dOURNALOF PUBLIC ECONOMICS

ELSEVfER Journal of Public Economics 61 (1996) 289-305

The marginal welfare cost of public funds: Theory aad estimates

Arthur Snow, Ronald S. Warren, Jr.* Department of Economics, University of Georgia. Athens, GA 30602. USA

Received August 1994; revised version received March 1995

Abstract

We derive a general analytic formula for the marginal welfare cost (MWC) of public funds without imposing restrictive assumptions on preferences or tech- nologies. The framework encompasses ,the various special cases previously consid- ered, and reconciles the disparities among the reported estimates of MWC. Unresolved differences in existing estimates are fully explained by appropriate interpretations of intended thought experiments and alternative assumptions about the elasticity of labor supply with respect to public spending. Further progress in measuring MWC requires empirical estimation of labor-supply functions that include government spending as an explanatory variable.

Keywords: Marginal welfare cost; Distortionary taxation; Public goods

JEL classification: H20; H41

1. Introduction

The opportunity cost of an additional dollar of tax revenue includes the marginal welfare cost (MWC) occasioned by the increase in distortionary :axation. Estimates of MWC produced by either computer simulations or analytical calculations have varied widely, and attempts to reconcile ,the disparities have been only partially successful. In this paper we advance ,the

*Corresponding author. Tel.: (706) 542-1311; fax: (706) 542-3376; e-mail: warren@rigel.econ.uga.edu.

0047-2727/96/$19.00 (~) 1996 Elsevier Science S.A. All fights reserved SSD! 0047-2727(95)01535-3

290 A. Snow. R.S. Warren. Jr. ; Journal of Public Economics 61 (1996) 289-305

analytical approach to estimating MWC, and use the results to explain the differences among previously reported estimates. By clearly identifying the determinants of MWC, our analysis reveals that alternative interpretations of distinct thought experiments and different implicit assumptions about the elasticity of labor supply with respect to public spending account for all of the unexplained variations.

In the analytical approach to estimating MWC pursued by Wildasin (1984), Browning (1987), and Mayshar (1991), distinct form~lae are derived for alternative thought experiments. These formulae are then evaluated under various assumptions about the values of key parameters, including labor-supply elasticities and average and marginal tax rates. In the simula- tion approach adopted by Stuart (1984), Ballard (1990), and Ballard and Fullerton (1992), estimates of MWC are generated numerically using models of the economy that employ explicit functional forms for preference relations and production technologies. The utility and production functions are calibrated by data to yield specific values for the share and elasticity parameters.

The principal contribution of this paper is the derivation of a single formula that yields exact estimates of MWC for any specified thought experiment, without imposing restrictive assumptions on preferences or technologies. Our general framework encompasses the various special cases previously considered, and reveals the precise influence of all relevant behavioral and institutional parameters. In particular, we show that previ- ously unresolved differences in estimates of MWC are explained by appropriate interpretations of the intended thought experiments and by alternative assumptions about the elasticity of labor supply with respect to public spending.

The disparities among estimates of MWC have been attributed to a variety of sources. Ballard (1990) and Mayshar (1991) emphasize the importance of distinguishing between differential-incidence analyses, in which tax revenue is held constant, and balanced-budget experiments, in which increased tax revenue finances a publicly provided good. However, neither Ballard nor Mayshar is able fully to reconcile his estimates of MWC with those reported by Browning (1987) or Wildasin (1984). We find that appropriate interpretations of alternative thought experiments resolve some of the variatior~., but conclude that differing assumptions about the response of labor supply to changes in public spending also play an important role.

Fullerton (1991) emphasizes that different measures of excess burden have been employed to estimate MWC, and also suggests that Browning's (1987) estimates fail to incorporate the 'revenue effect" identified by Atkinson and Stern (1974). Although the use of different excess burden measures does explain ~ome of the disparities, we demonstrate that the estimates reported by Browning do incorporate the revenue effect once they are associated with the appropriate thought experiment.

A. .Snow. R.S. Warren, Jr. / Journal of Public Economics 61 (1996) 289-305 291

We arrive at these conclusions by first developing a formula in which MWC is expressed in terms of the uncompensated elasticity of labor supply with respect to public spending, An alternative but equivalent formula is then derived in terms of the compensated labor-supply elasticity, These formulae confirm the insights of previous studies concerning the importance of certain behavioral elasticities and institutional parameters, while also revealing critical determinants of MWC previously overlooked.

The paper is organized as follows. Section 2 sets out a model of consumer behavior in which the individual takes as given the level of public spending, which is financed by distortionary wage taxation. In Section 3 we derive two equivalent formulae for MWC using, respectively, the uncompensated and compensated labor-supply functions, and compare our general formulae with those developed by Wildasin (1984), Browning (1987), Ballard (1990), and Mayshar (1991) for the various special cases they consider. In Section 4 the formulae are applied to calculate MWC, producing values equal or close to the approximations reported by Stuart (1984), Browning (1987), Ballard (1990), and Ballard and Fullerton (1992). Section 5 summarizes our principal results and provides concluding remarks.

2. Consumer behavior in fiscal equilibrium

We investigate an economy with n identical consumers, each of whom derives utility u(x, l. z ) from a composite private good, x, which serves as numeraire, leisure, I, and a publicly provided good, z. The utility function is assumed to be twice continuously differentiable and strictly quasieoncave. The indirect utility function, defined by

V ( w , z , 1) = m a x {u(x, I, z)[x + wl = I and 1 ~< T}. (1)

depends on the net-of-tax wage rate, w, the level of z, and income. 1, devoted to spending on x and 1. Each consumer is endowed with T units of time and a fixed amount of capital, K. The marginal product of labor in producing the composite private good is equal to the gros~,-of-tax wage rate W = d Y / d h =ft,(h, K), where h = T - 1 denotes hours of labor supply, Y is the per-capita output of the composite good, and fh is the marginal product of labor. The publicly provided good, z, is produced using the composite good at an average cos', of A C : and marginal cost of M C : , both of which may depend on z.

Public spending is financed by a tax on wages. The government's fiscal policy or tax-and-spending program (m, y. z, A) specifies four parameters:

(i) the marginal ad valorem tax rate, m, which determines the net-of-tax wage rate

w = (1 - m ) W ; (2)

292 A. Snow, R.S. Warren, Jr. / ,;6t~rnal of Public Economics 61 (1996) 289-305

(ii) the a m o u n t of wage income y exempt f rom taxa t ion , which de t e rmines the ave r age tax ra te

a = m(1 - y / W h ) ; (3)

(iii) the a m o u n t of the publ ic ly p rov ided good, z ; and (iv) the per -cap i ta a u t o n o m o u s income transfer , A. E a c h c o n s u m e r t rea ts the fiscal policy as exogenous and chooses a

p r iva te ly op t ima l consumpt ion bundle tha t solves the ut i l i ty max imiza t ion p r o b l e m s ta ted in (1), g iven the vir tual or ' full ' income

l = w T + m y + N + A , (4)

whe re N deno t e s the consumer ' s capi ta l income, Y - W h , which is t r ea ted as exogenous . The resul t ing labor-supply funct ion is g iven by

h = T - l ° (w, z, 1 ) , (5)

whe re l ° deno te s the u n c o m p e n s a t e d (ordinary) d e m a n d funct ion for le isure .

T h e mode l is c losed by assuming tha t per -capi ta tax r evenue ,

R = a W h , (6)

equa l s per -capi ta publ ic sector spending , so tha t the g o v e r n m e n t ' s budge t is ba lanced . Thus ,

z = P - R ( n / A C ~ ) , (7)

w h e r e P is the p ropor t ion of to ta l tax revenue devo ted to exhaus t ive publ ic spend ing , t Similar ly , t ransfer p a y m e n t s are g iven by

A = (1 - P ) " R / A C A , (8)

w h e r e A C a ~> 1 is the ave rage cost of t ransfer r ing a dol la r of tax revenue . U n i q u e a m o u n t s of the publ ic ly p rov ided good and per -capi ta t ransfers are a s s u m e d to satisfy re la t ions (6)- (8) and to depend d i f ferent iably on the p a r a m e t e r s (m, a, p).2

W e examine the effects of a marg ina l increase in t axa t ion accompan ied by an equa l increase in spending . To encompass a var ie ty of hypothe t ica l

We follow Ballard et al. (1985b) and use the term 'exhaustive" spending to refer to the resource-using public expenditures envisioned in a balanced-budget analysis as opposed to the income transfers implicit in a differential-incidence analysis, which use resources only because of administrative overhead. Also, we follow Wildasin (1984) and refer to "compensated" and 'ordinary" independence for cases where, respective!y, )he compensated and uncompensated elasticities of labor supply with respect to the publicly provided good equal zero.

-" Eqs. (2). (4), and (5) define labor supply h as a function of (m. y, z, A). Assuming that (3) identifies a one-to-one relation between a and y, we treat h as a function of (m. a, z, A). Eqs. (6)-(8) then yield labor supply in fiscal equilibrium as a function of (m. a. P).

A . Snow. R . S . Warren. Jr. / lourna l o f Public Economics 61 (1996) 289-305 293

spending reforms, we allow tax revenue to be allocated differently at the margin than it is on average. Thus , while P is the proportion of totai tax revenue spent on exhaustive spending, we let ~, which may differ from P, denote the proportion of marginal tax revenue spent in that manner. As a consequence, the incremental change in P is determined implicitly as d P = (~ , - P ) d R / R , while the change in the public provision of z is given by

d z = *r - d R ( n l M C ~ ) , (9)

and the change in income transfers is given by

dA = (1 - ~r). d R / M C A . (10)

where M C A ~> 1 is the marginal cost of transferring a dollar of tax revenue ) Our approach allows for the analysis of a variety of possible tax and

spending policies. First, the average tax rate, a, can differ from the marginal tax rate, m, because of the income exemption, y, as shown by Eq. (3). Second, the income transfer, m y , implicitly effected by the exemption, is independent of the explicit income transfer, A , as shown by the income relation (4). Thus , average and marginal tax rates can be adjusted in- dependent ly of both each other and the explicit transfer payment by changing the exemption level as needed to maintain (3).

Finally, as is customary, we specify a change in tax policy in terms of a change in the average tax rate, even though this variable is actually determined endogenously by the consumer 's choice of labor supply. In practice, a change in tax policy is formulated as a change in the exogenous exemption level. As indicated by (3), however, the relationship between the average tax rate and the exemption level is solely a matter of definition, and therefore a change in tax policy can be equivalently expressed in terms of either variable.

3. Marginal welfare cost

In this section, two equivalent formulae for M W C are derived by using, in turn, the ordinary and compensated labor-supply functions. We begin by analyzing the effect of the policy change [dm, da, dP] on utility, since this approach allows us to develop a general expression for M W C within a benefi t-cost framework. The total change in utility is given by

3 Since, by assumption, the total cost of z is independent of the wage rate, we can write the total cost function as TC(z) and the balanced-budget condition as TC(z) = P . hR. From the latter, with A C = = TC(z ) / z . we obtain Eq. (7), and with OTC(zl/Oz = M C : , we obtain M C : dz = - d P nR + P . n d R which, with d P = (¢t - P ) d R / R . yields Eq. (9).

294 A. Snow, R.S. Warren. Jr. ! Journal of Public Econo.,nics 61 (19961 289-305

d V ( w , z, l) = {V__dz + V t d A } + {Vwdw + Vidl } , (11)

where subscripts indicate partial differentiation. The terms in the first set of braces represent the direct effects of the changes in exhaustive spending and in income transfers determined by (9) and (10), respectively. The terms in the second set of braces represent the direct effects of the tax change and the combined effects of the tax-and-spending reform on labor supply.

The change in utility per dollar of tax revenue gives the marginal net benefit ( M N B ) of the policy change as

M N B = d V / V , d R . (12)

Expanding the r ight-hand side by using (9)-(11) yields:

M N B = {(V: d z / V , dR) + ( d A / d R ) } + {(I/,,, dw/V~, dR) + ( d l / d R ) }

= {Ir( '~ M R S / M C = ) + (1 - ~ r ) / M C a } - {1 + M W C } , (13)

where Y . M R S = n V : / V t is the marginal social value of the publicly provided good., M W C is the marginal welfare cost of public funds, defined by

M W C = - 1 - { ( V ~ d w / V t d R ) + ( d I / d R ) }

= - 1 + {Whda - [ m - (1 - 7 ) a ] W d h } / d R , (14)

and the parameter y = - ( d W / d h ) ( h / W ) = - hfhh/f~, is (minus) the elasticity of the marginal product of labor with respect to labor supply. ~ Thus, M N B is

the difference between the marginal benefit of a dollar 's worth of public spending, the first term within braces in (13), and the marginal cost of public funds, which is one plus M W C .

Marginal welfare cost can be evaluated using (14) once dR and dh are determined. The relationships between m , a, and R established by (3) and (6) imply the following change in tax revenue:

d R = W h d a + (1 - y ) a W d h . (15)

To derive the expression in braces on the second line of (14). we substitute dl = T dw + dmy+dN. V =-V~I. dLv=-Wdm~( l -m)dW, and dN=(f~, W ) d h - h d W = h d W to arrive at

V d'.:" * l/~ dl - -V~Wh dm - mV, h dW + V~ dmy.

Using (3) to substitute for myiWh = m - a . and the differential of (3}.

da = dm - (dmylWh) + (my/WZh:)(W dh + h dW).

to substitute for dmy - Wh dm = - W h da + (m~,';Wh)(W dh + h dWL and then substituting dW = - T dh/h and simplifying the result, we obtain the set of terms on the second line of (14).

A. Snow. R.S. Warren. Jr. / Journal of Public Economics 61 (1996) 2,~¢9-305 295

By substituting this expression into (14), we obtain:

M W C = - m d h / [ h d a + (1 - y )adh] , (16)

so that M W C can be evaluated once the balanced-budget change in labor supply, dh, is determined. 5

The two equivalent formulae for M W C are obtained by decomposing dh into tax and spending effects, and analyzing both uncompensated and compensated labor-supply responses to the change in public spending. Taking account of the Slutsky relation:

TI~. - n~. = - ( w h / l ) ~ , (17)

where ~ t = - l l ~ / h is the income elasticity of labor supply, and using Eq. (3) relating labor supply to the average and marginal tax rates. Eq. (16) yields:

n ~ . ( d m / d a ) - (n~. - n ~ ) - (1 - m ) B ° /a • [ 1 - m a / d m ~ ] (t8)

M W C = 1 -mm + rs~ 1_ r----m--- - (1 - y t)~n l ~ - - ) j -; .'7~ - v/','

where B ° is given by

n ° : ~ r ( n ' ~ / P ) ( A C = / M C :) + (1 - ,0(*7 ° - r/~, )a/(1 - r n ) M C a , (19)

and "O~ ° = - z l ~ / h is the uncompensated elasticity of labor supply with respect to z. Details of the derivation are given in the appendix.

As (19) indicates. B ° would be equal to zero except for the change in labor supply induced by the change in public spending, either through ordinary complementarity effects in the case of exhaustive spending (rt'-' # 0) or through income effects in the case of transfers ( 7 / ° - r/~,)# 0. Moreover. Eqs. (18) and (19) show that M W C is lower the stronger is the ordinary complementarity between labor supply and exhaustive public spending. Finally, we note that if the compensated wage elasticity of labor supply is zero (r~. = 0), then leisure and private spending are, using Samueison's (1974) terminology. Fisher-perfect complements so that wage taxation is equivalent to lump-sum taxation ( m = 0 ) . In this case, Eq. (18). when multiplied by m / m , reveals that M W C equals zero.

An alternative but equivalent expression for M W C is derived from (18) by using the compensated labor-supply function. Differentiating the identity I°(w, z , 1 ) = - l ' ( w , z , V ( w . z . 1)) with respect to z. where l ' is the compen- sated demand function for leisure, and expressing the result in terms of elasticities yields:

Snow and Warren (1989) provide a comprehensive analysis of the effects on labor supply of a tax increase that is accompanied by an equal increase in public spending.

296 A. Snow. R.S. Warren, Jr. / Journal of Public Economics 61 (1996) 289-305

77 = n~ + ( z M R S / I ) ~ , (20)

where ri: = - z l : / h is the compensated elasticity of labor supply with respect to z. Substituting for rt ° from (20) into (18) and (19), using (7) and (13), and rearranging terms yields:

c {ri~ - r i w ) M N B - (1 - m ) B /a M W C = r iw(dm/ da) + ~ o ¢ 1 - m ~ [ 1 - m a / d m \ ]

m + r i ~ [ _ 7 ~ - (1 - Y)-m-m [,--~-) J

(21)

where

B e = .n - ( r i~ /p ) (ACZ/MC z) (22)

is the compensated version of B °, and therefore contains no income effects. Eqs. (21) and (22) reveal that M W C is lower either when there is compensated complementarity between labor supply and exhaustive public spending (ri~ >0) or when the marginal net benefit of such spending is negative, assuming leisure is a normal good.

The formulae for M W C given in (18) and (21) are necessarily equivalent because the latter is derived from the former. The equivalence is most readily apparent when the income elasticity of labor supply is zero since, in that case, Eqs. (17) and (20) show that the compensated and ordinary labor-supply elasticities are equal, i.e. ri~ = rio and ri: = 17°.

The three terms in the numerator of formula (18) correspond to the distinct sources of welfare cost identified by Atkinson and Stern (1974). Using their terminology, the "distortionary effect' caused by substitution away from the tax base is captured by the first term. The second term is the "revenue effect' arising from income effects that expand the tax base and counteract the distortionary effect when leisure is normal. The third term encompasses the "spending' or 'budget' effects discussed by Diamond and Mirrlees (1971) and Lindbeck (1982), and reflects the influence of public spending on the size of the tax base. In formula (21), the revenue and spending effects appear in their compensated forms.

4. Relationships with previous stt~dies

In a special case investigated by Wildasin (1984) and Browning (1987), the tax system is proportional ( a - ~ m , d m / d a = 1, and hence y =0) , all tax revenue is spent on the publicly provided good z (P = 1 = ~r), and the

A . Snow. R .S . Warren, Jr. / Journal o f Public Economics 61 (1996) 289-305 297

pre-tax wage rate remains constant (T = 0). With these assumptions, Eq. (21) reduces to

I~=m m c + . . . . / • MWC _ l - m [71~ (.~-n.)MNB]-n,(AC MC ) ~rl c

1 - ~ _ m ~ I ]=p=z¢

(23)

This expression corresponds to the set of inequalities obtained by Wildasin in his relations (9).

Browning (1987, p. 18) further assumes that ". . .marginal government spending provides benefits that return taxpayers to their initial (i.e., before the tax and expenditure change) utility levels", implying that taxation is at an optimum (MNB = 0). Although Browning does r, at explicitly refer to an assumption of compensated independence (~/~ = 0), he does assume that the publicly provided good is a perfect substitute for income (V~ = VI), in which event Eq. (20) is valid only if 7/~ = 0. Under these assumptions, (23) reduces to Browning's formula (11), which is given by

MWC a=--m 0=1, l = p - T r O= M N S=,)~

m c m c (24)

when the proportional tax rate increases incrementally. By calculating MWC using (21) and (24) whh Browning's parameter values for m, dm/da, and ~ , we effectively reproduce the lower portion of his table 2, with the correspondence being exact in the case of proportional taxation.

Mayshar (1991) shows that the right-hand side of (24) is proportional to MWC for a thought experiment in which marginal tax revenue is spent on a lump-sum income transfer. While this coincidence is of some interest, our analysis confirms Wildasin's (1984) original interpretation of (24) as applying to the case of exhaustive public spending.

Mayshar (1991) derives an alternative expression for MWC in the case of a lump-sum income transfer. Assuming proportional taxation, a constant pre-tax wage rate, and no marginal transaction cost in transferring income (MC A = 1), formula (18) reduces to Mayshar's eq. (4). Under Browning's additional assumption that the uncompensated wage elasticity of labor supply equals zero (r/~ = 0), (18) further reduces to

MWC a ~_ ,,.n

0= 7

A 0 1 = MC . O=r)~

m =i_--c-~.;, (25)

- ~ 8 A . S n o w , R .S . Warren. Jr. ! Journal o f Public Economics 61 ( t 9 9 6 ) 2 8 9 - 3 0 5

which is Browning's eq. (10) when the proportional tax rate increases incrementally.

While Browning associates this formula with the case of exhaustive spending, Fullerton (1991) observes that the association is incorrect, since the "revenue effect' of taxation would be missing. However, when (25) is interpreted as applying to the tax/transfer case, the revenue effect is properly incorporated and exactly counterbalances the spending effect. |ndeed, using Browning's parameter values for m, dm/da, and 7/~., Eqs. (18) and (25) effectively reproduce the upper portion of his table 2. Once again, the correspondence is exact in the case of proportional taxation.

The formulae for MWC in (24) and (25) refer to alternative thought experiments. The former is relevant in a balanced-budget analysis pertaining to exhaustive spending, while the latter applies to the tax/transfer or differential-incidence analysis, when distortionary taxation replaces an equal amount of lump-sum taxation. Nonetheless, for ooth experiments, MWC consists entirely of the distortiouary effect, the revenue and spending effects being absent from (24) and offsetting in (25). Consequently, in both instances, MWC equals zero if and only if leisure ~a. ~ private spending are Fisher-perfect complements.

For exhaustive public spending, Mayshar (1991) re onsiders Wildasin's (1984) analysis in the case of ordinary independence. Under this assump- tion, (18) reduces to Mayshar's eq. (3). Alternatively, assuming proportional taxation, a constant pre-tax wage rate, and no income transfers, (18) reduces to

m o . m o

(26)

which corresponds to Wiidasin's eq. (7). When 77 ° = 0, (26) yields Mayshar's eq. (1):

MWC m o/(

]=P ,-r 0 - ~ o

(27)

Thus, as observed by Stiglitz and Dasgupta (1971) and Atkinson and Stern (1974) in the context of optimal public spending and taxation, and as emphasized by Fullerton (1991) and Ballard and Fuilerton (1992) in the

A. Snow. R.S. Warren, Jr. I Journal o f Public Economics 61 (1996) 239-3f~ 299

general case when spending and taxation are not optimal, M W C can be negative if there is ordinary independence and the uncompensated wage elasticity of labor supply is negative.

Formulae (24) and (27) both pertain to the case of exhaustive spending, but under different assumptions, The distortionary effect of wage taxation is isolated in (24), since the revenue and spending effects vanish under the assumptions of optimal public spending and compensated independence. By contrast, public spending need not be optimal in (27), but the spending effect vanishes under the assumption of ordinary independence, and the combined distortionary and revenue effects arc proportional to the un- compensated wage elasticity of labor supply when d m / d a = 1. Indeed, it follows from (18) and (19) that 7/° = 0 and d m / d a = 1 are jointly sufficient for M W C to be proportional to m~°~ when all marginal tax revenue is allocated to exhaustive public spending.

Stuart (1984) provides estimates of M W C for the case of ordinary independence from a computer simulation of a model using a C o b b - Douglas production function with constant returns to scale. Under the latter assumption, y = (1 - s)/cr, where s is labor's share of output for which Stuart uses 0.72, and o- is the elasticity of substitution which equals unity in the Cobb-Douglas case. Stuart obtains M W C = 0.072, while we obtain 0.076 using (18) and his parameter values. Stuart also reports a value of 0.20"/for M W C in the tax/transfer experiment, while our formula yields 0.212.

Ballard (1990) considers the case of ordinary independence in the context of a simulation exercise, but he assumes proportional taxation along with 7/~ =0.35, m =0.4 , y=0 .25 /0 .8 , and ~r/MC A =0.85. 6 These parameter values applied to formula (18) yield estimates that closely track the curve labelled 'Marginal Welfare Cost' in his fig. 1, illustrating M W C as a function of rl~. For the differential-incidence or tax/transfer analysis conducted by Ballard (1990), we are able to replicate exactly the curve labelled 'Marginal Excess Burden' in his fig. 1 by using his parameter values for r/~, m, and y in Eqs. (18) and (19) with ~r = 0 and M C '~ = 1.

In a case considered by Ba!lard (1990), Fullerton (1991), and Ballard and Fullerton (1992), existing taxation is distortionary but marginal tax revenue is collected lump sum. For this experiment, M W C can be evaluated by setting d m / d a equal to zero, reducing formula (18) to

M W C dm/da=O F m o c _ m B O / a ] / =

m c o [1 + -7.)] (28) e T h e values for y and ¢r are oblained from Ballard (1987). and we assume that M C A = I.

300 A. Snow, R.S. Warren. Jr. / Journal o f Public Economics 61 (1996) 289-305

When all marginal spending is exhaustive (zr = 1) and there is ordinary independence (r/~ = 0), a constant pre-tax wage rate, and perfectly inelastic labor supply, Eq. (28) reduces to

M W C dm/da=O = - ~ _ m T l ~ 1 + l_ - -Z~Tl~) . ~ = O , ~ = t

(29)

Although (27) and (29) both refer to the case of exhaustive spending with ordinary independence, so there is no spending effect in either experiment, in (29) the marginal tax revenue is collected by lump-sum taxation rather than by the distortionary taxation assumed in (27). Consequently, there is no distortionary effect in (29), but only a revenue effect which is propor- tional to (minus) the compensated wage elasticity of labor supply when the uncompensated wage elasticity equals zero.

Formula (29) confirms the conclusion reached by Fullerton (1991) that M W C can be negative when marginal tax revenue is collected lump sum. In addition, using Ballard's (1990) parameter values in our formula (28), we obtain estimates that effectively reproduce the curve for "Lump-Sum Tax Finance" in his fig. 2, illustrating M W C as a function of 7/~.

Eq. (28) affirms that M W C = 0 when there is no distortionary taxation and all revenue is collected lump sum (m = 0). Alternatively, when m is positive but all marginal tax revenue is spent on transfers (It = 0), the M W C of lump-sum taxation is proportional to the marginal transaction cost of transferring tax revenue, M C a - 1, since the numerator in (28) r~duces to

m o c ,4 / A 1 - m ( n ' - ~ ) ( M C - 1 ) MC .

In this case- M W C = 0 if M C A = 1, showing that lump-sum taxation has no excess burden whc-n the tax proceeds are rebated lump sum.

Ballard and Fullerton (1992) present estimates of M W C from simulations of increases in the tax on wages for both proportional and progressive tax structures, and increases in lump-sum taxation. Using our formulae, we obtain estimates that are identical to theirs in most cases, and in no instance do the estimates differ by more than one cent. 7

• We cannot replicate the [onrth column of Ballard and Fullerton's (1992) table 1, which reffers to a case in which "'the inframarginal [tax] rate does not change", since the average tax rate changes, but in a manner not reported. With dm/da = 3.2, formula (18) produces estimates close to theirs.

A . S n o w . R . S . Warren, Jr. / Journal o f Public Economics 61 (1996) 2 8 9 - 3 0 5 301

W e c o n c l u d e by o b s e r v i n g t ha t w h e n m a r g i n a l t ax r e v e n u e is s i m p l y

w a s t e d , M N B = - ( 1 + M W C ' ) , w h e r e M W C is o b t a i n e d u s i n g f o r m u l a

(18) w i t h B ° = 0, s ince t h e r e a re n o u n c o m p e n s a t e d s p e n d i n g ef fec ts . In t h e

case o f l u m p - s u m t a x a t i o n , Eq . (28) w i th B ° se t e q u a l to ze ro s h o w s t h a t

M W C is n e g a t i v e ( p o s i t i v e ) w h e n l e i su re is n o r m a l ( in fe r io r ) .

Table l

MWC values from

Source Assumptions Source Eq. (18)

Browning (1987) ~r = 1 *1~ = 0 M N,~ = 0 0.310 ~ 0.246 ~

Browning (1987) ~ = 0 if'. = 0 0.23@ 0.212 d

Stuart (1984) ¢r = 1 ¢/~ = 0 o = 0 0.072 0.076 ~

Stuart (1984) ~" = 0 o = 0 0.207 0.212 a

Bal[ard (1990) ¢r = 0.85 ¢/~ = 0 ~ = 0 0.001 f 0.030 ~

Ballard (1990) n" = 0 n~ = 0 0.195 0.197 ~

Ballard and Fullerton "rr = 1 7/= ° = 0 ~,/~ = -0.022 0.025 0.023' (1992)

Ballard (1990) lump-sum taxation "rr=l "1/_.°=0 r/~=O -0.130 -0.(D2'

Ballard and Fullerton lump-srrq taxation (1992) ¢r = 1 7/~ = 0 T/~ = -0.022 -0.078 -0.078 ~

From Browning (1987, eq. (11)),

( m + d m / 2 ~ ( d m / d a ) M W C =

l - m - (m + dm)r/~(dr~/d~j "

with dm/2 = 0.005 and Stuart's (1984) parameter values for ~ , = 0.2. m = 0.427. a = 0.273, and din~do = m / a . With din~2 = O. M W C = 0.304.

h From (21), which is derived from (18}, evaluated with B: = 0 and with Stuart's parameter values for r/i, m, a, din/do, and ~, = 0.28.

Reported by Fullerton (1991) and obtained from Browning (1987. ¢q. (10)). M W C = [(m + d m / 2 ) q : d m / d a ] / [ 1 - m], with din/2 = 0.005 and Stuart's parameter values, With d m / 2 = O, M W C = 0.233.

d From (18), using (19), with Stuart's parameter values and M C ~ = l . " From (18) with Stuart's parameter values. f Reported in Ballard (1987, fn. 7). s From (t8), using (19), with Ballard's (1990) parameter values for T/~ = 0.35, m = 0.4, aild

y = 0.25t0.8, and assuming M C ~ = I . h From (18) with Ballard's parameter values. 'From (18) w/th Ballard and Fullerton's (1992) parameter values for "y=0. m =0.43.

a = 0.27, d m / d a = re~a, and ~/~ = 0.09. ~From (28). which is d~rived from (18), evaluated with Ballard's parameter values.

From (28) with Ballard and Fuilerton's parameter values.

392 A. Snow. R.S. Warren. Jr. / Journal of Public Economics 61 (1996) 289-305

"Fable 1 summarizes these results by comparing selected values of M W C

previously reported in the literature, listed in the second column from the right, with estimates obtained from our formula, given in the right-most column, under the assumptions indicated in the middle columns. The small differences between our estimates and those reported by Stuart (1984), Ballard (1990), and Ballard and Fullerton (1992) can be attributed to their use of finite changes in taxation to approximate the infinitesimal changes to which our formulae, apply. Browning's (1987) estimates tend to be slightly above ours outside the case of proportional taxation, even when his formulae are evaluated at the margin, s Finally, the estimates reported by Ballard (1990) and Ballard and Fullerton (1992) are based on simulation procedures that incorporate changes in the average rate of taxation which they do not report. Our inability to account fully for such changes may explain some of the differences between our estimates of M W C and those obtained in these studies.

5. Conclusions

We use an analytical approach to reconcile the differences between previously reported estimates of M W C by identifying the exact influence of alternative parameter values and distinct thought experiments. Moreover, our formulae for M W C can be us,~d to evaluate alternative fiscal reforms once the values of a few key parameters are specified. Although some of the critical variables in our formulae are determined by the fiscal reform under study, others are institutional and behavioral parameters that must be estimated. Examples of the latter e, re the effective average and marginal tax rates and the elasticities of labor supply with respect to the wage rate, income, and exhaustive public spending.

Some attention has been accorded in previous literature to estimating effective tax rates on labor income. For example, Stuart (1984) and Ballard et al. (1985a,b) construct est imates of effective average and marginal tax rates, Pechman (1987) calculates effective average tax rates, and Browning and Johnson (1984) and McGrattan (1994) provide estimates ef effective marginal tax rates. However, Killingsworth (1983, 7p. 341-352) emphasizes that published empirical estimates of labor-supply functions implicitly

Browning (1987) evaluates his formulae for a small, finite increase in the marginal tax rate, dm = 0.01. When evaluated at the margin (din = 0). his formulae continue to yield estimates of M W C above ours outside the ease of proportional taxation.

A. Snow. R.S. Warren. Jr. / Journal o f Public Economics 61 (1996) 2,~9-305 303

assume ordinary independence between labor supply and exhaustive public spending. Not only does this practice potentially bias existing estimates of wage and income elasticities, it also precludes estimation of the elasticity of labor supply with respect to exhaustive public spending, Hence, further progress in using analytic formulae to calculate M W C awaits estimates of labor-supply functions that include government spending as an explanatory variable.

Acknowledgements

We thank Charles Ballard, Edgar Browning, Don Ful!erton, Charles Stuart, and Gregory. Wozniak for their helpful comments on an earlier version.

Appendix

This appendix provides details of the derivations of formulae (18) and (21). To derive formula (18), we first evaluate the change in labor supply, dh = - I ° ~ . d w - l ~ d z - l~dl, by substituting the Slutsky equation, l~ = l ~ - liT, the income change, d l = T d w + drny + d N + dA, with dN = - h d W , the spending changes, dz and dA given in (9) and (10), respectively, the revenue change, dR, given in (15), along with dw = - W d m + (1 - m ) d W and d W = - ~,Wdh/h, to obtain:

dh = Wl~[dm + (1 - m)~dh/h] - t~r (n / MRT)Wh[da + (1 - 3,)adh/h]

- l~{drny - W h d m + m W ~ d h

+ [(1 - 7r)/MCa]Wh[da + (1 - y)adh/h]} .

Next, we substitute dmy - W h d m = - Whda + ( m y / W h ) ( W d h + h d W ) and m y / W h = r n - a from Eq. (3), use the Slutsky relation (17). and the expressions for R and z given in (6) and (7), respectively, and rearrange t e r m s to a r r ive at

dh -- n~dm + [~/~. - "q:. + (I - m)B"/a]da

h ( 1 - m ) ( l + ¢ + . . . . . . m - - ~7 , ) Ir/~-r/~;~[ (i " y ) a ] - I I - m ) ( l - 7 ) B °"

304 A. Snow. R.S. Warren, Jr. I Journal of Public Economics 61 (1996) 289-305

w h e r e B ° is de f ined in E q . ( i9) . Final ly , us ing th is expres s ion to subs t i tu te f o r d h in E q . (16) a n d r e a r r a n g i n g t e r m s yie lds (18).

T o de r ive f o l m u l a (2 l ) , w e subs t i tu te fo r r/° f r o m (20) a n d fo r z f r o m (7) i n to (18) a n d (19) to ob t a in :

- q : ( d m / d a ) + (-q: - 7I]){~r(ZMRS/MC ~) + [(I - ~ ) , ' M C a] - 1} - (1 - m ) B ¢ / a

in the n u m e r a t o r . Subs t i t u t i ng M N B + M W C f r o m (13) f o r the t e r m in b r a c e s a n d t hen so lv ing the resu l t ing express ion fo r M W C yields (21).

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