TAXATION AND LONG-RUN GROWTH

Transcription

1 TAXATION AND LONG-RUN GROWTH by Lutz Hendricks Arizona State University First draft: May 1994 This draft: April 1996 Correspondence address: Arizona State University, Department of Economics, PO Box , Tempe, AZ Tel. (602) For helpful suggestions I am grateful to Andrew Abel, Michael Binder, Matthias Kahl, and Lee Ohanian.

2 Abstract A number of recent papers have investigated the growth effects of tax reforms in the context of neoclassical growth models where growth is due to human capital accumulation. Stokey and Rebelo (1995) show that the predicted growth effects disagree to a striking extent and are highly sensitive to the choices of several parameters about which little evidence exists. The purpose of this paper is to argue that the question should be reconsidered in the context of a life-cycle framework instead of the infinite horizon model used previously. Since human capital is not heritable, the infinite horizon case can no longer be derived as a reduced form of an altruistically linked dynasty of finitely lived individuals. Moreover, modeling agents as infinitely lived hides a fundamental asymmetry between human and physical capital: Since human capital cannot be sold or decumulated, agents must primarily use physical capital holdings to smooth consumption over the life-cycle and in particular to finance retirement consumption. As a consequence, changes in factor taxation mostly affect the stock of physical but not that of human capital. Correspondingly, our simulation results show that changes in flat rate factor taxation have little impact on long-run growth. In marked contrast to the previous literature, this result is remarkably robust to changes in the calibration and even to variations in the way human capital accumulation and intergenerational transfers are modeled. This strongly suggests that the large growth effects of taxation found previously overstate the true effect, perhaps by an order of magnitude. Much smaller effects are consistent with the observed stability of the U.S. growth trend in spite of dramatic increases in income tax rates after World War II. JEL Classification: O41. Keywords: Economic growth; taxation; human capital.

3 Introduction A number of recent papers have investigated the growth effects of tax reforms in the context of neoclassical growth models. The predicted growth effects disagree to a striking extent. For comparable tax reforms, Lucas (1990) finds negligible changes in the growth rate, while some of Jones, Manuelli, and Rossi s (1993) models predict that growth rates should increase by more than 8 percent per annum. Other studies find a range of intermediate values. Most remarkably, all of these papers rely on almost the same model: the standard neoclassical growth model with human capital accumulation as the engine of growth. This surprising result prompted Stokey and Rebelo (1995) to investigate the source of such differences. Their simulations show that the predictions of the neoclassical model are highly sensitive to the exact specification of a number of parameters, such as the intertemporal elasticity of substitution in the utility function, the depreciation rate of human capital, and the factor shares of the human capital production function. Variations in each of these parameters alone can alter the growth effects of taxation by a factor of at least two. Since estimates of all three parameters are rather imprecise, it is very difficult to determine the magnitude of growth effects with any precision. The purpose of this paper is to argue that the question should be reconsidered in the context of a life-cycle framework instead of the infinite horizon model used previously. There are two main reasons. First, much of the sensitivity of the infinite horizon model stems from assumption that aggregate per capita income must grow at the same rate as individual consumption. If there is no bequest motive, this condition is relaxed and predicted growth effects are much less sensitive. Secondly, even if there is a bequest motive, the life-cycle model does not have a reduced form representation with an infinitely lived representative agent. The reason is that human capital is not heritable. This introduces a fundamental asymmetry between physical and human capital that 1

4 is missed in the infinite horizon case: Human capital accumulation is necessarily subject to diminishing returns. This changes savings behavior qualitatively which is seen most directly in the case of exogenous leisure. The household first chooses education to maximize total life-cycle resources. It then picks a consumption profile, given total resources. Finally, non-human savings are determined residually so as to reconcile the diverging profiles of earnings and consumption expenditures. In contrast, both types of capital are treated entirely symmetrically in the infinite horizon case. As a consequence, the household adjusts mostly non-human savings in response to changes in factor taxation, and human capital does not respond at all to wealth effects. Correspondingly, our simulation results show that changes in flat rate factor taxation have little impact on long-run growth. In marked contrast to the previous literature, this result is remarkably robust to changes in the calibration and even to variations in the way human capital accumulation and intergenerational transfers are modeled. While most of the key parameters identified by Stokey and Rebelo play only a minor role in our model, we find that an apparent detail, the way taxation treats inputs to education, affects the growth effects of taxation quite significantly. 1 The assumption implicitly made in previous studies, that goods inputs to education are not tax-deductible, finds little support in the data and may lead to a severe overstatement of the impact of taxation on growth. This strongly suggests that the large growth effects of taxation found previously overstate the true effect, perhaps by an order of magnitude. Much smaller effects are consistent with the observed stability of the U.S. growth trend in spite of dramatic increases in income tax rates after World War II. 1 Trostel (1993) makes a similar point regarding the impact of taxation on the stock of human capital. 2

5 The rest of the paper is organized as follows. The next section argues that, in the presence of human capital, an explicit overlapping generations model has several advantages over the infinite horizon setup used so far. Section 3 presents such a model and characterizes the competitive equilibrium. Section 4 briefly outlines the computational algorithm used to solve the model and explains the choice of parameter values. Particular attention is paid to the tax treatment of inputs in education. Simulation results with an extensive sensitivity analysis are presented in section 5, and the final section concludes. The Overlapping Generations Framework The models used in the previous literature are generally based on the standard neoclassical twosector growth model, where human capital accumulation by an infinitely lived representative household is the engine of growth. In many applications, the assumption of infinite lifetimes can be justified as a reduced form representation of an overlapping generations model. However, in the presence of human capital such a reduced form does not exist. Since human capital is lost when its owner dies, the transfer of human capital across generations can never be complete. Of course, the assumption of infinite lifetimes may still be a useful approximation, if it yields approximately the same results as a fully spelled out overlapping generations model, perhaps with bequests. But the simulation results presented below clearly demonstrate that this is not the case. Growth effects are considerably smaller and more robust than the neoclassical model suggests. A major problem with the predictions of the infinite horizon problem is their sensitivity with respect to a number of parameters about which little conclusive evidence exists (Stokey and Rebelo 1995). There are two main reasons why an overlapping generations model alleviates this 3

6 problem: (i) The Euler-equation governing the intertemporal allocation of consumption is broken. (ii) The symmetry between human and physical capital is broken. The role of the Euler-equation in the sensitivity of predicted growth effects is illustrated easily in a simple model. 2 An infinitely lived household maximizes utility 1 σ ρt e 0 1 ct dt, σ where ρ is the discount factor and c is consumption of a single good. This implies directly that along a balanced growth path the Euler-equation $c r = ρ σ (1) must hold. Here, $c denotes the growth rate of c, and r is the after-tax interest rate. This is the root cause of the sensitivity problem: Taxes affect growth by changing the interest rate, which the Euler-equation translates into a change in the growth rate of consumption and, by the assumption of a representative agent, income. To predict the growth effects of taxation, it is necessary to determine exactly how taxes affect the interest rate, for which the tax treatment and magnitude of depreciation are crucial, and how the change in r gets translated into growth effects, for which σ is key. Unfortunately, estimates of σ differ substantially, 3 making it difficult to pin down growth effects with any precision. 4 2 This follows section I of Stokey and Rebelo (1995). 3 See, for instance, the evidence cited in Auerbach and Kotlikoff (1987). 4 A striking example is provided by Jones, Manuelli, and Rossi s (1993) model 2. For σ = 2, the growth rate changes by only two percent. However, lowering σ to 1.1 magnifies this change to as much as eight percent. 4

7 To explore this sensitivity problem further, it is useful to consider the general equilibrium properties of a tractable special case. 5 The constraints faced by the household are the accumulation conditions for physical and human capital &k = I δ k t 1t 1 t &h = I δ h, (2) t 2t 2 t and the budget constraint p I + p I + c q k q h T =. 1t 1t 2t 2t t 1t t 2t t t 0 There are three sectors producing physical capital, human capital and final goods at tax inclusive prices p 1t, p 2t, and 1, respectively. I st is investment in accumulation of factor s. The corresponding after-tax rental prices are q st. T t are lump-sum transfers. The stocks of human and physical capital are denoted by k and h, while their respective depreciation rates are δ k and δ h. Assume that the production functions exhibit constant returns in (k, h) and denote them by Ik = G( kk, hk) Ih = H( kh, hh) It is easy to show that a balanced growth path is characterized by G G h k Hh = (3) H k where subscripts denote partial derivatives. In addition, the rates of return of both types of capital must equal the interest rate. In the case of taxation gross of depreciation: ( 1 τ ) G ( 1 τ ) H δ δ 1 k 1 2 h 2 = r = r 5 This follows Stokey and Rebelo (1995). 5 (4)

8 Here τ s is the tax rate on factor income in sector s. Note the peculiar structure induced by the assumption of a representative agent: The production side alone, via (3) and (4), determines the factor input ratios in both sectors and the interest rate. Preferences translate this interest rate into the growth rate. In the simplest symmetric case all income is taxed at the common rate τ = τ 1 = τ 2 and depreciation rates are equal for both factors. Then input ratios are unaffected by taxes and the change in the interest rate is r = τ G k which implies a change in the growth rate of γ r τg = = k δ = τ + + ρ γ. (5) σ σ σ If, on the other hand, returns are taxed net of depreciation, (4) becomes ( 1 τ1)[ Gk δ1] = r ( 1 τ )[ H δ ] = r 2 h 2 such that the change in the growth rate is r τ( G γ = = k δ) ρ = τ γ +. (7) σ σ σ Comparing (5) with (7) reveals that growth effects must be sensitive to the tax treatment and magnitude of depreciation. The difference in the change in the growth rate between the two cases is τδ / σ which is of the same magnitude as τ γ, if the depreciation rate is around 0.1 as in the case of physical capital. However, the depreciation rate of human capital is much more controversial. Estimates used in the literature range from near zero to as much as 12 percent (Lord 1989). But even if the true value of the depreciation rate were known with certainty, it would be difficult to determine the appropriate value for the corresponding model parameter δ h 6 (6)

9 from it. For, in an infinite horizon model, δ h has to capture obsolescence of knowledge as well as the depreciation of human capital at death. In general, it is not obvious how estimates of parameters in the data translate into parameter values in the reduced form model. A life-cycle model breaks the artificial link between the growth rate and the interest rate implied by the Euler-equation. As a consequence, depreciation rates and, importantly, the intertemporal elasticity of substitution play a much smaller role and the predicted growth effects are substantially more robust. Of course, the Euler-equation is retained, if there is an operative bequest motive, even though the infinite horizon model still is not a correct reduced form. Which case represents reality more accurately is an empirical question. However, it turns out that results are less sensitive even if the assumption of a bequest motive is retained. This is due to a fundamental asymmetry between human and physical capital that is easily missed in an infinite horizon model. Note that both types of capital are perfectly symmetric in the model presented above. In a life-cycle model, on the other hand, the roles of human and physical capital are qualitatively different: With exogenous leisure, the household chooses education so as to maximize the present value of earnings net of education costs. This determines the total amount of wealth available for consumption. Non-human savings are then determined residually so as to reconcile the divergent earnings and consumption profiles. All consumption smoothing is financed with non-human savings. The reason for this difference is that human capital accumulation is subject to diminishing returns to private inputs. Spelling out the life-cycle makes this assumption inevitable. Wage profiles peaks late in life, revealing that education is continuing for decades. With constant returns to private inputs, in contrast, it would be optimal to follow a bang-bang policy and concentrate all schooling at the beginning of life. Econometric estimates confirm that returns to private inputs are substantially below unity. 7

10 Note that diminishing returns change the household s savings behavior qualitatively. Human capital accumulation responds less to changes in factor prices and not at all to wealth effects. This point, which is missed entirely in the previous literature, can be illustrated easily in a slight extension of the model presented above. Suppose the accumulation equation for human capital in the model presented above is replaced by ϕ 1 ϕ &h = I h δ h, (8) t 2t t 2 t where h t denotes the average level of human capital in the economy. The household takes this as given, although in equilibrium h t = h. t If ϕ < 1, a tax on factor incomes now affects k and h asymmetrically. It can be shown that for small ϕ the growth effects of taxation are small because the household primarily adjusts nonhuman wealth. The intuition is that the household equates the rates of return of both assets. A tax reduces both returns symmetrically. If there are strongly diminishing returns in human capital accumulation, a small reduction in I2 t / ht raises the rate of return to human capital accumulation above the given interest rate. Therefore, the household largely adjusts non-human wealth instead. The assumption of diminishing returns to human capital accumulation is not necessarily linked to a life-cycle setup. Note, however, that there will never be constant returns to private inputs in a life-cycle model, simply because human capital accumulation takes time. Since human capital is lost at death, the payoff from additional human capital diminishes as the household ages. Even if the household is allowed to purchase additional human capital at a constant cost, returns will effectively be diminishing. The optimal accumulation profile is then characterized by a period of full-time schooling followed by a switch to full-time work (a bang-bang solution). If the household chooses to acquire additional education, the schooling period must be longer. The shorter payoff period for the additional human capital implies a lower rate of return. 8

11 The simulation results presented below confirm these predictions. The after tax interest rate changes only little when taxes are reduced. Growth effects are small and remarkably robust to changes in the critical parameters (σ, δ k, δ h ). An additional set of crucial parameters relates to the human capital technology. If the share of taxed inputs in the production of human capital is small, labor taxation will affect the opportunity cost of inputs in roughly the same proportion as the value of the accumulated skills. This explains why there are next to no growth effects in Lucas (1990). Again, the direct evidence on these parameters is not very conclusive, but the life-cycle profiles of time spent on education can aid the calibration. Note also that it is not obvious how human capital can be the engine of sustained growth if it fully depreciates in finite time. Some mechanism must exist that transfers human capital between generations. One advantage of the overlapping generations framework is that it allows to spell out explicitly the assumptions made about this mechanism. The Model This section outlines an overlapping generations model that spells out explicitly how human capital is transferred between generations. Below, we will consider several variations of the basic structure and demonstrate that results are not sensitive to those variations. The model structure is similar to Auerbach and Kotlikoff (1987). The main modification is the addition of an education technology which allows households to invest time and goods in order to increase their stock of human capital. This makes the growth rate endogenous. At each date, the economy is inhabited by members of T generations, indexed by age i. Each generation, in turn, consists of a large number (π it ) of identical individuals who maximize the discounted present value of utility over their remaining life-time by choosing paths of 9

12 consumption, leisure, time and goods inputs to education, work time, and savings in the form of physical capital. There are no intergenerational transfers. Firms act as profit maximizers and price takers in all markets. They rent effective labor and physical capital from households to produce a single good which can be used for consumption, investment in human capital, or as physical capital in the production of goods in the next period. The government imposes taxes on consumption, capital and labor income and uses the revenues to finance government consumption and transfers to the households. The resulting deficit is financed by government bonds. In addition, there is a system of old age transfers, loosely modeled after the U.S. social security system, which are financed by payroll taxes. Markets are perfectly competitive and clear at all times. All agents have perfect foresight. Human capital is assumed to be general, not job-specific, and it is acquired through intentional investment only, not through learning-by-doing. 6 The following paragraphs spell out the elements of the model in more detail. We begin with the mechanism that transfers human capital between generations and which provides the basis for sustainable growth. The Growth Mechanism The literature on human capital divides an individual s learning life into three phases: Pre-school child-rearing and early schooling, later formal schooling, and learning during work time (on the job training, OJT). Modeling the last two phases present no particular problem, since previous studies have established a generally accepted framework which we can adopt. Conventionally, these two phases are combined into one, implicitly assuming that the technologies for schooling 6 Under suitable conditions, learning-by-doing gives rise to the same formulation as intentional human capital investment. See Rosen (1977) and Weiss (1986). 10

13 and OJT are identical. Schooling is then defined as a corner solution where the agent participates full-time in education (see the surveys by Hanushek 1979, 1986). When a household enters the model, it is endowed with an initial stock of human capital (h 1 ) 7, which is determined by the first phase of education. In each period it can invest effective time (v i h i ) and goods (x i ) to produce additional human capital according to ( ) hi+ 1 = ( 1 δ h) hi + G vihi, xi, (9) where δ h is the rate of depreciation of human capital. A conventional assumption is that human capital is self-productive; that is, the individual s current stock of human capital enters as a factor of production. The justification comes from studies indicating that learning is facilitated by previously learned material; see Bloom (1976). Empirical estimates of human capital production functions indicate that the returns to private inputs are diminishing. Both Heckman (1976) and Haley (1976) estimate scale elasticities roughly between 0.4 and However, balanced growth requires constant returns to the reproducible factors. Two solutions to this problem have been proposed. Welch (1970) suggest that schooling enhances the ability to learn on the job. 8 A more common approach is to assume a knowledge spillover such that average human capital (h t ) enters the production function G as an additional argument. 9 We follow the latter specification but investigate the idea of learning ability as part of the sensitivity analysis. 7 To ease notation, variables are only indexed by age. The implied date subscript is suppressed. 8 Similarly, Glomm and Ravikumar (1992) assume that the human capital endowment enters the learning technology, representing the ability to learn. 9 See, for instance, Lucas (1988). 11

14 Modeling the first phase of education is more difficult. In an infinite horizon model, human capital is treated in essentially the same way as physical capital. In particular, it is possible for an individual to accumulate unlimited amounts of both assets. One key question remains unanswered: How does human capital get transmitted between generations? Without a link between human capital of old and the endowment of young generations perpetual human capitaldriven growth is not feasible. The literature suggests two possible mechanisms that could determine the endowment h 1. The first is an intergenerational spillover where human capital is acquired as a by-product of some other activity; the second is an intentional investment of parents into their children s human capital. Presumably, both mechanisms play a role, and their importance varies with the age of the child, but there is not enough evidence to distinguish both empirically. If a spillover determines the endowment, balanced growth requires that the young household inherits a constant fraction of the human capital of some older generations: h1, t =ε Ht. The remaining problem is to identify the appropriate H t. Previous papers assumed that individuals inherit a fraction of the average human capital in the economy (Azariadis and Drazen 1990; Arrau 1992). We adopt the alternative specification that H t corresponds to parental human capital. This assumption is supported by evidence which suggests that family background is an important determinant of children's educational attainment. 10 Both versions probably lead to 10 Strong evidence is provided by Hanushek (1986) who reviews 147 studies on the determinants of educational success. Card and Krueger (1992) question this finding. However, their way of controlling for family background may bias the findings against the importance of family background. The same assumption is made in Lucas (1988) who argues: This is simply an instance of a general fact that... human capital accumulation is a social activity, involving groups of people in a way that has no counterpart in the accumulation of physical capital. [p. 19, emphasis in original]. 12

15 very similar outcomes because average human capital and that of a particular generation are highly correlated. Modeling the first phase as parental investment in the child s human capital is theoretically straightforward. The details are worked out in Becker and Tomes (1986). The main assumption is that early education can be described by an accumulation equation of the form of (9). Altruistic parents can leave bequests by financing their children s education or via a cash transfer. If there are no credit constraints, both are equivalent and lead to the same amount of human capital accumulation. While the theoretical issues are straightforward, the empirical implementation faces serious data problems. Because of the difficulty of observing the inputs to pre- and early schooling, essentially nothing is known about the functional form of the production function. In the absence of better evidence we assume that the technology specified for later stages is also valid here. Fortunately, results turn out to be insensitive with respect to the exact specification. Because of the uncertain specification we adopt the spillover case as our baseline. This will understate the effects of policy changes insofar as human capital during this phase is responsive to income incentives. In a later section, we will consider the possibility that parents can invest into changing ε and show that our results are robust to this assumption. Note how an infinite horizon model can hide all these issues by simply assuming that human capital can be accumulated in the same way as physical capital. The Household Problem A household is born at age 1 with an endowment of human capital, h 1. He chooses paths for consumption (c i ), leisure (l i ), time (v i ) and goods (x i ) invested into schooling for the T periods of his life to maximize 13

16 U T i = β u( ci, li). (10) i= 1 subject to x, v, l, c 0 i i i i i 1 l v 0 i i i hi+ 1 = ( 1 δh) hi + G( xi, vi, hi, hi) i = 1,..., T 1 (11) (12) h 1 given T { i ( τci ) i i ( i i ) i i[ ( )( i τli ) τli ]} 1 Ri c 1+ Tr w 1 l v h + x 1 1 Λ s + Γ Λ = 0 (13) i= 1 R i i = ( 1+ rs). s= 1 The first set of constraints represents nonnegativity requirements for consumption, leisure, and inputs to education. Inada conditions on the utility function and the human capital technology guarantee that the household will choose l i, c i > 0, so that two of the constraints will never bind. The stock of human capital evolves according to (12), while (13) is the budget constraint which deserves some explanation. The budget constraint requires that the present discounted value of the period-by-period budget deficits equal zero. Equivalently, the household s asset holdings at death are not allowed to be negative. The first term in (13) represents consumption expenditure including a consumption tax at rate τ ci. As income the household receives transfers of Tr i, which include social security payments, and earnings of w ( 1 l v ) h ( 1 τ ) Λ x, where 14 i i i i li i τl τw + τo is the combined rate of all labor income taxes as explained below. The fraction Λ of education inputs x i is paid for by lower earnings and is therefore fully deductible from labor income taxation. The remaining part, (1-Λ) x i, is paid for directly by the household. It is subsidized at rate s and a fraction Γ of it is tax deductible. It is exactly these details of the tax

17 treatment of investment in education that have usually been overlooked, but turn out to be quite important. In order to be compatible with balanced growth, preferences must be of the constant elasticity form. 11 ci uc (, l) = i i 1 σ ρ( 1 σ) li 1 σ A young individual enters the model at physical age 16. Before this time, he is assumed to be dependent on his parents and unable to make choices affecting his later life. The initial stock of human capital is modeled as inherited from the parents generation as explained above. After the agent has entered the model, he is assumed to be independent of his parents. He can allocate his time between work and further schooling or OJT. To finance initial consumption, the agent has access to perfect credit markets. Following the bulk of the existing literature, we assume that human capital is produced via a production function of the exponential form ψ ζ η Gvxhh (,,, ) = Bv h x h ξ. Balanced growth requires constant returns to the reproducible factors, i.e., ζ + η + ξ = 1. The first order conditions characterizing the solution to the household s problem are tedious, but straightforward and will not be detailed here. 11 The assumption of balanced growth is conventional, although dramatic shifts in time usage over the last decades suggest that it may be a poor justification for imposing structure on preferences. See Owen (1986). 15

18 Firms Firms maximize period-by-period profits, taking prices in factor and output markets as given. 12 The technology uses goods and effective labor to produce a single output according to Y α = AK L 1 α t t t where T Lt = hit( 1 lit vit) πit i= 1, is the total input of effective labor. Firms hire effective labor and physical capital up to the point where the value of their marginal products equals the gross factor prices. * * r = Y / K ; w = Y / L. t t t t t t Households then face the following factor prices after tax and depreciation: * t = ( 1 τkt) rt ( 1 κτkt) δk r w * t lt t = ( 1 τ ) w, where κ is the fraction of depreciation that is tax-deductible. Government The government levies flat rate taxes on labor (τ w ) and capital income (τ K ) to finance lump-sum transfers to households, education subsidies, and government consumption. An additional tax (τ o ) is levied on labor income past age T 1, which is called the retirement age. The latter tax captures the increasing marginal tax rate on labor income of older workers who receive social security benefits. Details of the social security system are given below. 12 The simulation results of Auerbach and Kotlikoff (1987) suggest that modeling the dynamic investment problem of the firm changes results only in minor ways. 16

19 The budget deficit is financed by issuing one period bonds (D t ). These are considered to be perfect substitutes for physical capital by households and have to pay the same after tax interest rate in equilibrium. This assumption is inevitable in a model without uncertainty. It makes it difficult, however, to assign a reasonable numerical value to this hybrid interest rate when the model is calibrated. The government flow budget constraint is given by Dt+ 1 = ( 1 + rt) Dt + Gt + TRt + st Xt Taxt, where X is aggregate investment in education, which is assumed to be tax-deductible, TR t denotes lump-sum transfers, paid in equal amounts to members of all cohorts, and G t is government consumption which does not affect household welfare or firm productivity. The tax revenue, Tax t, has several components. Capital income taxes yield τ [ ] * wage taxes is wl X( Λ+ ( Λ) Γ) τ wt t t t ( r * κδ ) K. The revenue from Kt t K t 1. The expression for τ o is similar, the only difference being that X t and L t are restricted to households of age greater than T 1. Under perfect capital markets, the government budget constraint can be expressed in intertemporal form: The present value of government spending cannot exceed the present value of tax revenues minus the initial stock of government debt. 1 R { Taxt Trt Gt st Xt} D1( r1) t= 1 t 1+ 0 Social Security No attempt is made to model the U.S. social security system in detail. 13 The point to be captured is the discrete change in incentives when the household reaches retirement age as evidenced by a significant drop in hours worked and labor force participation at the age of See, for instance, Feldstein and Samwick (1992). 17

20 The key features of the social security system captured in the model are a payroll tax on labor earnings (τ s ) and old-age transfers in proportion to average labor earnings up to age T 1. The social security budget is assumed to be separate from the government budget and must be balanced in each period. A key feature of the U.S. system of old age transfers is a high implicit tax rate on income for recipients of benefits. Means testing of benefits introduces a 33 percent marginal income tax rate on most recipients. In addition, for some households with higher incomes social security benefits are subject to income taxation. 14 All of these provisions are loosely captured by a flat additional wage tax (τ o ) in the present model. Its revenues are assumed to be part of the overall government budget. Equilibrium A competitive equilibrium is defined in the usual way: a feasible allocation and a sequence of factor prices, such that the household s problem is solved for each generation, firms maximize profits at each date, the budget constraints of the government and the Social Security Authority are satisfied, and markets clear. For the factor markets this means: T Lt = hit( 1 vit lit) π it, i= 1 T Kt = aitπit Dt. i= 1 Implementation 14 See Feldstein and Samwick (1992) for details and estimates of the implied marginal tax rates. 18

21 Solution Method The numerical algorithm used to solve for the equilibrium is based on the Gauss-Seidel procedure used in Auerbach and Kotlikoff (1987). The idea is to iterate over guesses of market and shadow prices for all periods. The household problems are solved conditional on those prices, and the resulting aggregates are used to update the guesses. The iterations terminate when the difference between updated and previous guesses has become sufficiently small. In the present paper, the household problem is complicated significantly by the human capital accumulation decision and the sometimes binding inequality constraints. As a consequence, the household problem must be solved iteratively, even for given shadow prices. Since none of the algorithms available in the literature is capable of solving a problem of this structure, a new algorithm had to be developed for this paper. 15 The basic structure consists of two nested Gauss- Seidel loops. The inner loop solves the household problems for the various generations alive at each date, given factor prices and tax rates which are determined in the outer loop. Calibration Observations to be matched: Parameters are chosen to replicate selected observations of U.S. post-war experience. Unless stated otherwise, the numbers are taken from the Statistical Abstracts of the United States, Because of the problems of finding an empirical counterpart for the model s interest rate, it is not used for calibration purposes. Two observations are replicated precisely. First, the annual growth rate of per capita gross national product from 1950 to 1992 was 1.7 percent. Secondly, based on Juster and Stafford 15 The algorithm is also capable of handling transition paths under rational expectations, although that feature is not used in this paper. In order to simplify the computations, previous studies (Davies and Whalley 1991 or Arrau 1992) assumed either myopic or steady state expectations during transitions or interior solutions to the household problem. 19

22 (1991, table 1) the cross-sectional average of the share of leisure is set to The remaining observations used in the calibration consist of the life-cycle profiles of market time, education time, wages and earnings. Since there is no hope to replicate such profiles exactly, parameters are chosen to achieve an overall fit with the data. The goodness of this fit together with additional aggregates described below are then used to evaluate the success of the model. The profiles of market time and the fraction of time spent on education are taken from Juster and Stafford (1985). 16 Since a member of one generation represents an average over the whole cohort, hours worked must reflect variations in annual hours as well as labor force participation rates. Estimates of age-wage profiles differ considerably. To be consistent with the assumption of balanced growth, we construct hypothetical cohort profiles from the cross-sectional estimates in Fullerton and Rogers (1993), assuming that wages grow at the balanced growth rate over the lifetime of a cohort. The resulting profile is less peaked than that of Owen (1986) based on actual cohort data, but significantly steeper than most of the cross-sectional profiles found in the literature. Below, we show that using cross-sectional profiles without growth adjustment makes little difference. Other observations to be matched, such as the population growth rate or the fiscal policy parameters of the initial balanced growth path, directly translate into parameter choices and are described below. 16 The data provided in Owen (1986) are similar, but are based on Census data which are known to have measurement error problems; see Heckman (1993). The data in Rosen (1982), while estimated quite differently, are similar as well. 20

23 Table 1. Calibrated Parameters Preferences σ = 1.5, ρ = 1.02, β = 0.99 Technology α = 0.3, δ K = Human capital technology B = 0.6, η = 0.2, ψ = 0.3, ζ = 0.5, δ h = 0.01, ε = 0.74 Tax policies τ k = 0.375, τ w = 0.20, τ o = 0.25, τ s = 0.12, τ c = 0.092, Tr/Y = 0.069, G/Y = 0.20, D/Y = 0.37 Demographics n = , tb = 28 Parameter choices Table 1 summarizes the parameters choices. A brief description of selected values follows; details are available in Hendricks (1994). The population growth rate is set to 1.24%. Parents give birth at age 28, and their offspring inherits a fraction of their human capital stock at age 38. The latter value is somewhat arbitrary, the idea being to choose an age at which children are usually raised by their parents. Results are not sensitive to this value. The income tax rate chosen for this study is constructed from the IRS Individual Income Tax Returns for The capital income tax is interpreted to encompass, in addition to personal income taxes, corporate profit taxes and a part of property taxes. The resulting tax rate of 37.5 percent is quite close to the more careful estimate in Kim (1992). The consumption tax rate is computed from Fullerton and Rogers (1993, table 3-6). Government consumption is 20 percent of net national product, as in Lucas (1990). The ratio of government debt to GDP is set to 0.37, the average ratio of government net financial assets to GDP during the period from 1960 to The level of transfers is adjusted to balance the government budget on the initial balanced growth 21

24 path. For simplicity, we assume that individuals of all ages receive the same amount of transfers at any given date. The parameter choices for the human capital technology combine results from econometric studies with information contained in the observed life-cycle profiles of wages, market time, education time and earnings. Heckman (1976) and Haley (1976) provide widely cited estimates of a human capital production function. In both studies, point estimates of the returns to private inputs are around η + ψ = 0.5 with confidence intervals ranging to approximately 0.7. Balanced growth, on the other hand, requires constant returns to reproducible factors. We therefore assume a human capital spillover of ξ = 1 ψ η. The implied spillover of ξ = 0.5 appears large; therefore we set ξ to the smallest value consistent with the estimated confidence interval for η + ψ (ξ = 0.3). The vast literature on schooling also suggests that such spillovers are important without providing a reliable estimate though; see Hanushek (1986). While there is reason for caution as empirical evidence in the area of human capital is generally controversial and fraught with data problems, the case for the presence of spillovers appears quite strong. For η some direct evidence exists. For schooling, Johnson (1978) estimates that the ratio of factor shares (ψ/η) is around six. Becker (1975) estimates the share of goods in private education costs to be around As usual in the area of human capital, the estimates should be viewed with some caution, but the consensus view in the literature seems to be that foregone earnings constitute the most important cost of education. At least, education should not be more capital intensive than production in general, which constrains η to be less than 0.3. This value is reduced further, if there is a human capital spillover (ξ > 0) which decreases the returns to private factors. We choose η = 0.2 as the preferred value for the base line calibration. Constant returns to the reproducible factors then requires ζ =

25 Human capital depreciation, which is very controversial in the growth literature, can be calibrated quite precisely on the basis of earnings and wage profiles. The timing of human capital accumulation and the age at which wages and earnings peak are largely determined by its value. Given δ h, the productivity parameter B is chosen to match the observed steepness of the lifecycle profiles. Little evidence is available concerning the tax treatment of goods inputs in human capital production. Trostel (1993) summarizes some previous estimates that allow to roughly determine the fraction of goods inputs to education that are not tax-deductible. The answer depends somewhat on whether child-rearing costs are included in human capital investments. For an analysis of the effects of taxation, their inclusion does not appear reasonable. 17 Mincer (1993, chapter. 9) estimates that between 33 and 40 percent of total expenditure on education take the form of OJT, where all goods inputs are paid for by lower wages and are therefore effectively tax-deductible. Of the remaining inputs, the data cited by Trostel suggest that between 60 and 80 percent are provided by the government in the form of subsidies to schooling, etc. The taxable fraction of goods inputs must therefore lie roughly between 8 and 27 percent. Given that about three quarters of education costs consist of time inputs, this leaves only between three and seven percent of total cost as not tax-deductible. Since our base model focuses on individuals past the age of 16 when most formal schooling is completed, we assume that all goods inputs in human capital production are paid for by lower earnings. The tax deductibility and subsidy parameters for education which is purchased directly are then irrelevant. The previous literature on growth effects of taxation universally made the 17 It would amount to assuming that expenditures on an infant s food or clothing are responsive to the tax rates it might face as an adult! 23

26 opposite assumption that none of the goods inputs are deductible. We will show below that this assumption partly explains why growth effects were usually found to be higher than in this paper. Model Evaluation This section compares the balanced growth path predicted by the base calibration with the data observed for the U.S. Figure 1 shows the age profiles for earnings, wages, market time (1-l) and time spent on job training (v). Each observation represents an average over a range of age cohorts. The model replicates most life-cycle profiles quite well. Only training time is initially somewhat too low. As in most life-cycle models, the age-asset profile is too peaked, as the household does most of his lifetime savings close to retirement age. It should be noted that this fit is achieved with a fairly small number of free parameters. Among preference parameters, ρ is set to replicate the leisure share leaving only σ free. The other free parameters concern the human capital technology. Since η and ξ are determined independently and ζ = 1 ξ η, this leaves only three parameters: B, ψ and δ h. Given the number of observations to be matched, the fit achieved by the model appears satisfactory. The balanced growth pre-tax interest rate of over 11 percent per annum appears high, although it is controversial whether such an interest rate should be considered unreasonable; see Feldstein and Summers (1979). The model succeeds in roughly replicating the composition of national income, as shown in Table 2, though the capital stock is a little too low relative to income. 24

27 Table 2. Composition of National Income Variable U.S. Data Model Consumption / GNP Capital / National Income Net Investment / National Income Gross Investment / National Income Source: Statistical Abstracts of the United States U.S. data are averages over

28 (a) Age-earnings profiles (b) Age-wage profiles Relative Earnings Data Model Age Relative Wage Data 0.5 Model Age (c) Profiles of market time 1.2 (d) Time spent on education 0.25 Relative Market Time Data Model Age Figure 1. Age profiles Relative Education Time Model Data Age

29 1.2 Relative Asset Holdings Data Model 0 < Age Figure 2. Asset holdings by age 27

30 The Long Run Growth Effects of Taxation This section reports simulation results for the standard experiment of eliminating all taxes. We begin by showing that the base line model under the preferred calibration yields fairly small growth effects. This result is then shown to be robust against various modifications of parameter values and model structure. We then proceed to investigate the importance of the tax treatment of inputs to education. These will be shown to be substantial, but do not affect the overall conclusions that growth effects in the overlapping generations model are significantly smaller than those found in comparable infinite horizon models. Growth Effects in the Base Line Model Consider the extreme experiment of eliminating all taxes, government spending, debt and transfers including social security benefits. Column (1) of Table 3 summarizes the resulting balanced growth path. The increase in the growth rate of per capita income of 0.5 percent lies at the lower end of what other authors have found. At the low extreme of the spectrum is Lucas (1990). Due to the special structure of his model, there are no growth effects at all, except for a very small effect due to labor supply. Jones, Manuelli, and Rossi (1993), on the other hand, find growth effects that are up to an order of magnitude larger than those of the present model This paper is often cited as an example of the extraordinary growth effects the neoclassical growth model can generate. It should be emphasized, however, that the magnitude of growth effects is not their primary concern. 28

31 Table 3. Alternative Tax Experiments Baseline Case (1) Elimin. Gov. (2) t w = 0.1 (3) s = 0.25 (4) s = 0.5 Growth rate, % Capital labor ratio After tax interest rate, % Leisure share Welfare gain, % n.a Despite the small growth effect, the impact on the static allocation is significant. Holding levels (h 1 ) fixed, consumption at each age is almost doubled. This is financed through higher earnings due to the increased wage rate and longer work time late in life. Savings in the form of physical capital also rise significantly, increasing the capital stock by 51 percent and the capital-labor ratio by 23 percent. An important and robust feature of this model is that the after tax interest rate responds very little despite the drastic changes in tax rates. As explained above, the interest elasticity of savings in the form of physical capital is quite high. Therefore, human capital and growth are insensitive to taxation. To provide an idea of the magnitudes involved in more realistic tax reforms, column (2) reports the effects of reducing the wage tax to 10 percent, financed by an increase in capital income taxes. The growth effect is again quite small (0.2 percent). Sensitivity Analysis In marked contrast to the results of infinite horizon models, growth effects in the present model are much less sensitive to changes in the calibration and model structure. 29

32 Alternative Parameter Values: None of the critical parameters identified by Stokey and Rebelo (1995) has a large impact on the growth effects of our model. Surprisingly, this is true even for the intertemporal elasticity of substitution (1 / σ), which is so crucial for the predictions of the infinite horizon model, and for the depreciation rate of human capital. In fact, setting δ h = 0.05 barely affects the change in the growth rate at all. The life-cycle profiles of earnings and wages show that this depreciation rate is already too high: Training drops to zero many years earlier than the data suggest. It appears safe to conclude that realistic values of δ h cannot generate significant changes in our results. Whereas in the infinite horizon model the change in the growth rate is highly sensitive to σ, it is almost completely unresponsive in our model. Reducing σ to 1.2 changes the growth effect only by a fraction of one percent. The most important source of sensitivity of previous results has thereby disappeared. The reason is that, since the Euler-equation (1) no longer holds, individual consumption growth and aggregate income growth are no longer identical and the artificial link between the interest rate and the growth rate is broken.this finding is consistent with the notion that savings respond mostly to the wealth effects caused by changes in the interest rate, not to substitution effects. Several other parameters also have only a small impact on our results. Eliminating the tax deductibility of physical capital depreciation has negligible effects. Also unimportant is the exact shape of the age-earnings and wage profiles. Growth effects increase by only 0.2% when the model is calibrated to replicate the much flatter profiles of Welch (1979) or Fullerton and Rogers (1993). Even combining all those changes into a calibration that deliberately overstates the changes in the growth rate does not produce a large change. In particular, setting δ h = 0.05, η = 0.3, σ = 1.5 and eliminating the tax deductibility of depreciation of physical capital results in a growth effect of 30