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TAX RATE UNCERTAINTY, LABOR SUPPLY AND SAVING IN A NONEXPECTED UTILITY MAXIMIZING MODEL 1. Introduction While taxes may be as certain as death, tax policies and subsequent tax rates are far from that. Thus it is not surprising that in recent years the issues of tax rate uncertainty and their economic impacts have received considerable attention. Hess (1993), for example, examines whether the tax rates are too volatile so that they fluctuate in excess of variations of economic fundamentals. Such uncertainty in income taxes may arise due to changes in tax codes or variability of effective tax rates due to the interaction of inflation with an unindexed tax system (Feldstein and Summers (1979)). Weiss (1976), Stiglitz (1982), Alm (1988), Watson (1992) and Kim, Snow and Warren (1995) among others have examined the impact of tax rate uncertainty on factor supply and/or saving. One limitation of most of these works has been that they have treated individual's labor supply and saving decisions separately (see for example, Alm (1988)). Only Kim, Snow and Warren (1995) provide a comprehensive analysis of simultaneous labor supply and saving decisions by an individual who is faced with tax rate uncertainty. As for the impact of greater tax rate uncertainty, all these papers conclude that the comparative statics effects of an increase in tax rate uncertainty on an individual's labor supply (and saving) depend on the individual's attitude towards risk. Weiss (1976) and Stiglitz (1982), for example, demonstrate that faced with greater tax rate uncertainty, an individual can decide to work more and save more due to his aversion to downside risk. Alm (1988) concludes that such comparative statics effects depend on the relative risk aversion function. In the context of a dynamic model that incorporates portfolio decisions, Watson (1992) also shows that the impact of increased tax rate uncertainty on an individual's saving depends on the measure of relative risk 1 aversion.Focusing on simultaneous labor supply and saving decisions by an individual, Kim, Snow and Warren (1995) determine that aan increase in tax rate uncertainty can lead the individual to save less and work more provided that the individual's attitude towards risk satisfies restrictions involving multivariate risk aversion and the coefficient of partial risk aversion. However, in a dynamic setting this general conclusion, that the comparative statics effect of a tax rate uncertainty on an individual's labor supply (and saving) depends on an individual's risk aversion, is misleading. It may arise due to the inability of the time additive von Neumann- Morgenstern preferences and the Expected Utility maximizing approach to distinguish between an individual's utility smoothing motive (represented by intertemporal substitution in consumption) and his risk aversion. In this paper we examine the comparative statics effect of tax rate uncertainty on labor supply and savings in a nonexpected utility maximizing framework. The payoff to this extension is that it helps us understand the separate roles of intertemporal substitution and risk aversion in determining the impact of tax rate uncertainty on saving and labor supply and thus provides additional insights into the results of Kim et al. We consider a nonexpected utility maximizing framework using Selden's (1978) Ordinal Certainty Equivalent (OCE) preferences which allow us to formally distinguish between an agent's utility smoothing motive and his degree of risk aversion. Basu and Ghosh (1993) examine the effect of interest rate uncertainty on optimal savings using the OCE preferences. Basu (1995) analyzes the impact of income tax rate uncertainty on an agent's capital accumulation decision using a two period model with OCE preferences. In both these papers the individual supplies only one elastic factor, i.e., saving, which is sensitive to changes in the interest rate. Labor supply is assumed to be inelastic. In this paper, we extend this framework to examine the effect of tax rate uncertainty on labor supply (and saving) when the individual simultaneously supplies two elastic factors, savings and labor. The 2 crucial difference between our approach and Kim et al’s is that we use the nonexpected utility maximizing framework which has the ability to distinguish between an individual's risk aversion and intertemporal substitution.1 In order to make a meaningful analysis of the effect of wage income tax rate uncertainty in a two period setting, we assume that all compensation of labor is received in the second period i.e., labor is a “credit good” as in Kim, Snow and Warren(1995). We first look separately at the effects of interest income tax rate uncertainty and wage income tax rate uncertainty on the individual’s joint decision of labor supply and saving. Separating interest income tax uncertainty from wage income tax can be justified by looking at the tax codes. There may be special savings incentives, which solely affect the tax on interest income without affecting tax on wage income. On the other hand, wage income tax uncertainty may solely emanate from taxes on the retirement contributions. However, interaction between an unindexed tax system and inflation may make the effective tax rate on both wage income and capital income uncertain. We, therefore, also investigate the effect of uncertainty in a comprehensive tax rate that affects simultaneously the return to savings and wage income. In an OCE framework, the functional form for the risk preference is crucial in determining the comparative statics results. In order to make the OCE model tractable two separate functional forms for the utility function are used in the paper. For analyzing the effect of interest tax uncertainty, a constant relative risk aversion class a la Selden (1978) and Weil (1990) is used. For analyzing the wage income tax rate uncertainty as well a single general tax rate uncertainty a constant absolute risk aversion class of risk preferences a la Weil (1993) is used. Our theoretical results, suggest that the effect of an interest income tax rate uncertainty on 1. For the purpose of our comparative statics analysis, a general non-time additive utility function may be used. However to obtain closed form solutions we have used an iso-elastic preference structure similar to Watson (1992) who used it in an expected utility maximizing framework. Even though we assume an iso-elastic preference, as it has been shown by Selden (1979), using use an ordinal certainty equivalence approach it still enables us to distinguish 3 labor supply depends on the individual’s elasticity of intertemporal substitution. Optimal labor supply increases in response to a perceived tax rate uncertainty only when the elasticity of intertemporal substitution is less than unity indicating a strong utility smoothing motive. If the elasticity of intertemporal substitution exceeds unity, optimal labor supply decreases in response to an increase in tax rate uncertainty. The impacts of increased tax rate uncertainty on optimal saving are also similar. The wage income tax rate uncertainty alone, however, unambiguously impacts labor supply in a positive manner by inducing the agent to work harder in the first period to recoup the loss of certainty equivalent wage income. Such an increase in work effort gives rise to some utility smoothing making it less imperative to save more in the first period. The effect of wage income tax uncertainty on saving is thus ambiguous. However, for plausible range of parameter values an increase in wage income tax rate uncertainty is likely to promote saving. On the other hand, an uncertainty in the general tax rate lowers the certainty equivalent wage as well as return to capital. Since the certainty equivalent interest rate is altered, the effect of a common tax rate uncertainty on both labor supply and saving again depends on the size of the intertemporal substitution elasticity. The plan of the paper is as follows. In section 2, we lay out a general two-period framework with OCE preferences for analysis of rate of return tax rate and wage income tax rate uncertainty. Section 3 deals with the comparative statics of interest income tax rate uncertainty. Section 4 focuses on the comparative statics of wage income tax rate uncertainty. In section 5, we investigate the effect of uncertainty in a single tax rate which affects the returns to savings and wage income uncertainty. Section 6 ends with concluding comments. 2. Tax Rate Uncertainty: A Two Period OCE Framework between risk aversion and intertemporal substitution in consumption. 4 Following the standard framework of labor - leisure choice models (for example, Block and Heineke (1975)) we consider an individual who lives for two periods. Labor is assumed to be perfectly divisible. In period 1 the individual works for (1- h1 ) hours, h1 being leisure and 1 the normalized time endowment. The individual does not work in period 2 (a widely held assumption in two period models). The total compensation for the individual is W (1 h1 ) where W is the wage rate. (1- ) portion of the wage compensation package is received in period 1, and the remaining portion is received in period 2 when the individual is retired, 0 1 2. This general formulation encompasses both the possibilities that labor may be viewed as a “cash good” ( = 0) as in Alm (1988), or a “credit good” ( = 1) as in Kim, Snow and Warren (1995). The wage compensation ~ received in period 2 is subject to an income tax with the (random) tax rate being . (In the rest of the paper we denote a random variable by a ~ above the variable.) In addition to a wage income, in period 1 the individual also receives a fixed exogenous income, Y. (It may also be assumed that the individual receives an exogenous income in period 2 as well. But it does not alter any of our results.) Saving in period 1 amounts to S1 (1 )W (1 h1 ) Y C1 (1) ~ where C1 is the level of consumption in period 1. This saving generates an asset income of S1 R for ~ ~ period 2, where R , the after tax gross rate of return on saving is given by 1+ r (1 t ) , with r being ~ the risk free rate of return on saving and t the random tax rate on return on saving. The individual does not have any bequest motive and hence period 2’s consumption is given by ~ ~ ~ C2 (1 )W (1 h1 ) Y C1 R W (1 h1 )(1 ) (2) 2 Parameter is the same as (1-) used in Kim, Snow and Warren (1996). We are greatful to an anonymous referee 5 or equivalently, ~ ~ ~ C 2 W (1 h1 ) Z (Y C1 ) R (2') where ~ ~ ~ Z (1 ) R (1 ) (3) The individual has OCE preferences a` la Selden (1978,1979). Thus the individual maximizes U (C1 , h1 ) (C 2 ) (4) where U (. , .) is a quasi-concave utility function with U C 0, U h 0, U CC 0 , U hh 0 (as usual the subscripts are used to denote appropriate partial derivatives). 1 is the utility discount factor. For notational convenience, U (C 2 ,1) is denoted by (C 2 ) with / (C 2 ) 0. C 2 is certainty equivalent level of period 2’s consumption, i.e. the nonstochastic level of consumption which ~ provides utility equal to the expected utility of the random consumption, C 2 .Thus C 2 is defined by ~ V (C 2 ) E V (C 2 ) or, ~ C 2 V 1 {E[V (C 2 )]} (5) where V (.) is a strictly concave function. It should be noted that in view of (5) the utility functional in (4) is not linear in probabilities. Furthermore, in such a nonexpected utility maximizing framework, individual’s risk aversion is distinguished from the elasticity of intertemporal substitution. Specifically, it is the curvature of V that determines the degree of individual’s risk for pointing out this similarity. 6 aversion while the curvature of U determines the extent of elasticity of intertemporal substitution. The individual maximizes (4) subject to (5) and (2). The first order conditions are given by ~ ~ / (C 2 ) E[V / (C 2 ) R] U 1 (C1 , h1 ) (6) / V (C 2 ) ~ ~ / (C 2 ) E[V / (C 2 )W Z ] U 2 (C1 , h1 ) (7) / V (C 2 ) where subscripts of U denote appropriate partial derivatives. In the next two sections we examine separately the impacts of rate of return and wage income tax uncertainty using specific parameterizations of the preference structure. 3.Uncertainty in Rate of Return Tax We first consider that the tax rate ~ imposed on the return on saving is random while wage t income tax rate is non-stochastic. W is viewed as the after (income) tax wage rate. Furthermore, we assume that the entire wage income is received in period 1, i.e., labor is a “cash good” with = 0. Consequently from (1) we can readily obtain S 1 = W(1 - h1 ) + Y - C 1 (8) ~ ~ Furthermore Z as defined in (3) reduces to R . Consequently, in view of (2) the consumption level in period 2 is determined to be C2 W (1 h1 ) Y C1 R ~ ~ (9) In order to examine the importance of intertemporal substitution and risk aversion in the individual’s optimal choice, let us consider the following specifications of the U and V functions: 7 1 1 C h U (C1 , h1 ) 1 1 0 (10) 1 1 1- C V( C 2 ) = 2 , > 0 (11) 1- Additive utility functions similar to (10) have been frequently used in the literature (for example, Altonji (1982)). It is customary in nonexpected utility maximizing approach to use specific utility functions (for example Weil (1990), vanWijnbergen (1992), Basu (1995)). These specifications can easily distinguish between the risk aversion parameter and the elasticity of intertemporal substitution parameter and at the same time help to determine closed form solutions. Specifically, is the reciprocal of elasticity of intertemporal substitution for consumption and the coefficient of relative risk aversion. Note that if = , our nonexpected utility maximizing framework reduces to the familiar expected utility maximizing framework. The first order conditions (6) and (7) now reduce to: C1 ~ ~ ˆ E RC2 C2 (12) h1 ~ ~ ˆ WE RC2 C2 (13) where certainty equivalent consumption in period 2 is given by, 1 ˆ ~ 1 C2 E C2 1 (14) To examine the effect of an tax rate uncertainty on labor supply and saving we further simplify the above first order conditions. Note that the certainty equivalent interest rate is given by 8 1 ˆ ~ R E R 1 1 (15) Taking log transform of (13) and (15) and noting that by virtue of (8) and (9) ˆ ˆ C2 = S1 R and ~ ~ C2 S1 R we obtain ˆ log h1 log W log S1 (1 ) log R (16) But from the first order conditions (12) and (13) we get 1 C1 W h1 Substituting into (8) one obtains 1 S1 W (1 h1 ) W h1 Y (17) which upon substitution in (13) yields 1 1 (1 ) log h1 log W log W (1 h1 ) W h1 Y ˆ log R (18) In order to analyze the impact of uncertainty in an OCE framework we use a methodology originally suggested by van Wijnbergen (1992). Following van Wijnbergen we capture the effect of uncertainty by an increase in the risk aversion parameter starting from a risk neutral position. In our OCE framework assuming = 0 eliminates all impact of uncertainty. The impact of an 9 increase in tax rate uncertainty on optimal labor supply and saving decisions can then be analyzed by increasing from a value of zero. Obviously the impact of an increase in will be zero if there were no uncertainty.3 In order to examine the impact of an increase in tax rate uncertainty we first consider its ˆ ˆ effect on the certainty equivalent after tax gross return on saving, R by differentiating R with ˆ respect to and evaluating the resulting derivative at = 0. It is through the changes in R that an increase in tax rate uncertainty affects labor supply and saving. ˆ The following lemma determines the impact of a tax rate uncertainty on R . Lemma 1. As tax rate uncertainty increases, the certainty equivalent after tax gross rate of ˆ return R declines. ˆ Proof: Log transformation of the expression of R in (15) yields ˆ log R 1 1 ~ log E R 1 Differentiating with respect to and simplifying one obtains 1 ˆ dR 1 ~1 ~ ~ 1 E R 1 log R log E R ~ (19) ˆ R d (1 ) 2 1 E R 1 Noting that ˆ ~ R E(R) 0 we get 3 In a similar vein Basu (1995) analyzes the effect of tax rate uncertainty on capital accumulation in an overlapping generation model with OCE preferences. 10 ˆ dR ~ ~ ~ ~ E ( R ) log E ( R ) E ( R log R ) (20) d 0 ~ ~ ~ ~ Now let g ( R ) be defined as g ( R ) = R log R . It is easy to check that, ~ 1 g R ~ R 0 ~ ~ i.e., g ( R ) is convex in R . By Jensen’s inequality6 ~ ~ ~ ~ E R log R E( R ) log E( R ) ˆ dR Hence 0 d 0 An important caveat is in order here regarding our method of analyzing the effect of uncertainty as stated in Lemma 1. We are analyzing he effect of uncertainty by increasing the risk aversion parameter from zero. This principle of analyzing the effect of uncertainty differs from the standard notion of a mean preserving spread used by Selden (1978) and others. These two approaches may not be necessarily equivalent. However, in the context of a rate of return uncertainty, it turns out that these approaches are indeed equivalent. In the appendix we prove that even if the increase in risk is characterized by a mean preserving spread of the distribution of ~ ˆ R , R declines. We can now readily determine the effect of an increase in the rate of return risk on individual's labor supply decision. From (17) and (18) it is straightforward to show, 3 Jensen’s inequality states that if F ( ) is a convex function, then E[F( )]>F [E ( )] for non-degenerate . The inequality is reversed if F is a concave function 11 (1 ) ( 1 h1 ) = (21) ˆ ˆ 1 W R 1 R + 1 + W h1 S 1 (1 h1 ) Consequently 0 according as 1. In view of (17) similar results are obtained for ˆ R optimal saving, S1.7 The results regarding the effects of uncertainty in the rate of return tax on an individual's labor supply and saving are summarized in the following proposition. Proposition 1. The direction of the effect of a rate of return tax rate uncertainty on the level of optimal labor supply and saving depends only on the elasticity of intertemporal substitution; if the elasticity of intertemporal substitution is large with < 1 (small with > 1 ) such post tax rate of return uncertainty reduces (an increase)in labor supply and saving.8 These results have clear intuitive explanations. Uncertainty in the rate of return tax lowers the certainty-equivalent rate of return to saving. A higher value of means a lower intertemporal substitution in consumption which induces agents to smooth utility by working harder as well as saving more. Although there is a countervailing substitution effect of a lower certainty equivalent return to savings inducing agents to save less and work less, for a higher value of , the utility smoothing effect outweighs the substitution effect.. As for savings our results are 6. The critical role of intertemporal elasticity substitution in determining the impact of an increase in risk in nonexpected utility models has been examined elsewhere. For example, Selden (1979), Weil (1990) and Basu and Ghosh (1993) highlight the role of intertemporal substitution elasticity in determining optimal saving decision under uncertainty. 12 consistent with Selden (1978). Selden, however, does not take into account the effect of post- tax rate of return uncertainty on labor supply. Direction of the comparative statics effect of uncertainty on labor supply is the same as for savings because in the present model labor supply is a device of consumption as well utility smoothing. By working harder the agents can generate more interest income in the second period. Savings and labor supply thus move in the same direction in response to a post-tax rate of return uncertainty. The results obtained here are based on a non-expected utility functional, which clearly disentangles risk aversion from intertemporal substitution. If =, the model reduces to an expected utility model and the optimal saving rule turns out to be the same as obtained by Levhari and Srinivasan (1969). The property of optimal labor supply then depends on the relative risk aversion parameter. Our results on labor supply resemble Alm (1988) who using a model where the consumer exclusively saves or works shows that labor supply increases if the third derivative of the utility function is positive. Kim, Snow and Warren (1995) find that given nonincreasing absolute risk aversion, if the coefficient of relative risk aversion is large (in excess of two) , Alm's result holds. Kim, Snow and Warren (1995) also show that even when the agent supplies both saving and labor supply , under the same restriction on the risk aversion parameter, labor supply increases in response to an increase in risk in the interest tax rate. It is not possible to reduce our OCE non- expected utility functional to Alm or Kim et al. as a special case. However, their results on labor supply are similar in spirit to the special case of expected utility here (where =). In the context of our model with specific functional form, we also find that when the risk aversion parameter is large (exceeding unity) we obtain similar results as Alms (1988) and Kim et al. (1995).9 9 The issue arises whether in the special =, we are picking up the risk aversion or intertemporal substitution effects. This is actually an empirical question. Kocherlakota (1990) makes a persuasive argument that if an econometrician fits an expected utility functional to the data where the "true" preference is nonexpected utility, he 13 However, the principal contribution of this paper is to depart from the expected utility framework by separating risk aversion from intertemporal substitution and explore the implications for labor supply in the presence of income tax uncertainty. Since intertemporal substitution is critical in determining the direction of the comparative statics effect as per Proposition 1, a natural question arises about the empirically plausible magnitude of . Although there is no clear consensus about this issue, several estimates (such as Hall (1988), Epstein and Zin (1993)) have pointed to to be well above 1 meaning a low elasticity of intertemporal substitution. Consequently, in view of the above proposition one may infer that an increase in rate of return tax uncertainty is likely to increase labor supply and saving.10 4. Income Tax Uncertainty In this section we turn to wage income tax uncertainty. We assume that R , the after tax gross return on saving is nonstochastic. As in the previous section we consider a two period model. In period 1 the individual works for (1-h1) hours for a non-stochastic wage rate W. Following Kim, Snow and Warren we, however, assume that the individual receives the entire wage compensation in period 2. In other words, 1 and that labor is viewed as a pure “credit good”.11 From (1) it follows that saving in period 1 amounts to S1 Y C1 (22) will actually be estimating risk aversion not intertemporal substitution. 7. Epstein and Zin have also found the relative risk aversion coefficient to be close to 1. Such a difference between the values of and also justifies the use of a nonexpected utility maximizing approach. 14 Furthermore, (3) reduces to ~ ~ Z (1 ) (23) ~ Consequently, in view of (2') the random consumption level in period 2, C 2 is given by, ~ ~ C2 S1 R W (1 h1 ) (24) ~ ~ where W W Z . Notice that the wage income tax uncertainty makes the consumption uncertainty additive in the second period as opposed to multiplicative in the case of rate of return tax uncertainty. With a constant relative risk aversion risk preference, it is difficult to obtain analytical result with additive uncertainty. Following Weil (1993) we characterize the individual’s attitude towards risk from now on by a Constant Absolute Risk Aversion (CARA) utility function..12.13 Thus s the function V(.) that captures the individual’s attitude towards risk is given by: e ~ ~ c2 V (C2 ) A e A > 0, (25) where 0 is the risk aversion parameter. Certainty equivalent level of period 2 consumption is such that ˆ 2 A e C E A e C ~ 2 (26) 8. Our comparative statics results remain unchanged even if 0 1. 9. Notice that unlike CRRA preferences, it is not possible to reduce this preference structure to an expected utility maximizing framework principally because the proportional risk aversion is no longer a constant here. It depends on the level of wealth. 10. Using the same CARA preference structure in an OCE framework, Basu (1996) analyzes the effect of a proportional tax on the validity of Ricardian Equivalence. 15 Using (24) and simplifying we get ˆ C 2 S1 R log E e W (1h ) ~ 1 Consequently, C 2 S1 R Q (27) where 1 ~ Q(h1 , ) log E e W (1h ) 1 (28) is the certainty equivalent wage income for a given labor supply. For future reference note that ~ ~ Q E (We W (1 h ) ) 1 Q1 ( ) ~ (29) h1 E (e W (1 h ) ) 1 The individual maximizes 1 C1 h1 1 1 C2 1 1 1 subject to (22), (27) and (28). As before 0 is the inverse of the elasticity of intertemporal substitution and 0 1 the utility discount factor. The first order conditions are now C1 R C2 (30) h1 C2 W1 (31) 16 where W1 Q1 Substituting (30) and (31) in (22) yields 1 1 S1 Y h1 R W 1 (32) Finally from log transformation of (30) we get 1 log h1 const. log C2 log W1 (33) As explained in the previous section, an increase in the income tax rate risk is captured through an increase in the risk aversion parameter. Thus, to examine the impact of an increase in income tax rate risk on labor supply we differentiate h1 with respect to and evaluate the resulting derivative at 0 . From (33) it can be shown that 1 1 h1 1 ~ 1 ~ (1 h1 )Var (W ) 1 R (1 h1 ) 2 Var (W ) 1 <0 (34) E ~ 0 W 2C 2 where 1 ~ EW 1 ~ 1 R E (W ) >0 (35) h 1 C2 C2 (details of the derivation of (34) and (35) are provided in the appendix). The impact of an increase in the risk associated with the wage income tax rate on labor supply (= (1-h1)) is 17 summarized in proposition 2. Proposition 2. If individual's preference is of a constant absolute risk aversion class, as wage income tax rate risk increases labor supply increases. An increase in wage income tax rate uncertainty induces the household to work harder to recoup the loss of the certainty-equivalent after tax wage income. This result holds irrespective of the magnitude of the elasticity of intertemporal substitution in consumption. Our result differs from Kim et al. (1995) who find that the effect of wage income tax uncertainty on labor supply depends on the curvature of the utility function. This difference in results is due to the specific functional form assumed here for the utility function. As for the impact on saving, first note that the saving, S1 and the certainty equivalent income, Q are inversely related. To see why, use (22), (27) and (30) to obtain the following closed form for the optimal consumption policy rule. 1 1 1 1 R Y R Q C1 1 1 (36) 1 R An increase in Q unambiguously raises the current consumption C1 because it raises the permanent income of the consumer. Saving, S1 decreases as the certainty-equivalent after tax income Q rises. Thus in order to characterize the effect of income tax rate on savings, it is crucially important to determine the comparative statics effect of an increase in wage income tax risk on Q. Now we differentiate Q with respect to and evaluate the derivative at 0 to characterize the effect of wage income tax rate risk on Q. 18 dQ h1 Q Q1 (37) d 0 0 0 Because of (29) and (34), the first term is positive. In the appendix (see equation A6), we have shown that the second term reduces to Q 1 h1 Var (W ) 1 2 ~ (38) 0 2 which is negative. An increase in the wage income tax rate uncertainty has thus two opposing effects on Q. For a given labor supply, it lowers the certainty-equivalent income. Such a decline reflects a pure “risk effect” of an increase in tax rate uncertainty, which is captured by the second term in (37). On the other hand, in order to smooth utility in response to increased income tax uncertainty the household works harder by cutting back leisure time. This tends to raise the certainty-equivalent income, Q. This countervailing effect which we call the “utility smoothing effect” is captured by the first term in (37). The net effect of wage income tax rate uncertainty on Q and therefore, on saving is ambiguous. Saving will rise if the former “risk effect” outweighs the latter “utility smoothing effect.” Relative strengths of these two opposing effects depends on the parameter values. To see clearly the role of the preference parameters in determining the comparative statics effect of tax rate uncertainty effect, substitute (29), (34) and (38) into (37) and simplify terms to obtain the following expression for dQ / d . ~ (1 h1 )Var (W ) 1 h1 1 1 1 dQ 1 R 2h (39) d 0 1 (Details of the derivation are shown in the appendix). 19 1 h1 1 Notice that the sign of dQ / d crucially depends on the sign of which in turn 2h1 depends critically on the size of the elasticity of substitution (1 / ) and initial proportion of work and leisure (1 h1 ) / h1 . For larger value of (i.e., for lower value of the intertemporal elasticity of substitution) the term in the square bracket is likely to be positive making saving respond positively to increased tax rate uncertainty. As discussed earlier there is no clear consensus about the exact magnitude of the elasticity of substitution. Epstein and Zin (1993) find that the elasticity of substitution in general is less than one. Their GMM estimations with nondurable consumption and services data tend to suggest that the elasticity of substitution hovers around .2. Hall (1988) also finds that the elasticity of substitution is around .1. A reasonable lower bound for (1 h1 ) / h1 may be around .5 which means approximately one third of the time is spent on work. Thus for empirically plausible values of the elasticity of substitution and the proposition of time spent on work, saving is likely to increase in response to increase in wage income tax uncertainty. 5. Effect of a Single Tax Rate Uncertainty We now consider a comprehensive income tax that is imposed on both wage and interest income. As in the previous section, a la` Kim et al we assume that labor is a credit good (with = 1) and that wage compensation is received in period 2. For notational convenience we define period 2’s income as M S1r W (1 h1 ) (40) where r is the non-stochastic rate of return on saving and S1 is defined in (22). M is subject to a 20 ~ comprehensive income tax where the random tax rate is given by ( 1). 14 Consequently, period 2’s consumption level is given by : ~ ~ C2 S1 (1 )M (41) As in the previous section we characterize the individual’s attitude toward risk by a CARA utility ˆ function represented by (25). Certainty equivalent consumption C 2 is now determined to be ˆ C 2 S1 Q * (42) where Q* 1 log E e (1 ) M ~ is certainty equivalent net income. For future reference note the following properties of Q* Qs * Q * rE 1 e (1 ) M ~ ~ S1 ~ E e (1 ) M (43) Qh * Q * WE 1 e (1 ) M ~ ~ h1 ~ E e (1 ) M (44) First order conditions for the individual’s optimizing problem now reduce to C1 ˆ C 2 1 Q S * (45) h1 ˆ C2 W2 * (46) where W2 * Qh * . 14 We did not allow the first period income to be affected by this comprehensive tax rate. The reason is that the all the uncertainty in tax rate is revealed in the second period not in the first period, which is consistent with the OCE nonexpected utility framework. The key feature of the OCE framework is that the uncertainty unfolds in the second period not in the first period. That is why, Selden (1978) calls a prospect "certain x uncertain" meaning it is certain in the first period and uncertain in the second period. 21 Substituting (45) and (46) in (22) we obtain W 1 1 * S1 Y h1 1 QS * 2 (47) As before, the effect of uncertainty regarding the tax rate is captured by increasing the risk aversion parameter from the level =0. Using (45) and (46) and applying similar procedures as in section 4, one obtains: ~ 1 ~ dh1 1 h1 W [1 E ] ~ ) (1 r[1 E ]) WM Var 1 M 2 Var ( ) ~ ~ d * C 2 ~ rM Var ( 1 r[1 E ] ˆ ~ W [1 E ] ˆ 2C 2 (48) where 1 1 1 r ~ ~ W [1 E ] W [1 E ] * 0 (49) h1 Cˆ Cˆ 2 2 ^ ~ and C 2 = S1 (1 E ) M . (details of the derivation of (48) and (49) are relegated to the appendix). h1 In general, the sign of as obtained above is ambiguous. The ambiguity arises here because an increase lowers the certainty equivalent after tax rate of return on savings giving rise to opposing income and intermporal substitution effects on labor supply as seen in section 3. We next show that for sufficiently large value of , both labor supply and saving increase in response to tax rate uncertainty. To see this use (48) to note that as 1 ~ h1 W [1 E ] dh1 ~ 0. Consequently, 0. As for the impact of uncertainty on ˆ C 2 1 r[1 E ] d saving note from (49) 22 1 1 dS1 dh * 1 (1 QS ) (W2 ) d d 1 * Q S * 1 h1 1 QS * W2 (50) 1 W2 1 * h * 1 1 Q S W2 * Evaluating the expression at 0 and using ( ), ( ) and ( ) we obtain dh ~ 1 1 r 1 E W 1 E dS1 ~ 1 1 d d h 1 ~ 1 1 r 1 E W 1 E rM Var 1 ~ ~ (51) h 1 ~ 1 1 r 1 E W 1 E WM Var 1 ~ ~ dS1 dh As , the second and third terms on the right hand side approach zero and 1. d d These results are summarized in proposition 3. Proposition 3. If individuals have a constant absolute risk aversion preference and if indicating a very low elasticity of intertemporal substitution, a comprehensive income tax rate uncertainty raises labor supply as well as savings. Although no direct analog to Kim et al. (1995) can be established because of different functional form specification for the utility function, our results are similar to Kim et al. in regard to savings. Kim et al. finds that for a large risk aversion parameter (exceeding 2) saving responds positively to a comprehenstive tax rate uncertainty. In our framework with non- expected utility functional, which dissociates risk aversion from intertemporal substitution, we find that it is the intertemporal substitution parameter, that has to be sufficiently large to make saving increase in response to a comprehensive tax rate uncertainty. 5. Conclusion 23 In this paper, we analyze the effect of tax rate uncertainty on labor supply and saving when the agent both works and saves. Previous theoretical studies highlight the importance of risk aversion in determining the labor supply and saving of the individual in response to a greater tax rate uncertainty. We use the OCE framework which enables us to distinguish between the agent's attitude toward risk and his utility smoothing motive. We examine the effects of rate of return tax rate uncertainty as well as the wage income tax rate uncertainty on labor supply and saving. A higher uncertainty in the rate of return tax rate raises labor supply and saving if the elasticity of intertemporal substitution is less than unity. On the other hand, a higher wage income tax rate uncertainty unambiguously increases labor supply because of the agent’s drive to recoup loss of certainty equivalent after tax wage income by working harder. The effect on saving again depends critically on the size of the elasticity of substitution. The effect of comprehensive tax rate uncertainty on labor supply and saving again depends on the size of the elasticity of intertemporal substitution. These theoretical results highlight the role of intertemporal substitution elasticity as a decisive factor in determining the effect of income tax rate uncertainty on labor supply and savings. For empirically plausible size of the intertemporal substitution elasticity, our theoretical analysis suggests that saving and labor supply would respond positively to higher income tax rate uncertainty. Appendix Result. For an increase in tax rate uncertainty that is characterized by a mean preserving spread of ~ , the certainty equivalent after tax gross rate of return, R declines. t ˆ Proof: The certainty equivalent after tax gross rate of return on saving is defined as: = EQ(~) 1 1 ˆ R = E R1- ~ 1- t 1- 24 where Q = R1- . Noting that Q// (~) = - r 2 (1 - ) R -(1+ ) it can be verified that Q is a concave ~ t ~ (convex) function of ~ if < 1(> 1) . Using Jensen's inequality it can be shown that with t increase in risk EQ(~) falls when < 1 , but rises when > 1. It immediately follows that with t ˆ increase in risk, R declines. Derivation of (34), (35) and (38). Differentiating (33) with respect to we get 1 h1 ˆ 1 dC 2 1 dW1 (A1) ˆ h1 C 2 d W1 d Since W1 = - Q1, from (29) it is clear that ~ W1 0 E (W ) (A2) Also note that dW1 ~ ~ ~ (1 h1 ) E (W 2 ) (1 h1 )[ E (W )]2 (1 h1 )Var (W ) (A3) d 0 Now, rewrite (28) as 1 Q log Z where Z E e W (1h ) 1 (A4) 25 Consequently, Q 1 log Z Q2 2 log Z (A5) 0 But Q2 0 0 Applying L’Hpital’s rule to (A5) one can evaluate the limit of Q2 as 2 log Z LimQ2 Lim 2 0 0 2 Lim ~ ~ 1 ~ (1 h1 ) 2 E[(W 2 )e W (1 h ] E[e W (1 h ] {E[We W (1 h ) ]}2 1 ~ ~ 1 0 E[e ~ W (1 h1 ) 2 ] 2 (A6) 1 ~ (1 h1 ) 2 Var (W ) 2 This also proves (38). Now, dQ h Q1 1 Q2 (A7) d Since Q1 = -W1, using (A3) and (A6) we get dQ ~ h 1 ~ E (W ) 1 (1 h1 ) 2 Var (W ) (A8) d 0 2 From (27) and (33), using (A2) and (A8) one obtains ˆ dC 2 1 1 ~ 1 ~ 1 ~ h 1 ~ h 1 ~ h1 R ( EW ) (1 h1 )Var (W ) R ( EW ) 1 E (W ) 1 (1 h1 ) 2 Var (W ) d 0 2 (A9) Substituting (A3) and (A9) in (A1) we get 26 1 ~ ~ E (W ) 1 1 1 h1 1 1 1 ~ ~ R ( EW ) h1 R ( EW ) (1 h1 )Var (W ) ˆ h1 C 2 ˆ C2 ˆ C 2 (A10) 1 ~ 1 ~ ~ (1 h1 )Var (W ) ˆ (1 h1 ) Var (W ) 2 ( EW ) 2C 2 Using the first order condition (31), the first two terms on the right hand side of (A10) can be h1 further simplified to yield as in (34) where (as defined in (35)) is given by the square 0 ˆ bracketed term on the left hand side of (A10). Note that C 2 is also evaluated at 0 , which using L’Hpital’s rule can be determined to be ˆ ~ C2 S1 R (1 h1 ) E (W ) 0 Derivation of (39) Substituting (34) in (A8) and simplifying we get 1 ~ 1 1 ~ (1 h1 )Var (W ) 1 (1 h1 ) (1 h1 ) R ( EW ) 1 dQ 1 R 2 h (A11) d ˆ 0 1 2C 2 h Substituting for 1 from the first order condition (31) we can further simplify (A11) to yield (39). Cˆ 2 27 Derivation of (48) Differentiate (48) with respect to to get 1 dh1 ˆ 1 dC 2 1 dW2 * (A.12) ˆ d W2 * d h1 d C 2 Also, from (42) one obtains ˆ dC 2 dS1 * dS1 * dh1 QS Qh Q * (A.13) d d d d Q * where Q * . Substituting (47) in (A.13) and simplifying we get h 1 ˆ 1 QS (W 1 * 1 1 dC 2 dh1 * 1 QS (1 QS ) 1 QS * * * * W2 d d 2 W2 1 Q W 1 1 * h1 * * dh1 W2 Q * * d S 2 (A.14) Using (43) and (44) next evaluate Qs * and Qh * at 0 to get Qs * 0 r 1 E 0 ~ (A.15) Qh * 0 W 1 E 0 ~ (A.16) Using (43) and (44) we can also calculate 28 QS * ~ rM Var ( ) (A.17) 0 W2 Q * * ~ h WM Var ( ) (A.18) Also, note that in view of (43) 0 * Q 0 0 Using L’Hôpital’s rule and following the same procedure as outlined in (A6) we obtain 1 ~ M 2 Var ( ) * Q (A.19) 0 2 Now, substituting ( ), ( ) and ( ) in ( ) and using ( ) one obtains 1 1 * dh1 1 (1 QS ) (W2 ) * * W2 d h1 ˆ C2 C2 ˆ W W 1 1 QS W2 1 1 * * * h h Q 1 1 QS 1 1 QS * * * * Cˆ 2 ˆ C 2 2 ˆ C2 2 Further simplification yields (48). References Alm, James. (1988). "Uncertain Tax Policies, Individual Behavior, and Welfare," American Economic Review, 78, 237-245. Altonji, Joseph G. (1982). 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