ALTERNATIVE MEASURES OF THE MARGINAL COST OF FUNDS.

BROWNING, EDGAR K. ; GRONBERG, TIMOTHY ; LIU, LIQUN 等

LIQUN LIU [*]

This article reconciles the use of two different marginal cost-of-funds (MCF) measures by resorting to alternative marginal benefit measures. It demonstrates how the alternative MCF measures can be properly applied to two classic problems in expenditure analysis: local cost-benefit project evaluation and the second-best public good level question. Relative strengths of the two MCF approaches in addressing the two problems are identified. (JEL H21, H43)

1. INTRODUCTION

It is widely appreciated that the welfare cost of taxation should play a role in the normative analysis of government expenditures. Efficiency in government spending involves balancing the marginal benefits and marginal costs of expenditures. The marginal cost of spending in part depends on the welfare costs (also known as excess burdens or deadweight losses--we will use the term welfare costs) that result from raising the revenue that finances the expenditures. To determine the marginal cost of government spending, we multiply the direct resource cost of the expenditure by a term generally referred to as the marginal cost of funds (MCF). The MCF is the cost of raising an additional dollar of revenue, which includes the direct cost (the dollar of revenue actually raised) plus any additional welfare costs resulting from the change in the tax structure.

Martin Feldstein emphasized the importance of MCF in a recent paper appropriately titled "How Big Should Government Be?" (Feldstein [1997]). He correctly stresses that the answer to this question involves comparing the benefits of incremental government spending with the MCF and concludes that the economics profession should devote more attention to informing policy makers of the importance of MCF and providing estimates of its magnitude. Feldstein's own estimate of MCF is $2.65 (Feldstein [1999]), implying that the distortionary cost of raising revenue is amazingly high--raising an additional dollar of revenue carries with it an efficiency loss that is more than one and a half times (1.65) as large. In sharp contrast, there are a number of estimates in the literature on MCF that are strikingly lower, so low as to suggest that MCF should play a minor role in the evaluation of government spending programs. For example, Stuart [1984] reports an estimate of $1.07, and Ballard and Fullerton [1992] suggest that i n a variety of cases MCF is less than $1.10.

These disparate estimates of the magnitude of MCF are worrisome because they have contradictory implications regarding the importance of the distortionary cost of taxation. But a major reason for the differences in these estimates is simply definitional (Fullerton [1991] and Ballard and Fullerton [1992]): Feldstein is using a definition of MCF that depends primarily on the compensated effects of taxation, whereas the other estimates use a definition of MCF that depends primarily on uncompensated effects. The definition based on compensated effects always yields a larger estimate of MCF. [1] This raises the obvious question: Which definition is the appropriate one to use in the evaluation of government expenditures? That is the subject of the present paper.

In this paper we investigate the appropriate measure of MCF for a special case that has been widely discussed in the literature--when the government expenditure is to finance the provision of a public good that enters consumers' utility functions in a separable fashion from private goods and leisure. (That the appropriate definition of MCF depends on exactly how the expenditure affects consumer utility was emphasized by Wildasin [1984] and Ballard and Fullerton [1992].) Recent papers that help clarify the role of the separability assumption include Mayshar [1991], Snow and Warren [1996], and Ahmed and Croushore [1996]. Furthermore, we restrict our attention to the case in which the public good is financed by a tax on wage income. In this case, much of the literature on MCF emphasizes that the definition of MCF should be based on the uncompensated wage elasticity of labor supply, in contrast to Feldstein's measure. Indeed, when the uncompensated elasticity is zero, the MCF of a proportional wage tax is equal to one according to this literature, suggesting that the distortionary cost of taxation is irrelevant in the evaluation of government expenditures.

Within this economic environment, we consider the two competing measures of MCF Several economists working in the framework that Ballard and Fullerton [1992] refer to as Pigou-Harberger-Browning tradition have suggested the use of [MCF.sub.1], defined simply as one plus marginal total welfare cost, a measure that depends primarily on compensated effects. Other economists working in the framework developed by Stiglitz and Dasgupta [1971] and Atkinson and Stern [1974] have promoted application of [MCF.sub.2], defined simply as one plus marginal tax leakage, a measure that primarily depends upon uncompensated effects. Despite attempts by several authors to offer intuitive explanations (Fullerton [1991], Ballard and Fullerton [1992]), the difference between the two positions on MCF remains puzzling.

In this paper we solve the puzzle by showing that the two different MCF measures are both equally valid measures of MCF for purposes of conducting a cost-benefit analysis of government spending. This holds despite the fact that the former version of MCF yields a numerical measure that can be much larger than the alternative version. The key to the analysis is understanding that the marginal benefits of government spending must be measured in different ways under the two approaches. This was first pointed out by Triest [1990], but unfortunately he did not prove the result (or perhaps did not explain it clearly enough), as witness to the fact that more recent work continues to propose different MCF measures without specifying the appropriate corresponding marginal benefit measures. Although when correctly employed, both approaches yield the same results, we identify their relative advantages for local cost-benefit and global second-best level analyses.

II. TWO MEASURES OF THE MARGINAL COST OF FUNDS

In what has become the prototypical cost-benefit/project evaluation environment, [2] H identical consumers possess preferences u(C, 1) + g(G), [3] where C is the numeraire private good, 1 is leisure, and G is a public good. Each consumer faces the full-income budget constraint wT + N = C + wl, with w = w(1 - t) where w is the fixed gross wage, t is a proportional tax rate on labor income, T is the total time endowment, and N is non-labor income. Maximization yields demand functions C(w, N), l(w, N), and indirect utility V = U(w, N) + g(G). The government budget balance condition requires that H . R(w) = H(w - w)L(w, N) = [P.sub.G]G, where L(w, N) = T - l(w, N) is the representative individual labor supply function and [P.sub.G] is the constant marginal rate of transformation between C and G.

Starting from any arbitrary initial level of public good provision, consider a marginal increase in G financed by an increase in proportional income tax revenue. The balanced-budget increase in G will increase utility for the representative consumer if

(1) dV/dG = ([partial]U/[partial]w)(dw/dG) + g'(G) [greater than] 0.

From the government budget balance condition we have

(2) dw/dG = [P.sub.G]/[HR'(w)].

Combining equations (1) and (2), the appropriate cost-benefit criterion for this economy is

(3) Hg'(G) [greater than] [P.sub.G][-([partial]U/[partial]w)/R'(w)].

When judging alternative approaches to the applied measurement of the costs and benefits of a public goods project, satisfaction of equation (3) is the relevant benchmark. The left-hand side (LHS) of equation (3) is H times the addition to the representative individual's utility from one more unit's public good consumption, and the right-hand side (RHS) is the marginal production cost times the utility loss per unit revenue from an increase in the wage tax rate. Both sides are measured in units of utility and so need to be evaluated at some price vector to express the comparison in numeraire units. We will show how the choice between alternative MCFs boils down to the choice of reference price vector.

Within this framework of separable public goods, there exist two principal competing measures of the marginal cost of public funds, the factor used to adjust the direct cost of public goods project within a distortionary financing environment. Both measures are intended to convert the utility loss in equation (3) to numeraire units. When the net wage rate decreases from w to w' due to an increase in the wage tax rate, the first MCF measure is based on the difference between the equivalent variation measure of the utility loss when the net wage rate decreases from w to w' and the equivalent variation measure of the utility loss when the net wage rate decreases from w to w, that is, [N - e(w, U(w', N))] - [N - e(w, U(w, N))] = -[e(w, U(w', N)) - e(w, U(w, N))], where function e(w, U) is the expenditure function associated with the private utility function. The second MCF measure utilizes the equivalent variation measure of the utility loss when the net wage rate decreases from w to w', that is, [N - e(w, U(w', N))] = -[e(w, U(w', N)) - e(w, U(w, N))]. Hence,

(4) [MCF.sub.1] = [lim.sub.w' [right arrow] w] -[e(w, U(w, N)) - e(w, U(w, N))]/R(w') - R(w)

= [partial]e(w, U(w, N))/[partial]U [partial]U(w, N)/[partial]w/R'(w)

(5) [MCF.sub.2] = [lim.sub.w' [right arrow] w] -[e(w, U(w', N)) - e(w, U(w, N))]/R(w') - R(w)

= [partial]e(w, U(w, N))/[partial]U [partial]U(w, N)/[partial]w/R'(w)

Each of the MCF expressions in equations (4) and (5), when multiplied by the marginal rate of transformation [P.sub.G], generates a corresponding measure of the marginal social cost of an additional unit of public good provision. Note that [MCF.sub.1] uses the pretax wage in its evaluation of the marginal expenditure of utility, whereas [MCF.sub.2] uses the current post-tax wage.

It has been a general practice in the MCF literature to decompose an MCF measure into the form of one plus a term that indicates the indirect cost of raising one more dollar of tax revenue. This decomposition serves to distinguish further between the two MCF measures studied here. [MCF.sub.1] is grounded in the welfare cost (deadweight loss/excess burden) tradition of Pigou, Harberger, and Browning, as it can be shown that [MCF.sub.1] = 1 + MTWC, where MTWC is the marginal total welfare cost of taxation, which Browning [1987] defines as "the ratio of the change in total welfare costs to the change in tax revenue produced when tax rates are varied in some specified way." To see this, note that the measure of total welfare costs based on equivalent variation [4] is TWC(R) = N - e(w, U(w, N) - R. Hence, [MCF.sub.1] =1 + MTWC.

The second measure, [MCF.sub.2], first introduced in Stiglitz and Dasgupta (1971), can also be decomposed into one plus a meaningful term. By noting R(w) = (w - w)L(w, N) and applying Roy's identity, equation (5) can be simplified to

(6) [MCF.sub.2] = 1 + -(w - w)[partial]L(w, N)/[partial]w/R'(w)

where the second term on RHS has an interpretation of the tax revenue leakage per dollar of increased tax revenue.

We illustrate the two MCF measures from a representative household perspective in Figures 1a and 1b. In the separable case, changes in the quantity of the public good do not affect the indifference curves in numeraire-leisure space, so a two-dimensional treatment is possible. In both Figures 1a and 1b, the composite private good, C, is measured vertically and leisure horizontally; the nonlabor income is assumed to be zero. The pretax budget constraint is QT. With tax rate t (corresponding to wage rate w), the budget constraint is QT; and the worker is in equilibrium at point E. Line qq, which passes through E and is parallel to QT, is the equal-revenue line. To evaluate [MCF.sub.1] and [MCF.sub.2], we consider the effects of a small increase in the tax rate to t' (corresponding to wage rate w' [less than] w)--shown here as a large, discrete increase for clarity. The budget constraint becomes Q'T, and the new equilibrium is point E'. In Figure 1a, the total welfare cost of taxation before the tax increase is simply the difference between the equivalent variation of the tax, which identifies the total burden of the tax and is shown by distance AB (where hh is parallel to QT and tangent to indifference curve U), and the amount of actual tax revenue, AE. Thus the total welfare cost before the tax increase is equal to EB.

Drawing kk parallel to QT and tangent to U', we find that the total welfare cost associated with the higher tax rate equals E'F. MTWC is then naturally defined as the change in total welfare cost divided by the additional per capita tax revenue raised, or MTWC = [lim.sub.w' [right arrow] w](E'F - EB)/HE', where point H identifies the intersection of qq and vertical line D(T - L'). So

(7) [MCF.sub.1] = [lim.sub.w' [right arrow] w] DF - AB/HE' = 1 + MTWC

It should be noted that this measure is not totally dependent on the compensated elasticity of labor supply. The denominator, the actual change in tax revenue, depends on the uncompensated elasticity, while the numerator does depend on the compensated effects along the two indifference curves. Note also that it is possible for MTWC to be less than zero and for [MCF.sub.1] to be less than one, even with this definition. [5] This is most easily seen by imagining that U' is kinked at point E' (zero compensated elasticity), implying a zero total welfare cost when the tax rate is t'. This implies that MTWC is negative and [MCF.sub.1] is less than one. Obviously, this situation can never arise with any of the commonly used utility functions, but it is possible.

Turning to the definition of [MCF.sub.2], we once again consider the effects of increasing the tax rate from t to t' in Figure 1b. A different equivalent variation measure, taking the already distorted net wage along QT as the basis, is used to evaluate the effect on the taxpayer's welfare. The equivalent variation of the change in the net wage rate is identified by drawing jj parallel to QT and tangent to U'. The equivalent variation measure of the loss in utility is then given by JK, and the marginal cost of funds is simply

(8) [MCF.sub.2] = [lim.sub.w' [right arrow] w] JK/HE' = 1 + [lim.sub.w' [right arrow] w] JH/HE',

where the second equality follows due to the fact that [lim.sub.w' [right arrow] w] E'K'/HE' = 0.

Note that JH/HE' has an interpretation of tax revenue leakage per dollar because JH is the revenue lost at the initial tax rate due to the reduced labor supply caused by the increase in the tax rate, that is, JH = (L - L') (w - w).

In comparing these two alternative definitions, it is clear that they give rise to quite different numerical values for MCF. For example, when the uncompensated elasticity is zero, [MCF.sub.2] equals one regardless of the compensated elasticity or the level of the pre-existing tax rate. In sharp contrast, [MCF.sub.1] will typically be higher the larger the compensated elasticity is (given a zero uncompensated elasticity) and the greater the preexisting tax rate. The difference in the level and determinants of MCF identified by the two approaches can therefore be considerable.

[MCF.sub.1] is no less than [MCF.sub.2] as long as leisure is a normal good. This general quantitative relationship between the magnitudes of two MCF measures can be shown most clearly with Figure 2, which identifies [MCF.sub.1] and [MCF.sub.2] using labor supply curves.

In Figure 2, [L.sup.u] is the uncompensated labor supply curve. E and E' are two equilibria corresponding respectively to net wage rates w and w'. [[L.sup.c].sub.E] and [[L.sup.c].sub.E'], are the compensated labor supply curves respectively passing through E and E'. Note that [[L.sup.c].sub.E'] is to the right of [[L.sup.c].sub.E] as long as leisure is a normal good. Area GE'DFEH measures the increase in the total welfare cost due to the tax rate increase, and area GBEH represents the tax revenue leakage from reduced labor supply caused by the tax rate increase. Net tax revenue increase equals area w' E' Bw-area BEHG. So, according to our previous analysis,

[MCF.sub.1] = 1 + [lim.sub.w' [right arrow] w] GE'DFEH/w'E'Bw - BEHG

[MCF.sub.2] = 1 + [lim.sub.w' [right arrow] w] GBEH/w'E'Bw - BEHG

From this figure, it is obvious that [MCF.sub.1] [greater than or equal to] [MCF.sub.2].

III. TWO PROPOSITIONS ON MCF

We will now demonstrate that, when matched with an appropriate measure of marginal social benefits, the two MCF approaches yield an identical (and correct) cost-benefit criterion. Consider two alternative measures of marginal benefit for the public good

(9) [MB.sub.1] = [[partial]e(w, U(w, N))/[partial]U]g' (G)

(10) [MB.sub.2] = [[partial]e(w, U(w, N))/[partial]U]g' (G)

The two measures represent two alternative evaluations of the marginal benefit of G in terms of C. The utility level for evaluation, U(w, N), is identical, but the marginal utility of consumption will, in general, differ (since compensated choices of C and L will generally differ between the w and w price environments).

PROPOSITION 1. Within the propotypical public goods economic environment, the [MCF.sub.1]/[MB.sub.1], and [MCF.sub.2]/[MB.sub.2] approaches to local cost-benefit analysis are equivalent.

Proof. If we combine [MCF.sub.1] and [MB.sub.1], the project evaluation criterion is H [cdotp] [MB.sub.1] [greater than] [P.sub.G][MCF.sub.1], or

(11) [partial]e(w, U(w, N))/[partial]U Hg'(G)

[less than] [partial]e(w, U(w, N))/[partial]U

x[P.sub.G][ - [partial]U(w, N)/[partial]w/R'(w)].

Since [partial]e(w, U(w, N))/[partial]U [greater than] 0, equation (11) is equivalent to equation (3).

If we combine [MCF.sub.2] and [MB.sub.2] the project cost-benefit test is

(12) [partial]e(w, U(w, N))/[partial]U Hg'(G)

[greater than] [partial]e(w, U(w, N))/[partial]U

x[P.sub.G][- [partial]U(w, N)/[partial]w / R'(w)].

Again, because [partial]e(w, U(w, N))/[partial]U [greater than] 0, equation (12) is also equivalent to equation (3). Q.E.D.

The equivalence result here is a formal demonstration (for the case of separable utility and proportional income taxation) of the equivalence arguments forwarded by Triest [1990]. Our demonstration serves to clarify further the role of the reference price vector in the cost-benefit calculation. As is clear from equation (3), any common marginal expenditure of utility factor could be used to monetize the increased marginal utility from the additional public good and the marginal surplus loss from the increase in taxes to finance the public good expansion. The two prime candidates, [partial]e(w, U(w, N))/[partial]U and [partial]e(w, U(w, N))/[partial]U differ only with respect to the reference price vectors, w and w.

Because both [MCF.sub.1] and [MCF.sub.2] can, in principle, be correctly applied to public goods project evaluation, it is worthwhile to consider briefly what we view as the major advantages of the two approaches. For practical implementation, the [MCF.sub.2] approach has one significant advantage over the [MCF.sub.1] approach. It is that the required marginal benefit measure, [MB.sub.2], is evaluated at realized prices (i.e., at the distorted preproject equilibrium). All of the appropriate survey (contingent valuation) or econometric methods (such as hedonic or travel cost methods) yield marginal benefits in terms of the current cum-tax prices ([MB.sub.2]) and thus must be compared to costs using [MCF.sub.2]. Adapting valuation methodologies to uncover [MB.sub.1] seems problematic, because it would require evaluating benefits in prices of the no-tax equilibrium.

Although [MCF.sub.2] is the preferable measure for applied local cost-benefit analysis, the [MCF.sub.1] measure provides advantages for theoretical global analysis of second-best public goods levels. In particular, we present the following proposition ordering optimal public goods levels under alternative tax financing systems. [6]

PROPOSITION 2. In the benchmark public goods environment, assume the total utility function u(C, l) + g(G) is strictly quasiconcave in (C, l, G). If the public good can be financed by either tax system A or B and [[MCF.sup.A].sub.1](R) [greater than] [[MCF.sup.B].sub.1](R) for all R, then [[G.sup.*].sub.B] [greater than] [[G.sup.*].sub.A], where [[G.sup.*].sub.j] is the optimal level of G under tax system j (j = A or B).

Proof. After collecting R from the consumer through tax system j(j = A, B), the private utility becomes, according to the definition of total welfare costs (TWC), U(w), N - [TWC.sub.j](R) - R). The government chooses R to maximize U(w, N - [TWC.sub.j](R) - R) + g(HR). The first-order condition (FOC) for this problem is

(1 + [MTWC.sub.j](R)) [partial]U(w,N - [TWC.sub.j](R) - R)/[partial]N

= H [cdotp] g'(HR).

Note that the strict quasiconcavity of the total utility function implies that the private utility function satisfies that [partial]U(w, N)/[partial]N is nonincreasing in N for any w. [7]

Since [MTWC.sub.A](R) [greater than] [MTWC.sub.B](R) for all R, we have [TWC.sub.A](R) [greater than] [TWC.sub.B](R) for all R, and hence

[partial]U(w, N - [TWC.sub.A](R) - R)/[partial]N

[greater than or equal to] [partial]U(w, N - [TWC.sub.B](R) - R)/[partial]N

So the left side of the FOC for tax system A (when j = A) is larger than the left side of the FOC for tax system B (when j = B). Hence, [[G.sup.*].sub.A] [less than] [[G.sup.*].sub.B]. Q.E.D.

If the marginal cost of funds, as measured by [MCF.sub.1], is uniformly lower under one tax system, then the optimal level of the public goods will be greater. This result holds for comparison of first-best and second-best situations as well as comparison between two second-best situations. Nonintersecting [MCF.sub.2] schedules do not, however, provide a sufficient condition for ranking optimal public goods levels.

Proposition 2 validates one element of the original Pigovian logic (as characterized in Atkinson and Stern [1974]) concerning second-best public goods provision. If a second-best tax system is uniformly more marginally distortionary, that is, has a positive MTWC for all possible positive public goods levels, then the second-best optimum level of G will lie below the first-best level. The unqualified proposition that distortionary taxation necessarily implies lower optimal provision of public goods fails to hold, however, since [MCF.sub.1] can be less than one within second-best settings.

We can illustrate the point of Propositions 1 and 2 using a familiar graphical construct. Let [[MB.sup.*].sub.1](G) and [[MB.sup.*].sub.2](G) denote the equilibrium value of the marginal benefit of G for any (exogenous) level of G, as measured by either equation (9) or (10). [8] The second-best optimum, conditional on proportional income tax financing, occurs at the intersection of the marginal social benefit (MSB) schedule, either H . [[MB.sup.*].sub.1] or H . [[MB.sup.*].sub.2], and the corresponding marginal social cost (MSC) schedule, [P.sub.G] . [MCF.sub.1](G) or [P.sub.G] . [MCF.sub.2](G).

Figure 3 shows how Proposition 1 is illustrated graphically. A public good is produced at a constant MRT of [P.sub.G]. Using the approach based on the traditional concept of MWC, the second-best optimum is shown by the intersection of H [cdotp] [[MB.sup.*].sub.1] and [P.sub.G] [cdotp] [MCF.sub.1] at point B. With the approach based on the marginal surplus measure, the second-best optimum is shown by the intersection of H [cdotp] [[MB.sup.*].sub.2] and [P.sub.G] [cdotp] [MGF.sub.2] at point A, which lies directly below point B and identifies the same level of the public good. The important point is that at the second-best optimum, the difference between the two MCF measures is always matched by an equal difference in the two [MB.sup.*] measures. Part of the relevant distorting effect of the tax under the second approach has the effect of reducing [MB.sup.*], whereas with the first approach all of the relevant distorting effect is incorporated on the cost side in MCF. Heuristically, the first approach may be more in line with the way economists cl assify cost side versus benefit side influences, but substantively the two approaches yield the same result.

Figure 4 provides an illustration of Proposition 2. In this case we compare the financing of the public good with lump sum and labor income taxation using the first approach. With lump-sum financing, the marginal cost of funds, [[MCF.sup.L].sub.1], is simply equal to one, so marginal social cost is equal to [P.sub.G]. Let [[MB.sup.L*].sub.1](G) denote the equilibrium value of the marginal benefit under lump-sum financing. If under proportional income tax financing [MCF.sub.1](G)[greater than] 1 for all G[greater than] 0, then the second-best optimum level of the public good will be less than the first-best optimum at [G.sub.0]. [9]

IV. TWO INSTRUCTIVE EXAMPLES

The case of Cobb-Douglas preferences over private goods, u(C, l) = [C.sup.1-[alpha][l.sup.[alpha]]], is commonly referenced in the existing marginal cost of funds literature. It provides a useful benchmark example for our purposes here as well.

First, if consumers possess preferences [C.sup.1-[alpha][l.sup.[alpha]]] + g(G) and have zero nonlabor income, then proportional income tax financing yields [MCF.sub.2](G) = 1. Although marginal social cost is then equal to the marginal rate of transformation, [P.sub.G], just as in the lumpsum tax case, the optimal level of public good provision under income taxation is lower than under lump-sum taxation (Atkinson and Stern [1974]). The distortionary effect of the tax matters. Within the [MCF.sub.2] framework the distortionary effect of the tax leads to a leftward shift in the [MB.sup.*] schedule, that is, [[MB.sup.L*].sub.2](G) [greater than] [[MB.sup.*].sub.2](G) for all G [greater than] 0, resulting in a reduction in optimal public good provision. Within the [MCF.sub.1] framework, the [MB.sup.*] schedule under the distortionary tax is identical to that under the lump-sum tax, that is, [[MB.sup.L*].sub.1](G) = [[MB.sup.*].sub.1](G) for all G, but the marginal distortionary effect is incorporated directly via the marginal welfar e cost calculations and [MCF.sub.1](G) [greater than] 1 for all G [greater than] 0. Although (by Proposition 1) the same second-best optimal level of G is identified by both approaches, it seems that the fundamental role of the distortionary effect of the tax in the determination of optimal public goods provision is clarified by the [MCF.sub.1] approach but obscured within the [MCF.sub.2] framework.

The fundamental importance of distortionary effects (as embodied in the measure of [MCF.sub.1]) in determining the (second-best) optimal level of public good provision can also be demonstrated within a stylized CobbDouglas example by analyzing the impact of a ceteris paribus change in compensated labor supply elasticity. With N = 0 and Cobb-Douglas preferences, of the form u = [w.sub.[alpha]/[[(1 - [alpha]).sup.1-[alpha]][[alpha].sup.[alpha]]][C.sup.1-[alpha][l.sup.[ alpha]]], the compensated elasticity of labor supply is equal to [alpha] and the uncompensated elasticity of labor supply is zero, independent of [alpha]. For any [alpha] [greater than] 0, the first-best level of G is also independent of [alpha]. [10] It is easily shown that for [[alpha].sub.1] [greater than] [[alpha].sub.2] [MCF.sub.1](G, [[alpha].sub.1] [greater than] [MCF.sub.1] (G, [[alpha].sub.2]) for all G [greater than] 0. By Proposition 2, G([[alpha].sub.1]) [less than] G([[alpha].sub.2]). The larger compensated elasticity implies higher ma rginal total welfare costs, higher marginal social costs when measured by [P.sub.G] [cdotp] [MCF.sub.1](G), and a smaller optimal level of public spending. Because the [MCF.sub.2] approach depends only on the uncompensated elasticity, marginal social costs as measured by [P.sub.G] [cdotp] [MCF.sub.2](G) remain constant at [P.sub.G]. Using [MCF.sub.2] will, of course, locate the same smaller optimal value of G for a larger [alpha], since the larger distortionary effect of the tax will shift down the [[MB.sup.*].sub.2](G) schedule.

V. CONCLUSION

The economics literature has emphasized two seemingly contradictory approaches to the measurement of the marginal cost of funds. In this paper we have reconciled these two approaches by showing that both are correct when they are used with the appropriate (and different) measures of the marginal benefits from government spending. Any preference for one measure over the other must therefore reflect something other than the logical validity of the measure. For applied local cost benefit work, one obvious consideration is how actual benefits are measured. Because all relevant empirical methodologies will yield estimates corresponding to [MB.sub.2], the proper measure of the marginal cost of funds is [MCF.sub.2].

However, this does not mean the traditional concept of (marginal) welfare cost of taxation is not critical in understanding the optimal level of public good provision. If there is a unifying principle underlying these seemingly divergent measures of MCF, it is this: Distortionary effects are important. In the case of [MCF.sub.1], this is obvious, whereas in the case of [MCF.sub.2], it is less obvious. In both cases, the magnitude of compensated effects is a crucial determinant of the second-best optimal level of government spending.

(*.) We would like to thank Don Fullerton, two anonymous referees, and seminar participants at University of Texas for helpful comments and suggestions.

Browning: Alfred F. Chalk Professor, Department of Economics, Texas A&M University, College Station, Tex. 77843. Phone 1-409-845-7355, Fax 1-409-847-8757, E-mail ekb603@myriad.net

Gronberg: Professor, Department of Economics, Texas A&M University, College Station, Tex. 77843. Phone 1-409-845-8849, Fax 1-409-847-8757, E-mail tjg@econ.tamu.edu

Liu: Assistant Research Scientist, Private Enterprise Research Center, Texas A&M University, College Station, Tex. 77843. Phone 1-409-845-7723, Fax 1-409-845-6636, E-mail 1-liu@tamu.edu

(1.) The difference in definitions is not the only reason for the different MCFs reported. Stuart [1984] and Ballard and Fullerton [1992] develop estimates based only on the labor supply distortions of taxes, whereas Feldstein considers other distortions in addition to labor supply effects.

(2.) The framework here parallels that found in Mayshar [1990].

(3.) This form of separability is often referred to as "strong separability." The analysis and results in this paper can be extended to the situation of "weak separability" in which the preferences for the three goods can be represented by f(u(C,1), G).

(4.) Kay [1980] shows that the total welfare cost measure based on equivalent variation is superior to other measures in the sense that utility maximization is equivalent to TWC minimization when TWC is measured based on equivalent variation.

(5.) The MTWC here is different from the MWC in Ballard and Fullerton [1992], which depends only on the compensated labor supply elasticity. Although both measures are intended to measure the utility loss caused by an increase in the tax rate, there are two major differences between them. First, in the definition of MTWC here, the additional revenue raised with an increase in the tax rate is assumed to be used to provide a public good that is separable in the utility function, whereas in Ballard and Fullerton, the additional revenue is assumed to be lump sum rebated to the taxpayer. Second, the MTWC here is strictly based on the traditional concept of total welfare cost (indeed, MTWC here is the derivative of the TWC with respect to R), whereas the compensated version of MWC in Ballard and Fullerton is not directly related to the concept of TWC. Also note that the compensated version of MCF (i.e., one plus Ballard and Fullerton's MWC), which is also due to Browning [1976, 1987], has been shown to be the appro priate MCF measure for the situation in which the benefits from public expenditures can be regarded as perfect substitutes for private consumption (Wildasin [1984]; Ballard and Fullerton [1992]).

(6.) A more thorough and general treatment of the marginal welfare cost approach to the second-best level problem is found in Gronberg and Liu [1998].

(7.) Proof of this result is available on request.

(8.) Note that w in equations (9) and (10) is now a function of G as determined by the government budget constraint H(w - w)L(w, N) = [P.sub.G]G.

(9.) In Figure 4, we draw [[HMB.sup.*].sub.1] uniformly below [[HMB.sup.L*].sub.1] under the condition that [partial]U(w, N)/[partial]N is nonincreasing in N (see Proposition 2). When [partial]U(w, N)/[partial]N is constant in N, such as for any Constant Elasticity of Substitution (CES) or Cobb-Douglas (with coefficients summing to one), the two MSB schedules will be identical. In more general cases, the difference between MSB schedules arises because the income effect of the infra-marginal welfare cost of the wage tax affects the marginal valuation of the public good.

(10.) For comparability of various second best levels when [alpha] changes, we use the total utility specification [w.sup.[alpha]]/[[(1-[alpha]).sup.1-[alpha]][[alpha].sup.[alpha]]][C. sup.1-[alpha][l.sup.[alpha]]] +g(G) to make the first best of G independent of [alpha].

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