Optimal taxation with deferred compensation.

Warren, Ronald S., Jr.

I. Introduction

The traditional theory of optimal taxation assumes that wage and interest incomes are received and taxed at different times. For example, Atkinson and Sandmo [2] and King [9] derived formulae for optimal wage and interest tax rates in the standard two-period life-cycle model in which a representative consumer/taxpayer receives wage income in the first (working) period and saves out of this post-tax wage income. The net return from saving accrues and is taxed as interest income in the second (retirement) period.

In practice, however, a substantial portion of present wage and salary compensation is paid in the form of (expected) future pension benefits and is therefore tax-deferred. Table I presents data on total civilian wages and salaries and various tax-deferred pension contributions taken from individual income tax returns for 1988 in the United States. These data show that approximately 12 percent of wage income is comprised of tax-deferred pension contributions.

Table 1. Tax-Deferred Pension Contributions in 1988 (in millions
of dollars)

Wages and Salaries 2,337,984(a)
Tax-Deferred Pension Contributions 276,756
 Employer contributions to OASI(b) 120,813
 Employer contributions to private
 plans(c) 75,185
 Employer and government contributions to
 railroad retirement(d) 3,099
 Employer contributions to federal, state,
 and local government pension funds(d) 59,152
 Employee contributions to IRA and Keough
 plans(a) 18,509

Sources:

(a.) U.S. Department of Treasury, Internal Revenue Service, Individual Income Tax Returns 1988, Publication 1304, September 1991, Table A.

(b.) U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States 1991, Table 589, p. 361.

(c.) U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States 1990, Table 677, p. 413, and Table 696, p. 429. Calculated as the ratio of employer costs for pensions to wages and salaries per hour (0.38/10.02) times total, private-industry wages and salaries (1,982,500).

(d.) Statistical Abstract, Table 590, p. 361.

In this paper we derive rules for optimal taxation in an alternative two-period setting in which not only interest income but also a part of wage income is deferred. As in the standard model, we assume that interest income accrues in the second period because investors bear temporal risk, so that the return to saving is received only after the resolution of uncertainty about the productivity of capital. However, we assume that incentive contracting in the form of deferred compensation is required to deal with moral hazard and adverse selection in the labor market. For example, Lazear [10] argued that deferred compensation can mitigate the moral hazard problem arising from costly monitoring of the effort and productivity of workers. Salop and Salop [17] showed that pension-type arrangements may act as a sorting device which reduces adverse selection costs when there is asymmetric information about worker productivity.(1) These imperfections dictate that some portion of the return to labor supply be deferred until the verifiable results of work effort have been observed.

When taxes are incorporated into this framework, the returns from a portion of labor supply and all of saving are received and taxed contemporaneously. As a result, taxation distorts different margins of choice in our model and may lead to a substantially different optimal tax structure. We calculate optimal tax rates for a golden-rule economy in which government debt policy maintains the steady-state capital-labor ratio. For a plausible set of values for the compensated elasticities of consumption and labor supply, and realistic assumptions about the government's revenue requirement and the fraction of wage income that is tax-deferred, we find that optimal tax rates on interest income and on non-deferred wage income are not very sensitive to the presence or absence of deferred compensation. However, the deferred compensation is optimally taxed at a rate substantially above the rate on non-deferred wage income.

In the following section we set out a partial equilibrium, two-period model of consumer behavior and present the marginal conditions for utility-maximizing consumption and labor supply. In section III we derive expressions for optimal, golden-rule tax rates on deferred and non-deferred wages and on interest income that maximize the welfare of a representative individual subject to the government's budget constraint. Section IV contains a comparison of our formulae for optimal taxation with those of Atkinson and Sandmo [2] and King [9]. In section V, we use empirically relevant values for the compensated own-price elasticities of consumption and labor supply, the government's revenue requirement, and the proportion of wage income that is tax-deferred to illustrate the implications of our approach for the calculation of optimal tax rates along a golden-rule path. Section VI summarizes our principal results and provides concluding remarks.

II. The Consumer's Problem

We assume that individuals are identical and live for two periods. In the ith period (i = 1, 2), the individual consumes [c.sub.i] units of a composite consumption good, which serves as numeraire, and [l.sub.i] units of leisure (non-market) time. The consumer is endowed with M units of the numeraire good in period 1 and T units of time in each period. The individual is assumed to be retired in the second period so that [l.sub.2] = T.

A proportion a of the return to supplying labor during the first period is received during that period and can be saved or consumed. A share 1 - [Alpha] of the return to first-period labor is deferred compensation that is received in the second period, along with the gross return to first-period saving.2 Thus, the present-value budget constraint facing a representative consumer is

(1) [c.sub.1] + [pc.sub.2] + [wl.sub.1] = M + wT,

where

(2) p = 1/[1 + r(1 - [t.sub.r])]

is the price of second-period consumption, r is the one-period interest rate, [t.sub.r] is the tax rate on interest income, and

(3) w = [[Alpha](1 - [t.sub.w]) + (1 - [Alpha])(1 - [??.sub.w])p]w

is the after-tax wage rate as a function of the before-tax wage rate w and the tax rates [??.sub.w] on non-deferred wages and [t.sub.w] on deferred wages.

We assume that the consumer's utility function is continuous, increasing, and concave, and that utility depends separably on the level of government spending g, which in each period is a constant amount per person. The consumer's utility function can thus be written U([c.sub.1], [l.sub.1], [c.sub.2], T), as if it were independent of government spending.

The government finances the exogenous expenditure level g through contemporaneous taxes on interest and wage incomes. The government's budget constraint is

(4) g = [Alpha] [t.sub.w] wh + [(1 - [Alpha]) [??.sub.w]wh + [t.sub.r]rs]/(1 + n),

where n is the fixed rate of population growth, h = (T - l.sub.1]) denotes labor supplied in the first period, and

(5) s = [pc.sub.2] - (1 - [Alpha])(1 - [t.sub.w]) pwh

denotes first-period saving. By substituting for s from (5) into (4), the government's budget constraint can be written

(6) g = [Theta] wh + [t.sub.r] [rpc.sub.2]/(l + n),

where

(7) [Theta] = [Alpha] [t.sub.w] + (1 - [Alpha])[[??.sub.w] - [t.sub.r] rp (1 - [??.sub.w])]/(1 + n)

is the effective average tax rate on wage income.

The consumer maximizes utility by choosing a consumption plan ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) that meets the individual's budget constraint (1) and satisfies the first-order conditions

(8) [U.sub.1] - (1/p) [U.sub.2] = 0

(9) [U.sub.l] - (w/p) [U.sub.2] = 0,

where [U.sub.1] = [Delta] U/[Delta] [c.sub.1], [U.sub.2] = [Delta] U/[Delta] [c.sub.2], and [U.sub.l] = [Delta] U/[Delta] [l.sub.1]. The demands ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) are assumed to be differentiable functions of the government's choice variables, [t.sub.r], [t.sub.w], and [??.sub.w], given the interest rate r and wage rate w.

To capture the role of deferred compensation in mitigating moral hazard, we assume that labor's marginal product and, hence, the wage rate increase when [Alpha] decreases. In addition, we assume that the equilibrium level of [Alpha] is set in the labor market to maximize the utility of the representative consumer conditional on the government's tax and spending policies. Therefore, (dV/dw) (dw/d [Alpha]) = 0, where V(p, w, M + wT) denotes the indirect utility function. Thus, from (3) we obtain

(10) [[Alpha](1 - [t.sub.w]) + (1 - [Alpha])(l - [??.sub.w])p][w.sub.[Alpha]] ([Alpha]) + [1 - [t.sub.w] - (1 - [??.sub.w])p]w([Alpha]) = 0

as the equilibrium condition determining [Alpha]. Note that, while r remains constant when government adjusts the tax policy, w responds to the market's choice of [Alpha] which, in turn, is influenced by the government's choice of tax rates.

III. Optimal Taxation

The optimal tax problem is solved by choosing tax rates [t.sub.r], [t.sub.w], and [??.sub.w] to maximize the indirect utility function V(p, w, M + wT) subject to the government's budget constraint (6), taking r, n, and g as parameters. The first-order conditions for a welfare optimum are

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Lambda] is the marginal utility of income, [micro] is the Lagrange multiplier for the government's budget constraint, and R is total tax revenue given on the right-hand side of (6).

Combining conditions (11) and (12) to eliminate [Lambda] and [micro] and substituting for the derivatives using the Slutsky relations, we find that income effects cancel out, leaving only substitution effects.(3) Expressing the result in terms of elasticities yields

(14) [rt.sub.2](- [[Sigma].sub.22] + [[Sigma].sub.h2])/(1 + n) = [[Theta]w([[Sigma].sub.hh] - [[Sigma].sub.2h])/w] + [(r - n)s/(1 + n)[pc.sub.2]] + ([Delta] R/[Delta] [Alpha])[([Delta] [Alpha]/[Delta]] [t.sub.r]) - ([Delta] [Alpha]/[Delta] [t.sub.w])rps/[Alpha]wh]/r[p.sup.2] [c.sup.2]

where [[Sigma].sub.22] and [[Sigma].sub.2h] denote the compensated elasticities of demand for [c.sub.2] with respect to p and w, respectively, while [[Sigma].sub.h2] and [[Sigma].sub.hh] denote the compensated elasticities of labor supply with respect to p and w, respectively. Similarly combining conditions (11) and (13) yields

(15) [rt.sub.r] (- [[Sigma].sub.22] + [[Sigma].sub.h2])/(1 + n) = [[Theta]w([[Sigma].sub.hh] - [[Sigma].sub.2h])/w] + ([Delta] R/[Delta] [Alpha])[([Delta] [Alpha]/[Delta] [t.sub.r]) - ([Delta] [Alpha]/[Delta] [??.sub.w]) rs/wh (1 - [Alpha])]/[rp.sup.2] c.sub.2].

Finally, one obtains

(16) [[Alpha].sub.wh](n - r)/(1 + n) = ([Delta] R/[Delta] [Alpha])[[Alpha]([Delta]/[Alpha]) - p(1 - [Alpha])([Delta] [Alpha])/[Delta] [t.sub.w])]/p(1 - [Alpha])

by combining (12) and (13).

Rules for the optimal taxation of labor and capital incomes must account for the effect of taxation on the saving rate, capital accumulation, and the steady-state capital-labor ratio. We follow King [9] and focus on the special case in which government is assumed to use debt finance to place the economy on the golden-rule path with r = n. Thus, the capital-labor ratio is held fixed, regardless of the tax structure, by appropriate government debt policy. Hence, we rule out by assumption the dynamic efficiency losses that would arise if debt policy were not available to achieve golden-rule growth. In other words, the optimal tax rates we calculate are, by construction, independent of the capital-labor ratio.

Along the golden-rule path, r equals n and (16) implies that [Delta] R/[Delta] [Alpha], or the bracketed term multiplying it, must equal zero. Using the equilibrium condition (10) for the market's choice of [Alpha] it is straightforward to show that the term in brackets has the same sign as

(17) [(Alpha)/(1 - [Alpha])] - [Alpha]([w.sub.[Alpha]]/w) + 1 + [Alpha]([w.sub.[Alpha]]/w) = 1/(1 - [Alpha])

and hence cannot equal zero.(4) Thus, when tax policy is at an optimum, [Delta] R/[Delta] [Alpha] equals zero, indicating that optimal tax rates on the golden-rule path are set so that marginal adjustments in the market's choice of [Alpha] have no effect on total tax revenue.

We use the right-hand side of (6) to evaluate [Delta] R/[Delta] [Alpha], and substitute for [Delta] [Theta]/[Delta] [Alpha] = ([t.sub.w] - [Theta])/ (1 - [Alpha]) obtained from (7) and for [w.sub.[Alpha]] from condition (10) characterizing the equilibrium [Alpha]. Setting the result equal to zero and using (7) to eliminate [??.sub.w], we arrive at

(18) [t.sub.w] = [Theta] (1 + n)/(1 + [Alpha] n)

when r = n. Thus, when [Alpha] = 1, (18) shows that [t.sub.w] = [Theta], as expected. However, when [Alpha] is less than one, the optimal golden-rule tax rate on non-deferred wages exceeds the optimal effective average tax rate on all wages.

With r equal to n and [Delta] R/[Delta] [Alpha] equal to zero, conditions (14) and (15) are identical and, using equations (3) and (7) for w and 0, can be written as

(19) [rt.sub.r]([[Sigma].sub.2] - [[Sigma].sub.h2])/(1 + n) = - [Theta] (-[Sigma].sub.hh] + [[Sigma].sub.2h])/[[Theta] - (1 + [Alpha]n)/(1 + n)].

This condition along with (18) determines the optimal relative tax rates on deferred and nondeferred wage income and on interest income in a golden-rule economy, while the absolute levels of the tax rates depend on the exogenously determined revenue requirement.

To focus on the role of the own-price elasticities of labor supply and consumption in determining the optimal tax rates, we assume that the cross-price elasticities, [[Sigma].sub.h2] and [[Sigma].sub.2h], are equal to zero. In the limiting case of a perfectly inelastic compensated labor-supply curve ([[Sigma].sub.hh] = 0), condition (19) implies that the optimal tax on interest income is zero. Then, condition (18) and equation (7) defining [Theta] imply [t.sub.w] = [??.sub.w]/(1 + n). Alternatively, if the compensated demand for second-period consumption is perfectly inelastic ([[Sigma].sub.22] = 0), then the optimal value for [Theta] is zero and (18) implies [t.sub.w] = 0. In this case only interest income and deferred wage income are taxed, with the optimal tax rates satisfying the relation

(20) [rt.sub.r]/[(1 + r(1 - [t.sub.r])] = [??.sub.w]/(1 - [??.sub.w]).

In intermediate cases, optimal relative tax rates depend on the relative magnitudes of the compensated own-price elasticities of consumption and labor supply, and all tax rates are positive.

IV. Comparisons with Previous Studies

The rules for optimal taxation derived in the previous section differ from those obtained by Atkinson and Sandmo [2] and King [9], and reviewed by Sandmo [18], by incorporating tax-deferred compensation as a component of wage income. These differences are highlighted by examining two polar cases: the first, which corresponds to the case considered in previous studies, assumes that no wage income is tax-deferred ([Alpha] = 1), and the second assumes that all wage income is tax-deferred ([Alpha] = 0).

The relation in (14) is relevant when no wages are tax-deferred (a = 1) and reduces to

(21) [rt.sub.r](- [[Sigma].sub.22] + [[Sigma].sub.h2])/(1 + n) = [[t.sub.w]/(1 - [t.sub.w])]([[Sigma].sub.hh] - [[Sigma].sub.2h]) + (r - n)/(1 + n).

This condition was obtained by Atkinson and Sandmo [2] and King [9]. On the golden-rule path (r = n), condition (21) is identical to our equation (19) once (18) is taken into account. Hence, the relative taxes on capital and non-deferred wage incomes are unaffected by the presence of deferred labor compensation. The absolute levels of taxation are affected, however, since the government's revenue requirement (6) depends on [Theta], which incorporates [t.sub.w].

When all wages are tax-deferred ([Alpha] = 0), the relation in (15) is relevant and reduces to

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This condition differs from (21) in two respects. First, compensated wage elasticities enter both sides of the relation in (22). Second, the interest tax is discounted by the gross-of-tax interest rate in (21) and by the net-of-tax interest rate in (22).

The differences in the formulae for optimal tax rates can be traced to the fact that taxation distorts different margins of choice depending on whether wage income is or is not tax-deferred. The distortionary effects of taxation are revealed by the first-order conditions for the consumer's constrained utility maximum given in equations (8) and (9). These equations yield the following tangency conditions when no wages are tax-deferred:

(23) [U.sub.2]/[U.sub.1] = p

(24) [U.sub.l]/[U.sub.1] = (1 - [t.sub.w])w.

Together, these conditions imply that interest taxation distorts only the trade-off between consumption and saving, while wage taxation distorts only the labor-leisure trade-off. By contrast, when all wages are tax-deferred, one obtains the following tangency conditions:

(25) [U.sub.2]/[U.sub.1] = p

(26) [U.sub.l]/[U.sub.1] = (1 - [t.sub.w])pw.

These relations imply that a wage tax distorts only the labor-leisure trade-off, whereas an interest tax distorts both the consumption-saving and labor-leisure margins. Hence, the tax deferral of wage income generates a distortionary effect from interest taxation that is not present when no wages are tax-deferred and that alters the prescription for optimal taxation.

The implications of the tax-deferred status of wage income for optimal tax rules are clearest along the golden-rule path, where r equals n, and when the compensated cross-price elasticities are zero ([[Sigma].sub.h2] = 0 = [[Sigma].sub.2h]). Under these assumptions, if no wages are tax-deferred, then equation (21) applies and reduces to

(27) r[t.sub.r](-[[Sigma].sub.22])/(1 + r) = [[t.sub.w]/(l-[t.sub.w])][Sigma.sub.hh].

This formula provides a Ramsey rule for the standard model of optimal taxation, showing that optimal tax rates are inversely proportional to the compensated own-price elasticities. By contrast, if all wage income is tax-deferred, then the Ramsey rule must be modified to account for the implicit interest that accrues to the deferred wages in the second period. Specifically, when all wages are tax-deferred, equation (22) applies and reduces to

(28) r[t.sub.r],(-[[Sigma].sub.22] + [[Sigma].sub.hh])/[1 + r(1-[t.sub.r])] = [[t.sub.w]/(1-[t.sub.w)][[Sigma.sub.hh].

This formula reveals that the optimal tax rate on deferred wage income satisfies the standard Ramsey rule, but that the optimal tax rate on interest income is inversely proportional to the sum of the compensated own-price elasticities of consumption and labor supply, reflecting the two margins of choice that are distorted by the interest-income tax when all wages are tax-deferred.

V. Empirical Implications

To arrive at operational statements of the rules for optimal taxation, we assume that the compensated cross-price elasticities [[Sigma].sub.h2] and [[Sigma].sub.2h] equal zero in condition (19). We also recast the public budget constraint (6) by recognizing that net investment equals the savings of the working generation minus the consumption of the retired generation. The latter is assumed to equal the capital stock, so there are no bequests and M = 0. Hence, we have

(29) wh - [c.sub.1] - kd = nkh

where k denotes the capital-labor ratio. Combining (29) with the consumer's budget constraint (1) yields

(30) p[c.sub.2] = (1 + n)kh.

Substituting for p[c.sub.2]/(1 + n) from (30) into (6) and dividing by aggregate income we arrive at

(31) g = [Theta](1 - [Kappa]) + [t.sub.r][Kappa]

where [Kappa] denotes capital's share of income and g denotes government's share of total spending.

Condition (19) and the government's budget constraint (31) can be solved for optimal values of [Theta] and [t.sub.r] once the parameters r,[Alpha], [Kappa], g, [[Sigma].sub.22], and [[Sigma.sub.hh] are specified. Equation (18) can then be used to determine the optimal value for [t.sub.w] on the golden-rule path and, finally, equation (7) yields the optimal value for [t.sub.w].

Following King [9], we assume that the relevant time period is a generation and that the interest rate (r) is unity. The proportion of wage income not ax-deferred ([Alpha]) is calculated from data reported in Table I to be 0.882. The data used to calculate capital's share of income ([Kappa]) and government's share of spending (g) for 1988 are taken from Table B-25 and Tables B-83/B-84, respectively, in the Economic Report of the President, February 1995. Using the ratio of total employee compensation to national income we obtain [Kappa] = 0.270, and from the sum of federal, state, and local expenditures net of transfer payments divided by national income we arrive at g = 0.256.

Estimation of the interest elasticity of future consumption ([[Sigma.sub.22]) has been the subject of considerable controversy. Most researchers report estimates of the intertemporal elasticity of substitution between present and future consumption, denoted by 1/[Gamma]. Using the Hicks-Allen adding-up conditions, we find that -[[Sigma].sub.22] = [[[Beta].sub.1]/([[Beta.sub.1] + [Beta.sub.2])]/[Gamma], where [[Beta].sub.i] denotes the share of the budget devoted to the ith period's consumption.(5) Weber [22] presented evidence indicating that 1/[Gamma] lies between 0.56 and 0.75. Friend and Blume [3] suggested that 1/[Gamma] = 0.50, and Grossman and Shiller [5] assumed that 1/[Gamma] = 0.25. From a survey of previous research and from his own estimates, Summers [20] inferred that 1/[Gamma] [approximately equals to] 0.33. More recently, Hall [6] concluded from four sets of econometric results that 1/[Gamma] lies between zero and 0.2. Finally, using panel data on households, Runkle [16] found that 1/[Gamma] = 0.45. Although these estimates range widely over the unit interval and are frequently imprecise statistically, we use a value of 0.4 for 1/[Gamma] to approximate the mid-range of the reported estimates. Recall that, in this model, periods one and two represent working and retirement years, respectively. If we assume that an individual works forty years and is retired ten years, then, [[Beta].sub.1] = 4[[Beta].sub.2] and -[[Sigma].sub.22] = 0.32.

Estimation of the compensated wage elasticity of labor supply has been no less controversial. MaCurdy [12] reported values ranging between 0.14 and 0.35 for male labor supply, with standard errors of 0.07 and 0.16, respectively. Altonji [1] presented estimates that are centered around 0.27, with standard errors of about two-thirds of this value. Pencavel [15, 92] summarized the evidence by concluding that the average estimate of the compensated wage elasticity of male labor supply is 0.2.

Evaluating (19) and (31) with r = 1, [Alpha] = 0.882, [Kappa] = 0.270, g = 0.256, [[Sigma].sub.22] = - 0.32, and [[Sigma].sub.hh] = 0.2, we obtain [Theta] = 0.212 and [t.sub.r] = 0.376. As a consequence, (18) yields [t.sub.w] = 0.226 and (7) yields [t.sub.w] = 0.371. The optimal golden-rule tax rate on capital income (0.376) is close to the effective marginal tax rate estimated by Mendoza, Razin, and Tesar [13] for the benchmark year of 1988 (0.407), and to the rate implied by Fullerton's [4] estimate of 0.313 for the annual effective marginal tax rate on capital, which translates to a life-cycle tax rate ([t.sub.r]) of 0.387.(6) These estimates suggest that, given the assumed level of government spending, the effective tax rate on capital income in the U.S. would be approximately optimal if the economy were on the golden-rule path. In contrast, the estimated effective marginal tax rate on labor income reported by Mendoza, Razin, and Tesar [13] (0.285) is considerably higher than the optimal rate we calculate (0.226) for non-deferred wages in a golden-rule economy. Empirical estimates of the effective marginal tax rate on deferred wages are not available. Hausman and Poterba [7] and Long [11], in simulation analyses of the effect of tax deferment on savings, assume that deferred wages are taxed at a rate lower than the rate applied to non-deferred wages, and use a value of 0.16 for [t.sub.w] which is substantially below the optimal golden-rule tax rate we calculate (0.371).

When the optimal tax rates are recalculated using the same parameter values but assuming [Alpha] = 1 so that no income is tax-deferred, we obtain [Theta] = [t.sub.w] = 0.217 and [t.sub.r] = 0.361 in place of [t.sub.r] = 0.376, [t.sub.w] = 0.226, and [t.sub.w] = 0.371. We conclude that, along the golden-rule path and for the parameter values specified, the optimal tax rates on capital income and on non-deferred labor income are not very sensitive to the presence or absence of deferred compensation. However, deferred wage income is optimally taxed at a rate substantially above the optimal rate for non-deferred wage income. As a consequence, if optimal tax rates were calculated by erroneously treating all wage income as if it were non-deferred, then the tax rates for capital income and non-deferred wage income would nonetheless be close to optimal. However, the overall rate structure would be suboptimal, since deferred and non-deferred wage income would be inappropriately taxed at the same rate.

VI. Summary and Concluding Remarks

Previous research has shown that deferred compensation can ameliorate moral hazard and adverse selection problems arising in long-term implicit contracts in the labor market. As a practical matter, contributions to defined-benefit pension plans serving this incentive function are subject to tax deferment, and constitute a substantial portion of wage and salary compensation in the U.S. We present rules for the optimal taxation of wage and interest incomes when part of the return to labor accrues in the future on a tax-deferred basis.

Using a plausible set of values for compensated own-price elasticities of consumption and labor supply, and empirically relevant assumptions about the government's revenue requirement and the portion of wage income that is tax-deferred, we show that the optimal golden-rule tax rates on interest income and on non-deferred wage income are not very sensitive to the presence or absence of deferred compensation. However, these same calculations indicate that deferred wage income is optimally taxed at a rate substantially higher than the tax rate on non-deferred wage income. These results highlight the importance of the previously neglected distinction between deferred and non-deferred labor income for calculating optimal income tax rates.

Appendix

We first state the Slutsky relations used in deriving equations (14)-(16) from the first-order conditions (11)-(13). We then discuss the derivation of the expression given in the text for relating the interest elasticity of future consumption [[Sigma].sub.22] to the intertemporal elasticity of substitution 1/[Gamma].

The relevant Slutsky relations used in deriving (14)-(16) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sub.12] and [S.sub.11] denote, respectively, the compensated changes in [l.sub.1], when p and w increase, [S.sub.22] and [S.sub.21] denote the same for [c.sub.2], and I denotes income.

To derive the expression relating [[Sigma.sub.22] to 1/[Gamma], we observe that the formula for the intertemporal elasticity of substitution between present and future consumption is

1/[Gamma] = [([U.sub.1]/[U.sub.2])/([c.sub.2]/[c.sub.1])][d(c.sub.2)/ (c.sub.1)/d([U.sub.1])/[U.sub.2)]

where [U.sub.1] / [U.sub.2] = 1/p. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Hicks-Allen adding-up conditions require that the share-weighted compensated own- and cross-price elasticities sum to zero; that is, [[Beta].sub.1[Sigma]12]+ [[Beta].sub.2[Sigma]12] + [[Beta].sub.1[Sigma]12] = 0. Thus, when 0 = [[Sigma].sub.h2] = (-1/h)[[Sigma.sub.12], [[Sigma.sub.12] = (-[Beta].sub.2]/[[Beta].sub.1])[[Sigma].sub.22] and 1/[Gamma] = -[[Sigma].sub.22] ([[Beta].sub.1] + [[Beta].sub.2])/[[Beta].sub.1]. (*) We gratefully acknowledge the valuable comments of an anonymous referee.

(1.) Parsons [14] provides a detailed survey of the theory and practice of incentive pay. Strictly speaking, only defined-benefit pensions can play a role in incentive contracts designed to ameliorate moral hazard and adverse selection problems in the labor market. However, in the U.S. the vast majority of pension wealth is accumulated as a defined benefit, rather than as a defined contribution. Specifically, the US. Department of Labor [21] reported that approximately eighty percent of workers with private pensions are covered under a defined-benefit formula. In addition, of course, the old-age, survivors and disability insurance component of Social Security is essentially a defined-benefit plan.

(2.) Snow and Warren [19] analyzed a related two-period model in which all income from labor supply and saving accrues in the second, future period ([Alpha] = 0) in order to study the effects of an increase in price level uncertainty on labor supply and saving. Kim, Snow, and Warren [8] used a similar model to examine the effects of uncertainty about tax rates on interest and wage incomes.

(3.) The relevant Slutsky relations are given in the Appendix.

(4.) The first-order necessary condition (10) characterizing the market's choice of [Alpha] yields the comparative statics equations

[Delta] [Alpha]/[Delta] [??.sub.w] = - p[1 - (1 - [Alpha]) [w.sub.[Alpha]/]/D

and

[Delta] [Alpha]/[Delta] [t.sub.w] = (1 + [Alpha] [w.sub.[Alpha]/w)/D

where D is negative given the second-order sufficient condition. Substituting these expressions into the right-hand side of (16) yields (17) as the numerator with -D as the denominator. (5.) This expression, relating the interest elasticity of future consumption [[Sigma.sub.22] to the intertemporal elasticity of substitution 1/[Gamma], is derived in the Appendix.

(6.) If the annual marginal tax rate on interest income is [t.sub.[alpha]], and the annual interest rate is i, then the m-year marginal tax rate [t.sub.r], that leaves the consumer with the same present value of after-tax interest income satisfies the formula

1 + [[(1 + i).sup.m] - 1](1-[t.sub.r]) = [[1 + (1-[t.sub.alpha])i].sup.m].

When m equals 40 and i equals 0.0175, the 40-year interest rate (r) equals the assumed value of unity. Using these values and an annual tax rate [t.sub.alpha] = 0.313, the formula yields [t.sub.r] = 0.387.

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