International Journal of Industrial Organization 30 (2012) 399–402

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

Tax incidence under imperfect competition: Comment Philip J. Reny a,⁎, Simon J. Wilkie b, Michael A. Williams c a b c

Department of Economics, University of Chicago, United States Department of Economics, University of Southern California, United States Competition Economics LLC, Emeryville, CA, United States

a r t i c l e

i n f o

Article history: Received 22 June 2011 Received in revised form 13 April 2012 Accepted 17 April 2012 Available online 23 April 2012

a b s t r a c t Delipalla and O'Donnell (2001) contains a formula for the incidence of specific and ad valorem taxes in a conjectural variation oligopoly model with potentially asymmetric firms. The formula is incorrect. We derive the correct formula and provide a discussion of the error and its implications for empirical studies of passthrough. © 2012 Elsevier B.V. All rights reserved.

JEL classification: H22 L13 Keywords: Tax incidence Market power Conjectural variations Conduct parameter

1. Introduction Delipalla and O'Donnell (2001), henceforth D–O, contains a formula for the incidence of specific and ad valorem taxes in a conjectural variation model of an oligopolistic market for a homogeneous good with potentially asymmetric firms. The formula is incorrect. 1 Given the recent increase in interest in cost pass-through, which bears directly on tax incidence, and in the conjectural variation model (see Weyl and Fabinger (2009) and Jaffe and Weyl (2012)), 2 and given also the potential significance of the error to empirical pass-through studies, it is worthwhile to derive the correct formula, which we do here. In Section 2, we present the conjectural variation model and describe the error in D–O. In Section 3, we derive the correct formula and provide the conditions under which the D–O formula, despite the

⁎ Corresponding author. E-mail addresses: [email protected] (P.J. Reny), [email protected] (S.J. Wilkie), [email protected] (M.A. Williams). 1 D–O extends the analysis in Delipalla and Keen (1992) to the case of asymmetric firms. In the special case of symmetric firms, Delipalla and Keen (1992) derive the correct tax incidence formula. 2 The conjectural variations model provides “a useful framework for empirical investigations into the exercise of market power and the ‘competitiveness’ of an industry” Church and Ware (2000, p. 273). See also Dixit (1986), Bresnahan (1989), and Church and Ware (2000) for interpretations and empirical uses of the conjectural variations model. More recently, see Majumdar et al. (2011) for empirical applications, and Jaffe and Weyl (2012) for a theoretical application, to merger review. 0167-7187/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2012.04.001

error, turns out to be correct. The significance of the error is discussed in Section 4.

2. The conjectural variation model with ad valorem and specific taxes The N-firm conjectural variation model includes, for each firm i, a nonnegative conduct parameter, λi, that specifies the rate at which firm i expects total output to change per unit change in its own output. 3 The conduct parameter is usually interpreted as the “reduced form” coefficient determined by the equilibrium of an underlying repeated game, see Cabral (1995). Specifically, if qi is the output of firm i, and Q is total industry output, then firm i conjectures that, dQ ¼ λi : dqi

ð2:1Þ

When all λi = 0, market behavior is competitive. A value of λi = 1 for all firms corresponds to Cournot behavior, while in the symmetric case of identical cost functions λi = N for all firms corresponds to market-share collusion. 3 Nonnegativity ensures that, in equilibrium, no firm's marginal cost exceeds the price.

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Given an ad valorem tax of v and a specific tax of σ, a profitmaximizing firm chooses qi to maximize ½ð1−vÞP ðQ Þ−σ qi −ci ðqi Þ;

ð2:2Þ

where ci(qi) is the firm's cost function, Q is the total market quantity, and P(Q) is the inverse demand function. We assume throughout that all functions are differentiable and that c′i(qi) ≥ 0 and P′(Q) ≤ 0. Given the conjectures in Eq. (2.1), firm i's first-order condition for profit maximization is h i ′ ′ ð1−vÞ P ðQ Þ þ λi P ðQ Þqi −c i ðqi Þ−σ ¼ 0:

ð2:4Þ

and use Eq. (2.4) to substitute out all the dqi/dv terms. This allows them to solve for dQ/dv and leads quite directly to their expression for dP/dv. 7 That Eq. (2.4) need not hold can be seen by noting that Q = ∑ qi i implies that dQ ¼ ∑ dq . Therefore, if Eq. (2.4) were true, we would dv dv conclude that 8 ∑

1 ¼ 1; λi

Pass‐throughðv0 Þ ¼

ð2:3Þ

In equilibrium, each firm maximizes its profits given the output of the others. Hence, Eq. (2.3) holds for each firm i.4 Since the effect on price of changing a specific tax can be derived from the effect on price from changing an ad valorem tax, we will henceforth focus on the latter.5 To derive their ad valorem tax incidence formula, Delipalla and O'Donnell, after dividing Eq. (2.3) by λi and summing over i, differentiate the resulting equation with respect to the tax, v. In the course of performing this comparative statics exercise, they evidently incorrectly assume that Eq. (2.1) is an identity. 6 Combining this error i i dQ with the fact that dq ¼ dq , they evidently conclude that dv dQ dv dqi 1 dQ ; ¼ dv λi dv

The numerator in Eq. (3.1) is the additional amount paid by the consumer per unit of the good purchased and the denominator is the additional tax revenue collected per unit purchased. As is standard, we say that there is full-shifting (of the tax) if the right-hand side of Eq. (3.1) equals 1, undershifting when it is less than 1, and overshifting when it is greater than 1. The pass-through at the point v0 is defined by taking the limit of Eq. (3.1) as v → v0. Hence,

ð2:5Þ

si c′ ðs Q Þ þ σ 1 þ i i ¼ εðQ Þ ð1−vÞP ðQ Þλi λi

0¼

1 dsi ε′ ðQ Þ si P ðQ Þ d logP ðQ Þ − εðQ Þ dv ε2 ðQ Þ P ′ ðQ Þ dv   ″ c i ðsi Q Þ dsi si P ðQ Þ d logP ðQ Þ þ Qþ ′ ð1−vÞPðQ Þλi dv  P ðQ Þ dv 

 ′ c′ i ðsi Q Þ þ σ P ′ ðQ Þ c i ðsi Q Þ þ σ P ðQ Þ d logP ðQ Þ 1 þ − 2 : dv λi ð1−vÞ P ðQ Þλi P ′ ðQ Þ P ðQ Þ ð1−vÞ2

If a tax is changed, possibly from zero, the pass-through is the percentage of the resulting change in tax revenue paid by consumers. Formally, the pass-through for an ad valorem tax change from v0 to v is defined by, pðvÞ−pðv0 Þ ; vpðvÞ−v0 pðv0 Þ

ð3:3Þ

where each qi is the equilibrium quantity produced by firm i, Q is total output in equilibrium, si = qi/Q is firm i's equilibrium market share, and ε(Q) = − P(Q)/QP′(Q) is the (positive) elasticity of market demand evaluated at the equilibrium level of total output. The equilibrium quantities, qi(v) and Q(v), are functions of the ad valorem tax, v. In particular, p(v) = P(Q(v)). However, to keep the notation manageable, we will continue to write qi and Q rather than qi(v) and Q(v). Differentiating Eq. (3.3) with respect to v and using dQ/dv = (P(Q)/P′(Q))(d log P(Q)/dv) yield

3. Tax pass-through in an oligopoly setting

Pass‐throughðv0 ; vÞ ¼

ð3:2Þ

As in D–O, we henceforth assume that each λi is strictly positive so that division by λi is well-defined. However, all of our expressions have well-defined limits as any of the λi converge to zero and these limiting expressions are those that would result from solving the system when those λi are in fact zero in Eq. (2.3). In particular, results for the competitive case are obtained by considering the limit of our results as all the λi converge to zero. Computing pass-through begins with the first-order condition, Eq. (2.3), which can be rewritten as

which, of course, need not be true. For example, in the Cournot model, λi = 1 for all i.

p′ ðv0 Þ d logpðv0 Þ=dv ¼ : pðv0 Þ þ v0 p′ ðv0 Þ 1 þ v0 d logp ðv0 Þ=dv

Solving for dsi/dv gives ð3:1Þ

where p(v) is the equilibrium price of the good when the ad valorem tax is v. 9

dsi ai d logP ðQ Þ c′ i ðqi Þ þ σ 1 − ¼ dv bi P ðQ Þλi ð1−vÞ2 dv bi

ð3:4Þ

where 4

We assume throughout that pure strategy equilibria exist and satisfy the firms' first-order conditions (namely Eq. (2.3)) with equality. 5 See the Remark below for how to compute the former from the latter. 6 But Eq. (2.1) is not an identity because firm i's conduct parameter λi merely specifies how firm i conjectures that other firms' quantities will react to deviations from i's equilibrium level output. In contrast, when conducting a comparative statics exercise – e.g., a change in the ad valorem tax, v – none of the qi ever deviate from their equilibrium values and hence there is no opportunity for Eq. (2.1) to come into play. 7 A detailed derivation of the pass-through formula in D–O is not provided. Instead it is stated (D–O p. 890) that the derived formula for dP/dv “…is immediate on applying the implicit function theorem to [the first-order condition].” As we will see, obtaining the correct formula is not so immediate. 8 The maintained (but incorrect) assumption in D–O that Eq. (2.1) holds implies that dQ/dv is nonzero. 9 The value of the specific tax is held fixed and so we suppress the dependence of the price on σ.

ai ¼

c″ i ðqi Þqi ε ðQ Þ þ c′ ðqi Þ þ σ ε′ ðQ Þ q − P ðQ Þλi ð1−vÞ εðQ Þ i

and bi ¼

c″ i ðqi ÞQ 1 : þ P ðQ Þλi ð1−vÞ εðQ Þ

At this stage, it is useful to introduce the following notation. Let ηi ðqi ; σ Þ ¼

′

c i ðqi Þ þ σ qi c″ i ðqi Þ

P.J. Reny et al. / International Journal of Industrial Organization 30 (2012) 399–402

denote firm i's elasticity of supply if it were a price taker, let mi ðQ ; qi ; σ Þ ¼

pass-through at any point is dP dσ

ð1−vÞP ðQ Þ c′ i ðqi Þ þ σ

denote the elasticity with respect to quantity of the elasticity of dei mand. Then, because ∑ si = 1 implies ∑ ds ¼ 0, summing Eq. (3.4) dv over i gives,   a d logP ðQ Þ 1 1 ∑ i ; ¼ ∑ dv 1−v mi λi bi bi so that, ð3:5Þ

where, A¼∑

1=ε si ηi =ε ¼∑ ; mi λi bi 1 þ mi λi si ηi =ε

and

  dP = 1 þ v0 dσ , and in equilibrium

where ð3:8Þ

is a tax-adjusted average gross mark-up.10 Since the bi are non-negative when cost functions are weakly convex, and because individual firm tax-adjusted markups are always greater than unity, μ is at least unity under convex costs. Consequently, under the hypotheses of Theorem, at the margin the pass-through from a specific tax exceeds the pass through from an ad valorem tax and so the latter tax is preferred by consumers for raising any given amount of tax revenue. The pass-through formula in D–O is correct in two cases. The first case is when all firms have the same marginal cost functions and all the λi are equal to N. In this case, which corresponds to perfect market-share collusion, equilibrium is always such that qi/Q = 1/ N = 1/λi, and so Eq. (2.4) happens to hold. The second case is when marginal costs, though possibly distinct, are constant. Indeed, after dividing Eq. (2.3) by λi and summing over i, qi appears only as an argument of firm i's marginal cost function. Thus, when taking the derivative of the resulting equation with respect to v, the term dqi/dv is multiplied by c″1(qi). So if c″1(qi) = 0, the dqi/dv term is eliminated and the erroneous Eq. (2.4) has no effect. 4. Conclusion



B¼∑

dP dσ

! 1=λi bi μ ¼ ð1−vÞP ðQ Þ= σ þ ∑c i ðqi Þ : ∑ 1=λj bj

Qε′ ðQ Þ εðQ Þ

d logP ðQ Þ 1 A ¼ ; dv 1−v A þ B

¼μ

dlogP , dv

′

denote firm i's tax-adjusted gross mark-up, and let ξðQ Þ ¼ −

¼

∑1=ðλi bi Þ 1 1−v ∑ai =bi

401



si 1 þ ξmi λi si ηi =ε : 1 þ mi λi si ηi =ε

Hence, substituting Eq. (3.5) into Eq. (3.2) yields the desired result for pass-through at a point v, namely, Pass‐throughðvÞ ¼

A ; A þ ð1−vÞB

ð3:6Þ

where si, mi, ηi, and ε are functions of the equilibrium quantity qi produced by firm i, or the equilibrium market quantity Q, or both. Consequently, in equilibrium, each of these also depends upon all of the conduct parameters (i.e., all the λj) as well as the market demand function and all the cost functions. Eq. (3.6) immediately yields the following result. Theorem. If each firm's cost function is weakly convex, i.e., c″i(qi) ≥ 0, and if market demand satisfies Marshall's second law, i.e., ε′(Q) ≤ 0, then Pass-through(v) is non-negative and no larger than 1. Hence, under the stated conditions, overshifting of an infinitesimal tax change is not possible. Defining F(v) = (vp(v) − v0p(v0))/(v1p(v1) − v0p(v0)), the fundamental theorem of calculus yields, v

Pass‐throughðv0 ; v1 Þ ¼ ∫v10 Pass‐throughðvÞ dF ðvÞ:

ð3:7Þ

Under the hypotheses of Theorem, p(v) is increasing in v and therefore F(⋅) is a cumulative distribution function. Pass-through (v0,v1) is then a weighted average of Pass-through (v) for v ∈ [v0, v1]. Thus, we obtain the following corollary of Theorem. Corollary. If each firm's cost function is weakly convex, i.e., c″1(qi)≥0, and if market demand satisfies Marshall's second law, i.e., ε′(Q)≤0, then for distinct v0 and v1, Pass-through (v0,v1) is non-negative and no larger than 1. Hence, under the stated conditions, overshifting of a discrete tax change is not possible. Remark. The analysis for a specific tax is similar to that of an ad valorem tax. In particular, holding fixed the ad valorem tax at v0, the specific-tax

The correction pointed out here is important for at least two reasons. First, if conduct parameters are known, the pass-through formula in D–O can result in predicted pass-through rates that differ dramatically from correct pass-through rates. Moreover, the incorrect formula can move pass-through rates in the wrong direction as the conduct parameter approaches the competitive level (e.g., with an undershifted tax, the pass-through rate from the incorrect formula can fall toward zero as the market becomes more competitive). Conversely, if as in D-O, one first estimates pass-through rates directly and then applies those estimates to the incorrect formula to infer conduct parameters, on can be led, to either dramatically understate or dramatically overstate the competitiveness of the market. 11 Consequently, the methodological contribution for estimating market conduct via tax incidence as suggested in D–O is flawed. 12 Second, while the present (corrected) theory is consistent with the possibility of overshifting, our Theorem and Corollary demonstrate that the conditions under which overshifting of an ad valorem tax occurs in the standard oligopoly model are unusual and generally neither used in economic theory nor found in empirical work.13 For example, with 10 The formula for μ in D–O just above Eq. (2.9) is incorrect for the same reasons as previously pointed out and so the specific-tax pass-through formula in D–O is also incorrect. 11 See Reny et al. (2011) for examples illustrating each of the points raised in this paragraph. 12 In addition to the two cases already identified where D–O's tax incidence formula are correct, D–O's method for estimating market conduct is valid, for example, when firms have identical cost functions. In this case, even though D–O's formulae for the incidence of ad valorem and specific taxes remain incorrect, D–O's formula for the ratio of the two is correct, and only the ratio is used in D–O's method to estimate market conduct. 13 Hamilton (2009) shows in a differentiated products model with multiproduct firms that an ad valorem tax imposed on all of a firm's products may be overshifted even when ε′(Q)b0. This occurs in Hamilton's model because higher excise taxes on all products – assumed to be mutual substitutes – can, in long-run and general equilibrium, lead to a reduction in product variety, thereby reducing competition and incentivizing firms to increase the prices of remaining products even further. The short-run, single-product analysis conducted here and in much of the pass-through literature is, of course, a partial equilibrium analysis valid conditional on holding fixed the prices and quantities of other products. Whether the long run and general equilibrium effects of changes in the prices and quantities of other products will attenuate or exaggerate the undershifting predicted from the partial equilibrium analysis will depend, in particular, on whether those other products are produced by the same firm and whether those other products are complements or substitutes to the product subject to the tax.

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constant elasticity of demand, an ad valorem tax is overshifted if and only if marginal costs are decreasing. On the other hand, with a demand curve that satisfies ε′(Q) b 0, such as a linear demand curve, marginal costs must be falling at an even faster rate. That is, under either of these typical market demand hypotheses, only a “sufficiently natural” monopoly would overshift an ad valorem tax. Similarly, if marginal costs are constant, there is overshifting if and only if ε′(Q) > 0. If marginal costs are increasing, then the demand elasticity must be increasing at an even faster rate for overshifting to occur. However, demand functions that exhibit the property ε′(Q) > 0 rarely arise in either theoretical or empirical applications. Indeed, we are not aware of any empirical study that has found a demand curve for which ε′(Q) > 0. With such stringent, and evidently unusual, conditions for overshifting to occur, the standard conjectural variation homogenous good model of oligopoly, even with potentially asymmetric firms, does not provide a likely justification for empirical studies that claim to find overshifting of an ad valorem tax. 14,15 Acknowledgment We thank Brijesh Pinto for helpful comments and Reny gratefully acknowledges support from the National Science Foundation (SES-0922535).

14 On the other hand, under the hypotheses of Theorem 1, there is overshifting of a specific tax when all the λiai are less than unity, which occurs, in particular, when elasticity of demand and all marginal costs are constant. 15 D–O employ reduced-form regressions using data on prices and taxes to estimate tax incidence directly. Consequently D–O's empirical tax-incidence estimates are independent of their theoretical error in analyzing the oligopoly model. Interestingly, D–O's empirical tax incidence estimates support our theoretical prediction of the rarity of overshifting of an ad valorem tax. Indeed, except for countries where the conjectural variations oligopoly model may be ill-suited (i.e., France, Italy, Portugal, and Spain, which have state-run monopolies) or in which the data are problematic (Greece and Luxemburg), D–O find no statistically significant evidence supporting overshifting of an ad valorem tax (D–O, Tables 3 and 5).

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