tax rates and labor supply in fiscal equilibrium

TAX RATES AND LABOR SUPPLY IN FISCAL EQUILIBRIUM ARTHUR SNOW and RONALD S. WARREN, JR.*

A central tenet of supply-side economics is that a balanced- budget reduction in the marginal tax rate on wage income increases aggregate labor supply. In contrast, the orthodox Keynesian analysis concludes that the relationship between tax rates and the economy-wide supply of labor is theoretically ambiguous. Our analysis of a general model reveals that these two propositions are associated, respectively, with the special assumptions of “compensated independence” and “ordinary independence” between leisure and public spending.

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; 19861 and Ehrenberg and Smith [1988, 179-801, descends from an older literature represented by the works of Friedman [ 1949; 19541, Goode [ 19491, 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 [ 19651 and subsequently elaborated upon by Lindbeck [1982], Fullerton [1982], Hansson 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 [ 19861 assumes a preference structure which implies that

* University of Georgia. We thank Frank C. Wykoff and an anonymous referee for helpful comments on an earlier version of this paper.

511 Economic Inquiry VO~. XXVII, July 1989. 511-520

512 ECONOMIC INQUIRY

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 account- ing 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 plan of the paper is as follows. Section I1 contains a model of labor supply in the presence of wage taxation and public spending. In section 111, 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 (I), a pure private good ( x ) which serves as numeraire, and a publicly provided good (2 ) . The utility function u(l ,x ,z) is assumed to be twice con- tinuously 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 2

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-

1. Thus, our analysis does not take issue with the well-known proposition discussed by Ful- lerton [1982, 71, Lindbeck [1982, 4861, Hausman and Ruud [1984, 247-2481, and Betson and Greenberg [ 19861, among others, that models with heterogeneous individuals can exhibit aggregate income effects arising from the purely distributional consequences of a tax and spending change. Our intention, instead, is to explore the nature of the aggregate income effects that may be present in a “representative consumer” model.

2. Malcomson [1986] has emphasized that income effects may arise in a representative consumer model from nonlinearities in technology or the tax structure. We abstract from these sources of income effects.

SNOW & WARREN: TAX RATES AND LABOR SUPPLY 513

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 simplifications dictate that the government’s fiscal budget is balanced which, in turn, requires

where h 5 T - 1 denotes an individual’s labor supply. The traditional approach to tax analysis is adopted by assuming that gov-

ernment 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

A4 = w[+ x = wT, (3)

where A4 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

(4) 1 R = W

and the budget constraint (3), where R’ = U j / U x (i = 1 or z) denotes the marginal rate of substitution of the ith good for n~mera i r e .~ The consumer’s uncom- pensated demand function for leisure is denoted 1 ‘(w, z, M). The associated indirect utility function is V(w, z, M) and the compensated demand for leisure is TL(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 par- ticular tax rate t are an equilibrium quantity of the publicly provided good z( t ) and a supply of labor h(t) E T - 1(t) such that

and

z( l ) = (n/P)tWh(t). (6)

3. Throughout the paper subscripts are used to denote partial derivatives.


111. ANALYSIS

To analyze the effect of a balanced-budget tax increase on aggregate labor supply, I ( ( ) and z ( f ) 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 h =T& - l l L, yields

dl/dt = (-WT - Whl ) + 1 dz /d t . (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, 1 :, and the change in spending, dz/dt.

In the special case of “ordinary independence,” 1 = 0 and therefore public spending (z) is irrelevant to the labor-leisure choice. Ordinary independence occurs when the marginal rate of substitution of 1 for x is independent of z; that is, when Rf = 0.4 This case is considered by Gahvari [ 19863 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, = -VM I , yields the following expres- sion for the change in utility:

4. Specifically, I := R L /D, where D = WR f > 0 is the determinant of the Jacobian of the two- equation system given by the first-order condition (4) and the personal budget constraint (3). We emphasize that R := 0 does not imply R = 0, although with a separable utility function, such as a Cobb-Douglas, R 1. = 0 and R f = 0 by definition.

SNOW &WARREN: TAX RATES AND LABOR SUPPLY

dV/dt = Vr + V, d d d t = -WhVM + V, dZ/dt.

Solving for dz/dt,

dz/dt = Wh/p -k (dV/dt)/V,,

where p = R’ = u, /u, = V, / V M is assumed to be positive. Differentiating the identity lz(w, z , M) I Tz[w, z , V(w, z, M)] yields

515

(8)

(9)

Substituting from (9) and (10) into (7). leads to the decomposition

(11) dl/dt = - W7; + (dV/dt - Vt) <ci:/V,) + 7; dV/dt.

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 - Vr = V, dz/dt , affects labor supply through a compensated (Hicksian) cross effect, 7;. 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 T : = O . 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 accom- panied 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, 4481 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

which, combined with (7) and (lo), yields

dl/dr = [- WTf, - Whlh + Wh(n/P) 1 : I / [ 1 + tW(n/P)l :I (1 3a)

dl/dt = [-WTL + Wh@:/p) + Wh(l : / p P ) (np - P)]/[l+ tW(n/P)l:]. (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 (1 i = 0), (13a) reduces to (7). For departures from ordinary independence, however, (13b) shows that the im- pact of a nonzero ordinary cross effect depends on the difference between the marginal social value of the change in spending (np = RZ) and its marginal social cost ( ~ 1 , ~

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. Re- placing 1 : with 1 k and setting P/n = p = 1,

(14) dl/dt = -WT?[ 1 + tW1 h] > 0.

Thus, as Hansson and Stuart [1983, 5871 and Lindbeck [1982, 4781 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, = VM and P/n = p = 1 in (8) and (12) to obtain

6

5. The Bowen-Samuelson condition R'= P will not necessarily hold at optimum because of the distortionary wage taxation. Wildasin [1984; 19851 discusses optimal departures from this condition. His analysis also highlights the two focal cases of ordinary and compensated independence between leisure and the public good.

6. Kay [I9801 presents a correct analysis of excess burden using a similar model.


Hence, in the tax-transfer case there is an income 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 Hansson 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 and contradictory senses in which leisure and public spending can be considered complements or substitute^.^ For example, equation (10) shows that compensated independence between leisure and public spending implies ordinary complementarity (I : > 0), in which case (7) reveals that the spending change has only an income effect. Alter- natively, ordinary independence implies that leisure and public spending are compensated substitutes (7: > 0) 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, 4841 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” (p. 482). Inspection of equation (13b) shows that if leisure and public spending are complementary in a compensated sense (7; > 0), 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. 8

7. The different definitions of complementarity are explored by Samuelson [1974]. As his discussion reveals, these definitions do not always accord with one’s intuition about complementarity relations.

8. Specifically, the conceptual experiment holds constant all prices and quantities, while the compensated cross elasticity of leisure demand with respect to public spending increases. Of course, not oll demand elasticities can remain constant in this experiment, given equations (10) and the Hicks-Allen adding-up conditions, qIp + fib+ qzp,= 0 and axqxr + alqo + anz, = 1 (where the a’s are budget shares and I = x + wl + p z = w T + p z IS virtual income). To ensure that these relations are satisfied, income elasticities and the compensated cross-price elasticity of x with z are allowed to vary, while compensated own-price elasticities are held constant.


Winston [1965, 671 argued that the critical issue is “...whether public goods are more complementary to marketed goods or to leisure,” Samuelson [1974, 12721 made this notion of more complementary precise by proposing that the statement “z is more complementary with I than with x” means that receiving increments Al and Az together is more highly valued than receiving Ar and Az together, when Ar, Al, and Az, received separately have the same value. Samuelson showed that this is equivalent to R > 0 which, in turn, is equivalent to I > 0 , Equation (13a) thus illustrates Winston’s point that when leisure and public spending are complementary in an ordinary sense (I > 0), the ordinary cross effect of the spending change on the demand for leisure is positive.

More recently, Bohanon and Van Cott [1986, 2981 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 choice.” 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 (I and x ) are unrelated to public spending in a compensated sense (in partic- ular ’7: = 0). However, the decisive feature of Fisher-perfect complementarity is that the private goods are independent of public spending in an ordinary sense (I = O).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), 7: = 0 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, 2421 observes, “[Mlany 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 [ 19741 suggested a Cobb-Douglas utility function. With these preferences, utility is separable so there is ordinary independence (I = 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 theoretical analyses have arrived at various propositions regarding the effect of a balanced-budget tax change on aggregate labor supply.

9. Note that in the case of Fisher-perfect complementarity between private and public spending = 0 and equation (10) holds for any p . 1


On the one hand, it has been argued that-for the economy as a whole-there is only a substitution effect, so that labor supply unambiguously increases in response to a wage-tax cut. Alternatively, it has been shown that with separable 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 conclusions are valid, 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.

REFERENCES

Bailey, Martin J. “The Marshallian Demand Curve.’’ Journal of Political Economy, June 1954, 2 5 5 4 1.

Betson, David M. and David Greenberg. “Labor Supply and Tax Rates: Comment.” American

Bohanon, Cecil E. and T. Norman Van Coa. “Labor Supply and Tax Rates: Comment.” American

Ehrenberg, Ronald G. and Robert S . Smith. Modern Labor Economics, 3rd ed. Glenview, 1L:

Friedman, Milton. “The Marshallian Demand Curve.” Journal of Political Economy, December

Economic Review, June 1986, 551-56.

Economic Review, March 1986, 277-79.

Scott, Foresman and Company, 1988.

1949, 463-95. . “The Marshallian Demand Curve: A Reply.” Journal of Political Economy, June 1954,

261-66. Fullerton, Don. “On the Possibility of an Inverse Relationship between Tax Rates and Government

Gahvari, Firouz. “Labor Supply and Tax Rates: Comment.” American Economic Review, March

G d e , Richard. “The Income Tax and the Supply of Labor.” Journal of Political Economy,

Gwartney, James and Richard Stroup. “Labor Supply and Tax Rates: A Correction of the Record.”

. ‘‘Labor Supply and Tax Rates: Reply.” American Economic Review, March 1986,284-85. Hansson, Ingemar and Charles Stuart. ”Taxation, Government Spending, and Labor Supply: A

Hausman, Jerry and Paul Ruud. “Family Labor Supply with Taxes.” American Economic Review,

Kay, John A. “The Deadweight Loss from a Tax System.” Journal of Public Economics, February

Lindbeck, Assar. “Tax Effects Versus Budget Effects on Labor Supply.” Economic Inquiry, October

Malcomson, James M. “Some Analytics of the Laffer Curve.’’ Journal oJPublic Economics, April

Samuelson, Paul A. “Complementarity - An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory.” Journal of Economic Literature, December 1974, 1255-89.

Scitovsky, Tibor. Weyare and Competition. Chicago: Richard Irwin, 1951.

Revenues.” Journal of Public Economics, October 1982, 3-22.

1986. 280-83.

October 1949, 428-37.

American Economic Review, June 1983, 446-51.

Diagrammatic Exposition.” Economic Inquiry, October 1983, 584-87.

May 1984, 242-48.

1980, 11 1-19.

1982, 473-89.

1986, 263-19.


Wildasin, David E. “On Public Good Provision with Distortionary Taxation.” Economic Inquiry, April 1984, 22143.

. “Errata.” Economic Inquiry, January 1985, 185.

Winston, Gordon. “Taxes, Leisure and Public Goods.” Economica, February 1965, 65-69.

tax rates and labor supply in fiscal equilibrium

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