Introduction to Economic Growth. Daron Acemoglu MIT Department of Economics

Transcription

1 Introduction to Economic Growth Daron Acemoglu MIT Department of Economics January 2006

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3 Contents I Introduction 1 1 StylizedFactsofEconomicGrowthandDevelopment A Quick Look at the Facts Interpretation TheAgenda The Solow Growth Model TheBasicModelinDiscreteTime TheProductionStructure Endowments FundamentalLawofMotionoftheSolowModel Definition of Equilibrium Equilibrium Without Population Growth and Technological Progress TransitionalDynamicsintheSolowModel TheSolowModelinContinuousTime From Difference to DifferentialEquations The Fundamental Equation of the Solow Model in Continuous Time AFirstLookatSustainedGrowth iii

4 2.3 SolowModelwithTechnologicalProgress Balanced Growth NeutralTechnologicalProgress TheSteady-StateTechnologicalProgressTheorem The Solow Growth Model with Technological Progress: Continuous Time The Solow Model and the Data Growth Accounting Solow Model and Cross-Country Income Differences SolowModelwithHumanCapital ProblemswiththeMankiw,RomerandWeilApproach The Macro Mincer Approach (Bils-Klenow-Rodriguez-Hall-Jones) An Alternative Approach to Estimating Productivity Differences (Trefler) Fundamental Determinants of Differences in Income FromProximatetoFundamentalCauses Hypotheses Europe sexpansionandcolonialoriginsofinstitutions II Neoclassical Growth 95 5 Towards Neoclassical Growth Representative Consumer Problem Formulation Welfare Theorems iv

5 5.4 OptimalGrowthinDiscreteTime OptimalGrowthinContinuousTime Dynamic Programming and Optimal Growth BriefReviewofDynamicProgramming Digression: Technical Details ContractionMappings Application of Contraction Mappings to Dynamic Programming BacktotheFundamentalsofDynamicProgramming Basic Equations DynamicProgrammingVersustheSequenceProblem OptimalGrowthinDiscreteTime CompetitiveEquilibriumGrowth Brief Review of Optimal Control Finite-HorizonOptimalControl TheFundamentalProblem VariationalArguments SimplifiedMaximumPrinciple Generalizations Limitations Infinite-HorizonOptimalControl The Basic Problem: Necessary and SufficientConditions LackofTransversalityConditions Discounted Infinite-HorizonOptimalControl v

6 8 The Neoclassical Growth Model Preferences,TechnologyandDemographics Characterization of Equilibrium Definition of Equilibrium TheConsumerProblem Equilibrium Prices OptimalGrowth Steady-State Equilibrium Transitional Dynamics TechnologicalChangeandtheCanonicalNeoclassicalModel The Role of Policy Quantitative Evaluations Policy Differences Extensions VariantsoftheNeoclassicalModel Growth with Overlapping Generations Problems of Infinity OverlappingGenerationsandOveraccumulation Demographics,PreferencesandTechnology ConsumptionDecisions Equilibrium More Specific Utility Functions Pareto Optimality RoleofSocialSecurityinCapitalAccumulation vi

7 9.3.1 FullyFundedSocialSecurity UnfundedSocialSecurity Recitation Material: Stochastic Growth The Brock-Mirman Model Application: Risk, DiversificationandGrowth The Environment Equilibrium Dynamics Efficiency Implications InefficiencywithAlternativeMarketStructures III Endogenous Growth First-Generation Models of Endogenous Growth AK Model Revisited Demographics,PreferencesandTechnology Equilibrium TransitionalDynamics The Role of Policy The Extended AK Model Growth with Externalities PreferencesandTechnology Equilibrium ParetoOptimalAllocations vii

8 12 Multiple Equilibria and the Process of Development MultipleEquilibriaFromAggregateDemandExternalities PreferencesandTechnology Equilibrium HumanCapitalAccumulationwithImperfectCapitalMarkets ASimpleCaseWithNoBorrowing TheGalorandZeiraModel Learning-by-Doing,StructuralChangeandNon-BalancedGrowth Demographics,PreferencesandTechnology Equilibrium Interdependence and Growth in the Open Economy HumanCapitalandTechnology(Nelson-Phelps) Trade and Technology Diffusion TheBasicKrugmanModel Understanding the EffectsofTrade Trade,SpecializationandtheWorldIncomeDistribution TheModel Equilibrium Implications GrowthwithFactorPriceEqualization IV Endogenous Technological Change Expanding Variety Models TheLab-EquipmentModelofGrowthwithProductVarieties viii

9 Demographics,PreferencesandTechnology DigressiononContinuousTimeValueFunctions CharacterizationofEquilibrium Definition of Equilibrium Steady State TransitionalDynamics ParetoOptimalAllocations PolicyintheEndogenousTechnologyModel Growth with Knowledge Spillovers TheRoleofCompetitionPolicy Growth without Scale Effects Models of Quality Competition BaselineModel Pareto Optimality Directed Technical Change Basics and Definitions Definitions BasicModel Implications EquilibriumTechnologyBias:SomeMoreGeneralResults EndogenousLabor-AugmentingTechnologicalChange Demographics,PreferencesandTechnology ConsumerandFirmDecisions AsymptoticandBalancedGrowthPaths ix

10 TheBalancedGrowthPath TransitionalDynamics Policy Implications Recitation Material: Appropriate Technology DifferencesinCapital-LaborRatios(Atkinson-Stiglitz) The Role of Human Capital (Acemoglu-Zilibotti) AModel Implications Calibration Epilogue: Political Economy of Growth ThinkingofInstitutionsandGrowth TheImpactofInstitutions Modeling Institutional Differences Institutions in Action ASimpleModelofNon-GrowthEnhancingInstitutions Baseline Model EconomicEquilibrium InefficientPolicies Revenue Extraction FactorPriceManipulation Revenue Extraction and Factor Price Manipulation Combined PoliticalConsolidation Subgame Perfect Versus Markov Perfect Equilibria LackofCommitment Holdup x

11 TechnologyAdoptionandHoldup Inefficient Economic Institutions ModelingPoliticalInstitutions DictatorshipoftheMiddleClass Democracy Inefficiency of Political Institutions and Inappropriate Institutions InstitutionalChangeandPersistence xi

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13 Part I Introduction 1

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15 We start with a quick look at the stylized facts of economic growth and the most basic model of growth, the Solow growth model. The purpose is to both prepare us for the analysis of more modern models of economic growth with forward-looking behavior and explicit capital accumulation and technological progress, and also give us a way of mapping the simplest model to the data. I will also discuss differences between proximate and fundamental causes of economic growth and development. 3

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17 Chapter 1 Stylized Facts of Economic Growth and Development 1.1 AQuickLookattheFacts There are very large differences in income per capita or output per worker across countries today. Countries at the top of the world income distribution are thirty times as rich as countries at the bottom in PPP adjusted dollars. For example, in 2000, GDP per capita in the United States was $32500 (valued at 1995 $ prices). In contrast, income per capita is much lower in many other countries: $9000 in Mexico, $4000 in China, $2500 in India, $1000 in Nigeria, and much much lower in some other sub-saharan African countries such as Chad, Ethiopia, Mali (all figures adjusted for purchasing power parity). The gap is larger when there is no PPP adjustment. The next figure shows a cross-sectional look at these income-level differences in the year

18 Should we care about cross-country income differences? The answer is a big yes. High income levels reflect high standards of living. It is true that together with economic growth, pollution increases and individual aspirations may also increase so that the same bundle of consumption may no longer make an individual as happy. But at the end of the day, when one compares an advanced, rich country with a less-developed one, there are striking differences in the quality of life, standards of living and health. In fact, it is even difficult for us to imagine the burden of poverty at the levels experienced by countries in sub-saharan Africa. There is little doubt that the consumption level, living standards and health level of richer countries are appreciably higher than those with lower income per capita. These gaps represent big welfare differences. Understanding how some countries can be so rich while some others are so poor is one of the most important, perhaps the most important, challenges facing social science. 6

19 How could a country be 30-times or so richer than another? The answer lies in differences in growth rates. Take two countries, A and B, with the same level of income to start with. Imagine that country A has 0% growth per capita, so its income per capita remains constant, while country B grows at 2% per capita. In 200 years time country B will be more than 52 times richer than country A. Therefore, the United States is considerably richer than Nigeria because it has grown steadily over an extended period of time, while Nigeria has not. In fact, even in the historically-brief postwar era, we see tremendous differences in growth rates across countries. This is shown in the next picture for the postwar era: This picture shows how East Asian tigers have grown at much higher rates than the rest of the world over the past 40 years, while a number of countries in sub-saharan Africa and Central America have experienced negative growth. 7

20 However, the substantial growth differences in the postwar era do not mean that these growth differences are responsible for the current differences in income levels. For one thing, it may be precisely the poor countries that are growing faster. For instance, Hong Kong, South Korea, Singapore and Taiwan were substantially poorer than the United States and Western Europe in For another, these growth differences may be small relative to those necessary to cause large per capita income level differences. The next question is therefore when this growth gap opened up. The answer is that much of the divergence happened during the 19th century and early 20th century. There are striking growth differences during the postwar era, but the world income distribution has been more or less stable, with a slight tendency towards becoming more unequal. For example, despite some big growth successes and disasters, countries that were rich in 1960 are very very likely to be rich today. A regression of log income per worker in 1990 on log income per worker in 1960 gives the following relationship: ln y1990 = ln y1960 (0.48) (0.06) (1.1) with R 2 =0.78. The next figure shows this relationship diagrammatically. 8

21 If we look at output or income per worker, the overall shape of the world income distribution has been relatively stable in the postwar period. There is certainly no narrowing of income gaps. Instead, there is a small but notable increase in the dispersion of incomes. This isshowninthenextfigure which depicts the standard deviation of log income per capita in the world and the ratio of the income of the five richest to the five poorest countries in the world. 9

22 Moreover, there is also a pattern of stratification, whereby some of the middle-income countries of the 1960s appear to have joined either the low-income or the high-income club. This is shown in the next figure: 10

23 The above statements refer to the unconditional distribution that is, they refer to whether the income gap between two countries increases or decreases irrespective of these countries characteristics. Alternatively, we can look at the conditional distribution (e.g., Barro and Sala-i-Martin, 1992). Here the picture is one of conditional convergence: in the postwar period, the income gap between countries that share the same characteristics typically closes over time (though it does so quite slowly). How do we capture conditional convergence? Consider a typical Barro growth regression: g t,t 1 = β ln y t 1 + Xt 1 0 α + ε t (1.2) where g t,t 1 is the annual growth rate between dates t 1andt, y t 1 is per capita income at date t 1andX is a set of variables that the regression is conditioning on (in theory, the determinants of steady state income and/or growth). When no covariates are included, this 11

24 regression leads to a positive or zero estimate of β, reiterating the absence of unconditional convergence as shown in the estimation of equation (1.1) above. In fact, without covariates, this is really identical to the regression equation (1.1), since g t,t 1 ' ln y t ln y t 1, so equation (1.2) can be written as ln y t ' (1 + β)lny t 1 + ε t, which is identical to (1.1) above. The estimate of (1 + β) in (1.1) equal to 1 implies that β ' 0, thus no unconditional convergence. But when X t 1 includes some human capital-related variables such as years of schooling or life expectancy, β is estimated to be approximately -0.02, indicating that the income gap between countries that have the same human capital endowment has been typically narrowing over the postwar period, roughly at the rate of 2 percent a year. If we look at a longer period, for example, from 1870 to today, the pattern is quite different, however. Here, there is divergence. The income gap between countries was much smaller during the 19th century than today. Pritchett illustrates this point using data from Angus Maddison and deriving an absolute lower bound on country incomes due to subsistence. He argues that $250 in terms of 1985 purchasing power parity is a practical lower bound below which the death rate would be extremely high. This suggests that in 1870, theu.s. wasatmosteighttimesasrichasthe poorest country in the world, while it is over 30 times as rich today. Therefore there has been significant divergence over the past 130 years. This is illustrated in the next figure: 12

25 If we go even further back, the pattern may be one of reversal: Acemoglu, Johnson and Robinson (2002) show that in 1500, among the societies that were later to be colonized by European powers, those that were relatively prosperous are today relatively poor. How do we measure/proxy economic prosperity in 1500? It turns out that urbanization rates and population density are good proxies for prosperity during preindustrial periods (and urbanization rates are also good proxies even today). A variety of evidence shows that in 1500 the Mughals, Aztecs and Incas were much more urbanized and densely settled than the civilizations in North America, New Zealand and Australia. Today the U.S., Canada, New Zealand and Australia are orders of magnitude richer than the countries now occupying the territories of the Mughal, Aztec and Inca Empires, such as India, Ecuador or Peru. Therefore, among this set of countries there was a pattern of reversal, whereby those that were relatively prosperous in 1500 have become relatively poor today. The reversal is not confined to this set of countries, and is more 13

26 widespread among the former European colonies. This is shown in the next two figures, the first using urbanization, the second population density as proxies for prosperity in 1500: 10 USA CAN AUS SGP HKG NZL Log GDP per capita, PPP, CHL ARG URY VEN BRA PRY GUY DOM JAM PHL HTI MYS COL PAN CRI ECU BLZ PER GTM IDN SLV BOL LKA HND NIC PAK VNM IND LAO BGD MEX TUN DZA EGY MAR Urbanization in CAN AUS SGP USA HKG NZL Log GDP per capita, PPP, ARG BWA BRA NAM SUR GUY VEN ZAF CHL LCA DOM GRDECU TUN DMA PER BLZ DZA VCT GTM PRY JAM SWZ PHL IDN CPV MAR BOL SLV AGO ZWE HND LKA CMR NIC GIN COG MRTCOM CIV LSO GHA SEN GMB SDN PAK IND HTI CAF TGOVNM LAOKEN BEN UGA NPL ZMB TCD MDG ZAR NGA BFA BGD NER MLI ERI BDI MWI MOZ RWA Log Population Density in 1500 BHS BRB MYS KNA GAB COLTTO PAN CRI TZA MEX SLE ETH EGY 14

27 When did this reversal take place? Consistent with the discussion from Pritchett s paper above, the evidence suggests that the reversal among the former European colonies took place during the 19th century as well. Up to the late 18th century, previously prosperous places continued to be somewhat more prosperous. It was the age of industrialization, the 19th century, when previously less-prosperous former colonies became rapidly urbanized, industrialized and increased their GDP per capita. The next two pictures give a sense of these processes: Timing of the Reversal 25 Urbanization in excolonies with low and high urbanization in 1500 (averages weighted within each group by population in 1500) low urbanization in 1500 excolonies high urbanization in 1500 excolonies 15

28 Reversal, Industrialization and Divergence Industrial Production Per Capita, UK in 1900 = 100 (from Bairoch) US Australia Canada New Zealand Brazil Mexico India 1.2 Interpretation This discussion points to the following set of facts and questions that are central to an investigation of the determinants of long-run differences in income levels and growth: 1. The major pattern to be explained is why there are such large differences in income per capita and worker productivity across countries. This immediately takes us to questions of why some countries grow (or have grown) while other countries have failed to grow and stagnated. 2. The relative stability of the postwar income distribution has suggested to many economists that we should look for differences across countries leading to very large permanent 16

29 differences in income, but not necessarily large permanent differences in growth rates in the recent decades. This is based on the following reasoning: with substantially different long-run growth rates (as in models of endogenous growth, where countries that invest at different rates grow at different rates), we should expect significant divergence. We saw above that despite some widening between the top and the bottom, the cross-country distribution of income across the world is relatively stable. So this reasoning might have some merit. Furthermore, economists with this view argue that the finding of conditional convergence suggests the presence of transitional dynamics taking countries towards their steady state values as in the basic Solow and neoclassical models. 3. Nevertheless, we have seen that there there is still some notable (though perhaps not so large) divergence in the world income distribution. Clearly, countries have not settled into a stationary world income distribution. It is important to understand why even in this age of free-flow of technology some countries are growing faster than others. Equally puzzling is how the very large income differences we observe today can persist in this age of free-flow of technology, trade and financial integration. 4. Moreover, the divergence from the 19th century to today suggests that we might want to look for a set of theories where the large differences in income per capita, at least to some extent, reflect technological or institutional changes that took place during the 19th and early 20th centuries. For example, some countries may have taken advantage of industrialization opportunities, while other societies have failed to do so, or may have only started adopting technologies very late. We therefore need theories which can shed light on why certain societies may fail to take advantage of better technologies. 17

30 5. The reversal (among the former European colonies) suggests that theories that emphasize differences in (economic and perhaps political) institutions or social organization or more generally man-made factors as key determinants of economic performance may be more promising than theories emphasizing fixed environmental factors such as geography or climate. (With such environmental factors as the main determinants of income differences, we should expect countries that were relatively rich 200 or 500 years ago to be also relatively rich today i.e., persistence not a reversal). More ambitiously, we may want to investigate whether and why certain characteristics that make countries richer at some point contribute to their relative poverty during other episodes. Alternatively, we may want to see what type of shocks could cause a reversal in the relative incomes of countries over long periods. 1.3 The Agenda In the rest of the class, we will look at models that can help us understand the mechanics of economic growth. This means understanding a variety of models that underpin the way economists think about the process of capital accumulation, technological progress, and productivity growth. Only by understanding these mechanics can we have a framework for thinking about the causes of why some countries are growing and some others are not, and why some countries are rich and some others are not. Therefore, the approach will be two pronged: on the one hand, we want to understand the mathematical structure of these models as well as possible; on the other, we want to understand what these models and others have to say about which key parameters or key economic processes are different across countries and why. 18

31 Chapter 2 The Solow Growth Model 2.1 The Basic Model in Discrete Time The Production Structure We start with the simplest growth model, sometimes referred to as the Solow-Swan model after two economists who developed versions of it, or simply as the Solow growth model after our own Bob Solow, who was awarded the Nobel prize for his contributions to growth theory. This is a closed economy, with a unique final good. The economy is in discrete time running to infinite horizon, so that time is indexed by t =0, 1, 2,... Time periods here can correspond to days, weeks, or years. So far we do not need to take a position on this. The economy is inhabited by a large number of households, and for now we are going to make relatively few assumptions on the households because in this baseline model, they will not be optimizing. To fix ideas, you may want to assume that all households are identical, so that the economy admits arepresentativeconsumer. We return to what this 19

32 assumption of the representative consumer involves below. As an aside, you should know from basic general equilibrium theory that most economies do not admit a representative consumer, in fact the celebrated Debreu-Mantel-Sonnenschein theorem states that we can say relatively little about the preferences of a consumer obtained by aggregating a number of well-behaved neoclassical consumers. But much of macroeconomics (unfortunately) ignores this basic theorem, and works with representative consumers. In many situations this can be justified on the basis of parsimony. Here I will adopt the same defense and for much of this course I will limit myself to models with representative consumers. Heterogeneity of preferences, abilities and income are in fact quite important to understand the process of economic growth, but many of these topics are beyond the scope of this class. The key assumption of the Solow model will be that each household saves an exogenous fraction s of their income. Much of the neoclassical growth theory is about understanding exactly how much individuals save and how capital accumulates. In the basic model this is taken as exogenous. The other key agents in the economy are firms. Let us assume that the economy also admits an aggregate production function for the unique final good Y (t) =F [K (t),l(t),a(t)] (2.1) where Y (t) is the total amount of production of the final good, K (t) is the capital stock, L (t) is total employment and A (t) is technology. The capital stock here denotes the quantity of machines used in production. Both the capital stock and technology are taken to be single indices, and at some level, they are treated as black boxes we will later discuss how such models can be extended to think of multiple types of technologies and capital goods. For now, the important assumption is that technology is free, it is publicly available as a 20

33 non-excludable, non-rival good. Thus the firm does not have to pay for it. As an aside, you might want to note that some authors use x t or K t when working with discrete time and reserve the notation x (t) ork (t) for continuous time. Since I will go back and forth between continuous time and discrete time, I use the latter notation all throughout, except when discussing dynamic programming where the subscripts are the usual notation. Throughout, I will drop time dependence when this causes no confusion, but include it when there is any chance of such confusion. The production function F : R 3 R is, for simplicity, assumed to be twice continuously differentiable and increasing in all of its arguments, and to be strictly concave in K and L. In particular, we have: Assumption 1 (Continuity, Differentiability, Positive Marginal Products, Concavity and Constant Returns to Scale) F is twice continuously differentiable in K and L, andsatisfies F K (K, L, A) F(K, L, A) K > 0, F L (K, L, A) F(K, L, A) L > 0, F KK (K, L, A) 2 F (K, L, A) K 2 < 0, F LL (K, L, A) 2 F (K, L, A) L 2 < 0. Moreover, F exhibits constant returns to scale in K and L. All of the components of Assumption 1 are important. It specifies that marginal products are positive (thus ruling out some production functions), but more importantly that there are diminishing returns both to capital and labor, i.e., F KK < 0andF LL < 0. We will see below that the degree of diminishing returns to capital will play a very important role in many of the results of the basic growth model. 21

34 The other important assumption is that of constant returns to scale. Recall that F exhibits constant returns to scale in K and L if it is linearly homogeneous (homogeneous of degree 1) in these two variables. More specifically: Definition 1 Let z R K for some K 1. The function g (x, y, z) is homogeneous of degree m in x R and y R if and only if g (λx, λy, z) =λ m g (x, y, z) for all λ R + and z R K. Linearly homogeneous (constant returns to scale) production functions are particularly useful because of the following theorem: Theorem 1 (Euler s theorem) Suppose that g : R K+2 R is continuously differentiable in x R and y R, with partial derivatives denoted by g x and g y and is homogeneous of degree m in x and y. Then mg (x, y, z) =g x (x, y, z) x + g y (x, y, z) y for all x R, y R and z R K. Moreover, g x (x, y, z) and g y (x, y, z) are themselves homogeneous of degree m 1 in x and y. Proof. We have that g is continuously differentiable and g (λx, λy, z) =λ m g (x, y, z). (2.2) Differentiate both sides of equation (2.2) with respect to λ, which gives mλ m 1 g (x, y, z) =g x (λx, λy, z) x + g y (λx, λy, z) y 22

35 for any λ. Setting λ = 1 yields the first result. To obtain the second result, differentiate both sides of equation (2.2) with respect to x: λg x (λx, λy, z) =λ m g x (λx, λy, z). Dividing both sides by λ establishes the desired result. Throughout this course we are going to assume that all factor markets are competitive. Until we come to models of endogenous technological change, we will further assume that product markets are also competitive, so ours will be a prototypical competitive general equilibrium model. Moreover, as noted above, we will work with aggregate production functions as a representation of underlying production structure of the economy. This would be the case, for example, when the economy consists of a large number of firmsallhavingaccesstothesame constant returns to scale production function, for example F above. In that case, there is no difference between assuming an aggregate production function or working with a large number of firms competing for factors of production. Notice, however, that the assumption of an aggregate production function could be quite restrictive. In particular it rules out heterogeneity of productivity among firms, and it also creates problems when there are non-constant returns to scale (can you see what would go wrong with decreasing returns to scale?) Endowments Let us imagine that all factors of production are owned by households. In particular, households own all of the labor, which they supply inelastically. If there is population growth, this can be thought of as existing households becoming larger, or new households being born. 23

36 For our purposes here this does not matter. The households also own the capital stock of the economy, and we take their initial holdings of capital, K (0), as given (as part of the description of the environment), and this will determine the initial condition of the dynamical system we will be analyzing. For now how this initial capital stock is distributed among the households is not important. The more important point is that the households will rent their capital to firms. Let the rental price of capital be denoted by R (t) and the rental price of labor by w (t). Then in competitive markets a representative firm is solving the problem of profit maximizing. Another important set of issues involves how to think of capital. There are many different ways of conceptualizing capital, and some of them are beyond the scope of this course. Loosely speaking, we want to think of capital as corresponding to machines. But for now let us make the rather heroic assumption that capital is essentially the same as the final good. So the economy consists of corn, and it can use some amount of this corn as input into producing further corn. Then K (0) is the amount of corn that individual households have at the beginning of period t = 0, which they can eat or rent to firms to enable them to produce further corn. [...These types of models are sometimes referred to as putty-putty, since capital is totally malleable both before and after it is designated as capital. Alternatives include putty-clay models where corn can be used as capital, but once it is in place, it becomes fixed and it cannot be turned back into consumption goods, and certain features of it, for example, at which capital-labor ratio it can be used, cannot be changed...] Given this structure, there is a natural choice of numeraire in this economy which is to normalize the price of the final good in each period to 1. Recall that we always have to choose a numeraire, but here we are making a normalization in each period. But this is 24

37 without loss of any generality, because the interest-rate between periods will play the role of relative prices. This discussion should already alert you to a central fact: you should think of all of the models we are going to be talking about as general equilibrium economies, wheredifferent commodities correspond to the same good at different dates. Recall from basic general equilibrium theory that the same good at different dates (or in different states or in different localities) is a different commodity. Therefore, in almost all of the models that we will study in this course, there will be an infinite number of commodities (because time runs to infinity). This raises a number of special issues in the theory of general equilibrium which wewilltouchonaswegoalong. Now returning to our treatment of the basic model, the next important assumption is that capital depreciates. We assume that this depreciation takes an exponential form. This means that capital depreciates (exponentially) at the rate δ, so that out of 1 unit of capital this period, only 1 δ is left for next period. This depreciation in general stands for the wear and tear of the machinery, as well as, in more realistic models, the replacement of old machines by new machines. For now it is treated as a black box. The importance of this for a household is that, combined with the normalization of the price of the final goods to 1, it implies that the rate of return faced by the household will be r (t) =R (t)+1 δ. Recall that every unit of capital can be eaten now or rented to firms. In the latter case, the household will receive R (t) units of good as the rental price, but will get back only 1 δ units of the capital, since the rest has depreciated. This implies that the individual has given up one unit of commodity dated t 1forr (t) units of commodity dated t. 25

38 Now let us consider the problem of a representative firm. This firm will maximize profits, which implies max F [K(t),L(t),A(t)] w (t) L (t) R (t) K (t). (2.3) L(t),K(t) A couple of features are worth noting: 1. I set up the problem in terms of aggregate variables. This is without loss of any generality given the representative firm (or the existence of aggregate production function). 2. There is nothing multiplying the F term, since the price of the final good has been normalized to This way of writing the problem already imposes competitive factor markets, since the firm is taking the prices of labor and capital, w (t) andr (t), as given. 4. This is a concave problem, since F is concave (though not necessarily strictly so). The first-order necessary conditions of the firm s problem (combined with differentiability of F ) imply that the competitive factor returns are equal to their marginal products: w (t) =F L [K(t),L(t),A(t)]. (2.4) and R (t) =F K [K(t),L(t),A(t)]. (2.5) An immediate corollary of Theorem 1 combined with competitive factor markets is: Proposition 1 In equilibrium, firms make no profits, and in particular, Y (t) =w (t) L (t)+r (t) K (t). 26

39 Proof. This follows immediately from Theorem 1 for the case of m = 1, i.e., constant returns to scale. This result is convenient, since it implies that firms make no profits, so, in contrast to the basic general equilibrium theory, the ownership of firms does not need to be specified. All we need to know is that firms are profit-maximizing entities. In addition to these standard assumptions on the production function, in growth theory we often impose the following additional boundary conditions, referred to as Inada conditions. Assumption 2 (Inada conditions) F satisfies the Inada conditions lim K (K, L, A) K 0 = and lim K (K, L, A) =0for all L>0 and all A K lim L (K, L, A) L 0 = and lim F L (K, L, A) =0for all K>0 and all A. L Fundamental Law of Motion of the Solow Model Finally, we can write the law of motion of the capital stock of the economy. Recall that K depreciates exponentially at the rate δ, so that the law of motion of the capital stock is given by K (t +1)=(1 δ) K (t)+i (t), (2.6) where I (t) is investment at time t. From national income accounting for a closed economy, we have Y (t) =C (t)+i (t)+g(t), (2.7) where C (t) is consumption and G (t) is government spending. For now, we take G (t) 0, so that national income is divided between consumption and investment. Therefore, using 27

40 (2.1), (2.6) and (2.7), feasible dynamic allocations in this economy would have to satisfy K (t +1) F [K (t),l(t),a(t)] + (1 δ) K (t) C (t). The question is to determine the equilibrium dynamic allocation among the set of feasible dynamic allocations. Here the behavioral rule of the constant savings rate simplifies the structure of equilibrium considerably. It is important that the constant savings rate is a behavioral rule, it is not derived from a well-defined utility function. This means that any welfare comparisons based on the Solow model have to be taken with a grain of salt. We have no idea what the utility function of the individuals are. First note that given G (t) 0 (and the closed economy assumption), aggregate investment is equal to savings, S (t) =I (t) =Y (t) C (t). Now recall that individuals are assumed to save a constant fraction s of their income, i.e., S (t) =sy (t), (2.8) so that they consume the remaining 1 s fraction of their income: C (t) =(1 s) Y (t) (2.9) Thus combining (2.1), (2.6) and (2.8), we have the key dynamic (difference) equation of the Solow growth model: K (t +1)=sF [K (t),l(t),a(t)] + (1 δ) K (t). (2.10) In the Solow growth model, the equilibrium is essentially described by this equation together with laws of motion for L (t) anda (t). 28

41 2.1.4 Definition of Equilibrium The Solow model is a mixture of an old-style Keynesian model and a modern dynamic macroeconomic model. Households do not optimize when it comes to their savings/consumption decisions. Instead, their behavior is captured by a behavioral rule. Butfirms maximize and factormarketsclear. Thusitisusefultostartdefining equilibria in the way that is customary in modern dynamic macro models. Definition 2 In the basic Solow model for a given sequence of {L (t),a(t)} t=0 and an initial capital stock K (0), an equilibrium path is a sequence of capital stocks, output levels, consumption levels, wages and rental rates {K (t),y (t),c(t),w(t),r(t)} t=0 such that K (t) satisfies (2.10), Y (t) is given by (2.1), C (t) is given by (2.9), and w (t) and R (t) are given by (2.4) and (2.5) Equilibrium Without Population Growth and Technological Progress We can make more progress by exploiting the constant returns to scale nature of the production function. To do this, let us make some further assumptions: 1. Let us assume that population is constant and individuals supply labor inelastically, so that L (t) =L. 2. Let us also assume that there is no technological progress, so that A (t) =A. We will relax these assumptions later. For now, let us define the capital-labor ratio of the economy as k (t) K (t) L. 29

42 Then using the constant returns to scale assumption we have that income per capita, y (t) Y (t) /L, isgivenby y (t) = F K (t) L, 1,A f (k (t)). (2.11) In other words, with constant returns to scale, income per capita is simply a function of the capital-labor ratio. Given Theorem 1, we also have R (t) = f 0 (k (t)) > 0and w (t) = f (k (t)) k (t) f 0 (k (t)) > 0. (2.12) The fact that both of these factor prices are positive follows from Assumption 1, which imposed that the first derivatives of F with respect to capital and labor are always positive (with more general production functions, zero factor prices are possible over certain ranges). Given this, we can divide both sides of (2.10) by L andobtainasimplerdifference equation k (t +1)=sf (k (t)) + (1 δ) k (t). (2.13) Since this difference equation is derived from (2.10), it also can be referred to as the equilibrium difference equation of the Solow model, in that it describes the equilibrium behavior of the key object of the model, the capital-labor ratio, and the other equilibrium quantities can be obtained from the capital-labor ratio k (t). At this point, we can also define a steady-state equilibrium for this model without technological progress and population growth. Definition 3 A steady-state equilibrium without technological progress and population growth is an equilibrium path in which k (t) =k for all t. 30

43 In other words, in the steady-state equilibrium the capital-labor ratio remains constant. Mostofthemodelswewillanalyzeinthiscoursewilladmitasteadystateequilibrium,and typically the economy will tend to this steady state equilibrium over time (but often never reach it in finite time). This is also the case for this simple model. This can be seen by plotting the difference equation which governs the equilibrium behavior of this economy, (2.13). The intersection of the right hand side with the 45 line gives the steady-state value of the capital-labor ratio k,whichsatisfies δ s = f (k ) k. (2.14) An alternative visual representation of the steady state is to view it as the intersection between a ray through the origin with slope δ (representing the function δk) and the function sf (k). The next figure shows this picture, which is also useful in seeing the level of consumption and investment in a single figure. The figure also shows that there exist another steady state (as long as f (0) = 0), whereby k = 0. This trivial steady state, without any consumption or production, is of no economic 31

44 interest, and as we will see the low, starting with any k (0) > 0, the economy will never tends to this trivial steady state. In what follows, with a slight abuse of terminology, we will state that there exists a unique steady state (thus ignoring the trivial steady state with k =0). Therefore, we have: Proposition 2 Consider the basic Solow growth model and suppose that Assumptions 1 and 2 hold. Then there exists a unique steady state where the capital-labor ratio is equal to k (0, ) and is given by (2.14), per capita output is given by y = f (k ) (2.15) and per capita consumption is given by c =(1 s) f (k ). (2.16) Proof. The preceding argument establishes that (2.14) is a steady state, i.e., a zero of the difference equation (2.13). To establish existence, note that from Assumption 2, lim k 0 f (k) /k = and lim k f (k) /k =0. Moreover,f (k) /k is continuous from Assumption 1, so there exists k such that (2.14) is satisfied. To see uniqueness, differentiate f (k) /k with respect to k, whichgives [f (k) /k] k = f 0 (k) k f (k) k 2 = w k < 0, (2.17) where the last equality uses (2.12). Since f (k) /k is everywhere decreasing, there can only exist a unique value k that satisfies (2.14). Equation (2.15) and (2.16) then follow by definition. So far the model is very parsimonious, and does not have many parameters. But what we are most interested in is to understand how cross-country differences in certain parameters 32

45 translate into differences in growth rates or income levels. This will be done in the next proposition. But before doing so, let us generalize the production function in one simple way, and assume that f (k) =a f (k) so that a is a shift parameter, with greater values corresponding to greater productivity of factors. This type of productivity is referred to as Hicks-neutral as we will see below, but for now it is just a convenient way of looking at the impact of productivity differences across countries. Since f (k) satisfies the regularity conditions imposed above, so does f (k). Proposition 3 Suppose Assumptions 1 and 2 hold and f (k) =a f (k). Denote the steadystate level of the capital-labor ratio by k (a, s, δ) and the steady-state level of output by y (a, s, δ) when the underlying parameters are given by a, s and δ. Then we have k (a, s, δ) a y (a, s, δ) a > 0, k (a, s, δ) s > 0, y (a, s, δ) s > 0 and k (a, s, δ) δ > 0 and y (a, s, δ) δ < 0 < 0. Proof. The proof follows immediately by writing f (k ) = δ k as, whichholdsforanopensetofvaluesofk. Now apply the implicit function theorem to obtain the results. For example, k s = δ (k ) 2 s 2 w > 0 where w = f (k ) k f 0 (k ) > 0. The other results follow similarly. Therefore, countries with higher savings rates and better technologies will have higher capital-labor ratios and will be richer. Those with greater (technological) depreciation, will 33

46 tend to have lower capital-labor ratios and will be poorer. All of the results in Proposition 3 are intuitive, and start giving us a sense of some important determinants of the capital-labor ratios and income 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 monotonic in the savings rate (think, for example, of the case where s = 1!). To obtain the steady state relationship between c and s, let us suppress the other parameters and write c (s) = (1 s) f (k (s)). = f (k (s)) δk (s) Now differentiating this expression with respect to s (again using the implicit function theorem), we have c (s) s =[f 0 (k (s)) δ] k s. Since from Proposition 3 we have k / s > 0, consumption can only be maximized when f 0 (k (s)) = δ. Moreover, when f 0 (k (s)) = δ, itcanbeverified that 2 c (s) / s 2 < 0, so f 0 (k (s)) = δ is indeed a local maximum. That f 0 (k (s)) = δ is also the global maximum follows from the following observations: s [0, 1], we have k / s > 0andmoreover, when s<s gold, f 0 (k (s)) δ>0bytheconcavityoff, so c (s) / s > 0 for all s<s gold, and by the converse argument, c (s) / s < 0foralls > s gold. Therefore, only s gold satisfies f 0 (k (s)) = δ and gives the unique global maximum of consumption per capita. The relationship between consumption and the savings rate takes the form plotted in the next figure. Consequently, we have established: 34

47 Proposition 4 In the basic Solow growth model, the highest level of consumption is reached for s gold, with the corresponding steady state capital level k gold such that f 0 k gold = δ. In other words, there exists a unique savings rate and the corresponding capital-labor ratio which will maximize steady-state consumption. This is shown in the next figure with the consumption-maximizing savings rate denoted by s gold and the corresponding consumption per capita by c gold : Below this savings rate, the society has too low a capital-labor ratio to maximize consumption, and above this rate, the capital-labor ratio is too high, i.e., individuals are investing too much and not consuming enough. This is the essence of what people refer to as dynamic inefficiency, which we will encounter in greater detail in models of overlapping generations. However, recall that there is no explicit utility function here, so statements about inefficiency have to be considered with caution and skepticism. In fact, the reason why such dynamic inefficiency will not arise once we endogenize consumption-saving decisions of 35

48 individuals will be apparent to many of you already Transitional Dynamics in the Solow Model Proposition 2 establishes a unique steady state equilibrium. Recall, however, that an equilibrium path does not refer simply to the steady state but to the entire path of capital stock, output, consumption and factor prices. To determine what this equilibrium path looks like we need to study the transitional dynamics of the equilibrium difference equation (2.13) starting from an arbitrary capital-labor ratio, k (0). Of special interest is the answer to the question of whether the economy will tend to this steady state starting from such an arbitrary capital-labor ratio, and how it will behave along the transition path. It is important to consider an arbitrary capital-labor ratio, since, as noted above, the total amount of capital at the beginning of the economy, K (0), is taken as a state variable, while for now, the supply of labor L is fixed. Therefore, at time t = 0, the economy starts with k (0) = K (0) /L as its initial value and then follows the law of motion given by the difference equation (2.13). Thus the question is whether the difference equation (2.13) will take us to the unique steady state. Before doing this, recall some definitions and key results from the theory of dynamical systems. Consider the nonlinear system of autonomous difference equations, x (t +1)=F (x (t)), (2.18) where x (t) R n and F : R n R n. Let x be a zero (equilibrium) of this system, which means a fixed point of the mapping F ( ), i.e., x = F (x ). Definition 4 An equilibrium point x is (locally) asymptotically stable if there exists an open set B (x ) 3 x such that for any solution {x (t)} t=0 to (2.18) with x (0) B (x ),we 36

49 have x (t) x. Moreover, x is globally asymptotically stable if for all x (0) R n,for any solution {x (t)} t=0, we have x (t) x. Theorem 2 Consider the following linear difference equation system x (t +1)=Ax (t) (2.19) with initial value x (0), wherex (t) R n for all t and A is an n n matrix. Suppose that all of the eigenvalues of A are strictly inside the unit circle (i.e., the absolute values of the real and complex parts are both less than 1). Then the difference equation (2.19) is globally asymptotically stable, in the sense that starting from any x (0) R n,theuniquesolution {x (t)} t=0 satisfies x (t) x where x is the steady state (zero) of the difference equation given by Ax = x. The proof of this theorem can be found in any textbook on dynamical systems, for example, David Luenberger Introduction to Dynamic Systems: Theory Models and Applications, John Wiley & Sons, 1979, and a version of it for differential equations is in Carl Simon and Lawrence Bloom Mathematics for Economists, Norton, Next let us return to be the nonlinear autonomous system (2.18). Unfortunately, much less can be said about nonlinear systems, but the following is a standard local stability result. Theorem 3 Consider the following nonlinear autonomous system x (t +1)=F [x (t)] (2.20) where F :R n R n and suppose that F is continuously differentiable, with initial value x (0). Let x beazeroofthissystem,i.e.,f (x )=x.define A = F (x ), 37

50 and suppose that all of the eigenvalues of A are strictly inside the unit circle. Then the difference equation (2.20) is locally asymptotically stable, in the sense that there exists an open neighborhood of x, B (x ) R n such that starting from any x (0) B (x ), we have x (t) x. Therefore, for nonlinear systems, we can have local stability results. corollary of these results is: An immediate Corollary 1 Let x (t) R, then the linear difference equation x (t +1) = ax (t) +b is asymptotically stable (in the sense that x (t) x = b/ (1 a)) if a < 1. Moreover, let g : R R be a continuous function, differentiable at x where g (x )=x. Then, the nonlinear difference equation x (t +1)=g (x (t)) is locally asymptotically stable if g 0 (x ) < 1. Now let us apply this result to (2.13): Proposition 5 Suppose that Assumptions 1 and 2 hold, then the equilibrium of the Solow growth model described by the difference equation (2.13) is asymptotically stable, and starting from any k (0) > 0, k (t) k. Proof. From (2.13), we have k (t +1)=sf (k (t)) + (1 δ) k (t), (2.21) withauniquezeroatk. Now recall that f ( ) is concave from Assumption 1 and satisfies f (0) = 0 from Assumption 2. For any strictly concave function, we have that f (k) >f(0) + kf 0 (k) =kf 0 (k), (2.22) 38

51 where the second line uses the fact that f (0) 0. Now linearizing (2.21) around k,wehave k (t +1)' [sf 0 (k )+(1 δ)] (k(t) k ). Since from (2.14), δk = sf (k ), (2.22) implies that δ = sf (k ) /k >sf 0 (k ), and thus [sf 0 (k )+(1 δ)] (0, 1), establishing local asymptotic stability for the Solow model from Corollary 1. Moreover, (2.21) also implies that for all k >k(0) > 0, we have k (t +1) k (t) > 0 and for all k (0) >k,wehavek(t +1) k (t) < 0. Consequently, the solution to (2.21), {k (t)} t=0 always approaches k, thus must be globally stable. This stability result is easier to see diagrammatically, which is shown in the next figure. The following corollary is then immediate: Corollary 2 Suppose that Assumptions 1 and 2 hold, and k (0) <k,then{w (t)} t=0 is an increasing sequence and {R (t)} t=0 is a decreasing sequence. If k (0) >k, the opposite results apply. Intuitively, if the economy starts with too little capital relative to its labor supply, there will be capital deepening (capital accumulation relative to labor), and as a result the marginal product of capital will fall given the diminishing returns to capital feature embedded in Assumption 1, and the wage rate will increase. Conversely, if it starts with too much capital, it will decumulate capital, and in the process the wage rate will decline and the rate of return to capital will increase. The next figure shows this process diagrammatically, emphasizing that the trade-off is between the replacement of the capital stock per effective labor due to depreciation (and perhaps population growth and technological change) and the capital to effective labor ratio: 39

52 Therefore, the Solow growth model has a number of nice properties; unique steady state, asymptotic stability, and simple and intuitive comparative statics. So far, it has no growth however. The steady state is the point at which there is no growth in the capital-labor ratio, no more capital deepening, and no growth in income per capita. The Solow model typically incorporates economic growth by allowing technological change. Before doing this, however, it is useful to look at the mapping between discrete time and continuous time. 40

53 2.2 The Solow Model in Continuous Time From Difference to Differential Equations Recall from the discussion above that the time periods could refer to days, weeks, months or years. In some sense, the time unit is not important. This suggests that perhaps it may be more convenient to look at dynamics by making the time unit as small as possible, i.e., by going to continuous time. The continuous time setup in general has a number of advantages, since some pathological results of discrete time disappear in continuous time (see Problem Set 1). Moreover, especially in the presence of uncertainty, continuous time models have more flexibility both in doing dynamics and for providing explicit form solutions. For us, they are useful particularly because a lot of growth theory is cast in continuous time. Let us start with a simple difference equation x (t +1) x (t) =g (x (t)). (2.23) This equation states that between time t and t + 1, the absolute growth in x is given by g (x (t)). Let us now consider the following approximation x (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,itgives (2.23). In-between it is a linear approximation, which should not be too bad if the distance between t and t +1 is not very large, so that g (x) ' g (x (t)) for all x [x (t),x(t +1)] (however, you should also convince yourself that this approximation could in fact be quite bad if you take a very nonlinear function g, for which the behavior changes significantly between x (t) andx (t + 1)). Now divide both sides of this equation by t, and take limits 41

54 to obtain x (t + t) x (t) lim = ẋ (t) ' g (x (t)), t 0 t as a differential equation representing the same dynamics as the difference equation (2.23) for the case in which the distance between t and t + 1 is small. Recall that here ẋ (t) denotes the time derivative x(t) / t The Fundamental Equation of the Solow Model in Continuous Time We can now repeat all of the analysis so far using the continuous time representation. Nothing has changed on the production side, so we continue to have (2.4) and (2.5) as the factor prices, but now these refer to instantaneous rental rates (i.e., w (t) istheflow of wages that the worker receives for an instant etc.). Savings are again given by S (t) =sy (t), while consumption is given by (2.9) above. Also, let us now introduce population growth into this model, and assume that the labor force L (t) grows proportionally, i.e., L (t) =exp(nt) L (0). (2.24) The purpose of doing so is that in many of the classical analyses of economic growth, population growth plays an important role, so it is useful to see how it affects things here. We are not introducing technological progress yet, which will be done below. 42

55 Recall that which implies that k (t) K (t) L (t), k (t) k (t) = K (t) K (t) n. The law of motion of the capital stock, from the limiting argument in the previous subsection, 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 scale properties of the production function, we obtain the fundamental law of motion of the Solow model in continuous time for the capital-labor ratio as k (t) =sf (k (t)) (n + δ) k (t), (2.25) Therefore we have: Definition 5 In the basic Solow model in continuous time with population growth at the rate n, no technological progress and an initial capital stock K (0), an equilibrium path is a sequence of capital stocks, labor, output levels, consumption levels, wages and rental rates [K (t),l(t),y (t),c(t),w(t),r(t)] t=0 such that K (t) satisfies (2.25), L (t) satisfies (2.24), Y (t) is given by (2.1), C (t) is given by (2.9), and w (t) and R (t) are given by (2.4) and (2.5). As before, a steady-state equilibrium involves k (t) remaining constant. As before, we will refer to the steady-state equilibrium capital-labor ratio as k. 43

56 It is easy to verify that the equilibrium differential equation (2.25) has a unique zero at k, which is given by a slight modification of (2.14) above to incorporate population growth: f (k ) k = n + δ. (2.26) s In other words, going from discrete to continuous time has not changed any of the basic economic features of the model, and again the steady state can be plotted in the familiar figure used above (now with the population growth rate featuring in there as well): We immediately obtain: Proposition 6 Consider the basic Solow growth model in continuous time and suppose that Assumptions 1 and 2 hold. Then there exists a unique steady state equilibrium where the capital-labor ratio is equal to k (0, ) and is given by (2.26), per capita output is given 44

57 by y = f (k ) and per capita consumption is given by Moreover, again let c =(1 s) f (k ). Then we have f (k) =a f (k). Proposition 7 Suppose Assumptions 1 and 2 hold and f (k) =a f (k). Denote the steadystate equilibrium level of the capital-labor ratio by k (a, s, δ, n) and the steady-state level of output by y (a, s, δ, n) when the underlying parameters are given by a, s and δ. Then we have k (a, s, δ, n) > 0, k (a, s, δ, n) > 0, k (a, s, δ, n) and k (a, s, δ, n) < 0 a s δ n y (a, s, δ, n) > 0, y (a, s, δ, n) > 0, y (a, s, δ, n) and y (a, s, δ, n) < 0. a s δ n The new result relative to the earlier comparative static proposition is that now a higher population growth rate, n, also reduces the capital-labor ratio and income per capita. The reason for this is simple. A higher population growth rate means there is more labor to use the existing amount of capital, which only accumulates slowly, and consequently the equilibrium capital-labor ratio ends up lower. This result implies that countries with higher population growth rates will have lower incomes per person (or per worker). The stability analysis is also unchanged. To do this in detail, we simply need to remember the equivalents of the above theorems for differential equations. In particular we have: 45

58 Theorem 4 Consider the following linear differential equation system ẋ (t) =Ax (t) (2.27) with initial value x (0), wherex (t) R n for all t and A is an n n matrix. Suppose that all of the eigenvalues of A have negative real parts. Then the differential equation (2.27) is asymptotically stable, in the sense that starting from any x (0) R n, x (t) x where x is the steady state (zero) of the system given by Ax =0. Theorem 5 Consider the following nonlinear autonomous differential equation ẋ (t) =F [x (t)] (2.28) where F : R n R n and suppose that F is continuously differentiable, with initial value x (0). Letx be a zero of this system, i.e., F (x )=0.Define A = F (x ), and suppose that all of the eigenvalues of A have negative real parts. Then the differential equation (2.28) is locally asymptotically stable, in the sense that there exists an open neighborhood of x, B (x ) R n such that starting from any x (0) B (x ), we have x (t) x. Corollary 3 Let x (t) R, then the linear difference equation ẋ (t) =ax (t) is asymptotically stable (in the sense that x (t) 0) ifa<0. Moreover, let g : R R be continuous and differentiable at x where g (x ) = 0. Then, the nonlinear differential equation ẋ (t) = g (x (t)) is a locally asymptotically stable if g 0 (x ) < 0. Finally, with continuous time, we also have another useful theorem: 46

59 Theorem 6 Let g : R R be a continuous function, and suppose that there exists a unique x such that g (x )=0. Moreover, suppose g (x) < 0 for all x>x and g (x) > 0 for all x<x. Then the nonlinear differential equation ẋ (t) =g (x (t)) is a (globally) asymptotically stable,andstartingwithanyx (0), x (t) x. Notice that the equivalent of Theorem 6 is not true in discrete time, and this will be illustrated by one of the problems in Problem Set 1. In view of these results, Proposition 5 immediately generalizes: Proposition 8 Suppose that Assumptions 1 and 2 hold, then the basic Solow growth model in continuous time with no population growth and technological change is asymptotically stable,andstartingfromanyk (0) > 0, k (t) k. Proof. The proof of stability is now simpler and follows immediately from Theorem 6 by noting that whenever k<k, sf (k) (n + δ) k>0andwhenever k>k, sf (k) (n + δ) k< 0. It is also useful at this point to look at one of the most common examples of the production function used in macroeconomics, the Cobb-Douglas production function: Example 1 Supposed the aggregate production function is given by F [K, L] =AK α L 1 α with 0 <α<1. You should remember from basic micro theory that the Cobb-Douglas production function is extremely special, in particular because it has an elasticity of substitution equal to 1 between capital and labor. This production function is very easy to work with, but it also has many special features that are far from general. It is a good vehicle to illustrate issues, but you should not think that all production functions are Cobb-Douglas! 47

60 One very important feature of the Cobb-Douglas production function is that factor shares are constant. It can be immediately calculated that, with competitive factor markets, we have the share of capital is constant irrespective of the capital-labor ratio: R (t) K (t) α K (t) = Y (t) = F K (K(t),L(t)) K (t) Y (t) = αa [K (t)]α 1 [L (t)] 1 α K (t) A [K (t)] α [L (t)] 1 α = α. Similarly, the share of labor is α L (t) =1 α. With this production function, we have that f (k) =Ak α, so the steady state is given again from (2.26) (with population growth at the rate n) as A (k ) α 1 = n + δ s or µ 1 sa k 1 α =, n + δ which is a very nice and simple interpretable form for the steady-state capital-labor ratio. Transitional dynamics are also straightforward in this case. In particular, we have: k (t) =sa [k (t)] α (n + δ) k (t) with initial condition k (0). To solve this equation, let x (t) k (t) 1 α,sotheequilibrium law of motion of the capital labor ratio can be written in terms of x (t) as ẋ (t) =(1 α) sa (1 α)(n + δ) x (t), 48

61 which is a linear differential equation, with a general solution x (t) = x (0) sa n + δ + sa n + δ or in terms of the capital-labor ratio ½ sa k (t) = n + δ + [k (0)] 1 α sa δ exp ( (1 α)(n + δ) t) ¾ 1 1 α exp ( (1 α)(n + δ) t). This solution illustrates that starting from any k (0), the equilibrium k (t) k =(sa/ (n + δ)) 1/(1 α), and in fact, the rate of adjustment is related to (1 α)(n + δ). This is intuitive: a higher α implies less diminishing returns to capital, which slows down dynamics. Similarly a smaller δ means less replacement of depreciated capital and a smaller n means slower population growth, both of those slowing down the adjustment of capital per worker and thus transitional dynamics A First Look at Sustained Growth Before discussing technological progress, it is useful to see how the model we have developed so far can generate sustained growth (without technological progress). The Cobb-Douglas example above already shows that when α is close to 1, adjustment of the capital-labor ratio back to its steady-state level can be very very slow. A very slow adjustment towards a steady-state has the flavor of sustained growth rather than the system settling down to a stationary point quickly. In fact, the simplest model of sustained growth essentially takes α = 1 in terms of the Cobb-Douglas production function above. To do this, let us relax Assumptions 1 and 2 (which do not allow α = 1), and suppose that F [K (t),l(t),a(t)] = AK (t), (2.29) 49

62 where A>0isaconstant. This is the so-called AK model, and in its simplest form output does not even depend on labor. The results I would like to highlight apply with a more general constant returns to scale production function, for example, F [K (t),l(t),a(t)] = AK (t)+bl(t), (2.30) but it is simpler to illustrate the main insights with (2.29), leaving the analysis of the richer production function (2.30) to Problem Set 1. With this production function, the fundamental law of motion of the capital stock is given by (again with population growth given by (2.24)): k (t) = sa δ n. k (t) Therefore, if sa δ n>0, there is sustained growth in the capital-labor ratio, and given (2.29), there is sustained growth in income per capita. This immediately establishes the following proposition: Proposition 9 Consider the Solow growth model with the production function (2.29) and suppose that sa δ n>0. Then in equilibrium, there is sustained growth of income per capita at the rate sa δ n. In particular, starting with a capital-labor ratio k (0) > 0, the economy has k (t) =exp((sa δ n) t) k (0) and y (t) =exp((sa δ n) t) Ak (0). This proposition not only establishes the possibility of endogenous growth, but also shows that in this simplest form, there are no transitional dynamics. The economy always grows 50

63 at a constant rate sa δ n, irrespective of what level of capital-labor issue it starts from. The next figure shows this equilibrium diagrammatically, denoting the growth rate of the economy (and the capital-labor ratio by γ K ): 2.3 Solow Model with Technological Progress Balanced Growth The models analyzed so far did not feature technological progress. We now introduce changes in A (t) to capture improvements in the technological know-how of the economy. There is little doubt that what human societies know to produce, and how efficiently they can produce them, has progressed tremendously over the past 200 years, and even more tremendously over the past 1000 or 10,000 years. An attractive way of introducing economic growth is to allow technological progress. The question is how to do this. At some level we will see that the production function F [K (t),l(t),a(t)] is too general to achieve our objective. In 51

64 particular, with this general structure, we may not have balanced growth. By balanced growth, we mean a path of the economy in which, while income per capita increases, the capital-labor ratio and the distribution of income between capital and labor is roughly constant. These are sometimes referred to as the Kaldor facts. The next picture, for example, shows the evolution of the share of capital in national income in the United States. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Labor and capital share in total value added Labor Capital Capital and Labor Share in the U.S. GDP. Despite fairly large fluctuations, there is no trend. This and the relative constancy of capital-output ratios until the 1970s have made many economists prefer models with balanced growth to those without. (Since the 1970s capital-output ratios may or may not be constant depending on how you measure them). Also for future reference, note that the capital share in national income is about 1/3, while the labor share is about 2/3. We are ignoring the share of land here as we did in the analysis so far: land is not a major factor of production. This 52

65 is clearly not the case for the poor countries, and we should think about how incorporating land into this picture changes the patterns. In any case, this pattern of factor distribution of income, combined with economists desire to work with simple models, often makes them choose an aggregate production function of the form AK 1/3 L 2/3 as an approximation to reality (especially since it ensures that factor shares are constant by construction). This production function does a good job in certain circumstances, but of course it is very special. For us, the most important characteristic of balanced growth is that it is much easier to handle than non-balanced growth. So it is an advantage to have models featuring balanced growth. In reality, growth has many non-balanced features. For example, the share of different sectors changes systematically over the growth process, with agriculture shrinking, manufacturing first increasing and then shrinking. Ultimately, we would like to have models that combine certain quasi-balanced features with these types of structural transformations embedded in them. These are interesting frontiers of research, but for this course, we will largely focus on models with balanced growth Neutral Technological Progress What are some convenient special forms of the general production function F [K (t),l(t),a(t)]? First we could have F [K (t),l(t),a(t)] = A (t) F [K (t),l(t)], so that technological progress simply multiplies output. This is known as Hicks-neutral technological progress. Intuitively, in this case if we think of the isoquants in the L-K space, technological progress simply corresponds to a relabeling of the isoquants (without any change in their shape). 53

66 Another alternative is to have capital-augmenting or Solow-neutral technological progress, in the form F [K (t),l(t),a(t)] = F [A (t) K (t),l(t)]. This is referred to as capital-augmenting progress, because a higher A (t) isequivalentto the economy having more capital. This type of technological progress corresponds to the isoquants shifting with technological progress in a way that they have constant slope at a given labor-output ratio. Finally, we can have labor-augmenting or Harrod-neutral technological progress F [K (t),l(t),a(t)] = F [K (t),a(t) L (t)], whereby an increase in technology increases output as if the economy had more labor. Equivalently, the slope of the isoquants are constant along rays with constant capital-output ratio. Of course, in practice technological change can be a mixture of these, so we could have a vector valued at index of technology and a production function that looks like F [K (t),l(t),a(t)] = A H (t) F [A K (t) K (t),a L (t) L (t)]. It turns out that, although all of these forms of technological progress look equally plausible ex ante, balanced growth forces us to oneofthesetypesofneutraltechnological progress. In particular, balanced growth necessitates that all technological progress be labor augmenting or Harrod-neutral. This is a very surprising result, and it is also somewhat troubling, since we have no idea why technological progress should take this form. We now stateandprovetherelevanttheoremhere. 54

67 2.3.3 The Steady-State Technological Progress Theorem A version of the following theorem was first proved by Uzawa in For simplicity and without loss of any generality, let us focus on continuous time models. The key elements of balanced growth, as suggested by the discussion above, are the constancy of factor shares and the constancy of the capital-output ratio, K (t) /Y (t). Since there is only labor and capital in this model, by factor shares, we mean α L (t) w (t) L (t) Y (t) and α K (t) R (t) K (t). Y (t) By Assumption 1 and Theorem 1, we have that α L (t)+α K (t) =1. The following theorem was first stated and proved by Uzawa. Here I present a version of Uzawa s proof along the lines of the more recent paper by Jones and Scrimgeour (2005), and then also give a more heuristic proof. Theorem 7 (Uzawa) Consider a growth model with a constant returns to scale aggregate production function F [K (t),l(t),a(t)] and capital accumulation equation K (t) =F [K (t),l(t),a(t)] C (t) δk (t). Suppose also that there is a constant growth rate of population, i.e., L (t) =exp(nt) L (0). If a balanced growth path exists with constant capital-output ratio and per capita growth rate, i.e., ẏ (t) /y (t) =g>0, and factor shares are nonzero and constant, i.e., α K (t) =α (x ) (0, 1) as t, then asymptotically, the production function can be represented as: Y (t) = F [K (t),a(t) L (t)], where * s denote asymptotic steady-state values, and Ȧ (t) A (t) = g. 55

68 Proof. Let us look at the following derivative v = = = log y (t) log (k (t) /y (t)) 1 log K(t) log Y (t) 1 1 ³ log F [K(t),L(t),A(t)] log K(t) 1 ³ FK [K(t),L(t),A(t)]K(t) F [K(t),L(t),A(t)] α K (t) = 1 α K (t), where the last line uses the definition of α K (t) Now let x (t) K (t) /Y (t), and by hypothesis asymptotically α K (t) =α (x )where x refers to the steady state value of K/Y, and the share of capital in national income is potentially a function of this capital-output ratio. Therefore, asymptotically, we have the following partial differential equation: log y (t) log x (t) = α (x ) 1 α (x ). Integrating both sides and noting that the right hand side does not depend on time, we have Z log y (t) =a (t)+ α (x ) dx 1 α (x ) x for some function a (t), which only depends on time. Taking exponents, we have y (t) =A (t) ξ (x ), ³ R where A (t) exp (a (t)) and ξ (x ) exp α(x ) dx. Notice, also, for future use that 1 α(x ) x from the inverse function theorem, ξ (x ) is invertible in the neighborhood of x,withinverse denoted by ξ 1 (y/a) 56

69 Since ξ (x )isconstantandẏ (t) /y (t) =g, wemusthavea (t) exp (gt) A (0). Finally, note that, by definition, k (t) = x (t) y (t), which implies asymptotically (in steady state) that k (t) A (t) = y (t) A (t) ξ 1 = f 1 y (t) A (t) µ y (t) A (t) or and thus y (t) A (t) = f Y (t) =A (t) L (t) f k (t), A (t) K (t) A (t) L (t) which, under constant returns to scale, is another way of writing, Y (t) = F [K (t),a(t) L (t)], completing the proof. For a more heuristic reasoning, consider production function of the form F [A K (t) K (t),a L (t) L (t)]. Balanced growth requires factor shares to be constant, which can only be the case when total capital inputs, A K (t) K (t), and total labor inputs, A L (t) L (t), grow at the same rate; otherwise, the share of either capital or labor will be increasing over time. Capital accumulation implies that K (t) will grow at the same rate as A L (t) L (t). Thus balanced growth can only be possible if A K (t) is asymptotically constant. There is one exception to this, which is the Cobb-Douglas production function, where we can have Y (t) =[A K (t) K (t)] α [A L (t)l(t)] 1 α 57

70 and both A K (t) anda L (t) could grow asymptotically, while maintaining balanced growth. However, notice that Theorem 7 does not require that Y (t) = F [K (t),a(t) L (t)], but that it should have a representation of the form Y (t) = F [K (t),a(t) L (t)]. It is quite straightforward to see that in this Cobb-Douglas example we can define A (t) = [A K (t)] α/(1 α) A L (t), and the production function can be represented as Y (t) =[K (t)] α [A(t)L(t)] 1 α, in other words, technological change can be represented as purely labor augmenting, which is what Theorem 7 requires. Notice finally that this theorem does not state that technological change has to be labor augmenting all the time. But it requires that it has to be labor augmenting asymptotically, i.e., along the balanced growth path. Based on these ideas, is possible to give the more heuristic proof of Theorem 7. Alternative Proof of Theorem 7: Suppose that Y (t) =A H (t) F [A K (t) K (t),a L (t) L (t)], and since we are interested in asymptotic states, suppose that A H (t), A K (t) anda L (t) are growing asymptotically at the rates g H, g K and g L. Normalize A H (0), A K (0) and A L (0) to1. Then we can write that asymptotically Y (t) = exp((g H + g K ) t) F 1, A L (t) L (t) K (t) A K (t) K (t) exp ((g H + g K ) t) f µ exp ((g L g K ) t) L (t) K (t) Now we also have K (t) K (t) = s Y (t) K (t) δ, and in steady state, according to the hypotheses of the theorem, we have Y (t) /K (t) constant, so K (t) /K (t) =g, i.e., capital grows at the same rate as total output. Combined 58.

71 with the hypothesis that L (t) =exp(nt) L (0), this then implies (for L(0) normalized to 1), Y (t) K (t) =exp((g H + g K ) t) f (exp ((g L g K + n g) t)). But from this equation Y (t) /K (t) can remain constant only under the one of the two following circumstances: 1. exp ((g H + g K ) t) is constant and exp ((g L g K + n g) t) is constant, i.e., g H = g K = 0, and g = g L + n. 2. exp ((g H + g K ) t) increases exactly at the same rate as f (exp ((g L g K + n g) t)) decreases, which is only possible when f (x) = x β for some β.then, if we impose Assumption 1 (or just CRS and positive marginal products) then we get β (0, 1). This completes the alternative proof of Theorem The Solow Growth Model with Technological Progress: Continuous Time Now we are ready to analyze the Solow growth model with technological progress. I will only present the analysis for continuous time (the discrete time case is equivalent). From Theorem 7, we know that the production function must take the form F [K (t),a(t) L (t)], with purely labor-augmenting technological progress asymptotically. For simplicity, let us assume that it takes this form throughout. Moreover, suppose that there is technological progress at the rate g, i.e., Ȧ (t) A (t) = g, (2.31) 59

72 andpopulationgrowthattheraten, L (t) L (t) = n. Again using the constant savings rate we have K (t) =sf [K (t),a(t) L (t)] δk (t). (2.32) The simplest way of analyzing this economy is again to express everything in terms of a normalized variable. Since effective units of labor are given by A (t) L (t), and F exhibits constant returns to scale in its two arguments (by virtue of exhibiting constant returns to scale in capital and labor), we can define k (t) K (t) A (t) L (t). (2.33) Now differentiating this expression with respect to time, we obtain k (t) k (t) = K (t) K (t) g n (2.34) The quantity of output per unit of effective labor can be written as ŷ (t) Y (t) A (t) L (t) K (t) = F A (t) L (t), 1 f (k (t)). Income per capita is y (t) Y (t) /L (t), i.e., y (t) =A (t)ŷ (t). 60

73 Now substituting for K (t) from (2.32) into (2.34), we have Now using (2.33), k (t) sf [K (t),a(t) L (t)] = δ g n. k (t) K (t) k (t) sf (k (t)) = δ g n, (2.35) k (t) k (t) which is very similar to the law of motion of the capital-labor ratio in the continuous time model, (2.25). An equilibrium in this model is defined similarly to before. Consequently, we have: Proposition 10 Consider the basic Solow growth model in continuous time, with Harrodneutral technological progress at the rate g and population growth at the rate n. Suppose that Assumptions 1 and 2 hold, and define the effective capital-labor ratio as in (2.33). Then there exists a unique steady state equilibrium where the effective capital-labor ratio is equal to k (0, ) and is given by f (k ) k = δ + g + n. s Per capita output and consumption grow at the rate g. The comparative static results are also similar to before, with the additional comparative static with respect to the initial level of the labor-augmenting technology, A (0) (since the level of technology later, A (t), is completely determined by A (0) given the assumption in (2.31)). Proposition 11 Suppose Assumptions 1 and 2 hold and let A (0) be the initial level of technology. Denote the balanced growth path level of effective capital-labor ratio by k (A (0),s,δ,n) 61

74 and the level of income per capita by y (A (0),s,δ,n,t) (the latter is a function of time since it is growing over time). Then we have k (A (0),s,δ,n) A(0) =0, k (A (0),s,δ,n) s > 0, k (A (0),s,δ,n) n < 0 and k (A (0),s,δ,n) δ < 0, and also y (A (0),s,δ,n,t) A(0) > 0, y (A (0),s,δ,n,t) s > 0, y (A (0),s,δ,n,t) n < 0 and y (A (0),s,δ,n,t) δ < 0, Finally, we also have very similar transitional dynamics. Proposition 12 Suppose that Assumptions 1 and 2 hold, then the Solow growth model with Harrod-neutral technological progress and population growth in continuous time is asymptotically stable, and starting from any k (0) > 0, theeffective capital-labor ratio converges to a steady-state value k, i.e., k (t) k. Therefore, the comparative statics and dynamics are very similar to the model without technological progress (and without population growth). The major difference, of course, is that now the model generates growth in income per capita, so can be mapped to the data much better. However the disadvantage is that this growth is driven entirely exogenously. The growth rate is exactly the same as the exogenous growth rate of the technology stock. The model does not specify where this technology stock comes from and how fast it grows. 62

75 Chapter 3 TheSolowModelandtheData One of the important uses of the aggregate production function approach and the basic Solow model is that they provide us with a simple vehicle to look at the data, both at growth over time and income-level differences (and growth rate differences) across countries. I start here with over-time changes, i.e., growth accounting, and then will move to the more important application for the purposes of this course, which involves looking at cross-country differences. 3.1 Growth Accounting Let us go back to the most general form of the aggregate production function given by (2.1), whereby Y (t) =F [K (t),l(t),a(t)]. 63

76 Differentiate this function with respect to time on both sides to obtain (dropping timedependence) Ẏ Y = F AA Ȧ Y A + F KK K Y K + F LL L Y L. Recalling the definition of factor shares above, and denoting g Ẏ/Y, g K K/K and g L L/L, and also defining x F AA Ȧ Y A as the contribution of technology to growth, we have x = g α K g K α L g L. This is the fundamental growth accounting equation. Thisequationletsusestimatethecon- tribution of technological progress to economic growth from factor shares, output growth, labor force growth and capital stock growth. This contribution from technological progress is also referred to as Total Factor Productivity (TFP) or sometimes as Multi Factor Productivity. In particular, denoting an estimate by ˆ, we have the estimate of TFP growth as: ˆx = g α K g K α L g L. If we are interested in Ȧ/A rather than x, we need to make further assumptions. For example, if we assume that the production function takes the standard labor-augmenting form Y (t) = F [K (t),a(t) L (t)], then we have Ȧ A = 1 [g α K g K α L g L ], α L 64

77 but this equation is not particularly useful, since Ȧ/A is not something we are inherently interested in. Much more interesting is precisely ˆx. In continuous time, this equation is exact. In practice, of course, instead of instantaneous changes, we look at changes over discrete time periods, for example over a year (or sometimes with the better data, perhaps over a quarter or a month). In this case, there is a problem, since over the time horizon in question, factor shares can change. It can be shown that this could lead to serious biases. The most common way of dealing with this is to use factor shares calculated as the average of the two points in time. Therefore in discrete time, for a change between times t and t +1,wehave ˆx t,t+1 = g t,t+1 ᾱ K,t,t+1 g K,t,t+1 ᾱ L,t,t+1 g L,t,t+1, where and ᾱ L,t,t+1 is defined similarly. ᾱ K,t,t+1 α K,t + α K,t+1 2 Applying this method, Solow found that much of economic growth over the 20th century was due to technological progress. This has been a landmark finding, focusing the attention of economists on sources of technology differences over time, across nations, across industries and across firms. Since then, many economists, most notably Dale Jorgensen, have attempted to reduce the amount due to the residual technology by adjusting for the quality of labor and capital inputs. This is still an active research area, partly because there are conceptual issues about how far one should go in adjusting the quality of inputs. For example, better computers can translate into more capital, reducing the TFP residual, but at the end of the day better computers are a result of better technology. We will return to these issues again below. 65

78 3.2 Solow Model and Cross-Country Income Differences We are now in a position to take the basic Solow model to the data. The simplest way of doing this is to follow the approach of Mankiw, Romer and Weil (1992). These authors basically estimated a cross country regression inspired by the above model. However, a basic estimation which does not take human capital into account proved to be inadequate. Therefore, Mankiw, Romer and Weil (1992) used an augmented Solow also incorporating human capital. I first develop this model briefly, and then look at the empirical evidence. Since our purpose here is to look at cross-country income differences, from the beginning, I present the model for a cross-section of countries. Here already there is a major (and at some level a very problematic assumption), adopted by many authors, among them Mankiw, Romer and Weil (1992), Barro (1991) and much of Barro and Sala-i-Martin (2004), which is that the world consists of a cross-section of countries which do not interact. In other words, these countries do not trade financial assets, goods, or there is no slow diffusion of technology across these countries. These countries inhabit the world, but they are all islands onto themselves. I start with this case of no interdependence, but interdependences arising from technology flows and international trade will be discussed below Solow Model with Human Capital Suppose that output in country j is given by Y j = K β j Hα j (A j L j ) 1 α β, (3.1) 66

79 where I have dropped time to simplify notation. We have α, β 0, α + β 1, j denotes country, Y is total output, H is human capital, L is labor, A is labor-augmenting technical change. The important assumption here is that human capital is taken to be a different factor of production rather than simply augmenting labor (i.e., equation (3.1) rather than Y j = Kj 1 α (A j H j ) α with H j interpreted as efficiency units of labor, see (3.6) below). In fact, this latter approach is much more in line with the Becker model of human capital, and writing the model in this way is not without loss of any generality (as we will see below). But before seeing why this is, we should solve the model. First, we can use the usual trick of the neoclassical growth model of transforming variables to per capita effective units: k j K j A j L j and h j and define y j Y j /L j as output per worker. Then H j A j L j, y j = A j k β j hα j. (3.2) Suppose also that population grows at a constant rate n j in country j. This model cannot be easily taken to the data because we have no idea what A j is. A key assumption of Mankiw, Romer and Weil (1992), which enables them to take the augmented-solow model to the data is the following: Common technology advances assumption: A j (t) =A j exp (gt). That is, countries may differ according to their technology level, but they share the same common technology growth rate, g. This is in part motivated by the relative stability of 67

80 the world income distribution discussed earlier. In the absence of this assumption, countries would grow at different rates, and the world income distribution would become more and more dispersed. Next, consider constant savings rates for human and physical capital, as a direct generalization of the standard Solow model: K j = s k j Y j δ k K j and Ḣj = s h j Y j δ h H j where δ s denote constant depreciation rates. Then µ k j = s k yj j n j + g + δ k k j (3.3) A µ j ḣ j = s h yj j n j + g + δ h h j. (3.4) A j As in our baseline models, in steady state, both k j and h j have to be constant. Thus setting k j =0andḣj = 0 in (3.3) and (3.4) and solving yields the following steady-state values of physical capital and human capital ratios to effective labor: k j = h j = Ã! 1 α Ã! α s k j n j + g + δ k s h j n j + g + δ h Ã! β Ã! 1 β s k j n j + g + δ k s h j n j + g + δ h 1 1 α β 1 1 α β. Now substituting back into (3.2) and taking logs, we obtain Ã! α ln y j =lna j + gt + 1 α β ln s h j β + n j + g + δ h 1 α β ln à s k j n j + g + δ k! (3.5) This is an equation which can be estimated using cross-country data if we have measures of s h j. In addition, we can use investment rates (investments/gdp) for s k j,populationgrowth 68

81 rates n j, and the standard depreciation rates for δ k. This is what Mankiw, Romer and Weil do (or they estimate a version of this with δ h = δ k ). They approximate s h j using the fraction oftheworkingagepopulationenrolledinschool[... isthisagoodproxyforinvestmentin human capital?...]. However, with all of these assumptions, equation (3.5) can still not be estimated, because the term ln A j is unobserved to the econometrician, and could be correlated with all of the other right hand side variables. Therefore implicitly, Mankiw, Romer and Weil make another crucial assumption, considerably stronger than the common technology advances assumption: Orthogonal technology assumption: A j = ε j A where ε j is orthogonal to all other country variables. With these assumptions, Mankiw, Romer and Weil estimate equation (3.5). The estimation is a success for the augmented-solow model. If human capital is not included, the fit is not very good and the estimates are not reasonable. This is shown in the next table. 69

82 Without human capital, the coefficient in front of the investment/gdp ratio should be β/(1 β), thus the estimate suggests β ' 0.6,whichisfartoohighbearinginmindthat given the factor distribution of income we expect the exponent of capital in the production function to be closer to 1/3. But for the augmented model with human capital, the fit isverygoodasshowninthe next table. Now the parameter estimates imply α 1/3, β 1/3 andr

83 At face value, these results provide strong support for the augmented Solow model. The estimate of α is consistent with a capital share of one-third in national income, and the R 2 implies that almost 80 percent of the differences in income per capita can be explained by investment decisions (human and physical capital differences) Problems with the Mankiw, Romer and Weil Approach But there are two major (and related) problems with this approach: 1. The orthogonal technology assumption is too strong. When A j varies across countries, 71

84 it will plausibly be correlated with our measures of s h j and s k j, so there will be an omitted variable bias leading to overestimates of α and β as well as an exaggeration of the R The coefficient on s h j is too large. To see this, recall that Mankiw, Romer and Weil use the fraction of the working age population enrolled in school. This variable ranges from 0.4 to over 12 in the sample of countries used for this regression. Their estimates therefore imply that a country with approximately 12 for this variable should have income per capita about 9 times that of a country with s h j other variables constant). = 1! (This is holding all More explicitly, the predicted log difference in incomes between these two countries is α (ln 12 ln (0.4)) 2.24, 1 α β and exp (2.24) 1 9 times. In practice, the difference in average years of schooling between any two countries over this time period is less than 12. The labor literature suggests that additional years of schooling is associated with a 6 to 10 percent increase in individual earnings (e.g., consider the individual level Mincer regression ln w i = Xiγ 0 + φe i where w is wage income, X i is a set of demographic controls, and E is years of schooling. Here φ is estimated to be between 0.06 and 0.1). This implies that a worker with one more year of schooling is typically about 6 to 10 percent more productive. So in the absence of human capital externalities, a country with 12 more years of average schooling should be at most twice as rich insteadof9timesasrich!evenallowingfor human capital externalities, one would need very very large human capital externalities 72

85 in order to get this type of results (existing estimates of human capital externalities, for example, Acemoglu and Angrist, 2000, show that they are rather small). To understand this last point, consider a simple competitive economy. Suppose that each firm has a production function y = k 1 α (Ah) α Firms face cost of capital r, and human capital is a function of schooling, with the standard exponential form h i =exp(φe i ). First-order condition from firm maximization gives r = (1 α)(ah/k) α. In other words, all workers, irrespective of their level of schooling, will work exactly at the same physical to human capital ratio. Wages are equal to marginal product, so w (h) =α (1 α) (1 α)/α Ar (1 α)/α h So wages are linear in human capital due to constant returns to scale. Taking logs of this equation, we end up with the standard log linear wage equation ln w i = cst + φe i, with the slope coefficient on education measuring the relationship between education and human capital. Now consider two economies with the same technology, the same interest rate (for example, open capital accounts), the same technology, but in one economy all workers have E 1 years of schooling, while in the other, they have E 2 >E 1 schooling. How large should the income gap between these two countries be? Using the fact that with the same interest rate, both economies will function at the same physical to human capital ratio, we immediately obtain Y i = A (1 α) (1 α)/α r (1 α)/α exp (φe i ), 73

86 Or, taking logs, we obtain that log Y 2 log Y 1 = φ (E 2 E 1 ). So if one economy has on average one year more of schooling, and φ is about 6 percent, its income should be 6 percent higher. In the data, there are much larger differences. For example a cross-country regression of income per capita on average years of schooling in 1985 gives log Y = E (0.027) In other words, the correlation between income and schooling is too strong relative to what we should expect on the basis of micro evidence. In particular, the effect of schooling on income is much larger than the 6-10 percent difference expected. This result is not simply explained by the fact that interest rates vary across countries. Notice that we can write r =(1 α) Y/K, so including the (log) capital output ratio would be one way to control for interest rate differences. In this regression, the log capital-output should have a coefficient of (1 α) /α, approximately0.5takingα as 2/3. Running this regression with 1985 data, we obtain log Y = E log K Y (0.033) (0.178) So, there is still a very large effect of education on income, and the quantitative effect of capital (as a proxy for interest rates) is plausible. This relationship between education and income may reflect human capital externalities. For example, we might have the productivity term, A, as a function of average human capital in the economy. In this case, the rate of return to human capital in the Mincer regressions would only reflect the private return that is, the increase in the individual s wage as a 74

87 function of his own human capital, holding average human capital constant. But regressions using aggregate data would capture the total effectofanincreaseinhumancapitalon income that is, the private plus the external effect of schooling. Therefore, one possibility is that there are large human capital externalities. However, as noted above, existing evidence indicates that human capital externalities are limited. The alternative interpretation of the patterns is that there are differences in technologies, A j s, and these are correlated withhumancapitaldifferences. Such a pattern of correlation may arise because human capital responds to technology, or because some third factor affects both human capital and technology The Macro Mincer Approach (Bils-Klenow-Rodriguez-Hall- Jones) A related approach is to use calibration/levels accounting rather than regression analysis and make use of the findings of Mincer (micro wage) regressions. This is the approach first taken by Bils and Klenow, and then by Klenow and Rodriguez and Hall and Jones. The advantage of the calibration approach is that the omitted variable bias underlying the estimates of Mankiw, Romer and Weil will be less important (since microlevel evidence is being used to anchor the contribution of human capital). The disadvantage is that certain assumptions on functional forms have to be taken much more seriously, and we explicitly have to assume no human capital externalities. Here let me follow Hall and Jones. Consider the following production function Y j = K 1 α j (A j H j ) α (3.6) with H j interpreted as efficiency units of labor. Assume the following Mincer-type relation- 75

88 ship between human capital and education H j = X E exp {φ (E)} L j (E) where φ (E) is the rate of return to E years of schooling and L j (E) isthenumberofindividuals in country j with E years of schooling. We can use different values for φ (E) and construct alternative estimates of H j. Hall and Jones (1999) use a piecewise linear specification for φ (E) based on work by Psacharapoulos from less developed countries (showing returns to earlier years of schooling that are greater than to higher education). Once we have aseriesforh j and one for K j, which can be constructed using standard perpetual inventory methods, we can construct predicted incomes, for example, as Ŷ j = K 1/3 j A US 2/3 t H j and compare these predicted incomes with actual incomes. Alternatively, we could back out country-specific technology terms (relative to the U.S.) as A j t A US t = Ã Y j t Y US t! 3/2 µ K US t K j t 1/2 µ H US t H j t Hall and Jones perform this exercise using output per worker rather than income per capita. They find: 1. Differences in physical and human capital still matter a lot, accounting for as much as 50 percent of the actual differences in output per worker. 2. But there are also significant productivity differences. 76

89 The next figure and the table show a summary of their results: 77

90 The conclusion of this calibration exercise is therefore very similar to the one that followed from the regression analysis presented in the previous section. Naturally, some of the assumptions of these calibration exercise can be relaxed. For example instead of assuming a Cobb-Douglas production function, one could do levels accounting. Essentially, this would involve ranking the countries according to their capitallabor ratio (or capital-output ratio), and then using the equivalent of the growth accounting equation above. In particular, we can write ˆx j,j+1 = g j,j+1 ᾱ K,j,j+1 g K,j,j+1 ᾱ Lj,j+1 g L,j,j+1, 78

91 where j stands for country, thus g K,j,j+1 is the proportional difference in capital stock between countries j and j +1, g L,j,j+1 is a proportional difference of labor supply between the two countries, and ˆx j,j+1 is the TFP difference. With this method, and taking one of the countries, for example the United States, as the base country, we can calculate relative technology differences across countries. Of course, for this we need to have good measures of factor shares in different countries which are not always available. 3.3 An Alternative Approach to Estimating Productivity Differences (Trefler) In the above approach, productivity/technology differences are obtained as residuals from a calibration exercise, so we have to trust the functional form assumptions used in this exercise. An alternative is to use additional data. This is what Trefler does to test an augmented version of the Heckscher-Ohlin approach to international trade. Although Trefler does not emphasize the implications of his findings for productivity differences, a byproduct ofhisanalysisisaseriesofestimatesfordifferences in factor productivities across countries. Trefler starts from the standard Heckscher-Ohlin model of international trade, but allows for factor-specific productivitydifferences across countries. Other than these factor-specific differences, all countries share the same technology (i.e., there are no differences in industry technologies) and share the same homothetic preferences (in particular, they allocate consumption expenditures across goods in the same manner). It is important that technology differences take the form of factor productivity differences. In particular, one unit of labor (or one college graduate) in the U.S. could be more productive than one unit of labor (or one college graduate) in Nigeria. The same applies to 79

92 capital. This specification of course is more general than the production function in (3.1), since capital-augmenting technology differences are allowed and the elasticity of substitution between different factors is not assumed to be equal to 1. A standard equation in international trade is that, in the absence of any trading frictions and with identical (or homothetic) preferences, the net export of factor f embedded in the exports of country j, X f j,is X f j = πf j V f j s j NX i=1 π f i V f i (3.7) where V f j is that endowment of factor f in country j, π f j is the factor productivity of factor f in country j, ands j is the share of country j in world consumption (this uses the assumption that all countries have the same homothetic preferences). N is the total number of countries. Given estimates of the net export of factor contents, the X f j s, equation (3.7) solves for a unique sequence of π f j s taking one of the countries as the base. So from this equation we canobtainanestimateofthedifferences in factor productivities. At this level, this may be viewed simply as an untested strong hypothesis. The major contribution of Trefler s paper is to note that if there is factor price equalization, we should also have w f j π f j = wf j 0 π f j 0 (3.8) for any pair of countries, j and j 0,wherew f j is the price of factor f in country j. Withdata on factor prices, we can therefore construct alternative series for π f j s.itturnsoutthatthe series implied by (3.7) and (3.8) are very similar, so there appears to be some validity to this approach. The following figure shows his estimates: 80

93 Given this validation, we can presume that there is some information in the numbers that Trefler obtains. These numbers imply that there are very large differences in labor productivity, and some substantial, but much smaller differences in capital productivity. For example,laborinpakistanis1/25thasproductiveaslaborintheunitedstates. Incontrast, capital productivity differences are much more limited than labor productivity differences. For example, capital in Pakistan is only half as productive as capital in the United States. 81

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95 Chapter 4 Fundamental Determinants of Differences in Income 4.1 From Proximate to Fundamental Causes The use of the Solow model and the production function approach illustrated how cross country income differences can be understood as resulting from physical capital differences, human capital differences and technology differences. These technology differences, themselves, may represent actual differences in the technologies used by countries, or other efficiency differences in the use of the factors. At this level, the framework we have does a very good job of helping us understand the proximate causes of income differences. The same procedure also helps us understand the proximate causes of the process of economic growth. However, the observation that a country is poorer than another because it has worse technology, less physical capital and less human capital immediately poses the next question: why does it have worse technology, less physical capital, less human capital? This question 83

96 is, in some sense, about the fundamental causes of income per capita (and growth) differences across countries. Growth theory is useful in highlighting the proximate causes, in providing us with a framework for thinking about the fundamental causes, and also in clarifying the mechanics of the process of growth, so that we can more carefully evaluate different theories and approaches. But we have to take this additional step of looking for fundamental causes, otherwise what we have learned will be only partial. 4.2 Hypotheses Why do some countries invest more in physical and human capital and possess better technologies? There are four sets of broad hypotheses: 1. Luck: some countries just turned out to be lucky. It is difficult to operationalize this approach, and at some level, it is quite similar to the other hypotheses, but less specific (one way of operationalizing it may be by using the multiple equilibrium models we will discuss below). A version of this hypothesis where such differences are transitory is clearly not supported by the evidence presented so far, which points out to very persistent differences over long periods. A version of this hypothesis where a small difference caused by luck may lead to large persistent differences is also difficult to reconcile with the data given the reversal documented above. So I will place less emphasis on the importance of luck. Nevertheless, some of the theories presented below will show how small differences in initial condi- 84

97 tions can lead to large ultimate differences. 2. Geography: This view is becoming very popular recently. It claims that differences in economic performance reflect, to a large extent, differences in geographic, climatic and ecological characteristics across countries. The most common is the view that climate has a direct effect on income through its influence on work effort. This idea dates back to Machiavelli and Montesquieu. Alfred Marshall (1890) similarly wrote: vigor depends partly on race qualities: but these, so far as they can be explained at all, seem to be chiefly due to climate. Gunnar Myrdal (1968): climate exerts everywhere a powerful influence on all forms of life, and that serious study of the problems of underdevelopment... should take into account the climate and its impacts on soil, vegetation, animals, humans and physical assets in short, on living conditions in economic development. The recent bestseller by Jared Diamond, Guns, Germs and Steel, suggests that the timing of the Neolithic revolution has had a long lasting effect by determining which societies were the first ones to develop strong armies, and technology. For example, he states that:...proximate factors behind Europe s conquest of the Americas were the differences in all aspects of technology. These differences stemmed ultimately from Eurasia s much longer history of densely populated... societies dependent on food production (1997, p. 358). Diamond argues that differences in the nature and history of food production, in turn, are due to the types of crops, domesticated animals, and the axis of agricultural technology diffusion indifferent continents, all of which are geographically determined characteristics. In the economics circles, Jeff Sachs has been pushing for this view. He argues that Certain parts of the world are geographically 85

98 favored. Geographical advantages might include access to key natural resources, access to the coastline and sea navigable rivers, proximity to other successful economies, advantageous conditions for agriculture, advantageous conditions for human health. (2000, p. 30). He further suggests that Tropical agriculture faces several problems that lead to reduced productivity of perennial crops in general and of staple food crops in particular (2000, p. 32), and that The burden of infectious disease is similarly higher in the tropics than in the temperate zones (2000, p. 32). Finally, Sachs argues that the greater population in the temperate areas over the past centuries led to more rapid advances in technologies appropriate for these areas relative to technologies necessary for development in the tropics (2001, p. 3 and 2000, pp ). The following figure shows the geographical distribution of income per capita, which is consistent with some geographic factors, such as climate) having an effect on the long-run distribution of income across countries: 86

99 3. Institutions : according to this view, differences in economic performance largely reflect differences in the organization of society. Societies that provide incentives and opportunities for investment will be richer than those that fail to do so. There are many versions of this hypothesis, some of them suggesting that institutions that support property rights and rule of law are important, others suggest that limited government, or equal opportunity, or specific government policies are important for investment and efficiency (of course, whether these policies are adopted is in turn determined by other factors). 4. Culture and social capital: this view instead emphasizes whether societies are able to engender the values conducive to entrepreneurship or cooperation among agents. 87