# Market substitution and the Pareto dominance of ad valorem taxation.

1. Introduction

Excise taxes take two basic forms: unit taxes based on quantity sold and ad valorem taxes based on sales value. While these two versions of excise taxes are equivalent in perfectly competitive markets, (1) in noncompetitive markets ad valorem taxation has been shown to welfare dominate unit taxation. (2) Ad valorem taxation also Pareto dominates unit taxation in monopoly and, in some cases, oligopoly markets, in the sense that, under an isorevenue constraint, replacing unit taxation with ad valorem taxation increases both consumer welfare and producer profits (Skeath and Trandel 1994). (3)

In this paper, we depart from the homogeneous product oligopoly markets assumed in previous literature by adopting a model of heterogeneous products developed by Dixit and Stiglitz (1977). First, we study the Pareto dominance of ad valorem taxation in short-run equilibrium, where a fixed number of firms produce imperfectly substitutable goods and engage in Bertrand competition. We define market demand as a weighted sum of firm quantities demanded and market price as a similarly weighted average of firm prices. By assuming that market demand is isoelastic with respect to market price, our framework allows the elasticity of substitution among the goods in the taxed market and the price elasticity of market demand to bear on the comparison of the two forms of excise taxes.

We find that, in the short run, ad valorem taxation always dominates unit taxation both in terms of consumer welfare and overall welfare (the precise meaning of overall welfare is made clear later). However, Pareto dominance of ad valorem taxation never exists if market demand is inelastic because, in this case, firms always earn lower profits under ad valorem taxation. Restricting our analysis to the case where market demand is elastic, we find that ad valorem taxation Pareto dominance is more likely the smaller the within-market elasticity of substitution or the larger the market demand elasticity. We also generalize a prior result for homogenous products: Increasing the number of firms in a market decreases the likelihood of ad valorem taxation Pareto dominance. Finally, we find that for a sufficiently large within-market elasticity of substitution, ad valorem taxation Pareto dominance is more likely the smaller the tax level, contrary to an existing result for the homogenous product case.

Given the results of previous literature that unit taxation tends to be welfare dominated by equal-revenue ad valorem taxation in noncompetitive environments, the existence of both types of taxes must be explained by nonoptimal behavior on the part of government. (4) One possible explanation for the coexistence of unit and ad valorem taxes, despite the welfare dominance of ad valorem taxation, is that a government only cares about the amount of revenue collected from each market, and the choice between these two taxes in a specific market is dictated by producers' interests, which are more concentrated than consumers' interests in general. Given this hypothesis concerning the political economy of choice between the two major forms of excise taxes, the results of this paper make testable predictions with respect to how the relative desirability of ad valorem taxation (from the perspective of producers) changes with several important characteristics of a market: the elasticity of substitution among goods in the market, the market demand elasticity, the number of firms in the market, and the level of taxation in the market.

Extending our analysis to long-run equilibrium, where entry and exit provides an additional market equilibrating mechanism and where firms always earn zero profits, we show that an equal-revenue switch from unit to ad valorem taxation has welfare effects on consumers through two channels. First, such a switch always lowers market price, which has a positive welfare effect. Second, such a switch may reduce the number of firms and, therefore, the range of consumer choice. However, we are able to show that the combined effect of lower market price and reduced range of choice always favors ad valorem taxation.

In spirit, our paper is similar to Anderson, de Palma, and Kreider (2001a, b), Kay and Keen (1983), and Keen (1998), who have also studied excise taxes in markets with horizontal product differentiation. (5) However, the short-run and long-run results obtained in this paper are complementary to these earlier studies in several important ways. First, the short-run results in Anderson, de Palma, and Kreider (2001b) focus on welfare comparisons of alternative forms of excise taxes and generally confirm the comparative efficiency advantage of ad valorem taxation previously found for markets with homogenous products. (6) We, on the other hand, focus on firms' comparative profitability under alternative tax regimes and use it to explain why unit taxation persists in some markets despite the efficiency advantage of ad valorem taxation. In particular, we link the comparative profitability under the two excise taxes to market parameters such as the number of firms in the taxed market and the market demand elasticity. For example, we find that ad valorem taxation generates lower profits when market demand is inelastic, perhaps explaining why unit taxation persists in markets featuring inelastic demand, such as the cigarette and gasoline markets.

Second, the long-run analysis of this paper also goes beyond these earlier studies, by combining both price and variety effects in assessing the relative long-run efficiency of the two excise taxes. Long-run welfare analyses of Kay and Keen (1983), Keen (1998), and Anderson, de Palma, and Kreider (2001b) point to the negative variety effect of ad valorem taxation in making a case for a long-run unit taxation efficiency advantage. (7) In their locational models of product differentiation, however, the price effect does not have any real efficiency role to play because the quantity demanded is either one or none for each consumer. In contrast, it is exactly the price effect that generates ad valorem taxation's long-run welfare dominance in studies featuring homogenous products. Using a different model of product differentiation in this paper, we take both price and variety effects into consideration and show that the price reduction benefits of ad valorem taxation can always sufficiently compensate for its variety disadvantage so that in the long run, ad valorem taxation welfare dominates equal-revenue unit taxation.

2. The Model

Demand Functions

Let the taxed market consist of a set of m [greater than or equal to] 2 goods {1,..., i,..., m} with each good produced by a single firm. Further, assume that there are n identical individuals in the economy. Following Dixit and Stiglitz (1977), let a representative individual's utility function be

(1) u([z.sub.1],...[z.sub.m];x) [equivalent to] [f[([z.sup.([theta]-1)/[theta].sub.1] + ... + [z.sup.([theta]-1)/[theta].sub.m]).sup.[theta]/([theta]-1)],x] [equivalent to] f(Z, x),

where [z.sub.i] is the individual's consumption of the ith firm's product, 1 < [theta] < [infinity] is the common elasticity of substitution among goods in the market, (8) and x is the individual's consumption of a composite numeraire good. It is easily shown that given total expenditure by all individuals on the m goods in the market, E, the total demand from the ith firm is

(2) [q.sub.i] = n[z.sub.i] = [m.sup.-1]E[p.sup.-[theta].sub.i][P.sup.[theta]-1],

where [p.sub.i] is the price of good i, and

(3) P = [([m.sup.-1][m.summation over i=1][p.sup.1-[theta].sub.i]).sup.1/(1-[theta])]

is a measure of "average market price."

Define market price as Equation 3 and note that this market price has the property that if [p.sub.i] = p, [for all]i, then P = p. Using utility function 1, we define market quantity demanded, Q, as

(4) Q = [m.sup.-1/([theta]-1)] [([m.summation over i=1] [q.sup.([theta]-1)/[theta].sub.i]).sup.[theta]/([theta]-1)],

where Q has the property that if [q.sub.i] = q, [for all]i, then Q = mq. Substituting Equation 2 into Equation 4 and using Equation 3, we have QP [equivalent to] E for arbitrary [p.sub.i] (i = 1, ..., m). Therefore, the above definitions of P and Q result in their product equaling total market expenditure.

In general, E is a function of P, with the exact functional form depending on the functional form of f(Z, x) in utility function 1. Specifically, given E, we have from Equation 2 that [z.sub.i] = [(mn).sup.-1]E[p.sup.-[theta].sub.i][P.sup.[theta]-1]. Substituting this into f (Z, x) and simplifying, we have f (Z, x) = f ([m.sup.1/([theta]-1)][P.sup.-1]E/n, x). E(P) is the solution to the problem of choosing E and x to maximize f ([m.sup.1/([theta]-1) [P.sup.-1]E/n, x), subject to (E/n) + x = Y, where E/n is total individual expenditure on goods in the taxed market and Y is individual income.

To facilitate tractable analytical treatment, we assume

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

so that E(P) has the form

E(P) = [bar.E][P.sup.-[eta]+1],

where K and [eta] are positive constants, and [bar.E] = n[K.sup.[eta][m.sup.([eta]-1)/([theta]-1)] is a coefficient independent of P or any individual [p.sub.i]. This assumption implies a constant-elasticity market demand,

Q = [bar.E][P.sup.-n],

with [eta] being the absolute value of the price elasticity of market demand. (9) The constant market demand elasticity assumption allows us to write the demand for firm i's product Equation 2 as

(5) [q.sub.i] = [bar.E][m.sup.-1][p.sup.-[theta].sub.i][P.sup.[theta]-[eta].

The two elasticities, [eta] and [theta], play important roles in this paper. We have the following assumptions and properties concerning their values.

ASSUMPTION 1. [theta] + [eta] > 2.

As we see further on, a positive market price in the oligopoly equilibrium requires that ([theta] - 1)m - [theta] + [eta] > 0, which is equivalent to [theta] + [eta] > 2, as indicated by the following property.

PROPERTY 1. ([theta] - 1)m - [theta] + [eta] > 0 for all m [greater than or equal to] 2 if and only if [theta] + [eta] >2.

PROOF: The equivalence between the two conditions is straightforward using the facts that ([theta] - 1)m - [theta] + [eta] is an increasing function of m for [theta] > 1 and that ([theta] - 1)m - [theta] + [eta] > 0 for m = 2 if and only if [theta] + [eta] > 2. QED.

ASSUMPTION 2. [theta] > [eta].

It seems reasonable to assume that any good in the taxed market is a closer substitute to the goods in the market as a group than any good outside the market. If this is true, then from the following property, [theta] > [eta].

PROPERTY 2. If [q.sub.i] is a closer substitute to Q than x is to Q for all i, then [theta] > [eta].

PROOF: From Equation 5, we have that the cross elasticity of demand for good i in the taxed market with respect to the market price P is

[differential][q.sub.i]/[differential]P P/[q.sub.i] = [theta] - [eta],

implying that [theta] - [eta] is a measure of substitutability between any [q.sub.i] and Q. Further, from the budget constraint P[Qn.sup.-1] + x = Y and Q = [bar.E][P.sup.-[eta]], the cross elasticity of demand for the numeraire good (goods outside the market as a group) x with respect to P is

[differential]x/[differential]P P/x = PQ/nx([eta] - 1).

One would normally expect that [q.sub.i] (a good in the market) is a closer substitute to Q than x (the composite of goods outside the market) is to Q. Therefore,

[theta] - [eta] > PQ/nx([eta] - 1).

It follows then that [theta] - [eta] > 0 because if [eta] < 1, [theta] - [eta] > 0 by virtue of the assumption that [theta] > 1, and if [eta] [greater than or equal to] 1, the right side of the preceding relation is greater than or equal to 0; therefore [theta] - [eta] > 0. QED.

As we show shortly, whether [eta] is larger or smaller than unity often determines the direction of the relative desirability of the two forms of excise taxation. The following property relates this condition to the substitutability and complementarity between the taxed market and the other market. As a result, we can say that while [theta] is a measure of within-market substitutability, [eta] is a measure of between-market substitutability.

PROPERTY 3. x and Q are gross substitutes if and only if [eta] > 1.

PROOF: From the budget constraint PQ[n.sup.-1] + x = Y and Q = [bar.E][P.sup.-n], we have that

[differential]x/[differential]P = ([eta] - 1)Q/n.

So x and Q are gross substitutes if and only if [eta] > 1. QED.

Short-Run Market Equilibrium

Let all firms in the market have constant marginal cost equal to c. Given the prices charged by other firms and a unit tax [tau], the first order condition for firm i's profit-maximizing pricing behavior is

(6) [q.sub.i] + ([p.sub.i] - [tau] -c) [differential][q.sub.i] / [differential][p.sub.i] = 0.

It follows from symmetry that all prices are equal in equilibrium. Using Equation 5 in solving Equation 6, the (Nash) equilibrium price and quantity under the unit tax [tau] are, respectively,

(7) [p.sup.[tau]] = [[theta]m - [theta] + [eta]/([theta] - 1)m - [theta] + [eta]] (c + [tau])

and

(8) [q.sup.[tau]] = [Em.sup.-1][([p.sup.[tau]]).sup.-[eta]].

Under an ad valorem tax t, where t is the tax rate on producer prices, a consumer price for firm i's product [p.sub.i] implies a corresponding producer price of [p.sub.i]/(1 + t). Thus, the first order condition for firm i's profit maximizing pricing problem, given other firms' prices, is

(9) [q.sub.i]/1 + t + ([p.sub.i]/1 + t - c) [differential][q.sub.i]/[differential][p.sub.i] = 0.

The (Nash) equilibrium price and quantity under the ad valorem tax t are, respectively,

(10) [p.sup.t] = [[theta]m - [theta] + [eta]/([theta] - 1)m - [theta] + [eta]] (c + ct)

and

(11) [q.sup.t] = [Em.sup.-1] [([p.sup.t]).sup.[eta]]

Note that according to Property 1, Assumption 1 ([theta] + [eta] > 2) guarantees that ([theta] - 1)m - [theta] + [eta] > 0. Therefore, from Equations 7 and 10, the equilibrium price under either tax regime is positive. On the other hand, consistent with the short-run analysis, we do not impose a condition here that guarantees positive profits for the firms in the market. In section 4, we analyze long-run equilibrium where entry and exit provide an additional equilibrating mechanism.

3. Short-Run Results

Denoting fixed cost as [C.sub.F] and using Equations 7 and 8, the equilibrium per-firm profit and government revenue under a unit tax regime are, respectively,

(12) [[pi].sup.[tau]] = ([p.sup.[tau]] - [tau] - c) [q.sup.[tau]] - [C.sub.F] = [E/[(c + [tau]).sup.[eta]-1]] {[[([theta] - 1)m - [theta] + [eta]].sup.[eta]-1]/ [[[theta]m - [theta] + [eta]].sup.[eta]]} - [C.sub.F]

and

(13) [R.sup.[tau]] = m[tau][q.sup.[tau]] = [E[tau]/[(c + [tau]).sup.[eta]]] [[([theta] - 1)m - [theta] + [eta]]/[theta]m - [theta] + [eta]].sup.[eta]].

In the same fashion, using Equations 10 and 11, the equilibrium per-firm profit and government revenue under the ad valorem tax are, respectively,

(14) [[pi].sup.t] = ([p.sup.t]/1 + t - c)[q.sup.t] - [C.sub.F] = [Ec/[(c + ct).sup.[eta]]] {[[([theta] - 1)m - [theta] + [eta]].sup.[eta]-1]/[[[theta]m - [theta] + [eta]].sup.[eta]] - [C.sub.F]

and

(15) [R.sup.t] = m([p.sup.t]t/1 + t)[q.sup.t] = [Ect/[(c + ct).sup.[eta]]] [[([theta] - 1)m - [theta] + [eta]/[theta]m - [theta] + [eta]].sup.[eta]-1].

For market demand elasticities greater than unity, the requirement that government always sets tax rates in the increasing portion of the total tax revenue function implies, from Equations 13 and 15, certain relations between [tau] and [eta] and between t and [eta], which are stated as the following assumption.

ASSUMPTION 3. For [eta] > 1,

[tau]/c < 1/[eta] - 1 and t < 1/[eta] - 1.

Note that if market demand is inelastic ([eta] [less than or equal to] 1), feasible tax rates are unrestricted. We adopt Assumption 3 throughout this paper. (10) Now consider the welfare implications of switching from a unit tax with a tax rate [tau] to an ad valorem tax that maintains the same level of revenue. The minimum ad valorem tax rate that generates at least the unit tax revenue, which we denote as [t.sup.[tau]], is the solution to

(16) [t.sup.[tau]] / [(1 + [t.sup.[tau]]).sup.[eta]] = [tau]/c / [(1 + [tau]/c).sup.[eta]] [([theta] - 1)m - [theta] + [eta] / [theta]m - [theta] + [eta]].

Because the bracketed term on the right side of Equation 16 is less than 1 for any finite 0, we have [t.sup.[tau]] < [tau]/c as long as total tax revenue is increasing in the tax rate at the original unit tax rate. (11) Thus, it follows from Equations 7 and 10 that [p.sup.t] < [p.sup.[tau]], and we have the following proposition.

PROPOSITION 1. Whenever market goods are heterogeneous (finite [theta]), for any unit tax, there exists an ad valorem tax that raises the same amount of revenue and generates higher welfare for consumers of the taxed goods.

That consumer surplus is increased when a unit tax is replaced with an equal-revenue ad valorem tax does not automatically imply that such a tax switch is Pareto improving or even overall welfare improving (however the overall welfare is defined), because firm owners may experience reduced income (as is clear from Proposition 2). Following the Pareto comparison approach as represented in Skeath and Trandel (1994), we assume in this paper that firm owners, who only consume the numeraire commodity so that their welfare is monotonic in profits, are a different group of people than the consumers of the taxed goods. Therefore, only if an equal-revenue replacement of a unit tax with an ad valorem tax raises both consumer surplus and firm profits, is such a change regarded as Pareto improving. In the rest of this section, the focus is on firms' comparative profitability under alternative forms of excise taxes. Nonetheless, in the Appendix, we show that ad valorem taxation always welfare dominates unit taxation in the short run in our model if consumers of the taxed goods are also the owners of the firms that produce these goods. (12)

Using Equations 12 and 14, we have that the difference in profits when a unit tax is replaced with an equal-revenue ad valorem tax is

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

For any given values of [theta], [eta], m and [tau]/c, the equal-revenue generating ad valorem tax rate [t.sup.[tau]] can be calculated from Equation 16. Then Equation 17 can be checked to see whether profits are higher under ad valorem taxation.

Begin by considering the special case where goods in the market taken as a group and all other goods (represented by the numeraire good x) are not gross substitutes (that is, [eta] [less than or equal to] 1 according to Property 1). Note first that price-setting firms have a stronger incentive to reduce their price under an ad valorem tax regime than under a unit tax regime because the tax paid on each unit of their product is proportional to the price charged. The effects on firms' profits of a lower price under ad valorem taxation are twofold. Lower prices reduce per unit profit but increase quantity demanded. Whether profits will be enhanced by switching to the ad valorem regime depends on the relative magnitude of these two effects. The smaller the market demand elasticity, the less likely the quantity effect will dominate the price effect. In particular, when [eta] [less than or equal to] 1, with a lower price, output rises, but total sales revenue falls, leaving firms with lower profits. Thus, we have the following proposition.

PROPOSITION 2. If [eta] < 1, then firms earn lower profits under ad valorem taxation than under unit taxation that generates the same tax revenue.

PROOF: We must show that, when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and finite [theta]. Because [t.sup.[tau]] < [tau]/c, [eta] [less than or equal to] 1 implies from Equation 17 that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. QED.

Proposition 2 says that, in the case in which the taxed market features an inelastic market demand, ad valorem Pareto dominance fails although consumers are always better off under ad valorem taxation than under equal-revenue unit taxation according to Proposition 1. Proposition 2 provides us with a basis for the existence of both forms of taxation that is not based on market or government imperfections. Specifically, for markets with inelastic demand, firm profits are greater with unit taxation. However, we do not expect many such markets to exist because such markets must be gross complements to all other goods. (13)

We know that ad valorem taxation Pareto dominance does not hold when the taxed market is a gross complement to all other goods ([eta] [less than or equal to] 1), but is there a combination of finite [theta] and [eta] > 1 that assures ad valorem taxation Pareto dominance? If so, is Pareto dominance more likely with smaller [theta] or larger [eta]? (14) The answer to both questions is yes and is reflected in the following fact and proposition.

FACT (Existence). Pareto dominance of ad valorem taxation holds for some [theta] < [infinity] and [eta] > 1.

Consider a hypothetical economy with [theta] = 3, [eta] = 2, m = 2, [tau]/c = 0.5. (15) From Equation 16, [t.sup.[tau] = 0.188 and from Equation 17, the ratio determining the relative advantage from the firms' perspective of ad valorem taxation is 1.063 > 1. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So, for this case, an equal-revenue replacement of a unit tax with an ad valorem tax will be Pareto improving. Given the existence of ad valorem Pareto dominance, we have the following proposition.

PROPOSITION 3.

(a) Other things being equal, ad valorem taxation is more likely to Pareto dominate equal-revenue unit taxation the larger the degree of product differentiation in the taxed market.

(b) Other things being equal, ad valorem taxation is more likely to Pareto dominate equal-revenue unit taxation the more elastic the market demand.

PROOF: (a) We want to prove that for fixed values of [eta] > 1, m, [tau]/c, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [theta] = [[theta].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[theta].sub.2] < [[theta].sub.1]. First, the right side of Equation 16 is increasing in [theta]. Second, the left side of Equation 16 is increasing in [t.sup.[tau]] under Assumption 2 that the revenue effect of an increase in tax rates is positive. Thus, [dt.sup.[tau]]/d[theta] > 0. Then, the left side ratio of Equation 17, [(1 + [tau]/c).sup.[eta]-1]/[(1 + [t.sup.[tau]).sup.[eta]], is decreasing in [theta]. Hence, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] when [theta] = [[theta].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] when [theta] = [[theta].sub.2] < [[theta].sub.1].

(b) We want to prove that for given values of [theta], m, [tau]/c, if [t.sup.[tau]] from Equation 16 decreases in [eta] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], for [eta] = [[eta].sub.1] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[eta].sub.2] > [[eta].sub.1]. Note that

(18) d { 1n [[(1+[tau]/c).sup.[eta]-1]/ [(1+[t.sup.[tau]).sup.[eta]]] / d[eta]} = 1n (1 + [tau]/c / 1 + [t.sup.[tau]]) - ([eta] / 1 + [t.sup.[tau]]) [dt.sup.[tau]] / d[eta],

which is positive if ([dt.sup.[tau]/d[eta]) < 0. Then, from Equation 17, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [eta] = [[eta].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[eta].sub.2] > [[eta].sub.1]. QED.

Skeath and Trandel (1994) established that a monopoly can earn a larger profit under an

ad valorem tax that generates the same level of tax revenue as a unit tax. So part (a) of Proposition 3 can be easily explained because each firm in the oligopoly market operates more like a monopoly as the elasticity of substitution among their products gets smaller. The intuition for part (b) of Proposition 3 is the same as that for Proposition 2.

For an oligopoly market with homogenous products and linear demand, Skeath and Trandel (1994) found that Pareto dominance of ad valorem taxation never holds when the number of firms in a market is sufficiently large, but always holds when the tax level exceeds a critical value. With product heterogeneity where firms engage in Bertrand competition, however, the first of these propositions continues to hold while the second does not.

PROPOSITION 4. (a) Other things being equal, ad valorem taxation is less likely to Pareto dominate equal-revenue unit taxation the larger the number of firms in the taxed market. (b) Other things being equal, ad valorem taxation is more likely to Pareto dominate equal-revenue unit taxation the lower the tax level.

PROOF: (a) We want to prove that for given values of [theta], [eta], and [tau]/c, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for m = [m.sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [m.sub.2] > [m.sub.1]. First, by signing its derivative with respect to m, the second term on the right side of Equation 16 can be shown to be increasing in m. Second, the left side of Equation 16 is increasing in [t.sup.[tau]] so long as revenues increase with tax rates. Thus, ([dt.sup.[tau]]/dm) > 0. Then from Equation 17, results in part (a) follow.

(b) We want to prove that for given values of [eta], m, c and sufficiently large [theta], if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [tau] = [[tau].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[tau].sub.2] < [[tau].sub.1]. First, from Equation 16,

[dt.sub.[tau] / d([tau]/c) = 1/[tau]/c - [eta]/1+[tau]/c / 1/[t.sup.[tau]] - [eta]/1+[t.sup.[tau]].

Also from Equation 16, [t.sup.[tau]] [right arrow] [tau]/c when [theta] [right arrow] [infinity]. Hence,

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Second,

d 1n [[(1+[tau]/c).sup.[eta]-1] / [(1+[t.sup.[tau]]).sup.[eta]]] / d([tau]/c) = [eta] - 1 / 1 + [tau]/c - [eta] / 1 + [t.sup.[tau]] [dt.sup.[tau]] / d([tau]/c)

which, from Equation 19 and that [t.sup.[tau]] < [tau]/c, is negative when [theta] is sufficiently large. So,

[(1 + [tau]/c).sup.[eta]-1] / [(1 + [t.sup.[tau]]).sup.[eta]]

decreases in [tau] for given c. Then, from Equation 17, results in part (b) follow. QED.

The first part of Proposition 4 confirms prior results regarding the effect of the number of firms in an oligopoly market on ad valorem taxation Pareto dominance. We can offer some intuition for why [theta] > [eta] guarantees that a larger number of firms in the market makes an equal-revenue switch from unit to ad valorem tax regimes less likely to increase firm profits. First, note that a major difference between unit and ad valorem taxation is that price-setting firms have a stronger incentive to lower their prices under an ad valorem tax regime because taxes are proportional to price. A larger number of firms in the market serves to strengthen this price incentive. Whether all firms in the market can benefit from individual firms' incentive to lower prices depends on whether the sales increase for individual firms comes from expanding market demand or from "stealing" other firms' sales, which in turn depends on the relative magnitudes of [eta] and [theta].

The second part of Proposition 4 says that the smaller the original unit tax rate, the more likely an equal-revenue switch from unit to ad valorem taxation will increase firm profits. Thus, when the goods in the market are sufficiently close substitutes, the lower the original level of a unit tax the more likely an equal-revenue ad valorem tax will Pareto dominate, exactly the opposite of previous homogeneous product linear demand results. Alternative but equally plausible assumptions are responsible for the differential results. Specifically, we assume Bertrand competition and a demand with constant price elasticity, while Skeath and Trandel (1994) use Coumot competition and linear demand. Our results here serve to show that the ad valorem Pareto dominance, with respect to the level of taxation, is sensitive to model specifications.

4. Long-Run Equilibrium: The Case of Endogenous Variety

The short-run results for a heterogeneous product market are more restrictive than for homogeneous markets because holding the number of firms constant, holds product variety constant. Because firms' profits are always zero in the long ran, the focus here is on consumer welfare. (16) When variety is endogenous, excise tax comparisons must account for both price and variety effects. Given utility function 1, for any given elasticity of substitution among goods in the market, consumers' welfare is determined by market price as well as market variety. Thus, an assessment of the welfare implications of the two excise taxes must take into account their impact on both market price and variety.

In this section we examine whether ad valorem taxation welfare dominance exists in the long run. (17) Our main long-run results are the following: (i) While a switch to ad valorem taxation reduces market price, consumer choice may also be reduced; (ii) ad valorem taxation welfare dominance always holds.

The approach taken in this section is a little different from that taken in previous sections. Rather than making a direct comparison between all unit taxation and all ad valorem taxation, we begin with a mix of both taxes and analyze the effects of substituting ad valorem for unit taxation at the margin. (18) When both a unit tax [tau] and an ad valorem tax t are imposed, the ith firm's problem is to choose [p.sub.i] to maximize its profits

[q.sub.i] = ([p.sub.i] - [tau]/1 + t - c) - [C.sub.F],

where [q.sub.i] is related to [p.sub.i] through Equation 5. The first order condition yields the equilibrium price

(20) p = (c + [tau] + tc)[[theta]m - [theta] + [eta]]/([theta] - 1)m - [theta] + [eta]

for all the goods in the market. (19)

The zero profit condition that determines the equilibrium number of firms is

[Em.sup.-1][p.sup.-[eta]](p - [tau]/1 + t - c) - [C.sub.F] = 0,

or, substituting E = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)] from the discussion before Equation 5,

(21) n[K.sup.[eta]][m([eta]-[theta])/([theta]-1)][[p.sup.-[eta]](p - [tau] - c - tc) - (1 + t)[C.sub.F] = 0

From Equations 20 and 21, the following comparative statics results, with respect to the effects of a change in either z or t on the equilibrium m and p, are obtained (See Appendix for a derivation):

[differential]m/[differential][tau] = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) ([eta] - 1),

[differential]p/[differential][tau] = - p/[OMEGA](c + [tau] + tc) ([theta]m + [eta] - 1)[([theta] - 1)m - [theta] + [eta]],

[differential]m/[differential]t = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) [[tau] + [eta]c(1 + t)]/(1 + t),

[differential]p/[differential]t = - p/[OMEGA](c + [tau] + tc) ([theta] - 1)m[[tau] + [eta]c(1 + t)] - c(1 + t)[OMEGA]/(1 + t),

where

[OMEGA] = -([theta] - 1)([theta] - [eta])(m - 1) - ([theta]m - [theta] + [eta])[([theta] - 1)m - [theta] + [eta]] < 0.

Hence, from Equation 22, [differential]p/[differential][tau] > 0, [differential]p/[differential]t > 0, [differential]m/[differential]t < 0, and [differential]m/[differential][tau] [greater than or equal to] 0 if and only if [eta] [less than or equal to] 1.

The effect of taxation on the equilibrium number of firms is a combination of two factors: an industry scale effect and a firm scale effect. A pure increase in ad valorem taxation reduces equilibrium market quantity. Although equilibrium firm size may go up or down, the market downsizing effect always dominates. As a result, an uncompensated increase in ad valorem taxation reduces the number of firms. On the other hand, a pure increase in the level of a unit tax decreases both equilibrium market quantity and equilibrium firm size. If the resulting market quantity decrease is small enough, it will be more than offset by the decrease in equilibrium firm scale, and the number of firms may actually rise. As it turns out, a market demand elasticity less than one is sufficient to ensure that the reduction in firm scale dominates the market scale reduction, so the number of firms rises in response to a pure increase in unit taxation.

Kay and Keen (1983) and Anderson, de Palma, and Kreider (2001 a) found, as we did here, that the long-run equilibrium consumer price increases in both tax rates. Using a locational model of product differentiation, Kay and Keen (1983) also found that an increase in ad valorem taxation reduces variety. As they explain, ad valorem taxation is akin to a grossing up of the fixed cost associated with each variety. While we confirm the negative effect of ad valorem taxation on variety, we find that unit taxation also has a variety effect and relate the variety effect of unit taxation to the elasticity of market demand.

For our purposes, we want to consider the revenue-neutral substitution of ad valorem taxation for unit taxation. In equilibrium, the market price p and the number of firms m are functions of tax rates [tau] and t, as implicitly determined by Equations 20 and 21. Therefore, the total tax revenue from the market can also be expressed as a function of these tax rates, which is

R([tau], t) = mq(p - p - [tau]/1 + t),

where q = [Em.sup.-1][p.sup.-[eta]]. Substituting E = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)] and the zero-profit condition

q(p - [tau]/1 + t - c) = [C.sub.F],

we arrive at

(23) R([tau], t) = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]](p - c) - m[C.sub.F].

Note that in the tax revenue expression 23, tax rates [tau] and t do not directly appear, and R([tau], t) is a function of [tau] and t through p and m. Such an expression of the tax revenue function is significant in simplifying some of the following derivations.

The effects of an increase in [tau] or t on the tax revenue are given by

(24) [differential]R/[differential][tau] = n[K.sup.[eta]][m.sup.([eta] - 1)/([theta] - 1)][p.sup.-[eta]][1 - [eta]([p-c)/p] [differential]p/[differential][tau] + n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)-1][p.sup.-[eta]][(p - c) [eta]- 1/ [theta]-1 - p-c-[tau]-tc/1+t] [differential]m/[differential][tau]

[differential]R/[differential]t = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]][1 - [eta]([p-c)/p] [differential]p/[differential]t + n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)-1][p.sup.-[eta]][(p - c) [eta]-1/[theta]-1 - p-c-[tau]-tc/1+t] [differential]m/[differential]t

both of which are assumed to be positive. (20) Substituting ad valorem taxation for unit taxation at the margin while keeping tax revenue unchanged requires that

(25) d[tau]/dt = - [differential]R/[differential]t/[differential]R/[differential][tau].

PROPOSITION 5. A revenue-constant substitution of ad valorem taxation for unit taxation always reduces market price, but reduces the number of firms as well if [eta] [less than or equal to] 1, that is, if the taxed market is not a gross substitute to the numeraire.

PROOF: The impact on the consumer price of a constant-revenue increase in ad valorem taxation (with an offsetting decrease in unit taxation) is given by

(26) dp/dt = [differential]p/[differential]t + [differential]p/[differential]t d[tau]/dt = [([differential]R/[differential][tau]).sup.-1] ([differential]R/[differential][tau] [differential]p/[differential]t - [differential]R/[differential]t [differential]p/[differential][tau]) = [([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)-1][p.sup.-[eta]][(p-c)[eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t]{[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]}.

In the Appendix, it is shown that

(27) (p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t < 0

and

(28) [differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau] = - p([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/(1 + t)[OMEGA][(c + [tau] + tc).sup.2]([theta] - n) [[tau] + c(1 + t)] > 0.

Therefore, dp/dt < 0. That is, market price falls as unit taxation is replaced with ad valorem taxation. On the other hand, the impact on the number of firms of a revenue-constant move toward ad valorem taxation is determined by

dm/dt = [differential]m/[differential]t + [differential]m/[differential][tau] d[tau]/dt.

Because [differential]m/[differential]t, d[tau]/dt < 0, [differential]m/[differential][tau] [greater than or equal to] 0 is sufficient (but not necessary) for ensuring that dm/dt < 0, which is the case if [eta] [less than or equal to] 1. QED.

When variety matters in the long-run equilibrium of a heterogenous product market, a lower consumer price under ad valorem taxation by itself cannot guarantee its welfare dominance, and the variety dimension must also be considered in determining its relative efficiency. Then, exactly how is the relative long-run efficiency of the two tax regimes determined based on their effects on the equilibrium price and variety? When there are m firms in the taxed market and each firm charges a price of p, individual welfare, as we established in section 2, is given by

(29) f([m.sup.1/([theta]-1)][p.sup.-1] E/n, Y - E/n),

where E/n is individual expenditure on goods in the taxed market and Y is individual income. From Equation 29 it follows that a consumer welfare index that combines both price and variety effects is [m.sup.1/([theta]-1)][p.sup.-1]. Thus, for an equal-revenue switch from unit to ad valorem taxation to increase consumer welfare in the long ran, the price (reduction) effect must be sufficiently large to overcome any possible negative variety effect. The following proposition says that this is always the case in our model.

PROPOSITION 6. Ad valorem taxation always welfare dominates unit taxation in the long run.

PROOF: Because [m.sup.1/([theta]-1)][p.sup.-1] is the welfare index that combines both price and variety considerations, the welfare impact of an equal-revenue switch from unit to ad valorem taxation can be assessed by

(30) d([m.sup.1/([theta]-1)][p.sup.-1])/dt = [m.sup.1/([theta]-1)-1]/([theta] - 1)p dm/dt - [m.sup.1/([theta]-1)]/[p.sup.2] dp/dt,

where dp/dt is given by Equation 26 and

dm/dt = [differential]m/[differential]t + [differential]m/[differential][tau] d[tau]/dt = [([differential]R/[differential][tau]).sup.-1] ([differential]R/[differential][tau] [differential]m/[differential]t - [differential]R/[differential]t [differential]m/[differential][tau]) = - [([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.[eta]-1/[theta]-1][p.sup.-[eta]][1 - [eta](p - c)/p] {[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]}.

Substituting dp/dt and dm/dt, Equation 30 becomes

d([m.sup.1/([theta]-1)][p.sup.-1])/dt = [([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.([eta] - [theta]+1)/ ([theta]-1)][p.sup.-[eta]-2]{- p/[theta] - 1 [1 - [eta](p-c)/p] - [(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t]} {[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]} =[([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.([eta]-[theta]+1)/([theta]-1)][p.sup.-[eta]-2][-c/[theta] - 1 + p - c - [tau] - tc/1 + t] {[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]}.

From Equation 28,

[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/ [differential][tau] > 0.

Further,

-c/[theta]-1 + p - c - [tau] - tc/1 + t = - c/[theta] - 1 + (c + [tau] + tc)m/[([theta] - 1)m - [theta] + [eta]](1 + t) = -c(1 + t)[([theta] - 1)m - [theta] + [eta]] + ([theta] - 1)(c + [tau] + tc)m/([theta] - 1)[([theta] - 1)m - [theta] + [eta]](1 + t) = c(1 + t)([theta] - [eta]) + ([theta] - 1)[tau]m/([theta] - 1)[([theta] - 1)m - [theta] + [eta]](1 + t) > 0.

So, d([m.sup.1/([theta]-1)][p.sup.-1])/dt > 0. That is, an equal-revenue substitution of ad valorem taxation for unit taxation always increases consumer welfare in the long run. QED.

Proposition 6 is complementary to Kay and Keen (1983) and Anderson, de Palma, and Kreider (2001b), both of whom used versions of a locational model of product differentiation to study the relative efficiency of ad valorem and unit taxes. In Kay and Keen (1983), variety is excessive in the absence of taxation. So they showed that, initially, the ad valorem tax, which is variety reducing in locational models, should be used to bring about the optimal level of product variety, but any additional revenue should be raised using the unit tax, which is variety neutral in locational models. For similar reasons, Anderson, de Palma, and Kreider (2001b) found that ad valorem taxation is welfare dominated by unit taxation in a locational model in which the equilibrium variety is optimal in the absence of taxation. However, the locational models used in these studies, while allowing variety to play a role in long-run welfare analysis, do not allow a quantity and/or price effect to play any role in the analysis. On the other hand, the established long-run welfare dominance of ad valorem taxation for homogenous product markets is entirely based on the quantity and/or price advantage of ad valorem taxation. While Anderson, de Palma, and Kreider (2001b) suggested that allowing the quantity and/or price effect may reverse the results they obtained and give back the ad valorem tax its efficiency advantage, they did not provide any formal analysis to substantiate this valuable point. This paper confirms that point and generalizes their findings.

5. Concluding Remarks

There is a recent trend to study surplus incidence (in contrast to more traditional price incidence) of excise taxation. (21) Related to this topic is the investigation of Pareto superiority of ad valorem taxation in noncompetitive markets. In these Pareto comparisons, the effect of (equal-revenue) switching between tax regimes on both consumer and producer surpluses are examined, providing a more complete picture of who loses or gains from a change in tax structure, enhancing our understanding of why unit taxes are imposed in some markets and ad valorem taxes in others. Using the elasticity of substitution parameters in the utility function, we define market demand and price when a market consists of heterogeneous products. We can then bring the within-market and between-market substitutability to bear on the comparison of unit and ad valorem taxation. As a result, we show conditions under which consumers and firms would prefer one type of excise taxation to another.

Our results address several interesting aspects of excise taxation. In the short run, when the number of firms, and by construction product variety, is fixed, the conditions for Pareto dominance of ad valorem taxation, while similar to those derived assuming homogeneous product oligopoly markets, are different in certain critical aspects. First, when the market consists of heterogeneous products and there exists a unit tax rate such that an equal-revenue ad valorem tax is Pareto dominant, any smaller unit tax rate has a corresponding equal-revenue Pareto dominant ad valorem tax. More importantly, our work emphasizes the critical importance of between-market substitutability. In particular, when goods in the taxed market have a complementary relation to all other goods, implying that market demand is inelastic, ad valorem taxation Pareto dominance fails in the sense that firms earn lower profits. The nature of this ad valorem Pareto dominance failure is important because it rests on the effect on profits of a switch from unit to ad valorem taxation. While such a change makes consumers better off in that it results in lower price, firm owners are worse off because profits are lower. Thus, firms in markets with inelastic demand will prefer unit taxation, perhaps explaining the persistence of unit taxation in such markets as gasoline and cigarettes.

While our short-run results are interesting, the real advantage of our approach is that we allow for long-run adjustment in the number of firms and, therefore, in product variety. In the long run, the effect of taxation on general welfare depends not just on the final consumer price, but on the equilibrium number of firms as well. Again, the between-market substitutability (the price elasticity of the market demand for the taxed goods) plays an important role here. An equal-revenue substitution of ad valorem for unit taxation reduces variety as long as the goods in the taxed market have a complementary relation to the other goods (i.e., market demand is inelastic). Nonetheless, we have been able to show that ad valorem taxation always welfare dominates unit taxation in the long run.

Appendix

Short-Run Welfare Dominance (in Total Welfare) of Ad Valorem Taxation

Note that one cannot argue for the short-run ad valorem total welfare dominance by simply looking at the sum of consumers' and producers' surpluses and be satisfied with the fact that price is lower and output is higher, and, therefore, consumer surplus plus profits is higher under ad valorem taxation than under equal-revenue unit taxation. First, if consumers and firm owners are two different sets of people, as these previous studies have implicitly assumed, the sum of consumer and producer surpluses is not an unambiguous welfare indicator. In this case, one must separately consider the welfare of consumers and producers (firm owners), as we did in the present paper. Second, if consumers are also firm owners, then a comparison of their welfare under two forms of excise taxes can be unambiguously made. In this case, however, profits must be explicitly added to consumers' income in determining final consumer utility under each tax. With regard to the second scenario, we have the following proposition which upholds the ad valorem taxation welfare dominance (in total welfare).

PROPOSITION 1'. For the purpose of making a short-run welfare comparison, assume consumers of the taxed goods own the firms that produce these goods. Whenever market goods are heterogeneous (finite 0), for any unit tax, there exists an ad valorem tax that raises the same amount of revenue and generates higher welfare for individuals as both consumers and firm owners.

PROOF: The proof consists of two steps. The first step is to show that adding profits to consumers' income does not alter the finding that equilibrium price is lower, and equilibrium quantity higher, under ad valorem taxation than under equal-revenue unit taxation. Within the assumptions of this model, the equilibrium price and quantity of each firm under two alternative tax regimes are given by Equations 7 to 8, and 10 to 11, regardless of whether consumers' income includes profits. Therefore, the relation between the unit tax rate and the equal-revenue ad valorem tax rate--Equation 16--still holds. As a result, price is lower and quantity is higher under ad valorem taxation than under equal-revenue unit taxation.

The second step is to show that consumer (as both consumers and firm owners) welfare is improved as the tax regime switches from unit taxation to equal-revenue ad valorem taxation, given that the price is lower (and quantity higher) under the latter tax regime. To show this, it is sufficient to demonstrate that consumers have more money left after buying the original quantity with the new, lower price, taking into account the difference in profits under the two tax regimes. Denote (p, Q) as the price-(aggregate) quantity pair under the original unit tax regime, and (p', Q') as the price-quantity pair under the equal-revenue ad valorem tax regime. We have shown in the first step that p' < p and Q' > Q. Under unit taxation, the money left (for all consumers) after buying the equilibrium quantity of each firm's product is

nY + (pQ - R - cQ - m[C.sub.F]) - pQ,

where R is tax revenue paid. Under equal-revenue ad valorem taxation, on the other hand, the money left after buying the old quantity (the equilibrium quantity under the unit taxation) of each firm's product is

nY + (p'Q' - R - cQ' - m[C.sub.F]) - p'Q,

which is larger than the earlier expression by (p' - c)(Q' - Q) > 0. QED.

Derivation of Comparative Statics Results in Equation 22

Using logarithmic terms, equilibrium conditions 20 and 21 can be expressed as

n p = ln(c + [tau] + tc) + ln([theta]m - [theta] + [eta]) - ln[([theta] - 1)m - [theta] + [eta]]

ln(n[K.sup.[eta]]) + [eta] - [theta]/[theta] - 1 ln m - [eta] ln p + ln(p - c - [tau] - tc) = ln(1 + [tau]) + ln (C.sub.F).

Taking derivatives with respect to [tau] in the above two equations, we have

1/p [differential]p/[differential][tau] = 1/c + [tau] + tc + [[theta]/[theta]m - [theta] + [eta] - [theta] - 1/([theta] - 1)m - [theta] + [eta]] [differential]m/[differential][tau]

[eta] - [theta]/[theta] - 1 1/m [differential]m/[differential][tau] + 1/p - c - [tau] - tc ([differential]p/[differential][tau] - 1) = 0.

Solving for [differential]p/[differential][tau] and [differential]m/[differential][tau], we have

[differential]m/[differential][tau] = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) x ([eta] - 1)

[differential]p/[differential][tau] = - p/[OMEGA](c + [tau] + tc) x ([theta]m + [eta] - 1)[([theta] - 1)m - [theta] + [eta]].

Similarly, taking derivatives with respect to t in the two equations in (A1), we have

1/p [differential]p/[differential]t = c/c + [tau] tc + [[theta]/[theta]m - [theta] + [eta] - [theta] - 1/([theta] - 1)m - [theta] + [eta]] [differential]m/[differential]t

[eta] - [theta]/[theta] - 1 1/m [differential]m/[differential]t - n/p [differential]p/[differential]t + 1 /p - c [tau] - tc ([differential]p/[differential]t - c) = 1/ 1 + t.

Solving for [differential]p/[differential]t and [differential]m/[differential]t, we have

[differential]m/[differential]t = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) x [[tau] + [eta]c(1 + t)/(1 + t)

[differential]p/[differential]t = - p/[OMEGA](c + [tau] + tc) x ([theta] - 1)m[[tau] + [eta]c(1 + t)] - c(1 + t) [OMEGA]/(1 + t)

PROOF OF EQUATION 27. From Equation 24,

(A2) [differential]R/[differential]t = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]][1 - [eta](p - c)/p] [differential]p/[differential]t + n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]][(p - c) [eta] - 1/ [theta] - 1 - p - c - [tau] - tc/1 + t] [differential]m/[differential]t.

To prove

(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t < 0,

consider the following two situations, while keeping in mind that, from Equation 22, [differential]p/[differential]t > 0 and [differential]m/[differential]t < 0.

(a) 1 - [eta](p - c)/p < 0.

In this case, the assumption [differential]R/[differential]t > 0 implies

(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t < 0.

(b) 1 - [eta](p - c)/p [greater than or equal to] 0.

In this case, [eta] [less than or equal to] p/(p - c). Then

(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t [less than or equal to] (p - c) p/p - c - 1/[theta] - 1 - p - c - [tau] - tc/1 + t = c/[theta] - 1 - p - c - [tau] - tc/1 + t = c/[theta] - 1 - (c + [tau] + tc)m/(1 + t)[([theta] - 1)m - [theta] + [eta]] = - c(1 + t)([theta] - [eta]) + ([theta] - 1) [tau]m/([theta] - 1)(1 + t)[([theta] - 1)m - [theta] + [eta]] < 0.

DERIVATION OF EQUATION 28. From Equation 22,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

References

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(1) However, the two versions of excise taxes are not equivalent in a perfectly competitive market when product quality is endogenous. For examples of incorporating quality into comparative commodity tax analysis, see Barzel (1976), Bohanon and Van Cott (1991), Kay and Keen (1991), Cremer and Thisse (1994), and Liu (2003). See also Saving (1982) for a general discussion of the product quality and market structure. Throughout the discussion of this paper, however, we assume that the quality of each firm's product is exogenously determined.

(2) Suits and Musgrave (1953) showed for the monopoly case that when the same revenue is raised, equilibrium output and thus welfare is greater under an ad valorem tax than under a unit tax. Similar welfare comparison results for the oligopoly market were established by Delipalla and Keen (1992).

(3) The Pareto dominance of ad valorem taxation in a monopoly market was first found by Skeath and Trandel (1994). Assuming a homogeneous product and linear demand in an oligopoly market, Skeath and Trandel (1994) also demonstrated that the Pareto dominance of ad valorem taxation holds when the tax level exceeds a critical value but never holds when the number of firms in a market is sufficiently large.

(4) For example, federal telephone and air transportation taxes and state and local public utility taxes are ad valorem, while national gasoline taxes and state liquor and cigarette taxes are unit.

(5) Also see Cremer and Thisse (1994) for an analysis of excise taxes in a market with vertical product differentiation.

(6) The focus of Kay and Keen (1983) and Keen (1998) is on the long run in which firms always earn zero profits. On the other hand, although the incidence analysis of both forms of excise taxes in Anderson, de Palma, and Kreider (2001a) includes their effects on profits, it is not a differential incidence analysis in which one form of excise tax is substituted for another with the total tax revenue unchanged. As a result, it does not address firm comparative profitability under the alternative tax regimes.

(7) Anderson, de Palma, and Kreider (2001a) considered the long-ran price effects but not the variety effects of the two forms of excise taxes.

(8) The constraint [theta] > 1, first introduced by Dixit and Stiglitz (1977) to motivate a desire for variety, is imposed here to ensure an equilibrium for firms' profit maximization problem. The welfare role of [theta] emphasized by Dixit and Stiglitz is discussed in section 4 where entry and exit is endogenous.

(9) From this assumption the price elasticity of total expenditures on goods produced in the oligopoly market is constant at [eta] - 1 and total expenditures decrease (increase) in P if [eta] - 1 is positive (negative).

(10) Note that a revenue (or welfare) maximizing government would not raise tax rates to a point where revenue is decreasing in the tax rates.

(11) If [theta] = [infinity] (i,e., all goods in the oligopoly markets are identical), Equation 16 implies [t.sub.[tau]] = [tau]/c. Therefore, from Equations 7, 8, 10, and 11, the equilibriums for equal revenue ad valorem and unit taxes are identical. This equivalence result stands in marked contrast to previous nonequivalence results for homogenous oligopoly markets [Suits and Musgrave (1953), Delipalla and Keen (1992), and Skeath and Trandel (1994)] and results from the fundamental difference between Bertrand competition and Coumot competition in modeling oligopoly behavior. Bertrand price competition by oligopoly firms producing identical products leads to the perfect competition outcome (marginal cost pricing) regardless of tax regime.

(12) It has been unanimously found that ad valorem taxation has an efficiency advantage in the short run. See, for example, Delipalla and Keen (1992) for the case of product homogeneity and Anderson, de Palma, and Kreider (2001b) for the case of product heterogeneity.

(13) We regard gross complementarity as a rare case because, while possible in a two good world, it is unrealistic in a world where the other good consists of the composite of all goods not in the single heterogeneous market, because the income effect of a change in P is proportional to the ratio of expenditures on Q and all other goods. Viewing the consumer's utility maximization problem as a choice between Q/n and x subject to P[Qn.sup.-1] + x = Y, the Slutsky equation gives us [[epsilon].sub.xP] = [[epsilon].sub.xP]|[sub.u] - ([differential]x/[dif (QP/nx)--where [[epsilon].sub.xP], [[epsilon].sub.xP]|[sub.u] are, respectively, uncompensated and compensated elasticities of x with respect to P--which is positive for sufficiently small QP/nx.

(14) Note that the experiment with changes in [eta] must be conducted within the range where at the original [tau], tax revenue is increasing in the tax rate so that [tau]/c < 1/([eta] - 1).

(15) Note these parameter values satisfy Assumptions 1 to 3.

(16) If existing firms earn positive (negative) short-run profits following the regime switch, they will earn positive (negative) profits during the entire transition. Thus, the short-run results concerning firms' profitability presented in Propositions 2, 3, and 4, can be directly generalized to account for transitional profits.

(17) Delipalla and Keen (1992) demonstrate that ad valorem taxation welfare dominates unit taxation in the long run in homogenous product oligopoly markets.

(18) The reason for doing so is that with an additional endogenous variable m and a nonlinear relation among variables, it is impossible to explicitly solve for p and m under either tax regime. On the other hand, the marginal approach here does not rely on an explicit solution to the initial equilibrium.

(19) Note Equation 20 is the general form of Equation 7 or 10.

(20) This assumption is similar to Assumption 3 in spirit. However, with endogeneity of both p and m, it cannot be easily boiled down to a simple up-bound on the tax rates.

(21) For examples, see Hines, Hlinko, and Lubke (1995), Trandel (1999), and Anderson, de Palma, and Kreider (2001a).

Liqun Liu * and Thomas R. Saving ([dagger])

* Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231, USA; E-mail: lliu@tamu.edu.

([dagger]) Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231, USA; E-mail:

t-saving@tamu.edu; corresponding author.

We want to thank Andy Rettenmaier, Laura Razzolini, and several anonymous referees for very helpful comments and suggestions.

Received November 3, 2003; accepted February 3, 2005.

Excise taxes take two basic forms: unit taxes based on quantity sold and ad valorem taxes based on sales value. While these two versions of excise taxes are equivalent in perfectly competitive markets, (1) in noncompetitive markets ad valorem taxation has been shown to welfare dominate unit taxation. (2) Ad valorem taxation also Pareto dominates unit taxation in monopoly and, in some cases, oligopoly markets, in the sense that, under an isorevenue constraint, replacing unit taxation with ad valorem taxation increases both consumer welfare and producer profits (Skeath and Trandel 1994). (3)

In this paper, we depart from the homogeneous product oligopoly markets assumed in previous literature by adopting a model of heterogeneous products developed by Dixit and Stiglitz (1977). First, we study the Pareto dominance of ad valorem taxation in short-run equilibrium, where a fixed number of firms produce imperfectly substitutable goods and engage in Bertrand competition. We define market demand as a weighted sum of firm quantities demanded and market price as a similarly weighted average of firm prices. By assuming that market demand is isoelastic with respect to market price, our framework allows the elasticity of substitution among the goods in the taxed market and the price elasticity of market demand to bear on the comparison of the two forms of excise taxes.

We find that, in the short run, ad valorem taxation always dominates unit taxation both in terms of consumer welfare and overall welfare (the precise meaning of overall welfare is made clear later). However, Pareto dominance of ad valorem taxation never exists if market demand is inelastic because, in this case, firms always earn lower profits under ad valorem taxation. Restricting our analysis to the case where market demand is elastic, we find that ad valorem taxation Pareto dominance is more likely the smaller the within-market elasticity of substitution or the larger the market demand elasticity. We also generalize a prior result for homogenous products: Increasing the number of firms in a market decreases the likelihood of ad valorem taxation Pareto dominance. Finally, we find that for a sufficiently large within-market elasticity of substitution, ad valorem taxation Pareto dominance is more likely the smaller the tax level, contrary to an existing result for the homogenous product case.

Given the results of previous literature that unit taxation tends to be welfare dominated by equal-revenue ad valorem taxation in noncompetitive environments, the existence of both types of taxes must be explained by nonoptimal behavior on the part of government. (4) One possible explanation for the coexistence of unit and ad valorem taxes, despite the welfare dominance of ad valorem taxation, is that a government only cares about the amount of revenue collected from each market, and the choice between these two taxes in a specific market is dictated by producers' interests, which are more concentrated than consumers' interests in general. Given this hypothesis concerning the political economy of choice between the two major forms of excise taxes, the results of this paper make testable predictions with respect to how the relative desirability of ad valorem taxation (from the perspective of producers) changes with several important characteristics of a market: the elasticity of substitution among goods in the market, the market demand elasticity, the number of firms in the market, and the level of taxation in the market.

Extending our analysis to long-run equilibrium, where entry and exit provides an additional market equilibrating mechanism and where firms always earn zero profits, we show that an equal-revenue switch from unit to ad valorem taxation has welfare effects on consumers through two channels. First, such a switch always lowers market price, which has a positive welfare effect. Second, such a switch may reduce the number of firms and, therefore, the range of consumer choice. However, we are able to show that the combined effect of lower market price and reduced range of choice always favors ad valorem taxation.

In spirit, our paper is similar to Anderson, de Palma, and Kreider (2001a, b), Kay and Keen (1983), and Keen (1998), who have also studied excise taxes in markets with horizontal product differentiation. (5) However, the short-run and long-run results obtained in this paper are complementary to these earlier studies in several important ways. First, the short-run results in Anderson, de Palma, and Kreider (2001b) focus on welfare comparisons of alternative forms of excise taxes and generally confirm the comparative efficiency advantage of ad valorem taxation previously found for markets with homogenous products. (6) We, on the other hand, focus on firms' comparative profitability under alternative tax regimes and use it to explain why unit taxation persists in some markets despite the efficiency advantage of ad valorem taxation. In particular, we link the comparative profitability under the two excise taxes to market parameters such as the number of firms in the taxed market and the market demand elasticity. For example, we find that ad valorem taxation generates lower profits when market demand is inelastic, perhaps explaining why unit taxation persists in markets featuring inelastic demand, such as the cigarette and gasoline markets.

Second, the long-run analysis of this paper also goes beyond these earlier studies, by combining both price and variety effects in assessing the relative long-run efficiency of the two excise taxes. Long-run welfare analyses of Kay and Keen (1983), Keen (1998), and Anderson, de Palma, and Kreider (2001b) point to the negative variety effect of ad valorem taxation in making a case for a long-run unit taxation efficiency advantage. (7) In their locational models of product differentiation, however, the price effect does not have any real efficiency role to play because the quantity demanded is either one or none for each consumer. In contrast, it is exactly the price effect that generates ad valorem taxation's long-run welfare dominance in studies featuring homogenous products. Using a different model of product differentiation in this paper, we take both price and variety effects into consideration and show that the price reduction benefits of ad valorem taxation can always sufficiently compensate for its variety disadvantage so that in the long run, ad valorem taxation welfare dominates equal-revenue unit taxation.

2. The Model

Demand Functions

Let the taxed market consist of a set of m [greater than or equal to] 2 goods {1,..., i,..., m} with each good produced by a single firm. Further, assume that there are n identical individuals in the economy. Following Dixit and Stiglitz (1977), let a representative individual's utility function be

(1) u([z.sub.1],...[z.sub.m];x) [equivalent to] [f[([z.sup.([theta]-1)/[theta].sub.1] + ... + [z.sup.([theta]-1)/[theta].sub.m]).sup.[theta]/([theta]-1)],x] [equivalent to] f(Z, x),

where [z.sub.i] is the individual's consumption of the ith firm's product, 1 < [theta] < [infinity] is the common elasticity of substitution among goods in the market, (8) and x is the individual's consumption of a composite numeraire good. It is easily shown that given total expenditure by all individuals on the m goods in the market, E, the total demand from the ith firm is

(2) [q.sub.i] = n[z.sub.i] = [m.sup.-1]E[p.sup.-[theta].sub.i][P.sup.[theta]-1],

where [p.sub.i] is the price of good i, and

(3) P = [([m.sup.-1][m.summation over i=1][p.sup.1-[theta].sub.i]).sup.1/(1-[theta])]

is a measure of "average market price."

Define market price as Equation 3 and note that this market price has the property that if [p.sub.i] = p, [for all]i, then P = p. Using utility function 1, we define market quantity demanded, Q, as

(4) Q = [m.sup.-1/([theta]-1)] [([m.summation over i=1] [q.sup.([theta]-1)/[theta].sub.i]).sup.[theta]/([theta]-1)],

where Q has the property that if [q.sub.i] = q, [for all]i, then Q = mq. Substituting Equation 2 into Equation 4 and using Equation 3, we have QP [equivalent to] E for arbitrary [p.sub.i] (i = 1, ..., m). Therefore, the above definitions of P and Q result in their product equaling total market expenditure.

In general, E is a function of P, with the exact functional form depending on the functional form of f(Z, x) in utility function 1. Specifically, given E, we have from Equation 2 that [z.sub.i] = [(mn).sup.-1]E[p.sup.-[theta].sub.i][P.sup.[theta]-1]. Substituting this into f (Z, x) and simplifying, we have f (Z, x) = f ([m.sup.1/([theta]-1)][P.sup.-1]E/n, x). E(P) is the solution to the problem of choosing E and x to maximize f ([m.sup.1/([theta]-1) [P.sup.-1]E/n, x), subject to (E/n) + x = Y, where E/n is total individual expenditure on goods in the taxed market and Y is individual income.

To facilitate tractable analytical treatment, we assume

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

so that E(P) has the form

E(P) = [bar.E][P.sup.-[eta]+1],

where K and [eta] are positive constants, and [bar.E] = n[K.sup.[eta][m.sup.([eta]-1)/([theta]-1)] is a coefficient independent of P or any individual [p.sub.i]. This assumption implies a constant-elasticity market demand,

Q = [bar.E][P.sup.-n],

with [eta] being the absolute value of the price elasticity of market demand. (9) The constant market demand elasticity assumption allows us to write the demand for firm i's product Equation 2 as

(5) [q.sub.i] = [bar.E][m.sup.-1][p.sup.-[theta].sub.i][P.sup.[theta]-[eta].

The two elasticities, [eta] and [theta], play important roles in this paper. We have the following assumptions and properties concerning their values.

ASSUMPTION 1. [theta] + [eta] > 2.

As we see further on, a positive market price in the oligopoly equilibrium requires that ([theta] - 1)m - [theta] + [eta] > 0, which is equivalent to [theta] + [eta] > 2, as indicated by the following property.

PROPERTY 1. ([theta] - 1)m - [theta] + [eta] > 0 for all m [greater than or equal to] 2 if and only if [theta] + [eta] >2.

PROOF: The equivalence between the two conditions is straightforward using the facts that ([theta] - 1)m - [theta] + [eta] is an increasing function of m for [theta] > 1 and that ([theta] - 1)m - [theta] + [eta] > 0 for m = 2 if and only if [theta] + [eta] > 2. QED.

ASSUMPTION 2. [theta] > [eta].

It seems reasonable to assume that any good in the taxed market is a closer substitute to the goods in the market as a group than any good outside the market. If this is true, then from the following property, [theta] > [eta].

PROPERTY 2. If [q.sub.i] is a closer substitute to Q than x is to Q for all i, then [theta] > [eta].

PROOF: From Equation 5, we have that the cross elasticity of demand for good i in the taxed market with respect to the market price P is

[differential][q.sub.i]/[differential]P P/[q.sub.i] = [theta] - [eta],

implying that [theta] - [eta] is a measure of substitutability between any [q.sub.i] and Q. Further, from the budget constraint P[Qn.sup.-1] + x = Y and Q = [bar.E][P.sup.-[eta]], the cross elasticity of demand for the numeraire good (goods outside the market as a group) x with respect to P is

[differential]x/[differential]P P/x = PQ/nx([eta] - 1).

One would normally expect that [q.sub.i] (a good in the market) is a closer substitute to Q than x (the composite of goods outside the market) is to Q. Therefore,

[theta] - [eta] > PQ/nx([eta] - 1).

It follows then that [theta] - [eta] > 0 because if [eta] < 1, [theta] - [eta] > 0 by virtue of the assumption that [theta] > 1, and if [eta] [greater than or equal to] 1, the right side of the preceding relation is greater than or equal to 0; therefore [theta] - [eta] > 0. QED.

As we show shortly, whether [eta] is larger or smaller than unity often determines the direction of the relative desirability of the two forms of excise taxation. The following property relates this condition to the substitutability and complementarity between the taxed market and the other market. As a result, we can say that while [theta] is a measure of within-market substitutability, [eta] is a measure of between-market substitutability.

PROPERTY 3. x and Q are gross substitutes if and only if [eta] > 1.

PROOF: From the budget constraint PQ[n.sup.-1] + x = Y and Q = [bar.E][P.sup.-n], we have that

[differential]x/[differential]P = ([eta] - 1)Q/n.

So x and Q are gross substitutes if and only if [eta] > 1. QED.

Short-Run Market Equilibrium

Let all firms in the market have constant marginal cost equal to c. Given the prices charged by other firms and a unit tax [tau], the first order condition for firm i's profit-maximizing pricing behavior is

(6) [q.sub.i] + ([p.sub.i] - [tau] -c) [differential][q.sub.i] / [differential][p.sub.i] = 0.

It follows from symmetry that all prices are equal in equilibrium. Using Equation 5 in solving Equation 6, the (Nash) equilibrium price and quantity under the unit tax [tau] are, respectively,

(7) [p.sup.[tau]] = [[theta]m - [theta] + [eta]/([theta] - 1)m - [theta] + [eta]] (c + [tau])

and

(8) [q.sup.[tau]] = [Em.sup.-1][([p.sup.[tau]]).sup.-[eta]].

Under an ad valorem tax t, where t is the tax rate on producer prices, a consumer price for firm i's product [p.sub.i] implies a corresponding producer price of [p.sub.i]/(1 + t). Thus, the first order condition for firm i's profit maximizing pricing problem, given other firms' prices, is

(9) [q.sub.i]/1 + t + ([p.sub.i]/1 + t - c) [differential][q.sub.i]/[differential][p.sub.i] = 0.

The (Nash) equilibrium price and quantity under the ad valorem tax t are, respectively,

(10) [p.sup.t] = [[theta]m - [theta] + [eta]/([theta] - 1)m - [theta] + [eta]] (c + ct)

and

(11) [q.sup.t] = [Em.sup.-1] [([p.sup.t]).sup.[eta]]

Note that according to Property 1, Assumption 1 ([theta] + [eta] > 2) guarantees that ([theta] - 1)m - [theta] + [eta] > 0. Therefore, from Equations 7 and 10, the equilibrium price under either tax regime is positive. On the other hand, consistent with the short-run analysis, we do not impose a condition here that guarantees positive profits for the firms in the market. In section 4, we analyze long-run equilibrium where entry and exit provide an additional equilibrating mechanism.

3. Short-Run Results

Denoting fixed cost as [C.sub.F] and using Equations 7 and 8, the equilibrium per-firm profit and government revenue under a unit tax regime are, respectively,

(12) [[pi].sup.[tau]] = ([p.sup.[tau]] - [tau] - c) [q.sup.[tau]] - [C.sub.F] = [E/[(c + [tau]).sup.[eta]-1]] {[[([theta] - 1)m - [theta] + [eta]].sup.[eta]-1]/ [[[theta]m - [theta] + [eta]].sup.[eta]]} - [C.sub.F]

and

(13) [R.sup.[tau]] = m[tau][q.sup.[tau]] = [E[tau]/[(c + [tau]).sup.[eta]]] [[([theta] - 1)m - [theta] + [eta]]/[theta]m - [theta] + [eta]].sup.[eta]].

In the same fashion, using Equations 10 and 11, the equilibrium per-firm profit and government revenue under the ad valorem tax are, respectively,

(14) [[pi].sup.t] = ([p.sup.t]/1 + t - c)[q.sup.t] - [C.sub.F] = [Ec/[(c + ct).sup.[eta]]] {[[([theta] - 1)m - [theta] + [eta]].sup.[eta]-1]/[[[theta]m - [theta] + [eta]].sup.[eta]] - [C.sub.F]

and

(15) [R.sup.t] = m([p.sup.t]t/1 + t)[q.sup.t] = [Ect/[(c + ct).sup.[eta]]] [[([theta] - 1)m - [theta] + [eta]/[theta]m - [theta] + [eta]].sup.[eta]-1].

For market demand elasticities greater than unity, the requirement that government always sets tax rates in the increasing portion of the total tax revenue function implies, from Equations 13 and 15, certain relations between [tau] and [eta] and between t and [eta], which are stated as the following assumption.

ASSUMPTION 3. For [eta] > 1,

[tau]/c < 1/[eta] - 1 and t < 1/[eta] - 1.

Note that if market demand is inelastic ([eta] [less than or equal to] 1), feasible tax rates are unrestricted. We adopt Assumption 3 throughout this paper. (10) Now consider the welfare implications of switching from a unit tax with a tax rate [tau] to an ad valorem tax that maintains the same level of revenue. The minimum ad valorem tax rate that generates at least the unit tax revenue, which we denote as [t.sup.[tau]], is the solution to

(16) [t.sup.[tau]] / [(1 + [t.sup.[tau]]).sup.[eta]] = [tau]/c / [(1 + [tau]/c).sup.[eta]] [([theta] - 1)m - [theta] + [eta] / [theta]m - [theta] + [eta]].

Because the bracketed term on the right side of Equation 16 is less than 1 for any finite 0, we have [t.sup.[tau]] < [tau]/c as long as total tax revenue is increasing in the tax rate at the original unit tax rate. (11) Thus, it follows from Equations 7 and 10 that [p.sup.t] < [p.sup.[tau]], and we have the following proposition.

PROPOSITION 1. Whenever market goods are heterogeneous (finite [theta]), for any unit tax, there exists an ad valorem tax that raises the same amount of revenue and generates higher welfare for consumers of the taxed goods.

That consumer surplus is increased when a unit tax is replaced with an equal-revenue ad valorem tax does not automatically imply that such a tax switch is Pareto improving or even overall welfare improving (however the overall welfare is defined), because firm owners may experience reduced income (as is clear from Proposition 2). Following the Pareto comparison approach as represented in Skeath and Trandel (1994), we assume in this paper that firm owners, who only consume the numeraire commodity so that their welfare is monotonic in profits, are a different group of people than the consumers of the taxed goods. Therefore, only if an equal-revenue replacement of a unit tax with an ad valorem tax raises both consumer surplus and firm profits, is such a change regarded as Pareto improving. In the rest of this section, the focus is on firms' comparative profitability under alternative forms of excise taxes. Nonetheless, in the Appendix, we show that ad valorem taxation always welfare dominates unit taxation in the short run in our model if consumers of the taxed goods are also the owners of the firms that produce these goods. (12)

Using Equations 12 and 14, we have that the difference in profits when a unit tax is replaced with an equal-revenue ad valorem tax is

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

For any given values of [theta], [eta], m and [tau]/c, the equal-revenue generating ad valorem tax rate [t.sup.[tau]] can be calculated from Equation 16. Then Equation 17 can be checked to see whether profits are higher under ad valorem taxation.

Begin by considering the special case where goods in the market taken as a group and all other goods (represented by the numeraire good x) are not gross substitutes (that is, [eta] [less than or equal to] 1 according to Property 1). Note first that price-setting firms have a stronger incentive to reduce their price under an ad valorem tax regime than under a unit tax regime because the tax paid on each unit of their product is proportional to the price charged. The effects on firms' profits of a lower price under ad valorem taxation are twofold. Lower prices reduce per unit profit but increase quantity demanded. Whether profits will be enhanced by switching to the ad valorem regime depends on the relative magnitude of these two effects. The smaller the market demand elasticity, the less likely the quantity effect will dominate the price effect. In particular, when [eta] [less than or equal to] 1, with a lower price, output rises, but total sales revenue falls, leaving firms with lower profits. Thus, we have the following proposition.

PROPOSITION 2. If [eta] < 1, then firms earn lower profits under ad valorem taxation than under unit taxation that generates the same tax revenue.

PROOF: We must show that, when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and finite [theta]. Because [t.sup.[tau]] < [tau]/c, [eta] [less than or equal to] 1 implies from Equation 17 that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. QED.

Proposition 2 says that, in the case in which the taxed market features an inelastic market demand, ad valorem Pareto dominance fails although consumers are always better off under ad valorem taxation than under equal-revenue unit taxation according to Proposition 1. Proposition 2 provides us with a basis for the existence of both forms of taxation that is not based on market or government imperfections. Specifically, for markets with inelastic demand, firm profits are greater with unit taxation. However, we do not expect many such markets to exist because such markets must be gross complements to all other goods. (13)

We know that ad valorem taxation Pareto dominance does not hold when the taxed market is a gross complement to all other goods ([eta] [less than or equal to] 1), but is there a combination of finite [theta] and [eta] > 1 that assures ad valorem taxation Pareto dominance? If so, is Pareto dominance more likely with smaller [theta] or larger [eta]? (14) The answer to both questions is yes and is reflected in the following fact and proposition.

FACT (Existence). Pareto dominance of ad valorem taxation holds for some [theta] < [infinity] and [eta] > 1.

Consider a hypothetical economy with [theta] = 3, [eta] = 2, m = 2, [tau]/c = 0.5. (15) From Equation 16, [t.sup.[tau] = 0.188 and from Equation 17, the ratio determining the relative advantage from the firms' perspective of ad valorem taxation is 1.063 > 1. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So, for this case, an equal-revenue replacement of a unit tax with an ad valorem tax will be Pareto improving. Given the existence of ad valorem Pareto dominance, we have the following proposition.

PROPOSITION 3.

(a) Other things being equal, ad valorem taxation is more likely to Pareto dominate equal-revenue unit taxation the larger the degree of product differentiation in the taxed market.

(b) Other things being equal, ad valorem taxation is more likely to Pareto dominate equal-revenue unit taxation the more elastic the market demand.

PROOF: (a) We want to prove that for fixed values of [eta] > 1, m, [tau]/c, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [theta] = [[theta].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[theta].sub.2] < [[theta].sub.1]. First, the right side of Equation 16 is increasing in [theta]. Second, the left side of Equation 16 is increasing in [t.sup.[tau]] under Assumption 2 that the revenue effect of an increase in tax rates is positive. Thus, [dt.sup.[tau]]/d[theta] > 0. Then, the left side ratio of Equation 17, [(1 + [tau]/c).sup.[eta]-1]/[(1 + [t.sup.[tau]).sup.[eta]], is decreasing in [theta]. Hence, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] when [theta] = [[theta].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] when [theta] = [[theta].sub.2] < [[theta].sub.1].

(b) We want to prove that for given values of [theta], m, [tau]/c, if [t.sup.[tau]] from Equation 16 decreases in [eta] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], for [eta] = [[eta].sub.1] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[eta].sub.2] > [[eta].sub.1]. Note that

(18) d { 1n [[(1+[tau]/c).sup.[eta]-1]/ [(1+[t.sup.[tau]).sup.[eta]]] / d[eta]} = 1n (1 + [tau]/c / 1 + [t.sup.[tau]]) - ([eta] / 1 + [t.sup.[tau]]) [dt.sup.[tau]] / d[eta],

which is positive if ([dt.sup.[tau]/d[eta]) < 0. Then, from Equation 17, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [eta] = [[eta].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[eta].sub.2] > [[eta].sub.1]. QED.

Skeath and Trandel (1994) established that a monopoly can earn a larger profit under an

ad valorem tax that generates the same level of tax revenue as a unit tax. So part (a) of Proposition 3 can be easily explained because each firm in the oligopoly market operates more like a monopoly as the elasticity of substitution among their products gets smaller. The intuition for part (b) of Proposition 3 is the same as that for Proposition 2.

For an oligopoly market with homogenous products and linear demand, Skeath and Trandel (1994) found that Pareto dominance of ad valorem taxation never holds when the number of firms in a market is sufficiently large, but always holds when the tax level exceeds a critical value. With product heterogeneity where firms engage in Bertrand competition, however, the first of these propositions continues to hold while the second does not.

PROPOSITION 4. (a) Other things being equal, ad valorem taxation is less likely to Pareto dominate equal-revenue unit taxation the larger the number of firms in the taxed market. (b) Other things being equal, ad valorem taxation is more likely to Pareto dominate equal-revenue unit taxation the lower the tax level.

PROOF: (a) We want to prove that for given values of [theta], [eta], and [tau]/c, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for m = [m.sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [m.sub.2] > [m.sub.1]. First, by signing its derivative with respect to m, the second term on the right side of Equation 16 can be shown to be increasing in m. Second, the left side of Equation 16 is increasing in [t.sup.[tau]] so long as revenues increase with tax rates. Thus, ([dt.sup.[tau]]/dm) > 0. Then from Equation 17, results in part (a) follow.

(b) We want to prove that for given values of [eta], m, c and sufficiently large [theta], if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [tau] = [[tau].sub.1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any [[tau].sub.2] < [[tau].sub.1]. First, from Equation 16,

[dt.sub.[tau] / d([tau]/c) = 1/[tau]/c - [eta]/1+[tau]/c / 1/[t.sup.[tau]] - [eta]/1+[t.sup.[tau]].

Also from Equation 16, [t.sup.[tau]] [right arrow] [tau]/c when [theta] [right arrow] [infinity]. Hence,

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Second,

d 1n [[(1+[tau]/c).sup.[eta]-1] / [(1+[t.sup.[tau]]).sup.[eta]]] / d([tau]/c) = [eta] - 1 / 1 + [tau]/c - [eta] / 1 + [t.sup.[tau]] [dt.sup.[tau]] / d([tau]/c)

which, from Equation 19 and that [t.sup.[tau]] < [tau]/c, is negative when [theta] is sufficiently large. So,

[(1 + [tau]/c).sup.[eta]-1] / [(1 + [t.sup.[tau]]).sup.[eta]]

decreases in [tau] for given c. Then, from Equation 17, results in part (b) follow. QED.

The first part of Proposition 4 confirms prior results regarding the effect of the number of firms in an oligopoly market on ad valorem taxation Pareto dominance. We can offer some intuition for why [theta] > [eta] guarantees that a larger number of firms in the market makes an equal-revenue switch from unit to ad valorem tax regimes less likely to increase firm profits. First, note that a major difference between unit and ad valorem taxation is that price-setting firms have a stronger incentive to lower their prices under an ad valorem tax regime because taxes are proportional to price. A larger number of firms in the market serves to strengthen this price incentive. Whether all firms in the market can benefit from individual firms' incentive to lower prices depends on whether the sales increase for individual firms comes from expanding market demand or from "stealing" other firms' sales, which in turn depends on the relative magnitudes of [eta] and [theta].

The second part of Proposition 4 says that the smaller the original unit tax rate, the more likely an equal-revenue switch from unit to ad valorem taxation will increase firm profits. Thus, when the goods in the market are sufficiently close substitutes, the lower the original level of a unit tax the more likely an equal-revenue ad valorem tax will Pareto dominate, exactly the opposite of previous homogeneous product linear demand results. Alternative but equally plausible assumptions are responsible for the differential results. Specifically, we assume Bertrand competition and a demand with constant price elasticity, while Skeath and Trandel (1994) use Coumot competition and linear demand. Our results here serve to show that the ad valorem Pareto dominance, with respect to the level of taxation, is sensitive to model specifications.

4. Long-Run Equilibrium: The Case of Endogenous Variety

The short-run results for a heterogeneous product market are more restrictive than for homogeneous markets because holding the number of firms constant, holds product variety constant. Because firms' profits are always zero in the long ran, the focus here is on consumer welfare. (16) When variety is endogenous, excise tax comparisons must account for both price and variety effects. Given utility function 1, for any given elasticity of substitution among goods in the market, consumers' welfare is determined by market price as well as market variety. Thus, an assessment of the welfare implications of the two excise taxes must take into account their impact on both market price and variety.

In this section we examine whether ad valorem taxation welfare dominance exists in the long run. (17) Our main long-run results are the following: (i) While a switch to ad valorem taxation reduces market price, consumer choice may also be reduced; (ii) ad valorem taxation welfare dominance always holds.

The approach taken in this section is a little different from that taken in previous sections. Rather than making a direct comparison between all unit taxation and all ad valorem taxation, we begin with a mix of both taxes and analyze the effects of substituting ad valorem for unit taxation at the margin. (18) When both a unit tax [tau] and an ad valorem tax t are imposed, the ith firm's problem is to choose [p.sub.i] to maximize its profits

[q.sub.i] = ([p.sub.i] - [tau]/1 + t - c) - [C.sub.F],

where [q.sub.i] is related to [p.sub.i] through Equation 5. The first order condition yields the equilibrium price

(20) p = (c + [tau] + tc)[[theta]m - [theta] + [eta]]/([theta] - 1)m - [theta] + [eta]

for all the goods in the market. (19)

The zero profit condition that determines the equilibrium number of firms is

[Em.sup.-1][p.sup.-[eta]](p - [tau]/1 + t - c) - [C.sub.F] = 0,

or, substituting E = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)] from the discussion before Equation 5,

(21) n[K.sup.[eta]][m([eta]-[theta])/([theta]-1)][[p.sup.-[eta]](p - [tau] - c - tc) - (1 + t)[C.sub.F] = 0

From Equations 20 and 21, the following comparative statics results, with respect to the effects of a change in either z or t on the equilibrium m and p, are obtained (See Appendix for a derivation):

[differential]m/[differential][tau] = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) ([eta] - 1),

[differential]p/[differential][tau] = - p/[OMEGA](c + [tau] + tc) ([theta]m + [eta] - 1)[([theta] - 1)m - [theta] + [eta]],

[differential]m/[differential]t = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) [[tau] + [eta]c(1 + t)]/(1 + t),

[differential]p/[differential]t = - p/[OMEGA](c + [tau] + tc) ([theta] - 1)m[[tau] + [eta]c(1 + t)] - c(1 + t)[OMEGA]/(1 + t),

where

[OMEGA] = -([theta] - 1)([theta] - [eta])(m - 1) - ([theta]m - [theta] + [eta])[([theta] - 1)m - [theta] + [eta]] < 0.

Hence, from Equation 22, [differential]p/[differential][tau] > 0, [differential]p/[differential]t > 0, [differential]m/[differential]t < 0, and [differential]m/[differential][tau] [greater than or equal to] 0 if and only if [eta] [less than or equal to] 1.

The effect of taxation on the equilibrium number of firms is a combination of two factors: an industry scale effect and a firm scale effect. A pure increase in ad valorem taxation reduces equilibrium market quantity. Although equilibrium firm size may go up or down, the market downsizing effect always dominates. As a result, an uncompensated increase in ad valorem taxation reduces the number of firms. On the other hand, a pure increase in the level of a unit tax decreases both equilibrium market quantity and equilibrium firm size. If the resulting market quantity decrease is small enough, it will be more than offset by the decrease in equilibrium firm scale, and the number of firms may actually rise. As it turns out, a market demand elasticity less than one is sufficient to ensure that the reduction in firm scale dominates the market scale reduction, so the number of firms rises in response to a pure increase in unit taxation.

Kay and Keen (1983) and Anderson, de Palma, and Kreider (2001 a) found, as we did here, that the long-run equilibrium consumer price increases in both tax rates. Using a locational model of product differentiation, Kay and Keen (1983) also found that an increase in ad valorem taxation reduces variety. As they explain, ad valorem taxation is akin to a grossing up of the fixed cost associated with each variety. While we confirm the negative effect of ad valorem taxation on variety, we find that unit taxation also has a variety effect and relate the variety effect of unit taxation to the elasticity of market demand.

For our purposes, we want to consider the revenue-neutral substitution of ad valorem taxation for unit taxation. In equilibrium, the market price p and the number of firms m are functions of tax rates [tau] and t, as implicitly determined by Equations 20 and 21. Therefore, the total tax revenue from the market can also be expressed as a function of these tax rates, which is

R([tau], t) = mq(p - p - [tau]/1 + t),

where q = [Em.sup.-1][p.sup.-[eta]]. Substituting E = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)] and the zero-profit condition

q(p - [tau]/1 + t - c) = [C.sub.F],

we arrive at

(23) R([tau], t) = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]](p - c) - m[C.sub.F].

Note that in the tax revenue expression 23, tax rates [tau] and t do not directly appear, and R([tau], t) is a function of [tau] and t through p and m. Such an expression of the tax revenue function is significant in simplifying some of the following derivations.

The effects of an increase in [tau] or t on the tax revenue are given by

(24) [differential]R/[differential][tau] = n[K.sup.[eta]][m.sup.([eta] - 1)/([theta] - 1)][p.sup.-[eta]][1 - [eta]([p-c)/p] [differential]p/[differential][tau] + n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)-1][p.sup.-[eta]][(p - c) [eta]- 1/ [theta]-1 - p-c-[tau]-tc/1+t] [differential]m/[differential][tau]

[differential]R/[differential]t = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]][1 - [eta]([p-c)/p] [differential]p/[differential]t + n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)-1][p.sup.-[eta]][(p - c) [eta]-1/[theta]-1 - p-c-[tau]-tc/1+t] [differential]m/[differential]t

both of which are assumed to be positive. (20) Substituting ad valorem taxation for unit taxation at the margin while keeping tax revenue unchanged requires that

(25) d[tau]/dt = - [differential]R/[differential]t/[differential]R/[differential][tau].

PROPOSITION 5. A revenue-constant substitution of ad valorem taxation for unit taxation always reduces market price, but reduces the number of firms as well if [eta] [less than or equal to] 1, that is, if the taxed market is not a gross substitute to the numeraire.

PROOF: The impact on the consumer price of a constant-revenue increase in ad valorem taxation (with an offsetting decrease in unit taxation) is given by

(26) dp/dt = [differential]p/[differential]t + [differential]p/[differential]t d[tau]/dt = [([differential]R/[differential][tau]).sup.-1] ([differential]R/[differential][tau] [differential]p/[differential]t - [differential]R/[differential]t [differential]p/[differential][tau]) = [([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)-1][p.sup.-[eta]][(p-c)[eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t]{[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]}.

In the Appendix, it is shown that

(27) (p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t < 0

and

(28) [differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau] = - p([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/(1 + t)[OMEGA][(c + [tau] + tc).sup.2]([theta] - n) [[tau] + c(1 + t)] > 0.

Therefore, dp/dt < 0. That is, market price falls as unit taxation is replaced with ad valorem taxation. On the other hand, the impact on the number of firms of a revenue-constant move toward ad valorem taxation is determined by

dm/dt = [differential]m/[differential]t + [differential]m/[differential][tau] d[tau]/dt.

Because [differential]m/[differential]t, d[tau]/dt < 0, [differential]m/[differential][tau] [greater than or equal to] 0 is sufficient (but not necessary) for ensuring that dm/dt < 0, which is the case if [eta] [less than or equal to] 1. QED.

When variety matters in the long-run equilibrium of a heterogenous product market, a lower consumer price under ad valorem taxation by itself cannot guarantee its welfare dominance, and the variety dimension must also be considered in determining its relative efficiency. Then, exactly how is the relative long-run efficiency of the two tax regimes determined based on their effects on the equilibrium price and variety? When there are m firms in the taxed market and each firm charges a price of p, individual welfare, as we established in section 2, is given by

(29) f([m.sup.1/([theta]-1)][p.sup.-1] E/n, Y - E/n),

where E/n is individual expenditure on goods in the taxed market and Y is individual income. From Equation 29 it follows that a consumer welfare index that combines both price and variety effects is [m.sup.1/([theta]-1)][p.sup.-1]. Thus, for an equal-revenue switch from unit to ad valorem taxation to increase consumer welfare in the long ran, the price (reduction) effect must be sufficiently large to overcome any possible negative variety effect. The following proposition says that this is always the case in our model.

PROPOSITION 6. Ad valorem taxation always welfare dominates unit taxation in the long run.

PROOF: Because [m.sup.1/([theta]-1)][p.sup.-1] is the welfare index that combines both price and variety considerations, the welfare impact of an equal-revenue switch from unit to ad valorem taxation can be assessed by

(30) d([m.sup.1/([theta]-1)][p.sup.-1])/dt = [m.sup.1/([theta]-1)-1]/([theta] - 1)p dm/dt - [m.sup.1/([theta]-1)]/[p.sup.2] dp/dt,

where dp/dt is given by Equation 26 and

dm/dt = [differential]m/[differential]t + [differential]m/[differential][tau] d[tau]/dt = [([differential]R/[differential][tau]).sup.-1] ([differential]R/[differential][tau] [differential]m/[differential]t - [differential]R/[differential]t [differential]m/[differential][tau]) = - [([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.[eta]-1/[theta]-1][p.sup.-[eta]][1 - [eta](p - c)/p] {[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]}.

Substituting dp/dt and dm/dt, Equation 30 becomes

d([m.sup.1/([theta]-1)][p.sup.-1])/dt = [([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.([eta] - [theta]+1)/ ([theta]-1)][p.sup.-[eta]-2]{- p/[theta] - 1 [1 - [eta](p-c)/p] - [(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t]} {[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]} =[([differential]R/[differential][tau]).sup.-1] n[K.sup.[eta]][m.sup.([eta]-[theta]+1)/([theta]-1)][p.sup.-[eta]-2][-c/[theta] - 1 + p - c - [tau] - tc/1 + t] {[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/[differential][tau]}.

From Equation 28,

[differential]m/[differential][tau] [differential]p/[differential]t - [differential]m/[differential]t [differential]p/ [differential][tau] > 0.

Further,

-c/[theta]-1 + p - c - [tau] - tc/1 + t = - c/[theta] - 1 + (c + [tau] + tc)m/[([theta] - 1)m - [theta] + [eta]](1 + t) = -c(1 + t)[([theta] - 1)m - [theta] + [eta]] + ([theta] - 1)(c + [tau] + tc)m/([theta] - 1)[([theta] - 1)m - [theta] + [eta]](1 + t) = c(1 + t)([theta] - [eta]) + ([theta] - 1)[tau]m/([theta] - 1)[([theta] - 1)m - [theta] + [eta]](1 + t) > 0.

So, d([m.sup.1/([theta]-1)][p.sup.-1])/dt > 0. That is, an equal-revenue substitution of ad valorem taxation for unit taxation always increases consumer welfare in the long run. QED.

Proposition 6 is complementary to Kay and Keen (1983) and Anderson, de Palma, and Kreider (2001b), both of whom used versions of a locational model of product differentiation to study the relative efficiency of ad valorem and unit taxes. In Kay and Keen (1983), variety is excessive in the absence of taxation. So they showed that, initially, the ad valorem tax, which is variety reducing in locational models, should be used to bring about the optimal level of product variety, but any additional revenue should be raised using the unit tax, which is variety neutral in locational models. For similar reasons, Anderson, de Palma, and Kreider (2001b) found that ad valorem taxation is welfare dominated by unit taxation in a locational model in which the equilibrium variety is optimal in the absence of taxation. However, the locational models used in these studies, while allowing variety to play a role in long-run welfare analysis, do not allow a quantity and/or price effect to play any role in the analysis. On the other hand, the established long-run welfare dominance of ad valorem taxation for homogenous product markets is entirely based on the quantity and/or price advantage of ad valorem taxation. While Anderson, de Palma, and Kreider (2001b) suggested that allowing the quantity and/or price effect may reverse the results they obtained and give back the ad valorem tax its efficiency advantage, they did not provide any formal analysis to substantiate this valuable point. This paper confirms that point and generalizes their findings.

5. Concluding Remarks

There is a recent trend to study surplus incidence (in contrast to more traditional price incidence) of excise taxation. (21) Related to this topic is the investigation of Pareto superiority of ad valorem taxation in noncompetitive markets. In these Pareto comparisons, the effect of (equal-revenue) switching between tax regimes on both consumer and producer surpluses are examined, providing a more complete picture of who loses or gains from a change in tax structure, enhancing our understanding of why unit taxes are imposed in some markets and ad valorem taxes in others. Using the elasticity of substitution parameters in the utility function, we define market demand and price when a market consists of heterogeneous products. We can then bring the within-market and between-market substitutability to bear on the comparison of unit and ad valorem taxation. As a result, we show conditions under which consumers and firms would prefer one type of excise taxation to another.

Our results address several interesting aspects of excise taxation. In the short run, when the number of firms, and by construction product variety, is fixed, the conditions for Pareto dominance of ad valorem taxation, while similar to those derived assuming homogeneous product oligopoly markets, are different in certain critical aspects. First, when the market consists of heterogeneous products and there exists a unit tax rate such that an equal-revenue ad valorem tax is Pareto dominant, any smaller unit tax rate has a corresponding equal-revenue Pareto dominant ad valorem tax. More importantly, our work emphasizes the critical importance of between-market substitutability. In particular, when goods in the taxed market have a complementary relation to all other goods, implying that market demand is inelastic, ad valorem taxation Pareto dominance fails in the sense that firms earn lower profits. The nature of this ad valorem Pareto dominance failure is important because it rests on the effect on profits of a switch from unit to ad valorem taxation. While such a change makes consumers better off in that it results in lower price, firm owners are worse off because profits are lower. Thus, firms in markets with inelastic demand will prefer unit taxation, perhaps explaining the persistence of unit taxation in such markets as gasoline and cigarettes.

While our short-run results are interesting, the real advantage of our approach is that we allow for long-run adjustment in the number of firms and, therefore, in product variety. In the long run, the effect of taxation on general welfare depends not just on the final consumer price, but on the equilibrium number of firms as well. Again, the between-market substitutability (the price elasticity of the market demand for the taxed goods) plays an important role here. An equal-revenue substitution of ad valorem for unit taxation reduces variety as long as the goods in the taxed market have a complementary relation to the other goods (i.e., market demand is inelastic). Nonetheless, we have been able to show that ad valorem taxation always welfare dominates unit taxation in the long run.

Appendix

Short-Run Welfare Dominance (in Total Welfare) of Ad Valorem Taxation

Note that one cannot argue for the short-run ad valorem total welfare dominance by simply looking at the sum of consumers' and producers' surpluses and be satisfied with the fact that price is lower and output is higher, and, therefore, consumer surplus plus profits is higher under ad valorem taxation than under equal-revenue unit taxation. First, if consumers and firm owners are two different sets of people, as these previous studies have implicitly assumed, the sum of consumer and producer surpluses is not an unambiguous welfare indicator. In this case, one must separately consider the welfare of consumers and producers (firm owners), as we did in the present paper. Second, if consumers are also firm owners, then a comparison of their welfare under two forms of excise taxes can be unambiguously made. In this case, however, profits must be explicitly added to consumers' income in determining final consumer utility under each tax. With regard to the second scenario, we have the following proposition which upholds the ad valorem taxation welfare dominance (in total welfare).

PROPOSITION 1'. For the purpose of making a short-run welfare comparison, assume consumers of the taxed goods own the firms that produce these goods. Whenever market goods are heterogeneous (finite 0), for any unit tax, there exists an ad valorem tax that raises the same amount of revenue and generates higher welfare for individuals as both consumers and firm owners.

PROOF: The proof consists of two steps. The first step is to show that adding profits to consumers' income does not alter the finding that equilibrium price is lower, and equilibrium quantity higher, under ad valorem taxation than under equal-revenue unit taxation. Within the assumptions of this model, the equilibrium price and quantity of each firm under two alternative tax regimes are given by Equations 7 to 8, and 10 to 11, regardless of whether consumers' income includes profits. Therefore, the relation between the unit tax rate and the equal-revenue ad valorem tax rate--Equation 16--still holds. As a result, price is lower and quantity is higher under ad valorem taxation than under equal-revenue unit taxation.

The second step is to show that consumer (as both consumers and firm owners) welfare is improved as the tax regime switches from unit taxation to equal-revenue ad valorem taxation, given that the price is lower (and quantity higher) under the latter tax regime. To show this, it is sufficient to demonstrate that consumers have more money left after buying the original quantity with the new, lower price, taking into account the difference in profits under the two tax regimes. Denote (p, Q) as the price-(aggregate) quantity pair under the original unit tax regime, and (p', Q') as the price-quantity pair under the equal-revenue ad valorem tax regime. We have shown in the first step that p' < p and Q' > Q. Under unit taxation, the money left (for all consumers) after buying the equilibrium quantity of each firm's product is

nY + (pQ - R - cQ - m[C.sub.F]) - pQ,

where R is tax revenue paid. Under equal-revenue ad valorem taxation, on the other hand, the money left after buying the old quantity (the equilibrium quantity under the unit taxation) of each firm's product is

nY + (p'Q' - R - cQ' - m[C.sub.F]) - p'Q,

which is larger than the earlier expression by (p' - c)(Q' - Q) > 0. QED.

Derivation of Comparative Statics Results in Equation 22

Using logarithmic terms, equilibrium conditions 20 and 21 can be expressed as

n p = ln(c + [tau] + tc) + ln([theta]m - [theta] + [eta]) - ln[([theta] - 1)m - [theta] + [eta]]

ln(n[K.sup.[eta]]) + [eta] - [theta]/[theta] - 1 ln m - [eta] ln p + ln(p - c - [tau] - tc) = ln(1 + [tau]) + ln (C.sub.F).

Taking derivatives with respect to [tau] in the above two equations, we have

1/p [differential]p/[differential][tau] = 1/c + [tau] + tc + [[theta]/[theta]m - [theta] + [eta] - [theta] - 1/([theta] - 1)m - [theta] + [eta]] [differential]m/[differential][tau]

[eta] - [theta]/[theta] - 1 1/m [differential]m/[differential][tau] + 1/p - c - [tau] - tc ([differential]p/[differential][tau] - 1) = 0.

Solving for [differential]p/[differential][tau] and [differential]m/[differential][tau], we have

[differential]m/[differential][tau] = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) x ([eta] - 1)

[differential]p/[differential][tau] = - p/[OMEGA](c + [tau] + tc) x ([theta]m + [eta] - 1)[([theta] - 1)m - [theta] + [eta]].

Similarly, taking derivatives with respect to t in the two equations in (A1), we have

1/p [differential]p/[differential]t = c/c + [tau] tc + [[theta]/[theta]m - [theta] + [eta] - [theta] - 1/([theta] - 1)m - [theta] + [eta]] [differential]m/[differential]t

[eta] - [theta]/[theta] - 1 1/m [differential]m/[differential]t - n/p [differential]p/[differential]t + 1 /p - c [tau] - tc ([differential]p/[differential]t - c) = 1/ 1 + t.

Solving for [differential]p/[differential]t and [differential]m/[differential]t, we have

[differential]m/[differential]t = ([theta] - 1)m[([theta] - 1)m - [theta] + [eta]][[theta]m - [theta] + [eta]]/[OMEGA](c + [tau] + tc)([theta] - [eta]) x [[tau] + [eta]c(1 + t)/(1 + t)

[differential]p/[differential]t = - p/[OMEGA](c + [tau] + tc) x ([theta] - 1)m[[tau] + [eta]c(1 + t)] - c(1 + t) [OMEGA]/(1 + t)

PROOF OF EQUATION 27. From Equation 24,

(A2) [differential]R/[differential]t = n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]][1 - [eta](p - c)/p] [differential]p/[differential]t + n[K.sup.[eta]][m.sup.([eta]-1)/([theta]-1)][p.sup.-[eta]][(p - c) [eta] - 1/ [theta] - 1 - p - c - [tau] - tc/1 + t] [differential]m/[differential]t.

To prove

(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t < 0,

consider the following two situations, while keeping in mind that, from Equation 22, [differential]p/[differential]t > 0 and [differential]m/[differential]t < 0.

(a) 1 - [eta](p - c)/p < 0.

In this case, the assumption [differential]R/[differential]t > 0 implies

(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t < 0.

(b) 1 - [eta](p - c)/p [greater than or equal to] 0.

In this case, [eta] [less than or equal to] p/(p - c). Then

(p - c) [eta] - 1/[theta] - 1 - p - c - [tau] - tc/1 + t [less than or equal to] (p - c) p/p - c - 1/[theta] - 1 - p - c - [tau] - tc/1 + t = c/[theta] - 1 - p - c - [tau] - tc/1 + t = c/[theta] - 1 - (c + [tau] + tc)m/(1 + t)[([theta] - 1)m - [theta] + [eta]] = - c(1 + t)([theta] - [eta]) + ([theta] - 1) [tau]m/([theta] - 1)(1 + t)[([theta] - 1)m - [theta] + [eta]] < 0.

DERIVATION OF EQUATION 28. From Equation 22,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

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Trandel, Gregory A. 1999. Producers lose: The relative percentage reductions in surplus due to an ad valorem commodity tax. Public Finance Review 27:96-104.

(1) However, the two versions of excise taxes are not equivalent in a perfectly competitive market when product quality is endogenous. For examples of incorporating quality into comparative commodity tax analysis, see Barzel (1976), Bohanon and Van Cott (1991), Kay and Keen (1991), Cremer and Thisse (1994), and Liu (2003). See also Saving (1982) for a general discussion of the product quality and market structure. Throughout the discussion of this paper, however, we assume that the quality of each firm's product is exogenously determined.

(2) Suits and Musgrave (1953) showed for the monopoly case that when the same revenue is raised, equilibrium output and thus welfare is greater under an ad valorem tax than under a unit tax. Similar welfare comparison results for the oligopoly market were established by Delipalla and Keen (1992).

(3) The Pareto dominance of ad valorem taxation in a monopoly market was first found by Skeath and Trandel (1994). Assuming a homogeneous product and linear demand in an oligopoly market, Skeath and Trandel (1994) also demonstrated that the Pareto dominance of ad valorem taxation holds when the tax level exceeds a critical value but never holds when the number of firms in a market is sufficiently large.

(4) For example, federal telephone and air transportation taxes and state and local public utility taxes are ad valorem, while national gasoline taxes and state liquor and cigarette taxes are unit.

(5) Also see Cremer and Thisse (1994) for an analysis of excise taxes in a market with vertical product differentiation.

(6) The focus of Kay and Keen (1983) and Keen (1998) is on the long run in which firms always earn zero profits. On the other hand, although the incidence analysis of both forms of excise taxes in Anderson, de Palma, and Kreider (2001a) includes their effects on profits, it is not a differential incidence analysis in which one form of excise tax is substituted for another with the total tax revenue unchanged. As a result, it does not address firm comparative profitability under the alternative tax regimes.

(7) Anderson, de Palma, and Kreider (2001a) considered the long-ran price effects but not the variety effects of the two forms of excise taxes.

(8) The constraint [theta] > 1, first introduced by Dixit and Stiglitz (1977) to motivate a desire for variety, is imposed here to ensure an equilibrium for firms' profit maximization problem. The welfare role of [theta] emphasized by Dixit and Stiglitz is discussed in section 4 where entry and exit is endogenous.

(9) From this assumption the price elasticity of total expenditures on goods produced in the oligopoly market is constant at [eta] - 1 and total expenditures decrease (increase) in P if [eta] - 1 is positive (negative).

(10) Note that a revenue (or welfare) maximizing government would not raise tax rates to a point where revenue is decreasing in the tax rates.

(11) If [theta] = [infinity] (i,e., all goods in the oligopoly markets are identical), Equation 16 implies [t.sub.[tau]] = [tau]/c. Therefore, from Equations 7, 8, 10, and 11, the equilibriums for equal revenue ad valorem and unit taxes are identical. This equivalence result stands in marked contrast to previous nonequivalence results for homogenous oligopoly markets [Suits and Musgrave (1953), Delipalla and Keen (1992), and Skeath and Trandel (1994)] and results from the fundamental difference between Bertrand competition and Coumot competition in modeling oligopoly behavior. Bertrand price competition by oligopoly firms producing identical products leads to the perfect competition outcome (marginal cost pricing) regardless of tax regime.

(12) It has been unanimously found that ad valorem taxation has an efficiency advantage in the short run. See, for example, Delipalla and Keen (1992) for the case of product homogeneity and Anderson, de Palma, and Kreider (2001b) for the case of product heterogeneity.

(13) We regard gross complementarity as a rare case because, while possible in a two good world, it is unrealistic in a world where the other good consists of the composite of all goods not in the single heterogeneous market, because the income effect of a change in P is proportional to the ratio of expenditures on Q and all other goods. Viewing the consumer's utility maximization problem as a choice between Q/n and x subject to P[Qn.sup.-1] + x = Y, the Slutsky equation gives us [[epsilon].sub.xP] = [[epsilon].sub.xP]|[sub.u] - ([differential]x/[dif (QP/nx)--where [[epsilon].sub.xP], [[epsilon].sub.xP]|[sub.u] are, respectively, uncompensated and compensated elasticities of x with respect to P--which is positive for sufficiently small QP/nx.

(14) Note that the experiment with changes in [eta] must be conducted within the range where at the original [tau], tax revenue is increasing in the tax rate so that [tau]/c < 1/([eta] - 1).

(15) Note these parameter values satisfy Assumptions 1 to 3.

(16) If existing firms earn positive (negative) short-run profits following the regime switch, they will earn positive (negative) profits during the entire transition. Thus, the short-run results concerning firms' profitability presented in Propositions 2, 3, and 4, can be directly generalized to account for transitional profits.

(17) Delipalla and Keen (1992) demonstrate that ad valorem taxation welfare dominates unit taxation in the long run in homogenous product oligopoly markets.

(18) The reason for doing so is that with an additional endogenous variable m and a nonlinear relation among variables, it is impossible to explicitly solve for p and m under either tax regime. On the other hand, the marginal approach here does not rely on an explicit solution to the initial equilibrium.

(19) Note Equation 20 is the general form of Equation 7 or 10.

(20) This assumption is similar to Assumption 3 in spirit. However, with endogeneity of both p and m, it cannot be easily boiled down to a simple up-bound on the tax rates.

(21) For examples, see Hines, Hlinko, and Lubke (1995), Trandel (1999), and Anderson, de Palma, and Kreider (2001a).

Liqun Liu * and Thomas R. Saving ([dagger])

* Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231, USA; E-mail: lliu@tamu.edu.

([dagger]) Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231, USA; E-mail:

t-saving@tamu.edu; corresponding author.

We want to thank Andy Rettenmaier, Laura Razzolini, and several anonymous referees for very helpful comments and suggestions.

Received November 3, 2003; accepted February 3, 2005.

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Author: | Saving, Thomas R. |
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Publication: | Southern Economic Journal |

Geographic Code: | 1USA |

Date: | Oct 1, 2005 |

Words: | 10143 |

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