ X. Properties ofvector spaces will be discussed further in Section 10.1 below.Definition A.27. Aset X is read more..

##### Page - 1161

Introduction to Modern Economic Growthwhere G : X⇒ Y and f : X ×Y → R. Suppose that f is continuous and G is convex-valued,compact-valued and continuous at x.Then(1) If f is quasi-concave, then Π (x) = arg maxy∈Y {f (x, y): y ∈ G (x)} is nonempty,compact-valued, upper hemi-continuous, has closed graph and is convex-valued atx.(2) If f is strictly quasi-concave in a neighborhood of x,then Π (x) is a singleton.(3) If f satisﬁes the conditions in part 2 everywhere in X,then Π (x) is a read more..

##### Page - 1162

Introduction to Modern Economic Growth(2) If f is strictly quasi-convex, then in addition, Π (x) is a singleton.(3) If f satisﬁes the conditions in part 2 everywhere, then Π (x) is a continuous single-valued mapping.What makes continuous correspondences and thus Theorems A.13 and A.14 particularlyinteresting in many applications is the following simple result.Fact A.16. Let (X, dX) and (Y, dY ) be metric spaces and consider the continuous concavefunction g : Y → R. Then the set-valued read more..

##### Page - 1163

Introduction to Modern Economic GrowthA.7. Diﬀerentiation, Taylor Series and the Mean Value TheoremIn this and the next section, I brieﬂydiscuss diﬀerentiation and some important resultsrelated to diﬀerentiation that are useful for the analysis in the text. The material in thissection should be more familiar, thus I will be somewhat more brief in my treatment. In thissection, the focus is on a real-valued function of one variable f : R→ R. Functions of severalvariables and read more..

##### Page - 1164

Introduction to Modern Economic GrowthProof. See Exercise A.24.¤It is also useful to note that diﬀerentiability over some set X0 does not imply continuousdiﬀerentiability. The following example illustrates this point.Example A.13. Consider the function f such that f (x)= x2 sin (1/x) for all x 6=0 andf (0) = 0.It can be veriﬁed that f is continuous and diﬀerentiable, with derivative f0 (x)=2x sin (1/x) − cos (1/x) and f0 (0) = 0.But clearly, limx↓0 f0 (x) 6=0.Higher order read more..

##### Page - 1165

Introduction to Modern Economic Growthexists andeither limx→cf (x)= limx→cg (x)= 0 or limx→cf (x)= limx→cg (x)= ∞,then we have thatlimx→cf (x)g (x)=limx→cf0 (x)g0 (x).Proof. See Exercise A.26.¤The ﬁnal result in this section are the Taylor Theorem and the resulting Taylor Seriesapproximation to diﬀerentiable real-valued functions. For this theorem, let the nth derivativeof a real-valued function f be denoted by f (n) (e.g., f0 = f (1),etc.).Theorem A.20. (Taylor’s Theorem I) read more..

##### Page - 1166

Introduction to Modern Economic Growth(2) Suppose that f :[a, b] → R is a C∞ (real analytical) function. Thenf (y)= f (x)+ limn→∞nXk=0f (k) (x)k!(y − x)k .Proof. See Exercise A.27.¤A somewhat more useful corollary, which was used in the text, isCorollary A.3. Suppose that f :[a, b] → R is twice continuously diﬀerentiable andconcave. Then for any x, y ∈ [a, b],f (y) ≤ f (x)+ f0 (x)(y − x) .Proof. By Theorem A.20, f (y)= f (x)+ f0 (x)(y − x)+ f00 (z)(y − x)2 /2 for some read more..

##### Page - 1167

Introduction to Modern Economic Growththat(A.3)limh→0¯¯¯¯φ(x+h)−φ(x)−J(x)hkhk¯¯¯¯=0,where h ∈ X is a vector, and khk is the usual Euclidean norm of the vector h.We say thatthe mapping φ is a diﬀerentiable at x if the above limit exists and deﬁnes the unique J (x).Inthis case, the derivative of φ (x) is denoted by φ0 (x)= J (x). Once again, the derivative is alinear operator; it depends on x, but it assigns the value J (x) h to any vector h such that x+hin X0. The read more..

##### Page - 1168

Introduction to Modern Economic GrowthA general mapping φ : X → Y ,where Y is a subset of RKY can then be thought of asconsisting of KY real-valued functions of several variables, φ1 (x),...,φKY (x).We can deﬁnethe partial derivatives of each of these functions in a similar fashion and denote them byφjk (x). The Jacobian can then be written asJ (x)=⎛⎜⎜⎜⎜⎜⎜⎝φ11 (x)·· · φ1KX (x)···········φKY1(x) ·· · φKYKX (x)⎞⎟⎟⎟⎟⎟⎟⎠.Higher-order read more..

##### Page - 1169

Introduction to Modern Economic Growtheconomic analysis. Consider a mapping φ : X → X for X ⊂ RKX . One obvious question iswhether this mapping will have an inverse φ−1 : X → X. If for some subset X0 of X, φ issingle-valued, has an inverse φ−1, and also its inverse is also a single valued, then we say thatit is one-to-one.Theorem A.22. (The Inverse Function Theorem) Consider a C1 mapping φ : X →X for X ⊂ RKX . Suppose that the Jacobian of φ, J (x) evaluated at some interior read more..

##### Page - 1170

Introduction to Modern Economic GrowthA.9. Separation TheoremsIn this section, I will brieﬂy discuss the separation of convex disjoint sets using linearfunctionals (or hyperplanes). These results form the basis of the Second Welfare Theorem,provided in Theorem 5.7 in Chapter 5. They also provide the basis of many important resultsin constrained optimization (see Section A.10).For this section, we take X be a vector space (linear space). Recall from Section A.6)that this implies: if x, y ∈ X read more..

##### Page - 1171

Introduction to Modern Economic Growth(5) For X nonempty, consider the discrete metric d (x, y)= 1 if x 6= y and d (x, y)= 0if x = y. The metric space (X, d) is not a normed vector space.When the norm is understood implicitly, we refer to X as a normed vector space.Definition A.32. Let X be a normed vector space. Then φ : X → R is a linearfunctional on X if for any x, y ∈ X and any scalars λ and μ, we haveφ (λx + μy)= λφ (x)+ μφ (y) .Linear functionals on normed vector spaces have read more..

##### Page - 1172

Introduction to Modern Economic GrowthTherefore,|φ(x)| =¯¯¯¯φµδxkxk¶¯¯¯¯·kxkδ<ε · kxkδ= M kxk,with M = ε/δ, completing the proof.¤The smallest M that satisﬁes |φ(x)| ≤ M kxk for all x ∈ X is deﬁned as the norm of thelinear functional φ and is sometimes denoted by kφk. Theorem A.24 therefore implies that acontinuous linear functional has a ﬁnite norm.Definition A.33. Let X be a normed vector space. The space of all continuous linearfunctionals on X is the read more..

##### Page - 1173

Introduction to Modern Economic Growthis dominated by a semi-norm p (x), i.e., f (x) ≤ p (x) for all x ∈ M , then there is an extensionΦ of φ to the entire X such that Φ is a continuous linear functional on X, Φ (x)= φ (x) forall x ∈ M and Φ (x) ≤ p (x) for all x ∈ X. This theorem therefore establishes that normedvector spaces are “abundant” in linear functionals. More important for our purposes, it alsoimplies Theorem A.25. Since its proof is not particularly useful for our read more..

##### Page - 1174

Introduction to Modern Economic GrowthA.10. Constrained OptimizationMany of the problems we encountered in this book are formulated as constrained optimiza-tion problems. Chapters 6, 7, and 16 dealt with dynamic (inﬁnite-dimensional) constrainedoptimization problems. Complementary insights about these problems can be gained by us-ing the separation theorems of the previous section. Let me illustrate this here by focusing onﬁnite-dimensional optimization problems. In particular, suppose that read more..

##### Page - 1175

Introduction to Modern Economic Growthwhere b< 0 again means that each element of the N -dimensional vector b is negative. Y 1 isclearly convex. Moreover, the quasi-concavity of f and the convexity of g ensure that Y 2 isalso convex.By the hypothesis that x∗ is a solution to (A.5), the two sets are disjoint. Then TheoremA.26 implies that there exists a hyperplane separating these two sets. In other words, thereexists a nonzero vector η ∈ RN +1 such thatη · y1 ≤ c ≤ η · y2 for all read more..

##### Page - 1176

Introduction to Modern Economic Growthwhich establishes the ﬁrst inequality in (A.6). To establish the second inequality, again usethe complement the slackness condition and the fact that g (x∗) ≤ 0 to obtainL(x∗,λ∗)= f (x∗) ≤ f (x∗) − λ · g (x∗)= L(x∗,λ) for all λ ∈ RN+ ,which completes the proof of the ﬁrst part.(Part 2)Suppose to obtain a contradiction that (A.6) holds, but x∗ is not a solution to(A.5). This implies that there exists x0 ∈ X with g (x0) ≤ 0 read more..

##### Page - 1177

Introduction to Modern Economic Growthand the complementary slackness conditionλ∗ · g (x∗)=0holds.Proof. (Sketch)The constraint qualiﬁcation condition ensures that there exists a N1 +M -dimensional manifold at x∗,deﬁned by the equality and active inequality constraints. Sinceg and h are diﬀerentiable, this manifold is diﬀerentiable at x∗.Let vε (x) denote a feasibledirection along this manifold for small ε ∈ RK , in particular, such that x∗ ± εvε (x∗ + ε)remains read more..

##### Page - 1178

Introduction to Modern Economic Growth(2) Prove the generalization of this inequality for K = ∞.Exercise A.2. Using Minkowski’s inequality show that the metric spaces in Example A.1part 1 satisfy the triangle inequality.Exercise A.3. Show that the sup metric d∞ (f, g)= supx∈X |f (x) − g (x)| on C(X) inExample A.1 satisﬁes the triangle inequality.Exercise A.4. Using the deﬁnition of equivalent metrics in Deﬁnition A.4, show that if dand d0 are equivalent metrics on X, and a subset read more..

##### Page - 1179

Introduction to Modern Economic Growth(2) Show that if G1 (x) and G2 (x) are continuous, G1 (x) ∩ G2 (x) is not necessarilycontinuous. [Hint: consider G1 (x)=(−∞,x] and G2 (x)= {a,b} for some a 6= b].Exercise A.19. Prove Fact A.14.Exercise A.20. Prove Fact A.16.Exercise A.21.(1) Prove Part 2 of Theorem A.11.(2) Prove that the setF =n[y(t)]∞t=0: ˙x (t)= g (x (t) ,y (t)) with x (0) = x0 and limt→∞x (t) ≥ x1ois compact in the product topology if it is bounded in the sense that read more..

##### Page - 1180

Introduction to Modern Economic GrowthExercise A.32. Show that x =(1, 1, 1,...) is an interior point of +∞.[Hint: consider zε =(1 + ε, 1+ ε, ...),and show that z ∈ Nε (x) ⊂+∞].Exercise A.33. For the mapping g : X → RN for some X ⊂ RK , construct the set G ={x:g (x) ≤ 0}. Show that even when each component of g is not a convex function, the set Gcan be convex.Exercise A.34. Consider the problem of maximizing x subject to the constraint that x2≤ 0.Show that there exists a read more..

##### Page - 1181

##### Page - 1182

CHAPTER BReview of Ordinary Diﬀerential EquationsIn this chapter, I give a very brief overview of some basic results on diﬀerential equationsand also include a few results on diﬀerence equations. I limit myself to results that are usefulfor the material covered in the body of the text. In particular, I provide the backgroundfor the major theorems on stability, Theorems 2.2, 2.3, 2.4, 2.5, 7.18, and 7.19, which werepresented and then extensively used in the text. I will also provide some read more..

##### Page - 1183

Introduction to Modern Economic Growthis a solution to the equationAv= 0is the zero vector v=(0,..., 0).If Ais invertible, then there exists A−1 such thatA−1A = I.Conversely, if there exists a nonzero solution vor if det A=0,then Ais singular and doesnot have an inverse.Let a, b ∈ R and deﬁne the imaginary number i = √−1.Then χ = a + bi is a complexnumber.A complex number ξ is an eigenvalue of Aifdet (A−ξI)=0,or in other words, if by subtracting ξ times the identity matrix from read more..

##### Page - 1184

Introduction to Modern Economic Growthdenoted byRbaf(x)dx,canbedeﬁnedasfollows. Divide the interval [a, b] into N equal-sizedsegments, and letxj ≡ a + jb − aNfor j =0, 1,...,N.Then, form the Riemann sumb − aNN −1Xj=0f (xj) .The Riemann integral is obtained by taking arbitrarily ﬁner partitions of this interval, andis thus equal to(B.1)Z baf (x) dx =limN →∞b − aNNXj=0f (xj) .In fact, there are more general ways of creating partitions. When the Riemann integral iswell-deﬁned, read more..

##### Page - 1185

Introduction to Modern Economic GrowthTheorem B.3. (Integration by Parts) Let f :[a, b] → R and g :[a, b] → R continuousfunctions and let F :[a, b] → R and G :[a, b] → R be diﬀerentiable functions such that F0 (x)=f (x) and G0 (x)= g (x) for all x ∈ [a, b].ThenZ baF (x) g (x) dx = F (b) G (b) − F (a) G (a) −Z baG (x) f (x) dx.Theorem B.4. (Leibniz’s Rule) Let f (x, y) be continuous in x on [a, b] and diﬀeren-tiable in y at y0 and suppose that the functions a (y) and b (y) are read more..

##### Page - 1186

Introduction to Modern Economic Growthmeaning that time is not a separate argument. Alternatively, if it cannot be written thisway, it is a nonautonomous equation. In addition to ﬁrst-order diﬀerential equations, we canconsider, second order or nth order equations, for example,d2x (t)dt2= gµdx(t)dt,x (t) ,t¶,or(B.3)dnx(t)dtn= gµdn−1x(t)dtn−1,...,dx (t)dt,x (t) ,t¶.Iwillfocus on ﬁrst-order equations, since higher-order equations can always be transformedinto a system of ﬁrst-order read more..

##### Page - 1187

Introduction to Modern Economic GrowthandZ adt = at + c1,where c0 and c1 are constants of integration. Now taking exponents on both sides, the solutionto (B.5) is obtained as(B.6)x (t)= c exp (at) ,where c is a constant of integration combining c0 and c1 (in fact, c =exp (c1 − c0)). Diﬀer-entiating this equation, one easily obtains (B.5) and veriﬁes that (B.6) is indeed a solutionto (B.5). If (B.5) is speciﬁed as an initial value problem, then we also have a boundarycondition, which, read more..

##### Page - 1188

Introduction to Modern Economic Growthis the unique steady state. Inspection of (B.10) immediately shows that x (t) will approachthe steady-state value x∗ as t increases if a< 0 and it will diverge away from it if a> 0.Thisis naturally what we would expect from Theorem 2.4, which states that the steady state isasymptotically stable if a< 0.Finally, let us consider the most general case of the ﬁrst-order linear diﬀerential equation,that given in (B.4). Once again, with an analysis read more..

##### Page - 1189

Introduction to Modern Economic Growthtakes the formx(t)=nXj=1cj exp¡ξjt¢vξj,where vξ1,..., vξndenote the eigenvectors corresponding to the eigenvalues ξ1,...,ξn andc1,..., cn denote the constants of integration.Proof. The proof follows by diagonalizing the matrix A.In particular, since Ahas ndistinct real eigenvalues, recall thatP−1AP = D,where Dis a diagonal matrix with the eigenvalues ξ1,...,ξn on the diagonal andP=¡vξ1,...,vξn¢ is the matrix of the eigenvectors corresponding read more..

##### Page - 1190

Introduction to Modern Economic Growthdiﬀerentiable in its ﬁrst argument and satisﬁes(B.15)ddtΦ(t, s)= A(t) Φ(t, s) and Φ(t, t)= Ifor all t and s.The state-transition matrix is useful because it enables us to express the solutions to homoge-neous systems and then derive the solutions to (B.14) from the solutions to the correspondinghomogeneous systems. In particular, if ˆx(t) is a solution to the homogeneous system(B.16)˙x(t)= A(t) x(t) ,then it is straightforward to verify that (see read more..

##### Page - 1191

Introduction to Modern Economic GrowthTherefore,˙x(t) ≡ddtx(t)= A(t) Φ(t, 0) x0 + A(t)Z t0Φ(t, s) B(s) ds + B(t)= A(t) x(t)+ B(t) ,completing the veriﬁcation that (B.19) satisﬁes (B.14) with initial condition x(0) = x0.¤B.4. Stability for Nonlinear Diﬀerential EquationsSystems of nonlinear diﬀerential equations can be analyzed in the neighborhood of thesteady state by using Taylor’s Theorem (Theorem A.21). In particular, consider the systemof nonlinear autonomous diﬀerential read more..

##### Page - 1192

Introduction to Modern Economic GrowthB.5. Separable and Exact Diﬀerential EquationsWe cannot obtain explicit solutions to nonlinear diﬀerential equations in general (thoughexistence of solutions can be guaranteed under some conditions as shown in the next sec-tion). Nevertheless, two important special classes of diﬀerential equations, separable andexact diﬀerential equations, often enable us to derive explicit solutions. I start with separablediﬀerential equations. A diﬀerential read more..

##### Page - 1193

Introduction to Modern Economic GrowthAnother example, which is more relevant for economic applications, is given in ExerciseB.8.Next, consider a diﬀerential equation of the form (B.21) again, and suppose that thefunction g can be written asg (x (t) ,t) ≡G1 (x (t) ,t)G2 (x (t) ,t),whereG1 (x (t) ,t)=∂F (x (t) ,t)∂tand G2 (x (t) ,t)= −∂F (x (t) ,t)∂x,then (B.21) deﬁnes an exact diﬀerential equation. In particular, in this case, we can write˙x (t)=G1 (x (t) ,t)G2 (x (t) ,t)= read more..

##### Page - 1194

Introduction to Modern Economic GrowthB.6. Existence and Uniqueness of SolutionsInitial value problems generally enable us to establish the existence and uniqueness ofsolutions under relatively weak conditions. In fact, there are many related existence theorems.I will state the most basic existence theorem here, which extends the original theorem byPicard. Consider a ﬁrst-order diﬀerential equation(B.24)˙x (t)= g (x (t) ,t)deﬁnedonsomeinterval D ⊂ R, i.e., deﬁned for all t ∈ D. read more..

##### Page - 1195

Introduction to Modern Economic GrowthwhereG: X× D → Xand t ∈ D ⊂ R.Definition B.2. The system of ﬁrst-order diﬀerential equation (B.24) satisﬁes the Lip-schitz condition over the strip S = X× D if there exists a real number L< ∞ suchthat¯¯G(x,t)−G¡x0,t¢¯¯≤L¯¯x−x0¯¯for all x, x0 ∈ X and for all t ∈ D.Theorem B.10. (Picard’s Theorem II)Suppose that Gis continuous in all of itsarguments and satisﬁes the Lipschitz condition in Deﬁnition B.2. Then, there read more..

##### Page - 1196

Introduction to Modern Economic Growththen every solution ˜x (t) to the perturbed initial value problem˙x (t)= ˜g (x (t) ,t) with x (0) = ˜x0satisﬁes|˜x(t) − x (t)| <ε for all t ∈ D.Proof. See Walter (1991, Chapter 3.12).¤This theorem can also be extended to systems of diﬀerential equations.B.8. Diﬀerence EquationsSolutions to diﬀerence equations have many features that are common with the solutionsto diﬀerential equations. For example, the simple ﬁrst-order diﬀerence read more..

##### Page - 1197

Introduction to Modern Economic Growthwhere vξ1,..., vξndenote the eigenvectors corresponding to the eigenvalues ξ1,...,ξn andc1,..., cn denote constants determined by the initial conditions.Proof. The proof again follows by diagonalizing the matrix A.Recall that since Ahasn distinct real eigenvalues, we have P−1AP = D,where Dis a diagonal matrix with theeigenvalues ξ1,..., ξn on the diagonal and P=¡vξ1,...,vξn¢isthematrixoftheeigenvectorscorresponding to the eigenvalues. Let z(t) read more..

##### Page - 1198

Introduction to Modern Economic GrowthTheorem B.16. (Existence and Uniqueness of Solutions to Diﬀerence Equa-tions) Consider the system of ﬁrst-order nonlinear diﬀerence equations(B.29)x(t +1) = G(x (t)) ,where x(t) ∈ Rn, n is an integer, and G:Rn → Rn is an arbitrary mapping. Suppose that theinitial condition is speciﬁed as x(0) = x0. Then (B.29) has a unique solution for all t ∈ N.Proof. Given x0, x(1) is uniquely deﬁned as G(x0). Proceeding iteratively, we deter-mine a unique read more..

##### Page - 1199

Introduction to Modern Economic GrowthExercise B.9. Prove Theorem B.15.Exercise B.10. Consider the nth order diﬀerence equationx (t + n)= H (x (t + n − 1) ,..., x (t) ,t) ,where H : Rn → R. Prove that if the initial values x (0) ,x (1) ,...,x (n − 1) are speciﬁed, thisequation has a unique solution for any t.1178 read more..

##### Page - 1200

CHAPTER CBrief Review of Dynamic GamesThis chapter provides a very brief overview of some basic deﬁnitions, results and notationfor inﬁnite-horizon dynamic games. The reader is already assumed to be familiar with basicgame theory, and the notions of Nash Equilibrium and Subgame Perfect Nash Equilibriumin ﬁnite games. A review of these notions as well as much of the material covered here canbe found in standard graduate game theory textbooks such as Fudenberg and Tirole (1994),Myerson read more..

##### Page - 1201

Introduction to Modern Economic Growthconsistent with the types of models analyzed in the text, the action set of each player Ai (k)is only a function of the state variable k and not of calendar time.Each player has an instantaneous utility function ui (a (t) ,k (t)) whereui : A × K → Ris assumed to be continuous and bounded. This notation emphasizes that each player’s payoﬀdepends on the entire action proﬁle in that period (and not on past actions) and also on acommon vector of state read more..

##### Page - 1202

Introduction to Modern Economic Growthset of all potential histories at time t be denoted by Ht. It should be clear that any elementht ∈ Ht for any t corresponds to a subgame of this game.1Let a (pure) strategy for player i at time t beσi (t): Ht−1 × K → Ai,i.e., a mapping that determines what to play given the entire past history ht−1 and the currentvalueofthe statevariable k (t) ∈ K. This is the natural speciﬁcation of a strategy for timet given that ht−1 and k (t) entirely read more..

##### Page - 1203

Introduction to Modern Economic GrowthPerhaps the most popular alternative concept often used in dynamic games is that ofMarkov Perfect Equilibrium (MPE). The MPE diﬀers from the SPE in only conditioning onthe payoﬀ-relevant “state”. The motivation comes from dynamic programming where, as wehave seen, an optimal plan is a mapping from the state vector to the control vector. MPE canbe thought of as an extension of this reasoning to game-theoretic situations. The advantageof the MPE read more..

##### Page - 1204

Introduction to Modern Economic GrowthLet us next deﬁne:Definition C.2. A Markov Perfect Equilibrium (MPE) is a proﬁle of Markovian strate-gies ˆσ∗ =(ˆσ∗1,..., ˆσ∗N ) ∈ˆS such that the extension of these strategies satisfy ˆσ0∗i (t) ∈BR¡ˆσ0∗−i(t)|ht−1,k(t)¢forall¡ht−1,k(t)¢∈Ht−1×K,foralli∈N and for all t =0, 1,...Therefore, the only diﬀerence between MPE and SPE is that in the former atten-tion is restricted to Markovian strategies. It is important read more..

##### Page - 1205

Introduction to Modern Economic GrowthThis theorem basically implies that in dynamic games, we can check whether a strategy isa best response to other players’ strategy proﬁle by looking at one-stage deviations, keepingthe rest of the strategy proﬁle of the deviating player as given. The uniform boundednessassumption can be weakened to require “continuity at inﬁnity”, which essentially means thatdiscounted payoﬀs converge to zero along any history (and this assumption can also be read more..

##### Page - 1206

Introduction to Modern Economic Growthconstruct the strategy ˆσ∗i for each player i ∈ N such that ˆσ∗i (k)= ˆσ∗(i,k),i.e., ˆσ∗i : K → ∆ (Ai).This strategy proﬁle ˆσ∗ is Markovian. Consider the extension of ˆσ∗ to ˆσ0∗ as above, i.e.,ˆσ0∗i¡k,ht−1¢=ˆσ∗i(k) for all ht−1 ∈ Ht−1, k (t) ∈ K, i ∈ N and t. Then, by construction,given ˆσ0∗−i,itisimpossibletoimprove over ˆσ0∗iwith a deviation at any k ∈ K,thusTheoremC.1 implies that read more..

##### Page - 1207

Introduction to Modern Economic GrowthFinally, a well-known theorem for subgame perfect equilibria from repeated gamesalso generalizes to dynamic games. Let p (a | σ) be the probability distribution over theequilibrium-path actions induced by the strategy proﬁle σ, with the usual understandingthatRa∈Ap(a|σ)da=1forallσ∈S,whereAisasetofadmissibleaction proﬁles. Witha slight abuse of terminology, I will refer to p (a | σ) as the equilibrium-path action inducedby strategy σ. Then, read more..

##### Page - 1208

Introduction to Modern Economic Growthi in all subgames, and in response, the best that player i can doistoplayanequilibriumstrategy.By deﬁnition of a SPE and the minimum equilibrium payoﬀ of player i deﬁnedin(C.3),we haveUdit+1¡k(t+1),ht|σ∗−i¢≥UNi(k(t+1)).The preceding two inequalities implyUci [t]¡k(t),ht−1|σ∗¢≥ maxai(t)∈Ai(k)Eui¡ai(t),a−i¡σ∗−i¢|k(t),ht−1¢+βUNi(k(t)).Therefore,wecanconstructσ∗∗, whichisidenticaltoσ∗ exceptreplacingU di [t read more..

##### Page - 1209

Introduction to Modern Economic Growthas the minmax payoﬀ in this stage game. Let V ∈ RN be the set of feasible per period payoﬀsfor the N players, with vi corresponding to the payoﬀ to player i (so that discounted payoﬀscorrespond to vi/ (1 − β))Thenwehave:Theorem C.7. (The Folk Theorem for Repeated Games) Suppose that {Ai}i∈Nare compact. Then for any v ∈ V such that vi >mi for all i ∈ N,there exists ¯β ∈ [0, 1) suchthat for all β> ¯β, v can be supported as the read more..

##### Page - 1210

Introduction to Modern Economic Growththat the following grim strategy proﬁle implements (C,C) at every date: for both players, thestrategyistoplayCif ht includes only (C,C) and play D otherwise.Why this strategy combination is not a MPE is also straightforward to see. The grimstrategy ensures cooperation by conditioning on past history, i.e., it conditions on whethersomebody has defected at any point in the past. This history is not payoﬀ relevant for thefuture of the game given the action read more..

##### Page - 1211

Introduction to Modern Economic Growth(3) Next focus on “continuous” and symmetric MPE, where each agent will pursue astrategy of consuming cN (K) when the capital stock is K. Given symmetry, thisimplies that when all other agents are pursuing this strategy and agent i choosesconsumption c, this will imply a savings level ofS = AK − NcN (K) − c.Using this observation, write the value function of an individual as(C.5)VN (K)=maxS≤AK−NcN (K)©log¡AK−NcN(K)−S¢+βVN(S)ª.Explain read more..

##### Page - 1212

CHAPTER DList of TheoremsIn this appendix, I list the theorems presented in various diﬀerent chapters for reference.Many of these theorems refer to mathematical results used in diﬀerent parts of the book.Some of them are economic results that are more general and widely applicable than theresults I labeled “propositions”. To conserve space, I do not list additional mathematicalresults given in Lemmas, Corollaries and Facts.Chapter 2Theorem 2.1: Euler’s Theorem.Theorem 2.2: Stability read more..

##### Page - 1213

Introduction to Modern Economic GrowthTheorem 6.4: Concavity of the Value Function.Theorem 6.5: Monotonicity of the Value Function.Theorem 6.6: Diﬀerentiability of the Value Function.Theorem 6.7: The Contraction Mapping Theorem.Theorem 6.8: Applications of the Contraction Mapping Theorem.Theorem 6.9: Blackwell’s Suﬃcient Conditions for a Contraction.Theorem 6.10: Suﬃciency of Euler Equations and the Transversality Condition.Chapter 7Theorem 7.1: Variational Necessary Conditions for an read more..

##### Page - 1214

Introduction to Modern Economic GrowthChapter 16Theorem 16.1: Equivalence of Sequential and Recursive Formulations.Theorem 16.2: Principle of Optimality in Stochastic Dynamic Programming.Theorem 16.3: Existence of Solutions in Stochastic Dynamic Programming.Theorem 16.4: Concavity of the Value Function.Theorem 16.5: Monotonicity of the Value Function in State Variables.Theorem 16.6: Diﬀerentiability of the Value Function.Theorem 16.7: Monotonicity of the Value Function in Stochastic read more..

##### Page - 1215

Introduction to Modern Economic GrowthTheorem A.13: Berge’s Maximum Theorem.Theorem A.14: Properties of Maximizers under Quasi-Concavity.Theorem A.15: Properties of Minimizers under Quasi-Convexity.Theorem A.16: Kakutani’s Fixed Point Theorem.Theorem A.17: Brouwer’s Fixed Point Theorem.Theorem A.18: Mean Value Theorems.Theorem A.19: L’Hospital’s Rule.Theorem A.20: Taylor’s Theorem and Taylor Approximations.Theorem A.21: Taylor’s Theorem for Functions of Several Variables.Theorem read more..

##### Page - 1216

Introduction to Modern Economic GrowthTheorem B.15: Solution to Systems of Linear Diﬀerence Equations with Constant Coef-ﬁcients.Theorem B.16: Existence and Uniqueness of Solutions to Diﬀerence Equations.Appendix Chapter CTheorem C.1: One-Stage Deviation Principle.Theorem C.2: Existence of Markov Perfect Equilibria in Finite Dynamic Games.Theorem C.3: Existence of Subgame Perfect Equilibria in Finite Dynamic Games.Theorem C.4: Relationship between Markov and Subgame Perfect read more..

##### Page - 1217

##### Page - 1218

References (incomplete)Abramowitz, Moses (1957) “Resources an Output Trends in the United States since1870.” American Economic Review, 46, pp. 5-23.Abreu, Dilip (1998) “On the Theory of Inﬁnitely Repeated Games with Discounting.”Econometrica, 56, pp. 383-396.Acemoglu, Daron (1996) “A Microfoundation For Social Increasing Returns in HumanCapital Accumulation.” Quarterly Journal of Economics, 111 (3), pp 779-804.Acemoglu, Daron (1997a) “Training and Innovation in an Imperfect Labor read more..

##### Page - 1219

Introduction to Modern Economic GrowthAcemoglu, Daron (2008) “Innovation by Incumbents and Entrants.” MIT EconomicsDepartment Working Paper.Acemoglu, Daron, Philippe Aghion and Fabrizio Zilibotti (2006) “Distance to Frontier,Selection, and Economic Growth.” Journal of the European Economic Association, 4(1), pp.37-74.Acemoglu, Daron and Josh Angrist (2000) “How Large are Human Capital Externalities?Evidence from Compulsory Schooling Laws.” NBER Macroeconomics Annual 2000. MITPress, read more..

##### Page - 1220

Introduction to Modern Economic GrowthAcemoglu, Daron and James A. Robinson (2000b) “Political Losers as a Barrier to Eco-nomic Development.”American Economic Review, 90 (3), 126-130.Acemoglu, Daron and James A. Robinson (2001) “A Theory of Political Transi-tions.”American Economic Review, 91(2), 938-963.Acemoglu, Daron and James A. Robinson (2006a) Economic Origins of Dictatorship andDemocracy, New York; Cambridge University Press.Acemoglu, Daron and James A. Robinson (2006b) read more..

##### Page - 1221

Introduction to Modern Economic GrowthAllen, Robert C. (1992) Enclosure and the Yeoman, New York; Oxford University Press.Allen, Robert C. (2004) “Agriculture During the Industrial Revolution, 1700-1850.” inRoderick Floud and Paul A. Johnson (editors) Cambridge Economic History of ModernBritain, Cambridge University Press, Cambridge UK, pp. 96-116.Aliprantis, Charalambos and Kim Border (1999) Inﬁnite Dimensioal Analysis: A Hitch-hiker’s Guide. Springer-Verlag, New York, NY.Alesina, read more..

##### Page - 1222

Introduction to Modern Economic GrowthAutor, David, Lawrence Katz and Alan Krueger (1998) “Computing Inequality: HaveComputers Changed the Labor Market?” Quarterly Journal of Economics, 113, pp. 1169-1214.Bairoch, Paul (1988) Cities and Economic Development: From the Dawn of History tothe Present. (translated by Christopher Braider) University of Chicago Press, Chicago, IL.Balasko, Y. and Karl Shell (1980) “The Overlapping-Generations Model I: The Case ofPure Exchange without Money.” read more..

##### Page - 1223

Introduction to Modern Economic GrowthBasu, Kaushik (1997) Analytical Development Economics: The Less Developed EconomyRevisited. The MIT Press, Cambridge, MA.Basu, Susanto and David Weil (1998) “Appropriate Technology and Growth.” QuarterlyJournal of Economics, 113(4), pp. 1025-1054.Bates, Robert (1981) Markets and States in Tropical Africa. University of CaliforniaPress, Berkeley, CA.Baumol, William J., (1986) “Productivity Growth, Convergence, and Welfare: What theLong-Run Data Show.” read more..

##### Page - 1224

Introduction to Modern Economic GrowthBencivenga, Valerie and Bruce Smith (1995) “Unemployment, Migration and Growth.”Working Paper 95-17, Center for Analytic Economics, Cornell University.Benhabib, Jess and Mark M. Spiegel (2000) “The Role of Financial Development inGrowth and Investment.” Journal of Economic Growth, 5, pp. 341-360.Benhabib, Jess and Mark M. Spiegel (2005) “Human Capital and Technology Diﬀusion.”in Philippe Aghion and Steven Durlauf (editors) Handbook of Economic read more..

##### Page - 1225

Introduction to Modern Economic GrowthBorder, Kim (1989) Fixed Point Theorems in Economics. Cambridge University Press,Cambridge, UK.Borjas, George J. (1992) “Ethnic Capital and Intergenerational Mobility.” Quarterly Jour-nal of Economics, 107, pp. 123-150.Boserup, Ester (1965) The Conditions of Agricultural Progress. Aldine Publishing Com-pany, Chicago.Bourguignon, Francois and Christian Morrison (2002) “Inequality Among World Citizens:1820-1992.” American Economic Review, 92, pp. read more..

##### Page - 1226

Introduction to Modern Economic GrowthCarrol, Christopher, Byung-Kun Rhee and Changyong Rhee (1994) “Are There CulturalEﬀects on Saving? Some Cross-Sectional Evidence.” Quarterly Journal of Economics, 109,pp. 685-699.Caselli, Francesco (2005) “Accounting for Cross-Country Income Diﬀerences.” in PhilippeAghion and Steven Durlauf (editors) Handbook of Economic Growth, North Holland, Ams-terdam, pp. 680-743.Caselli, Francesco and Wilbur John Coleman (2001) “Cross-Country Technology read more..

##### Page - 1227

Introduction to Modern Economic GrowthCoatsworth, John H. (1993) “Notes on the Comparative Economic History of Latin Amer-ica and the United States.”in Walter L. Bernecker and Hans Werner Tobler eds. Develop-ment and Underdevelopment in America: Contrasts in Economic Growth in North and LatinAmerica in Historical Perpsective, Walter de Gruyter, New York.Coe, David T. and Elhanan Helpman (1995) “International R&D Spillovers.” EuropeanEconomic Review, 39, pp. 857-887.Cohen, Wesley M. read more..

##### Page - 1228

Introduction to Modern Economic GrowthDiamond, Jared M. (1997) Guns, Germs and Steel: The Fate of Human Societies.W.W.Norton & Co., New York NY.Diamond, Peter (1965) “National Debt in a Neoclassical Growth Model.” American Eco-nomic Review, 55, pp. 1126-1150.Diamond, Peter, Daniel McFadden and Miguel Rodriguez (1978) “Measurement of Elas-ticity of Factor Substitution and Bias of Technical Change.” In Fuss, Melvyn and DanielMcFadden (editors) Production Economics: A Dual Approach to read more..

##### Page - 1229

Introduction to Modern Economic GrowthDunne, Timothy, Mark J. Roberts and Larry Samuelson (1989) “The Growth and Failureof US Manufacturing Plants.” Quarterly Journal of Economics, 104(4), pp. 671-698.Durlauf, Steven (1991) “Nonergodic Economic Growth.”Review of Economic Studies, 60,pp. 349-366.Durlauf, Steven (1996) “A Theory of Persistent Income Inequality.” Journal of EconomicGrowth, 1, pp. 75-94.Durlauf, Steven and Paul A. Johnson (19950 “Multiple Regimes and read more..

##### Page - 1230

Introduction to Modern Economic GrowthEngerman, Stanley L. (1981) “Notes on the Patterns of Economic Growth in the BritishNorth America Colonies in the Seventeenth, Eighteenth and Nineteenth Centuries” in Dis-parities in Economic Development since the Industrial Revolution, Paul Bairoch and MauriceLevy-Leboyer, eds., St. Martin’s Press, 1981, New York.Engerman, Stanley and Kenneth Sokoloﬀ (1994) “Factor Endowments, Institutions, andDiﬀerential Paths of Growth among New World read more..

##### Page - 1231

Introduction to Modern Economic GrowthFunk, Peter (2002) “Induced Innovation Revisited.”Economica, 69, 155-171.Gabaix, Xavier (2004) “Zipf’s Law for Cities: An Explanation.” Quarterly Journal ofEconomics, 114, pp. 739-767.Galenson, David W. (1996) “The Settlement and Growth of the Colonies: Population,Labor and Economic Development.”in Stanley L. Engerman and Robert E. Gallman eds. TheCambridge Economic History of the United States, Volume I, The Colonial Era, CambridgeUniversity read more..

##### Page - 1232

Introduction to Modern Economic GrowthGlomm, Gerhard and B. Ravikumar (1992) “Public vs. Private Investment in HumanCapital: Endogenous Growth and Income Inequality.” Journal of Political Economy, 100(4),pp. 818-834.Greenwood, Jeremy and Zvi Hercowitz (1991) “The Allocation of Capital and Time overthe Business Cycle.” Journal of Political Economy, 99, pp. 1188-1214.Greenwood, Jeremy, Zvi Hercowitz and Per Krusell (1997) “Long-Run Implications ofInvestment-Speciﬁc Technological read more..

##### Page - 1233

Introduction to Modern Economic GrowthHalkin, Hubert (1974) “Necessary Conditions for Optimal Control Problems with InﬁniteHorizons.” Econometrica, 42, pp. 267-272.Hall, Bronwyn (1987) “The Relationship between Firm Size and from Growth in the USManufacturing Sector.” JournalofIndustrialEconomics, 20, pp. 583-606.Hall, Robert E. (1978) “Stochastic Implications of the Life-Cycle - Permanent IncomeHypothesis: Theory and Evidence.” Journal of Political Economy, 86(6), read more..

##### Page - 1234

Introduction to Modern Economic GrowthHenderson, J. Vernon (1988) Urban Development: Theory, Fact, and Illusion.OxfordUniversity Press, Oxford, UK.Hendricks, Lutz (200) “How Important is Human Capital for Development? Evidencefrom Immigrant Earnings.” American Economic Review, 92(1), pp. 198-219.Heston, Allen, Robert Summers and Bettina Aten (200) Penn World Tables Version6.1. Downloadable Data Set. Center for International Comparisons at the University ofPennsylvania.Hicks, John (1932) The read more..

##### Page - 1235

Introduction to Modern Economic GrowthIrwin, Douglas and Peter Klenow (1994) “Learning-by-Doing Spillovers in the Semicon-ductor Industry.” Journal of Political Economy, 102(6), pp. 1200-1227.Jayaratne, Jay and Philip Strahan (1996) “The Finance-Growth Nexus: Evidence fromBank Branch Deregulation.” Quarterly Journal of Economics, 111, pp. 639-670.Jones, Charles I. (1995) “R&D-Based Models of Economic Growth.” Journal of PoliticalEconomics, 103, pp. 759-784.Jones, Charles I. read more..

##### Page - 1236

Introduction to Modern Economic GrowthJovanovic, Boyan and Saul Lach (1989) “Entry, Exit & Diﬀusion with Learning by Doing.”American Economic Review, 79(4), 690-699.Jovanovic, Boyan and Yaw Nyarko (1996) “Learning by Doing and the Choice of Tech-nology.” Econometrica, 64, pp. 1299-1310.Judd, Kenneth (1985) “On the Performance of Patents” Econometrica, 53, pp. 567-585.Judd, Kenneth (1998) Numerical Methods in Economics, MIT Press, Cambridge.Kaldor, Nicholas (1957) “Alternative read more..

##### Page - 1237

Introduction to Modern Economic GrowthKlenow, Peter J (1996) “Industry Innovation: Where and Why?” Carnegie-RochesterConference Series on Public Policy, 44, pp. 125-150.Klenow, Peter J. and Andres Rodriguez-Clare (1997) “The Neoclassical revival in GrowthEconomics: Has It Gone Too Far?.”NBER Macroeconomics Annual, 73-103.Klenow, Peter J and Anders Rodriguez-Clare (2005) “Externalities and Growth.” inPhilippe Aghion and Steven Durlauf (editors) Handbook of Economic Growth,North read more..

##### Page - 1238

Introduction to Modern Economic GrowthKrussel, Per and Anthony Smith (1998) “Income and Wealth Heterogeneity in the Macro-economy.” Journal of Political Economy, 106(5), pp. 867-896.Krusell, Per and José-Víctor Ríos-Rull (1996) “Vested Interests in a Theory of Stagnationand Growth.”Review of Economic Studies, 63, 301-330.Krusell, Per and José-Víctor Ríos-Rull (1999) “On the Size of Government: PoliticalEconomy in the Neoclassical Growth Model.”American Economic Review, 89, read more..

##### Page - 1239

Introduction to Modern Economic GrowthLevine, Ross and David Renelt (1992) “A Sensitivity Analysis of Cross-Country GrowthRegressions.” American Economic Review, 82, pp. 942-963.Levine, Ross and Sara Zervos (1998) “Stock Markets, Banks, and Economic Growth.”American Economic Review, 88, pp. 537-558.Lewis, William Arthur (1954) “Economic Development with Unlimited Supplies of La-bor.“Manchester School of Economics and Social Studies, 22, pp. 139-191.Lieberman, M. B. (1984) “The read more..

##### Page - 1240

Introduction to Modern Economic GrowthMcDaniel, Timothy (1991) Autocracy, Modernization and Revolution in Russia and Iran,Princeton; Princeton University Press.McEvedy, Colin and Richard Jones (1978) Atlas of World Population History,New York;Facts on File.Maddison, Angus (2001) The World Economy: A Millennial Perspective. DevelopmentCentre, Paris.Maddison, Angus (2003) The World Economy: Historical Statistics.CD-ROM. OECD,Paris.Malthus, Thomas R. (1798) An Essay on the Principle of read more..

##### Page - 1241

Introduction to Modern Economic GrowthMitch, David (1983) “The Role of Human Capital in the First Industrial Revolution.”inJoel Mokyr ed. The British Industrial Revolution: An Economic Perspective, San Francisco;Westview Press.Mokyr, Joel (1990) The Lever of Riches: Technological Creativity and Economic Progress.Oxford University Press, New York.Mokyr, Joel (1993) “ Introduction” The British Industrial Revolution,edited by JoelMokyr, Westview Press, Boulder Colorado.Morck, Randall, read more..

##### Page - 1242

Introduction to Modern Economic GrowthNorth, Douglass C. and Barry R. Weingast (1989) “Constitutions and Commitment:Evolution of Institutions Governing Public Choice in Seventeenth Century England, Journalof Economic History, 49, 803-832.Nurske, Ragnar (1958) Problems of Capital Formation in Underdeveloped Countries. Ox-ford University Press, New York.Obstfeld, Maurice (1994) “Risk-Taking, Global Diversiﬁcation, and Growth.” AmericanEconomic Review, 84, pp. 1310-1329.Ok, Efe (2007) Real read more..

##### Page - 1243

Introduction to Modern Economic GrowthPontryagin, Lev S. et al (1962) The Mathematical Theory of Optimal Processes.Inter-science Publishers, New York, NY.Postan, M. M. (1966) “Medieval Agrarian Society in its Prime: England.”in M.M. Postaned. The Cambridge Economic History of Europe, London; Cambridge University Press.Pratt, John W (1964) “Risk Aversion in the Small and in the Large.” Econometrica,32(1-2), pp. 122-136.Prescott, Edward (1998) “Needed: A Theory of Total Factor read more..

##### Page - 1244

Introduction to Modern Economic GrowthRamey, Garey and Valerie Ramey (1995) “Cross-Country Evidence of the Link BetweenVolatility and Growth.” American Economic Review, 88, pp. 1138-1151.Ramsey, Frank (1928) “A Mathematical Theory of Saving.” Economic Journal, 38, pp.543-559.Ricardo, David (1817) On the Principles of Political Economy and Taxation. CambridgeUniversity Press, Cambridge, UK.Rivera-Batiz, Luis A. and Paul M. Romer (1991) “Economic Integration and EndogeousGrowth.” read more..

##### Page - 1245

Introduction to Modern Economic GrowthRothschild, Michael and Joseph Stiglitz (1971) “Increasing Risk II: Its Economic Conse-quences.” Journal of Economic Theory, 3(1) pp. 66-84.Rothstein, Paul (1991) “Representative Voter Theorems.”Public Choice, 72, pp. 193-212.Royden, Halsey (1994) Real Analysis. Macmillan, New York, NY.Rudin, Walter 91976) Introduction to Mathematical Analysis. McGraw-Hill, New York,NY.Rybczynski, T. M. (1955) “Factor Endowment and Relative Commodity Prices.” read more..

##### Page - 1246

Introduction to Modern Economic GrowthSeierstad, Atle and Knut Sydsaeter (1987) Optimal Control Theory with Economic Ap-plications. El Sevier Press, Amsterdam, Holland.Shapley, L. (1953) “A Value for n-Person Games.” In Kuhn, H. and A. Tucker, eds.,Contributions to the Theory of Games. Princeton University Press, Princeton, NJ.Shell, Karl (1967) “A Model of Inventive Activity and Capital Accumulation.” in KarlShell, (editor), Essays on the Theory of Optimal Economic Growth, MIT Press, read more..

##### Page - 1247

Introduction to Modern Economic GrowthSutherland, William (1975) Introduction to Metric and Topological Spaces. ClarendonPress, Oxford, UK.Sutton, John (1997) “Gibrat’s Legacy.” Journal of Economic Literature, 35, pp. 40-59.Sutton, John (1998) Technology and Market Structure:Theory and History. MIT Press,Cambridge, MA.Swan, Trevor W. (1956) “Economic Growth and Capital Accumulation.” EconomicRecord, 32, pp. 334-361.Tawney, R.H. (1926) Religion and the Rise of Capitalism: A Historical read more..

##### Page - 1248

Introduction to Modern Economic GrowthVentura, Jaume (1997) “Growth and Independence” Quarterly Journal of Economics,112, pp. 57-84.Ventura, Jaume (2005) “A Global View of Economic Growth.” in Philippe Aghion andSteven Durlauf (editors) Handbook of Economic Growth, North Holland, Amsterdam, pp. ???Vernon, Raymond (1966) International Investment and International Trade in Product-Cycle.” Quarterly Journal of Economics, 80, pp. 190-207.Walter, Wolfgang (1991) Ordinary Diﬀerential read more..

**Write Your Review**