Income uncertainty and optimal redistribution.

1. Introduction

Redistribution policy under a utilitarian criterion aims to transfer dollars from one group in the population to another if the (weighted) utility gain to the recipients exceeds the utility cost to the donors. Under diminishing marginal utility of income, part of any exogenous change in one group's income should be shared with the other group. For example, if taxpaying workers suffer a negative shock to their mean earnings, then part of this utility shock should be spread to nonworkers in the form of lower transfer payments, and vice versa for the case of a positive shock.

This article demonstrates the counterintuitive result that, under reasonable preference representations, a policymaker may respond optimally to a negative utility shock by redistributing income away from the harmed group, thus exacerbating the group's utility loss. In particular, under risk aversion, a mean-preserving increase in the variance of income typically has an effect on well being analogous to a decrease in mean income. (1) It may nevertheless be welfare enhancing to transfer income away from those suffering an increase in income uncertainty even though this policy compounds their welfare loss.

The main ideas can be demonstrated in a simple framework. For illustration, the analysis will focus on redistribution from taxpaying workers to nonworkers. The central result is that a mean- preserving increase in earnings uncertainty among taxpayers can increase optimal transfer amounts to nonworkers (increasing optimal income replacement rates) even though the redistribution exacerbates the taxpayers' utility loss from the increase in uncertainty.

2. Model

Consider a population of n individuals comprised of [n.sub.1] nonworkers receiving transfers T and [n.sub.2] = n - [n.sub.1] taxpaying workers earning random before-tax wage earnings x [member of] [[x.sup.-], [x.sup.+]]. Let the cumulative distribution function of x be given by F(*, [sigma]), assumed to be continuously differentiable with respect to x and the uncertainty parameter [sigma]. The mean of x is denoted [mu]. A worker's random consumption level is denoted C = (1 - t)x, where t is a proportional payroll tax. Labor hours are assumed fixed in the theoretical framework, but this restriction is relaxed in a numerical example below. (2)

Let social welfare be given by a weighted utilitarian function,

[psi] = [phi][n.sub.1]U(T) + (1 - [phi])[n.sub.2]E[U(C)],

where U(T) is the utility of transfer recipients, E[U(C)] is the expected utility of taxpayers, and [phi] and 1 - [phi] are the social weights. Utility is assumed to be increasing in consumption at a diminishing rate, implying risk aversion.

The government's objective is to choose an eamings tax t and transfer amount T that maximizes [psi] such that its budget is balanced on average: [n.sub.1]T = [n.sub.2]t[mu] - G, where G is the government's required revenue for uses other than redistribution. (3) The government solves

max T, t [phi] = [phi][n.sub.1]U(T) + (1 - [phi])[n.sub.2]EU[(1 - t)x] - [lambda]([n.sub.1]T - [n.sub.2]t[mu] + G), (1)

where [lambda] is the shadow value of decreasing the government's revenue requirement by a dollar. Differentiating (1) separately with respect to T and t and combining results yields the necessary condition for an optimum,

[phi]U'([T.sup.*]) = (1 - [phi])E[(x/[mu])U'([C.sup.*])]. (2)

Of interest is the effect of an increase in earnings risk on the optimal size of the transfer amount. The Rothschild--Stiglitz (1971) definition of a mean-preserving increase in uncertainty is used in this analysis, a definition more general than the often-employed mean-variance definition. (4) Let p = -U"/U' and r = Cp denote the usual measures of absolute and relative risk aversion (ARA and RRA), respectively. Then differentiating both sides of Equation (2) with respect to the uncertainty parameter [sigma] yields the following proposition (see Appendix for the proof).

PROPOSITION 1. The optimal transfer amount [T.sup.*] rises with earnings uncertainty [sigma] if and only if [partial][rho]/[partial]C > [[rho].sup.2][(r - 2)/r], or equivalently, if and only if r < 2 + (C/[rho])([partial][rho]/[partial]C).

The optimal income replacement rate for nonworkers therefore rises with earnings uncertainty if taxpayers' absolute risk aversion declines sufficiently slowly (or rises) with consumption. For sufficiently slowly declining ARA, taxpayers' expected marginal utility of consumption falls with their degree of earnings uncertainty. With lower expected marginal utility, a dollar becomes less valuable to a taxpayer than prior to the increase in uncertainty, thus reducing the cost of redistribution. (5) Note that decreasing absolute risk aversion (DARA) requires that r < 2 for transfers to rise with earnings uncertainty. (6) In utility notation, the inequality in the proposition can be expressed as U"' < - 2U"/C. Although the optimal transfer amount and replacement rate depend on the social weights [phi] and 1 - [phi], the condition for when they rise with earnings uncertainty does not.

Examples

The proposition is illustrated with three examples. One of the most common preference representations featured in applied work is quadratic utility (e.g., Gale and Scholz 1994).

Example 1: Quadratic (Mean-Variance) Utility: U(C) [alpha]C - [C.sup.2], C < [alpha]/2. With quadratic preferences, the optimal transfer level always rises with the degree of earnings risk because U" < 0 and U"' = 0. (7) Although popular, the quadratic specification is often criticized because it implies rising absolute risk aversion and rules out "prudence" U"' > 0 (see Kimball 1990). The second example examines the case of constant absolute risk aversion.

Example 2: Constant Absolute Risk Aversion: U(C) - exp(- [rho]C). Because [partial][rho]/[partial]C = 0 in this case, transfers increase with earnings uncertainty if and only if relative risk aversion is less than two. This example provides a boundary case for when transfers can rise with earnings risk under DARA.

The third example considers another popular specification in which absolute risk aversion declines with consumption. Numerical calculations were conducted to allow for endogenous labor hours, L, which a worker chooses optimally after learning the realization of the random after-tax wage rate. The policymaker anticipates the taxpayers' labor supply behavior when setting optimal transfer policy. The following specification allows for a variety of plausible shapes for the labor supply function (Barzel and McDonald, 1973; Cowell 1981).

Example 3: U(C, L) = 1/[delta](1 - [gamma])[[(C + k)[(1 - L).sup.[delta]]].sup.1-[gamma]] for [delta], [gamma] > 0 with C (1 - t)wL. Absolute risk aversion always falls with consumption, while relative risk aversion is falling/constant/rising with consumption depending on whether k (which can be thought of as either a utility parameter or nonstochastic unearned income) is negative/zero/positive. A worker's optimal number of labor hours and consumption are given by

[L.sup.*] = 1/1 + [delta] [ 1 - [delta]k/(1 - t)w] and [C.sup.*] = 1/1 + [delta] [(1 - t)w - [delta]k],

respectively (L = 0 for nonworkers). Considering first the special case where k = 0 (constant relative risk aversion with locally inelastic labor hours), it is easy to show analytically that C[U.sub.ccc] + 2[U.sub.cc] has the same sign as 1 - [gamma]. The optimal transfer amount thus rises with earnings uncertainty if and only if [gamma] < 1, corresponding to relative risk aversion less than one. Log utility represents the limiting case [gamma] = 1, where transfers are invariant to the degree of earnings uncertainty.

Numerically calculated critical values for other combinations of parameters in which transfers are invariant to earnings uncertainty are plotted in Figure 1 when the fraction of nonworkers is 0.1. A worker faces a wage rate [w.sub.0] = 1 under certainty and a wage rate of either [w.sub.1] = 0.5 or [w.sub.2] = 1.5 with equal probability in the uncertainty case. Combinations of k and [gamma] on a [delta]-curve represent critical values where the optimal transfer level is the same for the certain and uncertain earnings cases. Values of k and [gamma] below a [delta]-curve represent combinations where transfers rise under uncertainty, while values above the curve represent combinations where transfers fall. For instance, at the combination {[gamma] = 1.2, [delta] = 1, k = 0.15}, with social weight [phi] = 0.25, it can be shown that the optimal transfer amount rises 7%, from 0.199 to 0.214, when earnings become uncertain. The optimal tax rate rises from 0.052 to 0.056, and the optimal replacement rate rises from 0. 50 to 0.54. Holding constant t and T at their optimal levels under the certainty case, the introduction of uncertainty reduces workers' expected utility from -6.288 to -6.385. Although the workers have been harmed by the uncertainty while the nonworkers have not, the optimal policy response requires the policymaker to further harm the workers: the higher replacement rate after uncertainty further reduces workers' expected utility from -6.385 to -6.389 while increasing nonworkers' utility from -6.907 to -6.808. (8)

Although there is a strong consensus in the literature that absolute risk aversion typically declines in consumption (see, e.g., Levy 1994), there is no consensus about whether relative risk aversion is likely to be less than or greater than one or about whether it is likely to be increasing or decreasing in consumption. Many studies (e.g., Stiglitz 1970; Rothschild and Stiglitz 1971) state their results in terms of whether r is less than, equal to, or greater than one. Arrow (1971, p. 98) hypothesized that "broadly speaking, the relative risk aversion must hover around one" and shows (p. 111) that if utility is bounded from below, then r approaches a value less than one as consumption goes to zero. Thus, he shows that if utility is bounded from below and r is everywhere constant or decreasing in consumption, then r is everywhere less than one. (9)

There is a wide range of estimated values for relative risk aversion. While many studies in the literature estimate 1 < r < 2 or r > 2, others estimate r < 1. In the former categories, Weber (1975) estimates values of r between 1.3 and 1.8, with Szpiro (1986) similarly estimating values between 1.2 and 1.8. Weber (1970), Friend and Blume (1975), Davies (1981), Summers (1982), Barsky et al. (1997), and Hall (1988) suggest that average r exceeds two, with Friedman (1973) estimating a value close to 10. In contrast, Otrok (2001) recently estimates r to be 0.72 with a 90% confidence interval of [0.54, 0.95]. (10)

Schluter and Mount (1976) report estimates from 0.05 to 3, with an average value of 0.8. Hansen and Singleton's (1982) estimates fall between 0.35 and 1. Five out of six of Hansen and Singleton's (1983) estimates, ranging from 0.17 to 1.36, are also smaller than one. Chang (1990) and Rust and Phelan (1997) estimate r to be very close to 1, at 1.03 and 1.07, respectively. Using a Bayesian approach, Harrison (1990) estimates r to be 0.55. (11)

3. Conclusion

This analysis has shown that a policymaker may respond optimally to an exogenous shock in the economy by redistributing income away from those directly harmed by the shock. In particular, it may be welfare enhancing to redistribute income away from risk-averse taxpayers who suffer an increase in the variance of their earnings. From an efficiency perspective, the direction of redistribution following an increase in uncertainty depends on the degree to which absolute risk aversion declines with consumption. For populations or subpopulations in which absolute risk aversion declines sufficiently slowly with consumption, transfers should flow away from those facing the most uncertainty (all else equal). Such taxpayers' expected marginal utilities of income fall with the amount of uncertainty, thus reducing their welfare cost of parting with a dollar. The argument is distinct from Eaton and Rosen's (1980a, b) conclusion that the taxation of variable wages can improve welfare by reducing the amount of uncertainty in the economy; the present argument holds even under a system of pure lump sum redistribution with no wage tax component. (12) As a natural extension of this analysis, the efficient distribution of program benefits may depend on the variances as well as the means of recipients' incomes.

Appendix

To prove the result in Proposition 1, rewrite condition (2) as

[phi]U'(T) = (1 - [phi]) [[integral].sup.[x.sup.+].sub.x] x/[mu]U'(C) dF.

Differentiating both sides with respect to [sigma] implies

[phi]U"(T)[partial]T/[partial][sigma] = (1 - [phi])[[[integral].sup.[x.sup.+].sub.x] x/[mu]U'(C) d[F.sub.[sigma]] - [partial]t/[partial][sigma]E([x.sup.2]/[mu]U"(C))]

or

sgn([partial]T/[partial][sigma]) = sgn[(1 - [phi] [[integral].sup.[x.sup.+].sub.x] x/[mu]U'(C) d[F.sub.[sigma]]/[phi]U"(T) + (1 - [phi])[n.sub.1]/[n.sub.2]E([x.sup.2]/[[mu].sup.2]U"(C))] = -sgn [[integral].sup.[x.sup.+].sub.x] xU'(C) d[F.sub.[sigma]] (A1)

using [partial]t/[partial][sigma] = ([n.sub.1]/[n.sub.2])(1/[mu]([partial]T/[partial][sigma] from the government's revenue constraint. Twice integrating by parts obtains

[[integral].sup.[x.sup.+].sub.x] xU'(C) d[F.sub.[sigma] = - [[integral].sup.[x.sup.+].sub.x] [CU"(C) + U'(C)][F.sub.[sigma]]dx = (1 - t) [[integral].sup.[x.sup.+].sub.x] [CU'"(C) + 2U"(C)][phi](x)dx

using the Rothschild-Stiglitz conditions [F.sub.[sigma]]([x.sup.-], [sigma]) = [F.sub.[sigma]]([x.sup.+], [sigma]) = 0 for the first equality and [[integral].sup.[x.sup.+].sub.x] [F.sub.[sigma]](x, [sigma]) dx = 0 for the second equality (see also footnote 4). At both steps, the integrand is differentiated with respect to the random variable x. The function [phi](x) [equivalent to] [[integral].sup.x.sub.x] [F.sub.[sigma]]([zeta], [sigma]) d[zeta] is positive using the Rothschild-Stiglitz condition that [[integral].sup.a.sub.x] [F.sub.[sigma]](x, [sigma]) dx [greater than or equal to] 0 for all a [member of] [[x.sup.-], [x.sup.+]]. Hence, sgn([partial]T/[partial][sigma]) = - sgn[CU"(C) + 2U"(C))]. Using [partial][rho]/[partial]C = -(U'"U' - [U".sup.2])/[U'.sup.2] and substituting the resulting value of U" into the prior result obtains sgn([partial]T/[partial][sigma]) = sgn([partial][rho]/[partial]C - [[rho].sup.2](r - 2)/r).

[FIGURE 1 OMITTED]

Received August 2001; accepted February 2002.

(1.) Many factors in the economy affect earnings uncertainty even conditional on being employed; such as the form of compensation (e.g., salary vs. commissions vs. self-employment), health status, uncertain promotions, or the degree of unpredictability about real wages during times of inflation. A large literature has studied the implications of such wage rate uncertainty for work behavior and optimal taxation. See, e.g., Block and Heineke (1973), Tressler and Menezes (1980), Eaton and Rosen (1980a, b), de Meza (1984), Hamilton (1987), Dardanoni (1988), Grossberg (1989), Ormiston and Schlee (1994), Cremer and Gahvari (1995), Kreider (1998), and Hartwick (2000).

(2.) Numerical testing showed that the main conclusion is robust to the endogeneity of labor supply and the inclusion of a lump sum transfer from workers to nonworkers. It does not appear possible to derive clean theoreticat results when allowing for endogenous labor hours because that adds numerous extra terms in the comparative statics (even after applying the Envelope Theorem), most of which cannot be reasonably signed. For example, there is no consensus in the literature about the sign of the first derivative of labor hours with respect to [sigma] (even for homothetic indifference curves; see Hartwick 2000) let alone for the required higher order derivatives and cross-derivatives with other parameters. The main result also continues to hold when the fraction of taxpayers [n.sub.2]/n is endogenously determined as a function of the relative value of working. Note that most public insurance programs limit the degree of moral hazard by imposing eligibility restrictions that are outside the scope of this model (e.g., work history requirements, time limits on eligibility, job search requirements, etc.).

(3.) Alternately. tax receipts can be considered certain if wage risk is independent across a large number of workers.

(4.) They define the random variable X to be more uncertain than the random variable Z if and only if their respective densities have the same means but there is more weight in the tails of the density of X. They show formally and graphically that an increase in [sigma] is a Rothsclsild--Stiglitz mean-preserving increase in risk if and only if [[integral].sup.[x.sup.+].sub.[x.sup.-]] [F.sub.[sigma]](x, [sigma]) dx = 0 and [[integral].sup.a.sub.[x.sup.-]] [F.sub.[sigma]](x, [sigma]) dx [greater than or equal to] 0

(5.) Consider a strictly concave utility function. The average of the slopes (marginal utilities) at [C.sub.0] - 1 and [C.sub.0] + 1 may be lower than the slope at [C.sub.0]. Moreover, these slopes are weighted by x/[micro] in Equation (2) with the weighted average of the slopes lower than she unweighted average.

(6.) I thank an anonymous referee for pointing Out this result, which led so the second inequality in the proposition.

(7.) For this specification, it is easy to directly derive the relationship between optimal transfers and the variance of x (where the variance is a special case of the Rothschild-Stiglitz uncertainty measure). The value of U'x in Equation (2) is given by [alpha]x - 2(1 - t)[x.sup.2] so that E[U'x) = [alpha][mu] - 2(1 - t)([[sigma].sup.2] + [[mu].sup.2], where [[sigma].sup.2] is the variance of x. Then differentiating both sides of (2) for this case with respect to [[sigma].sup.2] and using [partial]t/[partial][[sigma].sup.2] = ([n.sub.1]/[n.sub.2])(1/[mu])([partial]T/[partial][[sigma].sup.2]) from the government's revenue constraint implies

[partia]T/[partial][[sigma].sup.2] = 2(1 - pphi])[n.sub.2](1 - t)[mu]/-[phi][[sigma].sub.2]U"(T)[[mu].sup.2] + 2(1 - [phi])[n.sub.1]([[sigma].sup.2] + [[mu].sup.2] > 0.

(8.) In the certainty case, labor hours equal 0.42 with an uncompensated elasticity of 0.19. In the uncertainty ease, labor hours adjust to 0.34 when the wage is low and 0.45 when the wage is high. The effect of uncertainty on the optimal transfer amount is very similar when labor hours are chosen before the resolution of the uncertainty.

(9.) Theoretical studies that assume relative risk aversion is less than one include Lucas (1972), Azariadis (1978), Kihlstrom and Laffont (1979), Grossman (1981), and Newbery and Stiglitz (1982); those assuming that relative risk aversion exceeds one include Champernowne (1969) and Eaton (1981).

(10.) Because Otrok (2001) incorporates habit persistence in this model, his baseline estimate is not a clean estimate of relative risk aversion. However, he finds evidence of only weak habit persistence, and his estimate of r is little affected by the inclusion of such persistence. In sensitivity analysis, seven out of eight of his alternative estimates remain less than one, with the exception being an estimate of 1.03 (nearly log utility).

(11.) In intertemporal asset pricing models, the coefficient of relative risk aversion is equal to the reciprocal of the elasticity of intertemporal substitution (EIS) for homothetic time and state separable preferences. These studies often estimate r < i (e.g., Schluter and Mount 1976). Kocherlakota (1990) argues that such estimates should indeed be interpreted as estimates of the truer r and not as estimates of the EIS. Ermini (1991) and others have argued that the estimated value of r in consumptionbased asset-pricing models tends to be overestimated due to inappropriate assumptions about the consumption-generating mechanism. Gabaix and Laibson (2001) argue that estimates of r generated from consumption Euler equations are biased upward, perhaps many-fold, because they do not account for decision lags.

(12.) It can be shown that the set of cases in which transfers rise with uncertainty under lump sum taxation is a subset of the cases in which they rise under a wage tax.

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Brent Kreider *

* Department of Economics, Iowa State University, Ames, IA 50011-1070, USA; E-mail [e-mail].

I have benefitted from conversations with Jim Andreoni, Joydeep Bhattacharya, Steve Coate, David Cutler, Maxim Engers, Ed Glaeser, Jon Gruber, Jonathan Hamilton, Tom Nechyba, Ed Olsen, Chris Otrok, Holger Sieg, Jon Skinner, Koleman Strumpf, Alex Tabarrok, and three anonymous referees. I also thank seminar participants at Iowa State, the University of Virginia, and the University of Wisconsin for helpful comments. The Bankard Fund for Political Economy provided generous financial support.