Multi-Dimensional Pass-Through and Welfare Measures under Imperfect Competition
MMulti-Dimensional Pass-Through and Welfare Measuresunder Imperfect Competition ∗ Takanori Adachi † Michal Fabinger ‡ December 11, 2018
Abstract
This paper provides a comprehensive analysis of welfare measures when oligopolisticfirms face multiple policy interventions and external changes under general forms of marketdemands, production costs, and imperfect competition. We present our results in terms oftwo welfare measures, namely, marginal cost of public funds and incidence, in relation to multi-dimensional pass-through . Our arguments are best understood with two-dimensionaltaxation where homogeneous firms face unit and ad valorem taxes. The first part of the paperstudies this leading case. We show, e.g., that there exists a simple and empirically relevantset of sufficient statistics for the marginal cost of public funds, namely unit tax and advalorem pass-through and industry demand elasticity. We then specialize our general settingto the case of price or quantity competition and show how the marginal cost of public fundsand the pass-through are expressed using elasticities and curvatures of regular and inversedemands. Based on the results of the leading case, the second part of the paper presents ageneralization with the tax revenue function specified as a general function parameterizedby a vector of multi-dimensional tax parameters. We then argue that our results are carriedover to the case of heterogeneous firms and other extensions. ∗ We are grateful to Yong Chao, Germain Gaudin, Makoto Hanazono, Hiroaki Ino, Konstantin Kucheryavyy,Laurent Linnemer, Carol McAusland, Babu Nataha, and Glen Weyl as well as conference and seminar participantsfor helpful discussions. Adachi and Fabinger acknowledge a Grant-in-Aid for Scientific Research (C) (15K03425)and a Grant-in-Aid for Young Scientists (A) (26705003) from the Japan Society for the Promotion of Science,respectively. Adachi also acknowledges financial support from the Japan Economic Research Foundation. Anyremaining errors are solely ours. † School of Economics, Nagoya University, 1 Furo-cho, Chikusa, Nagoya 464-8601, Japan; [email protected]. ‡ Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan. Fabingeris also a research associate at CERGE-EI, Prague, Czechia; [email protected]. a r X i v : . [ q -f i n . E C ] D ec Introduction
In many economic situations, it is important to understand the impact of changes in governmentpolicies, market conditions, or production technology. Both the impact on the economic equi-librium and the associated change in the welfare of individual economic agents are of interest tothe policymaker. A convenient framework for studying these questions is laid out in Weyl andFabinger (2013) for the case of constant changes in marginal cost such as a unit tax. In this paper,we provide a substantial generalization of this framework under general forms of market demands,production costs, and imperfect competition. We allow not only for changes in costs proportionalto the value of the goods such as an ad-valorem tax but also for much more general interventionssuch as different kinds of market regulation. In addition, we provide a new perspective on thecase of heterogeneous firms.The range of possibilities for governments’ intervention is much richer than just specific and ad-valorem taxes, on which the existing literature has focused (see Section 2), especially if we considerregulations of various kinds. Governments often intervene in the marketplace by restricting sales.Examples include restrictions of sales on Sundays and holidays in many European countries, orrestrictions of sales of alcohol, both in terms of time of sales and in terms of locations where salesare allowed. Governments also often regulate the labor market. Examples include restrictions onthe number of working hours or stipulation of a minimum wage. Besides these, governments alsoimpose reporting requirements that influence the degree to which non-compliant firms misreporttheir marketplace data to minimize their tax bill.Given that there are many possible interventions, it is convenient to summarize the interven-tions of interest in a (multi-dimensional) vector. The impact of infinitesimal changes in theseinterventions on prices then corresponds to a vector (or more generally a matrix), which may betermed multi-dimensional pass-through . The relative size of its components then provides insightinto the relationship between the impact of individual interventions, for example, the impact ofspecific vs. ad-valorem taxes. We show that the multi-dimensional pass-through is an importantdeterminant of the welfare effects of various kinds of interventions. The usefulness of pass-through in welfare analysis has been verified by related studies such as Cowan (2012); elasticities and curvatures of demand and inversedemand when the setting is specialized to price (differentiated Bertrand) or quantity (pure ordifferentiated Cournot) competition. We also provide numerical examples. Our results also applywithout a change to symmetric oligopoly with multi-product firms. Throughout the analysis, weallow for non-zero levels of unit and ad valorem taxes. However, we also discuss some additionalsimplifications that appear when instead the initial level of taxes is zero.Inspired by the results of this setting of two-dimensional taxation, the second part of the papergeneralizes our model in two directions (Sections 4, 5, and 6). First, we allow for interventions thatare more general than just specific and ad valorem taxes. Second, we introduce firm heterogeneity.We find that these more general relationships have a form that still relatively simple and succinct.This substantially expands the applicability of our results.From both theoretical and empirical standpoints, it is desirable to be able to understand thewelfare properties of oligopolistic markets with a fairly general type of competition. In real-worldsituations, firms’ behavior may not simply be categorized into either the idealized price competitionor the idealized quantity competition. Price competition does not allow for any friction in scalingproduction levels up or down, yet in reality there tend to be substantial frictions, such as thoserelated to financial constraints or the labor market. Quantity competition implies that the firm
Miller, Remer, and Sheu (2013); Weyl and Fabinger (2013); Gaudin and White (2014); MacKay, Miller, Remer, andSheu (2014); Adachi and Ebina (2014a,b); Chen and Schwartz (2015); Gaudin (2016); Cowan (2016); Alexandrovand Bedre-Defolie (2017); Mr´azov´a and Neary (2017); and Chen, Li and Schwartz (2018). See also Ritz (2018)for an excellent survey of theoretical studies on pass-through and pricing under imperfect competition. In thecontext of antitrust analysis, Froeb, Tschantz, and Werden (2005) theoretically compare the price effects when nosynergies in cost reduction realize when they are passed through as a form of price reduction. See also Alexandrovand Koulayev (2015) for discussions on the role of pass-through in antitrust analysis. conduct index . Besides working with a more general type of competition, it is also useful to relax the assump-tion of constant marginal costs that often appears in the literature. Production technologies oftenhave a non-trivial structure, and so does the internal organization of the firm. For example, ifa firm decides to operate at a larger scale, it may take advantage of technological and logisticaleconomies of scale, but at the same time, it may face more severe principal-agent problems as topmanagers have to delegate responsibilities to lower-level managers. The interplay between theseforces can lead to a non-trivial dependence of the marginal cost of production on the scale of theoperation. A notable benefit from our general framework is that one does not necessarily have toassume constant marginal cost in conducting welfare assessment in a precise manner.In this spirit, the aim of Weyl and Fabinger’s (2013) study is to analyze imperfect competitionin a way that does not rely on limiting functional form assumptions. It is followed by Fabingerand Weyl (2018), who propose using further flexible functional forms to make economic modelsor their parts solvable in closed form. The present paper goes in a different direction: we focuson more general market interventions and exogenous market changes without requiring that themodels or their parts be solved in closed form. As mentioned above, ad valorem taxes are notstudied in Weyl and Fabinger (2013). In this paper, we provide a more general framework thatallows not only for ad valorem taxes but for other interventions in a general manner. Our conduct index is a generalization of what is known as “conduct parameter” in the empirical industrialorganization literature, where it is supposed to be constant for any level of industry output as a target of estimation(see, e.g., Bresnahan 1989; Genesove and Mullin 1998; Nevo 1998; Corts 1999; Delipalla and O’Donnell 2001; andShcherbakov and Wakamori 2017). It has also been successfully applied to more general situations, such as selectionmarkets (Mahoney and Weyl 2017) or supply chains (Gaudin 2018). In this paper, we generalize this concept byallowing it to vary across different levels of output: we thus opt for the term “conduct index” to make it explicitthat it is a variable. In a less general setting, such “conduct index” was used by Weyl and Fabinger (2013), whereit was still referred to as “conduct parameter”. d different tax parameters and discusses the implications of these general results. We also discussother types of government/exogenous interventions that are suitable for our framework. Then, inSection 5 we generalize our formulas to include heterogeneous firms. Finally, Section 6 generalizesour previous results to include changes in both production costs and taxes. Section 7 concludesthe paper. We begin with a canonical setting where firms face two types of taxation: unit and ad valorem taxes. This issue has been in the central part of the existing literature on government interventions.The welfare cost of taxation has been extensively studied at least since Pigou (1928). The majorityof the studies simply assume perfect competition (with zero initial taxes). As is widely known,under perfect competition, unit tax and ad valorem tax are equivalent, and whether consumers orproducers bear more is determined by the relative elasticities of demand and supply (Weyl andFabinger, 2013, p. 534). The initial attempt to relax the assumption of perfect competition startedwith an analysis of homogeneous -product oligopoly under quantity competition, i.e., Cournotoligopoly. Notably, Delipalla and Keen (1992), Skeath and Trandel (1994), and Hamilton (1999) See, e.g., Vickrey (1963), Buchanan and Tullock (1965), Johnson and Pauly (1969), and Browing (1976) forearly studies. A study of unit and ad valorem taxation under imperfect competition with homogeneous productsdates back to Delipalla and Keen (1992). See Auerbach and Hines (2002) and Fullerton and Metcalf (2002) forcomprehensive surveys for this field. differentiated oligopoly under price competition. Inparticular, Anderson, de Palma, and Kreider (2001a) find that whether the after-tax price forfirms and their profits rise by a change in ad valorem tax depends importantly on the ratio of thecurvature of the firm’s own demand to the elasticity of the market demand. In this section, we extend Anderson, de Palma, and Kreider’s (2001a) setting and results in anumber of important directions. First, we consider a “general” mode of competition, captured bythe conduct index, including both quantity and price competition. Second, we provide a completecharacterization of tax burdens that enables one to quantitatively compare consumers’ burden withproducers’ burden, whereas Anderson, de Palma, and Kreider’s (2001a) focus only on the effectiveprices for consumers and producers’ profits. Third, while Anderson, de Palma, and Kreider’s(2001a) assume constant marginal cost, we allow non-constant marginal cost and show how thisgeneralization makes a difference in our general formulas. Fourth, we further generalize the initialtax level. When they analyze the effects of a unit tax, Anderson, de Palma, and Kreider (2001a)assume that ad valorem tax is zero, and vice versa. In contrast, we allow non-zero initial taxes inboth dimensions. Finally, and importantly, we generalize these results to the case of a very generaltype of taxation, as well as to production cost changes. This opens up the possibility to study awider range of interventions/taxes and to derive convenient sufficient statistics for characteristics,including welfare characteristics, of the markets of interest. This curvature is denoted ε m in their notation, whereas we instead use α F below, and this elasticity is denoted ε DD in their notation, whereas we instead use (cid:15) below. Our framework is in line with the “sufficient-statistics” approach to connecting structural and reduced-formmethods, as advocated by Chetty (2009), which has been successful in empirical economics. For example, in thestudy by Atkin and Donaldson (2016), the pass-through rate provides a sufficient statistic for welfare implicationsof intra-national trade costs in low-income countries, without the need for a full demand estimation. Similarly,Ganapati, Shapiro, and Walker (2018) examine the welfare effects of input taxation, where a unit tax is levied on theinput. These effects are related to the effects of unit taxes on output, but not identical. See also Fabra and Reguant(2014); Shrestha and Markowitz (2016); Stolper (2016); Hong and Li (2017); Duso and Sz¨ucs (2017); Gulati,McAusland, and Sallee (2017); and Muehlegger and Sweeney (2017) for studies with the same spirit. In contrast,Kim and Cotterill (2008) is among the first studies that structurally estimate cost pass-through in differentiatedproduct markets, followed by Bonnet, Dubois, Villas-Boas, and Klapper (2013); Bonnet and R´equillart (2013);Campos-V´azquez and Medina-Cortina (2015); Miller, Remer, Ryan, and Sheu (2016); Conlon and Rao (2016);Miller, Osborne, and Sheu (2017); and Griffith, Nesheim, and O’Connell (2018). .1 Setup Below, we study oligopolistic markets with n symmetric firms and a general (first-order) modeof competition and the resulting symmetric equilibria. Our discussion applies to single-productfirms as well as to multi-product firms if intra-firm symmetry conditions are satisfied, as discussedin Appendix A.1.12. For simplicity of exposition, we use terminology corresponding to single-product firms here, and later we discuss how to interpret the results in the case of multi-productfirms.Formally, the demand for firm j ’s product q j = q j ( p , ..., p n ) ≡ q j ( p ) depends on the vector ofprices p ≡ ( p , ..., p n ) charged by the individual firms. The demand system is symmetric and thecost function c ( q j ) is the same for all firms. We assume that q j ( · ) and c ( · ) are twice differentiableand conditions for the uniqueness of equilibrium and the associated second-order conditions aresatisfied. We denote by q ( p ) the per-firm industry demand corresponding to symmetric prices: q ( p ) ≡ q j ( p, ..., p ). The elasticity of this function, defined as (cid:15) ( p ) ≡ − pq (cid:48) ( p ) /q ( p ) > price elasticity of industry demand, should not be confused with the elasticity of theresidual demand that any of the firms faces. We also use the notation η ( q ) = 1 /(cid:15) ( p ) | q ( p )= q forthe reciprocal of this elasticity as a function of q . For the corresponding functional values, whenwe do not need to specify explicitly their dependence on either q or p , we use η interchangeablywith 1 /(cid:15) .As mentioned above, we introduce two types of taxation: a unit tax t and an ad valorem tax v ,with firm j ’s profit being π j = (1 − v ) p j ( q j ) q j − tq j − c ( q j ). At symmetric quantities the governmenttax revenue per firm is R ( q ) ≡ tq + vp ( q ) q , and we denote by τ ( q ) the fraction of firm’s pre-tax Although for brevity we speak of a general mode of competition, we consider only “first-order” competition, inthe sense of the firms making decisions based on marginal cost and marginal revenue. This excludes, for example,the possibility of vertical industries, an important issue left for future research. The elasticity (cid:15) here corresponds to (cid:15) D in Weyl and Fabinger (2013, p. 542). Note that q (cid:48) ( p ) = ∂q j ( p ) /∂p j +( n − ∂q j ( p ) /∂p j (cid:48) | p =( p,...,p ) for any two distinct indices j and j (cid:48) . We will define the firm’s elasticity and otherrelated concepts in Section 3. This specification corresponds to a two-dimensional (tax) instrument ( t, v ), which is a special case of multi-dimensional instruments. For example, if the cost function had an additional technology parameter z , we coulddescribe the situation using a three-dimensional instrument ( t, v, z ) . In Section 4, we introduce a framework formulti-dimensional pass-through. For now, we specialize to the two-dimensional case of specific and ad valoremtaxes, which are very common: for example, in the United States both types of taxes are imposed on the sales ofsoda, alcohol and cigarettes. τ ( q ) ≡ R ( q ) /pq = v + t/p ( q ).We now introduce the conduct index θ ( q ), which measures the industry’s competitiveness (a lower θ corresponds to a fiercer level of competition). Here, it is determined independently of the cost sidebut potentially can change for different values of the industry’s output, q . Then, the symmetricequilibrium condition is written as1 η ( q ) p ( q ) (cid:18) p ( q ) − t + mc ( q )1 − v (cid:19) = θ ( q ), (1)where mc ( q ) ≡ c (cid:48) ( q ) is the marginal cost of production. We denote by θ the functional value of θ ( q ) at the equilibrium quantity. This is also understood as the elasticity-adjusted Lerner index :the markup rate [ p − ( t + mc ) / (1 − v )] /p should be adjusted by the industry-wide elasticity to reflectthe competitiveness in the industry, where ( t + mc ) / (1 − v ) is interpreted as the effective marginalcost. We emphasize here that once the conduct index is introduced, one is able to describeoligopoly in a synthetic manner, without specifying whether it is price or quantity setting, orwhether it exhibits strategic substitutability or complementarity.
The marginal cost of public funds , i.e., the marginal social welfare loss associated with raisingadditional tax revenue, is a crucial characteristic that a policymaker needs to take into accountwhen designing an optimal system of taxes. In the special case of linear demand and constantmarginal cost, H¨ackner and Herzing (2016, p. 147) explain that as long as the initial level of As already noted in Footnote 2 above, θ ( q ) is a generalization of conduct parameter in the sense that it isa function of q rather than a constant for any q . Hence, Equation (1) should not be interpreted as an equationthat defines θ ( q ). For our analysis we introduce θ ( q ) in an implicit manner: θ ( q ) is a function independent ofthe cost side of the problem such that Equation (1) is the symmetric first-order condition of the equilibrium. Forspecific types of ( " first-order " ) competition, such as those discussed in Section 3, it is possible to derive explicitexpressions for θ ( q ) that can replace our implicit definition. Presumably, it is natural to assume θ (cid:48) ( · ) <
0: a smalleramount of production is associated with a smaller degree of competitiveness in the industry due to other reasonsthan non-cooperative and simultaneous pricing (modeled here), such as (unmodeled) tacit collusion. However, thisrestriction is not necessary for the following analysis. Also, note that θ ( q ) > , Accordingly, one can write the modified Lerner rule under ( v, t ) as ( p − t + mc − v ) /p = ηθ, which implies therestriction on θ : θ ≤ (cid:15) . In the absence of other considerations, the marginal cost of public funds should be equalized across markets inorder to maximize social welfare.
M C t = θρ t , where ρ t isthe unit-tax pass-through rate (the marginal effect of unit taxes on prices), and θ is the conductindex. For ad valorem taxes, H¨ackner and Herzing (2016) provide a similar formula. They argue,however, that if we let the initial level of taxes be non-zero, those formulas are no longer valid.For this reason, they are forced to analyze the magnitude of the marginal cost of public funds ona case-by-case basis using explicit solutions to specific models.This situation represents a puzzle. If there are simple formulas for the marginal cost of publicfunds that were valid at zero taxes, is there no compact generalization of these expressions in thecase of non-zero taxes? If there is no such generalization, that would be an obstacle to empiricalwork, since we would have to make additional modeling assumptions before obtaining empiricalestimates of the marginal cost of public funds. Our paper provides a solution to this problem. Inparticular, Proposition 1 below presents formulas for the marginal cost of public funds that arevalid even when the initial level of (both ad valorem and unit) taxes is non-zero. They are a bitlonger than M C t = θρ t , but still very manageable. They also represent a starting point for thetopics discussed in the rest of the paper. These results with a non-zero initial taxes being allowed,which are differentiated from Weyl and Fabinger (2013) and H¨ackner and Herzing (2016), shouldbe useful if one needs to evaluate the marginal cost of taxation when some tax has been alreadyimplemented .The marginal welfare cost M C t or M C v of raising government revenue by the unit tax t or thead valorem tax v , i.e. the marginal cost of public funds associated with such a tax, is defined as M C t ≡ − (cid:18) ∂R∂t (cid:19) − ∂W∂t , M C v ≡ − (cid:18) ∂R∂v (cid:19) − ∂W∂v , where W is the social welfare per firm, which includes consumer surplus, producer surplus, andgovernment tax revenue. We define the unit tax pass-through rate ρ t and the ad valorem tax ass-through semi-elasticity ρ v as: ρ t = ∂p∂t , ρ v = 1 p ∂p∂v . Consider an infinitesimal change in the unit tax, with the initial tax level ( t, v ). As mentionedin the introduction, in the special case of zero initial taxes, linear demand, and constant marginalcost, H¨ackner and Herzing (2016, p. 147) show that
M C t = θρ t and M C v = θρ v , noting that atnon-zero initial taxes the formula no longer applies. In the absence of such formula, they wereforced to study the marginal cost of public funds on a case-by-case basis, for different specificationsof demand and cost.Intuitively, there are at least two reasons why θρ t fails to be an accurate measure of the marginalcost of public funds when a unit tax is raised. First, the expression is simply proportional to θ ,but when v is large, the firms sell at prices that are too high from the social perspective not mainlybecause of a lack of competitiveness, but primarily because the tax effectively raises their perceivedcost. When v is large, we would expect the marginal cost of public funds to be less sensitive to θ , for a given value of ρ t . Second, the expression θρ t does not explicitly feature the level of theunit tax t . However, a situation where t is large and mc small is very different from a situationwhere t is small and mc large, even if the equilibrium prices and quantities are the same. In theformer case, raising additional tax revenue is quite harmful, since firms’ production cuts will notsubstantially decrease the total technological (i.e., pre-tax) cost of production. In the latter case,raising additional tax revenue is less harmful since it leads to reduced total technological cost.Based on this intuition, we would expect the marginal cost of public funds to be an increasingfunction of t . Thus, we are led to find a generalization of the formula
M C t = θρ t and M C v = θρ v that wouldbe applicable even at non-zero initial taxes. It turns out that it is possible to identify a formula Note that H¨ackner and Herzing (2016) use the symbol ρ v for the ad valorem tax pass-through rate ∂p/∂v ,which corresponds to pρ v in our notation. An analogous argument applies for θρ v and the marginal cost of public funds of tax v . In the sense of making the change t → t + ∆ t , and simultaneously c ( q ) → c ( q ) − q ∆ t in order to keep q , θ ,and ρ t at some fixed values. Proposition 1.
Marginal cost and total of public funds for unit and ad valoremtaxations.
Under symmetric oligopoly with a possibly non-constant marginal cost, the marginalcost of public funds associated with a unit tax may be expressed as
M C t = (1 − v ) θ + (cid:15) τ ρ t + v − (cid:15) τ , and the marginal cost of public funds associated with an ad valorem tax may be expressed as M C v = (1 − v ) θ + (cid:15)τ ρ v + v − (cid:15)τ . The proposition is proven in Appendix A.1.1 and the intuition behind it is discussed in detailin Appendix A.1.2. Here in the main text we just include Figure I, which documents that theseexpressions for the marginal cost of public funds
M C t and M C v evaluated at realistic values oftaxes and other economic variables are very different from the values of the expressions θρ t and θρ v (discussed above) that would be equal to M C t and M C v if taxes were zero. We define the incidence I t of unit taxation as the ratio of changes dCS in (per-firm) consumersurplus and changes dP S in (per-firm) producer surplus induced by an infinitesimal increase dt in the unit tax t . The incidence I v of ad valorem taxation is defined analogously. We obtain thefollowing succinct results for the incidence of taxation at non-zero unit and ad valorem taxes. Proposition 2.
Incidence of taxation.
Under symmetric oligopoly with a general type ofcompetition and with a possibly non-constant marginal cost, the incidence of unit taxes I t and ofad valorem taxes I v is given by I t = 1 ρ t − (1 − v ) (1 − θ ) , I v = 1 ρ v − (1 − v ) (1 − θ ) . M C and the naive expression θρ discussed just before Proposition 1, plotted as a function of combinations of the conduct index θ ,the pass-through ρ , and the industry demand elasticity (cid:15) . The figures on the left correspond toinfinitesimal changes in unit taxation: ρ stands for ρ t and M C stands for
M C t . The numericalvalues were chosen to be t = 0 , v = 0 . , τ = 0 .
2. The figures on the right correspond toinfinitesimal changes in ad valorem taxation: ρ stands for ρ v and M C stands for
M C v . Thenumerical values were chosen to be t/p = 0 . , v = 0 , τ = 0 .
2. The top figures correspond to θ = 0 .
3, the middle figures correspond to (cid:15) = 2, and the bottom figures correspond to ρ = 1.11he proposition is proven in Appendix A.1.3, and we discuss it in detail in Appendix A.1.4.Note that in the case of zero ad valorem tax, the expression for I t reduces to Weyl and Fabinger’s(2013, p. 548) Principle of Incidence 3. Next we show how ρ t and ρ v are related in the followingproposition. Proposition 3.
Relationship between pass-through of ad valorem and unit taxes.
Under symmetric oligopoly with a possibly non-constant marginal cost, the pass-through semi-elasticity ρ v of an ad valorem tax may be expressed in terms of the unit tax pass-through rate ρ t ,the conduct index θ , and the industry demand elasticity (cid:15) as ρ v = (cid:18) − θ(cid:15) (cid:19) ρ t . (2)The proposition is proven Appendix A.1.5, and Appendix A.1.6 provides a detailed discussion.Combined with Proposition 1, it is consistent with the well-known result that unit tax and advalorem tax are equivalent in the welfare effects under perfect competition: if θ = 0, then ρ t = ρ v .Under imperfect competition, ρ t > ρ v , and M C t > M C v . This provides another look of Anderson,de Palma, and Kreider’s (2001b) result that unit taxes are welfare-inferior to ad valorem taxes. By combining Propositions 1 and 3, we find that
M C t and M C v can be expressed without thedegree of competitiveness θ . Proposition 4.
Sufficient statistics for marginal costs of public funds.
Under sym-metric oligopoly with a possibly non-constant marginal cost, the unit pass-through rate ρ t , the advalorem pass-through semi-elasticity ρ v , and the elasticity (cid:15) of industry demand (together with thetax rates and the fraction τ of the firm’s pre-tax revenue collected by the government in the formof taxes) serve as sufficient statistics for the marginal cost of public funds both with respect to unittaxes and ad valorem taxes. In particular: M C t = (1 − v + τ ) ρ t − (1 − v ) ρ v v − (cid:15)τ ) ρ t (cid:15), M C v = (1 − v + τ ) ρ t − (1 − v ) ρ v v − (cid:15)τ ) ρ v ρ v ρ t (cid:15). Under Cournot competition, Equation (6.13) of Auerbach and Hines (2002) coincides with Equation (2) above.Proposition 3 shows that this equation holds more generally. We thank Germain Gaudin for pointing this out. θ as θ = (1 − ρ v /ρ t ) (cid:15) .Substituting this into the relationships in Proposition 1 then gives the desired result. For a detaileddiscussion of related intuition, see Appendix A.1.7.As the last result presented in this section, the following proposition shows how the two formsof pass-through are characterized. Proposition 5.
Pass-through under symmetric oligopoly.
Under symmetric oligopoly witha general (first-order) mode of competition and with a possibly non-constant marginal cost: ρ t = 11 − v (cid:2) − τ − v (cid:15)χ (cid:3) − ( η + χ ) θ + (cid:15)q ( θη ) (cid:48) , where the derivative is taken with respect to q and χ ≡ mc (cid:48) q/mc is the elasticity of the marginalcost with respect to quantity. Further, ρ v = (cid:15) − θ (1 − v ) (cid:15) (cid:2) − τ − v (cid:15)χ (cid:3) − ( η + χ ) θ + (cid:15)q ( θη ) (cid:48) . The proof and a related discussion are in Appendix A.1.8. Further, we discuss a relationshipwith Weyl and Fabinger (2013) in Appendix A.1.9, provide a comparison of perfect and oligopolis-tic competition in Appendix A.1.10, and show the applicability of these results to exchange ratechanges in Appendix A.1.11.
So far, we have discussed local, i.e. infinitesimal, changes in surplus measures ( CS , P S , R , W ).For larger changes in some tax T , it is desirable to have an understanding of global changes insurplus measures. Consider surplus measures A and B . Their finite changes ∆ A = (cid:82) T T dA ( T ) dT dT and ∆ B = (cid:82) T T dB ( T ) dT dT induced by a tax change from T = T to T = T are related to theirincidence ratios Θ AB ≡ dA ( T ) dT / dB ( T ) dT . In particular, ∆ A/ ∆ B is a weighted average of Θ AB over theinterval ( T , T ): ∆ A ∆ B = (cid:90) T ∈ ( T ,T ) Θ AB dw ( T ,T ) B ( T ) , dw ( T ,T ) B ( T ) ≡ dB ( T ) dT dT / (cid:82) T T dB ( T (cid:48) ) dT (cid:48) dT (cid:48) is a weight normalized to one on the correspondinginterval: (cid:82) T T dw ( T ,T ) B ( T ) = 1. The weight is positive as long as dB ( T ) dT has the same sign as (cid:82) T T dB ( T ) dT dT , which is typically satisfied in useful applications. In many useful cases A and B arezero at infinite T . Then A ( T ) B ( T ) = (cid:90) T ∈ ( T , ∞ ) Θ AB dw ( T , ∞ ) B ( T ) . For example, CS ( T ) P S ( T ) = (cid:90) T ∈ ( T , ∞ ) I T dw ( T , ∞ ) P S ( T ) , W ( T ) R ( T ) = (cid:90) T ∈ ( T , ∞ ) M C T dw ( T , ∞ ) R ( T ) . In the case of a per-unit tax, we obtain CS ( t ) P S ( t ) = (cid:82) ∞ t I T q dt (cid:82) ∞ t q dt . In this section, we show that for price competition and quantity competition in differentiatedoligopoly, our general expressions of the marginal cost of public funds and pass-through leadto expressions in terms of demand primitives such as the elasticities and the curvatures , and themarginal cost elasticity χ defined above. Throughout this section, we assume that firms’ conductis simply described by one-shot Nash behavior, without any other further possibilities such as tacitcollusion. As seen below, this assumption enables one to express the conduct index in terms ofdemand and inverse demand elasticities, using Equation (1) directly (see Subsection 3.2). We also Note that in this case dw ( t, ∞ ) P S = q dt/P S = q dt/ (cid:82) ∞ t q dt The question of whether quantity- or price-setting firms are more appropriate depends on the nature of compe-tition. As Riordan (2008, p. 176) argues, quantity competition is a more appropriate model if one depicts a situationwhere firms determine the necessary capacity for production. However, price-setting firms are more suitable if firmsin the industry of focus can quickly adjust to demand by changing their prices. Although the real-world case ofcompetition is not as clear-cut as this, as we have emphasized in Introduction, we argue below that it is possibleto provide another useful characterization for the marginal costs of public funds and the pass-through rates byspecifying the mode of competition.
Direct demand.
Following Holmes (1989, p. 245), we define the own price elasticity (cid:15) F ( p ) and the cross price elasticity (cid:15) C ( p ) of the firm’s direct demand by (cid:15) F ( p ) ≡ − pq ( p ) ∂q j ( p ) ∂p j | p =( p,...,p ) , (cid:15) C ( p ) ≡ ( n − pq ( p ) ∂q j (cid:48) ( p ) ∂p j | p =( p,...,p ) , where j and j (cid:48) is an arbitrary pair of distinct indices. These are related to the industry demandelasticity (cid:15) ( p ) by (cid:15) F = (cid:15) + (cid:15) C . Next, we define the curvature of the industry’s direct demand α ( p ), the own curvature α F ( p ) of the firm’s direct demand and the cross curvature α C ( p ) of thefirm’s direct demand : α ( p ) ≡ − pq (cid:48)(cid:48) ( p ) q (cid:48) ( p ) , α F ( p ) ≡ − p (cid:18) ∂q j ( p ) ∂p j (cid:19) − ∂ q j ( p ) ∂p j , α C ( p ) ≡ − ( n − p (cid:18) ∂q j ( p ) ∂p j (cid:19) − ∂ q j ( p ) ∂p j ∂p j (cid:48) , where again the derivatives are evaluated at p = ( p, ..., p ), and j and j (cid:48) is an arbitrary pair ofdistinct indices. These curvatures satisfy α = ( α F + α C ) (cid:15) F /(cid:15) and are related to the elasticity of (cid:15) F ( p ) by p (cid:15) (cid:48) F ( p ) /(cid:15) F ( p ) = 1 + (cid:15) ( p ) − α F ( p ) − α C ( p ) (see Appendix A.2.1 for the derivation and arelated discussion). Inverse demand.
We introduce analogous definitions for inverse demand. We define the ownquantity elasticity η F ( q ) and the cross quantity elasticity η C ( q ) of the firm’s inverse demand as η F ( q ) ≡ − qp ( q ) ∂p j ( q ) ∂q j | q =( q,...,q ) , η C ( q ) ≡ ( n − qp ( q ) ∂p j (cid:48) ( q ) ∂q j | q =( q,...,q ) , Holmes (1989) shows this for two symmetric firms, but it is straightforward to verify this relation more generally.See the equation in Footnote 7 above. Note that the equation (cid:15) F = (cid:15) + (cid:15) C simply means that the percentage ofconsumers who cease to purchase firm j ’s product in response to its price increase is decomposed into (i) those whono longer purchase from any of the firms ( (cid:15) ) and (ii) those who switch to (any of) the other firms’ products ( (cid:15) C ).Thus, (cid:15) F measures the firm’s own competitiveness : it is decomposed into the industry elasticity and the degreeof rivalry. In this sense, these three price elasticities characterize “first-order competitiveness,” which determineswhether the equilibrium price is high or low, but one of them is not independently determined from the other twoelasticities. The curvature α F ( p ) here corresponds to α ( p ) of Aguirre, Cowan, and Vickers (2010, p. 1603). j and j (cid:48) . These satisfy η F = η + η C . We define the curvature of theindustry’s inverse demand σ ( q ), the own curvature σ F ( q ) of the firm’s inverse demand and the cross curvature σ C ( q ) of the firm’s inverse demand by: σ ( q ) ≡ − qp (cid:48)(cid:48) ( q ) p (cid:48) ( q ) , σ F ( q ) ≡ − q (cid:18) ∂p j ( q ) ∂q j (cid:19) − ∂ p j ( q ) ∂q j , σ C ( q ) ≡ − ( n − q (cid:18) ∂p j ( q ) ∂q j (cid:19) − ∂ p j ( q ) ∂q j ∂q j (cid:48) , where again the derivatives are evaluated at q = ( q, ..., q ) and the indices j and j (cid:48) are distinct.These curvatures represent an oligopoly counterpart of monopoly σ ( q ) of Aguirre, Cowan, andVickers (2010, p. 1603). They satisfy the relationship σ = ( σ F + σ C )( η F /η ) and are related tothe elasticity of η F ( q ) by q η (cid:48) F ( q ) /η F ( q ) = 1 + η ( q ) − σ F ( q ) − σ C ( q ) (see Appendix A.2.2 for thederivation and a related discussion). In the case of price competition, the conduct index θ is θ = (cid:15)/(cid:15) F = 1 / ( η(cid:15) F ), which is verified bycomparing the firm’s first-order condition with Equation (1). The marginal cost of public fundsand the incidence are obtained by substituting these expressions into those of Propositions 1 and2. Proposition 6.
Pass-through under price competition.
Under symmetric oligopoly withprice competition and with a possibly non-constant marginal cost: ρ t = 11 − v
11 + (1 − α/(cid:15) F ) (cid:15)(cid:15) F + (cid:16) − τ − v − (cid:15) F (cid:17) (cid:15)χ ,ρ v = 11 − v − /(cid:15) F + (1 − α/(cid:15) F ) (cid:15)(cid:15) F − + (cid:16) − τ − v (cid:15) F (cid:15) F − − (cid:15) F − (cid:17) (cid:15) χ . The identity η F = η + η C means that as a response to firm j ’s increase in its output, the industry as awhole reacts by lowering firm j ’s price ( η ). However, each individual firm (other than j ) reacts to this firm j ’soutput increase by reducing its own output. This counteracts the initial change in the price ( η C < j ( η F ) is smaller than η , which does not take into account strategicreactions. Note here that 1 /η F , not η F , measures the industry’s competitiveness. Thus, these three quantityelasticities characterize “first-order competitiveness,” which determines whether the equilibrium quantity is highor low. θ is given by θ = η F /η , whichis, again, verified by comparing the firm’s first-order condition with Equation (1). Again, themarginal cost of public funds and the incidence are obtained by substituting these expressionsinto those of Propositions 1 and 2. For the proof and a related discussion, see Appendix A.2.5and A.2.6. Proposition 7.
Pass-through under quantity competition.
Under symmetric oligopoly withquantity competition and with a possibly non-constant marginal cost: ρ t = 11 − v
11 + η F η − σ + (cid:0) − τ − v − η F (cid:1) χη , ρ v = 11 − v (1 − η F )1 + η F η − σ + (cid:0) − τ − v − η F (cid:1) χη . Although our formulas are general, it is illustrative to work through a few simple examples. Belowwe provide two parametric examples with n symmetric firms and constant marginal cost: χ = 0.In this case, the pass-through expressions are simplified to ρ t = 1(1 − v ) (cid:104) (cid:16) − α(cid:15) F (cid:17) θ (cid:105) , ρ v = (cid:15) F − (cid:15) F (cid:110) (1 − v ) (cid:104) (cid:16) − α(cid:15) F (cid:17) θ (cid:105)(cid:111) under price competition, where θ = (cid:15)/(cid:15) F , and ρ t = 1(1 − v ) (cid:2) (cid:0) − σθ (cid:1) θ (cid:3) , ρ v = 1 − η F (1 − v ) (cid:2) (cid:0) − σθ (cid:1) θ (cid:3) under quantity competition, where θ = η F /η . One is the case wherein each firm faces the following linear demand , q j ( p , ..., p n ) = b − λp j + µ (cid:80) j (cid:48) (cid:54) = j p j (cid:48) , where b > mc and λ > ( n − µ ≥
0, implying that all firms produce substitutes and µ substitutability (firms are effectively monopolists when µ = 0). , Undersymmetric pricing, the industry’s demand is thus given by q ( p ) = b − [ λ − ( n − µ ] p . The inversedemand system is given by p j ( q j , q − j ) = λ − ( n − µ ( λ + µ ) [ λ − ( n − µ ] ( b − q j ) + µ ( λ + µ ) [ λ − ( n − µ ] (cid:34)(cid:88) j (cid:48) (cid:54) = j ( b − q j (cid:48) ) (cid:35) , implying that p ( q ) = ( b − q ) / [ λ − ( n − µ ] under symmetric production. Obviously, both thedirect and the indirect demand curvatures are zero: α = 0 , σ = 0. Thus, the pass-through ratesare simply given by ρ t = 1(1 − v ) (1 + θ ) , ρ v = (cid:15) F − (cid:15) F (1 − v ) (1 + θ )under price competition, where θ = [ λ − ( n − µ ] /λ , and (cid:15) F = λ ( p/q ) (where p and q are theequilibrium price and output under price setting), and ρ t = 1(1 − v ) (1 + θ ) , ρ v = 1 − η F (1 − v ) (1 + θ )under quantity competition, where θ = [ λ − ( n − µ ] / ( λ + µ ) and η F = { [ λ − ( n − µ ]( q/p ) } / { ( λ + µ )[ λ − ( n − µ ] } (where q and p are the equilibrium output and price under quantity setting).Now, from Propositions 1 and 2, the marginal cost of public funds and the incidence are givenby M C t = (1 − v ) θ + (cid:15) τ − v ) θ − (cid:15) τ , M C v = (1 − v ) θ + (cid:15)τ (1 − v )(1+ θ ) (cid:15) F − + v − (cid:15)τ These linear demands are derived by maximizing the representative consumer’s net utility, U ( q , ..., q n ) − (cid:80) nj =1 pq j , with respect to q , ..., and q n . See Vives (1999, pp. 145-6) for details. In our notation below, the demand in symmetric equilibrium is given by q j ( p j , p − j ) = b − λp j + µ ( n − p − j ,whereas it is written as q j ( p j , p − j ) = α γ ( n − − γ ( n − − γ )[1 + γ ( n − p j + γ ( n − − γ )[1 + γ ( n − p − j in H¨ackner and Herzing’s (2016) notation, where γ ∈ [0 ,
1] is the parameter that measures substitutability be-tween (symmetric) products. Thus, if our ( b, λ, µ ) is determined by b = α/ [1 + γ ( n − λ = [1 + γ ( n − / { (1 − γ )[1 + γ ( n − } , and µ = γ/ { (1 − γ )[1 + γ ( n − } , given H¨ackner and Herzing’s (2016) ( α, γ ), thenour results below can be expressed by H¨ackner and Herzing’s (2016) notation as well. Note here that our formula-tion is more flexible in the sense that the number of the parameters is three. This is because the coefficient for theown price is normalized to one: p j ( q j , q − j ) = α − q j − γ ( n − q − j , which is analytically innocuous, and H¨acknerand Herzing’s (2016) γ is the normalized parameter. (cid:15) = [ λ − ( n − µ ] (cid:16) pq (cid:17) η = λ − ( n − µ (cid:16) qp (cid:17) (cid:15) F = λ (cid:16) pq (cid:17) η F = λ − ( n − µ ( λ + µ )[ λ − ( n − µ ] (cid:16) qp (cid:17) θ = (cid:15)/(cid:15) F = 1 − ( n − (cid:0) µλ (cid:1) θ = η F /η = λ − ( n − µλ + µ α = 0 σ = 0(b) Logit DemandPrice setting Quantity setting (cid:15) = β (1 − ns ) p η = β (1 − ns ) p (cid:15) F = β (1 − s ) p η F = − ( n − sβ (1 − ns ) p θ = (cid:15)/(cid:15) F = − ns − s θ = η F /η = 1 − ( n − sα = (2 ns − ns − ns p σ = − ns − ns I t = 12(1 − v )[1 − ( n − µ/λ )] , I v = (cid:15) F − − v )[2 − (cid:15) F (1 − θ )]under price competition, with (cid:15) = [ λ − ( n − µ ]( p/q ) is additionally provided, where p and q arethe equilibrium price and output under price setting, and M C t = (1 − v ) θ + η τ − v ) θ − η τ , M C v = (1 − v ) θ + η τ (1 − v )(1+ θ )1 − η F + v − η τI t = λ + µ − v )[ λ − ( n − µ ] , I v = 1 − η F (1 − v )[ η F + (2 − η F ) θ ]under quantity competition, with 1 /η = [ λ − ( n − µ ]( p/q ) is additionally provided, where p and q are the equilibrium price and output under quantity setting. Thus, it suffices to solve forthe equilibrium price and output under both settings to compute the pass-through rate and themarginal cost of public funds for all four cases.Table I (a) summarizes the key variables that determine the pass-through rates and themarginal costs of public funds. It is verified that under both price and quantity competition, ∂θ/∂n < ∂θ/∂µ <
0. To focus on the roles of these two parameters, n and µ , which directlyaffect the degree of competition, we employ the following simplification to compute the ratio p/q in equilibrium: b = 1, mc = 0, and λ = 1 (see Appendix A.2.7 for the actual expressions of the19quilibrium prices and outputs under price and quantity competition).The top two panels in Figure II illustrate how ρ t and ρ v behave as the number of firms ( n ; theleft) or the sustainability parameter ( µ ; the left) increases, with the superscript denoting price ( P )or quantity ( Q ) setting. Similarly, the middle and the bottom panels draw M C t and M C v , and I t and I v , respectively. It is observed that the ad valorem tax pass-through rates are close to zerobecause in this case both (cid:15) F and η F are close to 1. As competition becomes fiercer, both ρ Pt and ρ Qt become larger, although the discrepancy also becomes larger. In the case of linear demand, thedifference in the mode of competition does not yield a significant difference in each of the threemeasures. As is verified by Anderson, de Palma, and Kreider (2001b), the ad valorem tax is moreefficient than the unit tax: the dashed lines in the two middle panels lie below the solid lines.This ranking is related inversely to the pass-through and the incidence: as the pass-through orthe incidence becomes larger, the marginal cost of public funds becomes smaller. The next parametric example is the logit demand . Each firm j = 1 , ..., n faces the followingdemand: s j ( p ) = exp( δ − βp j ) / [1 + (cid:80) j ´ =1 ,...,n exp( δ − βp j ´ )] ∈ (0 , δ is the (symmetric)product-specific utility and β > responsiveness to the price. We define s = 1 − (cid:80) j =1 ,...,n s j < β and n , we assume that δ = 1 and mc = 0. Because ∂s j ( p ) /∂p j | p =( p,...,p ) = − βs (1 − s ), the first-order conditions for the symmetric equilibrium price and the market share satisfy p − t/ (1 − v ) = 1 / [ β (1 − s )] and s = exp(1 − βp ) / [1 + n exp(1 − βp )]. If p and s are solved Here, q j ( p , ..., p n ) is derived by aggregating over individuals who choose product j (the total number ofindividuals is normalized to one): individual i ’s net utility from consuming j is given by u ij = δ − βp j + ˜ ε ij ,whereas u i = ˜ ε i is the net utility from consuming nothing, and ˜ ε i , ˜ ε i , ..., ˜ ε in are independently and identicallydistributed according to the Type I extreme value distribution for all individuals. See Anderson, de Palma, andThisse (1992, pp. 39-45) for details. We work in terms of market share variables s j and s , instead of q j and q , whichis consistent with the usual notation in the industrial organization literature. n ρ t , ρ v ρ tP ρ vP ρ tQ ρ vQ ( below ) μ ρ t , ρ v ρ tP ρ vP ρ tQ ρ vQ ( below ) n MC t , MC v MC tP MC vP MC tQ MC VQ μ MC t , MC v MC tP MC vP MC tQ MC VQ n I t , I v I tP I vP I tQ I VQ μ I t , I v I tP I vP I tQ I VQ Figure II: Pass-through rate (top), marginal cost of public funds (middle), and incidence (bottom)with linear demand. The horizontal axes on the left and the right panels correspond to the numberof firms ( n ) and the substitutability parameter ( µ ), respectively.21igure III: Pass-through rates (top), marginal costs of public funds (middle), and incidence (bot-tom) with logit demands. The horizontal axes on the left and the right panels are the number offirms ( n ) and the price coefficient ( β ), respectively.22umerically, then (cid:15) , (cid:15) F , θ and α can also be numerically computed. Next, we consider the inversedemands under quantity competition. Then, as in Berry (1994), firm j ’s inverse demand is givenby p j ( s ) = [ δ − log( s j /s )] /β , which implies that ∂p j ( s ) /∂s j | s =( s,...,s ) = − [1 − ( n − s ] / [ βs (1 − ns )].Thus, the first-order conditions for the symmetric equilibrium price and the market share satisfy p − t/ (1 − v ) = [1 − ( n − s ] / [ β (1 − ns )] and p = [1 − log( s/ [1 − ns ])] /β . Then, as above, η , η F , θ and σ are computed by numerically solving the first-order conditions for p and s . Interestingly, itis verified that in symmetric equilibrium under quantity setting, ∂p/∂n = 0: the equilibrium priceis the same irrespective of the number of firms, whereas the individual market share is decreasingin the number of firms: ∂s/∂n <
0. On the other hand, both the equilibrium price and marketshare are decreasing in the price coefficient, β .Figure III illustrates the pass-through rate, the marginal cost of public funds, and the incidence,in analogy with Figure II. The right panels now show the variables’ dependence on the pricecoefficient β . Overall, as in the case of linear demand, an increase in the ad valorem tax has a smallimpact on these measures for each of n and β , whereas an increase in the unit tax has a large effect.However, there are important differences between the cases of linear and logit demand. First, theunit tax pass-through under quantity competition ρ Qt is decreasing in the number of firms. Tounderstand this, compare the difference in the denominators of ρ Pt = 1 / { (1 − v ) [1 + (1 − α/(cid:15) F ) θ ] } and ρ Qt = (1 − v ) [1 + θ − σ ]. As θ decreases (i.e., as competition becomes fiercer), the secondterm in the denominator of ρ Pt decreases, and thereby ρ Pt increases as n increases. However, θ − σ increases as θ decreases, and thus ρ Qt decreases . This difference in the denominators is alsoreflected in the fact that I Qt is decreasing in n as well. Naturally, M C Qt is decreasing in n as in thecase of linear demand because 1 /ρ Qt becomes larger (see the formulas in Proposition 1). Second,while the pass-through rate and the incidence increase as β increases, the marginal cost of publicfunds is also increasing in contrast to the case of linear demands. The reason is that the effecton M C of decreases in θ is weaker than the effect of the increase in (cid:15) : the industry’s demandbecomes elastic quickly as consumers become more sensitive to a price increase. It can be verified that s j ( · ; p − j ) is convex as long as s j < / ∂ s j /∂p j = − β ( ∂s j /∂p j )(1 − s j ) > ∂ π j /∂p j = − βs j <
0. In symmetric equilibriumwith δ = 1 and mc = 0, the largest market share is attained as 1 / ( n + 1) when the equilibrium price is zero, whichimplies that the market share of the outside goods s is no less than each firm’s market share: s > s . Multi-Dimensional Pass-Through Framework
Now, we generalize our previous results to a more general specification of taxation that involvesmultiple tax instruments. We define two different types of pass-through vectors: (i) the pass-through rate vector and (ii) pass-through quasi-elasticity . We study their properties and show thatthey play a central role in evaluating welfare changes in response to changes in taxation.
Consider a tax structure under which a firm’s tax payment is expressed as φ ( p, q, T ), where T ≡ ( T , ..., T d ) is a d -dimensional vector of tax instruments so that the firm’s profit in symmetricequilibrium is written as π = pq − c ( q ) − φ ( p, q, T ). Note that the argument so far is a specialcase of two dimensional pass-through: φ ( p, q, T ) = tq + vpq , where T = ( t, v ). The componentsof the (per-firm) tax revenue gradient vector ∇ φ ( p, q, T ) are φ T (cid:96) ( p, q, T ) ≡ ∂φ ( p, q, T ) ∂T (cid:96) . Here, as in other parts of the paper, we use the symbol ∇ for the d -dimensional gradient withrespect to T . The arguments p and q in φ ( p, q, T ) are treated as fixed for the purposes of takingthis gradient. We also denote by f a vector components φ T (cid:96) ( p, q, T ) /q . We denote the equilibriumprice function by p (cid:63) ( T ) and its gradient, the pass-through rate vector , by ˜ ρ ≡ ∇ p (cid:63) ( T ) . Further,we use the components of the f and ˜ ρ to define the pass-through quasi-elasticity vector as ρ ≡ (cid:0) ρ T , ..., ρ T d (cid:1) , ρ T (cid:96) ≡ ˜ ρ T (cid:96) f T (cid:96) = qφ T (cid:96) ( p, q, T ) ∂p (cid:63) ∂T (cid:96) . Note that the components of ρ are all dimensionless. We define the ( first-order ) price sensitivity ν of the tax revenue and the ( first-order ) quantity sensitivity τ of the (per-firm) tax revenue as To be precise, φ ( p, q, T ) represents a simplified notation for a function φ ( p, q, T , ..., T d ) with d + 2 arguments. Unlike the inverse demand function p ( q ), the function p (cid:63) ( T ) takes the vector of taxes as arguments, and itsfunctional value is the price in the resulting equilibrium. ν ( p, q, T ) ≡ q φ p ( p, q, T ) , τ ( p, q, T ) ≡ p φ q ( p, q, T ) , and their derivatives are ν T (cid:96) ( p, q, T ) ≡ ∂ν ( p, q, T ) ∂T (cid:96) , τ T (cid:96) ( p, q, T ) ≡ ∂τ ( p, q, T ) ∂T (cid:96) . The analogous definitions for the second-order sensitivities are: ν (2) ( p, q, T ) ≡ pq ∂ φ ( p, q, T ) ∂p , τ (2) ( p, q, T ) ≡ qp ∂ φ ( p, q, T ) ∂q , κ ( p, q, T ) ≡ ∂ φ ( p, q, T ) ∂p ∂q . The first-order and second-order sensitivities are dimensionless, as are the components of ρ . Inthis section, we keep the same definition of the elasticities (cid:15) and η as before. We introduce the conduct index θ as a function, independently of the cost-side of the oligopolygame, so that in equilibrium the following condition holds:[1 − τ − (1 − ν ) η θ ] p = mc. (3)In the case of unit and ad valorem taxation, this definition reduces to the conduct index definedearlier (Equation 1), where τ = v + t/p and ν = v : this is the reason why we can keep using τ below. In principle, there are many possible definitions that agree with the earlier definitionin the case of unit and ad valorem taxation. However, we find the specification of Equation 3particularly convenient. We now establish the following relationship for the relative size of pass-through vector component.25 roposition 8.
The pass-through rates and quasi-elasticities satisfy ˜ ρ T (cid:96) ˜ ρ T (cid:96) (cid:48) = τ T (cid:96) (cid:48) − ν T (cid:96) (cid:48) η θτ T (cid:96) − ν T (cid:96) η θ , ρ T (cid:96) ρ T (cid:96) (cid:48) = f T (cid:96) (cid:48) f T (cid:96) τ T (cid:96) − η θ ν T (cid:96) τ T (cid:96) (cid:48) − η θ ν T (cid:96) (cid:48) . The proposition is proven in Appendix A.3.1. Since the components have known proportions,we can write them using a common factor pρ (0) as˜ ρ T (cid:96) = ( τ T (cid:96) − ν T (cid:96) η θ ) pρ (0) , ρ T (cid:96) = pf T (cid:96) ( τ T (cid:96) − ν T (cid:96) η θ ) ρ (0) , (4)with the factor ρ (0) determined in the following proposition. Proposition 9.
The value of the factor ρ (0) introduced in Equation (4) is given by: ρ (0) = 1 − κ + (cid:15)τ (2) + (1 − τ ) (cid:15)χ + (cid:2) ν − κ + ην (2) + ( ω − η − χ ) (1 − ν ) (cid:3) θ, (5) where ω ≡ q ( ηθ ) (cid:48) / ( ηθ ) , with the prime denoting a derivative with respect to the quantity q. The proof is in Appendix A.3.2. Now, we establish the general formulas for the marginal cost of public fund and incidence inthe multi-dimensional pass-through framework. Welfare component changes in response to aninfinitesimal change in taxes can be found as follows. The (per-firm) consumer surplus change in If the denominators are zero, the fractions become ill-defined. In that case, of course, the statement does notapply. If φ ( p, q, T ) = tq + vpq , then τ = ( t + vp ) /p = t/p + v and ν = vq/q = v. First, ρ t = q∂φ/∂t ˜ ρ t = qq ˜ ρ t = ˜ ρ t , and ρ v = q∂φ/∂v ˜ ρ t = qpq ˜ ρ t = p ˜ ρ t . Next, ρ t = pq∂φ/∂t (cid:2) τ t − ν t (cid:0) θ(cid:15) D (cid:1)(cid:3) ρ (0) = ρ (0) because τ t = 1 /p and ν t = 0. Then,1 ρ (0) = [(1 − κ (cid:124) (cid:123)(cid:122) (cid:125) ) =1 − v + (cid:15) D τ (2) (cid:124) (cid:123)(cid:122) (cid:125) =0 + (1 − τ ) (cid:18) (cid:15) D (cid:15) S (cid:19) ] + ν − κ + ην (2) (cid:124) (cid:123)(cid:122) (cid:125) =0 + (cid:18) ω − (cid:15) D − (cid:15) S (cid:19) (1 − ν (cid:124) (cid:123)(cid:122) (cid:125) ) =1 − v θ = (1 − v ) (cid:26)(cid:20) − τ − v (cid:18) (cid:15) D (cid:15) S (cid:19)(cid:21) − (cid:18) (cid:15) D + 1 (cid:15) S (cid:19) θ + (cid:15) D q ∂ ( θ/(cid:15) D ) ∂q (cid:27) since κ ( p, q, T ) ≡ ∂ φ ( p,q, T ) ∂p ∂q = v, τ (2) ( p, q, T ) ≡ qp · ∂ φ ( p,q, T ) ∂q = 0 , and ν (2) ( p, q, T ) ≡ pq · ∂ φ ( p,q, T ) ∂p = 0 . dT (cid:96) in the tax T (cid:96) is dCS = − qdp = − q ˜ ρ T (cid:96) dT (cid:96) , which means that in vector notation, q ∇ CS = − ˜ ρ . The change in (per-firm) producer surplus is dP S = d ( pq − c ( q ) − φ ( p, q, T )) = (cid:2) φ T (cid:96) ( p, q, T ) − (1 − ν ) (1 − θ ) ˜ ρ T (cid:96) (cid:3) dT (cid:96) , where we utilize Equation (3) to eliminate the marginal cost. In vector notation, this is q ∇ P S =(1 − ν ) (1 − θ ) ˜ ρ − f , since f = q ∇ φ ( p, q, T ). The change in tax revenue is dR = φ p ( p, q, T ) dp + φ q ( p, q, T ) dq + φ T (cid:96) ( p, q, T ) dT (cid:96) = (cid:2) φ T (cid:96) ( p, q, T ) − ( (cid:15)τ − ν ) ˜ ρ T (cid:96) (cid:3) dT (cid:96) . In vector notation, q ∇ R = f − ( (cid:15)τ − ν ) ˜ ρ . Finally, for the change in social welfare, we have dW = ( p − mc ) dq = [ (cid:15)τ + θ (1 − ν )] ˜ ρ T (cid:96) dT (cid:96) . In vector notation, q ∇ W = − [ (cid:15)τ + θ (1 − ν )] ˜ ρ .Note that the welfare components CS ( T ) , P S ( T ) , R ( T ), and W ( T ) = CS ( T ) + P S ( T ) + R ( T ) are all treated as functions of taxes only and represent the equilibrium outcomes. This isdifferent from the tax revenue function φ ( p, q, T ), which has also p and q as arguments and whichis specified by the government irrespective of the equilibrium. We summarize these findings in thefollowing proposition. Proposition 10.
The tax gradients of consumer surplus, producer surplus, tax revenue, and socialwelfare with respect to the taxes all belong to a two-dimensional vector space spanned by f and ˜ ρ .The precise linear combinations of f and ˜ ρ are q ∇ CS = − ˜ ρ , q ∇ P S = (1 − ν ) (1 − θ ) ˜ ρ − f , q ∇ R = f + ( ν − (cid:15)τ ) ˜ ρ , q ∇ W = − [(1 − ν ) θ + (cid:15)τ ] ˜ ρ . These relationships, considered component-wise, immediately imply the following results forwelfare change ratios and generalize Propositions 1 and 2. Proposition 11.
The marginal cost of public funds of a tax T (cid:96) , M C T (cid:96) = ( ∇ W ) T (cid:96) / ( ∇ R ) T (cid:96) , is M C T (cid:96) = (1 − ν ) θ + (cid:15)τ ρ T(cid:96) + ν − (cid:15)τ . The incidence of this tax, I T (cid:96) = ( ∇ CS ) T (cid:96) / ( ∇ P S ) T (cid:96) , equals: I T (cid:96) = 1 ρ T(cid:96) − (1 − ν ) (1 − θ ) . Similarly, the social incidence, SI T (cid:96) = ( ∇ W ) T (cid:96) / ( ∇ P S ) T (cid:96) , equals: SI T (cid:96) = (1 − ν ) θ + (cid:15)τ ρ T(cid:96) − (1 − ν ) (1 − θ ) . The results of the previous subsection contain our results for ad valorem and unit taxes as specialcases, but provide much greater generality, since the taxes (government interventions) may bespecified in a very flexible way.
Weyl and Fabinger’s (2013) results under symmetric oligopoly can be interpreted as special casesof the present results. In particular, Weyl and Fabinger’s (2013) analysis considers either unit Remember that the T (cid:96) component of the vector f is φ T (cid:96) ( p, q, T ) /q =˜ ρ T (cid:96) /ρ T (cid:96) . T = ˜ q of the form: φ ( p, q, ˜ q ) = ˜ q p + c ( q − ˜ q ) − c ( q ). Then, the firm’s profit isgiven by: pq − c ( q ) − φ ( p, q, ˜ q ) = p ( q − ˜ q ) + c ( q − ˜ q ) . The firm, therefore, has the same profit function as in the case of exogenous competition ˜ q inWeyl and Fabinger (2013). Proposition 11 above (specialized to constant marginal cost and zeroinitial ˜ q ) then implies the social incidence result in Principle of Incidence 3 in Weyl and Fabinger(2013, p. 548).Similarly, the relationships between pass-through of unit taxes and of exogenous competitionare implied by the general result of Proposition 8 for the tax specification T = t , T = ˜ q , φ ( p, q, t, ˜ q ) = tq + ˜ q p + c ( q − ˜ q ) − c ( q ) . To obtain the absolute size of the two types of pass-through, one can straightforwardly use Propo-sition 9. More generally, φ is extended as φ = c ( q − ˜ q ) + v ( q − ˜ q ) p ( q ) + (1 − v )˜ qp ( q ) + t ( q − ˜ q ) − c ( q ) , where an ad valorem tax is also considered. As an example, one can think of a government whichprocures goods from abroad and supplies them to the market in order to lower domestic prices.In the special case of a monopolist with constant marginal cost, the mathematics allow foranother interesting interpretation: It is isomorphic to the case of “depreciating licenses” in Weyland Zhang (2017). Depreciating licenses correspond to a tax scheme where the owner of an assetannounces a reservation price at which she is willing to sell it and gets taxed a fixed fraction ofthat prices. Another agent in the economy may buy the asset at the announced price. The ownerthen faces a tradeoff between announcing a low price and paying low taxes and announcing a high29rice in order to be able to keep the asset and derive utility from it. The optimization problemthen leads to exactly the same mathematical form as the problem of a monopolist with constantmarginal cost facing exogenous competition. We include a more detailed explanation in AppendixA.3.3. Governments often regulate when, where, and to whom products may be sold. For example, thereare restrictions on weekend sales, store locations, etc. A simple way of modeling this situation is toassume that due to the restrictions a firm loses a fixed proportion of its customers. If the absenceof the regulation and taxation, the profit function is p ( q ) q − c ( q ). The new profit function willbe (1 − v ) p ([1 + κ ] q ) q − tq − c ( q ), where 1 − / (1 + κ ) is the fraction of customers lost. The onlychange is in the argument of the inverse demand function: for the firm to sell quantity q , eachremaining customer needs to buy (1 + κ ) times more than in the absence of the regulation, andthe price would have to be correspondingly lower. This change may be described using as: φ ( p, q, t, v, κ ) = (1 − (1 − v ) h ( q, κ )) pq + tq, h ( q, κ ) ≡ p ( q ) − p ((1 + κ ) q ) p ( q ) . For demand with constant elasticity (cid:15) , h ( q, κ ) = 1 − (1 + κ ) − /(cid:15) , independently of q . Tax evasion, clearly, is a very important problem in many situations since economic agents do notalways strictly follow the law (Choi, Furusawa, and Ishikawa 2017).For simplicity, consider a firm that needs to pay an ad valorem tax v ˜ pq , where ˜ p is the pricereported to the government and may differ from the true price p . We capture the cost associatedwith deceiving the government by introducing a concealment cost of the form:14 λ p − ζ (˜ p − p ) q − ξ . We thank Glen Weyl for suggesting this relationship between Weyl and Zhang (2017) and our analysis. p to minimize the sum of these two additional costs: v ˜ pq + 14 λ p − ζ q − ξ (˜ p − p ) . The corresponding first-order condition implies ˜ p = p − λvp ζ q ξ , which gives the effective additionalcost pqv − λv p ζ q ξ .φ ( p, q, t, v, λ ) = tq + pqv − λv p ζ q ξ ˜ φ ( p, q, t, v, λ ) = tq + pqv − λv p ζ q ξ The government needs to pay additional enforcement cost inversely related to λ , which needs tobe remembered in the welfare analysis. In this section, we extend our results to the case of n heterogeneous firms (i.e. asymmetric firms),where each firm i controls a strategic variable σ i , which could be, for example, the price or quantityof its product. We allow for the tax function φ i ( p i , q i , T ) to depend explicitly on the identity of thefirm; we write f i T (cid:96) ( p i , q i , T ) = q i ∂∂T (cid:96) φ i ( p i , q i , T ) for its derivative with respect to tax T (cid:96) . Similarly,the sensitivities τ i ( p i , q i , T ), ν i ( p i , q i , T ), etc., now also have the firm index i . The marginal cost mc i ( q i ) of firm i is also allowed to depend on the identity of the firm, and we denote its elasticity χ i ( q i ) ≡ q i mc (cid:48) i ( q i ) /mc i ( q i ) . We define the pricing strength index ψ i ( q ) of firm i to be a function independent of the cost sideof the economic problem such that the first-order condition for firm i is: { − τ i ( p i ( q ) , q i , T ) − ψ i ( q ) [1 − ν i ( p i ( q ) , q i , T )] } p i ( q ) = mc ( q i ) .
31n the special case of symmetric firms, this definition reduces to ψ i = η θ .We express the pass-through rate in terms of these pricing strength indices. Specifically, thepass-through rate is an n × d matrix ˜ ρ with rows ˜ ρ T (cid:96) ≡ ∂ p /∂T (cid:96) and elements ˜ ρ i T (cid:96) = ∂p i /∂T (cid:96) . Itis shown that the pass-through rate equals˜ ρ T (cid:96) = b − . ι T (cid:96) , (6)where the factors on the right-hand side are defined as follows. The matrix b is an n × n matrix,independent of the choice of T (cid:96) , with elements b ij = (cid:2) − κ i − (cid:0) − ν i − ν (2) i (cid:1) ψ i (cid:3) δ ij − (1 − ν i ) ψ i Ψ ij + (cid:8) τ (2) i + ν i ψ i − κ i + [1 − τ i − (1 − ν i ) ψ i ] χ i (cid:9) (cid:15) ij , where δ ij is the Kronecker delta, and (cid:15) ij = − p i q i ∂q i ( p ) ∂p j , Ψ ij = p i ψ i ∂ψ i ( q ( p )) ∂p j . For each tax T (cid:96) , ι T (cid:96) is an n -dimensional vector with components ι i T (cid:96) = p i ∂τ i ( p i , q i , T ) ∂T (cid:96) − p i ψ i ∂ν i ( p i , q i , T ) ∂T (cid:96) . In the case of symmetric firms and at symmetric prices, the pass-through rate expression inEquation (6) agrees with the expression represented by Equations (4) and (5) in Section 4. To generalize the notion of pass-through quasi-elasticity to the case of heterogeneous firms, wedefine the pass-through quasi-elasticity matrix ρ as an n × d matrix with elements ρ i T (cid:96) = 1 f i T (cid:96) ( p i , q i , T ) ∂p i ∂T (cid:96) , To confirm this agreement, note that at symmetric prices, (cid:80) nj =1 Ψ ij = − (cid:15)ω . Note also that (cid:15) ii ( p ) | p =( p,...,p ) = (cid:15) F ( p ) , and for j (cid:54) = i , (cid:15) ij ( p ) | p =( p,...,p ) = − n − (cid:15) C ( p ). ρ T (cid:96) . In the following, for each i , (cid:15) i is an n -dimensional vector with its j -th component equal to (cid:15) ij . Forthe tax gradients of welfare components corresponding to individual firms we obtain:1 q i ∇ CS i = − e i . ˜ ρ , q i ∇ P S i = (1 − ν i ) ( e i − ψ i (cid:15) i ) . ˜ ρ − f i , q i ∇ R i = f i + ( ν i e i − τ i (cid:15) i ) . ˜ ρ , q i ∇ W i = − [ τ i + ψ i (1 − ν i )] (cid:15) i . ˜ ρ . The corresponding gradients of total welfare components are then obtained by adding up contri-butions from individual firms. For example, ∇ CS = (cid:80) ni =1 ∇ CS i . Denoting the total quantity as Q ≡ (cid:80) ni =1 q i , this means that Q ∇ CS is a weighted average of − e i . ˜ ρ , with the weights proportionalto q i . The arguments for the other welfare components are similar. This generalizes Proposition10 above.We can also consider ratios of welfare changes corresponding to some tax T (cid:96) : M C i T (cid:96) = [ τ i + (1 − ν i ) ψ i ] (cid:15) ρi T (cid:96) ρ i T(cid:96) + ν i − τ i (cid:15) ρi T (cid:96) ,I i T (cid:96) = 1 ρ i T(cid:96) − (1 − ν i ) (1 − ψ i (cid:15) ρi T (cid:96) ) ,SI i T (cid:96) = [ τ i + (1 − ν i ) ψ i ] (cid:15) ρi T (cid:96) ρ i T(cid:96) − (1 − ν i ) (1 − ψ i (cid:15) ρi T (cid:96) ) , where (cid:15) ρi T (cid:96) ≡ (cid:15) i . ˜ ρ T (cid:96) / ˜ ρ i T (cid:96) = (cid:15) i . ρ T (cid:96) /ρ i T (cid:96) . The ratios of the corresponding total welfare changes willbe weighted averages of these firm-specific ratios. The weights correspond to the sizes of thedenominators times q i . For example, M C T (cid:96) will lie between min i M C i T (cid:96) and max i M C i T (cid:96) . The33ame reasoning also holds for the other ratios. This generalizes Proposition 11 above.
For heterogeneous firms, we introduce the conduct index of firm i so that θ i = − (cid:80) nj =1 { p j [1 − τ j ( p j , q j , T )] − mc ( q j ) } dq j dσ i (cid:80) nj =1 [1 − ν j ( p j , q j , T )] q j dp j dσ i holds. In the special case of only unit taxation, this definition reduces to Weyl and Fabinger’s(2013, p. 552) Equation (4). In the special case of symmetric firms the definition reduces to ourEquation (3) with θ i = θ .The conduct index θ i is closely connected to the pricing strength index ψ i , but not as closelyas it would be in the case of symmetric oligopoly. Using the definitions of the indices, it is shownthat θ i = − (cid:80) nj =1 (1 − ν j ) ψ j p j dq j dσ i (cid:80) nj =1 (1 − ν j ) q j dp j dσ i . For symmetric oligopoly, this equation reduces simply to θ = (cid:15)ψ. The conduct index is used to express welfare component changes in response to infinitesimalchanges in taxes. The relationships are a bit more complicated than in the case of using the pricingstrength index: they can be expressed as follows. We define the price response to an infinitesimalchange in the strategic variable σ k of firm j as ζ ij = dp i dσ j . Since the vectors ζ i , ζ i , ... , ζ in form a basis in the n -dimensional vector space to which ˜ ρ i T (cid:96) fora given (cid:96) belongs, we can write ˜ ρ i T (cid:96) as a linear combination of them for some coefficients λ i T (cid:96) :˜ ρ i T (cid:96) = n (cid:88) j =1 λ j T (cid:96) ζ ij . dCSdT (cid:96) = − n (cid:88) i =1 q i ˜ ρ i T (cid:96) = − n (cid:88) j =1 (cid:32) n (cid:88) i =1 q i ζ ij (cid:33) λ j T (cid:96) ,dP SdT (cid:96) = − n (cid:88) i =1 f i T (cid:96) ( p i , q i , T ) − n (cid:88) j =1 ˆ ζ j (1 − θ j ) λ j T (cid:96) , where we used the notation ˆ ζ j ≡ n (cid:88) i =1 [1 − ν i ( p i , q i , T )] q i ζ ij . These surplus change expressions represent a generalization of the surplus expressions in Weyland Fabinger’s (2013) Section 5.
In the case of oligopoly in the form of aggregative games, where all other firms’ actions aresummarized as an aggregator in each firm’s profit, we can further manipulate the above formulasfor pricing strength and conduct indices. We identify the firm’s strategic variable σ i with anaction a i ≡ σ i the firm can take, which contributes to an aggregator A = (cid:80) ni =1 a i . The prices andquantities are functions of just two arguments: p i ( A, a i ) and q i ( A, a i ). Their derivatives that takeinto account the dependence of A on the action of firm i are dq j dσ i = ∂q j ∂a i + ∂q j ∂A and dp j dσ i = ∂p j ∂a i + ∂p j ∂A .The firm’s first-order condition is:0 = (cid:18) ∂p j ∂a i ( A, a i ) + ∂p j ∂A ( A, a i ) (cid:19) q i ( A, a i ) ( ν i ( p i ( A, a i ) , q i ( A, a i ) , T ) −
1) + (cid:18) ∂q j ∂a i ( A, a i ) + ∂q j ∂A ( A, a i ) (cid:19) ( mc ( q i ( A, a i )) + p i ( A, a i ) ( τ i ( p i ( A, a i ) ,q i ( A, a i ) , T ) − , which gives us a relatively simple expression for the pricing strength index: ψ i ( A, a i ) = − q i ( A, a i ) p i ( A, a i ) ∂p j ∂a i ( A, a i ) + ∂p j ∂A ( A, a i ) ∂q j ∂a i ( A, a i ) + ∂q j ∂A ( A, a i ) . Here, we consider a setup in Anderson, Erkal, and Piccinin’s (2016) Section 2. θ i = n (cid:88) j =1 w j γ j ( A, a i ) γ j ( A, a j ) , where w i is a normalized version of unnormalized “weights” ˜ w j , w i ≡ ˜ w i (cid:80) nj =1 ˜ w j , ˜ w j ≡ (1 − ν j ) q j ( A, a j ) (cid:18) ∂p j ∂a i ( A, a i ) + ∂p j ∂A ( A, a i ) (cid:19) , and γ j ( A, a i ) ≡ ∂q j ∂a i ( A, a i ) + ∂q j ∂A ( A, a i ) . These simplified formulas would be used for further analysis of pass-through and welfare in ag-gregative oligopoly games.
In the previous sections, we have studied changes in taxation, but not changes in production costs.Here, we generalize our main results to incorporate both taxation and production costs. As shownbelow, the generalization to production cost changes turns out to be straightforward. These moregeneral formulas may be applied to a range of economic situations such as cost changes due toexchange rate movements or movements in the world prices of commodities.The additional cost to the firm is denoted φ ( p, q, T ) as before, but the tax bill of firm i , denoted˜ φ ( p, q, T ), is different, in general. Here T is a vector of interventions (by the government or byexternal circumstances), which may or may not include traditional taxes. We recover the previouscase of only taxation by setting ˜ φ ( p, q, T ) = φ ( p, q, T ). If all of the additional cost to the firmcomes from the production side, we have ˜ φ ( p, q, T ) = 0. In general, φ ( p, q, T ) − ˜ φ ( p, q, T ) is theproduction part of the additional cost φ ( p, q, T ).36 .1 Symmetric firms In addition to the notation used in the previous section, we define ˜ f = q ∇ ˜ φ ( p, q, T ). First, weobtain a generalization of the formulas for the tax gradients of welfare components in Proposition10. The equilibrium outcome depends only on the additional cost φ ( p, q, T ) and not on its splitbetween taxes and production costs. For this reason, the formulas for consumer and producersurplus will be unchanged. The government revenue and therefore also total social welfare dependson ˜ φ ( p, q, T ). In the formula for the gradient of government revenue, f will be replaced by ˜ f , andthe formula for social welfare will be adjusted to reflect this difference. Hence, the generalizationof the results in Proposition 10 is: 1 q ∇ CS = − ˜ ρ , q ∇ P S = (1 − ν ) (1 − θ ) ˜ ρ − f , q ∇ R = ˜f + ( ν − (cid:15)τ ) ˜ ρ , q ∇ W = − [(1 − ν ) θ + (cid:15)τ ]˜ ρ + ˜f − f . We further define g T (cid:96) ≡ ˜ f T (cid:96) /f T (cid:96) , which represents the fraction of an increase in additional cost( φ ) to the firm (due to a change in the tax parameter T (cid:96) ) that is collected by the government inthe form of taxes (˜ φ ). In other words, g T (cid:96) is the government’s share in increases of the additionalcosts induced by marginal changes in T (cid:96) . If φ is a pure tax, then g T (cid:96) = 1, and if φ is a pureproduction cost with no tax component, then g T (cid:96) = 0. By taking ratios of the components of thetax gradients above, we obtain a generalization of Proposition 11: The marginal cost of publicfunds associated with intervention T (cid:96) , M C T (cid:96) = ( ∇ W ) T (cid:96) / ( ∇ R ) T (cid:96) , is M C T (cid:96) = − g T(cid:96) ρ T(cid:96) + (1 − ν ) θ + (cid:15)τ g T(cid:96) ρ T(cid:96) + ν − (cid:15)τ . I T (cid:96) = ( ∇ CS ) T (cid:96) / ( ∇ P S ) T (cid:96) , equals: I T (cid:96) = 1 ρ T(cid:96) − (1 − ν ) (1 − θ ) . Similarly, the social incidence, SI T (cid:96) = ( ∇ W ) T (cid:96) / ( ∇ P S ) T (cid:96) , equals: SI T (cid:96) = − g T(cid:96) ρ T(cid:96) + (1 − ν ) θ + (cid:15)τ ρ T(cid:96) − (1 − ν ) (1 − θ ) . The adjustments to our formulas needed to generalize the results of Subsection 5.2 are analogousto the case of symmetric firms we just discussed. For each firm i , we define ˜ f i = q ∇ ˜ φ i ( p, q, T ) . For the welfare gradients, we obtain: 1 q i ∇ CS i = − e i . ˜ ρ , q i ∇ P S i = (1 − ν i ) ( e i − ψ i (cid:15) i ) . ˜ ρ − f i , q i ∇ R i = ( ν i e i − τ i (cid:15) i ) . ˜ ρ + ˜f i , q i ∇ W i = − [ τ i + ψ i (1 − ν i )] (cid:15) i . ˜ ρ + ˜f i − f i . Similarly, for each firm i , we define g i T (cid:96) ≡ ˜ f i T (cid:96) /f i T (cid:96) . For the firm-specific welfare change ratios,we obtain: M C i T (cid:96) = − g i T(cid:96) ρ i T(cid:96) + ( τ i + (1 − ν i ) ψ i ) (cid:15) ρi T (cid:96) g i T(cid:96) ρ i T(cid:96) + ν i − τ i (cid:15) ρi T (cid:96) ,I i T (cid:96) = 1 ρ i T(cid:96) − (1 − ν i ) (1 − ψ i (cid:15) ρi T (cid:96) ) ,SI i T (cid:96) = − g T(cid:96) ρ i T(cid:96) + ( τ i + (1 − ν i ) ψ i ) (cid:15) ρi T (cid:96) ρ i T(cid:96) − (1 − ν i ) (1 − ψ i (cid:15) ρi T (cid:96) ) . Concluding Remarks
In this paper, we characterize the welfare measures of taxation and other external changes inoligopoly with a general specification of competition, market demand and production cost. Forsymmetric oligopoly, we first derive formulas for marginal welfare losses from unit and ad valoremtaxation,
M C t and M C v , using the unit tax pass-through rate ρ t and the ad valorem tax pass-through semi-elasticity ρ v (Proposition 1) as well as the formulas for tax incidence, I t and I v (Proposition 2). We then show that ρ v can be expressed in terms of ρ t (Proposition 3). Theserelationships are used to derive sufficient statistics for M C t and M C v (Proposition 4). The pass-through is also characterized, generalizing Weyl and Fabinger’s (2013) formula (Proposition 5). Inthe case of price or quantity competition, we explain how ρ t and ρ v can be written only in termsof the demand elasticities, the demand curvatures, and the marginal cost elasticity (Propositions6 and 7). We have discussed the relationships to other quantities of interest, as well as illustrativespecial cases.The second part of the paper extends the results beyond the two-dimensional taxation problem.Specifically, we show that the previous results have a very natural generalization to a generalspecification of the tax revenue function as a function parameterized by a vector of tax parameters(Propositions 8, 9, 10, and 11). We further discuss an extension of our analysis to the case ofasymmetric oligopoly, where the firms face different costs and possibly also different taxes (Section5). , In addition, we provide a generalization of our results to the case of changes in bothproduction costs and taxes (Section 6).As already mentioned above, it would be possible to extend our analysis to the case of supply By allowing (constant) asymmetric marginal costs, Anderson, de Palma, and Kreider (2001b) show that underquantity competition with homogeneous products (i.e., Cournot competition), ad valorem taxation is still preferableto unit taxation, although they were not able to verify if the same conclusion held under quantity competitionwith product differentiation. However, Anderson, de Palma, and Kreider (2001b) discuss a specific demand system(with perfectly inelastic individual demand) under which unit taxation is preferable to ad valorem taxation if therequired tax revenue is sufficiently high. We conjecture that one could obtain further generalization by allowingthe conduct index θ to be firm-specific. See also Zimmerman and Carlson (2010) for a parametric analysis ofasymmetric firms. Interestingly, Tremblay and Tremblay (2017) study tax incidence in an asymmetric duopoly where one firmcompetes in price and the other firm competes in quantity, focusing on unit taxation. The pass-through rates canbe different for the two identical firms (in terms of demand and cost): the quantity-competing firm has a higherpass-through rate than the price-competing firm has. This is in contrast with the result that the pass-through rateunder price competition is generally higher under quantity competition. In addition, our methodologycould be utilized to study other important issues of pricing in general such as the welfare effects ofoligopolistic third-degree price discrimination (Adachi and Fabinger 2018). One may also study,for example, advertising pass-through (Draganska and Vitorino 2017). Free-riding, because ofthe spillover effect, may be more or less serious, depending on the conduct index and other relatedindices. Furthermore, it would be of interest to develop flexible, but analytically solvable examplesalong the lines of Fabinger and Weyl (2018).
A Appendix
A.1 Proofs and discussions for Section 2
A.1.1 Proof of Proposition 1
Using Equation (1) to substitute for mc , we first obtain a useful expression for the markup: p − mc = t + pv + p (1 − v ) ηθ . Now consider an infinitesimal change dt in the unit tax thatinduces a change dp in the equilibrium price and a change dq in the equilibrium quantity. Theseare related by dt = dp/ρ t = − η p dq/ ( q ρ t ). The corresponding change in social welfare per firmis dW = ( p − mc ) dq = t dq + vp dq + (1 − v ) pηθ dq , and the change in tax revenue per firm is dR = ( t + vp ) dq + vq dp + q dt = ( t + vp ) dq − vpη dq − ηp dq/ρ t . Combining these relationshipsgives the result M C t = − dWdR = − t + vp + (1 − v ) pηθt + vp − vpη − ρ t pη = (1 − v ) ηθ + tp + v ρ t η + vη − tp − v = (1 − v ) θ + (cid:15)τ ρ t + v − (cid:15)τ . Next, consider an infinitesimal change dv in the ad valorem tax that induces a change dp inthe equilibrium price and a change dq in the equilibrium quantity, related by dv = dp/ ( pρ v ) = Among many others, Ballard, Shoven, and Whalley (1985) study this issue for perfectly competitive markets. The firm’s demand can be modeled as q j = q j ( p , ..., p n ; a , ..., a n ), where a j is firm j ’s investment in advertising. η dq/ ( qρ v ). The change in social welfare per firm is again dW = ( p − mc ) dq = t dq + vp dq +(1 − v ) pηθ dq . The change in tax revenue per firm can be written as ( t + vp ) dq + vq dp + pq dv =( t + vp ) dq − vpη dq − pη dq/ρ v . Combining these relationships leads to the result M C t = − dWdR = − t + vp + (1 − v ) pηθt + vp − vpη − ρ v pη = (1 − v ) ηθ + tp + v ρ v η + vη − tp − v = (1 − v ) θ + (cid:15)τ ρ v + v − (cid:15)τ . A.1.2 Intuition behind Proposition 1
The intuition behind Proposition 1 for the case of unit taxation can explained as follows. Theargument for ad valorem taxation is analogous. First, the firm’s per-output profit margin is de-composed into two parts: (1) tax payment, t + vp = pτ and (2) surplus from imperfect competition,(1 − v ) pηθ . Under imperfect competition, the effects of an increase in unit tax, dt , on the socialwelfare can be written as dW = ( p − mc ) dq , which implies that the firm’s per-output profit marginserves as a measure for welfare change . On the other hand, the effects of an increase in unittax, dt , on the tax revenue are: dR = q dt (cid:124)(cid:123)(cid:122)(cid:125) (1) > + vq dp (cid:124) (cid:123)(cid:122) (cid:125) (2) > + ( t + vp ) dq (cid:124) (cid:123)(cid:122) (cid:125) (3) < , where term (1) expresses (direct) gain, multiplied by the output q , and term (2) shows (indirect)gain, due to the associated price increase, multiplied by vq , whereas term (3) is the part thatexhibits (indirect) loss from the output reduction for both unit tax revenue and ad valorem taxrevenue. Now recall that dp = ρ t dt and pηdq = − qdp . Thus, qdt = qdp/ρ t = − ( pη/ρ t ) dq and vqdp = − ( vqp/q ) ηdq = − ( vpη ) dq , which implies that dR = − ( pη/ρ ) dq − ( vpη ) dq + ( t + vp ) dq = [( − pη/ρ t ) (cid:124) (cid:123)(cid:122) (cid:125) (1) > + ( − vpη ) (cid:124) (cid:123)(cid:122) (cid:125) (2) > + ( t + vp ) (cid:124) (cid:123)(cid:122) (cid:125) (3) < ] . The welfare change dW = ( p − mc ) dq is further decomposed into: dW = − qdp + { pdq + qdp − [ qdt + vqdp + ( t + vp ) dq ] − mc · dq } + [ qdt + vpdp + ( t + vp ) dq ]= − qdp (cid:124) (cid:123)(cid:122) (cid:125) dCS + { [(1 − v ) p − t ] dq + [(1 − v ) dp − dt ] q − mc · dq (cid:124) (cid:123)(cid:122) (cid:125) } + dP S [ qdt + vpdp + ( t + vp ) dq (cid:124) (cid:123)(cid:122) (cid:125) ] , dR where dP S can be further simplified (see below). M C t are expressed as follows: M C t = (1 − v ) ηθ + τ (cid:124) (cid:123)(cid:122) (cid:125) welfare loss expressed by the profit margin (cid:18) ρ t + v (cid:19) η (cid:124) (cid:123)(cid:122) (cid:125) revenue gain + ( − τ ) (cid:124) (cid:123)(cid:122) (cid:125) revenue loss . A.1.3 Proof of Proposition 2
The impact of a change dt in the tax t on consumer surplus (per firm) is dCS = − qdp = − qρ t dt .The impact on producer surplus is dP S = d [(1 − v ) pq − c ( q ) − tq ] = − q dt + (1 − v ) p dq + (1 − v ) qdp − mc dq − t dq, ⇔ dP S = − qdt + (1 − v ) qρ t dt + [(1 − v ) p − mc − t ] dq. Substituting for mc from Equation (1) as mc = (1 − v ) (1 − ηθ ) p − t gives dP S = − qdt + (1 − v ) qρ t dt + (1 − v ) ηθpdq = − qdt + (1 − v ) qρ t dt − (1 − v ) θqdp, ⇔ dP S = − qdt + (1 − v ) qρ t dt − (1 − v ) θqρ t dt = − [1 − (1 − v ) (1 − θ ) ρ t ] q dt. The reciprocal of the incidence ratio is1 I t = dP SdCS = (1 − v ) (1 − θ ) qρ t − q − qρ t = 1 ρ t − (1 − v ) (1 − θ ) . For infinitesimal changes in ad valorem taxes, we proceed analogously. The change in consumersurplus is dCS = − qdp = − qpρ v dv. For the change in producer surplus we have dP S = d ((1 − v ) pq − c ( q ) − tq ) = − pq dv + (1 − v ) p dq + (1 − v ) qdp − mc dq − t dq. dP S = − pq dv + (1 − v ) p dq + (1 − v ) qdp − mc dq − t dq,dP S = − pq dv + (1 − v ) qpρ v dv − (1 − v ) θqpρ v dv = [(1 − v ) (1 − θ ) ρ v − qp dv. The reciprocal of the incidence ratio then becomes1 I t = dP SdCS = (1 − v ) (1 − θ ) ρ v q − q − qρ v = 1 ρ v − (1 − v ) (1 − θ ) . A.1.4 Intuition behind Proposition 2
The intuitive reasoning behind Proposition 2 can be provided as follows. First, the effects of anincrease in unit tax, dt , on the producer surplus can be decomposed into the following five parts: dP S = [( − q dt ) (cid:124) (cid:123)(cid:122) (cid:125) (1) < + (1 − v ) p dq (cid:124) (cid:123)(cid:122) (cid:125) (2) < ] + [(1 − v ) q dp (cid:124) (cid:123)(cid:122) (cid:125) (3) > + ( − mc dq ) (cid:124) (cid:123)(cid:122) (cid:125) (4) > + ( − t dq ) (cid:124) (cid:123)(cid:122) (cid:125) (5) > ] , where term (1) shows the (direct) loss from an increase in unit tax: the tax increase multiplied bythe output q , and term (2) is another (indirect) loss from a reduction in production, multiplied bythe ad valor em tax adjusted unit price (1 − v ) p , whereas term (3) corresponds to the (direct) gainfrom the associated price increase, mitigated by (1 − v ), due to the ad valorem tax, multiplied bythe output q , and finally terms (4) and (5) are (indirect) gains from cost savings by the outputreduction, dq , and from unit tax saving by the output reduction, dq , respectively. Note here thatthe equation above is rewritten as dP S = [ − q dt (cid:124) (cid:123)(cid:122) (cid:125) (1) < + (1 − v ) q dp (cid:124) (cid:123)(cid:122) (cid:125) (3) > ] + [(1 − v ) p − ( mc + t ) (cid:124) (cid:123)(cid:122) (cid:125) marginal cost ] dq. mc + t , is equal to the marginal benefit,(1 − v ) p [1 − ηθ ], which implies dP S = [ − q dt (cid:124) (cid:123)(cid:122) (cid:125) (1) < + (1 − v ) q dp (cid:124) (cid:123)(cid:122) (cid:125) (3) > ] + [(1 − v ) p ] ηθ dq. Under perfect competition, part (2) is equal to the sum of parts (4) and (5), and thus only parts(1) and (3) survive. However, under imperfect competition, the marginal cost is less than (1 − v ) p ,thus part (2) is greater than the sum of parts (4) and (5). The third term in the equation abovenow expresses the difference between part (2) and the sum of parts (4) and (5). Now, recall that dp = ρ t dt and pηdq = − qdp . Thus, dP S = [ − q dt + (1 − v ) qρ t dt ] − (1 − v ) qθ dp = [ − q dt + (1 − v ) qρ t dt ] − (1 − v ) qθρ t dt = [ − − v ) ρ t − (1 − v ) θρ t ] q dt = [ − (cid:124)(cid:123)(cid:122)(cid:125) (1) < + (1 − v )(1 − θ ) ρ t (cid:124) (cid:123)(cid:122) (cid:125) (3) −{ (2) − [(4)+(5)] } ≷ ] q dt. On the other hand, dCS = − ρ t ( qdt ). Thus, while it is always the case that dCS <
0, it is possiblethat dP S > A.1.5 Proof of Proposition 3
Let us consider a simultaneous infinitesimal change dt and dv in the taxes t and v that leaves theequilibrium price (and quantity) unchanged, which requires the effective marginal cost ( t + mc ) / (1 − v )in Equation (1) to remain the same. This implies the comparative statics relationship ∂∂t (cid:18) t + mc − v (cid:19) dt + ∂∂v (cid:18) t + mc − v (cid:19) dv = 0 ⇒ dt − v + t + mc (1 − v ) dv = 0 ⇒ dt = − t + mc − v dv. Note that here we do not need to take derivatives of mc even though it depends on q , simplybecause by assumption the quantity is unchanged. The total induced change in price, which One can also define social incidence by SI t ≡ dW/dP S and SI v in association with a small change in t and v , respectively. Hereafter, we focus on M C t and M C v as measures of welfare burden in society, and I t and I v as measures of loss in consumer welfare. We provide general formulas for social incidence in the context ofmulti-dimensional pass-through after Section 4. dp = ρ t dt + ρ v p dv , must equal zero in this case, implying theresult ρ t dt + ρ v p dv = 0 ⇒ − t + mc − v ρ t dv + ρ v p dv = 0 ⇒ ρ v = (1 − ηθ ) ρ t ⇒ ρ v = (cid:15) − θ(cid:15) ρ t . A.1.6 Intuition behind Proposition 3
To understand this proposition (3) intuitively, note that to keep prices and quantities constant, ∆t and ∆v must satisfy: t + ∆t + mc − ( v + ∆v ) = t + mc − v . Thus, the relative ∆t that must be offset by a reduction − ∆v equal to ( t + mc ) / (1 − v ): ∆t = − ( t + mc ) ∆v/ (1 − v ), which, together with ρ t dt + ρ v p dv = 0, leads to ( t + mc ) ρ t / [(1 − v ) p ] = ρ v .Now, recall the Lerner rule: 1 − t + mc (1 − v ) p (cid:124) (cid:123)(cid:122) (cid:125) per-price marginal cost = ηθ, which implies that (1 − ηθ ) ρ t = ρ v , as Proposition 3 claims. Now, θ/(cid:15) = 1 − ρ v /ρ t implies that ρ v ≤ ρ t ≤ (1 − /(cid:15) ) ρ v . A.1.7 Intuition behind Proposition 4
To gain a perspective in Proposition 4, recall from Proposition 1 that
M C t = (1 − v ) ηθ + τ (cid:124) (cid:123)(cid:122) (cid:125) welfare loss expressed by the profit margin (cid:18) ρ t + v (cid:19) η (cid:124) (cid:123)(cid:122) (cid:125) revenue gain + ( − τ ) (cid:124) (cid:123)(cid:122) (cid:125) revenue loss . M C t = (1 − v ) (cid:18) − ρ v ρ t (cid:19) + τ (cid:124) (cid:123)(cid:122) (cid:125) welfare loss expressed by the profit margin (cid:18) ρ t + v (cid:19) η (cid:124) (cid:123)(cid:122) (cid:125) revenue gain + ( − τ ) (cid:124) (cid:123)(cid:122) (cid:125) revenue loss . Of course, it is true that θ is expressed by the empirical measures such as θ = (1 − ρ v /ρ t ) (cid:15) . Forexample, in the case of the assumption of Cournot competition, researchers often may observe thenumber n of firms and conclude that the value of conduct index is θ = 1 /n . However, even in thecase of homogeneous products, the “true” conduct may be higher than 1 /n due to such reasonsas collusion. Proposition 4 above circumvents this difficulty in estimating
M C t and M C v . Conversely, one would be able to estimate θ using the proposition above once (cid:15) , ρ t , and ρ v areestimated. A.1.8 Proof of Proposition 5
Here we provide a proof of Proposition 5, as well as related intuitive arguments. Consider thecomparative statics with respect to a small change dt in the per-unit tax t . Following Weyl andFabinger (2013, p. 538), we define ms ≡ − p (cid:48) q : this is the negative of marginal consumer surplus.Then, the Learner condition becomes: p − t + mc − v (cid:124) (cid:123)(cid:122) (cid:125) markup = θ ms (cid:124) (cid:123)(cid:122) (cid:125) CS , See Miller and Weinberg (2017) for an empirical study of the possibility of oligopolistic collusion in a differentmanner from directly estimating the conduct parameter. Similarly, the incidence of a unit tax is expressed as1 I t = 1 ρ t − (1 − v ) (cid:20) (1 − (cid:15) ) + ρ v ρ t (cid:15) (cid:21) , and analogously for the case of an ad valorem tax. CS is consumer surplus for the infra-marginal consumers. Importantly, θms measures howmuch consumer surplus rises for a small increase in output, and it is largest under monopoly. Nowconsider a small change in unit tax expressed by dt >
0. Then, in equilibrium, dp − dt + dmc − v = d ( θ ms ) ⇔ (1 − v )[ dp (cid:124)(cid:123)(cid:122)(cid:125) > − d ( θ ms ) (cid:124) (cid:123)(cid:122) (cid:125) < ] (cid:124) (cid:123)(cid:122) (cid:125) change in marginal benefit = dt (cid:124)(cid:123)(cid:122)(cid:125) > + dmc (cid:124)(cid:123)(cid:122)(cid:125) < (cid:124) (cid:123)(cid:122) (cid:125) change in effective marginal cost Thus, using dt = dp/ρ t , the equation is rewritten as ρ t = 1(1 − v ) [ dp + ( − d ( θ ms ))] (cid:124) (cid:123)(cid:122) (cid:125) (1) >
0: revenue increase + ( − dmc ) (cid:124) (cid:123)(cid:122) (cid:125) (2) >
0: cost savings dp.
Now, consider term (1). Note first d ( θms ) = ( θms ) (cid:48) dq so that d ( θms ) = − q(cid:15) ( θms ) (cid:48) dp/p because by definition dq = − q(cid:15)dp/p . Here, for a small increase dt > d ( θ ms ) (cid:124) (cid:123)(cid:122) (cid:125) < = − q(cid:15) (cid:124)(cid:123)(cid:122)(cid:125) > ( θ ms ) (cid:48) dpp (cid:124)(cid:123)(cid:122)(cid:125) > so that ( θ ms ) (cid:48) >
0. By definition, ms ≡ − p (cid:48) q = ηp . Thus, d ( θms ) = − q(cid:15) ( θηp ) (cid:48) dp/p . Now notethat ( θηp ) (cid:48) = ( θη ) (cid:48) p + ( θη ) p (cid:48) . Thus, d ( θ ms ) = − q(cid:15) [( θη ) (cid:48) p + ( θη ) p (cid:48) ] dpp ⇔ d ( θ ms ) = − q(cid:15) ( θη ) (cid:48) dp + ( − q(cid:15) ( θη ) p (cid:48) dp/p ) = [ θη − q(cid:15) ( θη ) (cid:48) ] dp > . Next, consider term (2). A change in the marginal cost, dmc , is expressed in terms of dp by dmc = − [(1 − v ) θη + 1 − τ ] χ(cid:15) dp <
0. To see this, note first that dmc = χmc · ( dq/q )= − ( χ(cid:15) mc ) ( dp/p ). Then, mc in this expression can be eliminated rewriting p − θ ms = ( mc + t ) / (1 − v ) ⇒ mc = (1 − v ) ( p + θqp (cid:48) ) − t = (1 − v ) (1 − θη ) p − t , which leads to dmc = − [(1 − v ) (1 + θη ) − t/p ] χ(cid:15) dp . Then, in terms of the per-unit revenue burden, τ ≡ v + t/p , that is, dmc = − [(1 − v ) (1 − θη ) − + v ] χ(cid:15) dp = − [ − (1 − v ) θη + 1 − τ ] χ(cid:15) dp . Finally, using the expressions for dmc and d ( θms ), ρ t = dp (1 − v ) [ dp − d ( θ ms )] − dmc = 1(1 − v ) [(1 − θη ) + ( θη ) (cid:48) (cid:15)q ] (cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (1 − τ ) (cid:15)χ − (1 − v ) θχ (cid:124) (cid:123)(cid:122) (cid:125) cost savings . ⇔ ρ t = 11 − v − θη ) + ( θη ) (cid:48) (cid:15)q ] (cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − θ + 1 − τ − v (cid:15) (cid:21) χ (cid:124) (cid:123)(cid:122) (cid:125) cost savings . A.1.9 Relationship to Weyl and Fabinger (2013)
It can be verified that our formula for ρ t above is a generalization of Weyl and Fabinger’s (2013,p. 548) Equation (2): ρ = 11 + (cid:15) D − θ(cid:15) S + θ(cid:15) θ + θ(cid:15) ms , where (cid:15) θ ≡ θ/ [ q · ( θ ) (cid:48) ], (cid:15) ms ≡ ms/ [ ms (cid:48) q ] ( ms ≡ − p (cid:48) q is defined in the proof of Proposition 5 justabove), and (cid:15) D and (cid:15) S here are our (cid:15) and 1 /χ , respectively. First, the denominator in our formulais rewritten as:1 − ( η + χ ) θ + (cid:15)q ( θη ) (cid:48) + 1 − τ − v (cid:15)χ = 1 + − τ − v (cid:15) D − θ(cid:15) S + θ(cid:15) θ + θ · (cid:18) − (cid:15) D + η (cid:48) (cid:15) D q (cid:19) because ( θη ) (cid:48) (cid:15)q = ( θ (cid:48) η + θη (cid:48) ) (cid:15)q = (cid:20) θq(cid:15) θ η + θη (cid:48) (cid:21) (cid:15)q = θ(cid:15) θ + θη (cid:48) (cid:15)q. Next, since η = − qp (cid:48) /p , it is verified that η (cid:48) = −{ p (cid:48) p + qpp (cid:48)(cid:48) − q [ p (cid:48) ] } /p , implying that η (cid:48) (cid:15) D q = p (cid:48) p + qpp (cid:48)(cid:48) − q [ p (cid:48) ] p · pp (cid:48) q · q = 1 (cid:15) D + (cid:18) p (cid:48)(cid:48) p (cid:48) q (cid:19) , where 1 + p (cid:48)(cid:48) q/p is replaced by 1 /(cid:15) ms because ms ≡ − p (cid:48) q and thus ms (cid:48) = − ( p (cid:48)(cid:48) q + p (cid:48) ). Then, it isreadily verified that1 − ( η + χ ) θ + (cid:15)q ( θη ) (cid:48) + 1 − τ − v (cid:15)χ = 1 + − τ − v (cid:15) D − θ(cid:15) S + θ(cid:15) θ + θ(cid:15) ms .
48n summary, Weyl and Fabinger’s (2013, p. 548) original Equation (2) is generalized to ρ = 11 − v
11 + − τ − v (cid:15) D − θ(cid:15) S + θ(cid:15) θ + θ(cid:15) ms with non-zero initial ad valorem tax, which is equivalent to our formula for ρ t : ρ t = 11 − v
11 + − τ − v (cid:15)χ − ( η + χ ) θ + (cid:15)q ( θη ) (cid:48) . A.1.10 Comparison of perfect and oligopolistic competition
One can further interpret the formula for ρ t in comparison to the case of perfect competition (withzero initial taxes), when the unit tax pass-through rate is given by (see Weyl and Fabinger 2013,p. 534): ρ t = 1 / (1 + (cid:15)χ ). An analogous argument can be made for ρ v as well.First, through the term − ( η + χ ) θ in the denominator of ρ t in Proposition 5, as competitive-ness becomes fiercer (i.e., a lower θ ) the pass-through rate ρ t lowers , that is, the pass-throughbecomes smaller as the degree of competition becomes closer to perfect competition. This isinterpreted as the negative effect of competitiveness on the pass-through rate, via the first-ordercharacteristics of demand and supply, captured by η and χ , respectively. However, through the other term (cid:15)q ( θη ) (cid:48) = − (cid:15)q ( − θη ) (cid:48) , competitiveness raises the pass-through rate ρ t . To see this, suppose that η is close to a constant. Then, − (cid:15)q ( − θη ) (cid:48) = − q ( − θ ) (cid:48) ,which implies that a larger ( − θ ) (cid:48) ≡ − ∂θ/∂q > higher value of the pass-through rate. With an abuse of notation, this situation is interpreted as the case when − ∂q/∂θ issmall: the effect of imperfect competition on the output reduction is small, implying less distor-tion , an important feature if the degree of competition is close to perfect competition.The argument so far is clearer if θ , as often assumed, is a constant. Then, the second term is Here, with fixed θ , the denominator becomes smaller, and thus, the pass-through rate becomes larger as thedemand becomes inelastic (i.e., η becomes larger, although η cannot be too large; recall the restriction, η < /θ )or the supply becomes inelastic (i.e., χ becomes larger). − (cid:15)q ( − θη ) (cid:48) = − (cid:15)qθ ( − η ) (cid:48) = θ ( η + 1 /(cid:15) ms ) so that ρ t = 11 − v (cid:2) − τ − v (cid:15)χ (cid:3) + (cid:16) (cid:15) ms − χ (cid:17) θ . Thus, if the marginal cost is constant ( χ = 0), ρ t becomes larger as the degree of competitionbecomes closer to perfect competition. Here, with fixed θ , the pass-through rate is also larger as1 /(cid:15) ms becomes smaller. Recall that 1 /(cid:15) ms = ( ∆ms/ms ) / ( ∆q/q ) measures how quick the marginalsurplus lowers as a response to a decrease in output q . Thus, a lower 1 /(cid:15) ms is associated with lessdistortion . Overall, Weyl and Fabinger’s (2013, p. 548) Equation (2) and our formula for ρ t showhow it is influenced by the industry’s competitiveness which is captured by the conduct index. A.1.11 Application to exchange rate changes
Let us also point out that the exchange rate pass-through can be included naturally in ourframework of Section 2. Suppose that domestic firms in a country of interest use some im-ported inputs for production. For concreteness, let us specify the profit function of firm j as π j = [(1 − v ) p j − t ] q j − (1 + a e ) c ( q j ), where the constant coefficient a measures the importanceimported inputs and e > π j = (1 + ae ) (cid:20)(cid:18) − v ae p j − t ae (cid:19) q j − c ( q j ) (cid:21) . Since the first factor on the right-hand side is con-stant, the firm will behave as if its profit function was simply ˜ π j = (cid:2) (1 − ˜ v ) p j − ˜ t (cid:3) q j − c ( q j ), with˜ v ≡ ( v + ae ) / (1 + ae ) and ˜ t ≡ t/ (1 + ae ). By utilizing the explicit expressions for the derivatives ∂ ˜ v/∂e = ( a − v ) / (1 + ae ) and ∂ ˜ t/∂e = − at/ (1 + ae ) , one can analyze the effect of a change inthe exchange rate e on social welfare. Note that this is simply interpreted as the cost pass-through as well (see the references in Footnote 42 for empirical studies). Alternatively, one may use theresults of Section 6 to study the consequences of exchange rate movements. See, e.g., Feenstra (1989); Feenstra, Gagnon, and Knetter (1996); Yang (1997); Campa and Goldberg (2005);Hellerstein (2008); Gopinath, Itskhoki, and Rigobon (2010); Goldberg and Hellerstein (2013); Auer and Schoenle(2016); and Chen and Juvenal (2016) for empirical studies of exchange rate pass-through. .1.12 Oligopoly with multi-product firms Here, we argue that the results obtained in Sections 2 and 3 can be extended to the case ofmulti-product firms just by a reinterpretation of the same formulas (without modifying them). Assume there are n p product categories, and the demand for firm j ’s k -th product is given by q jk = q jk ( p , p , .., p n ), where p j = ( p j , ..., p jk , ..., p jK ) for each j = 1 , , ..., n . The firms aresymmetric, and for each firm, the product it produces are also symmetric. The firm’s profit perproduct is π j = 1 n p n p (cid:88) k =1 ((1 − v ) p jk q jk − tq jk − c ( q jk )) . We work with an equilibrium in which any firm j sets a uniform price p j for all of its products: p jk = p j , and consequently sells an amount q j of each of them: q jk = q j . In this case, the profitper product equals π j = (1 − v ) p j q j − tq j − c ( q j ) , which is formally the same as for single-productfirms. For this reason, we can identify the prices p j and quantities q j of Section 2 with the prices p j and quantities q j introduced here in this paragraph. The discussion in Section 2 was generaland applies to this case of symmetric oligopoly with multi-product firms as well. We can use thesame definitions for the variables of interest, including the industry demand elasticity (cid:15) and theconduct index θ .The definitions and results for the cases of price competition and quantity competition dis-cussed Section 3 are also applicable here. It may be useful to translate some of the most importantvariables of that discussion into product-level variables. For derivatives of the direct demand sys- Lapan and Hennessy (2011) study unit and ad valorem taxes in multi-product Cournot oligopoly. Alexan-drov and Bedre-Defolie (2017) also study cost pass-through of multi-product firms in relation to the Le Chate-lier–Samuelson principle. See, e.g., Armstrong and Vickers (2018) and Nocke and Schutz (2018) for recent studies of multi-productoligopoly. For brevity, we do not explicitly discuss the standard conditions for the existence and uniqueness of non-cooperative Nash equilibria of the different underlying oligopoly games. ξ ≡ ∂q jk ∂p jk , ξ , ≡ ∂q jk ∂p jk (cid:48) ,ξ ≡ ∂q jk ∂p jk , ξ , ≡ ∂q jk ∂p jk ∂p jk (cid:48) , ξ , ≡ ∂q jk ∂p jk (cid:48) , ξ , , ≡ ∂q jk ∂p jk (cid:48) ∂p jk (cid:48)(cid:48) , ˜ ξ ≡ ∂q jk ∂p jk ∂p j (cid:48) k ˜ ξ , ≡ ∂q jk ∂p jk ∂p j (cid:48) k (cid:48) , ˜ ξ , ≡ ∂q jk ∂p jk (cid:48) ∂p j (cid:48) k (cid:48) , ˜ ξ , , ≡ ∂q jk ∂p jk (cid:48) ∂p j (cid:48) k (cid:48)(cid:48) , where the derivatives are evaluated at the fully symmetric point, where any p jk equals the commonvalue p . For specific choices of the demand system, these derivatives can be closely related. Forexample, if the substitution pattern between two goods produced by two different firms does notdepend on the identity of the goods, then ˜ ξ = ˜ ξ , = ˜ ξ , = ˜ ξ , , . In terms of these derivatives,we can write (cid:15) F = − pq (cid:0) ξ + ( n p − ξ , (cid:1) ,(cid:15) = − pq (cid:16) ξ + ( n p − ξ , + ( n −
1) ˜ ξ + ( n −
1) ( n p −
1) ˜ ξ , (cid:17) ,α F = p q (cid:15) F (cid:0) ξ + ( n p − (cid:0) ξ , + ξ , + ( n p − ξ , , (cid:1)(cid:1) ,α C = ( n − p q (cid:15) F (cid:16) ˜ ξ + ( n p −
1) (˜ ξ , + ˜ ξ , + ( n p −
2) ˜ ξ , , ) (cid:17) . These can be substituted into the results of Proposition 6 to find the pass-through and the marginalcost of public funds under price competition.For the inverse demand system the analogous definitions are ζ ≡ ∂q jk ∂p jk , ζ , ≡ ∂q jk ∂p jk (cid:48) ,ζ ≡ ∂q jk ∂p jk , ζ , ≡ ∂q jk ∂p jk ∂p jk (cid:48) , ζ , ≡ ∂q jk ∂p jk (cid:48) , ζ , , ≡ ∂q jk ∂p jk (cid:48) ∂p jk (cid:48)(cid:48) , ˜ ζ ≡ ∂q jk ∂p jk ∂p j (cid:48) k ˜ ζ , ≡ ∂q jk ∂p jk ∂p j (cid:48) k (cid:48) , ˜ ζ , ≡ ∂q jk ∂p jk (cid:48) ∂p j (cid:48) k (cid:48) , ˜ ζ , , ≡ ∂q jk ∂p jk (cid:48) ∂p j (cid:48) k (cid:48)(cid:48) . In this notation, the first subscript counts the derivatives with respect to the relevant price with index k ,the second subscript counts the derivatives with respect to the price with index k (cid:48) distinct from k , and the thirdsubscript counts derivatives respect to the price with index k (cid:48)(cid:48) distinct from both k and k (cid:48) . Further, ξ correspondsto derivatives with respect to prices charged by the same firm j , while ˜ ξ corresponds to derivatives with respect toprices charged by firm j and some other firm j (cid:48) . η F = − qp (cid:0) ζ + ( n p − ζ , (cid:1) ,η = − qp (cid:16) ζ + ( n p − ζ , + ( n −
1) ˜ ζ + ( n −
1) ( n p −
1) ˜ ζ , (cid:17) ,σ F = q p η F (cid:0) ζ + ( n p − (cid:0) ζ , + ζ , + ( n p − ζ , , (cid:1)(cid:1) ,σ C = ( n − q p η F (cid:16) ˜ ζ + ( n p −
1) (˜ ζ , + ˜ ζ , + ( n p −
2) ˜ ζ , , ) (cid:17) . can be substituted into the results of Proposition 7 to find the pass-through and marginal cost ofpublic funds under price competition. A.2 Proofs and discussions for Section 3
A.2.1 Relationship between elasticities and curvatures under the direct demandsystem
This relationship can be verified as follows. The elasticity of the function (cid:15) F ( p ) equals the sum ofthe elasticities of the three factors it is composed of:1 (cid:15) F ( p ) p ddp (cid:15) F ( p ) = 1 p p ddp p + q ( p ) p ddp q ( p ) + (cid:18) ∂q j ( p ) ∂p j (cid:19) − | p =( p,...,p ) p ddp (cid:18) ∂q j ( p ) ∂p j | p =( p,...,p ) (cid:19) . The first elasticity on the right-hand side equals 1, the second elasticity equals (cid:15) ( p ), and the thirdelasticity equals − α F ( p ) − α C ( p ), since p ddp ∂q j ( p ) ∂p j | p =( p,...,p ) = p ∂ q j ( p ) ∂p j | p =( p,...,p ) + ( n − p ∂ q j ( p ) ∂p j ∂p j (cid:48) | p =( p,...,p ) . Note that α is weakly positive ( weakly negative ) if the industry demand is convex ( concave ),and α F is weakly positive (weakly negative) if the demand as a function of firm j ’s own price isconvex (concave). Hence, both α and α F measure the degree of convexity in the demand functionfor an industry-wide price change and for an individual firm’s price change, respectively. Note alsothat ∂ ( ∂q j /∂p j ) /∂p j (cid:48) in α C measures the effects of firm j ’s price change on how many consumersrival j (cid:48) loses if it raises its price. If this is negative (positive), then firm j (cid:48) loses more (less)53onsumers by its own price increase for a higher value of p j . Thus, because ∂q j /∂p j (cid:48) is positive inthe expression for α C , a higher α C also indicates more competitiveness in the industry. It is alsoexpected that the industry is more competitive if α and α F are higher. In effect, the equilibriumprice is characterized by (cid:15) F . However, a policy change around equilibrium is also affected bythe curvatures, which measure “second-order competitiveness” around the equilibrium. However,Proposition 6 in the text shows that α is the only curvature that determines the pass-throughrates. A.2.2 Relationship between elasticities and curvatures under the inverse demandsystem
In analogy with Appendix A.2.1, the elasticity of the function η F ( q ) is the sum of the elasticitiesof the three factors it is composed of, which are equal to 1, η ( q ), and − σ F ( q ) − σ C ( q ).Now, σ is weakly positive ( weakly negative ) if the industry’s inverse demand is convex ( con-cave ), and σ F is weakly positive (weakly negative) if the inverse demand as a function of firm j ’sown output is convex (concave). Here, concavity, not convexity, is related to a sharp reductionin price in response to an increase in firm j ’s output. Thus, − σ and − σ F measure “second-ordercompetitiveness” of the industry, which characterizes the responsiveness of the equilibrium outputwhen a policy is changed. Note also that ∂ ( ∂p j /∂q j ) /∂q j (cid:48) in σ C measures the effects of firm j ’soutput increase on the extent of rival ( j (cid:48) )’s price drop if it increases its output. If this is negative(positive), then firm j (cid:48) expects a large (little) drop in its price by increasing its output for a highervalue of q j . Because ∂p j /∂q j (cid:48) is negative in the expression for σ C , a lower σ C or a higher − σ C indicates more competitiveness in the industry. In sum, while 1 /η F characterizes competitivenessthat determines the level of the equilibrium quantity, − σ , − σ F , and − σ C determine competitive-ness that characterizes the responsiveness of the equilibrium output by a policy change. However,similarly to the case of price competition, Proposition 7 in the text shows that σ is the onlycurvature that determines the pass-through rates.54 .2.3 Proof of Proposition 6 Since in the case of price setting θ = (cid:15)/(cid:15) F = 1 / ( η(cid:15) F ), we have ( η + χ ) θ = (1 + (cid:15)χ ) /(cid:15) F and( θη ) (cid:48) (cid:15)q = (cid:15)q ddq ( θη ) = (cid:15)q ddq ( (cid:15) − F ) = − (cid:15) − F (cid:15)q ddq (cid:15) F = (cid:15) − F p ddp (cid:15) F = (1 + (cid:15) − α(cid:15)/(cid:15) F ) /(cid:15) F , where in thelast equality we utilize the expression for the elasticity of (cid:15) F ( p ) and α F + α C = α(cid:15)/(cid:15) F fromSubsection 3.1. Substituting these into the expression for ρ t in Proposition 5 gives ρ t = 11 − v − (cid:15) F (1 + (cid:15)χ ) + (cid:15) F (cid:16) (cid:15) − α(cid:15)(cid:15) F (cid:17) + − τ − v (cid:15)χ , which is equivalent to the expression for ρ t in the proposition. Since for price setting θ = (cid:15)/(cid:15) F , therelationship in Proposition 3 implies ρ v = ( (cid:15) − θ ) ρ t /(cid:15) = ( (cid:15) F − ρ t /(cid:15) F , which leads to the desiredexpression for ρ v . A.2.4 Intuition behind Proposition 6
The intuition for ρ t in Proposition 6 is as follows. First, recall from Proposition 5 that ρ t = 11 − v − θη )+( θη ) (cid:48) (cid:15)q ] (cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − τ − v (cid:15) − θ (cid:21) χ (cid:124) (cid:123)(cid:122) (cid:125) cost savings . Then, with θ = (cid:15)/(cid:15) F , 1 − θη = 1 − /(cid:15) F , ( θη ) (cid:48) (cid:15)q = (1 + (cid:15) − α(cid:15)/(cid:15) F ) /(cid:15) F , the equality above isrewritten as ρ t = 11 − v (cid:20)(cid:18) − (cid:15) F (cid:19) + 1 + (cid:15) − α(cid:15)/(cid:15) F (cid:15) F (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − τ − v − (cid:15) F (cid:21) (cid:15)χ (cid:124) (cid:123)(cid:122) (cid:125) cost savings = 11 − v (cid:20) − α/(cid:15) F ) (cid:15)(cid:15) F (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − τ − v − (cid:15) F (cid:21) (cid:15)χ (cid:124) (cid:123)(cid:122) (cid:125) cost savings . To further facilitate the understanding the connection of this result for to Proposition 5,55onsider the case of zero initial taxes ( t = v = τ = 0). Then, Proposition 5 claims that ρ t = 11 + (cid:15)χ − θχ + [ − ηθ + (cid:15)q ( θη ) (cid:48) ] , whereas Proposition 6 shows that ρ t = 11 + (cid:15)χ − θχ + [ − (cid:15) · (cid:15)(cid:15) F + − α/(cid:15) F ) (cid:15)(cid:15) F ] = 11 + (cid:15)χ − θχ + (cid:16) − α(cid:15) F (cid:17) θ , because θ = (cid:15)/(cid:15) F . Here, the direct effect from − ηθ is canceled out by the part of the indirect effectfrom (cid:15)q ( θη ) (cid:48) . The new term, which appears as the fourth term in the denominator, shows howthe industry’s curvature affects the pass-through rate : as the demand curvature becomes larger(i.e., as the industry’s demand becomes more convex), then the pass-through rate becomes higher,although this effect is mitigated by the degree of competitiveness, θ . A.2.5 Proof of Proposition 7
In the case of quantity setting, θ = η F /η , so ( η + χ ) θ = (1 + χ/η ) η F and ( θη ) (cid:48) (cid:15)q = q ( η F ) (cid:48) /η =(1 + η − ση/η F ) η F /η , where in the last equality we utilize the expression for the elasticity of η F ( q ) and σ F + σ C = ση/η F from Subsection 3.1. Substituting these into the expression for ρ t inProposition 5 gives ρ t = 11 − v − (1 + η χ ) η F + η (cid:16) η − σηη F (cid:17) η F + − τ − v η χ , which is equivalent to the expression for ρ t in the proposition. Since θ = η F /η , Proposition3 implies ρ v = ( (cid:15) − θ ) ρ t /(cid:15) = (1 /η − η F /η ) ρ t η = (1 − η F ) ρ t , which can be used to verify theexpression for ρ v . 56 .2.6 Intuition behind Proposition 7 The intuition for ρ t in Proposition 6 is similar to the case of price competition. Recall again that ρ t = 11 − v − θη )+( θη ) (cid:48) (cid:15)q ] (cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − τ − v (cid:15) − θ (cid:21) χ (cid:124) (cid:123)(cid:122) (cid:125) cost savings . Then, θ = η F /η implies (1 /(cid:15) S − η ) θ = [(1 /(cid:15) S η ) − η F and ( θη ) (cid:48) ( q/η ) = q ( η F ) (cid:48) /η = (1 + η − σ F − σ C ) ( η F /η ). Thus, the equality above is rewritten as ρ t = 11 − v (cid:20) (1 − η F ) + 1 + η − ση/η F η η F (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − τ − v (cid:15) S η − η F (cid:15) S η (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) cost savings = 11 − v (cid:20) η F − σηη (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) revenue increase + (cid:20) − τ − v − η F (cid:21) (cid:15) S η (cid:124) (cid:123)(cid:122) (cid:125) cost savings . To further facilitate the understanding the connection of this result for to Proposition 5,consider the case of zero initial taxes ( t = v = τ = 0) again. Then, Proposition 7 shows that ρ t = 11 + (cid:15)χ − θχ + [ − η · η F η + (cid:16) η − ση F (cid:17) η F ] = 11 + (cid:15)χ − θχ + (cid:0) − σθ (cid:1) θ because θ = η F /η . Here, the term (1 − σ/θ ) θ demonstrates the effects of the industry’s inversedemand curvature, σ , on the pass-through rate : as the inverse demand curvature becomes larger(i.e., as the industry’s inverse demand becomes more convex), the pass-through rate becomeshigher. Interestingly, in contrast to the case of price competition, this effect is not mitigated bythe degree of competitiveness, θ . 57 .2.7 Equilibrium prices and outputs under price and quantity competition withthe linear demand The equilibrium price and output under price competition are obtained as p = 1 + t − v − ( n − µ , q = 1 − [1 − ( n − µ ] t − v − ( n − µ , and thus pq = 11 − [1 − ( n − µ ] t − v (cid:18) t − v (cid:19) , implying that (cid:15) = [1 − ( n − µ ] (cid:0) t − v (cid:1) − [1 − ( n − µ ] t − v , (cid:15) F = 1 + t − v − [1 − ( n − µ ] t − v . Similarly, the equilibrium price and output under quantity competition are given by p = − ( n − µ − ( n − µ + (1 + µ ) t − v − ( n − µ , q = (1 + µ ) 1 − [1 − ( n − µ ] t − v − ( n − µ , and thus pq = 11 − [1 − ( n − µ ] t − v (cid:18) − ( n − µ (1 + µ )[1 − ( n − µ ] + t − v (cid:19) , implying that η = 1 − [1 − ( n − µ ] t − v − ( n − µ µ + [1 − ( n − µ ] t − v , η F = 1 − [1 − ( n − µ ] t − v (1+ µ )[1 − ( n − µ ]1 − ( n − µ t − v . A.3 Proofs and discussions for Section 4
A.3.1 Proof of Proposition 8
Consider an infinitesimal tax change such that the equilibrium price (and therefore quantity) doesnot change: ˜ ρ · d T = 0 . Let us choose d T to have just two non-zero components: dT (cid:96) and dT (cid:96) (cid:48) .58his implies ˜ ρ T (cid:96) ˜ ρ T (cid:96) (cid:48) = − dT (cid:96) (cid:48) dT (cid:96) . (7)Since Equation (3) must hold both before and after the tax change, it must be the case that1 − τ − (1 − ν ) ηθ does not change, and in turn( − τ T (cid:96) + ν T (cid:96) ηθ ) dT (cid:96) + (cid:0) − τ T (cid:96) (cid:48) + ν T (cid:96) (cid:48) ηθ (cid:1) dT (cid:96) (cid:48) = 0 . Substituting for dT (cid:96) (cid:48) from this equation into Equation (7) and using the definition of pass-throughquasi-elasticities leads to the desired result. A.3.2 Proof of Proposition 9
The same type of reasoning as in the proof of Proposition 5 is useful in proving Proposition 9. Inparticular, comparative statics of Equation (3) with respect to a tax T (cid:96) leads to the desired resultafter utilizing the definitions above and eliminating marginal cost using, again, Equation (3). Thecalculation is a bit tedious but completely straightforward. A.3.3 Depreciating licenses
Here we discuss the relationship of exogenous competition to depreciating licenses in Weyl andZhang (2017). In the setup of Section 2 of Weyl and Zhang (2017), there are two agents, S and B (“seller” and “buyer”). Agent S holds an asset and declares a reservation value p , which influencesthe tax (“license fee”) ˜ qp the agent needs to pay to the government. Here ˜ q is the license taxrate (denoted τ in the original paper). Agent B may then purchase the asset at that price p fromagent S . The value for agent S is η + γ S , and for agent B it is η + γ B , for some common valuecomponent η . Here γ B is a random variable with CDF F ( γ B ) representing heterogeneity in B ’s value, which is not observed by S . As Weyl and Zhang (2017) show, the sale probability q (denoted the same way in the original paper) is then determined as the solution of P ( q ) = p , where P ( q ) ≡ F − (1 − q ) + η . Up to a constant, agent S ’s expected profit function (utility function) is The original paper considers η being determined by agent S at the very beginning. Here we focus on thesubgame after η has been determined. P ( q ) − η − γ S )( q − ˜ q ) or P ( q ) ( q − ˜ q ) − ( q − ˜ q ) mc , where we used the notation mc ≡ η + γ S . We recognize that this is exactly of the same form as the profit function in the case of monopolywith constant marginal cost mc subject to exogenous competition ˜ q and inverse demand function P ( q ). Nagoya UniversityUniversity of Tokyo
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