ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E16-10

Multi-Dimensional Pass-Through,
Incidence, and the Welfare Burden of
Taxation in Oligopoly(Previously titled: Pass-Through and the Welfare
Burden of Taxation in Differentiated Oligopoly)
byTakanori Adachi
Michal Fabinger

Revised version as of March 2017

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Multi-Dimensional Pass-Through, Incidence, and
the Welfare Burden of Taxation in Oligopoly∗
Takanori Adachi†

Michal Fabinger‡

February 18, 2017

Abstract
This paper studies welfare consequences of unit and ad valorem taxes inoligopoly with general demand, non-constant marginal costs, and a generalized type of competition. We present formulas providing connections between
marginal cost of public funds, tax incidence, unit tax pass-through, ad valorem tax pass-through, and other economic quantities of interest. We showthat there exists a simple, empirically relevant set of sufficient statistics forthe marginal cost of public funds, namely the pass-through and the industry
demand elasticity. Specializing to the case of price or quantity competition,
we show how marginal cost of public funds and pass-through may be expressedusing elasticities and curvatures of demand and inverse demand. These results apply also to symmetric oligopoly with multi-product firms. Finally,
we present a generalization with the tax revenue function specified as a general function parametrized by a vector of tax parameters. We define multidimensional generalizations of pass-through and show that they are crucial
for evaluating welfare changes in response to changes in taxation.

Keywords: Oligopoly; Unit tax; Ad valorem tax; Welfare; Pass-through; Elasticities; Curvatures; Sufficient statistics.
JEL classification numbers: D43; L11; L13.
∗

Adachi and Fabinger acknowledge a Grant-in-Aid for Scientific Research (C) (15K03425) anda Grant-in-Aid for Young Scientists (A) (26705003) from the Japan Society for the Promotionof Science, respectively. Adachi also acknowledges financial support from the Japan Economic
Research Foundation.
†
School of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan.
E-mail: adachi.t@soec.nagoya-u.ac.jp‡
Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan. E-mail: fabinger@e.u-tokyo.ac.jp. Fabinger is also a research associate at CERGE-EI,
Prague, the Czech Republic.

1

1

Introduction

The marginal cost of public funds, i.e. the marginal social welfare loss associatedwith raising additional tax revenue, is a crucial characteristic that a policymakerneeds to take into account when designing an optimal system of taxes.1 However, it
is somewhat surprising that relatively little is known about the general mechanismof how tax levies are passed on to final prices as well as their welfare consequencesin oligopoly, a ubiquitous feature of competition in the real-world economy.
In this paper, we aim to contribute to the general understanding of the welfare consequences of taxation in symmetric oligopolistic markets with a general(first-order) type of competition (of single-product or multi-product firms), with apossibly non-constant marginal cost, and with non-zero unit and ad valorem tax
rates. Specifically, we establish connections between welfare measures, namely themarginal cost of public funds for unit taxes and ad valorem taxes, and variables
that are easily interpretable from an empirical standpoint, namely the pass-throughof these taxes (i.e. the marginal change of prices induced by tax rate changes).2 In
particular, we show that with a general type of competition, there exists a simple
set of sufficient statistics that determines the marginal cost of public funds of unit
and ad valorem taxes, namely pass-through of these taxes and the industry demand
elasticity (in addition to the easily observed taxation levels).3
1

In the absence of other considerations, the marginal cost of public funds should be equalized
across markets in order to maximize social welfare.
2
The usefulness of pass-through in welfare analysis has been verified by related studies suchas Cowan (2012); Miller, Remer, and Sheu (2013); Weyl and Fabinger (2013); Gaudin and White(2014); MacKay, Miller, Remer, and Sheu (2014); Adachi and Ebina (2014a,b); Chen and Schwartz(2015); Gaudin (2016); Cowan (2016); and Alexandrov and Bedre-Defolie (2017). See also Ritz
(2017) for an excellent survey of theoretical studies on pass-through and pricing under imperfectcompetition.
3
The sufficient-statistics approach to connecting structural and reduced-form methods, as advocated by Chetty (2009), has been successful in empirical economics. For example, in the study
by Atkin and Donaldson (2016), the pass-through rate provides a sufficient statistic for welfare implications of intra-national trade costs in low-income countries, without the need for a full demand
estimation. Similarly, Ganapati, Shapiro, and Walker (2017) examine the welfare effects of input
taxation, where a unit tax is levied on the input. These effects are related to the effects of unit taxes2This result is a part of a larger set of relationships that link economic quantities
of interest. We derive succinct formulas that relate the marginal cost of publicfunds to pass-through of taxes of the same type. We also establish a relationshipthat connects pass-through of unit taxes and pass-through of ad-valorem taxes inthe same market. Further, we derive convenient expressions for values of unit andad valorem pass-through that are valid under a general type competition and havenot appeared in the previous literature.
In addition, specializing to price (differentiated Bertrand) competition and quantity (differentiated Cournot) competition, we show how marginal cost of public funds
and pass-through may be expressed using elasticities and curvatures of demand andinverse demand, and provide illustrative examples. Our results apply without change
also to fully symmetric oligopoly with multi-product firms. Throughout the analysis, we allow for finite levels of unit and ad valorem taxes. However, we also discuss
some additional simplifications that appear when instead the initial level of taxes is
zero.
Finally, we generalize our results to a significantly more general specificationof taxation that involves multiple tax parameters. We define two different typesof pass-through vectors: the pass-through rate vector and the pass-through quasielasticity vector. We study their properties and show that they are crucial forevaluating welfare changes in response to changes in taxation. Special cases involve
not only unit and ad valorem taxation, but also exogenous competition discussed
by Weyl and Fabinger (2013), as well as, for example, value-added tax, under which
the firm can deduct a portion of its costs from its profit for taxation purposes.
From both theoretical and empirical standpoints, it is desirable to be able tounderstand the welfare properties of oligopolistic markets with a general type of(first-order) competition. In real-world situations, the behavior of firms corresponds
on output, but not identical. See also Fabra and Reguant (2014); Shrestha and Markowitz (2016);
Duso and Szücs (2016); Stolper (2016); and Hong and Li (2017) for studies with the same spirit. In
contrast, structural studies with a full specification of demands include Kim and Cotterill (2008);
Bonnet, Dubois, Villas-Boas, and Klapper (2013); Bonnet and Réquillart (2013); Campos-Vázquez
and Medina-Cortina (2015); Griffith, Nesheim, and O’Connell (2015); Miller, Remer, Ryan, andSheu (2016); and Miller, Osborne, and Sheu (2017).

3

to neither the idealized price competition nor the idealized quantity competition.
Price competition does not allow for any friction in scaling production levels up or
down, yet in reality there tend to be substantial frictions, such as those related tofinancial constraints or the labor market. Quantity competition implies that the firm
will not be able to increase production levels when its competitor suddenly decides
to increase prices. In reality, such adjustment is feasible, since capacity utilization
is typically less than complete, and even if the firm is operating at full capacity,
boosting production levels is possible by overtime work or by hiring temporary
workers. Moreover, firms may behave to some extent in a collusive way. Although
the realities of firm competition may be complicated, it is possible to capture theiressence by working with a general type of competition, using the conduct index.4
Besides working with a general type of competition, it is also useful to relax the assumption of constant marginal costs that often appears in the literature.
Production technologies often have non-trivial structure, and so does the internalorganization of the firm. For example, if a firm decides to operate at a larger scale,it may take advantage of technological and logistical economies of scale, but at thesame time, it may face more severe principal-agent problems as top managers haveto 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 the operation.
This paper is related to the inspiring study by Häkner and Herzing (2016),
which motivates parts of this work. In the special case of linear demand, andconstant marginal cost, Häkner and Herzing (2016, p.147) explain that as long as
the initial level of taxes is zero, the marginal cost of public funds for unit taxation
equals M Ct = θρt , where ρt is the pass-through rate (the marginal effect of taxes
on prices), and θ, usually referred to as the conduct parameter, measures the degreeof non-competitiveness of the market (for example, in the case of monopoly θ = 1,
while under perfect competition θ = 0). For ad valorem taxes, Häkner and Herzing
4

For details of this approach, see Bresnahan (1989) and Weyl and Fabinger (2013). It has beensuccessfully applied also to more general situations, such as selection markets (Mahoney and Weyl(2017)) or supply chains (Gaudin (2016)).

4

(2016) provide a similar formula. They show, however, that if we let the initial levelof taxes be non-zero, those formulas are no longer valid. For this reason, they areforced to analyze the magnitude of the marginal cost of public funds on a case-bycase basis using explicit solutions to specific models.
This situation represents a puzzle. If there are simple formulas for the marginal
cost of public funds that were valid at zero taxes, is there no compact generalizationof these expressions in the case of non-zero taxes? If there is no such generalization,that would be an obstacle to empirical work, since we would have to make additionalmodeling assumptions before obtaining empirical estimates of the marginal costof public funds. Our paper provides a solution to this problem. In particular,
Propositions 1 and 2 present formulas for the marginal cost of public funds that are
valid even when the initial level of (ad valorem and unit) taxes is non-zero. They
are a bit longer than M Ct = θρt , but still very manageable. They also represent a
starting point for the topics 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 Häckner and Herzing (2016), should be useful if one needs to evaluate
the marginal cost of taxation when some tax has been already implemented.
The welfare cost of taxation has been extensively studied at least since Pigou(1928). The majority of the studies simply assume perfect competition.5 As is widely
known, under perfect competition, unit tax and ad valorem tax are equivalent, andwhether consumers or producers bear more is determined by the relative elasticitiesof demand and supply. The initial attempt to relax the assumption of perfect competition started with an analysis of homogeneous-product oligopoly under quantity
competition, i.e., Cournot oligopoly. Notably, Delipalla and Keen (1992), Skeathand Trandel (1994), and Hamilton (1999)) compare ad valorem and unit taxes insuch a setting. Then, Anderson, de Palma, and Kreider (2001a) extend these results
importantly to the case of differentiated oligopoly under price competition. In particular, Anderson, de Palma, and Kreider (2001a) find that whether the after-tax
5See, e.g., Vickrey (1963), Buchanan and Tullock (1965), Johnson and Pauly (1969), and Browing (1976) for early studies. A study of unit and ad valorem taxation under imperfect competition
with homogenous products dates back to Delipalla and Keen (1992).

5

price for firms and their profits rise by a change in ad valorem tax depends importantly on the ratio of the curvature of the firm’s own demand (εm in their notation,and αF in our notation below) to the elasticity of the market demand εDD in theirnotation, and  in our notation).
We extend Anderson, de Palma, and Kreider’s (2001a) setting and results in a
number of important directions. First, we consider the general mode of competition, captured by the conduct index, including both quantity and price competition.
Second, we provide a complete characterization of tax burdens that enables one toquantitatively compare consumers’ burden with producers’ burden, whereas Anderson, de Palma, and Kreider’s (2001a) focus only on the effective prices 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
this generalization makes a difference in our general formulas. Fourth, we further
generalize the initial tax level. When they analyze the effects of a unit tax, Anderson, de Palma, and Kreicher (2001a) assume that ad valorem tax is zero, andvice versa. In contrast, we allow non-zero initial taxes in both dimensions. Finally,
and importantly, we generalize these results to the case of a very general type oftaxation. This opens up the possibility to study a wide range of interventions/taxes
and to derive convenient sufficient statistics for characteristics, including welfare
characteristics, of the markets of interest.
In the next section, we study the problem of oligopoly with a general type ofcompetition. In Section 3, we specialize to the case of price or quantity competition.
Section 4 explains how our results apply to the case of oligopoly with multi-productfirms. Section 5 generalizes the results from unit and ad-valorem taxation to much
more flexible taxation parameterized by d different tax parameters and discusses theimplications of these general results. Section 6 concludes.

6

2

Taxation and Welfare in Symmetric Oligopoly

We study oligopolistic markets with n symmetric firms and a general (first-order)
mode of competition and the resulting symmetric equilibria.6 Our discussion applies
to single-product firms as well as to multi-product firms if intra-firm symmetry
conditions are satisfied, as discussed in Section 4. 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-product firms.
The demand for firm j’s product qj = qj (p1 , ..., pn ) ≡ qj (p) depends on thevector of prices p ≡ (p1 , ..., pn ) charged by the individual firms. The demand system
is symmetric and the cost function c(qj ) is the same for all firms. We assume thatqj (·) and c(·) are twice differentiable and conditions for the uniqueness of equilibrium
and the associated second-order conditions are satisfied.
We denote by q(p) the per-firm industry demand corresponding to symmetric prices: q(p) ≡ qj (p, ..., p). The elasticity of this function, defined as (p) ≡−pq 0 (p)/q(p) > 0 and referred to as the price elasticity of industry demand, should
not be confused with the elasticity of the residual demand that any of the firms
faces.7 We also use the notation η(q) = 1/ (p) |q(p)=q for the reciprocal of this elasticity as a function of q. For the corresponding functional values, when we do notneed to specify explicitly their dependence on either q or p, we use η interchangeably
with 1/.
We introduce two types of taxation: a unit tax t and an ad valorem tax v, withfirm j’s profit being π j = (1 − v)pj (qj )qj − tqj − c(qj ). At symmetric quantities thegovernment tax revenue per firm is R (q) ≡ tq + vp (q) q, and we denote by τ (q) the
fraction of firm’s pre-tax revenue that is collected by the government in the form of
6

Although for brevity we speak of a general mode of competition, we consider only “first-order”
competition, in the sense of the firms making decisions based on marginal cost and marginal
revenue. This excludes, for example, the possibility of each producer being composed of two
vertically related firms where the upstream firm sets prices for a relationship-specific intermediate
good, as in the usual double-marginalization problem specification.
7
The elasticity  here corresponds to D in Weyl and Fabinger (2013, p. 542). Note that q 0 (p) =∂qj (p)/∂pj + (n − 1)∂qj (p)/∂pj 0 |p=(p,...,p) for any two distinct indices j and j 0 . We will define thefirm’s elasticity and other related concepts in Section 3.

7

taxes: τ (q) ≡ R (q) /q = v + t/p (q). We define the conduct index θ (q) (or conduct
parameter ) as1
θ (q) =
η (q) p (q)t + mc (q)p (q) −
1−v


,

(1)

where mc(q) ≡ c0 (q) is the marginal cost of production, and we denote by θ its
functional value at the equilibrium quantity.8,9 The marginal welfare cost M Ct or
M Cv of raising government revenue by the unit tax t or the ad valorem tax v, i.e.
the marginal cost of public funds associated with such tax, are defined as

−1

−1
∂W
∂W
∂R
∂R
M Ct ≡ −,M Cv ≡ −
,
∂t
∂t
∂v
∂v
where W is the social welfare per firm, which includes consumer surplus, producer
surplus, and government tax revenue. We define the unit tax pass-through rate ρt
and the ad valorem tax pass-through semi-elasticity ρv as:10
ρt =

∂p
,
∂t

ρv =

1 ∂p
.
p ∂v

Consider an infinitesimal change in the unit tax, with the initial tax level (t, v).
As mentioned in the introduction, in the special case of zero initial taxes, linear
demand, and constant marginal cost, Häckner and Herzing (2016, p. 147) show that
M Ct = θρt . They further show that at non-zero initial taxes the formula no longer
applies. In the absence of such formula, they were forced to study the marginal costof public funds on a case-by-case basis, for different specifications of demand andcost.
We aim to find a generalization of the formula M Ct = θρt that would be applicable even at non-zero initial taxes. In this situation, the expression θρt as ameasure of marginal cost of public funds has two deficiencies. 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 because of a lack of competitiveness, but
8

The term “conduct parameter” originates from situations where it was supposed to be constant.
Since here we allow it to be variable, we opt for the term “conduct index”.
9More precisely, θ(q) is defined to be a function independent of the cost side of the economic
problem such that the symmetric equilibrium condition may be written in the form of Equation 1.
10Note that Häckner and Herzing (2016) use the symbol ρv for the ad-valorem tax pass-throughrate ∂p/∂v, which would be p ρv in our notation.

8

because the tax effectively raises their perceived cost. When v is large, we would
expect the marginal cost of public funds to be less sensitive to θ, for a given value ofρt . Second, the expression θρt does not explicitly feature the level of the unit tax t.
However, a situation where t is large and mc small is very different from a situation
where t is small and mc large, even if the equilibrium prices and quantities are thesame. In the former case, raising additional tax revenue is quite harmful, since firms’
production cuts will not substantially 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 increasing function of t.11
It turns out that it is possible to identify a formula with precisely these properties, as the following proposition shows.
Proposition 1. Marginal cost of public funds for unit taxation. Undersymmetric oligopoly with a possibly non-constant marginal cost, the marginal cost of
public funds associated with a unit tax may be expressed as
M Ct =(1 − v) θ +  τ.
1+ v − τρt

Proof. Using Equation 1 to substitute for mc, we first obtain a useful expression forthe markup: p − mc = t + pv + p(1 − v)ηθ. Now consider an infinitesimal change
dt in the unit tax that induces a change dp in the equilibrium price and a change
dq in the equilibrium quantity. These are related by dt = dp/ρt = −η p dq/ (q ρt ).
The corresponding change in social welfare per firm is 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/ρt . Combining these relationships gives
the result
M Ct = −(1 − v)ηθ + pt + v
dW
t + vp + (1 − v)pηθ(1 − v)θ + τ
=−
=
= 1
.
1
1
t
dR
t + vp − vpη − ρ pη
η + vη − p − v+ v − τ
ρ
ρ
t

t

t

11
In the sense of making the change t → t + ∆t, and simultaneously c (q) → c (q) − q∆t in orderto keep q, θ, and ρt at some fixed values.

9Figure 1: The ratio of the actual marginal cost of public funds M C and the naive
expression θρ discussed on p. 8, plotted as a function of combinations of the conductindex θ, the pass-through ρ, and the industry demand elasticity . The figures on
the left correspond to infinitesimal changes in unit taxation: ρ stands for ρt and M C
stands for M Ct . The numerical values were chosen to be t = 0, v = 0.2, τ = 0.2.
The figures on the right correspond to infinitesimal changes in ad valorem taxation:
ρ stands for ρv and M C stands for M Cv . The numerical values were chosen to bet/p = 0.2, v = 0, τ = 0.2. The top figures correspond to θ = 0.3, the middle figurescorrespond to  = 2, and the bottom figures correspond to ρ = 1.

10

For the welfare effects of an infinitesimal change in ad valorem tax we obtainthe following proposition.
Proposition 2. Marginal cost of public funds for ad valorem taxation.
Under symmetric oligopoly with a possibly non-constant marginal cost, the marginal
cost of public funds associated with an ad valorem tax may be expressed as
M Cv =(1 − v)θ + τ.
1
+ v − τ
ρv

Proof. In analogy with the previous proof, consider an infinitesimal change dv in thead valorem tax that induces a change dp in the equilibrium price and a change dq
in the equilibrium quantity, related by dv = dp/(pρv ) = −η dq/(qρv ). 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/ρv . Combining these relationships leads to the result
M Ct = −(1 − v)ηθ + pt + v
t + vp + (1 − v)pηθ(1 − v)θ + τ
dW
=−
=
= 1
.
1
1
t
dR
t + vp − vpη − ρ pη
η
+
vη
−
−
v
+
v
−
τ
ρ
p
ρ
v

v

v

Figure 1 documents that these expressions for the marginal cost of public funds
M Ct and M Cv evaluated at realistic values of taxes and other economic variables
are very different from the values of the expressions θρt and θρv (discussed on p. 8)that would be equal to M Ct and M Cv if taxes were zero.
For the incidence of taxation at non-zero unit and ad valorem taxes, we obtain
similarly succinct formulas. We define the incidence It of unit taxation as the ratio ofchanges dCS in (per-firm) consumer surplus and changes dP S in (per-firm) producer
surplus induced by an infinitesimal increase dt in the unit tax t. The incidence Iv isdefined analogously.
Proposition 3. Incidence of taxation. Under symmetric oligopoly with a general type of competition and with a possibly non-constant marginal cost, the incidenceof unit taxes It and of ad valorem taxes Iv is given by11=− (1 − v) (1 − θ) ,
It
ρt
11

1
1
=
− (1 − v) (1 − θ) .
Iv
ρv
Proof. The impact of a change dt in the tax t on consumer surplus (per firm) is
dCS = −qdp = −qρt dt. The impact on producer surplus isdP 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) qρt 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) qρt dt + (1 − v) ηθpdq = −qdt + (1 − v) qρt dt − (1 − v) θqdp,
dP S = −qdt + (1 − v) qρt dt − (1 − v) θqρt dt = ((1 − v) (1 − θ) ρt − 1) q dt.
The reciprocal of the incidence ratio isdP S(1 − v) (1 − θ) qρt − q
1
1
=
=
=
− (1 − v) (1 − θ) .
It
dCS
−qρt
ρt
Similarly, for infinitesimal changes in ad valorem taxes we proceed analogously. Thechange in consumer suprlus is dCS = −qdp = −qpρv dv. For the change in producer
surplus we havedP S = d ((1 − v) pq − c (q) − tq) = −pq dv +(1 − v) p dq +(1 − v) qdp−mc dq −t dq.
Manipulating the last four terms on the right-hand side in the same way as before
leads todP S = −pq dv + (1 − v) p dq + (1 − v) qdp − mc dq − t dq,
dP S = −pq dv + (1 − v) qpρv dv − (1 − v) θqpρv dv = ((1 − v) (1 − θ) ρv − 1) qp dv.
The reciprocal of the incidence ratio then becomes1
dP S(1 − v) (1 − θ) ρv q − q1===− (1 − v) (1 − θ) .
It
dCS
−qρv
ρv

In the case of zero ad valorem tax, the expression for It reduces to the one in
Weyl and Fabinger (2013). Next, we show how ρt and ρv are related in the following
proposition.
12

Proposition 4. Relationship between pass-through of ad valorem and unit
taxes. Under symmetric oligopoly with a possibly non-constant marginal cost, thepass-through semi-elasticity ρv of an ad valorem tax may be expressed in terms of theunit tax pass-through rate ρt , the conduct index θ, and the industry demand elasticity
 as
ρv =

−θ
ρt .


(2)

Proof. Let us consider a simultaneous infinitesimal change dt and dv in the taxes
t and v that leaves the equilibrium price (and quantity) unchanged, which requires
the effective marginal cost (t + mc) / (1 − v) in Equation 1 to remain the same. Thisimplies the comparative statics relationshipdt∂ t + mc
t + mc∂ t + mc
dt
t + mc
+ dv
=0⇒
+
dv.
2 dv = 0 ⇒ dt = −∂t 1 − v∂v 1 − v
1 − v (1 − v)
1−v

Note that here we do not need to take derivatives of mc even though it depends
on q, simply because by assumption the quantity is unchanged. The total inducedchange in price, which generally would be expressed as dp = ρt dt + ρv p dv, must
equal zero in this case, implying the result
ρt dt + ρv p dv = 0 ⇒ −−θt + mcρt dv + ρv p dv = 0 ⇒ ρv = (1 − ηθ) ρt ⇒ ρv =
ρ.
1−v
 t

The results above allow us to construct a convenient set of sufficient statistics
for the marginal cost of public funds.
Proposition 5. Sufficient statistics for marginal costs of public funds.
Under symmetric oligopoly with a possibly non-constant marginal cost, the unit passthrough rate ρt , the ad valorem pass-through semi-elasticity ρv , and the elasticity  ofindustry demand (together with the tax rates and the fraction τ of the firm’s pre-tax
revenue collected by the government in the form of taxes) serve as sufficient statistics
for the marginal cost of public funds both with respect to unit taxes and ad valorem
taxes. In particular:
M Ct =(1 − v + τ )ρt − (1 − v)ρv
,
1 + (v − τ )ρt13

M Cv =

(1 − v + τ ) ρt − (1 − v)ρv ρv.
1 + (v − τ ) ρv
ρt

Proof. Proposition 4 allows us to express the conduct index θ as θ = (1 − ρv /ρt ).
Substituting this into the relationships in the first two propositions gives the desired
result.
Note that to use these formulas, we do not need to make a guess about thedegree of competitiveness based on the number of firms. For example, in the case of
the assumption of Cournot competition, researchers often may observe the number
n of firms and conclude that the value of conduct index is θ = 1/n. The general formulas in the last proposition do no rely on such assumptions; the selected sufficient
statistics allow us to empirically estimate θ, i. e. allow us to measure the degree ofcartelization of those n firms.
Proposition 6. Pass-through under general symmetric oligopoly. Undersymmetric oligopoly with a general mode of competition and with a possibly nonconstant marginal cost:
ρt =

1
1
,
1 − v 1 − (η + χ) θ + q (θη)0 + 1−τχ
1−vwhere the derivative is taken with respect to q and χ ≡ mc0 q/mc. Further,
ρv =

1
−θ
.
(1 − v)  1 − (η + χ) θ + q (θη)0 + 1−τχ
1−v

Proof. Consider the comparative statics with respect to a small change dt in the perunit tax t. Totally differentiating the equation p−θ ms = (mc + t) / (1 − v) gives thecomparative statics equation dp−d (θ ms) = (dmc + dt) / (1 − v). Further, denoting
by prime the derivatives with respect to quantity, we have d (θ ms) = (θ ms)0 dq =−q (θms)0 dp/p = −q (θηp)0 dp/p = −q (θη)0 dp − pq θp0 dp = (θη − q (θη)0 )dp. Thechange in marginal cost can be written in terms of the marginal cost elasticity as
dmc = χ mc dq/q = −χ mc dp/p. In this expression, the marginal cost may beeliminated using p − θ ms = (mc + t) / (1 − v) ⇒ mc = (1 + v) (p − θqp0 ) − t =(1 − v) (1 + θη) p − t, which leads to dmc = −((1 − v) (1 + θη) − t/p)χ dp. In terms
14

of the per-unit-revenue tax burden τ ≡ v + t/p, this is dmc = −((1 − v) (1 + θη) −τ + v)χ dp = −((1 − v) θη + 1 − τ )χ dp. Substituting the resulting expressions ford (θ ms) and dmc into the comparative statics equation gives
ρt =

dp
1

=
,
0
(1−v) (dp−d (θ ms))−dmc(1−v) 1−θη+(θη) q +(1−v) θχ+(1−τ ) χ

which is equivalent to the expression for ρt in the proposition. The expression forρv then follows by Proposition 4.
Let us also point out that the exchange rate pass-through can be included naturally in our framework.12 Suppose that domestic firms in a country of interest use
some imported inputs for production. For concreteness, let us specify the profit function of firm j as π j = [(1 − v)pj − t]qj − (1 + a e)c(qj ), where the constant coefficienta measures the importance imported inputs and e > 0 is the exchange rate. Notice
 1−v


t
that the firm’s profit is rewritten as π j = (1+ae) 1+aepj − 1+ae
qj − c(qj ) . Since
the first factor on the right-hand side is constant, the firm will behave as if its profitfunction was simply π̃ j = (1 − ṽ)pj − t̃ qj − c(qj ), with ṽ ≡ (v + ae)/(1 + ae) andt̃ ≡ t/(1 + ae). By utilizing the explicit expressions for the derivatives ∂ṽ/∂e =(a − v)/(1 + ae)2 and ∂ t̃/∂e = −at/(1 + ae)2 , one can analyze the effect of a change
in the exchange rate e on social welfare. Note that this is simply interpreted as thecost pass-through as well (see the references in Footnote 12 for empirical studies).

3

Taxation and Welfare under Specific Types ofCompetition

In this section we show that for price competition and quantity competition indifferentiated oligopoly, our general expressions of the marginal cost of public funds
and pass-through lead to expressions in terms of demand primitives such as theelasticities and the curvatures, and the marginal most elasticity χ defined above.
We also provide parametric examples for these results.
12

See, e.g., Feenstra (1989); Feenstra, Gagnon, and Knetter (1996); Yang (1997); Campa andGoldberg (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.

15

3.1

Elasticities and curvatures of demand and inverse demand

Direct demand. Following Holmes (1989, p. 245), we define the own price elasticity of the firm’s demand by F (p) ≡ −(p/q(p)) ∂qj (p)/∂pj |p=(p,...,p) and the crossprice elasticity by C (p) ≡ (n − 1)(p/q(p)) ∂qj 0 (p)/∂pj |p=(p,...,p) for any distinct pairof indices j and j 0 . These are related to the industry demand elasticity (p) by
F (p) = (p) + C (p).13
We define the curvature of the industry’s direct demand α(p) ≡ −p q 00 (p)/q 0 (p),as well as the own curvature αF (p) of the firm’s direct demand and the cross curvature αC (p) of the firm’s direct demand :14αF (p) ≡ −p

∂qj (p)∂pjαC (p) ≡ − (n − 1) p

−1∂qj (p)∂pj∂ 2 qj (p)|p=(p,...,p) ,
∂p2j
−1

∂ 2 qj 0 (p)|p=(p,...,p) ,∂pj ∂pj 0

where j and j 0 is an arbitrary pair of distinct indices. These curvatures satisfy
α = (αF + αC )F /. They are related to the elasticity of F (p) by p 0F (p)/F (p) =1 +  (p) + αF (p) + αC (p).15
Inverse demand. We define the own quantity elasticity of the firm’s inverse
demand η F (q) ≡ −(q/p(q)) ∂pj (q)/∂qj |p=(p,...,p) and the the cross quantity elasticity
η C (q) ≡ (n − 1)(q/p(q)) ∂pj 0 (q)/∂qj |p=(p,...,p) . These satisfy η F (q) = η(q) + η C (q).
13

Holmes (1989) shows this for two symmetric firms, but it is straightforward to verify thisrelation more generally. See the equation in Footnote 7.
14
The curvature αF (p) here corresponds to α(p) of Aguirre, Cowan, and Vickers (2010, p. 1603).
15This relationship can be verified as follows. The elasticity of the function F (p) equals the sumof the elasticities of the three factors it is composed of:
1
d
1 d
d 1
p F (p) = p p + q (p) p+
F (p) dp
p dp
dp q (p)∂qj (p)∂pj

−1d|p=(p,...,p) pdp∂qj (p)|p=(p,...,p) .
∂pj

The first elasticity on the right-hand side equals 1, the second elasticity equals  (p), and the third
elasticity equals αF (p) + αC (p), sincep∂ 2 qj (p)∂ 2 qj (p)d ∂qj (p)|p=(p,...,p) = p
|
+
(n
−
1)
p
|p=(p,...,p) .
p=(p,...,p)dp ∂pj∂p2j∂pj ∂pj 0

16

We define the curvature of the industry’s inverse demand σ(q) ≡ −q p00 (q)/p0 (q),as well as the own curvature σ F (q) of the firm’s inverse demand and the crosscurvature σ C (q) of the firm’s inverse demand by:−1 2∂pj (q)∂ pj (q)σ F (q) ≡ −q
|q=(q,...,q) ,
∂qj
∂qj2

σ C (q) ≡ − (n − 1) q

∂pj (q)∂qj

−1∂ 2 pj 0 (q)|q=(q,...,q) ,∂qj ∂qj 0

for an arbitrary pair of distinct indices j and j 0 . These curvatures represent anoligopoly counterpart of monopoly σ(q) in Aguirre, Cowan, and Vickers (2010, p.
1603). They satisfy the relationship σ = (σ F + σ C )η F /η. They are related to theelasticity of η F (q) by q η 0F (q)/η F (q) = 1 + η (q) + σ F (q) + σ C (q).16

3.2

Expressions for pass-through and marginal cost of public funds

In the case of price competition, the conduct index θ is θ = /F = 1/(ηF ), which
may be verified by comparing the firm’s first-order condition with Equation 1.
Proposition 7. Pass-through and marginal cost of public funds underprice competition. Under symmetric oligopoly with price competition and with apossibly non-constant marginal cost:
ρt =

1
F
1 − v F +  − αF − αC +ρv =F − 11 .
1 − v F +  − αF − αC + 1−τ − 1 χ
1−v F

1−τ
 −1
1−v F



χ

,

For the marginal cost of public funds this implies:
M Ct =

1
1 + 1−v
F τ 
1
 − αF − αC + 1−v(1 − τ ) F +

1−τ
 −1
1−v F


1
1 + 1−v
F τ 
F − 1
M Cv =1
F
F +  − αF − αC + 1−v(F − 1) (v − τ ) +



χ

,

1−τ
 −1
1−v F



χ

.

16
In analogy with Footnote 15, 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).

17

In the case of constant marginal cost the pass-through expressions simplify to
ρt =

1
F
,
1 − v F +  − αF − αCρv =1
F − 1.
1 − v F +  − αF − αC

Proof. Since in the case of price setting θ = /F = 1/(ηF ), we have (η + χ) θ =
−2
−2 d
d
d
d
(1 + χ) /F and (θη)0 q = q dq(θη) = q dq(−1F ) = −F q dq F = F p dp F =(1 +  − αF − αC ) /F , where in the last equality we utilized the expression for theelasticity of F (p) from Subsection 3.1. Substituting these into the expression for ρtin Proposition 6 gives
ρt =

1
1
,
1
1
1 − v 1 − F (1 + χ) + F (1 +  − αF − αC ) + 1−τχ
1−vwhich is equivalent to the expression for ρt in the proposition. Since for price settingθ = /F , the relationship in Proposition 4 implies ρv = ( − θ) ρt / = (F − 1)ρt /F ,which leads to the desired expression for ρv . The implication for the marginal costof public funds follows by substituting these expressions into those of Propositions
1 and 2.
In the case of quantity competition, the conduct index θ is θ = η F /η, which
may be verified by comparing the firm’s first-order condition with 1.
Proposition 8. Pass-through and marginal cost of public funds underquantity competition. Under symmetric oligopoly with quantity competition andwith a possibly non-constant marginal cost:
ρt =

1
η
 ,
1 − v η + (1 + σ C + σ F ) η F + 1−τ
−
η
χ
F
1−v

ρv =

1
(1 − η F ) η .
1 − v η + (1 + σ C + σ F ) η F + 1−τ
−
η
χ
F
1−v

For the marginal cost of public funds this implies:
M Ct =

1
η F + 1−vτ ,1η + (1 + σ C + σ F ) η F + 1−v(v −  τ ) η + 1−τ− ηF χ1−v
1(1 − η F ) η F + 1−vτ
M Cv =.
1
η + (1 + σ C + σ F ) η F + 1−τ−ηχ + 1−v(1 − η F ) (v − τ ) η
F
1−v
18

In the case of constant marginal cost the pass-through expressions simplify to
ρt =

1
η
,
1 − v η + (1 + σ C + σ F ) η F

ρv =1(1 − η F ) η.
1 − v η + (1 + σ C + σ F ) η F

Proof. In the case of quantity setting, θ = η F /η, so (η + χ) θ = (1 + χ/η) η F and(θη)0 q = q (η F )0 /η = (1 + η + σ C + σ F ) η F /η, where in the last equality we utilized
the expression for the elasticity of η F (q) from Subsection 3.1. Substituting these
into the expression for ρt in Proposition 6 gives
ρt =

1
1
1
1 ,
1
1 − v 1 − (1 + η χ)η F + η (1 + η + σ C + σ F ) η F + 1−τχ1−v ηAfter a rearrangement this gives the expression for ρt in the proposition. Sinceθ = η F /η, Proposition 4 implies ρv = ( − θ) ρt / = (1/η − η F /η) ρt η = (1 − η F ) ρt ,which can be used to verify the expression for ρv . The implication for the marginal
cost of public funds follows by substituting these expressions into those of Propositions 1 and 2.
Cournot competition. Homogeneous-product Cournot competition is a very
simple special case, where θ = 1/n, η = n η F , and σ C = (n − 1) σ F .
Monopolistic competition. Monopolistic competition may be obtained bytaking the large n limit. As discussed in Weyl and Fabinger (2013, p. 544-546), inR
the case of quasi-linear utility of the form U ( u (qi ) di) − pi qi di, it may be shownthat θ = U 0 u00 /(U 00 (u0 )2 + U 0 u00 ). With the most typical specification u (q) = q β ,U (x) = xγ , this leads to a constant value of conduct index: θ = (1 − β) / (1 − βγ).
Then η = η F (1 − βγ) / (1 − β) .

3.3

Simple Parametric Examples

Tables 1 and 2 provide a summary of the variables of interest for linear demand andlogit demand, respectively.17
17

A discussion of the intuitive economic meaning of the formulas will be added to future versionsof this paper.

19Price settingP
qj (p1 , ..., pn ) = b − λpj + µ j 0 6=j pj 0
F = λpq
C = µ(n−1)pqθ = λ+µ−µnλλ(mc+t)bp = 2λ+µ−µn− (v−1)(2λ+µ−µn)bλq = 2λ+µ−µn+ λ(mc+t)(λ+µ−µn)(v−1)(2λ+µ−µn)λρt = − (v−1)(2λ+µ−µn)λ(mc+t)ρv = (v−1)(b(v−1)−λ(mc+t))
M Ct =

Quantity setting P
qj (p1 , ..., pn ) = b − λpj + µ j 0 6=j pj 0q(−λ−2µ+µn)η F = − p(λ+µ)(λ+µ−µn)q(µ−µn)η C = p(λ+µ)(λ+µ−µn)θ = λ+2µ−µnλ+µ(−λ−µ)(mc+t)b(λ−µ(n−2))+ (v−1)(2λ−µ(n−3))p = (2λ−µ(n−3))(λ+µ−µn)q = (λ+µ)(b(v−1)+(mc+t)(λ+µ−µn))(v−1)(2λ−µ(n−3))−λ−µρt = (v−1)(2λ−µ(n−3))
ρv = (v−1)(bκ2κ(λ+µ)(mc+t)(v−1)−κ(λ+µ)(mc+t))
M Ct =−(λ+µ−µn)(λpτ +q(1−v))bλτ +q(−λτ +µ(1−n)(v−1)+λ(v−2))

q(v−1)(λ−µ(n−2))−pτ (λ+µ)(λ+µ−µn)pτ (λ+µ)(λ+µ−µn)+µq(n(−v)+n+2v−3)+λq(v−2)
1
M Cv = λ+µ×(µ(n−2)q+κp(λ+µ)+λ(−q))(κ2 q(v−1)−κpτ (λ+µ))κp(q(µ(n−3)+µ(n−2)(τ −v)+λ(v−τ −2))+κpτ (λ+µ))+κ2 q 2 v

M Cv =−(λ+µ−µn)(λp−q)(λpτ +q(1−v))
λτ (b−q)(λp−q)+λq(−µ(n−1)p(v−1)+λp(v−2)+qv)

Table 1: Linear demand summary. Here κ ≡ λ − (n − 1) µ and κ2 ≡ λ − (n − 2)µ.

Price setting
exp(δ−βpj )qj (p1 , ..., pn ) = 1+P 0
j =1,...,n exp(δ−βpj 0 )s
pj (s1 , ..., sn ) = β1 (δ − log s0j )F = β(1 − s)pC = β(n − 1)p s
θ = 1−ns1−sQuantity setting
exp(δ−βpj )qj (p1 , ..., pn ) = 1+P 0
j =1,...,n exp(δ−βpj 0 )s
pj (s1 , ..., sn ) = β1 (δ − log s0j )sη F = 1−(n−1)s(δ − log 1−ns)−11−ns(n−1)ssη C = − 1−ns (δ − log 1−ns)−1
θ = 1 − (n − 1) ss

1−δ+s((δ−1)n+1)+(1−ns) log 1−ns
mc+t
=
1−v
β(ns−1)1
ρt = (1−v)(1+s (1−ns)−1 )1
p = mc+t+ β(1−s(p))
1−v
2

(s−1)
ρt = (1−v)(1−s−(n−1)s2)PTable 2: Logit demand summary. We define s0 = 1 − j=1,...,n sj < 1. Additional
expressions will be added to this table in future versions of this draft.

20

4

Oligopoly with Multi-Product Firms

In this section, we argue that the results above can be extended to the case of multiproduct firms just by a reinterpretation of the same formulas (without modifyingthem). Assume there are np product categories, and the demand for firm j’s k-th
product is given by qjk = qjk (p1 , p2 , .., pn ), where pj = (pj1 , ..., pjk , ..., pjK ) for eachj = 1, 2, ..., n.18 The firms are symmetric, and for each firm, the product it produces
are also symmetric. The firm’s profit per product is
np

1 X
πj =
((1 − v) pjk qjk − tqjk − c(qjk )) .
np k=1
We work with an equilibrium in which any firm j sets a uniform price pj for all of its
products: pjk = pj , and consequently sells an amount qj of each of them: qjk = qj .19
In this case the profit per product equals π j = (1 − v) pj qj − tqj − c(qj ), which isformally the same as for single-product firms. For this reason, we can identify theprices pj and quantities qj of Section 2 with the prices pj and quantities qj introduced
here in this paragraph. The discussion in Section 2 was general and applies to thiscase of symmetric oligopoly with multi-product firms as well. We can use the samedefinitions for the variables of interest, including the industry demand elasticity and the conduct index θ.
The definitions and results for the cases of price competition and quantity competition discussed Section 3 are also applicable here. It may be useful to translate
some of the most important variables of that discussion into product-level variables.
18

See, e.g., Nocke and Schutz (2016) for a recent treatment of multi-product oligopoly.
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.
19

21

For derivatives of the direct demand system, we introduce the notation20∂q∂q
ξ 2 ≡ ∂pjk
2 ,
jk
∂qjk
ξ̃ 2 ≡ ∂pjk ∂p 0
j k∂q

ξ 1 ≡ ∂pjk,
jk

ξ 0,1 ≡ ∂p jk0 ,∂qjk
ξ 1,1 ≡ ∂pjk ∂p
,
jk0
∂qjk
ξ̃ 1,1 ≡ ∂pjk ∂p 0 0 ,j k∂q
ξ 0,2 ≡ ∂p2jk ,jk0∂qjkξ̃ 0,2 ≡ ∂p 0 ∂p 0 0 ,
jk
j k

jk

∂q

jk
ξ 0,1,1 ≡ ∂p 0 ∂p
,
00
jk

jk

∂qjk
ξ̃ 0,1,1 ≡ ∂p 0 ∂p,
jk
j 0 k00

where the derivatives are evaluated at the fully symmetric point, where any pjk
equals the common value p. For specific choices of the demand system, these derivatives can be closely related. For example, if the substitution pattern between two
goods produced by two different firms does not depend on the identity of the goods,
then ξ̃ 2 = ξ̃ 0,2 = ξ̃ 1,1 = ξ̃ 0,1,1 . In terms of these derivatives, we can writeF = − pq ξ 1 + (np − 1) ξ 0,1 , = − pq ξ 1 + (np − 1) ξ 0,1 + (n − 1) ξ̃ 1 + (n − 1) (np − 1) ξ̃ 0,1 ,2αF = qpF ξ 2 + (np − 1) ξ 1,1 + ξ 0,2 + (np − 2) ξ 0,1,1 ,2αC = (n − 1) qpF ξ̃ 2 + (np − 1) (ξ̃ 1,1 + ξ̃ 0,2 + (np − 2) ξ̃ 0,1,1 ) .
These can be substituted into the results of Proposition 7 to find the pass-throughand the marginal cost of public funds under price competition.
For the inverse demand system the analogous definitions are∂qζ 2 ≡ ∂pjk
2 ,
jk

∂q

jk
ζ̃ 2 ≡ ∂pjk ∂p
0

j k

∂q

∂q

ζ 0,1 ≡ ∂p jk0 ,∂q

ζ 0,2 ≡ ∂p2jk ,,ζ 1 ≡ ∂pjk
jkjk
ζ 1,1 ≡ ∂pjk ∂p
,
0
jk

jk

∂q

∂q

jk
ζ 0,1,1 ≡ ∂p 0 ∂p
,
00

jk0
∂qjk

∂q

jk

jk

∂q

jk
jk
ζ̃ 1,1 ≡ ∂pjk ∂p.
, ζ̃ 0,2 ≡ ∂p 0 ∂p 0 0 , ζ̃ 0,1,1 ≡ ∂p 0 ∂p
0 00
0 0
j k

jk

j k

jk

j k

The relations
η F = − pq ζ 1 + (np − 1) ζ 0,1
η = − pq

ζ 1 + (np − 1) ζ 0,1 + (n − 1) ζ̃ 1 + (n − 1) (np − 1) ζ̃ 0,1
2
σ F = pqη ζ 2 + (np − 1) ζ 1,1 + ζ 0,2 + (np − 2) ζ 0,1,1 ,Fq2
σ C = (n − 1) p η ζ̃ 2 + (np − 1) (ζ̃ 1,1 + ζ̃ 0,2 + (np − 2) ζ̃ 0,1,1 ) .
F

can be substituted into the results of Proposition 8 to find the pass-through andmarginal cost of public funds under price competition.
20

In this notation, the first subscript counts the derivatives with respect to the relevant pricewith index k, the second subscript counts the derivatives with respect to the price with index
k 0 distinct from k, and the third subscript counts derivatives respect to the price with index k 00
distinct from both k and k 0 . Further,ξ corresponds to derivatives with respect to prices charged
by the same firm j, while ξ̃ corresponds to derivatives with respect to prices charged by firm j andsome other firm j 0 .

22

5Multi-Dimensional Pass-Through Framework

In this section we generalize our previous results to a significantly more general specification of taxation that involves multiple tax parameters. We define two different
types of pass-through vectors: the pass-through rate vector and the pass-throughquasi-elasticity vector. We study their properties and show that they play a central
role in evaluating welfare changes in response to changes in taxation.

5.1

Pass-through, conduct index, and welfare: a general discussion

5.1.1

Generalized pass-through and tax sensitivities

Consider a tax structure under which a firm’s tax payment may be expressed asφ(p, q, T), where T ≡ (T1 , ..., Td ) is a d-dimensional vector of tax parameters.21 The
components of the (per-firm) tax revenue gradient vectorf ≡ ∇φ(p, q, T)
are∂φ(p, q, T).
∂T`
Here, as in other parts of the paper, we use the symbol ∇ for the d-dimensionalφT` (p, q, T) ≡gradient with respect to T. The arguments p and q in φ(p, q, T) are treated as fixed
for the purposes of taking this gradient. We denote the equilibrium price function22
by p? (T) and its gradient, the pass-through rate vector, byρ̃ ≡ ∇p? (T) .
Further, we use the components of the f and ρ̃ to define the pass-through quasielasticity vector asρ ≡ ρT1 , ..., ρTd ,ρT` ≡ q

21

ρ̃T`
q
∂p?
=
.
fT`
φT` (p, q, T) ∂T`

To be precise, φ(p, q, T) represents a simplified notation for a function φ(p, q, T1 , ..., Td ) withd + 2 arguments.
22
Unlike the inverse demand function p (q), the function p? (T) takes the vector of taxes asarguments and its functional value is the price in the resulting equilibrium.

23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 ofthe (per-firm) tax revenue as follows:1ν(p, q, T) ≡ φp (p, q, T),q1τ (p, q, T) ≡ φq (p, q, T).
p

Their derivatives areν T` (p, q, T) ≡∂ν(p, q, T),∂T`τ T` (p, q, T) ≡∂τ (p, q, T).
∂T`

The analogous definitions for second-order sensitivities are:
ν (2) (p, q, T) ≡p ∂ 2 φ(p, q, T)
q ∂ 2 φ(p, q, T)∂ 2 φ(p, q, T)
.
,
τ
(p,
q,
T)
≡
,
κ(p,
q,
T)
≡
(2)
q
∂p2
p
∂q 2
∂p ∂q

The first-order and second-order sensitivities are dimensionless, just like the components of ρ. In this section we keep the same definition of the elasticities  and ηas before.
5.1.2

Generalized conduct index

We introduce the conduct index θ as a function independent of the cost-side of theoligopoly game such 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 conductindex defined earlier. In principle, there are many possible definitions that agreewith the earlier definition in the case of unit and ad-valorem taxation, but we foundthe specification of Equation 3 particularly convenient.
5.1.3

Relative size of the components of pass-through vectors

Proposition 9. The pass-through rates and quasi-elasticities satisfy 23
ρ̃T`τ T 0 − ν T`0 η θ
= `
,
ρ̃T`0
τ T` − ν T` η θ

ρT`
fT 0 τ T` − η θ ν T`
= `
.
ρT`0
fT` τ T`0 − η θ ν T`0

23
If the denominators are zero, the fractions become ill-defined. In that case, of course, thestatement does not apply.

24

Proof. Consider an infinitesimal tax change such that the equilibrium price (andtherefore quantity) does not change: ρ̃ · dT = 0. Let us choose dT to have just two
non-zero components: dT` and dT`0 . This implies
ρ̃T`
dT`0
=−
.
ρ̃T`0
dT`

(4)

Since Equation 3 must hold both before and after the tax change, we must be that1 − τ − (1 − ν) ηθ does not change, and in turn(−τ T` + ν T` ηθ) dT` + −τ T`0 + ν T`0 ηθ dT`0 = 0.
Substituting for dT`0 from this equation into Equation 4 and using the definition ofpass-through quasi-elasticities leads to the desired result.
Since the components have known proportions, we can write them using a common factor pρ(0) asρ̃T` = (τ T` − ν T` η θ) pρ(0),
ρT` =
5.1.4

pq
(τ T` − ν T` η θ) ρ(0).
fT `Absolute size of the components of pass-through vectors

Proposition 10. The value of the factor ρ(0) introduced above is given by the formula:1= 1 − κ + τ (2) + (1 − τ )χ + ν − κ + ην (2) + (ω − η − χ) (1 − ν) θ,ρ(0)where ω ≡ q (ηθ)0 / (ηθ) , with the prime denoting a derivative with respect to thequantity q.
Proof. The same type of reasoning as in the proof of Proposition 6 is useful here.
In particular, comparative statics of Equation 3 with respect to a tax T` leads
to the desired result after utilizing the definitions above and eliminating marginal
cost using, again, Equation 3. The calculation is a bit tedious, but completely
straightforward.
25

5.1.5

Welfare changes and their relationship to pass-through vectors

Welfare component changes in response to an infinitesimal change in taxes may
be found as follows. The consumer surplus change in response to an infinitesimalchange dT` in the tax T` isdCS = −qdp = −qρ̃T` dT` ,which means that in vector notation, 1q ∇CS = −ρ̃. The change in producer surplus(firm profit) is
dP S = d (pq − c (q) − φ(p, q, T)) = φT` (p, q, T) − (1 − ν) (1 − θ) ρ̃T` dT` ,where we utilized Equation 3 to eliminate marginal cost. In vector notation, this is1∇P S = (1 − ν) (1 − θ) ρ̃ − f , since f = ∇T φ(p, q, T).
q

The change in tax revenueisdR = φp (p, q, T)dp+φq (p, q, T)dq+φT` (p, q, T)dT` = φT` (p, q, T) − (τ − ν) ρ̃T` dT` .
In vector notation, 1q ∇R = f − (τ − ν) ρ̃. Finally, for the change in social welfare,
we havedW = (p − mc) dq = (τ + θ (1 − ν)) ρ̃T` dT` .
In vector notation, 1q ∇W = − (τ + θ (1 − ν)) ρ̃.
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 theequilibrium outcomes. This is different from the tax revenue function φ (p, q, T),
which has also p and q as arguments and which is specified by the governmentirrespective of the equilibrium. We summarize these findings in the following proposition.
Proposition 11. The tax gradients of consumer surplus, producer surplus, tax revenue, and social welfare with respect to the taxes all belong to a two-dimensional
vector space spanned by f and ρ̃. The precise linear combinations of f and ρ̃ are1∇CS = −ρ̃,q26

1∇P S = (1 − ν) (1 − θ) ρ̃ − f ,q1∇R = f + (ν − τ ) ρ̃,q1∇W = − ((1 − ν) θ + τ ) ρ̃.
qThese relationships, considered component-wise, immediately imply the following results for welfare change ratios.24
Proposition 12. The marginal cost of public funds of a tax T` , M CT` = (∇R)T` / (∇W )T` ,is
M CT` =

(1 − ν) θ + τ.
1
+ ν − τ
ρT`

The incidence of this tax, IT` = (∇CS)T` / (∇P S)T` , equals:
IT` =

1
.
− (1 − ν) (1 − θ)

1
ρT

`

Similarly, the social incidence, SIT` = (∇W )T` / (∇P S)T` , equals:
SIT` =1ρT(1 − ν) θ + τ.
− (1 − ν) (1 − θ)`

5.2

Pass-through, conduct index, and welfare: special cases

The results of the previous subsection contain our results for ad valorem and unit
taxes as special cases, but offer much greater generality, since the taxes (government
interventions) may be specified in a very flexible way. In fact, it may be shown thesymmetric oligopoly results in Weyl and Fabinger (2013) are also special cases ofthe present results. In particular, the analysis there considers either unit taxes orexogenous competition (an exogenous quantity supplied to the market). The caseof unit taxes are clearly included in the present results (and motivated this paper).
But it turns out that the case of exogenous competition is included as well. The
reasoning is as follows.
24

Remember that the T` component of the vector f is φT` (p, q, T) =qρ̃T` /ρT` .

27

Consider a tax T1 = q̃ of the form
φ (p, q, q̃) = q̃ p + c(q − q̃) − c(q).
The firm’s profit is thenpq − c (q) − φ (p, q, q̃) = p (q − q̃) + c(q − q̃).
The firm, therefore, has the same profit function as in the case of exogenous competition q̃ in Weyl and Fabinger (2013). Proposition 12 (specialized to constantmarginal cost and zero initial q̃) then implies the social incidence result in Principleof Incidence 3 in Weyl and Fabinger (2013, p. 548).
Similarly, the relationships between pass-through of unit taxes and of exogenous
competition are implied by the general result of Proposition 9 for the tax specification T1 = t, T2 = q̃,
φ (p, q, t, q̃) = tq + q̃ p + c(q − q̃) − c(q).
To obtain the absolute size of the two types of pass-through, one can straightforwardly use Proposition 10.

6Concluding Remarks

In this paper, we characterize the welfare burden of taxation and the tax incidence
in symmetric oligopoly with a general specification of competition, demand andcost. We first derive formulas for marginal welfare losses from unit and ad valorem
taxation, M Ct and M Cv , using the unit tax pass-through rate ρt and the ad valorem
tax pass-through semi-elasticity ρv . We show that ρv can be expressed in terms of
ρt . These relationships are used to derive sufficient statistics for M Ct and M Cv .
In the case of price or quantity competition, we explain how ρt and ρv can be
written only in terms of the demand elasticities, the demand curvatures, and themarginal cost elasticity. We discuss relationships to other quantities of interest, as
well as illustrative special cases. We also show that these results have a very natural28

generalizations to a general specification of the tax revenue function as a function
parametrized by a vector of tax parameters.
In follow-up work (or in future versions of this paper), we plan to discuss how
this framework generalizes to other markets. We will provide an extension of ouranalysis to the case of asymmetric oligopoly, where the firms face different costs andpossibly also different taxes.25,26 This discussion is analogous, but since it would
require a lengthy explanation, we did not include it in this version of the paper.
Similarly, we plan to extend the analysis to the case of supply chains (see Peitz andReisinger (2014)). Other possible directions include the case of two-sided platformcompetition (White and Weyl (2016)) or the case of the interactive effects of taxation
for multiple imperfectly competitive product markets.27,28 In addition, it would beof interest to develop flexible, but analytically solvable examples along the lines ofFabinger and Weyl (2016).
25

By allowing (constant) asymmetric marginal costs, Anderson, de Palma, and Kreider (2001b)show that under quantity competition with homogeneous products (i.e., Cournot competition), advalorem taxation is still preferable to unit taxation, although they were not able to verify if thesame conclusion held under quantity competition with 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 the required tax revenue
is sufficiently high. We conjecture that one could obtain further generalization by allowing the
conduct index θ to be firm-specific. See also Zimmerman and Carlson (2010) for a parametricanalysis of asymmetric firms.
26Interestingly, Tremblay and Tremblay (2016) study tax incidence in a duopoly where one firmcompetes in price and the other firm competes in quantity, focusing on unit taxation. The passthrough rates can be different for the two identical firms (in terms of demand and cost): thequantity-competing firm has a higher pass-through rate than the price-competing firm has. This is
in contrast with the result that the pass-through rate under price competition is generally higher.
27
Among many others, Ballard, Shoven, and Whalley (1985) study this issue for perfectly competitive markets.
28
In addition, our methodology could be utilized to study other important issues of pricing in general such as welfare effects of oligopolistic third-degree price discrimination (Aguirre,
Cowan, and Vickers (2010) is a study of monopoly). One may also study, for example, advertising pass-through (Draganska and Vitorino (2015)): the firm’s demand can be modeled as
qj = qj (p1 , ..., pn ; a1 , ..., an ), where aj is firm j’s investment in advertising. Free-riding, because of
the spillover effect, may be more or less serious depending on demand elasticities and curvatures.

29

References
Adachi, T., Ebina, T., 2014a. Double marginalization and cost pass-through: WeylFabinger and Cowan meet Spengler and Bresnahan-Reiss. Economics Letters, 122(2),170–175.
Adachi, T., Ebina, T., 2014b. Cost pass-through and inverse demand curvature
in vertical relationships with upstream and downstream competition. Economics
Letters, 124(3), 465–468.
Aguirre, I., Cowan, S., Vickers, J., 2010. Monopoly price discrimination and demand
curvature. American Economic Review, 100(4), 1601–1615.
Alexandrov, A. and Bedre-Defolie, Ö., 2017. LeChatelier-Samuelson principle ingames and pass-through of shocks. Journal of Economic Theory, March 2017
Anderson, S. P., de Palma, A., Kreider, B., 2001a. Tax incidence in differentiatedproduct oligopoly. Journal of Public Economics, 81(2), 173–192.
Anderson, S. P., de Palma, A., Kreider, B., 2001b. The efficiency of indirect taxes
under imperfect competition. Journal of Public Economics, 81(2), 231–251.
Anderson, S. P., de Palma, A., Thisse, J.-F., 1992. Discrete choice theory of productdifferentiation. The MIT Press.
Atkin, D., Donaldson, D., 2016. Who’s getting globalized? The size and implicationsof intra-national trade costs. Manuscript.
Auer, R. A., Schoenle, R.S., 2016. Market structure and exchange rate pass-through.
Journal of International Economics, 98, 60–77.
Ballard, C. L., Shoven, J.B., Whalley, J., 1985. General equilibrium computationsof the marginal welfare costs of taxes in the United States. American Economic
Review, 75(1), 128–138.
Berry, S. T., 1994. Estimating discrete-choice models of product differentiation.
RAND Journal of Economics, 25(2), 242–262.
30

Bonnet, C., Dubois, P., Villas-Boas, S.B., Klapper, D., 2013. Empirical evidence on
the role of nonlinear wholesale pricing and vertical restraints on cost pass-through.
Review of Economics and Statistics, 95(2), 500–515.
Bonnet, C., Réquillart, V., 2013. Tax incidence with strategic firms in the soft drink
market. Journal of Public Economics, 106, 77–88.
Bresnahan, T. F., 1989. Empirical studies of industries with market power. In R.
Schmalensee and R.D. Willig (eds.), Handbook of industrial organization, Vol. 2.
Amsterdam, The Netherlands: Elsevier Science Publishers B.V., 1011–1057.
Browning, E. K., 1976. The marginal cost of public funds. Journal of Political Economy, 84(2), 283–298.
Buchanan, J. M., Tullock, G., 1965. Public and private interaction under reciprocal externality. In J. Margolis (ed.), The public economy of urban communities,
Resources for the Future, 52–73.
Campa, J. M., Goldberg, L.S., 2005. Exchange rate pass-through into import prices.
Review of Economics and Statistics, 87(4), 679–690.
Campos-Vázquez, R. M., Medina-Cortina, E.M., 2015. Pass-through and competition: The impact of soft drink taxes using Mexican supermarket stores. Manuscript.
Chen, N., Juvenal, L., 2016. Quality, trade, and exchange rate pass-through. Journalof International Economics, 100, 61–80.
Chen, Y., Schwartz, M., 2015. Differential pricing when costs differ: A welfare
analysis. RAND Journal of Economics, 46(2), 442–460.
Chetty, R., 2009. Sufficient Statistics for Welfare Analysis: A Bridge between Structural and Reduced-Form Methods. Ann. Rev. Econ. 1:451–88.
Corts, K. S., 1999. Conduct parameters and the measurement of market power.
Journal of Econometrics, 88(2), 227–250.

31

Cowan, S., 2012. Third-degree price discrimination and consumer surplus. Journalof Industrial Economics, 6(2), 333–345.
Cowan, S., 2016. Welfare-increasing third-degree price discrimination. RAND Journal of Economics, 47(2), 326–340.
Delipalla, S., Keen, M., 1992. The comparison between ad valorem and specifictaxation under imperfect competition. Journal of Public Economics, 49(3), 351–367.
Delipalla, S., O’Donnell, O., 2001. Estimating tax incidence, market power, andmarket conduct: The European cigarette industry. International Journal of Industrial Organization, 19(6), 885–908.
Draganska, M., Vitorino, M. A., 2015. Advertising pass-through: How does advertising affect retail prices? Manuscript.
Duso, T., Szücs, F., 2016. Market power and heterogeneous pass-through in German
electricity retail. Manuscript.
Fabinger, M., Weyl, E. G., 2016. The average-marginal relationship and tractableequilibrium forms. Manuscript.
Fabra, N., Reguant, M., 2014. Pass-through of emissions costs in electricity markets.
American Economic Review, 104(9), 2872–2899.
Feenstra, R. C., 1989. Symmetric pass-through of tariffs and exchange rates under imperfect competition: An empirical test. Journal of International Economics,
27(1–2), 25–45.
Feenstra, R. C., Gagnon, J. E., Knetter, M. M., 1996. Market share and exchange
rate pass-through in world automobile trade. Journal of International Economics,
40(1–2), 187–207.
Ganapati, S., Shapiro, J. S., Walker, R., 2017. “The Incidence of Carbon Taxes
in U.S. Manufacturing: Lessons from Energy Cost Pass-Through.” Unpublishedmanuscript.
32

Gaudin, G., 2016. Pass-through, vertical contracts, and bargains. Economics Letters,
139, 1–4.
Gaudin, G., 2016. Vertical Bargaining and Retail Competition: What Drives Countervailing Power, Economic Journal
Gaudin, G., White, A., 2014. Unit vs. ad valorem taxes under revenue maximization.
Manuscript.
Genesove, D., Mullin, W. P., 1998. Testing static oligopoly models: Conduct andcost in the sugar industry, 1890-1914. RAND Journal of Economics, 29(2), 355–377.
Goldberg, P. K., Hellerstein, R., 2013. A structural approach to identifying thesources of local currency price stability. Review of Economic Studies, 80(1), 175–210.
Gopinath, G., Itskhoki, O., Rigobon, R., 2010. Currency choice and exchange ratepass-through. American Economic Review, 100(1), 304–336.
Griffith, R., Nesheim, L., O’Connell, M., 2015. Income effects and the welfare consequences of tax in differentiated product oligopoly. Manuscript.
Häckner, J., Herzing, M., 2016. Welfare effects of taxation in oligopolistic markets.
Journal of Economic Theory, 163, 141–166.
Hamilton, S. F., 1999. Tax incidence under oligopoly: a comparison of policy approaches. Journal of Public Economics, 71(2), 233–245.
Hellerstein, R. 2008. Who bears the cost of a change in the exchange rate? Passthrough accounting for the case of beer. Journal of International Economics, 76(1),
14–32.
Holmes, T. J., 1989. The effects of third-degree price discrimination in oligopoly.
American Economic Review, 79(1), 244–250.
Hong, G. H., Li, N. 2017. Market structure and cost pass-through in retail. Review
of Economics and Statistics, Forthcoming.

33

Johnson, D. B., Pauly, M. V., 1969. Excess burden and the voluntary theory of
public finance. Economica, 36(143), 269–276.
Kim, D., Cotterill, R. W., 2008. Cost pass-through in differentiated product markets:
The case of U.S. processed cheese. Journal of Industrial Economics, 56(1), 32–48.
Kimmel, S., 1992. Effects of cost changes on oligopolists’ profits. Journal of Industrial Economics, 40(4), 441–449.
MacKay, A., Miller, N. H., Remer, M., Sheu, G., 2014. Bias in reduced-form estimates of pass-through. Economics Letters, 123(2), 200–202.
Mahoney, N. and Weyl, E. G., 2017. Imperfect Competition in Selection Markets.
Review of Economics and Statistics
Miller, N. H., Osborne, M., Sheu, G., 2017. Pass-through in a concentrated industry:
Empirical evidence and regulatory implications. RAND Journal of Economics, 48(1),69–93.
Miller, N. H., Remer, M., Sheu, G., 2013. Using cost pass-through to calibratedemand. Economics Letters, 118(3), 451–454.
Miller, N. H., Remer, M., Ryan, C., Sheu, G., 2016. Pass-through and the predictionof merger price effects. Journal of Industrial Economics, Forthcoming.
Nevo, A., 1998. Identification of the oligopoly solution concept in a differentiatedproducts industry. Economics Letters, 59(3), 391–395.
Nocke, V., Schutz, N., 2016. Multiproduct-firm oligopoly: An aggregative games
approach. Manuscript.
Pigou, A. C., 1928. A study in public finance. Macmillan.
Peitz, M., Reisinger, M., 2014. Indirect taxation in vertical oligopoly. Journal ofIndustrial Economics, 62(4), 709–755.

34

Reny, P. J., Wilkie, S. J., Williams, M. A., 2012. Tax incidence under imperfectcompetition: Comment. International Journal of Industrial Organization, 30(5),399–402.
Riordan, M. H., 2008. Competitive effects of vertical integration. In P. Buccirossi(ed.), Handbook of antitrust economics, The MIT Press, 145–182.
Ritz, Robert A., 2017. Oligopolistic competition and welfare. In Luis Corchı̈¿œn andMarco Marini (eds.), Handbook of Game Theory & Industrial Organization, Edward
Elgar.
Shrestha, V., Markowitz, S., 2016. The pass-through of beer taxes to prices: Evidence from state and federal tax changes. Economic Inquiry, 54(4), 1946–1962.
Skeath, S. E., Trandel, G. A., 1994. A Pareto comparison of ad valorem and unit
taxes in noncompetitive environments. Journal of Public Economics, 53(1), 53–71.
Stolper, S. 2016. Who bears the burden of energy taxes? The critical role of passthrough. Manuscript.
Tremblay, M. J., Tremblay, V. J., 2016. Tax incidence and demand convexity in
Cournot, Bertrand, and Cournot–Bertrand models. Public Finance Review, Forthcoming.
Vickley, W. W., 1963. General and specific financing of urban services. In H.G.
Shaller (ed.), Public expenditure decisions in the urban community, Resources for
the Future, 62–90.
Vives, X., 1999. Oligopoly pricing: Old ideas and new tools. The MIT Press.
Weyl, E. G., Fabinger, M., 2013. Pass-through as an economic tool: Principle of incidence under imperfect competition. Journal of Political Economy, 121(3), 528–583.
White, A., Weyl, E. G., 2016. Insulated platform competition. Manuscript.
Yang, J. 1997. Exchange rate pass-through in US manufacturing industries. Review
of Economics and Statistics, 79(1), 95–104.
35

Zimmerman, P. R., Carlson, J. A., 2010. Competition and cost pass-through indifferentiated oligopolies. Manuscript.

36