Market substitution and the Pareto dominance of ad valorem taxation.

• Author: Liu, Liqun

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

   (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...