http://elsa.berkeley.edu/~chad/Handbook.html HANDBOOK OF ECONOMIC GROWTH PHILIPPE AGHION AND STEVEN DURLAUF, EDITORS ELSEVIER, 2005 Here is a list of chapters that are available online in working paper form. Thanks to Mark Feldman for reorganizing the list to match the order of the published handbook. Apologies for any errors or omissions please let me know about them and I will correct them promptly. Last updated March 14, 2006 by Chad Jones chad@econ.berkeley.edu. The published version of the Handbook of Economic Growth is here. (Your university must have a subscription to the Handbooks for this to work.) Ch 00 Reflections on Growth Theory (Not available online, to my knowledge.) Robert M. Solow Ch 01 Neoclassical Models of Endogenous Growth: The Effects of Fiscal Policy, Innovation and Fluctuations Larry E. Jones and Rodolfo E. Manuelli Ch 02 Growth with Quality-Improving Innovations: An Integrated Framework Philippe Aghion, Harvard University and Peter Howitt, Brown University Ch 03 Horizontal Innovation in the Theory of Growth and Development Gino Gancia, CREI and Fabrizio Zilibotti, Institute for International Studies and University College London. Ch 04 From Stagnation to Growth: Unified Growth Theory Oded Galor, Hebrew University at Jerusalem and Brown University Ch 05 Poverty Traps Costas Azariadis, UCLA and John Stachurski, University of Melbourne Ch 06 Institutions as the Fundamental Cause of Long-Run Growth Daron Acemoglu, MIT, Simon Johnson, MIT, and James Robinson, UC Berkeley Ch 07 Growth Theory through the Lens of Development Economics Abhijit Banerjee, MIT and Esther Duflo, MIT http://elsa.berkeley.edu/~chad/Handbook.html (1 of 3)8/17/2006 11:03:16 AM

http://elsa.berkeley.edu/~chad/Handbook.html Ch 08 Growth Econometrics Steven Durlauf, University of Wisconsin, Paul Johnson, Vassar College, and Jonathan Temple, University of Bristol Ch 09 Accounting for Cross-Country Income Differences Francesco Caselli, Harvard University Ch 10 Accounting for Growth in the Information Age Dale Jorgenson, Harvard University Ch 11 Externalities and Growth Peter Klenow, Stanford University, and Andres Rodriguez-Clare, IADB Ch 12 Finance and Growth: Theory and Evidence Ross Levine, University of Minnesota Ch 13 Human Capital and Technology Diffusion Jess Benhabib, NYU and Mark Spiegel, FRBSF Ch 14 Growth Strategies Dani Rodrik, Harvard University Ch 15 National Policies and Economic Growth: A Reappraisal William Easterly, NYU Ch 16 Growth and Ideas Charles I. Jones, University of California at Berkeley Ch 17 Long-term Economic Growth and the History of Technology Joel Mokyr, Northwestern Ch 18 General Purpose Technologies Boyan Jovanovic, NYU and University of Chicago and Peter L. Rousseau, Vanderbilt Ch 19 Technological Progress and Economic Transformation Jeremy Greenwood, University of Rochester and Ananth Seshadri, University of Wisconsin Ch 20 The Effects of Technical Change on Labor Market Inequalities Andreas Hornstein (Richmond Fed), Per Krusell (Princeton), and Gianluca Violante (NYU) Ch 21 A Unified Theory of the Evolution of International Income Levels Stephen Parente, University of Illinois and Edward Prescott, University of Minnesota Ch 22 A Global View of Economic Growth http://elsa.berkeley.edu/~chad/Handbook.html (2 of 3)8/17/2006 11:03:16 AM

http://elsa.berkeley.edu/~chad/Handbook.html Jaume Ventura, CREI Pampeu Fabra Ch 23 Trade, Growth and the Size of Countries Alberto Alesina, Harvard University, Enrico Spolaore, Brown University, and Romain Wacziarg, Stanford University Ch 24 Urbanization and Growth J. Vernon Henderson, Brown University Ch 25 Inequality, Technology, and the Social Contract Roland Benabou, Princeton University Ch 26 Social Capital Steven N. Durlauf and Marcel Fafchamps Ch 27 The Effect of Economic Growth on Social Structures Francois Bourguignon, The World Bank Ch 28 Economic Growth and the Environment: A Review of Theory and Empirics William Brock, University of Wisconsin and M. Scott Taylor, University of Wisconsin Back to my homepage. http://elsa.berkeley.edu/~chad/Handbook.html (3 of 3)8/17/2006 11:03:16 AM

Neoclassical Models of Endogenous Growth: The Effects of Fiscal Policy, Innovation and Fluctuations��� Larry E. Jones University of Minnesota and Federal Reserve Bank of Minneapolis Rodolfo E. Manuelli University of Wisconsin August, 2004 1 Introduction Despite its role as the centerpiece of modern growth theory, the Solow model is decidedly silent on some of its basic questions: Why is average growth in per capita income so much higher now than it was 200 years ago? Why is per capita income so much higher in the member countries of the OECD than in the less developed countries (LDC) of the world? The standard implementation of the Solow model really has no answers for these questions except, perhaps for differences, across time and across countries in the production possibility set. This is typically summarized by differences in Total Factor Productivity (TFP). The fundamental reasons for why TFP might be different in different countries, or in different time periods is left open for speculation. If these differences are supposed to be due to differences in innovations, it is not made clear why access to these innovations should be different, nor is it noted that these innovations themselves are economic decisions they have costs and bene ts, and are made by optimizing, private agents. This basic weakness in the Solow model (and its followers) was the driving force behind the development of the class of endogenous growth models. This literature has been wide and varied, with the models developed ranging from perfectly com- petitive, convex models to ones featuring a range of types of market failures (e.g., increasing returns, external effects, imperfectly competitive behavior by rms, etc.). But, a common feature that has been emphasized throughout is knowledge, or hu- man capital, and its production and dissemination. In come cases, this has been ���Draft of a chapter for the forthcoming Handbook of Economic Growth edited by S. Durlauf and P. Aghion. 1

directly treated in the modeling, in others it has been more tangential, an important consideration for quantitative development, but less so for qualitative work. That this focus is essential follows from the fact that the Solow model already accurately re ects the quantitative limits of using models with only physical capital. (That is, capitals share is determined by the data to put us in the Solow range, technolog- ically.) Although they differ in their details, in the end, what this class of models points to as differences in development are differences in social institutions across time and countries. Thus, countries that have weaker systems of property rights, or higher wasteful taxation and spending policies, will tend to grow more slowly. Moreover, these differences in performance can be permanent if these institutions are unchanging. As a corollary, those countries who developed these growth enhancing institutions more recently (and some still have not), have levels of income that are lower than those in which they were adopted earlier, even if current growth rates show only small differences. In this paper we limit ourselves to studying neoclassical models. By this we mean models with convex production sets, well behaved preferences and a market structure that is consistent with competitive behavior. Therefore, we do not review the large literature that adresses the role of externalities and non-competitive markets. As it turns out, most of the basic ideas behind this literature can be expressed in simple, convex models of aggregate variables without uncertainty. These are the models that are the rst focus of this chapter. They have proven both highly exible and easy to use. With them, we can give substance to statements like those above that property rights and other governmental institutions are key to long run growth rates in a society. Most of this branch of the literature is well known by now, and much of it appears on standard graduate macro reading lists. Accordingly, our discussion will be fairly brief.1 One important, and as of yet unresolved issue, is the size of the growth effects of cross-country differences in scal policy. Thus, our review of the standard convex model is complemented with a discussion of the more recent ndings about the quantitative effects of taxes (and government spending) on growth. Even though the theoretical effects of social institutions are well understood, this is less true of the recent work on perfectly competitive models of innovation, and so, comparatively more space is used to discuss that ongoing development. As a second focus, one issue that comes immediately to light in studying this class of models is the possibility that uncertainty per se might have an impact on long run performance. This points to the possibility that instability in property rights and institutions might change the incentives for investment. That is, how are the time paths of savings, consumption and investment affected by uncertainty in this class of models? How does this compare with how uncertainty affects decisions in the Solow model (i.e., Brock and Mirman 1Other authors have also presented comprehensive surveys of this literature (see Barro and Sala- i-Martin (1995), Jones and Manuelli (1997), and Aghion and Howitt (1998) for examples). Our aim is to complement those presentations, rather than repeat them, and hence, our focus is somewhat distinct. 2

(1972) vs. Cass (1965) and Koopmans (1965))? Much less is known about the answers to these questions at the present time and that knowledge that does exist is much less widespread. For this reason, we present a fairly detailed discussion of the properties of stochastic, convex models of endogenous growth. To this end, we study models in which technologies and policies are subject to random shocks. We characterize the effects of differential amounts of uncertainty on average growth. We show that increased uncertainty can increase or decrease average growth depending on both the parameters of the model and the source of the uncertainty. A separate, but related topic, is the business cycle frequency properties of these models. This is left to future work. In section 2, we lay out the basics of the class of neoclassical (i.e., convex) models of endogenous growth. We show how differences in social institutions across time and across countries can give rise to different performance, even over the very long run. We also lay out some of the interpretations of the model, including human capital investment and innovation and knowledge diffusion sectors, that lend richness to its interpretation. Section 3 discusses properties of the models when uncertainty is added, and shows how this can affect the long run growth rate of an economy. 2 Endogenous Growth: In nite Lifetimes Historically, the engine of growth as depicted in Solows seminal work on the topic (1956) was the assumption of exogenous technical change. Thus, initially, growth models aimed at being consistent with growth facts, but gave up on the possibility of explaining them. Moreover, this approach has weaknesses in two distinct areas. First, it is di���cult using the exogenous growth model to explain the observed long run differences in performance exhibited by different countries. Second, the productivity changes that are assumed exogenous in the Solow model are, in fact, the result of conscious decisions on the part of economic agents. If this is the case, it is then important to explore both the mechanism through which productivity changes as well as the factors that can give rise to the observed long run differences if we are to understand these phenomena. In this section we brie y review the basic optimal growth model as initially analyzed by Cass (1965) and Koopmans (1965). We then discuss the nature of the technologies consistent with endogenous growth and the role of scal policy in in uencing the growth rate. We conclude with an analysis of the role of innovation in the context of convex models of equilibrium growth. 2.1 Growth and the Solow Model In the simplest time invariant version of the Solow model, it can be shown that the per capita stock of capital converges to a unique value independent of initial conditions. It is then necessary to assume some exogenous source of productivity growth in order to account for long run growth. In Solow (1956), it is assumed that 3

labor productivity grows continually and exogenously. In response, the capital stock (assumed homogeneous over time) is continually increased allowing for a continual expansion in the level of output and consumption. The literature on endogenous growth has concentrated on replacing this assumed exogenous productivity growth by an endogenous process. If this change in productivity of labor is thought to arise from the invention of techniques consciously developed, the literature on endogenous growth can then be thought of as explicitly modeling the decisions to create this technological improvement (see Shell (1967) and (1973)). For this to go beyond a reinterpretation of the Solow treatment, it must be that the technology for discovering and developing these new technologies does not have itself a source of exogenous technological change. Because of this, these models all feature technologies that are time stationary. The consumer problem in the simple growth model is given by max {ct} X��� t=0 ��tu(ct) subject to ��� X t=0 pt(ct + xt) ��� W0 + X��� t=0 ptrtkt, (1a) kt+1 = (1 ��� ��k)kt + xt, (1b) where ct is the level of consumption, xt is investment, kt is the capital stock, pt is the price of consumption (relative to time 0), and rt is the rental price of capital, all in period t, and W0 is the present value of wealth net of capital income. The rst order condition for (an interior) solution to this problem is just u0(ct) = ��u0(ct+1)[1 ��� ��k + rt+1]. (2) If, as is standard in the literature, the instantaneous utility function, u(ct), is assumed strictly concave, growth de ned as a situation in which ct+1 ct requires ��[1 ��� ��k + rt+1] 1. (3) Condition (3) is fairly general, and must hold independently of the details of the production side of the economy. Thus, if the economy is going to display long run growth, the rate of return on savings must be su���ciently high. What determines the economys rate of return? In the standard Solow growth model and in many convex models rms can be viewed as solving a static prob- lem. More precisely, each rm maximizes pro ts given by ��t = max k,n c + x ��� rtk ��� wtn, 4

subject to c + x ��� F (k, n), where F is a concave production function that displays constant returns to scale. Since in equilibrium the household offers inelastically one unit of labor, the rental rate of capital must satisfy rt = f(kt), (4) where f(k) = F (k, 1), and k is capital per worker. It is now straightforward to analyze growth in the Solow model. The equilibrium version of (2) is just u0(ct) = ��u0(ct+1)[1 ��� ��k + f(kt+1)]. (5) If the productivity of capital is su���ciently low as the stock of capital per worker increases, then there is no long run growth. To see this, note that if limk������f 0(k) =r, fl with 1 ��� ��k+r fl 1, there exists a nite k��� such that 1 ��� ��k + f(k���) = 1. It is standard to show that the unique competitive equilibrium for this economy (as well as the symmetric optimal allocation) is such that the sequence of capital stocks {kt} converges to k���. Given this, consumption is also bounded. (Actually, it converges to f(k���) ��� ��kk���.) Can exogenous technological change solve the problem. The answer depends on the nature of the questions that the model is designed to answer. If one is content to generate equilibrium growth, then the answer is a clear yes. If, on the other hand, the objective is to understand how policies and institutions affect growth, then the answer is negative. To see this assume that technological progress is labor augmenting. Speci cally, assume that, at time t, the amount of effective labor is zt = z(1 + ��)t. In order to guarantee existence of a balanced growth path we assume that the utility function is isoelastic (see Jones and Manuelli (1990) for details), and given by u(c) = c1�����/(1�����). Let a ^ over a variable denote its value relative to effective labor. Thus, ct ��� ct/(z(1+ ��)t). In this case, the balanced growth version of (2) is (1 + ��)u0(t) c = ��u0(t+1)[1 c ��� ��k + f 0(kt+1)] where �� = ��(1 + ��)1�����.2 Standard arguments show that the equilibrium of this economy converges to a steady state ( c, k). Thus, this implies that, asymptotically, consumption is given by ct = (1 + ��)tzc. Thus, even though there is equilibrium growth, the growth rate is completely determined by the exogenous increase in labor augmenting productivity. 2Existence of a solution requires that ��(1 + ��)1����� 1, which we assume. 5

2.2 A One Sector Model of Equilibrium Growth As we argued before, the critical assumption that results in the economy not growing is that the marginal product of capital is low. The modern growth literature has emphasized the analysis of economies in which the marginal product of capital remains (su���ciently) bounded away from zero. This induces positive long-run growth in equilibrium. As we will show, how fast output grows in these models depends on a variety of factors (e.g., parameters of preferences). Because of this, these models have the property that the rate of growth is determined by the agents in the model. Throughout, there will be one common theme. This mirrors the point emphasized above, that for growth to occur, the interest rate (either implicit in a planning problem or explicit in an equilibrium condition) must be kept from being driven too low. This follows immediately from the discussion above. In terms of key features of the environment that are necessary to obtain endoge- nous growth there is one that stands out: it is necessary that the marginal product of some augmentable input be bounded strictly away from zero in the production of some augmentable input which can be used to produce consumption. Since we are dealing with convex economies, the arguments in Debreu (1954) apply to the environments that we study. Thus, in the absence of distortionary government policies, equilibrium and optimal allocations coincide. Thus, for ease of exposition, we will limit ourselves to analyzing planners problems. The planners problem in the basic one sector growth model is given by max {ct} X��� t=0 ��tu(ct), subject to ct + xt ��� F (kt,nt), kt+1 ��� (1 ��� ��k)kt + xt, where ct is per capita consumption, kt is the per capita stock of capital, xt is the (nonnegative) ow of investment, and nt is employment at time t. Since we assume that leisure does not yields utility, the optimal (and equilibrium) level of nt equals the endowment, which we normalize to 1. The Euler equation for this problem is just (5) given that. as before, we set f(k) = F (k, 1). It follows that if limk������ ��[1�����k+f(k)] 1, then lim supt ct = ���. Thus, there is equilibrium growth. This result does not depend on the assumption of just one capital stock. More precisely, in the case of multiple capital stocks, the feasibility constraint is just ct + XI i=1 xit ��� f(k1t, ..., kIt), kit+1 ��� (1 ��� ��ik)kit + xit. 6

In this case, the natural analogue of the assumption that the marginal product of capital is bounded is just that there is a homogeneous of degree one function a linear function that is a lower bound for the actual production function. However, it turns out that all that is required is that there exist a ray that has bounded marginal products. Formally, this corresponds to Condition 1 (G) Assume that f(k1,...,kI) ��� h(k1,...,kI), where h is concave, ho- mogeneous of degree one and C1 for all (k1, ..., kI) ��� R+. I Moreover, assume that there exists a vector k = (k1, ..., kI), k = 0, such that if ki 0, ��[1 ��� ��k + hi(k)] 1, i = 1, ..., I The basic result is the following (see Jones and Manuelli (1990)) Proposition 2 Assume that Condition G is satis ed. Then, any optimal solution {ct ���} is such that lim supt ct ��� = ���. As Jones and Manuelli (1990) show, the planners solution can be supported as a competitive equilibrium. An extension to multiple goods is presented by Kaganovich (1998) and it is based on similar insights. It is clear that Condition G does not rule out decreasing returns to scale. This, in turn implies that this class of models is consistent with a version of the notion of conditional convergence: relatively poor countries are predicted to grow faster than richer countries, with the consequent closing of the income gap. Put it differently, theory suggests that, with a nite amount of data, it is di���cult to distinguish an endogenous growth model from a Cass-Koopmans exogenous growth model. The main difference lies in the tail behavior of the relevant variables (output or consumption), and not in the balanced (or unbalanced) nature of the equilibrium path. 2.3 Fiscal Policy and Growth In this section we describe the effects of taxes and government spending on the long run growth rate. Consider the problem faced by a representative agent max X��� t=0 ��tu(ct, 1 ��� nt) subject to (1+�� c)ct +(1+�� x)ptxkt +(1+�� h)qt ��� wt(1����� n)(nctht +nktht)+(1����� k)rtkt +Tt +��t, where �� j represent tax rates, ct is consumption, xkt is investment in physical capital, qt are market goods used in the production of human capital, nitht is effective labor 7

the product of human capital and hours allocated to sector i, kt is the stock of capital, Tt is a government transfer, and ��t are net pro ts. Accumulation of human capital at the household level satis es ht+1 ��� (1 ��� ��h)ht + F h(qt, nhtht), where F h is homogeneous of degree one, concave and increasing in each argument. The economy has two sectors: producers of capital and consumption goods. Out- put of the capital goods industries satis es xt ��� F k(kkt, nktht), where F k is homogeneous of degree one and concave. Feasibility in the consumption goods industry is given by ct ��� F c(kct,nctht), where F c is increasing and concave. It is not necessary to assume that this production function displays constant returns to scale. It is illustrative to consider several special cases. Throughout, we assume that the utility function is of the form that is consistent with the existence of a balanced growth path. Speci cally, we assume that u(c, ) = (cv( ))1�����/(1�����). Moreover, since our emphasis is on the role of taxes and tax-like wedges between marginal rates of substitution and transformation, we assume that lump sum transfers, Tt, are adjusted to satisfy the government budget constraint. Case I: One Sector Model with Capital Taxation We assume that the con- sumer supplies one unit of labor inelastically. In this case F c = F k = Ak + F (k), where F (k) is strictly concave and limk������ F 0(k) = 0. For now we ignore human capital and set F h ��� 0. It follows that the balanced growth rate satis es ���� = ��[1 ��� ��k + 1 ��� �� k 1 + �� x A]. Thus, in this setting, increases in the effective tax on capital, (1 ��� �� k)/(1 + �� x) unambiguously decrease the equilibrium tax rate. Thus, unlike exogenous growth models, government policies affect the growth rate. Moreover, this simple example illustrates the size of the impact of changes in tax rates on the long run growth rate depend on the intertemporal elasticity of substitution 1/��. More precisely the elasticity of the growth rate with respect to �� k is given by ����� ����� k �� k �� = ��� 1 �� ��k 1+��x A 1 ��� ��k + 1�����k 1+��x A . 8

It follows that, other things constant, high values of the intertemporal elasticity of substitution result in large changes in predicted growth rates in response to changes in tax rates. Thus, even an example as simple as this one illustrates that the quantitative predictions of this class of models will heavily depend on the values of the relevant preference (and technology) parameters. Case II: Physical and Human Capital: Identical Technologies In this sec- tion we assume that F c = F k, and F h = q. This implies that all three goods investment, consumption and human capital are produced using the same tech- nology and, in particular, the same physical to human capital ratio. As in the previous section, �� k and �� x do not play independent roles. Thus, to simplify notation, we will set �� x = 0. However, the reader should keep in mind that increases in the tax rate on capital income are equivalent to increases in the tax rate on purchases of capital goods. In this case, the balanced growth conditions are ���� = ��[1 ��� ��k + (1 ��� �� k)Fk(��, n)] (6a) c h v0(1 ��� n) v(1 ��� n) = 1 ��� �� n 1 + �� c Fn(��, n) (6b) (1 ��� �� k)Fk(��, n) ��� ��k = 1 ��� �� n 1 + �� h Fn(��, n)n ��� ��h (6c) c h + (�� + ��k ��� 1) = F (��,n). (6d) There are several interesting points. First, increases in the tax rate on consump- tion goods (i.e. sales or value added taxes) are equivalent to increases in the tax rate on labor income. Second, the relevant tax rate to evaluate the return on human capital is (1 ��� �� n)/(1 + �� h). Thus, it is possible that increases in �� n as observed in the U.S. between the pre World War II and the post WWII periods if matched by decreases in �� h (corresponding, for example, to expansion in the quantity and quality of free public education) have no effect on the physical capital - human capital ratio, ��. Third, it is possible to show that increases in �� k,�� n,�� h or �� c result in lower growth rates. Last, without making additional assumptions about preferences and technol- ogy, it is not possible to sign the impact of changes in tax rates on other endogenous variables. Case III: Physical and Human Capital: Different Factor Intensities In this case, we assume that only human capital is used in the production of human capital. Thus, F h = Ahnhh. This is the technology proposed by Uzawa (1964) and popularized in this class of models by Lucas (1988). For simplicity, we only consider capital and labor taxes. The relevant steady state conditions are (6a), (6b), and (6d). However, (6c) becomes ���� = ��[1 ��� ��h + Ahn] (7) 9

In this version of the model, changes in labor income taxes, reduce growth through their impact on hours worked (relative to leisure). However, if total work time is inelastically supplied, i.e. v( ) ��� 1, the growth rate is pinned down by ���� = ��[1 ��� ��h + Ah]. Thus, in this setting (which corresponds to Lucas (1988) model without the ex- ternality, and to Lucas (1990)), taxes have no effect on growth. Increases in the tax rate on capital income simply change physical capital - human capital ratio and they leave the after tax rate of return unchanged. The reason for this extreme for of neu- trality is that even though taxes on labor income reduce the returns from education, they also reduce the cost of using time to accumulate human capital (the value of time decreases with increases in taxes), and the two changes are identical. Thus, the cost-bene t ratio of investing in education is independent of the tax code. 2.3.1 Quantitative Analysis of the Effects of Taxes Since the development of endogenous growth theory there have been several studies of the implications of substituting lump-sum taxes for a variety of distortionary taxes. Jones, Manuelli and Rossi (1993), analyze the optimal choice of distortionary taxes in several models of endogenous growth. In the case that physical and human capital are produced using the same technology and labor supply is inelastic, they nd that for parameterizations that make the predictions of the model consistent with observations from the U.S., the potential growth effects of drastically reducing (eliminating in most cases) all forms of distortionary taxation is quite high. For their preferred parameterization the increase in growth rates is about 3%. They study a version of the model in which F c = F k = F h., and the functions F k andF h are both of the Cobb- Douglas variety, but differ in the average productivity of capital. Jones, Manuelli and Rossi estimate the capital share parameter to be equal 0.36 in the consumption sector, and to be somewhere in the 0.40-0.50 range in the human capital production sector.3 They also allow labor supply to be elastic. Their ndings suggest that switching to an optimal tax code result in increases in yearly growth rates of somewhere between 1.5% and 2.0% per year. These are substantial effects. The third experiment that they consider involves the endogenous determination (by the planner) of the level of government consumption. In this case, they revert back to the one sector version of the model, and they explore not only the consequences of changing the intertemporal elasticity of substitution, but they allow for varying elasticity of substitution between capital and human capital. For their preferred characterization, they also nd growth effects of about 2% per year. Moreover, as in the other experiments, the predictions are quite sensitive to the details of the model 3Jones, Manuelli and Rossi (1993) calibrate this share. Since they study the sensitivity of their results to changes in other parameters (e.g. the intertemporal elasticity of substitution), the market goods share is not constant across experiments. 10

in particular, to the choice of the intertemporal elasticity of substitution, and the degree of substitutability between capital and human capital. Stokey and Rebelo (1995) undertake a thorough review of several models that estimate the growth impact of tax reform. They argue that in the U.S. tax rates in the post WWII period are signi cantly higher than in the pre WWII era. This conclusion is based on the increase in the revenue from income taxes as a fraction of GDP in the early 1940s. To reconcile the models with this evidence, they conclude that the human capital share in the production of human capital must be large, and that this sector must be lightly taxed. This description is close to the Case III above and, as argued before, it results in no growth effects4. Thus, in agreement with Lucas (1990) and using a very similar speci cation of the human capital production technology they conclude that changes in tax rates cannot have large growth effects. This conclusion depends on several assumptions. First, that the U.S. evidence shows an increase in the general level of taxes after WWII. Second, that even if there is a tax increase, the additional revenue is used to nance lump-sum transfers. Third, that the balanced growth path is a good description of the pre and post WWII economy. Measuring changes in the relevant marginal tax rates is a di���cult task. Barro and Sahasakul (1986) using tax records compute average marginal tax rates for the U.S. economy. Their estimates, consistent with the Stokey and Rebelo assumption, show an increase in the 1940s. Even though the evidence about changes in the tax rate consistently points to an increase, the implications of this result for the model are not obvious. Consider, rst, the uses of tax revenue. If, for example, additional income tax revenues (at the local level) are used to nance local publicly provided goods (e.g. education), then Tiebout-like arguments suggest that the tax effect of a tax increase is zero. In the U.S. a substantial increase in government spending corresponds to increases in expenditures on education and, hence, the possibility of individuals sorting themselves to buy the right bundle of publicly provided private goods cannot be ignored. A second quantitatively important public spending program in the post WWII era is Social Security. To the extent that bene ts are dependent on contributions, the statutory tax rate on labor income used to nance social security overstates the true tax rate.5 In this case, tax payments purchase the right to an annuity whose value is dependent on the payment. Finally, in a model with multiple tax rates an increase in a single tax does not imply, necessarily, a decrease in the growth rate. For the U.S. the evidence on the time path of capital income taxes is mixed. In a recent study, Mulligan (2003) argues that the tax rate on capital income has steadily fallen in the last 50 years. Similarly, Prescott and McGrattan (2003) and (2004) nd that a decrease in the tax rate on corporate income one form of 4The results are continuous in the parameters. Thus, for market goods share close to zero, as Stokey and Rebelo prefer, the growth effects are small. 5In a pay-as-you-go system, even if the share of total payments that an individual receives is sensitive to his contributions, the same effect obtains. 11

capital income is instrumental in explaining the increase in the value of corporate capital relative to GDP. Overall, we nd that the quantitative evidence on the time path of the relevant tax rates to be di���cult to ascertain. More work is needed, with an emphasis on matching model and data. The next section considers the effects of endogenous government spending and transitional effects. 2.3.2 Productive Government Spending A Simple Balanced Growth Result In this section we study a simple one sec- tor model that provides a role for productive government spending. Our discussion follows the ideas in Barro (1990). Assume that rm i0s technology is given by yit ��� AkithitGt �� �� 1����������, where kit and hit are the amounts of physical and human capital used by the rm, and Gt is a measure of productive public goods that rms take as given. The government budget constraint is balanced in every period, and it satis es Gt = �� krtKt + �� hwtHt, where �� k and �� h are the tax rates on capital and income, and rt and wt are rental prices. For simplicity we assume that the instantaneous utility function is given by u(c) = c1����� ��� 1 1 ��� �� . We also assume that the technologies to produce market goods and human capital are identical. In this case, it is immediate to show that the equilibrium is fully described by ��h ��� ��k = A1/(��+��)(���� k + ���� h])(1����������)/(��+��)[��(1 ��� �� h)����/(��+��) ��� ��(1 ��� �� k)�������/(��+��)], ���� = ��[1 ��� ��k + ��(1 ��� �� k)A1/(��+��)(���� k + ���� h)(1����������)/(��+��)�������/(��+��)], where �� is the physical capital - human capital ratio. Some tedious algebra shows that the growth rate is not a monotonic function of the tax rates. In general, there is no growth when taxes are either too low (not enough public goods are provided) or too high (the private returns to capital accumulation are too low). For intermediate values of the tax rates, growth is positive (if A is su���ciently high). Thus, in general, increases in tax rates need not result in lower growth if they are accompanied by changes in government spending. Thus, a variant of the model with endogenous government spending (or endogenous taxation and optimally chosen government spending) has potential to reconcile positive growth effects associated with the removal of distortions with the U.S. evidence. 12