Tax rates and labor supply in fiscal equilibrium.

Snow, Arthur ; Warren, Ronald S., Jr.

TAX RATES AND LABOR SUPPLY IN FISCAL EQUILIBRIUM

I. INTRODUCTION

If the government changes the tax on wage income, what happens to aggregate labor supply? This question is at the core of debates between proponents of supply-side and Keynesian approaches to government fiscal policy. It is unfortunate, therefore, that previous attempts to resolve the issue have obscured decisive assumptions regarding the preference relation between leisure and public spending.

One argument, exposited recently by Gwartney and Stroup [1983; 1986] and Ehrenberg and Smith [1988, 179-80], descends from an older literature represented by the works of Friedman [1949; 1954], Goode [1949], Scitovsky [1951], and Bailey [1954], and emphasizes the importance of a balanced-budget framework for addressing the question. This approach reveals the presence of an income effect caused by the change in government spending that must accompany the tax change. At optimum, according to this view, this income effect exactly offsets the income effect of the tax change so that only the substitution effect of the tax remains. As a consequence, a balanced-budget increase in the wage tax unambiguously decreases economy-wide labor supply, provided the increase in public spending is valued the same as the forgone private spending.

A different approach was initiated by Winston [1965] and subsequently elaborated upon by Lindbeck [1982], Fullerton [1982], Hanson and Stuart [1983], Bohanon and Van Cott [1986], and Gahvari [1986]. This approach stresses the importance of the preference relation between public spending and private spending, rather than the role of the public spending income effect, in determining the change in aggregate labor supply. As a special example, Gahvari [1986] assumes a preference structure which implies that public spending does not have any influence on labor supply so there is only the partial equilibrium effect of the tax. However, the general conclusion of this line of reasoning is that the theoretical ambiguity of the labor supply response arises from both tax and spending effects.

In this paper a simple, formal model is used to develop a careful accounting of the various income and substitution effects. The model is sufficiently general to permit a rigorous comparison of earlier studies and to expose implicit assumptions responsible for their conclusions. We show that the two approaches outlined above are associated, respectively, with the focal cases of "compensated independence" and "ordinary independence" between leisure and public spending.

The pan of the paper is as follows. Section II contains a model of labor supply in the presence of wage taxation and public spending. In section III, the effect on aggregate labor supply of a balanced-budget change in wage taxation is analyzed. Section IV provides an interpretation of several previous analyses of these issues. Section V contains a summary of our results.

II. THE MODEL

Consider an economy with n identical consumers who derive utility from leisure (l), a pure private good (x) which serves as numeraire, and a publicly provided good (z). The utility function u(l, x, z) is assumed to be twice continuously differentiable and strictly quasi-concave. Everyone is endowed with T units of time which are allocated either to labor (market work) or leisure (nonmarket activities). The marginal product of labor in producing the private good is the constant, real (gross-of-tax) wage rate W. Because agents are identical, we can confine attention to allocations of equal consumption. As a consequence, the production possibilities frontier, assumed to be linear, can be expressed in per capita terms as Wl+x+(P/n)z=WT, (1) where P is the constant marginal cost of z.

With identical agents and only two goods not publicly provided, it may be assumed without loss of generality that public spending is financed by a wage tax with constant (marginal and average) ad valorem rate t. The net-of-tax real wage rate is w = (1-t)W. We follow previous studies and abstract from inflation and unemployment, as well as from dynamic issues of capital accumulation, growth, and debt finance. These simplications dictate that the government's fiscal budget is balanced which, in turn, requires tWh=(P/n)z, (2) where h=T-l denotes an individual's labor supply.

The traditional approach to tax analysis is adopted by assuming that government is an exogenous agent that sets fiscal policy (t,z). Each individual takes this fiscal policy as given and chooses labor supply so as to maximize utility, subject to the personal budget constraint M=wl+x=wT, (3) where M is the individual's total spending on the private goods. Since fiscal policy is exogenous, individual labor supply is determined by the first-order condition R.sub.1=w (4) and the budget constraint (3), where R.sup.i=u.sub.i/u.sub.x (i=l or z) denotes the marginal rate of substitution of the ith good for numeraire. The consumer's uncompensated demand function for leisure is denoted l.sup.z(w, z, M). The associated indirect utility function is V(w, z, M) and the compensated demand for leisure is l.sup.z(w, z, u).

To close the model, it is assumed that, for the given fiscal policy, the government's budget constraint (2) is satisfied. Thus, associated with a particular tax rate t are an equilibrium quantity of the publicly provided good z(t) and a supply of labor h(t)=T-l(t) such that l(t)=l.sup.z[w, z(t), wT] (5) and z(t)=(n/P)tWh(t). (6)

III. ANALYSIS

To analyze the effect of a balanced-budget tax increase on aggregate labor supply, l(t) and z(t) are assumed to be differentiable and equations (5) and (6) are used to derive alternative decompositions of the effect of the tax increase on the representative consumer's demand for leisure.

Tax and Spending Effects

Consider first a decomposition that isolates the tax and spending (Lindbeck's "budget") effects. Differentiating (5) and using the Slutsky equation, l.sup..sub.=*.sup.sub.w-l.l.sup.z.sub.M, yields dl/dt=(-W*.sup.z.sub.w=Whl.sup.z.sub.M)+1.sup.z.sub.zdz-dt. (7)

The effect of the tax change alone on the consumption of leisure (the terms in parentheses) is composed of a positive substitution effect and, under the maintained assumption of the normality of leisure, an opposing (negative) income effect. Thus, in a partial equilibrium analysis which ignores the change in public spending, an increase in the wage tax decreases leisure demand (and therefore increases labor supply) if and only if the labor supply curve is backward-bending. The effect of the accompanying balanced-budget spending change, however, depends on the sign of the ordinary (Marshallian) cross effect, l.sup.z.sub.z, and the change in spending, dz/dt.

In the special case of "ordinary independence," l.sup.z.sub.z=0 and therefore public spending (z) is irrelevant to the labor-leisure choice. Ordinary independence occurs when the marginal rate of substitution of l for x is independent of z; that is, when R.sup.1.sub.z=0.sup.4 This case is considered by Gahvari [1986] and Hansson and Stuart [1983], who conclude that with separable utility there are only the opposing, partial equilibrium substitution and income effects of the tax change.

Substitution and Income Effects

We now derive a decomposition of the response of leisure demand to a tax increase that isolates the substitution and income effects of a combined tax and spending change. Differentiating the indirect utility function V(w, z, M) and using Roy's Identity, V.sub.w=-V.sub.m.l, yields the following expression for the change in utility: dV/dt=V.sub.t + V.sub.z.dz/dt = -WhV.sub.M + V.sub.z.dz/dt. (8) Solving for dz/dt, dz/dt = Wh/p + (dV/dt)/V.sub.z., (9) where p = R.sup.z [triple bond] u.sub.z./U.sub.x = V.sub.z./V.sub.M is assumed to be positive.

Differentiating the identity l.sup.z.(w,z, M) [triple bond] l.sup.z.[w,z, V(w, z, M)] yields l.sup.z/.sub.z = l.sup.z/.sub.z + V.sub.z.l.sup.z/.sub.u = l.sup.z/.sub.z + pl.sup.z/.sub.M. (10) Substituting from (9) and (10) into (7), leads to the decomposition dl/dt = -Wl.sup.z/.sub.w + (dV/dt - V.sub.t.) (l.sup.z/.sub.z./V.sub.z.) + l.sup.z/.sub.u.dV/dt. (11)

The first term on the right-hand side of (11) is the substitution effect of the tax change. In the second term, the income effect of the spending change alone, dV/dt - V.sub.t = V.sub.z dz/dt, affects labor supply through a compensated (Hicksian) cross effect, l.sup.z/.sub.z. The third term is the income effect associated with the combined tax and spending change. At an interior optimum, the increase in the public spending is valued the same as the forgone private spending and dV/dt = 0.

"Compensated independence" between leisure and public spending means that l.sup.z/.sub.z = 0. In this case, at optimum, there is only the substitution effect of the tax so that an increase in the wage tax unambiguously decreases labor supply. However, from an initial position away from optimum, the balanced-budget increase in public spending will decrease (increase) real income if the burden of the tax increase is greater (less) than the value of the additional public spending it finances. Thus, away from optimum, the substitution effect of the tax increase--which unambiguously reduces labor supply--is accompanied by income effects from the tax and spending changes which together counteract (reinforce) the substitution effect. This is the conclusion reached by Gwartney and Stroup [1983, 448] and our analysis affirms that it is valid in the special case of compensated independence between leisure and public spending.

The Total Effect

The two decompositions given above contain undetermined expressions for the change in public spending (dz/dt in equation (7)) and the change in real income (dV/dt in equation (11)). These undetermined expressions can be eliminated to arrive at a wholly endogenous statement of the balanced-budget effect on leisure demand that will prove useful for interpreting our results and relating them to previous studies. Differentiating the public budget constraint (6), yields dz/dt = (n/P)W(h-tdl/dt) (12) which, combined with (7) and (10), yields dl/dt = [-Wl.sup.z/.sub.w + Wh(l.sup.z/.sub.z./p) + Wh(l.sup.z/.sub.z/.sub.P.P) (np - P)]/[1 + tW(n/P)l.sup.z/.sub.z.]. (13b)

In contrast to the decompositions derived previously, which highlight spending and income effects and therefore the total value of public spending, equations (13) draw attention to the importance of the marginal value of this spending. With ordinary independence (l.sup.z/.sub.z = 0), (13a) reduces to (7). For departures from ordinary independence, however, (13b) shows that the impact of a nonzero ordinary cross effect depends on the difference between the marginal social value of the change in spending (np = [sigma] R.sup.z.) and its marginal social cost (P).

Finally, (13a) can be used to examine the labor supply response to a wage-tax increase that finances an equal per capita, lump-sum transfer. Replacing l.sup.z/.sub.z with l.sup.z/.sub.M and setting P/n = p = 1, dl/dt = -Wl.sup.z/.sub.w./[1 + tWl.sup.z/.Sub.M.] > 0. (14)

Thus, as Hansson and Stuart [1983, 587] and Lindbeck [1982, 478] point out, in the tax-and-transfer case the effect on labor supply of a wage-tax increase is unambiguously negative. However, the decline in labor supply is less than the substitution effect caused by the wage-tax change because of the income effect associated with the excess burden of distortionary taxation. To see that the marginal excess burden is positive in the tax-transfer case, i.e., that real income necessarily decreases as t increases, set V.sub.z.=V.sub.M and P/n=p=1 in (8) and (12) to obtain dV/dt=-tWV.sub.M.dl/dt<0.

Hence, in the tax-transfer case there is an icome effect, which encourages labor supply, and an opposing substitution effect, although the substitution effect is necessarily dominant. In the special case of an infinitesimal tax increase from an equilibrium without taxes (t=0), (14) and (15) reveal that there is only a substitution effect, as stressed by Hanson and Stuart [1983].

IV. INTERPRETATION OF PREVIOUS STUDIES

It is commonly asserted that the complementarity between leisure and public spending determines the effect of public spending on the demand for leisure. However, there are various contradictory senses in which leisure and public spending can be dconsidered complements or substitutes. For example, equation (10) shows that compensated independence between leisure and public spending implies ordinary complementarity (l.sup.z.sub.z.>0), in which case (7) reveals that the spending change has only an income effect. Alternatively, ordinary independence implies that leisure and public spending are compensated substitutes * and equation (7) shows that public spending is irrelevant to the labor-leisure choice. However, if there is neither compensated nor ordinary independence, then the wholly endogenous decompositions given in equations (13) must be used to account properly for the public spending effects.

Lindbeck [1982, 484] suggested that the spending effect of a balanced-budget tax increase enhances the demand for leisure" ...if the good provided by government is a complement to leisure...such as in the case of recreational facilities." The increase in public spending" ...would be expected to create a negative cross-substitution effect on labor supply". Inspection of equation (13b) shows that is leisure and public spending are complementary in a compensated sense * , the compensated cross effect of the spending change on the demand for leisure is positive. This conclusion is consistent with Lindbeck's remarks and implies that, with the degree of ordinary complementarity held constant, greater compensated complementarity decreases labor supply.

Winston [1966, 67] argued that the critical issue is" ...whether public goods are more complementary to marketed goods or to leisure." Samuelson [1974, 1272] made this notion of more complementary precise by proposing that the statement "z is more complementary with l than with x" means that receiving increments [delta]l and [delta]z together is more highly valued than receiving [delta]x and [delta]z together, when [delta]x, [delta]l, and [delta]z, received separately have the same value. Samuelson showed that this is equivalent to R.sup.l.sub.z.>0 which, in turn, is equivalent to l.sup.z.sub.z.>0. Equation (13a) thus illustrates Winston's point that when leisure and public spending are complementary in an ordinary sense (l.sup.z.sub.z.>0), the ordinary cross effect of the spending change on the demand for leisure is positive.

More recently, Bohanon and Van Cott [1986, 298] asserted that "...if government goods are unrelated to consumption of leisure and private goods in a Hicksian sense, using tax revenues to provide government goods has no effect on the income-leisure choise." The case consistent with this statement arises when public spending and real private spending (M) are Fisher-perfect complements (Leontief preferences). With this assumption, the private goods (l and x) are unrelated to public spending in a compensated sense (in particular * . However, the decisive feature of Fisher-perfect complementarity is that the private goods are independent of public spending in an ordinary sense (l.sup.z.sub.z.=0).sup.9 As equation (7) reveals, public spending is then irrelevant to the labor-leisure choice. At the opposite extreme, when public and private spending are Fisher-perfect substitutes (linear preferences), * and, also, P/n=p=1. As a result, the spending effect is simply an income effect and this case is equivalent to the tax-transfer analysis. As Wildasin [1984, 242] observes, "[M]any forms of government spending, such as some types of health, education, and housing outlays, might plausibly be argued to have this characteristic...". Finally, to represent the intermediate case of "Fisher independence," Samuelson [1974] suggested a Cobb-Douglas utility function. With these preference, utility is separable so there is ordinary independence (l.sup.z.sub.z.=0) between leisure and public spending. Lindbeck [1982] cites certain pure collective goods, such as national defense and law and order, as examples of public spending that may have this relation with leisure.

V. SUMMARY AND CONCLUDING REMARKS

Previous theorerical analyses have arrived at various propositions regarding the effect of a balanced-budget tax change on aggregate labor supply. On the one hand, it has been argued that--for the economy as a whole--there is only a subtitution effect, so that labor supply unambiguously increases in response to a wage-tax cut. Alternatively, it has been shown that with separate utility changes in public spending have no effect on the labor-leisure choice and there are only the (opposing) partial equilibrium income and substitution effects of a tax change. This analysis of a general model reveals that these conslusions are valide, respectively, in the special cases of compensated and ordinary independence between leisure and public spending. Finally, several claims concerning the role of complementarity in determining the theoretical relationship between tax rates and aggregate labor supply are reviewed and clarified.