- Email: [email protected]
Capital accumulation and taxation in a general equilibrium model with risky human capital
WILLIAM LORD
University of Maryland Baltimore County Baltimore, Maryland
PETER RANGAZAS Indiana University-Purdue Universityat Indianapolis Indianapolis, Indiana
Capital Accumulation and Taxation in a General Equilibrium Model with Risky Human Capital* We integrate life cycle precautionary saving and human capital investment by modeling both activities in a single general equilibrium setting. Allowing human capital investment to respond to earnings uncertainty significantly increases precautionary saving but not enough for simulated saving rates to reach historical levels in the U.S. In contrast to recent deterministic analyses, we show income taxation may actually stimulate human capital due to the insurance effect of taxing uncertain returns. The same insurance effect helps explain why conversion to a wage tax base may lead to reduced savings, in contrast to some recent claims.
1. Introduction Two popular trends in macroeconomics attempt to account for the effects of earnings uncertainty on saving and to explain human capital investment. This paper integrates the study of precautionary saving and human capital investment by modeling both activities in a single general equilibrium setting. Our purpose is to establish the quantitative implications of uncertain returns to endogenous human capital for physical and human capital accumulation under various fiscal policies. More specifically, we first use the model to examine the general equilibrium determinants of aggregate savings (Section 4). Recently, following the theoretical work of Leland (1968) and Sandmo (1970), attempts have been made to quantify a precautionary motive for saving due to income uncertainty in both partial (Skinner 1988; Caballero 1991; and Hubbard, *We are very thankful to Chris Carroll, Steve Russell, Bob Sandy, Phil Trostel, and two anonymous referees for insightful comments on earlier versions of the paper.
Journal of Macroeconomics, Summer 1998, Vol. 20, No. 3, pp. 509~531 Copyright © 1998 by Louisiana State University Press 0164-0704/98/$1.50
509
William Lord and Peter Rangazas Skinner and Zeldes 1993) and general equilibrium (Engen 1992). 1 We extend this literature by explaining how endogenous human capital interacts with precautionary saving. As a natural consequence of our analysis, we also extend the literature by clarifying the role of risk aversion in determining general equilibrium saving rates. Our second application focuses on the effects of income taxation on saving and human capital investment (Section 5). A recent simulation study by Trostel (1993) concludes that there is a significant negative impact of income taxation on human capital accumulation. However, his study assumes there is no earnings uncertainty. The same insurance effect of income taxation that has been shown to discourage saving (Barsky 1986) encourages human capital investment (Eaton and Rosen 1980). We confront the negafive effect highlighted by Trostel with the insurance effect in order to obtain an estimate of the overall effect of income taxation on life cycle human capital investment. We find that the insurance effect either offsets or completely dominates any negative effects on human capital investments. In contrast to Trostel, we conclude that redistributive income taxation has a modest positive effect on life cycle human capital production and the effective labor supply. Our third application is in the area of tax reform (Section 6). Engen (1992) points out that the presence of uncertain earnings challenges the idea that switching from an income to a wage tax base will result in a large increase in saving. This is because the effective tax rate on uncertain labor income is higher under wage taxation than it is under an equal-yield income tax. The riskiness of labor income is thus lowered as one moves from an income to a wage tax base. The relatively greater insurance effect under wage taxation implies a decline in precautionary saving. Engen found that, in a particular tax reform experiment, the decline in precautionary saving significantly lessens the increase in total saving resulting from an income to wage base tax reform. We find that, under the standard balanced-budget definition of tax reform, the decline in precautionary saving is large enough to cause total saving to decline. We also show that this negative effect on saving would be IThese attempts have been motivatedby the fact that, under the assumptionof perfect certainty,realisticallycalibratedlife cyclemodelsdo not generatesufficientlyhighsavingsrates to mimic reality (White 1978; Auerbaeh and Kotlikoff1987; Hubbard and Judd 1987; and Kotlikoff1988). It has been suggestedthat the absence of a bequest motiveis a major reason for the low savingrates comingfrom pure life cycle models (Kotlikoffand Summers 1981). However,augmentingthe lifecyclemodelwithbequestmotivesdoesnot raisethe overallsaving rate significantly,as bequest saving"crowdsout"lifecyclesaving(Lordand Rangazas1991;and Lord 1992). This leads to precautionarysavingas the next natural candidatefor raisingtile savingrate.
510
Capital Accumulation and Taxation
larger, significantly so in some cases, if human capital investment were not also discouraged by the reform.
2. The Model Our model is a standard three period overlapping generations model with the following additions necessitated by the introduction of uncertain adult human capital. In the first period of adulthood households devote resources to increase their human capital beyond the exogenous initial stock (denoted by H1). However, the investments have uncertain returns and generate a source of labor income uncertainty in the middle years of life. The employable human capital in period two is then a random variable whose level is affected by the amount of first period investment and is denoted by ~/2. Under the expected utility assumption, the agent's preferences are represented by the usual CES utility function in consumption (ci) u = (1/(1 - ~))~{c]-~ + (1 + ~)-tc~'-~ + (1 + 8 ) - 2 c ~ - ~ } ,
(1)
where ~ is the constant coefficient of relative risk aversion and 8 is the constant pure rate of time preference. The expectation E is taken over all states of nature affecting the level of H2 and thus the levels of c2 and c J The single period budget constraints facing the household are p c c l + a 1 + pgg + we = w H 1 + ~1,
(2a)
pcc2 + a2 = a1(1 + r) + wH2 + ~ ,
(2b)
pcc3 = a2(1 + r) + xa,
(2c)
where a is financial asset holdings, g is the goods input in human capital production, e is the effective time input in human capital production (fraction of time endowment, normalized to 1, times H1), Pc is the price of consumption, pg is the price of goods inputs, w is the net of tax wage rate, r is the net of tax interest rate, z is a lump sum government transfer. Both c and 2In addition to the standard expected utility specification, we also consider Weil's (1990) version of the Kreps-Porteus nonexpeeted utility function. The motivation for introducing a nonexpected utility specification will not become clear until after our discussion of aggregate saving rate determination in Section 4. For this reason we delay an explicit statement of these preferences until then.
511
William Lord and Peter Rangazas g are produced by the same Cobb-Douglas production function, but their prices may differ from one, and from each other, because of the structure of tax policy. The second period human capital stock is given by H 2 = I~H1 -I- h ,
(3)
where 13 is the fraction of H1 left after depreciation over time and h = uenlg ~2 is a Ben-Porath (1967) human capital production function with a random efficiency scalar u. We assume that u is uncorrelated across households, so that all the risk is individual rather than social in nature. The agent's problem is then to maximize (1) subject to (2) and (3). Formally, this is accomplished using the reeursive method of stochastic dynamic programming,
VI(H1, ") = max {(1/(1 - 7))cI - ~ + (1 + 5)-lEV2(H2, ")},
(4)
(g, e, Cl)
subject to (2) and (3), where V2 is the maximum value function from the beginning of period 2, and the "." in the value function denotes features of the environment that are exogenous to the individual agent such as factor prices and fiscal policy variables. The microeconomics of (4) have been analyzed by Sandmo (1969, 1970), Lehvari and Weiss (1974), and Eaton and Rosen (1980) for the case where ~2 = 0.3 The key results from this work for our purposes are: (i) an increase in wealth increases investment in human capital; (ii) increased risk (a wider spread of uw around a given mean) lowers human capital investment; (iii) if a household is neither borrowing nor lending, an increase in the interest rate lowers human investment and raises saving; (iv) a compensated increase in wage taxation raises human investment (the insurance effect). 4 aThe theoretical analysis of the studies cited was conducted in a two-period setting but applies directly to our three-period model as well. To see this, just note that with a CES utility function the maximum value function takes the same form as the utility function (Merton 1971). V2 can be written as [1 + (1 + 8)-1((1 + r)/(] + 5)) ~¢I r)]c~2-r/(1 - 7), where c2, = (ai(1 + r) + w H 2 + z~ + zd(1 + r))/pc(1 + ((1 + r)/(1 + 5))°(1 + r)-l). Substituting into (4)reduces the three-period problem into what is essentially a two-period problem. 4Eaton and Rosen do not derive this exact result. However, using their central lemma and following Levhan and Weiss (1974) it is easy to demonstrate the result in our model.
512
Capital Accumulation and Taxation For £3 > 0, the last result must be revised. With only a time input in human capital production, a lower net wage decreases the benefit and cost of human investment proportionally and therefore there is no incentive or substitution effect toward less human investment. When there is a goods input, an increase in the wage taxation lowers benefits more than cost and encourages less human investment as emphasized by Boskin (1975), Lord (1989), and Trostel (1993). This effect works in the opposite direction of the dominant insurance effect that leads to (iv). The revised (iv) is (iv)' if)~2 > 0, then a compensated increase in wage taxation has an ambiguous effect on human investment. These household level results will prove useful in interpreting the general equilibrium experiments in Sections 4 6 . The microeeonomic portion of the model is completed by assuming a specific form for the distribution function of u. We follow Barsky, Mankiw and Zeldes (1986) and choose a simple symmetric three-point distribution of the form u = (1 - x)O,
with probability rt ;
, with probability 1 - 2n ; (1 + x)0,
with probability n .
Note that the variance of u is 2n(xO)2 and the coefficient of variation is
x(2n) 1/2. As in Barsky, Mankiw and Zeldes (1986), we will use the observed coefficient of variation of earnings to help calibrate the distribution of u. Households are assumed to have rational expectations so that their subjective distribution of the states of nature coincide with the objective distribution o f U.
The model of the entire economy is obtained by specifying the distribution of household types, the aggregate production function used to produce consumption, physical capital and the goods invested in human capital, and the initial stance of fiscal policy. At any moment in time there are seven household-types involved in market trade. Young households are identical but there are three different types of middle-aged households and three different types of old households (corresponding to the three different outcomes for u). We assume that there are a continuum of households in each new generation so that the fractions of the population of middle-aged and old households correspond exactly to the probability of the three different outcomes for u. There is also population growth at rate n, so that the ratio of the population of each generation to its predecessor is i + n. The average or per capita value of a variable is obtained by averaging across households 513
William Lord and Peter Rangazas of a given generation (with weights based on rt) and then averaging across generations (with weights based on n). In the economy's initial position, following previous simulation studies, we assume that a proportional income tax is used to collect government revenue. The revenue is used to provide non-productive government goods that enter the utility function of agents in an additively separable fashion, and in some cases to provide a subsidy to the goods input in human capital production. Redistributive government taxes and transfers will be added in Section 5.
3. Calibration As in most simulation studies, the ranges for the parameters of the model are determined primarily by the available empirical evidence and, where appropriate, for the purpose of making comparisons to other simulation studies. Our general strategy for organizing the parameter settings is driven by the desire to highlight the role of uncertainty. Where natural, we group parameters into a case that maximizes the role of uncertainty, a case that minimizes its role, as well as several intermediate cases which mix the parameter settings from the boundary cases. Two parameters are particularly important in this respect. Skinner (1988) shows that 7 is critical in determining the share of savings due to precautionary motives. A high 7 means greater risk aversion and a greater tendency for households to alter behavior in order to avoid uncertainty. As indicated by results (iv) and (iv)' from Section 2, L2 is crucial in determining the effect of changes in the net wage rate on human capital investment. The greater the value of Z2, the more likely it is that an increase in the net wage rate will encourage more human capital accumulation rather than discourage it by increasing the riskiness of investment.
Length of Time Period There are three periods in the model, two working periods and a retirement period. The three period assumption is needed to simplify the computations of a model that includes three features not found in previous general equilibrium simulation studies: uncertainty, endogenous human capital, and nonexpected utility functions. As indicated below, the three-period assumption does not appear to be a substantive limitation for the questions we address. We calibrate the model under the assumption that each period represents twenty years. This gets the length of work life about right, but produces a somewhat lengthy retirement period. The lengthy retirement period means we bias saving toward finding too high a saving rate. Since one of our 514
Capital Accumulation and Taxation TABLE 1.
GeneralEquilibrium Saving Rates Certainty
y = 5, L2 = 0 y = 5, L2 = 0.15 7 = 1.25, L2 = 0 y = 1.25, ~2 = 0.15
Uncertainty
L
H
2.3 (2.9) 2.3 (2.9) 4.8 (5.0) 4.6 (4.8)
1.2 (3.3) 1.3 (3.1) 4.4 (5.0) 4.4 (5.0)
L
H
3.3
4.1
3.2
4.1
5.0
5.3
4.9
5.2
NOTES: L (H) is the low (high) uncertainty case with CV = 0.20 (0.40). The number in parentheses gives the saving rate when human capital investment is kept constant at its level under uncertainty. There are L and H cases with certainty because17, differs across the two cases to keep the ratio of wH 1 and wH~ approximately one.
principal conclusions is that the general equilibrium saving rate remains too low, even with precautionary saving, the bias is not of concern. In addition, as a practical matter, the bias does not appear to be very large. As reported in Table i of the next section, our saving rates with and without uncertainty are very comparable to those obtained in multi-period general equilibrium models. The annualized value for the rate of population growth is set to 1% in all eases. With twenty-year periods this implies n = 0.2202.
Household Preferences The preference parameters of the model are y and 8. As mentioned, the value of 7 is very important for our study. This parameter has been estimated using both macroeconomic and microeconomic data. Early macroeconomic studies (for example Hansen and Singleton 1983 and Mankiw, Rotemberg, and Summers 1985) found estimates of y that centered around 1. Using additional data and more appropriate estimation techniques, Hall (1988) found substantially larger values of y with lower bounds around 10. The microeconomic evidence has produced estimates of y no lower than 1, with an average around 2.5. 5 In several of the cases we set T equal to an 5The intertemporal elasticity of substitutions estimated in these studies are: Ghez and Becker (1975) 0.28, Friend and Blume (1975) 0.50, Farber (1978) 0.30, MaCurdy (1981) 0.28, Grossman and Shiller (1981) 0.21, Hurd (1989) 0.91, and Engen (1991) 0.30. The simple average is about 0.40, implying an estimate of 7 around 3.
515
William Lord and Peter Rangazas intermediate value suggested by the empirical evidence of 5. Trostel's (1993) survey of previous simulation studies led him to set y equal to 1.25. Based on the available empirical estimates, we feel this value is much too low, when viewed from the perspective of estimating the intertemporal elasticity of substitution. However, for purposes of comparison to Trostel and other simulation studies we also set '/equal to 1.25. The other preference parameter, 3, has no clear influence on the role that uncertainty plays in the model. We set 3 equal to an annualized value of 0.01, a common value used in simulation studies, for all cases. Our previous study of saving rates with altruistic bequests makes this assumption and we make it here to facilitate comparison. 6
Human Capital Production The parameters of the human capital production function are )~1, )~z, as well as x and n from the distribution of t~. As in Trostel (1993) we set the returns to scale, )~1 + ~,2, equal to 0.6. In several of the cases, we set equal )~2 to 0 in order to highlight the insurance effect associated with changes in the net wage rate. In addition, setting L2 equal to 0 can be defended by the fact that the majority of human investments in our model can be interpreted as on-the-job training which involves relatively small amounts of goods inputs. We also consider cases where we set )~2 equal to (1/3))~1 as in Trostel (1993). Trostel's assumption is based on estimates from formal education and serves as an upper bound on )~2 for the adult human investment in our model. Again following Trostel, we assume a 4% annual depreciation rate for the initial human capital stock. We follow Barsky, Mankiw and Zeldes (1986) by calibrating the distribution of v to equate the coefficient of variation in earnings in the data to the coefficient of variation for second period earnings in the model (CV). There are now several studies that provide sufficient estimates of the error process in individual earnings equations to compute the coefficient of variation in earnings (Lillard and Willis 1978; Hall and Mishkin 1982; Abowd and Card 1989; Engen 1991; Hubbard, Skinner and Zeldes 1993). 7 The 6Of course, a certain way to generate historical saving rates is to set the pure rate of time preference as low as is necessary. This would imply a negative value. Although there are some negative estimates of this parameter, most economists find negative values implausible. 7The reported estimates of the coefficient of variations for earnings is based on the following procedure. Empirical studies of earnings typically assume that earnings are lognormafiy distributed and that the error term of the log of earnings is an AR(1) process (see, for example Engen 1992). Under these assumptions the variance of the log of earnings is an approximataon to the coefficient of variation of earnings. (Actually an underestimate which is more accurate the smaller is the variance of log-earnings. For the variances found in the literature the approximation is very accurate.) The variance of log-earnings can be computed from the estimates of the parameters of the AR(1) process provided in the studies cited below.
516
Capital Accumulation and Taxation simple average of the coefficient of variation estimates from these studies is about 0.40. This average can be viewed as an upper bound on the variation in earnings associated with uncertain returns to adult human capital investment, since part of the variations in an individual's earnings are due to events that also affect the level and market value of general human capital attained during childhood (for example, spells of unemployment and certain types of disabilities). We set CV = 0.40, in what is referred to as the "highuncertainty" ease (H-ease). It seems reasonable to attribute at least half the variations in earnings to events that affeet the return to adult investments only (for example, an exogenous change in job or occupation that continues to reward elementary knowledge at the same rate, but that differs in the reward paid to advanced knowledge or a particular specialized skill.). We will set CV = 0.20, in what is referred to as the "low-uncertainty" ease (Lease). In ease L, we set n equal to 0.25 and x equal to 0.566, while n equals 0.33 and x equals 0.9 in ease H. s Since earnings peak just after age 40, we set ~ so that Will and wile are roughly equal.
Aggregate Goods Production Our treatment of goods production is entirely standard. We follow Trostel (1993) and many others and set labor's share equal to 0.25.
Fiscal Policy Variables In the initial baseline steady state we assume an income tax rate of 20% with all proceeds going to finance nonproductive government purchases and, when )~2 > 0, a subsidy to goods inputs in human capital production. The quantity of government purchases is roughly consistent with real world descriptive statistics (see, for example, Browning and Johnson 1984, 186; or Barro 1984, 300) and other simulation studies (see, for example, Lord and Rangazas 199i or Trostel 1993). The subsidy rate on goods inputs is set equal to 60% as in Trostel (1993). 9 As mentioned, we increase the income tax SUnreported sensitivity analyses show that the results are little affected by altering the mix of uncertainty, r¢ and x, for a given CV. 9The presence of the subsidy on the endogenous goods input raises a conceptual issue about the definition of the budget constraint. One approach is to implicitly include the goods subsidy into government purchases but to keep government purchases fixed at 20% income. The second approach treats the subsidy expenditures as additions to the government purchases that are set at 20% of income. Tax revenue and the level of purchases remain constant when balancedbudget tax reform experiments are conducted under the first approach, despite changes in the goods input, but implicit changes would occur in "other" government purchases. Under the second approach, tax reform would cause a change in the level of taxation and government purchases (including the subsidy on the goods input), whenever the goods input ehange. We feel the first approach is more consistent with other tax reform experiments and the numbers reported were generated under this approach. We did the experiments under the second approach and none of the conclusions found in the paper were altered.
517
William Lord and Peter Rangazas rate to 40% and add the redistributive component of the government in Section 5.
4. Saving Rates In recent years substantial attention has been devoted to assessing the contribution of preeantionary saving to aggregate saving (Skinner 1988; Cabellero 1991; Engen 1992; and Hubbard, Skinner and Zeldes 1993). The analysis by Skinner and Cabellero suggests that preeantionary savings can explain a very largefraction of total savings in partial equilibrium. Hubbard, Skinner and Zeldes (1993) find that uncertainty can raise aggregate saving levels to realistic values but again in partial equilibrium. Engen does use a general equilibrium framework to address the saving rate issue. However, he does not focus on the relationship between the specification and calibration of preferences and the contribution of precautionary saving to total saving in the general equilibrium portion of his analysis. As we explain below, genera] equilibrium considerations are critical in assessing the implications of different preference specifications. None of the studies allows for uncertain and endogenous human capital. Our strategy for isolating the role of uncertainty on saving is to calibrate initial baseline equilibria as realistically as possible; i.e. accounting for estimated levels of uncertainty. We then compute eounterfaetual equilibria where uncertainty is completely eliminated. When uncertainty is eliminated, in order to isolate the role of human capital per se, we consider equilibria where human capital is allowed to endogeneously respond and equilibria where human capital is frozen at initial baseline levels.
Preferences, Uncertainty, and Equilibrium Saving Table i presents the fraction of income devoted to saving (S/Y) under the assumptions from Sections 2 and 3. The three-period specification does not appear to be an important limitation, since the saving rates under certainty and uncertainty are not much different from otherwise similar models but with periods corresponding to a year (for example, compare to Lord and Rangazas 1991; in the certainty case, and Engen 1992 in the uncertainty case). As in Skinner (1988), Table 1 demonstrates that the precautionary motive can explain a very large fraction of total saving. For 7 = 5 and L2 = 0, the difference between the saving rates under H and with no uncertainty is 2.9 percentage points. 1° One can then interpret the precautionary 1°Noticethat the results are not affectedmuchby tile valueof ~2. Thisis becausethe partial equilibriumeffectsof uncertaintyon humancapitalinvestmentare independentof ~-2.The role of L2onlybecomesimportantwhenthe effectsof taxationare discussedin the nexttwo sections. This is whyresult (iv) from Section2, but not result (ii), had to be adjustedfor the ~2 > 0 case. 518
Capital Accumulation and Taxation motive as explaining over 70% (2.9/4.1) of total saving. It must be emphasized that the shares for precautionary saving just reported are very similar to those obtained by Skinner under comparable parameter settings for 7. However, Skinner does not indicate the total saving associated with the precautionary saving shares. We find when the fraction of precautionary saving is high the saving rate is relatively low, and when the saving rate is relatively high, precautionary saving is not contributing much to the total. Thus, the precautionary motive is doing little to enable life cycle models to generate more realistic saving rates. The inability of the precautionary motive to add much to total saving stems from the fact that aggregate saving is relatively high when the willingness of consumers to substitute consumption across time is high. However, this is also the situation where the degree of risk aversion is low and the effect of uncertainty is weak. To see this point explicitly first note that the standard CES utility function forces the rate of relative risk aversion (7) and the rate of intertemporal substitution (~) to be dependent (or = 1/7). In certainty models, so long as r > 5, the consumption profile is flatter and fife cycle saving smaller, the lower is or. However, with CES preferences, a lower willingness to substitute consumption over time implies greater risk aversion. With uncertainty, the consumption profile is steeper, the stronger is risk aversion, so the greater is precautionary saving. Thus, the common CES expected utility preferences imply that increasing the degree of risk aversion reduces life cycle saving at the same time it increases precautionary saving. At low levels of risk (as captured by the CV) the effect of changes in 7 on life cycle saving will dominate. If risk is sufficiently large, the increase in precautionary saving from higher 7 will be decisive. Our results indicate that for empirically realistic CVs the reduction in life cycle saving outweighs the rise in precautionary saving. For example, in case H where )~2 = 0.15, the saving rate is 5.2% for 7 = 1.25, but only 4.1% for 7 = 5. Only if the degree of CV were much larger would the saving rate increase with 7.11 11Engen's (1992) Table 2 computes partial equilibrium (treating the interest rate as exogenous) saving rates for different values of 7. In conjunction with our findings, his table highlights the importance of doing general equilibrium calculations to avoid misleading conclusions. For values o f r that are low, relative to his assumed rate of time preference of 4%, Engen finds that the saving rate increases with 7, contrary to our general equilibrium calculations. This is because when r is low relative to the rate of time preference, the effect of raising 7 (and lowering ~) on hfe cycle sa,ang is weak. This allows the corresponchng rise in precautionary saving to dominate the fall in life cycle saving. However, low values of r are inconsistent with Engen's parameter setting. For values of r closer to what Engen computes in his general equilibrium simulations, he also finds that an increase in 7 results in lower saving rates (consider his increase in 7 from I to 5, closest to our experiment, when r = 0.06).
519
William Lord and Peter Rangazas The discussion above indicates that the choice of CES utility functions may limit one's ability to identify the full role of precautionary saving because it forces an inverse relationship between 7 and r~. The nonexpected utility formulation of Kreps and Porteus is an interesting alternative to CES preferences as it allows a complete decoupling of 7 and r~. We introduce Weil's (1990) parametric version of the Kreps-Porteus nonexpected utility function in our three period life-cycle model by using the backward recursion approach described by Farmer (1990). Backward reeursion starting from the third period yields a utility function of the form [1
+
- W(I-4,)\
1 + (1 + X'~--l|± T t~ - -
1
-
y
+ (1
~{]t?ol_qb, l _T / l _q b
+ 6)(1 - y)
(6)
Now ~ = 1/7 is the intertemporal elasticity of substitution for deterministic consumption paths, qb is the distinct degree of relative risk aversion for timeless gambles, and c2, maximizes utility for the two-period problem beginning in the second period of life (see footnote 3). As before, first-order conditions can be obtained and the model can be calibrated and solved numerically to determine the saving rate. The most interesting case to consider is where both a and qb take on their upper bound values (a = 0.8 and qb = 5). This upper bound case maximizes the chances of simultaneously producing high aggregate saving with a large role for precautionary saving. Under these preferences, increasing risk aversion will not directly reduce intertemporal substitutability, so aggregate saving is expected to increase with 7. However, other types of life cycle offsets do occur due to general equilibrium interactions. An increase in y causes a partial equilibrium rise in saving that reduces r and raises w, serving to increase human capital investment via both an increased rate of return and higher wealth. Since more human capital investment increases expenditures when young and income in middle age, conventional savings are reduced. Recall from Table 1 that in case H with )~2 = 0.15, if y = 1.25, implying = 0.8, the saving rate was 5.2% in the presence of uncertainty. Now using the Kreps-Porteus preferences with ~ kept at 0.80, but with the now distinct degree of risk aversion qb set to 5, the saving rate increases to 6.1%. Since this case employs the high CV, as well as the high a and qb, it produces the highest saving rate for the range of parameters supported by empirical evidence. From 1950 to 1980, the average U.S. saving rate was around 8% (Auerbach and Kotlikoff 1987, 64). Thus, one still must conclude that augmenting a life cycle model with exogenous earnings uncertainty leaves a significant portion of total saving unexplained. 520
Capital Accumulation and Taxation This conclusion is reinforced by the recent econometric estimates of Epstein and Zin (1991). They estimate cr and ¢b using a nonexpected utility approach. The estimates for rr range between 0.20 and 0.80, but estimates for ¢b did not differ significantly from 1. The case of ~ = 0.8 and qb = 1 would generate the highest saving rate from their estimates. This parameter combination is very close to the combination rr = 0.8 and 7 = 1.25, reported in the last two rows of Table 1, that generates a saving rate of only 5%. For the range of available estimates of G and qb, the nonexpected utility specification does not significantly alter the quantitative conclusions reached under the standard expected utility assumption.
Human Capital and Saving We now return to Table 1 to consider how the influence of uncertainty on savings is affected by endogenous human capital. The elimination of uncertainty increases human capital investment through the insurance effect. This reduces conventional saving by shifting the timing of income to later periods and increasing investment outlays in the first period. This suggests that the response of human capital investment accentuates the reduction in saving when uncertainty is eliminated (i.e. increases precautionary saving). However, general equilibrium factor price movements will generate an effect on human capital that works in the opposite direction. Eliminating uncertainty creates a partial equilibrium reduction in saving that induces a general equilibrium increase in the interest rate and decrease in the wage rate. These factor price movements lower the marginal return to human capital and reduce human wealth for a given level of human capital, both of which cause human capital investment to decline. The decline in human capital investment raises the saving rate under perfect certainty and reduces the estimate of precautionary saving relative to the case where human capital is exogenous. Under our calibrations the insurance effect dominates the general equilibrium effect. Eliminating uncertainty reduces saving much more when human capital is endogenous. For example, when risk aversion is strong and uncertainty high (7 = 5 in case H), eliminating uncertainty reduces saving from 4.1% to just 3.1% when human capital is exogenous. Allowing for the response in human capital investment reduces the saving rate all the way to 1.3%. Thus, the importance of earnings uncertainty for saving is significantly magnified by endogenous human capital.
5. Redistributive Taxation
The effects of income taxation on human capital are complex. Even when abstracting from uncertainty and general equilibrium interactions, 521
William Lord and Peter Rangazas TABLE 2.
Effectsof Redistributive Taxation h
7 = 5,)~2 = 0 7 = 5, 7~. = 0.15 7 = 1.25, L2 = 0 7 = 1.25, L 2 = 0.15
E
S/Y
K
Y
V
k
24.9 4.9 -26.2 -29.8 -5.1 -6.2 -33.1 (12.6) (2.0)(-9.4.9)(-29.2)(-6.8)(-19.7)(-30.4) 18.1 3.6 -34.3 -30.5 -6.1 -6.7 -32.6 (4.2) (0.2) (-21.1) (-26.7) (-7.3) (-19.1)(-27.1) 14.0 1.8 -19.3 -23.4 -5.2 -0.9 -24.8 (9.3) (i.0) (-17.5) (-22.0) (-5.3) 3.6 -0.i -17.6 -22.6 -6.2 (-i.i) (0.0)(-15.3)(-20.3)(-6.0)
(-1.2)(-22.8) -1.2 -22.8 (-1.4)(-19.7)
NOTES: All numbers represent the percentage change in the variable due to the introduction of redistributive taxation. The H-case is presented with the L-case in parentheses.
theoretical reasoning fails to produce a clear conclusion. A higher rate of income taxation lowers the net of tax interest rate, raising the present value of returns to human capital investments and thereby encouraging more investment activity (Heckman 1976). A higher tax also lowers the net wage rate, which lowers the return to human capital except in the case where the sole cost of investment is forgone earnings (Boskin 1975). Trostel (1993) examines these opposing effects in a general equilibrium, infinitely-lived agent model. He finds large negative effects of taxation on human capital, implying that the wage effect dominates the interest rate effect. Trostel's study does not consider the interaction between uncertainty and income taxation. If there is uninsured earnings risk associated with human capital investment then the introduction of a redistributive income tax lowers the riskiness of human investment (result (iv) from Section 2). Barsky, Mankiw and Zeldes (1986) stress that the implicit insurance provided by income taxation reduces precautionary saving. In a more general model, this same effect also induces a portfolio shift away from saving and toward the now relatively safer human capital investment. If the insurance effect of income taxation on human investment is strong enough, the overall effect of income taxation on human capital will be positive. We assess the importance of the insurance effect by introducing redistributive taxation into our model. Browning and Johnson (1984) have shown that the redistributive component of the U.S. tax system in the 1970s can be closely represented by a demogrant policy. Under a demogrant policy all tax revenues taken in for the purpose of redistribution are paid out in equal lump-sum transfers to all members of the economy. Table 2 presents the steady-state effects of introducing a twenty percent income tax (driving the total income tax rate to 522
Capital Accumulation and Taxation 40%) where the proceeds are redistributed to all members of the economy in equal lump-sum fashion so as to balance the government's budget in each period. Table 2 presents the effects of redistributive taxation on human capital investment (h), the effective labor supply (E), the saving rate (S/Y), the capital stock (K), and income (Y). The numbers represent percentage changes in the variables across steady states for the H-case, with the L-case in parentheses. Table 2 demonstrates that the insurance effect is quantitatively quite important. Under all parameter settings but one, the production of human capital is increased after introduction of the tax. The one exception is where the role of the insurance effect is expected to be minimized; the L-case with 7 = 1.25 and ~,2 = 0.15 (recall L2 > 0 is critical in establishing that wage taxation lowers the return to human capital investment, an effect that works against the insurance effect). Even in this case the negative effect on human capital investment is quite small, - 1.1%. Under the same parameter setting, but with perfect certainty and variable hours of labor supply, Trostel found the long-run income tax elasticity for aggregate effective labor supply to be - 0.480. The corresponding income tax elasticity in our model is -0.001.12 In contrast, under the assumptions where the insurance effect is maximized, the H-case with 7 = 5 and ~2 = 0, the positive effect on human capital is 24.9% with an income tax elasticity for aggregate effective labor supply of 0.048. Thus, Trostel's conclusion of a large negative effect of income taxation on human capital appears to depend on his assumptions of perfect certainty and a positive wage elasticity for hours of work (an assumption based on estimates now viewed skeptically by some labor economists (Heckman 1993). There is more agreement between our results and those of Trostel concerning the effects of taxation on physical capital. Both studies find sizable negative effects of taxation on the stock of physical capital, although the magnitude of our declines are much less (about one-sixth of Trostel's). Since redistribution raises human capital investment in our model, the negative effects on saving would be weaker and the negative effects on output stronger were human capital not endogenous. Fixing human capital at its initial baseline levels, result in 10% to 25% smaller falls in the saving rate and 0% to 50% greater falls in output. The last column of Table 2 gives the effect of the policy on the steadystate utility of a representative household. Household's receive a direct wel12Since we are considering a 100% change in the income tax rate, our percentage change numbers can also be interpreted as tax elasticities and may be directly compared to Trostel's Table 2.
523
William Lord and Peter Rangazas
fare benefit from the insurance provided by income taxation. However, the policy also lowers wages and raises interest rates as the capital-labor ratio falls (see the next to last column of Table 2). When the economy is dynamically efficient, as is the case here, a drop in the capital-labor ratio and the associated effects on factor prices will cause a fall in household welfare (see, for example, Diamond 1965). For steady-state utility, the factor price effect dominates the direct insurance effect in every case. For households alive earlier in the transition to the new steady state the insurance effect may dominate, since the capital-labor ratio will not have yet fallen to its steadystate levels. In conclusion it does seem that redistributive taxation lowers the economy's physical capital stock, its ability to produce output, and steady-state welfare. However, redistributive income taxation has at least a modest positive effect on human capital accumulation.
6. Tax Reform There has been much attention paid to which tax base would be the best one for promoting capital accumulation and economic growth. One common misconception is that removing the tax on capital income and shifting the burden to labor income unambiguously increases saving and investment. The proponents of the pro-growth effects of wage-based taxation are focusing solely on the positive substitution effect associated with eliminating the tax on interest income. Buchanan (1959) and Feldstein (1978) have pointed out that there is another factor at work; the timing of tax collection will change. A shift from an income to a wage tax base pushes taxation forward in the life cycle and thus net income is pushed toward the later years of life. Such a shift will cause saving to fall when young, and thus aggregate saving will fall as well whenever n > 0. Thus, in life cycle models without uncertainty it is quite possible that the tax timing effect could dominate the positive substitution effect of removing the tax on interest income, causing saving to fall due to the shift toward the wage base. Allowing for uncertainty gives further reason to believe that the sole reliance on standard substitution effects to draw inferences concerning tax reform is misleading. The real after-tax return to human capital, measured in consumption units, is given by (w/pc)h - Owgh =- (1 - Zw - zy)/(1 + Zc)wgh, where wg is the gross wage rate, 0 is the fraction of the human capital return available for private consumption, % is the tax rate on wages, • y is the tax rate on income, and zc is the tax rate on consumption. Note that a lower value of 0 lowers the variance of (w/pc)h and thus lowers the variance of c2 generated by variation in h. In our model it is this covarianee between
524
Capital Accumulation and Taxation TABLE 3.
Tax Reform: Income to Wage Taxation L
y 1' y 3'
= = = =
5,)~ = 0 5, 9~2 = 0.15 1.25, 9~2 = 0 1.25, Le = 0.15
r
h
30.0 28.6 17.5 16.6
-21.7 -24.7 -13.1 -15.3
H S/Y
0
r
h
-16.4 -11.7 39.2 -13.2 -14.9 -12.1 38.8 -16.2 0.0 -8.8 20.2 -10.4 1.2 -9.0 19.4 -13.0
S/Y
0
-22.6 -12.3 -22.7 -12.7 -2.3 -9.1 -1.5 -9.3
NOTE: All numbers representthe percentagechangesin the variabledue to the tax reform.
c2 and h that affects the saving and investment plans in period one. Thus, the lower is 0, the greater is the insurance effect of taxation. A switch from an income to a wage tax will lower 0 since, everything else constant, % must be larger than zy to collect the same tax revenue. Thus, the switch to wage taxation will generate an insurance effect that will reduce saving and raise human capital investment. We examine the relative strengths of the opposing effeets mentioned above by comparing the steady state under wage taxation to the original baseline-steady state under income taxation. The percentage changes in the steady-state values of r, h, S/Y, and 0 due to the tax reform are found in Table 3. Table 3 reveals that the combination of income timing and insurance effects dominate the substitution effects of switching from income to wage taxation. The last column of the table shows that the percentage decreases in 0 are substantial; a necessary condition for a quantitatively important insurance effect. In all eases but one, the economy's saving rate declines. The saving rate stays roughly constant only when uncertainty and the household's aversion to it are at their lower bounds. With what we regard as the more realistic setting for 7, the saving rate falls dramatieally in both the H and the L eases. The fact the decline in the saving rate is sensitive to the degree of risk aversion and the amount of uncertainty suggests that the insurance effect, and not only the tax timing effect, is playing an important role in overturning the substitution effects. Engen (1992) also analyzes tax reform with uncertain earnings. He finds a switch from an income to a wage base results in greater saving. The difference in results are in a large part due to the differences in how the tax experiments were constructed. In his attempt to construct tax experiments that would allow meaningful welfare comparisons across steady states, Engen deviated from the standard practice of forcing the new tax rates to adjust so as to maintain a balanced budget in each period. Instead, he restriets tax
525
William Lord and Peter Rangazas rates to adjust so as to keep the present value of the tax burden for the representative household constant. When r > n, Engen's approach will result in lower rates of wage taxation than with the standard balanced budget approach. 13 This shows up in the fact that 0 decreases by only 1.75% when Engen switches from the income to the wage tax base (see his Table 5). In contrast, our experiments generate decreases in 0 from 9% to 13%. The larger the decrease in 0, the larger the insurance effect and the larger the negative effect on precautionary saving. Turning now to the effects on human capital, first note that the removal of the tax on interest income and the decreases in saving cause large increases in the net interest rate. The higher interest rates serve to discourage human capital investment. This interest rate effect dominates the insurance effect that encourages human capital investment as seen from the negative overall effects reported in Table 2. The decline in human capital investment helps to lessen the negative effects of the policy on saving rates. If human capital were fixed at initial steady values, the percentage decline in the saving rate would be 2 to 12 percentage points greater.14
7. Conclusion To summarize, the simulations of this paper have produced new findings concerning the macroeconomic implications of uncertainty and endogenous human capital formation; including three findings that challenge those available in the literature. First, the presence of endogenous human capital investment significantly increases the share of total saving that is due to precautionary motives (an increase in earnings uncertainty induces a shift away from human capital investment and toward saving). However, even 13To see this most simply, consider a two period life-cycle model with fixed factor prices and no human investment. The balanced budget, per young household, can be written as g = xyw + x~rgs/(1 + n), where g is the level of government purchases per young household and rg is the gross interest rate. The standard tax reform experiment requires XwW = g, while Engen constrains his wage tax rates (t~) to satisfy twW = xvw + ~vrgs/(1 + r). I f r > n, then g > zvw + zyrgs/(1 + r) and Zw > t,~. Engen's approach is a potentially useful analytical device in conducting normative evaluation of policy options, but the balanced budget rule is almost always assumed in the positive analysis of the long-run effects of fiscal policy (presumably because it comes closer to the budget rules that are followed in practice). 14The results in this section have to be qualified since they overstate the insurance effect from a switch to wage taxation in a world with risky physical capital. The presence of a stock market that helps to diversify away from the idiosyncratic risk of physical capital ownership suggests that the insurance effects of income taxation relative to wage taxation may not be that important.
526
Capital Accumulation and Taxation with human capital, we find that precautionary motives do not drive saving rates up to the historical average for the period 1950-1980, a result that is contrary to previous partial equilibrium studies. Second, the redistributive component of the income tax structure has a modest positive effect on life cycle human capital accumulation (the insurance effect of taxation outweighs the negative effect of a lower after-tax return to investment). In contrast, Trostel (1993) finds large negative effects on human capital accumulation in an infinitely-lived agent model with perfect certainty. Our result demonstrates the potential quantitative importance of idiosyncratic risk and shows how sensitive Trostel's conclusion is to a reasonable change in model specification. Third, removing the capital income tax by switching from an income to a wage base will reduce physical capital accumulation (precautionary saving falls enough to offset the rise in saving from a higher after-tax interest rate), rather than increase it as suggested by models that assume perfect certainty. Human capital formation is also discouraged by the reform (due to the rise in interest rates), helping to mediate the fall in saving (less human capital investment implies a portfolio shift into saving). These results demonstrate the importance of combining the two currently standard ways of modeling uncertainty and human capital formation in maeroeeonomies. More direct empirical work is needed to test how uncertainty affects saving, education, and on-the-job training decisions. Our simulations suggest that it may be difficult to draw reliable inferences about saving and human capital investment by treating them as independent behaviors. Received: July 1996 Final version June 1997
References Abowd, John M., and David Card. "On the Covarianee Structure of Earnings and Hours Changes." Econometrica 57 (March 1989): 411-46. Auerbach, Alan J., and Laurence J. Kotlikoff. Dynamic Fiscal Policy. Cambridge: Cambridge University Press, 1987. Barro, Robert. Macroeconomics. New York: John Wiley, 1984. Barsky, Robert N., Gregory Mankiw, and Stephen P. Zeldes. "Ricardian Consumers with Keynesian Propensities." American Economic Review 76 (September 1986): 676-91. Ben-Porath, Yoram. "The Production of Human Capital and the Life Cycle of Earnings." Journal of Political Economy (August 1967): 352-65.
527
William Lord and Peter Rangazas Boskin, Michael J. "Notes on the Tax Treatment of Human Capital." In Conference on Tax Research 1975. Washington: Department of the Treasury, 1975. Browning, Edgar K., and William R. Johnson. "The Trade-Off between Equality and Efficiency." Journal of Political Economy 92, no. 2 (April 1984): 175-203. Buchanan, James. "Saving and the Rate of Interest: A Comment." Journal of Political Economy 67, no. 1 (February 1959): 79-82. Cabellero, Ricardo. "Earnings Uncertainty and Aggregate Wealth Accumulation." American Economic Review 81 (September 1991): 859-71. Diamond, Peter A. "National Debt in a Neoclassical Growth Model." American Economic Review 55 (December 1965): 1126-50. Eaton, Jonathan, and Harvey S. Rosen. "Taxation, Human Capital, and Uncertainty." American Economic Review 70 (September 1980): 705-15. Engen, Eric M. "Precautionary Saving and the Structure of Taxation." April 1992. Mimeo. --. "Consumption and Saving in a Life-Cycle Model with Stochastic Earnings and Mortality Risk." UCLA, 1992. Mimeo. • "Estimation of a Stochastic Life-Cycle Model with Mortality Risk Using Panel Data." UCLA, 1991. Mimeo. Epstein, Larry G., and Stanley E. Zin. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis." Journal of Political Economy 99 (April 1991): 263-86. Farber, Henry S. "Individual Preferences and Union Wage Determination: The Case of the United Mine Workers." Journal of Political Economy 86 (October 1978): 923-42. Farmer, Roger. "RINCE Preferences." Quarterly Journal of Economics 105 (February 1990): 43-61. Feldstein, Martin S., "The Welfare Cost of Capital Income Taxation." Jourhal of Political Economy 86, no. 2 (1978): $29-$51. Friend, Irwin, and Marshall E. Blume. "The Demand for Risky Assets." American Economic Review 65 (December 1975): 900-22. Ghez, Gilbert R., and Gary S. Becker The Allocation of Time and Goods over the Life-Cycle. New York: Columbia University Press, 1975. Grossman, Sanford J., and Robert J. Shiller, "Determinants of the Variability of Stock Market Prices." American Economic Review 71, no. 2 (May 1981): 222-27. Hall, Robert E. "Intertemporal Substitution in Consumption." Journal of Political Economy 96, no. 2 (April 1988): 339-57. Hall, Robert E., and Fredric Mishkin. "The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households." Econometrica 50 (March 1982): 461-81. 528
Capital Accumulation and Taxation Hansen, Lars Peter, and Kenneth J. Singleton. "Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns." Journal of Political Economy 91 (April 1983): 249-65. Heckman, James J. "A Life-Cycle Model of Earnings, Learning, and Consumption." Journal of Political Economy 84, no. 4 (August 1976): $1144. --. "What Has Been Learned about Labor Supply in the Past Twenty Years?" American Economic Review 83 (May 1993): 116-21. Hubbard, R. Glenn, and Kenneth Judd. "Social Security and Individual Welfare: Precautionary Saving, Liquidity Constraints, and the Payroll Tax." American Economic Review 77 (September 1987) 630-47. Hubbard, R. Glenn, Jonathan Skinner, and Stephen P. Zeldes, "The Importance of Precautionary Motives for Explaining Individual and Aggregate Saving." Paper presented at the Carnegie-Rochester Public Policy Conference, April 1993. Hurd, Michael. "MoRality Risk and Bequests." Econometrica (July 1989): 779-814. Kotlikoff, Laurence J. "Intergenerational Transfers and Savings." Journal of Economic Perspectives 2 (Spring 1988): 41-58. Kotlikoff, Laurence J., and Lawrence H. Summers. "The Role of Intergenerational Transfers in Aggregate Capital Accumulation." Journal of Political Economy 2 (August 1981): 706-32. Levhari, D., and Yoram Weiss. "The Effect of Risk on Investment in Human Capital." American Economic Review 64 (December 1974): 950-63. Leland, Hayne E. "Saving and Uncertainty: The Precautionary Demand for Saving." Quarterly Journal of Economics 82 (1968): 465-73. Lillard, Lee A., and Robert J. Willis. "Dynamic Aspects of Earnings Mobility." Econometrica 46 (1978): 985-1012. Lord, William. "The Transition from Payroll to Consumption Receipts with Endogenous Human Capital." Journal of Public Economics (February 1989): 53-74. --. "Saving, Wealth and Exchange Bequest Motives." Canadian Journal of Economics 25 (August 1992): 743-53. Lord, William, and Peter Rangazas. "Savings and Wealth with Altruistic Bequests." American Economic Review (March 1991): 289-96. MaCurdy, Thomas E. "An Empirical Model of Labor Supply in a Life-Cycle Setting." Journal of Political Economy 89 (December 1981): 1059-85. Mankiw, N. Gregory, Julio j. Rotemberg, and Lawrence H. Summers. "Intertemporal Substitution in Macroeconomics." Quarterly Journal of Economics 100 (February 1985): 225-51. Merton, Robert C. "Optimum Consumption and Portfolio Rules in a Continuous Time Model." Journal of Economic Theory 3 (1971): 373-413. 529
William Lord and Peter Rangazas Sandmo, Agnar. "Capita/Risk, Consumption, and Portfolio Choice." Econometrica 37 (1969): 586-99. --. "The Effect of Uncertainty on Savings Decisions." Review of Economic Studies (July 1970): 353-60. Skinner, Jonathan. "Risky Income, Life Cycle Consumption, and Precautionary Saving." Journal of Monetary Economics 22 (1988): 237-55. Trostel, Philip. "The Effect of Taxation on Human Capital." Journal of Political Economy 101, no.2. (April 1993): 327-350. Well, Phillipe. "Nonexpected Utility in Macroeconomics." Quarterly Journal of Economics 105 (February 1990): 29-42. White, Betsy Butrill. "Empirical Tests of the Life-Cycle Hypothesis." American Economic Review 68 (September 1978): 546-60.
Appendix a e
=
CV= e
=
E= g= h= H= n
=
K= Pc
=
pg= r =
S= V= W X
= =
y
~= 7= 8= O= )~ =
530
financial asset holdings. consumption of goods and services. coefficient of variation for second period earnings. effective time input in human capital production. aggregate effective labor supply. goods input in human capital production. government purchases per young worker. net investment in human capital. human capital stock. rate of population growth. aggregate physical capital stock. net-of-tax price of consumption goods and services. net-of-tax price of goods input. net-of-tax interest rate. net national saving. maximum value function. net-of tax wage rate. determines the spread of the symmetric two-state realizations of v about its mean. net national product and income. undepreciated fraction of first-period human capital stock. constant coefficient of relative risk aversion with CES utility. pure rate of time preference. fraction of human capital return available for private consumption net of all taxes. output elasticity in human capital production.
Capital Accumulation and Taxation V = random efficiency scalar in h u m a n capital production. n = probability of the symmetric two-state realizations of v about its mean. = intertemporal elasticity of substitution. = lump sum government transfer. Zc = consumption tax rate. Zw = wage tax rate. qb = constant coefficient of relative risk aversion with non-expected utility. zy = income tax rate.
531
University of Maryland Baltimore County Baltimore, Maryland
PETER RANGAZAS Indiana University-Purdue Universityat Indianapolis Indianapolis, Indiana
Capital Accumulation and Taxation in a General Equilibrium Model with Risky Human Capital* We integrate life cycle precautionary saving and human capital investment by modeling both activities in a single general equilibrium setting. Allowing human capital investment to respond to earnings uncertainty significantly increases precautionary saving but not enough for simulated saving rates to reach historical levels in the U.S. In contrast to recent deterministic analyses, we show income taxation may actually stimulate human capital due to the insurance effect of taxing uncertain returns. The same insurance effect helps explain why conversion to a wage tax base may lead to reduced savings, in contrast to some recent claims.
1. Introduction Two popular trends in macroeconomics attempt to account for the effects of earnings uncertainty on saving and to explain human capital investment. This paper integrates the study of precautionary saving and human capital investment by modeling both activities in a single general equilibrium setting. Our purpose is to establish the quantitative implications of uncertain returns to endogenous human capital for physical and human capital accumulation under various fiscal policies. More specifically, we first use the model to examine the general equilibrium determinants of aggregate savings (Section 4). Recently, following the theoretical work of Leland (1968) and Sandmo (1970), attempts have been made to quantify a precautionary motive for saving due to income uncertainty in both partial (Skinner 1988; Caballero 1991; and Hubbard, *We are very thankful to Chris Carroll, Steve Russell, Bob Sandy, Phil Trostel, and two anonymous referees for insightful comments on earlier versions of the paper.
Journal of Macroeconomics, Summer 1998, Vol. 20, No. 3, pp. 509~531 Copyright © 1998 by Louisiana State University Press 0164-0704/98/$1.50
509
William Lord and Peter Rangazas Skinner and Zeldes 1993) and general equilibrium (Engen 1992). 1 We extend this literature by explaining how endogenous human capital interacts with precautionary saving. As a natural consequence of our analysis, we also extend the literature by clarifying the role of risk aversion in determining general equilibrium saving rates. Our second application focuses on the effects of income taxation on saving and human capital investment (Section 5). A recent simulation study by Trostel (1993) concludes that there is a significant negative impact of income taxation on human capital accumulation. However, his study assumes there is no earnings uncertainty. The same insurance effect of income taxation that has been shown to discourage saving (Barsky 1986) encourages human capital investment (Eaton and Rosen 1980). We confront the negafive effect highlighted by Trostel with the insurance effect in order to obtain an estimate of the overall effect of income taxation on life cycle human capital investment. We find that the insurance effect either offsets or completely dominates any negative effects on human capital investments. In contrast to Trostel, we conclude that redistributive income taxation has a modest positive effect on life cycle human capital production and the effective labor supply. Our third application is in the area of tax reform (Section 6). Engen (1992) points out that the presence of uncertain earnings challenges the idea that switching from an income to a wage tax base will result in a large increase in saving. This is because the effective tax rate on uncertain labor income is higher under wage taxation than it is under an equal-yield income tax. The riskiness of labor income is thus lowered as one moves from an income to a wage tax base. The relatively greater insurance effect under wage taxation implies a decline in precautionary saving. Engen found that, in a particular tax reform experiment, the decline in precautionary saving significantly lessens the increase in total saving resulting from an income to wage base tax reform. We find that, under the standard balanced-budget definition of tax reform, the decline in precautionary saving is large enough to cause total saving to decline. We also show that this negative effect on saving would be IThese attempts have been motivatedby the fact that, under the assumptionof perfect certainty,realisticallycalibratedlife cyclemodelsdo not generatesufficientlyhighsavingsrates to mimic reality (White 1978; Auerbaeh and Kotlikoff1987; Hubbard and Judd 1987; and Kotlikoff1988). It has been suggestedthat the absence of a bequest motiveis a major reason for the low savingrates comingfrom pure life cycle models (Kotlikoffand Summers 1981). However,augmentingthe lifecyclemodelwithbequestmotivesdoesnot raisethe overallsaving rate significantly,as bequest saving"crowdsout"lifecyclesaving(Lordand Rangazas1991;and Lord 1992). This leads to precautionarysavingas the next natural candidatefor raisingtile savingrate.
510
Capital Accumulation and Taxation
larger, significantly so in some cases, if human capital investment were not also discouraged by the reform.
2. The Model Our model is a standard three period overlapping generations model with the following additions necessitated by the introduction of uncertain adult human capital. In the first period of adulthood households devote resources to increase their human capital beyond the exogenous initial stock (denoted by H1). However, the investments have uncertain returns and generate a source of labor income uncertainty in the middle years of life. The employable human capital in period two is then a random variable whose level is affected by the amount of first period investment and is denoted by ~/2. Under the expected utility assumption, the agent's preferences are represented by the usual CES utility function in consumption (ci) u = (1/(1 - ~))~{c]-~ + (1 + ~)-tc~'-~ + (1 + 8 ) - 2 c ~ - ~ } ,
(1)
where ~ is the constant coefficient of relative risk aversion and 8 is the constant pure rate of time preference. The expectation E is taken over all states of nature affecting the level of H2 and thus the levels of c2 and c J The single period budget constraints facing the household are p c c l + a 1 + pgg + we = w H 1 + ~1,
(2a)
pcc2 + a2 = a1(1 + r) + wH2 + ~ ,
(2b)
pcc3 = a2(1 + r) + xa,
(2c)
where a is financial asset holdings, g is the goods input in human capital production, e is the effective time input in human capital production (fraction of time endowment, normalized to 1, times H1), Pc is the price of consumption, pg is the price of goods inputs, w is the net of tax wage rate, r is the net of tax interest rate, z is a lump sum government transfer. Both c and 2In addition to the standard expected utility specification, we also consider Weil's (1990) version of the Kreps-Porteus nonexpeeted utility function. The motivation for introducing a nonexpected utility specification will not become clear until after our discussion of aggregate saving rate determination in Section 4. For this reason we delay an explicit statement of these preferences until then.
511
William Lord and Peter Rangazas g are produced by the same Cobb-Douglas production function, but their prices may differ from one, and from each other, because of the structure of tax policy. The second period human capital stock is given by H 2 = I~H1 -I- h ,
(3)
where 13 is the fraction of H1 left after depreciation over time and h = uenlg ~2 is a Ben-Porath (1967) human capital production function with a random efficiency scalar u. We assume that u is uncorrelated across households, so that all the risk is individual rather than social in nature. The agent's problem is then to maximize (1) subject to (2) and (3). Formally, this is accomplished using the reeursive method of stochastic dynamic programming,
VI(H1, ") = max {(1/(1 - 7))cI - ~ + (1 + 5)-lEV2(H2, ")},
(4)
(g, e, Cl)
subject to (2) and (3), where V2 is the maximum value function from the beginning of period 2, and the "." in the value function denotes features of the environment that are exogenous to the individual agent such as factor prices and fiscal policy variables. The microeconomics of (4) have been analyzed by Sandmo (1969, 1970), Lehvari and Weiss (1974), and Eaton and Rosen (1980) for the case where ~2 = 0.3 The key results from this work for our purposes are: (i) an increase in wealth increases investment in human capital; (ii) increased risk (a wider spread of uw around a given mean) lowers human capital investment; (iii) if a household is neither borrowing nor lending, an increase in the interest rate lowers human investment and raises saving; (iv) a compensated increase in wage taxation raises human investment (the insurance effect). 4 aThe theoretical analysis of the studies cited was conducted in a two-period setting but applies directly to our three-period model as well. To see this, just note that with a CES utility function the maximum value function takes the same form as the utility function (Merton 1971). V2 can be written as [1 + (1 + 8)-1((1 + r)/(] + 5)) ~¢I r)]c~2-r/(1 - 7), where c2, = (ai(1 + r) + w H 2 + z~ + zd(1 + r))/pc(1 + ((1 + r)/(1 + 5))°(1 + r)-l). Substituting into (4)reduces the three-period problem into what is essentially a two-period problem. 4Eaton and Rosen do not derive this exact result. However, using their central lemma and following Levhan and Weiss (1974) it is easy to demonstrate the result in our model.
512
Capital Accumulation and Taxation For £3 > 0, the last result must be revised. With only a time input in human capital production, a lower net wage decreases the benefit and cost of human investment proportionally and therefore there is no incentive or substitution effect toward less human investment. When there is a goods input, an increase in the wage taxation lowers benefits more than cost and encourages less human investment as emphasized by Boskin (1975), Lord (1989), and Trostel (1993). This effect works in the opposite direction of the dominant insurance effect that leads to (iv). The revised (iv) is (iv)' if)~2 > 0, then a compensated increase in wage taxation has an ambiguous effect on human investment. These household level results will prove useful in interpreting the general equilibrium experiments in Sections 4 6 . The microeeonomic portion of the model is completed by assuming a specific form for the distribution function of u. We follow Barsky, Mankiw and Zeldes (1986) and choose a simple symmetric three-point distribution of the form u = (1 - x)O,
with probability rt ;
, with probability 1 - 2n ; (1 + x)0,
with probability n .
Note that the variance of u is 2n(xO)2 and the coefficient of variation is
x(2n) 1/2. As in Barsky, Mankiw and Zeldes (1986), we will use the observed coefficient of variation of earnings to help calibrate the distribution of u. Households are assumed to have rational expectations so that their subjective distribution of the states of nature coincide with the objective distribution o f U.
The model of the entire economy is obtained by specifying the distribution of household types, the aggregate production function used to produce consumption, physical capital and the goods invested in human capital, and the initial stance of fiscal policy. At any moment in time there are seven household-types involved in market trade. Young households are identical but there are three different types of middle-aged households and three different types of old households (corresponding to the three different outcomes for u). We assume that there are a continuum of households in each new generation so that the fractions of the population of middle-aged and old households correspond exactly to the probability of the three different outcomes for u. There is also population growth at rate n, so that the ratio of the population of each generation to its predecessor is i + n. The average or per capita value of a variable is obtained by averaging across households 513
William Lord and Peter Rangazas of a given generation (with weights based on rt) and then averaging across generations (with weights based on n). In the economy's initial position, following previous simulation studies, we assume that a proportional income tax is used to collect government revenue. The revenue is used to provide non-productive government goods that enter the utility function of agents in an additively separable fashion, and in some cases to provide a subsidy to the goods input in human capital production. Redistributive government taxes and transfers will be added in Section 5.
3. Calibration As in most simulation studies, the ranges for the parameters of the model are determined primarily by the available empirical evidence and, where appropriate, for the purpose of making comparisons to other simulation studies. Our general strategy for organizing the parameter settings is driven by the desire to highlight the role of uncertainty. Where natural, we group parameters into a case that maximizes the role of uncertainty, a case that minimizes its role, as well as several intermediate cases which mix the parameter settings from the boundary cases. Two parameters are particularly important in this respect. Skinner (1988) shows that 7 is critical in determining the share of savings due to precautionary motives. A high 7 means greater risk aversion and a greater tendency for households to alter behavior in order to avoid uncertainty. As indicated by results (iv) and (iv)' from Section 2, L2 is crucial in determining the effect of changes in the net wage rate on human capital investment. The greater the value of Z2, the more likely it is that an increase in the net wage rate will encourage more human capital accumulation rather than discourage it by increasing the riskiness of investment.
Length of Time Period There are three periods in the model, two working periods and a retirement period. The three period assumption is needed to simplify the computations of a model that includes three features not found in previous general equilibrium simulation studies: uncertainty, endogenous human capital, and nonexpected utility functions. As indicated below, the three-period assumption does not appear to be a substantive limitation for the questions we address. We calibrate the model under the assumption that each period represents twenty years. This gets the length of work life about right, but produces a somewhat lengthy retirement period. The lengthy retirement period means we bias saving toward finding too high a saving rate. Since one of our 514
Capital Accumulation and Taxation TABLE 1.
GeneralEquilibrium Saving Rates Certainty
y = 5, L2 = 0 y = 5, L2 = 0.15 7 = 1.25, L2 = 0 y = 1.25, ~2 = 0.15
Uncertainty
L
H
2.3 (2.9) 2.3 (2.9) 4.8 (5.0) 4.6 (4.8)
1.2 (3.3) 1.3 (3.1) 4.4 (5.0) 4.4 (5.0)
L
H
3.3
4.1
3.2
4.1
5.0
5.3
4.9
5.2
NOTES: L (H) is the low (high) uncertainty case with CV = 0.20 (0.40). The number in parentheses gives the saving rate when human capital investment is kept constant at its level under uncertainty. There are L and H cases with certainty because17, differs across the two cases to keep the ratio of wH 1 and wH~ approximately one.
principal conclusions is that the general equilibrium saving rate remains too low, even with precautionary saving, the bias is not of concern. In addition, as a practical matter, the bias does not appear to be very large. As reported in Table i of the next section, our saving rates with and without uncertainty are very comparable to those obtained in multi-period general equilibrium models. The annualized value for the rate of population growth is set to 1% in all eases. With twenty-year periods this implies n = 0.2202.
Household Preferences The preference parameters of the model are y and 8. As mentioned, the value of 7 is very important for our study. This parameter has been estimated using both macroeconomic and microeconomic data. Early macroeconomic studies (for example Hansen and Singleton 1983 and Mankiw, Rotemberg, and Summers 1985) found estimates of y that centered around 1. Using additional data and more appropriate estimation techniques, Hall (1988) found substantially larger values of y with lower bounds around 10. The microeconomic evidence has produced estimates of y no lower than 1, with an average around 2.5. 5 In several of the cases we set T equal to an 5The intertemporal elasticity of substitutions estimated in these studies are: Ghez and Becker (1975) 0.28, Friend and Blume (1975) 0.50, Farber (1978) 0.30, MaCurdy (1981) 0.28, Grossman and Shiller (1981) 0.21, Hurd (1989) 0.91, and Engen (1991) 0.30. The simple average is about 0.40, implying an estimate of 7 around 3.
515
William Lord and Peter Rangazas intermediate value suggested by the empirical evidence of 5. Trostel's (1993) survey of previous simulation studies led him to set y equal to 1.25. Based on the available empirical estimates, we feel this value is much too low, when viewed from the perspective of estimating the intertemporal elasticity of substitution. However, for purposes of comparison to Trostel and other simulation studies we also set '/equal to 1.25. The other preference parameter, 3, has no clear influence on the role that uncertainty plays in the model. We set 3 equal to an annualized value of 0.01, a common value used in simulation studies, for all cases. Our previous study of saving rates with altruistic bequests makes this assumption and we make it here to facilitate comparison. 6
Human Capital Production The parameters of the human capital production function are )~1, )~z, as well as x and n from the distribution of t~. As in Trostel (1993) we set the returns to scale, )~1 + ~,2, equal to 0.6. In several of the cases, we set equal )~2 to 0 in order to highlight the insurance effect associated with changes in the net wage rate. In addition, setting L2 equal to 0 can be defended by the fact that the majority of human investments in our model can be interpreted as on-the-job training which involves relatively small amounts of goods inputs. We also consider cases where we set )~2 equal to (1/3))~1 as in Trostel (1993). Trostel's assumption is based on estimates from formal education and serves as an upper bound on )~2 for the adult human investment in our model. Again following Trostel, we assume a 4% annual depreciation rate for the initial human capital stock. We follow Barsky, Mankiw and Zeldes (1986) by calibrating the distribution of v to equate the coefficient of variation in earnings in the data to the coefficient of variation for second period earnings in the model (CV). There are now several studies that provide sufficient estimates of the error process in individual earnings equations to compute the coefficient of variation in earnings (Lillard and Willis 1978; Hall and Mishkin 1982; Abowd and Card 1989; Engen 1991; Hubbard, Skinner and Zeldes 1993). 7 The 6Of course, a certain way to generate historical saving rates is to set the pure rate of time preference as low as is necessary. This would imply a negative value. Although there are some negative estimates of this parameter, most economists find negative values implausible. 7The reported estimates of the coefficient of variations for earnings is based on the following procedure. Empirical studies of earnings typically assume that earnings are lognormafiy distributed and that the error term of the log of earnings is an AR(1) process (see, for example Engen 1992). Under these assumptions the variance of the log of earnings is an approximataon to the coefficient of variation of earnings. (Actually an underestimate which is more accurate the smaller is the variance of log-earnings. For the variances found in the literature the approximation is very accurate.) The variance of log-earnings can be computed from the estimates of the parameters of the AR(1) process provided in the studies cited below.
516
Capital Accumulation and Taxation simple average of the coefficient of variation estimates from these studies is about 0.40. This average can be viewed as an upper bound on the variation in earnings associated with uncertain returns to adult human capital investment, since part of the variations in an individual's earnings are due to events that also affect the level and market value of general human capital attained during childhood (for example, spells of unemployment and certain types of disabilities). We set CV = 0.40, in what is referred to as the "highuncertainty" ease (H-ease). It seems reasonable to attribute at least half the variations in earnings to events that affeet the return to adult investments only (for example, an exogenous change in job or occupation that continues to reward elementary knowledge at the same rate, but that differs in the reward paid to advanced knowledge or a particular specialized skill.). We will set CV = 0.20, in what is referred to as the "low-uncertainty" ease (Lease). In ease L, we set n equal to 0.25 and x equal to 0.566, while n equals 0.33 and x equals 0.9 in ease H. s Since earnings peak just after age 40, we set ~ so that Will and wile are roughly equal.
Aggregate Goods Production Our treatment of goods production is entirely standard. We follow Trostel (1993) and many others and set labor's share equal to 0.25.
Fiscal Policy Variables In the initial baseline steady state we assume an income tax rate of 20% with all proceeds going to finance nonproductive government purchases and, when )~2 > 0, a subsidy to goods inputs in human capital production. The quantity of government purchases is roughly consistent with real world descriptive statistics (see, for example, Browning and Johnson 1984, 186; or Barro 1984, 300) and other simulation studies (see, for example, Lord and Rangazas 199i or Trostel 1993). The subsidy rate on goods inputs is set equal to 60% as in Trostel (1993). 9 As mentioned, we increase the income tax SUnreported sensitivity analyses show that the results are little affected by altering the mix of uncertainty, r¢ and x, for a given CV. 9The presence of the subsidy on the endogenous goods input raises a conceptual issue about the definition of the budget constraint. One approach is to implicitly include the goods subsidy into government purchases but to keep government purchases fixed at 20% income. The second approach treats the subsidy expenditures as additions to the government purchases that are set at 20% of income. Tax revenue and the level of purchases remain constant when balancedbudget tax reform experiments are conducted under the first approach, despite changes in the goods input, but implicit changes would occur in "other" government purchases. Under the second approach, tax reform would cause a change in the level of taxation and government purchases (including the subsidy on the goods input), whenever the goods input ehange. We feel the first approach is more consistent with other tax reform experiments and the numbers reported were generated under this approach. We did the experiments under the second approach and none of the conclusions found in the paper were altered.
517
William Lord and Peter Rangazas rate to 40% and add the redistributive component of the government in Section 5.
4. Saving Rates In recent years substantial attention has been devoted to assessing the contribution of preeantionary saving to aggregate saving (Skinner 1988; Cabellero 1991; Engen 1992; and Hubbard, Skinner and Zeldes 1993). The analysis by Skinner and Cabellero suggests that preeantionary savings can explain a very largefraction of total savings in partial equilibrium. Hubbard, Skinner and Zeldes (1993) find that uncertainty can raise aggregate saving levels to realistic values but again in partial equilibrium. Engen does use a general equilibrium framework to address the saving rate issue. However, he does not focus on the relationship between the specification and calibration of preferences and the contribution of precautionary saving to total saving in the general equilibrium portion of his analysis. As we explain below, genera] equilibrium considerations are critical in assessing the implications of different preference specifications. None of the studies allows for uncertain and endogenous human capital. Our strategy for isolating the role of uncertainty on saving is to calibrate initial baseline equilibria as realistically as possible; i.e. accounting for estimated levels of uncertainty. We then compute eounterfaetual equilibria where uncertainty is completely eliminated. When uncertainty is eliminated, in order to isolate the role of human capital per se, we consider equilibria where human capital is allowed to endogeneously respond and equilibria where human capital is frozen at initial baseline levels.
Preferences, Uncertainty, and Equilibrium Saving Table i presents the fraction of income devoted to saving (S/Y) under the assumptions from Sections 2 and 3. The three-period specification does not appear to be an important limitation, since the saving rates under certainty and uncertainty are not much different from otherwise similar models but with periods corresponding to a year (for example, compare to Lord and Rangazas 1991; in the certainty case, and Engen 1992 in the uncertainty case). As in Skinner (1988), Table 1 demonstrates that the precautionary motive can explain a very large fraction of total saving. For 7 = 5 and L2 = 0, the difference between the saving rates under H and with no uncertainty is 2.9 percentage points. 1° One can then interpret the precautionary 1°Noticethat the results are not affectedmuchby tile valueof ~2. Thisis becausethe partial equilibriumeffectsof uncertaintyon humancapitalinvestmentare independentof ~-2.The role of L2onlybecomesimportantwhenthe effectsof taxationare discussedin the nexttwo sections. This is whyresult (iv) from Section2, but not result (ii), had to be adjustedfor the ~2 > 0 case. 518
Capital Accumulation and Taxation motive as explaining over 70% (2.9/4.1) of total saving. It must be emphasized that the shares for precautionary saving just reported are very similar to those obtained by Skinner under comparable parameter settings for 7. However, Skinner does not indicate the total saving associated with the precautionary saving shares. We find when the fraction of precautionary saving is high the saving rate is relatively low, and when the saving rate is relatively high, precautionary saving is not contributing much to the total. Thus, the precautionary motive is doing little to enable life cycle models to generate more realistic saving rates. The inability of the precautionary motive to add much to total saving stems from the fact that aggregate saving is relatively high when the willingness of consumers to substitute consumption across time is high. However, this is also the situation where the degree of risk aversion is low and the effect of uncertainty is weak. To see this point explicitly first note that the standard CES utility function forces the rate of relative risk aversion (7) and the rate of intertemporal substitution (~) to be dependent (or = 1/7). In certainty models, so long as r > 5, the consumption profile is flatter and fife cycle saving smaller, the lower is or. However, with CES preferences, a lower willingness to substitute consumption over time implies greater risk aversion. With uncertainty, the consumption profile is steeper, the stronger is risk aversion, so the greater is precautionary saving. Thus, the common CES expected utility preferences imply that increasing the degree of risk aversion reduces life cycle saving at the same time it increases precautionary saving. At low levels of risk (as captured by the CV) the effect of changes in 7 on life cycle saving will dominate. If risk is sufficiently large, the increase in precautionary saving from higher 7 will be decisive. Our results indicate that for empirically realistic CVs the reduction in life cycle saving outweighs the rise in precautionary saving. For example, in case H where )~2 = 0.15, the saving rate is 5.2% for 7 = 1.25, but only 4.1% for 7 = 5. Only if the degree of CV were much larger would the saving rate increase with 7.11 11Engen's (1992) Table 2 computes partial equilibrium (treating the interest rate as exogenous) saving rates for different values of 7. In conjunction with our findings, his table highlights the importance of doing general equilibrium calculations to avoid misleading conclusions. For values o f r that are low, relative to his assumed rate of time preference of 4%, Engen finds that the saving rate increases with 7, contrary to our general equilibrium calculations. This is because when r is low relative to the rate of time preference, the effect of raising 7 (and lowering ~) on hfe cycle sa,ang is weak. This allows the corresponchng rise in precautionary saving to dominate the fall in life cycle saving. However, low values of r are inconsistent with Engen's parameter setting. For values of r closer to what Engen computes in his general equilibrium simulations, he also finds that an increase in 7 results in lower saving rates (consider his increase in 7 from I to 5, closest to our experiment, when r = 0.06).
519
William Lord and Peter Rangazas The discussion above indicates that the choice of CES utility functions may limit one's ability to identify the full role of precautionary saving because it forces an inverse relationship between 7 and r~. The nonexpected utility formulation of Kreps and Porteus is an interesting alternative to CES preferences as it allows a complete decoupling of 7 and r~. We introduce Weil's (1990) parametric version of the Kreps-Porteus nonexpected utility function in our three period life-cycle model by using the backward recursion approach described by Farmer (1990). Backward reeursion starting from the third period yields a utility function of the form [1
+
- W(I-4,)\
1 + (1 + X'~--l|± T t~ - -
1
-
y
+ (1
~{]t?ol_qb, l _T / l _q b
+ 6)(1 - y)
(6)
Now ~ = 1/7 is the intertemporal elasticity of substitution for deterministic consumption paths, qb is the distinct degree of relative risk aversion for timeless gambles, and c2, maximizes utility for the two-period problem beginning in the second period of life (see footnote 3). As before, first-order conditions can be obtained and the model can be calibrated and solved numerically to determine the saving rate. The most interesting case to consider is where both a and qb take on their upper bound values (a = 0.8 and qb = 5). This upper bound case maximizes the chances of simultaneously producing high aggregate saving with a large role for precautionary saving. Under these preferences, increasing risk aversion will not directly reduce intertemporal substitutability, so aggregate saving is expected to increase with 7. However, other types of life cycle offsets do occur due to general equilibrium interactions. An increase in y causes a partial equilibrium rise in saving that reduces r and raises w, serving to increase human capital investment via both an increased rate of return and higher wealth. Since more human capital investment increases expenditures when young and income in middle age, conventional savings are reduced. Recall from Table 1 that in case H with )~2 = 0.15, if y = 1.25, implying = 0.8, the saving rate was 5.2% in the presence of uncertainty. Now using the Kreps-Porteus preferences with ~ kept at 0.80, but with the now distinct degree of risk aversion qb set to 5, the saving rate increases to 6.1%. Since this case employs the high CV, as well as the high a and qb, it produces the highest saving rate for the range of parameters supported by empirical evidence. From 1950 to 1980, the average U.S. saving rate was around 8% (Auerbach and Kotlikoff 1987, 64). Thus, one still must conclude that augmenting a life cycle model with exogenous earnings uncertainty leaves a significant portion of total saving unexplained. 520
Capital Accumulation and Taxation This conclusion is reinforced by the recent econometric estimates of Epstein and Zin (1991). They estimate cr and ¢b using a nonexpected utility approach. The estimates for rr range between 0.20 and 0.80, but estimates for ¢b did not differ significantly from 1. The case of ~ = 0.8 and qb = 1 would generate the highest saving rate from their estimates. This parameter combination is very close to the combination rr = 0.8 and 7 = 1.25, reported in the last two rows of Table 1, that generates a saving rate of only 5%. For the range of available estimates of G and qb, the nonexpected utility specification does not significantly alter the quantitative conclusions reached under the standard expected utility assumption.
Human Capital and Saving We now return to Table 1 to consider how the influence of uncertainty on savings is affected by endogenous human capital. The elimination of uncertainty increases human capital investment through the insurance effect. This reduces conventional saving by shifting the timing of income to later periods and increasing investment outlays in the first period. This suggests that the response of human capital investment accentuates the reduction in saving when uncertainty is eliminated (i.e. increases precautionary saving). However, general equilibrium factor price movements will generate an effect on human capital that works in the opposite direction. Eliminating uncertainty creates a partial equilibrium reduction in saving that induces a general equilibrium increase in the interest rate and decrease in the wage rate. These factor price movements lower the marginal return to human capital and reduce human wealth for a given level of human capital, both of which cause human capital investment to decline. The decline in human capital investment raises the saving rate under perfect certainty and reduces the estimate of precautionary saving relative to the case where human capital is exogenous. Under our calibrations the insurance effect dominates the general equilibrium effect. Eliminating uncertainty reduces saving much more when human capital is endogenous. For example, when risk aversion is strong and uncertainty high (7 = 5 in case H), eliminating uncertainty reduces saving from 4.1% to just 3.1% when human capital is exogenous. Allowing for the response in human capital investment reduces the saving rate all the way to 1.3%. Thus, the importance of earnings uncertainty for saving is significantly magnified by endogenous human capital.
5. Redistributive Taxation
The effects of income taxation on human capital are complex. Even when abstracting from uncertainty and general equilibrium interactions, 521
William Lord and Peter Rangazas TABLE 2.
Effectsof Redistributive Taxation h
7 = 5,)~2 = 0 7 = 5, 7~. = 0.15 7 = 1.25, L2 = 0 7 = 1.25, L 2 = 0.15
E
S/Y
K
Y
V
k
24.9 4.9 -26.2 -29.8 -5.1 -6.2 -33.1 (12.6) (2.0)(-9.4.9)(-29.2)(-6.8)(-19.7)(-30.4) 18.1 3.6 -34.3 -30.5 -6.1 -6.7 -32.6 (4.2) (0.2) (-21.1) (-26.7) (-7.3) (-19.1)(-27.1) 14.0 1.8 -19.3 -23.4 -5.2 -0.9 -24.8 (9.3) (i.0) (-17.5) (-22.0) (-5.3) 3.6 -0.i -17.6 -22.6 -6.2 (-i.i) (0.0)(-15.3)(-20.3)(-6.0)
(-1.2)(-22.8) -1.2 -22.8 (-1.4)(-19.7)
NOTES: All numbers represent the percentage change in the variable due to the introduction of redistributive taxation. The H-case is presented with the L-case in parentheses.
theoretical reasoning fails to produce a clear conclusion. A higher rate of income taxation lowers the net of tax interest rate, raising the present value of returns to human capital investments and thereby encouraging more investment activity (Heckman 1976). A higher tax also lowers the net wage rate, which lowers the return to human capital except in the case where the sole cost of investment is forgone earnings (Boskin 1975). Trostel (1993) examines these opposing effects in a general equilibrium, infinitely-lived agent model. He finds large negative effects of taxation on human capital, implying that the wage effect dominates the interest rate effect. Trostel's study does not consider the interaction between uncertainty and income taxation. If there is uninsured earnings risk associated with human capital investment then the introduction of a redistributive income tax lowers the riskiness of human investment (result (iv) from Section 2). Barsky, Mankiw and Zeldes (1986) stress that the implicit insurance provided by income taxation reduces precautionary saving. In a more general model, this same effect also induces a portfolio shift away from saving and toward the now relatively safer human capital investment. If the insurance effect of income taxation on human investment is strong enough, the overall effect of income taxation on human capital will be positive. We assess the importance of the insurance effect by introducing redistributive taxation into our model. Browning and Johnson (1984) have shown that the redistributive component of the U.S. tax system in the 1970s can be closely represented by a demogrant policy. Under a demogrant policy all tax revenues taken in for the purpose of redistribution are paid out in equal lump-sum transfers to all members of the economy. Table 2 presents the steady-state effects of introducing a twenty percent income tax (driving the total income tax rate to 522
Capital Accumulation and Taxation 40%) where the proceeds are redistributed to all members of the economy in equal lump-sum fashion so as to balance the government's budget in each period. Table 2 presents the effects of redistributive taxation on human capital investment (h), the effective labor supply (E), the saving rate (S/Y), the capital stock (K), and income (Y). The numbers represent percentage changes in the variables across steady states for the H-case, with the L-case in parentheses. Table 2 demonstrates that the insurance effect is quantitatively quite important. Under all parameter settings but one, the production of human capital is increased after introduction of the tax. The one exception is where the role of the insurance effect is expected to be minimized; the L-case with 7 = 1.25 and ~,2 = 0.15 (recall L2 > 0 is critical in establishing that wage taxation lowers the return to human capital investment, an effect that works against the insurance effect). Even in this case the negative effect on human capital investment is quite small, - 1.1%. Under the same parameter setting, but with perfect certainty and variable hours of labor supply, Trostel found the long-run income tax elasticity for aggregate effective labor supply to be - 0.480. The corresponding income tax elasticity in our model is -0.001.12 In contrast, under the assumptions where the insurance effect is maximized, the H-case with 7 = 5 and ~2 = 0, the positive effect on human capital is 24.9% with an income tax elasticity for aggregate effective labor supply of 0.048. Thus, Trostel's conclusion of a large negative effect of income taxation on human capital appears to depend on his assumptions of perfect certainty and a positive wage elasticity for hours of work (an assumption based on estimates now viewed skeptically by some labor economists (Heckman 1993). There is more agreement between our results and those of Trostel concerning the effects of taxation on physical capital. Both studies find sizable negative effects of taxation on the stock of physical capital, although the magnitude of our declines are much less (about one-sixth of Trostel's). Since redistribution raises human capital investment in our model, the negative effects on saving would be weaker and the negative effects on output stronger were human capital not endogenous. Fixing human capital at its initial baseline levels, result in 10% to 25% smaller falls in the saving rate and 0% to 50% greater falls in output. The last column of Table 2 gives the effect of the policy on the steadystate utility of a representative household. Household's receive a direct wel12Since we are considering a 100% change in the income tax rate, our percentage change numbers can also be interpreted as tax elasticities and may be directly compared to Trostel's Table 2.
523
William Lord and Peter Rangazas
fare benefit from the insurance provided by income taxation. However, the policy also lowers wages and raises interest rates as the capital-labor ratio falls (see the next to last column of Table 2). When the economy is dynamically efficient, as is the case here, a drop in the capital-labor ratio and the associated effects on factor prices will cause a fall in household welfare (see, for example, Diamond 1965). For steady-state utility, the factor price effect dominates the direct insurance effect in every case. For households alive earlier in the transition to the new steady state the insurance effect may dominate, since the capital-labor ratio will not have yet fallen to its steadystate levels. In conclusion it does seem that redistributive taxation lowers the economy's physical capital stock, its ability to produce output, and steady-state welfare. However, redistributive income taxation has at least a modest positive effect on human capital accumulation.
6. Tax Reform There has been much attention paid to which tax base would be the best one for promoting capital accumulation and economic growth. One common misconception is that removing the tax on capital income and shifting the burden to labor income unambiguously increases saving and investment. The proponents of the pro-growth effects of wage-based taxation are focusing solely on the positive substitution effect associated with eliminating the tax on interest income. Buchanan (1959) and Feldstein (1978) have pointed out that there is another factor at work; the timing of tax collection will change. A shift from an income to a wage tax base pushes taxation forward in the life cycle and thus net income is pushed toward the later years of life. Such a shift will cause saving to fall when young, and thus aggregate saving will fall as well whenever n > 0. Thus, in life cycle models without uncertainty it is quite possible that the tax timing effect could dominate the positive substitution effect of removing the tax on interest income, causing saving to fall due to the shift toward the wage base. Allowing for uncertainty gives further reason to believe that the sole reliance on standard substitution effects to draw inferences concerning tax reform is misleading. The real after-tax return to human capital, measured in consumption units, is given by (w/pc)h - Owgh =- (1 - Zw - zy)/(1 + Zc)wgh, where wg is the gross wage rate, 0 is the fraction of the human capital return available for private consumption, % is the tax rate on wages, • y is the tax rate on income, and zc is the tax rate on consumption. Note that a lower value of 0 lowers the variance of (w/pc)h and thus lowers the variance of c2 generated by variation in h. In our model it is this covarianee between
524
Capital Accumulation and Taxation TABLE 3.
Tax Reform: Income to Wage Taxation L
y 1' y 3'
= = = =
5,)~ = 0 5, 9~2 = 0.15 1.25, 9~2 = 0 1.25, Le = 0.15
r
h
30.0 28.6 17.5 16.6
-21.7 -24.7 -13.1 -15.3
H S/Y
0
r
h
-16.4 -11.7 39.2 -13.2 -14.9 -12.1 38.8 -16.2 0.0 -8.8 20.2 -10.4 1.2 -9.0 19.4 -13.0
S/Y
0
-22.6 -12.3 -22.7 -12.7 -2.3 -9.1 -1.5 -9.3
NOTE: All numbers representthe percentagechangesin the variabledue to the tax reform.
c2 and h that affects the saving and investment plans in period one. Thus, the lower is 0, the greater is the insurance effect of taxation. A switch from an income to a wage tax will lower 0 since, everything else constant, % must be larger than zy to collect the same tax revenue. Thus, the switch to wage taxation will generate an insurance effect that will reduce saving and raise human capital investment. We examine the relative strengths of the opposing effeets mentioned above by comparing the steady state under wage taxation to the original baseline-steady state under income taxation. The percentage changes in the steady-state values of r, h, S/Y, and 0 due to the tax reform are found in Table 3. Table 3 reveals that the combination of income timing and insurance effects dominate the substitution effects of switching from income to wage taxation. The last column of the table shows that the percentage decreases in 0 are substantial; a necessary condition for a quantitatively important insurance effect. In all eases but one, the economy's saving rate declines. The saving rate stays roughly constant only when uncertainty and the household's aversion to it are at their lower bounds. With what we regard as the more realistic setting for 7, the saving rate falls dramatieally in both the H and the L eases. The fact the decline in the saving rate is sensitive to the degree of risk aversion and the amount of uncertainty suggests that the insurance effect, and not only the tax timing effect, is playing an important role in overturning the substitution effects. Engen (1992) also analyzes tax reform with uncertain earnings. He finds a switch from an income to a wage base results in greater saving. The difference in results are in a large part due to the differences in how the tax experiments were constructed. In his attempt to construct tax experiments that would allow meaningful welfare comparisons across steady states, Engen deviated from the standard practice of forcing the new tax rates to adjust so as to maintain a balanced budget in each period. Instead, he restriets tax
525
William Lord and Peter Rangazas rates to adjust so as to keep the present value of the tax burden for the representative household constant. When r > n, Engen's approach will result in lower rates of wage taxation than with the standard balanced budget approach. 13 This shows up in the fact that 0 decreases by only 1.75% when Engen switches from the income to the wage tax base (see his Table 5). In contrast, our experiments generate decreases in 0 from 9% to 13%. The larger the decrease in 0, the larger the insurance effect and the larger the negative effect on precautionary saving. Turning now to the effects on human capital, first note that the removal of the tax on interest income and the decreases in saving cause large increases in the net interest rate. The higher interest rates serve to discourage human capital investment. This interest rate effect dominates the insurance effect that encourages human capital investment as seen from the negative overall effects reported in Table 2. The decline in human capital investment helps to lessen the negative effects of the policy on saving rates. If human capital were fixed at initial steady values, the percentage decline in the saving rate would be 2 to 12 percentage points greater.14
7. Conclusion To summarize, the simulations of this paper have produced new findings concerning the macroeconomic implications of uncertainty and endogenous human capital formation; including three findings that challenge those available in the literature. First, the presence of endogenous human capital investment significantly increases the share of total saving that is due to precautionary motives (an increase in earnings uncertainty induces a shift away from human capital investment and toward saving). However, even 13To see this most simply, consider a two period life-cycle model with fixed factor prices and no human investment. The balanced budget, per young household, can be written as g = xyw + x~rgs/(1 + n), where g is the level of government purchases per young household and rg is the gross interest rate. The standard tax reform experiment requires XwW = g, while Engen constrains his wage tax rates (t~) to satisfy twW = xvw + ~vrgs/(1 + r). I f r > n, then g > zvw + zyrgs/(1 + r) and Zw > t,~. Engen's approach is a potentially useful analytical device in conducting normative evaluation of policy options, but the balanced budget rule is almost always assumed in the positive analysis of the long-run effects of fiscal policy (presumably because it comes closer to the budget rules that are followed in practice). 14The results in this section have to be qualified since they overstate the insurance effect from a switch to wage taxation in a world with risky physical capital. The presence of a stock market that helps to diversify away from the idiosyncratic risk of physical capital ownership suggests that the insurance effects of income taxation relative to wage taxation may not be that important.
526
Capital Accumulation and Taxation with human capital, we find that precautionary motives do not drive saving rates up to the historical average for the period 1950-1980, a result that is contrary to previous partial equilibrium studies. Second, the redistributive component of the income tax structure has a modest positive effect on life cycle human capital accumulation (the insurance effect of taxation outweighs the negative effect of a lower after-tax return to investment). In contrast, Trostel (1993) finds large negative effects on human capital accumulation in an infinitely-lived agent model with perfect certainty. Our result demonstrates the potential quantitative importance of idiosyncratic risk and shows how sensitive Trostel's conclusion is to a reasonable change in model specification. Third, removing the capital income tax by switching from an income to a wage base will reduce physical capital accumulation (precautionary saving falls enough to offset the rise in saving from a higher after-tax interest rate), rather than increase it as suggested by models that assume perfect certainty. Human capital formation is also discouraged by the reform (due to the rise in interest rates), helping to mediate the fall in saving (less human capital investment implies a portfolio shift into saving). These results demonstrate the importance of combining the two currently standard ways of modeling uncertainty and human capital formation in maeroeeonomies. More direct empirical work is needed to test how uncertainty affects saving, education, and on-the-job training decisions. Our simulations suggest that it may be difficult to draw reliable inferences about saving and human capital investment by treating them as independent behaviors. Received: July 1996 Final version June 1997
References Abowd, John M., and David Card. "On the Covarianee Structure of Earnings and Hours Changes." Econometrica 57 (March 1989): 411-46. Auerbach, Alan J., and Laurence J. Kotlikoff. Dynamic Fiscal Policy. Cambridge: Cambridge University Press, 1987. Barro, Robert. Macroeconomics. New York: John Wiley, 1984. Barsky, Robert N., Gregory Mankiw, and Stephen P. Zeldes. "Ricardian Consumers with Keynesian Propensities." American Economic Review 76 (September 1986): 676-91. Ben-Porath, Yoram. "The Production of Human Capital and the Life Cycle of Earnings." Journal of Political Economy (August 1967): 352-65.
527
William Lord and Peter Rangazas Boskin, Michael J. "Notes on the Tax Treatment of Human Capital." In Conference on Tax Research 1975. Washington: Department of the Treasury, 1975. Browning, Edgar K., and William R. Johnson. "The Trade-Off between Equality and Efficiency." Journal of Political Economy 92, no. 2 (April 1984): 175-203. Buchanan, James. "Saving and the Rate of Interest: A Comment." Journal of Political Economy 67, no. 1 (February 1959): 79-82. Cabellero, Ricardo. "Earnings Uncertainty and Aggregate Wealth Accumulation." American Economic Review 81 (September 1991): 859-71. Diamond, Peter A. "National Debt in a Neoclassical Growth Model." American Economic Review 55 (December 1965): 1126-50. Eaton, Jonathan, and Harvey S. Rosen. "Taxation, Human Capital, and Uncertainty." American Economic Review 70 (September 1980): 705-15. Engen, Eric M. "Precautionary Saving and the Structure of Taxation." April 1992. Mimeo. --. "Consumption and Saving in a Life-Cycle Model with Stochastic Earnings and Mortality Risk." UCLA, 1992. Mimeo. • "Estimation of a Stochastic Life-Cycle Model with Mortality Risk Using Panel Data." UCLA, 1991. Mimeo. Epstein, Larry G., and Stanley E. Zin. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis." Journal of Political Economy 99 (April 1991): 263-86. Farber, Henry S. "Individual Preferences and Union Wage Determination: The Case of the United Mine Workers." Journal of Political Economy 86 (October 1978): 923-42. Farmer, Roger. "RINCE Preferences." Quarterly Journal of Economics 105 (February 1990): 43-61. Feldstein, Martin S., "The Welfare Cost of Capital Income Taxation." Jourhal of Political Economy 86, no. 2 (1978): $29-$51. Friend, Irwin, and Marshall E. Blume. "The Demand for Risky Assets." American Economic Review 65 (December 1975): 900-22. Ghez, Gilbert R., and Gary S. Becker The Allocation of Time and Goods over the Life-Cycle. New York: Columbia University Press, 1975. Grossman, Sanford J., and Robert J. Shiller, "Determinants of the Variability of Stock Market Prices." American Economic Review 71, no. 2 (May 1981): 222-27. Hall, Robert E. "Intertemporal Substitution in Consumption." Journal of Political Economy 96, no. 2 (April 1988): 339-57. Hall, Robert E., and Fredric Mishkin. "The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households." Econometrica 50 (March 1982): 461-81. 528
Capital Accumulation and Taxation Hansen, Lars Peter, and Kenneth J. Singleton. "Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns." Journal of Political Economy 91 (April 1983): 249-65. Heckman, James J. "A Life-Cycle Model of Earnings, Learning, and Consumption." Journal of Political Economy 84, no. 4 (August 1976): $1144. --. "What Has Been Learned about Labor Supply in the Past Twenty Years?" American Economic Review 83 (May 1993): 116-21. Hubbard, R. Glenn, and Kenneth Judd. "Social Security and Individual Welfare: Precautionary Saving, Liquidity Constraints, and the Payroll Tax." American Economic Review 77 (September 1987) 630-47. Hubbard, R. Glenn, Jonathan Skinner, and Stephen P. Zeldes, "The Importance of Precautionary Motives for Explaining Individual and Aggregate Saving." Paper presented at the Carnegie-Rochester Public Policy Conference, April 1993. Hurd, Michael. "MoRality Risk and Bequests." Econometrica (July 1989): 779-814. Kotlikoff, Laurence J. "Intergenerational Transfers and Savings." Journal of Economic Perspectives 2 (Spring 1988): 41-58. Kotlikoff, Laurence J., and Lawrence H. Summers. "The Role of Intergenerational Transfers in Aggregate Capital Accumulation." Journal of Political Economy 2 (August 1981): 706-32. Levhari, D., and Yoram Weiss. "The Effect of Risk on Investment in Human Capital." American Economic Review 64 (December 1974): 950-63. Leland, Hayne E. "Saving and Uncertainty: The Precautionary Demand for Saving." Quarterly Journal of Economics 82 (1968): 465-73. Lillard, Lee A., and Robert J. Willis. "Dynamic Aspects of Earnings Mobility." Econometrica 46 (1978): 985-1012. Lord, William. "The Transition from Payroll to Consumption Receipts with Endogenous Human Capital." Journal of Public Economics (February 1989): 53-74. --. "Saving, Wealth and Exchange Bequest Motives." Canadian Journal of Economics 25 (August 1992): 743-53. Lord, William, and Peter Rangazas. "Savings and Wealth with Altruistic Bequests." American Economic Review (March 1991): 289-96. MaCurdy, Thomas E. "An Empirical Model of Labor Supply in a Life-Cycle Setting." Journal of Political Economy 89 (December 1981): 1059-85. Mankiw, N. Gregory, Julio j. Rotemberg, and Lawrence H. Summers. "Intertemporal Substitution in Macroeconomics." Quarterly Journal of Economics 100 (February 1985): 225-51. Merton, Robert C. "Optimum Consumption and Portfolio Rules in a Continuous Time Model." Journal of Economic Theory 3 (1971): 373-413. 529
William Lord and Peter Rangazas Sandmo, Agnar. "Capita/Risk, Consumption, and Portfolio Choice." Econometrica 37 (1969): 586-99. --. "The Effect of Uncertainty on Savings Decisions." Review of Economic Studies (July 1970): 353-60. Skinner, Jonathan. "Risky Income, Life Cycle Consumption, and Precautionary Saving." Journal of Monetary Economics 22 (1988): 237-55. Trostel, Philip. "The Effect of Taxation on Human Capital." Journal of Political Economy 101, no.2. (April 1993): 327-350. Well, Phillipe. "Nonexpected Utility in Macroeconomics." Quarterly Journal of Economics 105 (February 1990): 29-42. White, Betsy Butrill. "Empirical Tests of the Life-Cycle Hypothesis." American Economic Review 68 (September 1978): 546-60.
Appendix a e
=
CV= e
=
E= g= h= H= n
=
K= Pc
=
pg= r =
S= V= W X
= =
y
~= 7= 8= O= )~ =
530
financial asset holdings. consumption of goods and services. coefficient of variation for second period earnings. effective time input in human capital production. aggregate effective labor supply. goods input in human capital production. government purchases per young worker. net investment in human capital. human capital stock. rate of population growth. aggregate physical capital stock. net-of-tax price of consumption goods and services. net-of-tax price of goods input. net-of-tax interest rate. net national saving. maximum value function. net-of tax wage rate. determines the spread of the symmetric two-state realizations of v about its mean. net national product and income. undepreciated fraction of first-period human capital stock. constant coefficient of relative risk aversion with CES utility. pure rate of time preference. fraction of human capital return available for private consumption net of all taxes. output elasticity in human capital production.
Capital Accumulation and Taxation V = random efficiency scalar in h u m a n capital production. n = probability of the symmetric two-state realizations of v about its mean. = intertemporal elasticity of substitution. = lump sum government transfer. Zc = consumption tax rate. Zw = wage tax rate. qb = constant coefficient of relative risk aversion with non-expected utility. zy = income tax rate.
531