From each according to his surplus: Equi-proportionate sharing of commodity tax burdens

JOURNAL OF

PUBLIC ECONOMICS ELSEVIER

Journal of Public Economics 58 (1995) 417-428

From each according to his surplus" Equi-proportionate sharing of commodity tax burdens James R. Hines, Jr. a'*, John C. Hlinko a, Theodore J.F. Lubke b aJohn F. Kennedy School of Government, 79 John F. Kennedy Street, Harvard University, Cambridge, MA 02138, USA bNational Economic Council, The White House, Washington, DC 20500, USA

Received June 1993; final version received September 1994

Abstract This paper examines the incidence of commodity taxes, finding that, when demand and marginal cost schedules are linear, the burden of commodity taxation is distributed between buyers and sellers so that each suffers the same percentage reduction in pre-tax surplus. This equi-proportionate reduction in surplus is the outcome of commodity taxes set at any rate, and is unaffected by relative demand and supply elasticities. Hence, when demand and marginal cost schedules are linear, commodity taxes resemble flat-rate taxes imposed on market surplus. Similar results apply to nonlinear schedules within certain ranges. Keywords: Tax incidence; Commodity taxes; Consumer surplus J E L classification: H22

I. I n t r o d u c t i o n O n e o f t h e m o s t i m p o r t a n t f e a t u r e s o f a tax is t h e d i s t r i b u t i o n o f its b u r d e n s . T h e f u n c t i o n o f tax i n c i d e n c e t h e o r y is to i d e n t i f y t h e c u m u l a t i v e effect o f m a r k e t i n t e r a c t i o n s t h a t t r a n s f o r m a tax i m p o s e d o n o n e p a r t y into a s e r i e s o f p r i c e a n d i n c o m e c h a n g e s affecting m a n y p a r t i e s . T h e t h e o r y * Corresponding author. 0047-2727/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSD1 0047-2727(94)01489-2

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usually begins by considering the simple partial-equilibrium effects of perunit taxes imposed on sales of commodities. 1 H e r e , matters are relatively straightforward: buyers of a good bear the burden of a tax imposed on its sale insofar as the aggregate demand curve for the good is inelastic, or the supply curve is elastic; the converse holds for sellers, who feel taxes most keenly when d e m a n d curves are elastic or the supply curve is inelastic. F r o m this it is a short step to conclude that buyers or sellers of a taxed good are better off if their net demands are elastic. 2 T h e p u r p o s e of this p a p e r is to reconsider such an interpretation of tax incidence by calling attention to an elementary, but overlooked, property of the distribution of tax burdens between buyers and sellers. We find that, w h e n d e m a n d and marginal cost schedules are linear, the burden of c o m m o d i t y taxation is distributed between buyers and sellers so that each suffers the same p e r c e n t a g e reduction in pre-tax surplus. This equi-proportionate reduction in surplus is the consequence of commodity taxes set at any rate, and is unaffected by relative d e m a n d and supply elasticities. The result is valid for both competitive and noncompetitive markets. D e m a n d and supply elasticities not only determine the incidence of c o m m o d i t y taxes, but they also determine pre-tax consumer surplus and p r o d u c e r surplus. If d e m a n d and supply curves are linear and the d e m a n d curve is relatively inelastic, a tax on a commodity imposes a burden on consumers that is larger than its burden on sellers, but not any larger as a fraction of original surplus. Linear commodity taxes have an effect on competitive markets that is similar to the effect of flat-rate income taxes, which impose a much larger absolute burden on high-income taxpayers than on low-income taxpayers, but impose the same average burden on all income classes. Thus, it m a y be misleading to conclude that tax burdens are b o r n e excessively by those with inelastic net d e m a n d schedules.

2. Geometry of tax incidence with linear demand and linear marginal cost Fig. 1 illustrates the partial-equilibrium consequences of a specific commodity tax in a competitive equilibrium when demand and supply curves are linear. In the figure, the schedule A C E denotes the relevant portion of the d e m a n d curve, and the schedule H G E that of the supply curve. Prior to the imposition of the tax, quantity D E is sold at price O D . After a per-unit tax 1See, for example, Atkinson and Stiglitz (1980, 162 ft.) or Kotlikoff and Summers (1987, 1045 ff.). 2 For example, Seligman (1927, pp. 232-233) concludes, "In other words, the greater the elasticity of the demand, the more favorable - other things being equal - will be the situation of the consumer."

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419

LA

B

D F Demand D O

q

Fig. l. Tax incidence with linear demand and supply schedules.

o f C G is i m p o s e d , buyers face a net price o f O B , sellers a net price o f OF; at these prices, quantity B C is traded. Prior to the imposition of the tax, c o n s u m e r s e n j o y a surplus equal to the area of the triangle A D E ; post-tax c o n s u m e r surplus equals the area o f the triangle A B C . 3 T h e ratio o f c o n s u m e r surplus after the tax to c o n s u m e r surplus b e f o r e the tax is area ( A B C ) (1/2)(BC)(AB) (BC) area ( A D E ) - ( 1 / 2 ) ( D E ) ( A D ) - ( D E )

(AB) (AD) "

(1)

E q . (1) can be f u r t h e r simplified by noting that A B C and A D E are similar triangles: b o t h share the angle B A C , and b o t h have adjacent 90 ° angles on the vertical axis. T h e ratios o f the lengths o f c o r r e s p o n d i n g sides of similar triangles are equal, so ( A B ) / ( A D ) = ( B C ) / ( D E ) . H e n c e (1) implies: area ( A B C ) area ( A D E ) - [(BC)/(DE)]2 "

(2)

Similar reasoning can be used to evaluate the ratio o f post-tax p r o d u c e r surplus, area ( H F G ) in Fig. 1, to pre-tax p r o d u c e r surplus, area ( H D E ) .

3 The use of consumer surplus as a measure of consumer welfare offers the advantage of simplicity at the cost of some well-known limitations discussed by Willig (1976), McKenzie (1979), Chipman and Moore (1980), and Hausman (1981). The partial-equilibrium framework also sidesteps various general-equilibrium complications analyzed by McLure (1975), Homma (1977), and Diamond (1978).

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These two triangles are similar, since they share the angle GHF and both have 90° angles along the vertical axis. Hence the ratio of their areas equals: area (HFG) area (HDE)

(1/2)(FG)(HF) (FG) (1/2)(DE)(HD) - (DE)

(HF) (HD)

: [(FG)/(DE)I 2 .

(3)

The final step is to observe that (BC) = (FG), so the left-hand sides of (2) and (3) equal each other. After the tax is imposed, consumer surplus equals the same fraction of pre-tax consumer surplus that after-tax producer surplus equals of pre-tax producer surplus. Hence the lost consumer surplus represents the same fraction of original consumer surplus that lost producer surplus represents of original producer surplus. Monopolized markets also have the equi-proportionate burden sharing features of competitive markets. Fig. 2 illustrates the partial-equilibrium consequences of a specific commodity tax in a monopolized market with linear demand and marginal cost curves. In the figure, the schedule ACE denotes the relevant portion of the demand curve, and the schedule KI that of the marginal cost curve. Because the demand curve is linear, the marginal revenue curve (schedule A G I ) is also linear. Prior to the imposition of the tax, the monopolist produces quantity HI, which is sold for price OD. After imposition of a per-unit tax of JK, the monopolist faces an effective marginal cost curve of JG, and chooses to sell FG units of the good at price OB. Prior to the imposition of the tax, consumers enjoy surplus equal to area (ADE); consumer surplus after the tax equals area (ABC). Producers enjoy pre-tax surplus equal to area (DKIE); post-tax producer surplus equals area (BJGC). It is helpful to divide the pre-tax producer surplus into the triangle HKI and the rectangle DHIE, and the after-tax producer surplus into the triangle FJG and the rectangle BFGC. A

F H J K 0

Marginal Revenue q

Fig. 2. Tax incidence in a monopolized market with linear demand and marginal cost schedules.

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421

Using reasoning analogous to that applied to competitive markets, it can be shown that the ratio of consumer surplus after the tax to consumer surplus before the tax equals [BC/DE] 2. Additionally, because FJG and H K I are similar triangles, the ratio of the areas of the two producer surplus triangles can also be shown to equal [BC/DE] 2. The ratio of the areas of the two producer surplus rectangles equals: area ( B F G C ) area ( D H I E )

-

(BC)(CG) (DE)(EI) "

(4)

The triangle A G C is similar to the triangle A l E , since the angles A G C and A l E correspond and both triangles share the angle G A C . Hence, ( C G ) / ( E l ) = ( A G ) / ( A I ) . The triangles A F G and A H I also are similar, since they share the angle F A G and have adjacent 90 ° angles on the vertical axis. Accordingly, ( A G ) / ( A I ) = ( F G ) / ( H I ) . But, F G = B C and H I = DE. Thus, ( C G ) / ( E I ) = ( B C ) / ( D E ) and area ( B F G C ) area ( D H I E ) = [(BC)/(DE)]2 "

(5)

These calculations indicate that the ratio of after-tax consumer surplus to pre-tax consumer surplus equals the corresponding ratio for each component of producer surplus. Hence, as in a competitive market, commodity taxes in a monopolized market reduce consumer and producer surpluses by the same percentage.

3. Tax incidence in competitive and noncompetitive markets This section generalizes the results of Section 2 to identify the necessary and sufficient conditions for equi-proportionate sharing of tax burdens under a variety of market structures. This generalization requires two steps. The first step is to determine the effect that taxes must have on prices in order to generate equi-proportionate burden sharing. The second step is to identify market structures in which taxes have this effect. Consider a market in which a per-unit sales tax, ~-, is imposed on sellers. 4 T h e r e are n sellers producing identical goods with identical convex cost functions c(qi) , in which qi is firm i's output. Market output is denoted

4The text analyzes unit taxes; ad valorem taxes are equivalent to unit taxes in competitive markets, but not in noncompetitive markets, as Suits and Musgrave (1953) and Skeath and Trandel (1994) show. Barzel (1976) and Kay and Keen (1987) analyze other differences between the effects of unit and ad valorem taxes. It is easily verified that linear demand and marginal cost schedules are not sufficient for equi-proportionate sharing of ad valorem tax burdens in monopolistic markets.

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422 n

q = ~i=1 q~. We restrict attention to cases in which a unique market equilibrium is associated with each tax rate, and in which quantity and price are continuous functions of ~-, denoting the quantity corresponding to equilibrium as q(~-) and the price corresponding to inverse market demand as p ( q). Consumer surplus is a function of a commodity's price, which is itself a function of the tax rate; hence, consumer surplus has a reduced form C S ( z ) , and similarly, producer surplus has a reduced form PS('r). Define a to be the ratio of these surpluses in the absence of taxation: a =- C S ( O ) / P S ( O ) . If lost consumer surplus is the same fraction of original consumer surplus as lost producer surplus is of original producer surplus, then [CS(~')/CS(O)] = [PS(~')/PS(O)], implying that C S ( r ) = a P S ( r ) for all values of ~'. Differentiating, dCS(~')/d~" = a d P S ( r ) / d ~ ' ,

Vr,

(6)

which is a necessary condition for equi-proportionate sharing of tax burdens at all tax rates, and since CS(oo) = pS(oo) = 0, the constancy of a at all values of ~" is a sufficient condition as well. In order to complete the first step, it is necessary to determine the effect of taxes on prices implied by (6). From the envelope theorem, d C S ( ~ ' ) / d r = - q ( ' r ) ( d p / d . Q . To evaluate d P S ( r ) / d ' r , consider firm i's profits (rri): (7)

1ri = p ( q)q~ - c( q,) - 1"q~ .

Firm i's first-order condition for profit maximization guarantees that d R / dq~ = 0. The assumption that all firms have identical convex cost functions implies that q = nq~, so n dq~/d~" = dq/d~'. Differentiating (7) with respect to 7, imposing d~r~/dq~ = 0, and substituting d q / d r = n dq~/d'r, d~r~/d~" = q i [ p ' ( q ) ( n - 1 ) ( d q ~ / d r ) - 1]. Since d p / d r = p ' ( q ) n ( d q / d z ) ,

(8)

it follows that

d~i/d~" = q , [ ( d p / d r ) ( n - 1 ) / n - 1].

(9)

Producer surplus equals total profits, n~'~. Consequently, d P S ( r ) / d ' r = q ( r ) [ ( d p / d r ) ( n - 1 ) / n - 1]. Hence, the condition d P S ( r ) / d r = ot d C S ( ' r ) / d r implies q(~')[(dp/d~-)(n - 1 ) / n - 1] = - a q ( r ) ( d p / d z )

,

(10)

or

dp/d¢ = 1 / [ 1 + a - 1 / n ] .

(11)

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423

Since n and o~ a r e unchanging, (11) implies that the constancy of dp/d,r is a necessary and sufficient condition for tax burdens to be shared in proportion to original surpluses at all values of :r. T h e second step is to identify situations in which (11) is satisfied. Firm i's first-order condition characterizing the maximum of ~i in (7) over the choice of qi is

p(q) + qip'(q)(1 + A) - c'(qi) - z -- 0 ,

(12)

in which A = Ej~ i (dqJdqi) is firm i's conjectural variation, taken to be constant for all firms and at all tax rates. 5 Cases of constant conjectural variations include perfect competition (A -- - 1), Cournot oligopoly (A = 0), and monopoly (A = 0 and n - - 1 ) . Differentiating (12) with respect to ~yields

dp/d~" + p ' ( q ) ( 1 + A) dqi/dz + qi(1 + A)p"(q)n dqi/d-r - c"(q~) dq,/d~= 1.

(13)

Market equilibrium implies that d p / d z = p ' ( q ) d q / d r . n dqi/dr = (dp/dr)/p'(q), and (13) implies

Since q i = q / n ,

n

dp/d-r -- n + 1 + A + q(1 + A)p"(q)/p'(q) - c"(q~)/p'(q) "

(14)

If the demand and marginal cost schedules are linear, then p"(q) -- 0 and both c"(qi) and p ' ( q ) are constant, making dp/d': constant for all values of ~" and A. Hence, linear demand and marginal cost schedules imply that tax burdens are shared equi-proportionately for all market structures characterized by constant conjectures. This result appears in noncompetitive as well as competitive markets. Linear demand curves generate linear marginal revenue schedules in noncompetitive as well as competitive markets; if marginal cost is also linear, then dq/d~- is constant. Linear demand then implies that dp/d~" is constant, which, from (11), is responsible for the result. Eq. (14) reveals that linearity of demand and marginal cost are not necessary for equi-proportionate burden sharing, since constancy of the denominator of the right-hand side of (14) is sufficient. In a competitive market in which otherwise linear demand and marginal cost curves exhibit a range of symmetric inflection that maintains the constancy of d p / d r , tax burdens are shared equi-proportionately at all tax rates. Fig. 3 illustrates one example. s We are grateful to a referee for suggesting this extension of the model to cases of constant conjectures. Stern (1987) also analyzes tax incidence in settings with constant conjectural variations.

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Marginal Cost

Demand

D

o

q

Fig. 3. Nonlinear demand and marginal cost schedules with constant dp/dr.

4. Tax incidence in competitive markets with nonlinear demand and marginal cost schedules The properties of models with linear functions often approximate those of models with nonlinear functions. The purpose of this section is to identify the degree of nonlinearity in demand and supply schedules necessary to produce large differences between the percentage tax burdens of buyers and sellers in competitive markets. To do so, we calculate the largest possible difference created by nonlinear demand and supply functions that can be approximated with second-order expansions, finding that the difference is relatively small. Let CS(r) and PS(~') represent consumer surplus and producer surplus after the imposition of a tax at rate r. The tax burden on consumers is then [ C S ( 0 ) - CS(~-)], and the tax burden on producers is [ P S ( 0 ) - PS(7)]. We define A to be the absolute value of the difference between the buyer's tax burden, expressed as a fraction of pre-tax consumer surplus, and the seller's tax burden, expressed as a fraction of pre-tax producer surplus:

c s ( o ) - cs(.o A

=_

PS(O)_- PS(¢) PS(O)

"

(15)

If the tax rate is extremely low, then this difference is negligible, since the tax burdens are small. If the tax rate is extremely high, then this difference is zero, since both buyers and sellers bear burdens equal to 100% of their respective pre-tax surpluses. Hence it is only at an intermediate rate of taxation that these ratios might differ significantly.

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The difference defined in (15) can take any value between zero and one. In order to explore the degree of nonlinearity necessary to generate a large value of A, we examine second-order approximations to demand and supply schedules. Let p ( q ) be the inverse demand function, and mc(q) the marginal cost function. Without loss of generality, we normalize units so that the equilibrium quantity and price in the absence of taxation are both unity [p(1) = m c ( 1 ) = 1]. Consider a second-order Taylor approximation to the inverse demand function: p ( q ) ~ p ( 1 ) + p ' ( 1 ) ( q - 1) + p"(1)(q - 1)2/2.

(16)

Total consumer surplus [CS(0)] is the area between the inverse demand function and the price line: CS(O) =

foI [p(q)

- p ( 1 ) ] dq ~

fo1 [p'(1)(q

- 1) + p"(1)(q - 1)2/2] dq

= fo I ( [ p " ( 1 ) / 2 - p ' ( 1 ) ] + [p'(1) - p " ( 1 ) ] q + [p"(1)/2]q 2} d q .

(17) Evaluating the integral in (17) yields C S ( O ) = p " ( 1 ) / 6 - p ' ( 1 ) / 2 . Consider a tax at rate ~'* that raises the market price of the commodity from unity to p*, and reduces the quantity consumed from unity to q*. Fig. 4

l p,

mc(q)

1

,

i

p(q)

It

O

q*

I

q

Fig. 4. Maximal A with nonlinear demand and marginal cost schedules.

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depicts the effect of the tax on consumer and producer surplus. The tax reduces consumer surplus by an amount equal to CS(O) - CS(r*):

CS(O) - CS(.r*) = ( p * - 1)q* + = ( p * - 1)q* +

. [ p ( q ) - p ( 1 ) l dq

fo'

. [p'(1)(q-

1) + p " ( 1 ) ( q -

1)2/2] dq

= ( p * - 1)q* + fql. { [ p " ( 1 ) / 2 - p ' ( 1 ) ] + [ p ' ( 1 ) - p " ( 1 ) l q

+ [p"(1)/2]q 2} d q .

(18)

Evaluating the integral in (18), and using (16) to substitute ( p * - 1 ) = p ' ( 1 ) ( q * - 1) + p"(1)(q* - 1)2/2,

CS(O) - CS(r*) = p " ( 1 ) / 6 - p ' ( 1 ) / 2 + q*2[p'(1) - p"(1)]/2 + q*3p"(1)/3.

(19)

Combining (17) and (19), the tax reduces consumer surplus by the percentage: [CS(0) - CS(T*)]/CS(O) = 1 + q*Z[3p'(1) - 3p"(1)

+ 2q*p"(1)]/[p"(1) - 3 p ' ( 1 ) ] .

(20)

It is useful to define ~/to be the elasticity of the d e m a n d derivative at the no-tax point (q = 1): ~l=-p"(1)/p'(1). Making this substitution, (20) becomes [CS(0) - CS(r*)]/CS(O) = 1 + q ' 2 ( 3 - 3~7 + 2~q*)/07 - 3 ) .

(21)

T h e r e is an analogous second-order Taylor approximation to the percentage reduction in producer surplus for competitive suppliers with nonlinear marginal cost schedules: [PS(0) - PS(~*)I/PS(O) = 1 + q'Z(3 - 33, + 23,q*)/(3' - 3 ) ,

(22)

in which y =- mc"(1)/mc'(1) is the elasticity of the marginal cost derivative at the no-tax point. C o m b i n i n g (15), (21) and (22): A = [2q'2(1 - q * ) { [ y / ( y - 3)] -

[n/(n

-

3)]}1.

(23)

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Setting the derivative of the right-hand side of (23) with respect to q* equal to zero yields q * = 2/3 as the value that maximizes za. Consequently, the maximal value of ~ is A = (8/27)113'/(3' -- 3)] -- [,//(r/-- 3)]1.

(24)

It is possible to bound A by imposing that the demand schedule slope downward. From (16), p ' ( q ) = p ' ( 1 ) + p " ( 1 ) ( q - 1), and the requirement that p ' ( q ) <~O, Vq E [0, 1] implies that p'(1) ~ 0, implying 3, ~< 1. Imposing the constraints that r/~< 1 and 3' ~< 1, the value of A in (24) is maximized either by r / = 1 and lim3"---~-~, or by 3,=1 and limr/---~-~. Either of these substitutions yields A = 4/9. This section examines situations that can be analyzed using a second-order Taylor approximation to demand and marginal cost schedules, and in these situations A cannot exceed 4/9. Since this difference represents an upper bound, it suggests that the results obtained with linear schedules generally apply to cases in which schedules are nonlinear. It is, however, worth noting that differences approaching 4/9 may loom large in certain applications. There exist nonlinear functions for which second-order approximations are not very accurate, and if demand or marginal cost schedules take these forms, then the difference between the percentage tax burdens of buyers and sellers might exceed 4/9. In Fig. 4 it is clear that a more severe nonlinearity of the demand schedule around q* would raise [ C S ( 0 ) CS(-r*)]/CS(O) and thereby raise A. In practice, demand and marginal cost functions may be nonlinear in ranges (particularly those near zero quantity) that are seldom observed, making it difficult to determine the full extent of their curvatures. The results in this section indicate the degree of curvature necessary to invalidate the implications of linear functions analyzed earlier.

5. Conclusion

The finding that linear demand and marginal cost schedules imply that tax burdens are borne by buyers and sellers in equal proportion to their pre-tax surpluses suggests a simple interpretation of linear commodity taxes as flat-rate income taxes of a particular, market-specific, variety. The standard interpretation of commodity tax i n c i d e n c e - t h a t burdens are imposed on market participants with inelastic net d e m a n d s - m i s s e s the point that the same inelastic net demands imply that those market participants receive the greatest benefits from the existence of the market. Furthermore, as the

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r es u l t s d e s c r i b e d in S e c t i o n 4 i n d i c a t e , t h e c o n n e c t i o n b e t w e e n tax b u r d e n s a n d m a r k e t s u r p lu s e s is n o t l i m i t e d to m a r k e t s with l i n e a r d e m a n d a n d m a r g i n a l cost schedules.

Acknowledgements We thank Anthony Atkinson, Kathryn Dominguez, Richard Musgrave, G r e g o r y Trandel, Richard Zeckhauser, and two anonymous referees for h e l p f u l c o m m e n t s o n p r e v i o u s drafts. This" article r e p r e s e n t s t h e p e r s o n a l v i e w s o f t h e a u t h o r s . It d o e s n o t n e c e s s a r i l y r e p r e s e n t t h e v i ew s o f t h e N a t i o n a l E c o n o m i c C o u n c i l o r the C l i n t o n A d m i n i s t r a t i o n .

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