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Progression under the benefit approach to the theory of taxation
95
Economics Letters 8 (1981) 95-99 North-Holland Publishing Company
PROGRESSION UNDER THEORY OF TAXATION Daniel
THE BENEFIT *
APPROACH
TO THE
KOVENOCK
CJniuermt_v of Wtsconsin,
Efraim
SADKA
Received
30 July 1981
Mudison,
WI 53706,
USA
The implications for progression of the benefit approach to the theory of taxation are studied in this paper. It is concluded that the case for progression rests on the degree of substitutability between public and private goods.
1. Introduction In the last decade we have witnessed major contributions to the theory of taxation which were essentially based on the ability-to-pay approach. ’ One of the conclusions drawn from these studies is that the case for progression is doubtful. 2 The purpose of this note is to investigate the implications for progression of an alternative approach to the theory of taxation, the benefit principle, as highlighted by the Lindahl equilibrium concept for pure public goods. 3 We find that the case for progression rests on the degree of substitutability between public and private goods. * The research reported here is financially supported by NSF Grant DAR-7917376 and by funds granted to the Institute for Research on Poverty at the University of WisconsinMadison by the Department of Health and Human Services (formerly HEW) pursuant to the provisions of the Economic Opportunity Act of 1964. The opinions expressed are those of the authors. The authors wish to thank Gene Smolensky for useful comments on an earlier draft of this paper. ’ See Atkinson and Stiglitz (1980) for references. ’ See. for instance, Sadka ( 1976). 3 See Musgrave (1959, ch. 4) for a historical survey of the two approaches.
0165-1765/81/0000-0000/$02.75
0 1981 North-Holland
96
D. Kovenock,
E. Sadku / Progression under the benefit approach
2. The model Consider a continuum of agents, 4 each consuming two goods: x which is a pure public good and y which is a pure private good. The agents all have the same preference ordering over bundles (x, y) which is representable by a twice continuously differentiable and quasi-concave utility function u( ., e). Each agent is distinguishable by his/her real income (initial endowment) which consists of a certain quantity of the private good. These incomes are contained in the closed and bounded interval [I,, Z,], where I, > 0. For each Z E [I,, Z,], we denote by Z’(Z) the number of agents with income less than or equal to I. We refer to an agent with income Z as ‘agent Z ‘. We denote by p the fixed unit cost of the public good in terms of the private good. The economy is in a Lindahl equilibrium. The Lindahl price (in terms of the private good) paid by agent Z for the public good is denoted by p*(Z). Let x* > 0 be the equilibrium supply of the public good. Then agent Z consumes y*(Z) = Z-p*(Z)x* units of the private good. For the sake of simplicity, we assume that all have a positive consumption of the private good, i.e., y*(Z) > 0 for all Z E [I,, Zz]. Given this assumption, which excludes corner solutions for each agent’s utility maximization problem, the Lindahl equilibrium is characterized by the following two equations: s(x*,z-p*(z)x*) $*(I)
/ 11
dF(Z)
‘p*(z)
forallZE[Z,,Z,],
=p,
(1)
(2)
where S is the marginal rate of substitution (MRS) of the private good for the public good, i.e., S(x, y) = u,( X, y)/u,,(x, JJ), where subscripts stand for partial derivatives. The first of these two equations states that each agent equates his willingness-to-pay for the public good to the price he/she actually pays for it. The second one states that with our constantreturns-to-scale technology the price of the public good, as perceived by the producer [namely, j?p*( I) dF( Z)], must equal its unit cost (namely, P). Before turning
to the results, let us make one more assumption
which
4 We consider a continuum rather than a discrete number of agents in order to facilitate the use of calculus.
91
D. Kovenock, E. Sadka / Progression under the benefit upprouch is
that. the public good is a normal good. This means that the MRS of Y for x rises as y is increased and x is kept constant, or formally: S,( x, y ) > 0 everywhere.
3. The results Agent Z pays a price p*( I) per unit of the public
good. Thus, he/she
pays a total of T*(Z)dAfp*( Z)x* for that good. This amount can be viewed as the total tax that he/she pays under the benefit principle of taxation. We are interested in the question of how this tax varies with income I. Observe that it follows immediately from our normality assumption (namely, SY > 0) that p*(Z) and hence also T*(Z) increase in income. Thus, the rich pay higher taxes than the poor. However, a more interesting question is whether the rich pay a higher proportion of their incomes in taxes than the poor. In other words, does the benefit principle imply progressive taxation? Formally, if we let t*(Z) ‘zf T*(Z)/Z be the average tax rate, then we ask whether t* is increasing in Z. By totally differentiating (1) with respect to I, we obtain
S,(x*,y*(i)) S(Z) = 1 +x*S,(x*,y*(Z)) which implies
’
that
g(z)= ~[z~(z)-p*(z)] 1 1 . I Therefore,
we conclude
that
Thus, the tax progressivity under the benefit principle depends exclusively on the magnitude of the elasticity of the MRS with respect to y,
98
namely: y$,/S. It turns out that the latter expression is exactly the inverse of the elasticity of substitution between x and y when the utility function is homothetic. To see this, notice that in the latter case, we can write s(x,y)
=S(l,y/x)~fH(y/x).
(3)
Now, the elasticity of substitution (a) is defined as the elasticity of y/x with respect to the MRS. Hence, u ~ ’ will be equal to the elasticity of the MRS (i.e., H) with respect to y/x: -= 1 O(Y/X)
WY/X)Y -NY/X)
Using (3), it follows that S,.(X,Y)
(5)
= NY/X)/X.
Combining
(3) (4) and (5) yields
-= 1 D(Y/X)
sJx,r)v S(X,Y)
.
Thus, we have proved: Proposition. If the public good is normal, then the tax according to the benefit principle is positively related to income. When, furthermore, the utility function is homothetic, then the tax is progressive, proportional or regressive as the elasticity of substitution between the public and the private goods is less than, equal to or greater than unity. 4. Concluding
remarks
(1) As one does not expect a high degree of substitutability between the public and the private goods, ’ it seems that the case for progressive ’ For instance,
Maital (1973) presents
estimates
of (r which are around
2/3.
D. Kovenock,
E. Sdka
/ Progression
under the benefit upprouch
99
taxation under the benefit approach is quite strong. (2) The proposition of the preceding section provides a readily applicable test for progression. For instance, with the CES class of utility functions, u(x, y) = [(Y,x~ + ~~~y~]‘/p, one has only to see whether the parameter u = l/( 1 - p) is greater than, equal to or less than unity.
References Atkinson, A.B. and J.E. Stiglitz, York). Maital, S., 1973. Public goods and 41, 561-568. Musgrave, R.A.. 1959, The theory Sadka, E.. 1976. On progressive 931-935.
1980 Lectures
on public
income distribution:
economics
Some further
(McGraw-Hill.
New
results, Econometrica
of public finance (McGraw-Hill, New York). income taxation, American Economic Review
66, Dec..
Economics Letters 8 (1981) 95-99 North-Holland Publishing Company
PROGRESSION UNDER THEORY OF TAXATION Daniel
THE BENEFIT *
APPROACH
TO THE
KOVENOCK
CJniuermt_v of Wtsconsin,
Efraim
SADKA
Received
30 July 1981
Mudison,
WI 53706,
USA
The implications for progression of the benefit approach to the theory of taxation are studied in this paper. It is concluded that the case for progression rests on the degree of substitutability between public and private goods.
1. Introduction In the last decade we have witnessed major contributions to the theory of taxation which were essentially based on the ability-to-pay approach. ’ One of the conclusions drawn from these studies is that the case for progression is doubtful. 2 The purpose of this note is to investigate the implications for progression of an alternative approach to the theory of taxation, the benefit principle, as highlighted by the Lindahl equilibrium concept for pure public goods. 3 We find that the case for progression rests on the degree of substitutability between public and private goods. * The research reported here is financially supported by NSF Grant DAR-7917376 and by funds granted to the Institute for Research on Poverty at the University of WisconsinMadison by the Department of Health and Human Services (formerly HEW) pursuant to the provisions of the Economic Opportunity Act of 1964. The opinions expressed are those of the authors. The authors wish to thank Gene Smolensky for useful comments on an earlier draft of this paper. ’ See Atkinson and Stiglitz (1980) for references. ’ See. for instance, Sadka ( 1976). 3 See Musgrave (1959, ch. 4) for a historical survey of the two approaches.
0165-1765/81/0000-0000/$02.75
0 1981 North-Holland
96
D. Kovenock,
E. Sadku / Progression under the benefit approach
2. The model Consider a continuum of agents, 4 each consuming two goods: x which is a pure public good and y which is a pure private good. The agents all have the same preference ordering over bundles (x, y) which is representable by a twice continuously differentiable and quasi-concave utility function u( ., e). Each agent is distinguishable by his/her real income (initial endowment) which consists of a certain quantity of the private good. These incomes are contained in the closed and bounded interval [I,, Z,], where I, > 0. For each Z E [I,, Z,], we denote by Z’(Z) the number of agents with income less than or equal to I. We refer to an agent with income Z as ‘agent Z ‘. We denote by p the fixed unit cost of the public good in terms of the private good. The economy is in a Lindahl equilibrium. The Lindahl price (in terms of the private good) paid by agent Z for the public good is denoted by p*(Z). Let x* > 0 be the equilibrium supply of the public good. Then agent Z consumes y*(Z) = Z-p*(Z)x* units of the private good. For the sake of simplicity, we assume that all have a positive consumption of the private good, i.e., y*(Z) > 0 for all Z E [I,, Zz]. Given this assumption, which excludes corner solutions for each agent’s utility maximization problem, the Lindahl equilibrium is characterized by the following two equations: s(x*,z-p*(z)x*) $*(I)
/ 11
dF(Z)
‘p*(z)
forallZE[Z,,Z,],
=p,
(1)
(2)
where S is the marginal rate of substitution (MRS) of the private good for the public good, i.e., S(x, y) = u,( X, y)/u,,(x, JJ), where subscripts stand for partial derivatives. The first of these two equations states that each agent equates his willingness-to-pay for the public good to the price he/she actually pays for it. The second one states that with our constantreturns-to-scale technology the price of the public good, as perceived by the producer [namely, j?p*( I) dF( Z)], must equal its unit cost (namely, P). Before turning
to the results, let us make one more assumption
which
4 We consider a continuum rather than a discrete number of agents in order to facilitate the use of calculus.
91
D. Kovenock, E. Sadka / Progression under the benefit upprouch is
that. the public good is a normal good. This means that the MRS of Y for x rises as y is increased and x is kept constant, or formally: S,( x, y ) > 0 everywhere.
3. The results Agent Z pays a price p*( I) per unit of the public
good. Thus, he/she
pays a total of T*(Z)dAfp*( Z)x* for that good. This amount can be viewed as the total tax that he/she pays under the benefit principle of taxation. We are interested in the question of how this tax varies with income I. Observe that it follows immediately from our normality assumption (namely, SY > 0) that p*(Z) and hence also T*(Z) increase in income. Thus, the rich pay higher taxes than the poor. However, a more interesting question is whether the rich pay a higher proportion of their incomes in taxes than the poor. In other words, does the benefit principle imply progressive taxation? Formally, if we let t*(Z) ‘zf T*(Z)/Z be the average tax rate, then we ask whether t* is increasing in Z. By totally differentiating (1) with respect to I, we obtain
S,(x*,y*(i)) S(Z) = 1 +x*S,(x*,y*(Z)) which implies
’
that
g(z)= ~[z~(z)-p*(z)] 1 1 . I Therefore,
we conclude
that
Thus, the tax progressivity under the benefit principle depends exclusively on the magnitude of the elasticity of the MRS with respect to y,
98
namely: y$,/S. It turns out that the latter expression is exactly the inverse of the elasticity of substitution between x and y when the utility function is homothetic. To see this, notice that in the latter case, we can write s(x,y)
=S(l,y/x)~fH(y/x).
(3)
Now, the elasticity of substitution (a) is defined as the elasticity of y/x with respect to the MRS. Hence, u ~ ’ will be equal to the elasticity of the MRS (i.e., H) with respect to y/x: -= 1 O(Y/X)
WY/X)Y -NY/X)
Using (3), it follows that S,.(X,Y)
(5)
= NY/X)/X.
Combining
(3) (4) and (5) yields
-= 1 D(Y/X)
sJx,r)v S(X,Y)
.
Thus, we have proved: Proposition. If the public good is normal, then the tax according to the benefit principle is positively related to income. When, furthermore, the utility function is homothetic, then the tax is progressive, proportional or regressive as the elasticity of substitution between the public and the private goods is less than, equal to or greater than unity. 4. Concluding
remarks
(1) As one does not expect a high degree of substitutability between the public and the private goods, ’ it seems that the case for progressive ’ For instance,
Maital (1973) presents
estimates
of (r which are around
2/3.
D. Kovenock,
E. Sdka
/ Progression
under the benefit upprouch
99
taxation under the benefit approach is quite strong. (2) The proposition of the preceding section provides a readily applicable test for progression. For instance, with the CES class of utility functions, u(x, y) = [(Y,x~ + ~~~y~]‘/p, one has only to see whether the parameter u = l/( 1 - p) is greater than, equal to or less than unity.
References Atkinson, A.B. and J.E. Stiglitz, York). Maital, S., 1973. Public goods and 41, 561-568. Musgrave, R.A.. 1959, The theory Sadka, E.. 1976. On progressive 931-935.
1980 Lectures
on public
income distribution:
economics
Some further
(McGraw-Hill.
New
results, Econometrica
of public finance (McGraw-Hill, New York). income taxation, American Economic Review
66, Dec..