2010 - Basso Ross (Price Fixing - JINDEC)

MEASURING THE TRUE HARM FROM PRICE-FIXINGTO BOTH DIRECT AND INDIRECT PURCHASERS�

LeonardoJ. Bassow

ThomasW. Ross§

Legal actions by direct and indirect purchasers to recover damages fromprice-fixing, common in the United States for years, are now appearingin a number of other countries. Traditional measures of damages areflawed as measures of the true harm suffered and will often significantlyunderstate that true harm.This paper providesmeasures of the degree ofunderstatement of the true harm when traditional approaches are usedand shows how the size of the error depends on the degree ofcompetitiveness of downstream markets. The paper also providesmeasures of distribution of the true harm between direct and indirectpurchasers.

I. INTRODUCTION

INANUMBEROFCOUNTRIESITISPOSSIBLE for individuals and businesses harmedby the price-fixing activities of sellers to launch legal actions against thosesellers in an attempt to recover damages.1 Many of these actions are ‘classactions’ that combine the (frequently) small individual damage claims ofnumerous plaintiffs into one large action.2 While these kinds of actions are

r 2010 The Authors.The Journalof IndustrialEconomicsr2010Blackwell PublishingLtd. and theEditorialBoardof TheJournalof IndustrialEconomics.Published byBlackwell Publishing, 9600GarsingtonRoad, OxfordOX4 2DQ,UK, and 350Main Street,Malden,MA02148,USA.

895

THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821Volume LVIII December 2010 No. 4

�The authors wish to acknowledge helpful discussions and/or communications with JamesBrander, Tim Brennan, John Connor, Martin Hellwig, Jan Tuinstra, Philip Williams, RalphWinter and The Editor, as well as very constructive comments from two anonymous referees;the very capable research assistance of Jennifer Ng; and the financial support for this researchfrom the Social Sciences and Humanities Research Council of Canada, the Phelps Centre forthe Study ofGovernment andBusiness in the Sauder School of Business and Instituto SistemasComplejos de Ingenierıa, Grants P-05-004-F y FB0816.

wAuthors’ affiliations: Department of Civil Engineering, Universidad de Chile, Avda.Blanco Encalada 2002, Santiago, Chile.e-mail: [email protected].

§Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver,British Columbia, Canada V6T 1Z2.e-mail: [email protected].

1 For example, in the United States this is authorized by Section 4 of the Clayton Act (1914)15U.S.C.A. §§12–27; inCanada by Section 36 of theCompetitionAct (1986)R.S.C. 1985, c. 19(2nd Supp.); and in Australia by Sections 82 and 87 of the Trade Practices Act (1974). Clarket al. [2004] discuss the situation in a number of European countries. See also Connor [2007,pp. 82–88].

2 For example, the vitamin price-fixing conspiracies of the 1990’s have led to damage awardsor settlements in class actions of $1.17 billion in theUnited States, CDN$140million inCanadaand AUS$30.5 in Australia. Sources: U.S. case – Wall Street Journal (Eastern Edition), NewYork, N.Y., November 4, 1999, p. A.3; Canadian case –National Post, DonMills, ON, June 1,

mailto:[email protected]

mailto:[email protected]

not without their critics, the trend clearly seems to be toward continued andeven growing support for private actions including class actions. Forexample, the recent report from the Antitrust Modernization Commission[Chapter 3, 2007] did not recommend fundamental changes to the regimemost generous to private plaintiffs. In addition, many European countriesare considering expanding the scope of private enforcement.3

In these price-fixing suits, apart from establishing that the defendantswere indeed responsible for the harm to plaintiffs – often accomplished bysimple reference to earlier convictions or guilty pleas from actionsprosecuted by antitrust agencies – the most important step is establishingwhat harm was done to plaintiffs as a way to establish the appropriate levelof damages to be paid.For our purposes, we draw an important distinction between these terms.

Here, ‘harm’ represents losses in economic surplus to direct and indirectpurchasers as a result of the price-fixing and its consequent downstreameffects, while ‘damages’ is a legal term representing payments defendantsmust make to plaintiffs as a result of the harms suffered. In some cases thelaw (and courts) may equate damages with proven harm, but in other casesthey may not be exactly the same thing. For example, the law may notrecognize that some harms are recoverable as damages4 and in some casesthe lawmaymake the legal damages somemultiple of the actual harm.5 Thissaid, we recognize that most measures of damages will be based to someextent on estimate of the true harm suffered, and therefore there is value ingetting that harm measured correctly.The purpose of this paper is to explore the relationship between the true

economic harm caused by price-fixing and the traditional measures ofdamages applied by courts, which focus on the overcharge for transactedquantities of the price-fixedproduct.Weargue that the standardmeasures ofdamage – the overcharge – may represent substantial underestimates of thetrue total harm, that is, the sum of the harm suffered by direct purchasers(those who bought directly from a cartel member) and the harm suffered byindirect purchasers (those who bought from direct purchasers or from thosefurther down the distribution chain). To show this, we consider a simplemodel with two productive stages followed by final consumption (e.g.,

2005, pg. FP2; Australian case –Wall Street Journal (Eastern Edition), New York, N.Y., July18, 2006, p. B8.

3On this, see Verboven and van Dijk [2007]. On April, 3, 2008, the European Commissionpublished a White Paper (European Commission [2008]) describing proposals to improveprivate-access regimes under European antitrust law. On this report, see, e.g., Tolaini andMorfey [2008].

4 For example, as a result of the decision in the Illinois Brick case discussed below, harm toindirect purchasers are not recoverable as damages in Federal price-fixing class actions in theUnited States.

5Under U.S. antitrust law, successful plaintiffs are entitled to damages equivalent to threetimes the harm established.

896 LEONARDO J. BASSO AND THOMASW. ROSS

r 2010 The Authors.The Journal of Industrial Economics r 2010 Blackwell Publishing Ltd. and the Editorial Board of The Journal of IndustrialEconomics.

manufacturer-retailer-consumer) in which the price-fixing (involving a non-marginal increase in price) takes place at the first level. We then address twospecific questions: (i) howmuch greater or smaller is the true total harm fromprice-fixing than the damages estimated by traditional methods and whatdetermines the size of this error, if there is one; and (ii) what determines theshare of this true total harm borne by direct purchasers as opposed toindirect purchasers? To be clear, whenwe use the terms ‘true total harm’ and‘total harm’ we are referring to all surplus losses by others in the price-fixedproduct’s vertical distribution chain. These will include lost profits fromother firms in the chain and lost consumers’ surplus going to finalconsumers. Total harm, as we use the term, therefore, includes the effectson total surplus generated in the vertical chain, excluding the profit gains ofthe conspiring firms.Our results suggest that traditionalmeasures of damages,may capture less

than half the total harmdone by price-fixing, when the price-fixed-product isan input and downstream markets are not perfectly competitive. We alsoshow that the fraction of the harm borne by direct purchasers will depend toa considerable degree on factors that influence the level of competitiondownstream and can range in our model from zero to 67%; andthat reducing the damages awarded to direct purchasers, based on the factthat part of the harm was passed-through via higher prices to indirectpurchasers – as it has been claimed – may not be justified.As we discuss below, in some jurisdictions the overcharge to direct

purchasers is seen as a measure of only the harm suffered by those directpurchasers, while any subsequent ‘downstream overcharge’ (i.e., higherprices charged by direct purchasers to indirect purchasers) serves as ameasure of the harm to those indirect purchasers. In our final section,therefore, we consider such a regime, showing that the overcharge is also animperfect estimator of direct purchasers’ harm; and that adding to theovercharge damages associated with the downstream overcharge does notlead to duplicative recovery of damages, as some have alleged.We also showthat this regime does not restore equality between total damage awards andthe sum of true harm of direct and indirect purchasers.Our work is related to some other recent contributions in the economics

literature looking at how higher prices are passed-through, how we mightbetter assess the harms caused by price fixing, and how those harmsmight bedistributed. Hellwig [2007] also argues that traditional overcharge measuresunderstate the true harm to direct purchasers of price-fixed products andrecognizes that if the downstream purchasing market is not perfectlycompetitive there are additional social costs created.6 Kosicki and Cahill

6He does, however, consider some interesting questions such as whether the cartel should beviewed as ‘responsible’ (in the sense of legally liable) for the harms done when their customers

MEASURING THE TRUEHARMFROM PRICE-FIXING 897


[2006] study the rate of pass-through of (marginal) price increasesdue to cartel conduct for a variety of differently-shaped demand and costcurves.Verboven and van Dijk [2007] provide an analysis of cartel damages and

pass-through complementary to that found here; they decompose the lostprofits suffered by direct buyers after a marginal price increase by the cartelinto various components (including a direct cost effect and a pass-througheffect) and assess the relative size of these effects as functions of the nature ofcompetition in the market.7 Their focus is on the degree to which a passing-on defense may be justified, and, importantly, they consider cases in whichnot all competitors in the plaintiffs’ industry purchased from the price-fixingfirms. Aswe do in our final section here, they also illustrate conditions underwhich direct purchasers’ harm is greater than the overcharge, yet they do notcompare the overcharge to total harm, and do not consider discrete priceincreases by the cartel.8

Boone and Muller [2008] present a model that explains the share ofdownstream harm suffered by final consumers after a marginal priceincrease by input producers. They do not determine the magnitude of thetotal harm or the extent to which traditional measures of damagesunderestimate the total harm; however, they do consider some importantestimation issues that would arise in applying this work in actual cases. Theyconsider fairly general functional forms for their demand and cost functions,but do restrict attention to the sharing of harm created by marginal inputprice increases.Finally, problems similar to those studied here can arise when the price-

fixing takes place some stage(s) downstreamwhile upstreammarkets are notperfectly competitive. Since our purpose is to illustrate how far adrifttraditional measures of damages can be when there are multiple steps in thedistribution chain and other steps are not perfectly competitive, we havefocused on the simplest kind of vertical chain. For a model with multiplestages and in which collusion make take place upstream or downstream, seethe recent paper by Han et al. [2009]. Among their results they show that,within the context of their model (e.g., linear demand, identical constantmarginal costs), the effect on total welfare (which in their case includes theprofits of the price-fixing firms and so is not equivalent to our notion of totalharm) of collusion is independent of the stage of the distribution chain in

increase their downstream prices and how damage measurement should change if it isdetermined that they are not responsible.

7 In our analysis wewill refer to the direct cost effect as the traditional overchargemeasure ofdamages, or the overcharge to direct purchasers.

8Our analysis here involves both, the consideration of discrete increases in price due to thecollusion and a comparison of the overcharge to total harm (direct plus indirect purchasersharm).



which that collusion took place.9 However, the size of the initial cartelovercharge does depend on where in the vertical chain firms are colluding.While there is some overlap with parts of a number of these papers, we

believe ours is the only treatment to compare the size and distribution of truetotal harm produced by a discrete, cartel-induced, increase in price, to avariety of candidate measures of damages including (i) the simpleovercharge and (ii) the overcharge plus the ‘downstream overcharge;’ andto provide numerical examples to illustrate the potential magnitudes oferrors associated with using incorrect measures of harm.

II. MEASUREMENT AND DIVISION OF DAMAGES IN PRACTICE

II(i). The Overcharge as a DamageMeasure and its Relation to Actual Harm

The most common approach to calculating damages takes the differencebetween the cartel’s price and the price that would have existed but for thedefendants’ price-fixing activity (i.e., the ‘per-unit price overcharge’) andmultiplies it by the number of units purchased by direct purchasers.10 Thismeasure, most often referred to as the ‘overcharge’ (sometimes the ‘directovercharge’), is illustrated in Figure 1. Let DI represent the demand forthe price-fixed good, w1 be the collusive price and w0 be the price that would

F

G

T

X1 X0 DI

DF

B

R

Price

w0

Ap1

p0=w1

Quantity

Figure 1

Overcharge in a Two-Layer Vertical Chain of Production

9They also show that some of their results generalize to wider sets of assumptions aboutdemand and cost curves.

10 This is why the emphasis in most of the work done by economists is on providingtechniques to determine this ‘but for’ price. See, e.g., Baker andRubinfeld [1999], Connor [2001& 2007], Fisher [1980], Rubinfeld and Steiner [1984] and White [2001]. Some of these providegeneral advice and others discuss particular case applications.



have obtained without (or ‘but for’) the price-fixing. The amount of ‘harm’suffered (the damages to be awarded) is then calculated as the per-unitovercharge (w1�w0) multiplied by the amount of the product sold duringthe period of collusion (X1). This would be areaR (for ‘Rectangle’) in Figure1: the ‘overcharge.’ The damages to be awarded to any one buyer are thencalculated as the per-unit overcharge times the quantity that buyerpurchased during the period of collusion.There are two reasons to believe that area R may be a significant

underestimate of the collective harm done to direct and indirect purchasers,even if the but-for price is correctly estimated. First of all, it is well-recognized that area T (‘Triangle’) above also represents lost surplus – thesewould be gains to buyers for units they would have purchased at the lowerbut-for price but which are not purchased at the cartel price.11 Area T istypically not estimated andhence not included in themeasure of damages fortwo main reasons: it requires additional information (the but-for quantity)and it is in any case often viewed as small relative to area R.12

The second reason why R is a poor measure of the total harm is the mainconcern of this paper. While standard discussions on measuring harmimplicitly or explicitly assume the first (direct) purchasers are also the finalconsumers, this is seldom the case. Most often cartels raise prices to directpurchasers who distribute and resell the product or who use the product asan input into the production of some other product.13 When this happens itis unclear in general how the overcharge R, or even RþT, would be relatedto true total harm. We illustrate this in Figure 1 using an example fromBrander and Ross [2006]. Let DF now represent the linear demand for thefinal product and let DI be the derived demand for the cartel’s input. If thedownstream firm is a monopoly which only resells the product withoutincurring further costs, the input prices w0 and w1 would translate into finalgood prices of p0 and p1 and quantities of Xo and X1, respectively.14

While final consumers’ surplus (CS) was originally equivalent to areasAþBþF, it is only A with the higher retail prices resulting from the inputcartel prices. The downstreammonopolist was initially earning profits equal

11 See, e.g., Connor [2007, pp. 88–90], Clark et al. [2004] and Brander and Ross [2006].Of course, the area is not a perfect triangle if the demand curve is not linear, but it is neverthe-less frequently referred to as the ‘triangle of deadweight loss.’

12Of course, itmay not be so small. See, for example, Cowling andMueller [1978, p. 729]whoshow that, in our notation, T is half as large as R (when demand is linear, marginal costs areconstant and the cartel chooses the full monopoly price), with the implication that ignoring it ismissing one third of the harm.

13 Consider for example the samples of cartel cases reviewed in Connor [2007] and inLevenstein and Suslow [2006] which include cartels related to the production of vitamins,lysine, sorbates, aluminum, copper, steel, sugar, sulfur and tin.

14 The fact that p0 andw1 are the same is convenient (i.e., it reduces clutter on the graph) and isattributable to our assumptions of linearity and the absence of any retail costs other thanwholesale price.



to RþTþG, but with the cartel these fall to B. The total harm (‘H’) to thedownstream producer and final consumers then amounts toH5RþTþFþG, which is indeed greater than R and RþT.15 AreaTþFþG represent the social value of the lost output (i.e., the social cost ofthe price-fixing), while R is the transfer of profits from downstreammonopoly to upstream cartel.These considerations lead to the first of our three research questions,

which we undertake in Sections 4 and 5, after introducing the model inSection 3:

(i) How good or poor an estimate of total harm are the measuresused traditionally in price-fixing damage actions and on what factorsdoes this accuracy or lack thereof depend? In addressing this question,we will in general consider the error associated with using the simpleovercharge – areaR in Figure 1 – as this is themost common approach todetermining damages. However, we will also consider the remainingerrorwhenwe addback in the triangle of direct purchasers’ surplus losses– area T above – to show that even this correction, which is widelyrecognized as appropriate though not actually applied, is frequently notenough.

Our research builds on a literature that seeks to establish thecorrespondence between input and output market Marshallian surpluses.For example, Jacobsen [1979] studies input price changes and derives therelationship between changes in input market surplus, downstream firmprofits and downstream consumers’ surplus for the cases in which thedownstream industry is either perfectly competitive or completely mono-polized. Quirmbach [1984] extends these results by considering a homo-geneous product oligopoly downstream.16 Basso [2006] provides a unifyingframework that nests and generalizes the models from these two papers andprovides a starting point for our analysis here.

II(ii). Distribution of Harm and the Addition of Downstream Damages

The question of just who is entitled to sue and thereby receive damageawards has become somewhat controversial. In the United States, the keySupreme Court decisions in the Hanover Shoe and Illinois Brick casesestablished that only direct purchasers should have the right to recoverdamages fromprice fixing under federal law.17However, in response to these

15Hellwig [2007] shows this as well in a similar setup.16Quirmbach’s analysis is restricted to changes in a single price only, however. See Basso

[2006] for other references to this literature.17Hanover Shoe Inc. v. United Shoe Machinery Corp., 392 U.S. 481 (1968) and Illinois Brick

Co. v. Illinois, 431 U.S. 720 (1977). There were two key arguments made in support of thisapproach: first that dividing harm among the many levels of purchasers is always going to be a



decisions, a number of U.S. states enacted what are frequently referredto as ‘Illinois Brick repealer laws’ that restore the rights of residents of thosestates who are indirect purchasers to pursue actions to recover damagesunder state antitrust law.18 As a result many indirect purchasers haverecovered for harms suffered. Importantly, as well, indirect purchaser suitsmay return at the federal level: the U.S. Antitrust ModernizationCommission [2007], after considering arguments for and against allowingindirect purchaser actions, recommended (recommendation 47) thatCongress should enact a statute that would, among other things, ‘OverruleIllinois Brick andHanover Shoe to the extent necessary to allow both directand indirect purchasers to sue to recover for actual damages from violationsof federal antitrust law’.19

Whether the Supreme Court was correct in Illinois Brick in trying tolimit the rights to damages to direct purchasers has been an active topic ofdebate,20 however it seems likely that we will see more indirect purchaseractions in the future. This means that we need to know something about thedivision of harm. In practice, this issue has usually been dealt with byestimating the degree to which direct purchasers ‘pass through’ input priceincreases via higher output prices; this has drawn considerable attentionfrom economists.21 The idea is that if direct purchasers increased pricesignificantly in response to input price increases, this could mean that mostof the harm was absorbed by indirect purchasers, and that this then is thegroup requiring compensation.Allowing direct purchasers to receive the fullovercharge, after they passed on much of that overcharge, would lead toduplication in the recovery of damages when indirect purchaser suits for therecovery of downstream harms are allowed as well.22

This leads to our second and third research questions. The secondquestion – which we address in Section 4.3 and illustrate numerically inSection 5 – relates to how the total harm suffered downstream of the cartel isshared by direct and indirect purchasers:

difficult, time-consuming and uncertain process about which courts may ultimately not haveconfidence; and, second, that deterrence was better served by letting the direct purchasers suefor the full amount of damages suffered even if this meant they were overcompensated in somecases.

18 See, e.g., Kosicki and Cahill [2006] and the references on state laws cited therein.19However, damages in such actions ‘could not exceed the overcharges (trebled) incurred by

direct purchasers.’ Also, there is no evidence yet that the U.S. government will adopt theserecommendations of the Commission.

20 See, e.g., Page [1999] and the references therein.21 See, e.g., Verboven and van Dijk [2007].22 Some states have addressed what they perceive may be such duplication. For example, the

District of Columbia Statute says that ‘a defendant shall be entitled to prove as a partial orcomplete defense to a claim for damages that the illegal overcharge has been passedon tootherswho are themselves entitled to recover so as to avoid duplication of recovery of damages.’ Citedin Kosicki and Cahill [2006].



(ii) Once we know how much the true total harm is, what fraction of thatharm would reside with direct purchasers and what fraction would fallto the indirect purchasers (here all final consumers)?

To answer this, and in contrast to much past work by economists in thisarea, we do not focus primarily on measures of pass-through. While suchcascading price changes are certainly happening in our model, our firstpurpose here is really to look at the totality of harm (which depends onquantity changes as well as price changes) and to establish how thatquantum of harm breaks down between direct and indirect purchasers.However, in doing so, we are able to draw some conclusions regarding thevalidity of the pass-through defense (for both marginal and non-marginalprice increases by the cartel).The third research question arises as a result of the regime created in U.S.

states that allow indirect purchaser actions. In these states, direct purchasershave the right to recover damages based on the full overcharge (area R inFigure 1) while the downstream indirect purchasers have the right to recoveradditional damages associated with the ‘downstream overcharge’ (area B inthe case illustrated in Figure 1). In such a regime, the overcharge can beviewed as an attempt to capture the harm to direct purchasers only, and nottotal harm. As mentioned above, concerns have arisen in these U.S. states,that duplicative recovery of damagesmay be occurring. From our graphicalanalysis, however, it is far from clear that the sum of direct and indirectpurchaser awards determined by these overcharges (e.g., areasRþB) wouldresult in an overestimate of the actual total harm caused by the price-fixing,nor that it gives to each downstream agent an award equal to the harm itsuffered. Hence, after considering how much the traditional overcharge Runderestimates the true total harm from price fixing, in Section 6 we alsoexamine:

(iii) How much closer (than earlier measures) would the combination ofdamages equal to sum of the (direct) overcharge plus indirectovercharge come to the true harm and could it ever over-estimate thetrue harm? And how does this division of damages between direct andindirect purchasers compare to the allocation of true harms suffered byeach group?

III. THEMODEL

As indicated, we use a simple model in which the cartelized industry isselling to a downstream market of buyers who in turn resell the product orembed it in anewproduct for sale to final consumers. There is therefore, afterthe direct purchasers, only one tier of indirect consumers.



To begin, we define a system of n linear and symmetric (possibly)differentiated final good demands given by:

ð1Þ piðqÞ ¼ a� bqi � eXj 6¼i

qj

Where a, b, e are positive parameters and b � e. If b4 e then products aredifferentiated and the system can be inverted to obtain:

ð2Þ qiðpÞ ¼ A� Bpi þ EXj 6¼i

pj

where A, B and E are positive parameters, easily derived as functions ofa, b, e.Depending on whether downstream firms compete in Cournot or

Bertrand fashion, the profits of the i-th firm will be given by:

ð3Þ piðqÞ ¼ qipiðqÞ � ciðqi;wÞ or piðpÞ ¼ piq

iðpÞ � ciðqiðpÞ;wÞ

where w is the vector of prices of Z inputs, and ci is the cost function. Notethat we allow for changes in a vector of input prices, w, rather than a just asingle input price. This does not mean that all inputs have to be price-fixed,because w1 may be equal to w0 in all but one component wz, zAZ, where Zdenotes the set of inputs. Considering many inputs, however, allows us toexamine what happens when various inputs are price fixed.23 Of course, thesimple single input case is nested within our model. Note also that our costfunctions are, at the outset, quite general. They do not have to be identicalacross firms and we do not begin with a fixed proportions productionassumption downstream. Therefore, all patterns of input substitution arecaptured in our approach. Stronger assumptions will be applied later in thepaper to push the analysis further.To accommodate differentmodels of competition in one simple structure,

we will use the familiar ‘conduct parameter’ approach, sometimes alsoknown as the ‘conjectural variations’ approach. Because we recognize theproblemswith interpreting this parameter literally as a conjecture, we simplyuse it as an expositional tool to allow us to provide for different types ofmarket competition in one model. The innovation here is that we use thisapproach in a differentiated products environment.24 Let ci

i � @ci(qi,w)/@qiand consider the following equation, where v is the conduct parameter:,

ð4Þ a� qið2bþ evÞ � eXj 6¼i

qj � cii ¼ 0

23 For example, the vitamin cartel manipulated prices on a variety of vitamin products.24Many other authors have taken this type of approach. For the homogenous products case

see, for example, Church and Ware [2000, p. 273].



Quite clearly, if v5 0 then (4) becomes the first-order condition of firm i inthe Cournot game. We can also try to find a value of v that makes (4)identical to the first-order conditions of the Bertrand game. It is tedious yetstraightforward to show that v should be:

ð5Þ vBertrand ¼ �e

bðn� 1Þ

1þ e

bðn� 2Þ

This value satisfies the condition � 1 � vBertrand � 0, because 0 � e � b.In fact, the value will always be larger than –1, except in the case in whiche5 b, which coincides with what we know from the homogenous case.25

Finally, if the n cost functions are identical and v5 n� 1, then it is easy toshow that (4) represents the outcome of collusion (and this is independent ofwhether firms choose prices or quantities).26

Hence, by using the three different values of v, we shall be able to say whathappens exactlywhen competition is of the Bertrand type, theCournot type,or when there is a cartel downstream.

IV. RESULTS

We can now evaluate the differences between the commonmeasure of harm,the overcharge, which for convenience we continue to refer to as R, and thetrue harm (H) as measured by the loss of surplus by direct and indirectpurchasers. We begin by comparing the total harmH to RþT and we starthere for two reasons: First, it is widely recognized that area T should beadded to area R in assessing harm from price-fixing even if it is never in factdone and here wewant to show that even this measure of whatmany think isthe ‘true’ measure of harm can be far from accurate in a vertical chain.Second, it is analytically simpler to see the error associated with usingR as ameasure of total harm as the product of two separate errors: that associatedwith R being less than the true surplus loss in the input market (RþT), andthat associated with the fact that even the full input market harm is less thanthe total harm (H). That is, R/H can be seen as the product of two terms:R/H5 (RþT)/H � R/(RþT). It is then convenient to evaluate the size ofthese errors separately. Importantly, as we show below, the part of the errorgiven by (RþT)/Hwill (given our assumptions about costs and the shape of

25 It is easy to show that, as long as the firms’ cost functions are not ‘too concave,’ a uniqueequilibrium exists for either the Bertrand or the Cournot game.Moreover, if the cost functionswere identical, the equilibriumwould be symmetric. In this paper, the focus is on cost functionsthat are either linear or convex. Existence, uniqueness and symmetry of equilibrium under costsymmetry follow directly fromTheorems (2.7), (2.8) andRemark (17) in Vives [1999, p.42–43].Formal proofs are available from the authors upon request.

26 If, in addition, marginal costs are constant, the collusive andmonopoly outcome coincide.



the demand curve) be independent of the degree of success the cartel enjoysin raising price. The second part of the error, however, will depend on theextent to which the cartel price exceeds the non-collusive price.

IV(i). The Error Associated with Using RþT to Measure Total Harm

Let xiz(q,w) be firm i’s conditional demand for input zAZ, and Xz ¼Pni¼1 xiz be input z’s total demand. These are the derived demands which, if

integrated between w0 and w1, deliver area RþT as in Figure 1. Formally,when there are many inputs that were potentially price-fixed, we have:27

ð6Þ Rþ T ¼Z w1

w0

Xz2Z XzðwÞdwz

While Jacobsen [1979] and Quirmbach [1984] derived the relationshipbetween RþT and total downstream surplus for certain specific cases, themodel restrictions imposed preclude the use of their results in the moregeneral model we are examining. To derivemore general results, we build onthework of Basso [2006].A series of steps,mapped out inAppendixA to thispaper, lead us to the following condition:

ð7Þ Rþ T ¼ �Xnj¼1

pjðq�;wÞ þ CSðq�;wÞ� bþ ev

2

Xnk¼1ðq�kÞ

2

��w1

w0

The right hand side of (7) has three terms. The first is the effect on directpurchasers’ profits, the second is the effect on indirect purchasers’consumers’ surplus, denoted by CS. If that were all there was on the righthand side, RþT would be a perfect measure of the total harm. However,there is a third term – negative in sign since q�kðw1Þ<q�kðw0Þ – suggesting thatthe amount represented byRþTmay be less than the total harm suffered bydirect and indirect purchasers.Specifically, equation (7) shows that, unless b5 e and v5 � 1, (i.e.,

homogenous products and Bertrand competition) area RþT alwaysunderestimates the total harm. The difference between RþT and the totalharm is dependent on the difference between the squares of downstreamfirms’ production in equilibrium at the fixed-prices and at the but-for prices.Equation (7) is quite general (though it does assume linear demand), and

allows us to assess what happens in several cases by simply replacing thevarious model parameters with candidate values. But to obtain more clear-cut comparative statics from (7) is difficult because there are opposing direct

27Note that in the case of multiple price changes, (6) is a line integral and not simply the sumof many one-dimensional integrals of the sort illustrated in Figure 1, unless the input demandsare independent.



and indirect effects when parameters change.28 A simple way to derive moreclear-cut comparative statics, and hence to see how the parameters of themodel affect the magnitude of the mistake, is to assume symmetry in costfunctions. This induces a symmetric downstreamNash equilibrium inwhichq�k ¼ q � Q=n 8k. Note that, despite this symmetry, the (common) costfunction does not require a specific functional form. With this additionalassumption we are able to establish that (see Appendix A):

ð8Þ Rþ T ¼ �Xnj¼1

pj þ CS

e

bðn� 1� vÞ

1þ e

bðn� 1Þ

0B@

1CA

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}D

��

��

w1

w0

Where consumers’ surplus is given by:

ð9Þ CS ¼ Q2ðbþ eðn� 1ÞÞ2n

If the term labeled D in (8) were to equal 1, areas RþT would exactly equalthe reduction in downstreamprofits and consumers’ surplus.However, sincee � b and � 1 � v � n� 1 for the cases considered here, then 0 � D � 1.Thus, we see again that the area RþT will in general be less than the truetotal harm suffered by direct and indirect purchasers.While the understatement of the true total harm is our focus here, it is also

important to recognize two things: First, as long as D4 0, RþT willoverstate the harm to direct purchasers alone. Therefore, in a legal regime inwhich direct purchasers were only permitted to recover for the harm theyactually suffered, area RþT would in general over-compensate them.Second, whenever the downstream market is monopolized (n5 1,

monopoly downstream; e/b5 0, multiple monopolies of highly differen-tiated products downstream; or v5 n� 1, a cartel) then D is 0 and thereforethe harm measured by RþT is exactly the loss of profits suffered by thefirm(s) downstream.29 This shows that, while downstream customers doindeed suffer from higher retail prices, these are harms in addition to theRþT harm suffered by the direct purchasers. Put another way,RþT is the

28For example, if v increases (e.g., amove fromBertrand toCournot competition), the size ofthe error associated with the RþT measure increases ceteris paribus, but, in fact, q�k woulddecrease aswell, reducing both the size ofRþT and the size of the true harm.The same is true ifcompetition is effectively reduced by decreasing e.

29 This result is certainly not new to us (see, e.g., Jacobsen [1979]) but we are not aware of itsbeing acknowledged in price-fixing damage actions. Where the RþT area may overstate theharm that remained with direct purchasers is in cases in which there are rivals competingdownstream such that each benefits (somewhat) indirectly through higher input prices becausethey induce rivals to raise their retail prices. Hellwig [2006] discusses this as well.



true measure of the harm suffered by the direct purchasers in these cases –that is, the harm they are left with after accounting for the higher prices theywill themselves collect from their customers. It would be incorrect to reducethe damages they collect belowRþT on the theory that they passed on someof this harm.30

WhileD serves as onemeasure of the error associatedwith the use ofRþTto measure total true harm, a second measure may be more convenient forapplied purposes. This measure expresses the amount of the total harm‘missed’ by RþT as a percentage of the total true harm. In other words, itanswers the question: if we use RþT as our measure of damage, whatpercentage of the total harm are we missing? Let F be this percentage error.Then F is given by:

F ¼ 1� Rþ T

�Pn

j¼1 pj þ CS�� w1

w0

0B@

1CA

Replacing RþT by its expression in (8) we get:

ð10Þ F ¼ 1�

Pnj¼1

pj þ CSe=bðn� 1� vÞ1þ e=bðn� 1Þ

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

D

��

��

w1

w0Pnj¼1 pj þ CS

�� w1

w0

To have expressions that depend only on the parameter values, we need anexplicit expression for Q�, which cannot be obtained in general.31 What isneeded is a more explicit cost function. We use the following:

ð11Þ ciðqi;wÞ ¼ qiðaþ wÞ

With this cost function, we are now considering changes in a single inputprice, w. Also, this cost function features the fixed-proportion property inwhich one unit of input is needed for one unit of output and imposesconstant marginal costs for firms: ci

i(qi,w)5 aþw.32 While this is the moststraightforward functional form to use for our purposes, we have repeated

30 This case also helps to illustrate the difference between the question of the extent to whichhigher upstream prices are passed through to downstream (indirect) purchasers and the actualsharing of harm as measured by the overcharge to direct purchasers.

31Recall that expression (8) for RþT was obtained assuming symmetry of the costfunctions, sowe need to use the expression forCS that follows from this symmetry assumption,that is, equation (9).

32Of course, given the assumption of fixed proportions, the assumption that thoseproportions are 1:1 is without further loss of generality.



the analysis that follows using a quadratic cost function that generates anupward-sloping marginal cost curve. The qualitative results are not muchaltered and the results are presented in Appendix B.With the cost function in (11) and the first-order conditions given by (4), it

is easy to solve for the equilibrium. Noting that, in equilibrium, qi 5Q/n, weget

ð12Þ Q ¼ nða� a� wÞ2bþ eðvþ n� 1Þ

And the (unique) equilibrium price is:

ð13Þ p ¼ aðbþ evÞ þ ðbþ eðn� 1ÞÞðaþ wÞ2bþ eðvþ n� 1Þ

Note that, given our fixed-proportions assumption, we have actuallyobtained the derived demand for the input, that is,X(w) � Q(w), whereQ isgiven by (12).Total downstream profits are then given by:

ð14ÞXnj¼1

pj ¼ nðbþ evÞða� a� wÞ2

2bþ eðvþ n� 1Þ½ �2

As for consumer surplus, using its expression for the symmetric case andusing (12) we get:

ð15Þ CS ¼ Q2ðbþ eðn� 1ÞÞ2n

¼ nðbþ eðn� 1ÞÞða� a� wÞ2

2 2bþ eðvþ n� 1Þ½ �2

With all this, we can calculate, for example, the area under the demandcurve, using equation (8). We obtain:

Rþ T ¼ �n ða� a� w1Þ2 � ða� a� w0Þ2� �

2 2bþ eðvþ n� 1Þð Þ

So, nowwe are prepared to look at the percentage error,F. Using changes in(14) and (15) and entering them into (10) we obtain:

ð16Þ F ¼1þ e

bv

3þ e

bð2vþ n� 1Þ

which enables the calculationof the percentage error in a number of differentmarket situations by replacing parameter values. Notice one importantfeature of this measure of the error from using RþT to which we alludedearlier: it does not depend on the fixed price, w1, or the ‘but-for’ price, w0. Itdepends only on model parameter values. This is an extremely useful



property that tells us that the magnitude of the error (in percentage terms) isnot influenced by the degree of success the cartel enjoyed at raising price.We are now in a position to establish a number of comparative statics

results with respect to the size of this error:33

Proposition 1. The following will increase the value of F, effectivelyincreasing the error associated with using RþT as a measure of the totalharm produced by upstream collusive price increases: (i) decreasing theintensity of competition as captured by increases in the value of theparameter v, (e.g., by moving from Bertrand to Cournot competition);(ii) reducing the degree of product substitutability as measured by e/b; and(iii) reducing the number of firms competing in the downstream market,n. Also, F is bounded above by 1/3.

Proof. These results are easily established by differentiating the expressionforF given in (16)with respect to the various parameter values. This gives us:

@F@v¼

e

b

e

bðn� 1Þ þ 1

� �

3þ e

bð2vþ n� 1Þ

h i2>0

@F@ðe=bÞ ¼ �

ðn� 1� vÞ

3þ e

bð2vþ n� 1Þ

h i2<0

Also, clearly

@F@e<0

and@F@b

>0

@F@n¼ �

e

b1þ e

bv

� �

3þ e

bð2vþ n� 1Þ

h i2<0

Themonotonicity of the first derivative tells us that the error gets larger aswemove to less competitive models of competition (e.g., Bertrand to Cournot,or Cournot to collusion). The second and third derivatives indicate thathigher levels of substitutability and the presence of a larger number of

33We have performed a similar exercise with respect to the error measure D and thequalitative results were essentially the same. That is, all the comparative statics had the samesign. See Basso and Ross [2007] for full details.



downstream competitors will lead to smaller errors. Finally, given the signson these derivatives we can find the highest possible value for F by setting vto the highest possible level we consider (i.e., v5 n� 1) and e/b and n to theirlowest possible levels (i.e., 0 and 1, respectively). Applying these values in(16) we find that, in this case, F5 1/3. QED

Proposition 1 establishes, for themeasureF, that the error associatedwiththe use of RþT will be larger the less competitive the downstream market.Any parameter change that effectively decreases the intensity of competi-tion: larger v, or decreasing e/b or n, will result in area RþT being a poorerestimate of the true harm from price-fixing.

IV(ii). The Error Associated with Using the Overcharge R to Measure theTotal Harm

Since RþT already underestimates the true total harm, it is clear that bylooking at the overchargeR alone, themistakemust be greater. To assess theadditional error here, we first determine what fraction ofRþT is missed byusing only R – that is, we calculate the ratio of T to RþT:

ð17Þ T

Rþ T¼ w1 � w0

2ða� aÞ � ðw0 þ w1Þ

Notice that this fraction does depend on the two prices. Comparative staticsreveal that:

@½T=ðRþ TÞ�@a

¼ � 2ðw1 � w0Þ2ða� aÞ � ðw0 þ w1Þð Þ2

<0

@½T=ðRþ TÞ�@a

¼ 2ðw1 � w0Þ2ða� aÞ � ðw0 þ w1Þð Þ2

>0

@½T=ðRþ TÞ�@w1

¼ 2ða� a� w0Þ2ða� aÞ � ðw0 þ w1Þð Þ2

>0

Hence, the relative size of T and RþT will depend on the marginal cost ofdownstream firms, the intercept of final consumer demands, and both thebut-for and cartel prices. For example, if input providers had flat marginalcosts, and were to charge marginal cost under competition and themonopoly price under collusion, it can be easily shown that T would behalf theR area, and thus a third of the combinedRþT area.34 To see this, let

34 This is the result demonstrated byCowling andMueller [1978, p. 729] and referred to earlier.



w0 5 c. The demand for the input X(w) is given by the right hand-sideof equation (12) and therefore simple calculations lead to a the monopolyprice given by w1 5 (a� aþ c)/2. Replacing w0 and w1 in (16), leads toT

RþT ¼ 13.

Under these conditions, which give us TRþT ¼ 1

3, we can assess the overall

error made by using R as an estimate of the total true harm. Since R is only2/3 ofRþT and we know thatRþT can be as little as 2/3 of the true harm,the measure R may be as little as 4/9 of the true harm. Thus, the commonpractice of using the overcharge, i.e., area R, to measure the total harm todirect and indirect purchasers can result in missing more than 50% of theharm.Note that using equation (16) and the analysis above one could, in

principle, correct themeasures of harmobtained fromRþTor fromR alonesince, from F, one could calculate the appropriate multiplier to get a betterestimate of the true total harm. We provide tables with the actual values forthese multipliers for a number of cases below. We first summarize theseresults in the following Proposition:

Proposition 2. The following will, in addition to those conditions given inProposition 1, increase the error associated with using the overcharge (R) asa measure of the total harm produced by upstream collusive price increases:(i) a smaller intercept of final consumer demand (i.e., lower a); (ii) increasesin themagnitude of the downstreamfirms’marginal costs (i.e., higher a); and(iii) higher input prices charged by the cartel (i.e., higher w1).

IV(iii). How is the Harm Shared between Direct and Indirect Purchasers?

The analysis to this point has already shed some light on questions related tothe distribution of harm between direct and indirect purchasers. Forexample, we have already seen that the area RþT will in generaloverestimate the harm suffered by direct purchasers, and that the exceptionarises when the downstream markets are monopolies, in which case RþTexactly measures the harm suffered by direct purchasers (despite the factthey raise the price to final consumers)We obtain now more general expressions for the amount of the true total

harm to direct and indirect purchasers. If the harmwere correctlymeasured,the percentage C of that harm that will be suffered by direct purchaserswould be:

ð18Þ C ¼

Pnj¼1 p

j�� w1

w0Pnj¼1 pj þ CS

�� w1

w0



Evaluating changes using (14) and (15) andentering them into (18)weobtain:

ð19Þ C ¼2 1þ e

bv

� �3þ e

bð2vþ n� 1Þ

We see immediately that C � 2F, that is, the percentage of harm falling ondirect purchasers is exactly two times the percentage error made by usingRþT as the measure of harm rather than the true harm.As above, one can calculate the value of C for a number of different

market situations just by replacing parameter values but, becauseC5 2F, ineachmarket situation the resulting value ofCwill simply be double the valueof F. This implies that the comparative statics results are thereforequalitatively the same, and that C will take on a maximal value of 2/3,leading immediately to Proposition 3.

Proposition 3. The fraction of total harm that falls on direct purchasersincreases with increasing values of the conduct parameter v, and withdecreasing levels of product substitutability and numbers of firms. Themaximal value of the fraction of harm falling on direct purchasers is 2/3.

Proof. Follows directly from the proof of Proposition 1 and the fact thatC5 2F. QED

These results present another consistent pattern – as each of theseparameters changes in a way that can be interpreted as intensifyingcompetition, the share of the harm that remains with direct purchasers falls.The fraction that direct purchasers face ranges froma lowof zero (when theyare effectively perfectly competitive) to a high of 2/3 (when they areeffectively monopolies). Something important to note is that if one is able tocalculate the true harm, then C times that figure is the harm suffered bydirect purchasers even though they may raise their price when facing thecartel’s overcharge. No further pass-through type of analysis is required toallocate the harm properly.

V. SOME NUMERICAL EXAMPLES

To this point, we have shown that the errors associated with the overchargeRand RþT as measures of total harm, and the share of the true harm that willfall ondirect purchasers, depend largely onparameter values of themarket. Toillustrate the potential magnitudes of these errors, we provide a set of tablesbelow that combine various assumptions about market parameter values.



V(i). Error from Using RþT to Estimate the Total True Harm

Tables I.A and I.B illustrate the error associated by the use of RþT tomeasure total harm by providing the appropriate multiplier to be applied toRþT to get a measure of the true total harm. For example, with 5 Cournotfirms and a value of e/b5 0.3, the true harm will be 1.31 times the harmmeasured by RþT.

Table I.A

RþTvs.TrueHarm: Cournot

e/b

0.3 0.6 1

n 1 1.50 1.50 1.505 1.31 1.23 1.17

10 1.21 1.14 1.09

Total harm is ‘x’ times RþT where x are cell entries.

Table I.B

RþTvs.TrueHarm: Bertrand

e/b

0.3 0.6 1

n 1 1.50 1.50 1.505 1.27 1.13 1.00

10 1.17 1.06 1.00

Total harm is ‘x’ times RþT where x are cell entries.

Table II.A

Rvs.TrueHarm: Cournot

e/b

0.3 0.6 1

n 1 2.25 2.25 2.255 1.97 1.84 1.75

10 1.82 1.70 1.64

Total harm is ‘y’ times R where y are cell entries.

Table II.B

Rvs.TrueHarm: Bertrand

e/b

0.3 0.6 1

n 1 2.25 2.25 2.255 1.90 1.69 1.50

10 1.76 1.60 1.50

Total harm is ‘y’ times R where y are cell entries.



V(ii). Error from Using R to Estimate Total True Harm

To produce a similar table assessing the error from using the overcharge Ralone, we need some additional assumptions. Specifically, we assume flatmarginal costs of input production; a but-for price equal to marginal cost;and a fixed price equal to themonopoly price (which is a worst-case scenariofor downstream market power, tending to make these errors as large aspossible).Tables II.A and II.B help to illustrate just how very inaccurate the

overcharge – the most commonly applied measure of harm – can be as ameasure of the true total harm suffered by direct and indirect purchaserswhen there is market power downstream. Even with five competing firms, ife/b5 0.3, R will measure only about half of the total true harm.

V(iii). Percentage of Harm Suffered by Direct Purchasers

Finally, Tables III.A and III.B present examples to illustrate how the shareof true harm that is suffered by direct purchasers will depend on the values ofthe various parameters. Again, the share falling on direct purchasers canrange from zero to 2/3. For example, if five direct purchasers compete in aCournot fashion and e/b5 0.6, just over 37% of the true harm will be borneby direct purchasers.

VI. COMBINING DIRECT AND INDIRECT DAMAGE ACTIONS

For a legal environment, such as that which exists in some U.S. states, inwhich direct purchasers are permitted to recover damages based on the full

Table III.A

Fraction ofHarm toDirect Purchasers: Cournot

e/b

0.3 0.6 1

n 1 0.67 0.67 0.675 0.48 0.37 0.29

10 0.35 0.24 0.17

Cell entries correspond to fraction of true harm borne by direct purchasers.

Table III.B

Fraction ofHarm toDirect Purchasers: Bertrand

e/b

0.3 0.6 1

n 1 0.67 0.67 0.675 0.42 0.22 0.00

10 0.29 0.12 0.00

Cell entries correspond to fraction of true harm borne by direct purchasers.



overcharge they paid and then indirect purchasers are entitled to damagesbasedon the resulting effects ondownstreamprices (i.e., on the ‘downstreamovercharge’), we might consider three additional questions. First, if thetraditional overcharge is meant only to account for harm to directpurchasers, how good an estimate is it for the true surplus loss suffered bydirect purchasers? Second, if we combine these two damage actions, addingthe overcharges paid by direct and indirect purchasers, how close to the trueharm is this combined overcharge measure? And third, how does thisallocation of damages to direct and indirect purchasers compare to the ratioof true harms suffered?

VI(i). Direct Overcharge as an Estimate of Harm Only to DirectPurchasers

We can use the tables we have already created to answer this question. First,notice that the total true harm (H) is x times the direct overcharge (R inFigure 1), with x given by Tables II.A and II.B. The harm to directpurchasers is givenby y times the total true harmwith y givenbyTables III.Aand III.B. Therefore the harm to direct purchasers will be given by:

Xn

j¼1 pj

�� w1

w0¼ y H ¼ x y R

Hence, if x y4 1, the direct overcharge (R) underestimates the harmsuffered by direct purchasers. If x yo 1, the direct overcharge over-estimates the harm suffered by direct purchasers. The values of z5 x yfor the various parameter configurations are given in Tables IV.A and IV.B.So, as can be seen, R is a very poor estimator of harm to direct purchasers –underestimating harm when these purchasers have a lot of marketpower but greatly overestimating it when there is limited marketpower downstream. Notice that, contrary to a popular (non-economic)argument, a direct purchaser monopolist by virtue of its own marketpower is not more easily able to ‘pass on’ the harm to its own customers.In particular, and as expected given our previous results, R here fallswell short (missing by a third) of compensating a monopolist directpurchaser.35

35 Intuition for this result was discussed above. In Figure 1 (where a number of simplifyingassumptions were made) the monopolist earns RþTþG5 2R before the cartel, sinceT5G5R/2, but earns onlyB5R/2 after the price increase. That is whyRþTwas exactly theharm of the monopolist, but R falls short because it misses area T. Van Dijk and Verboven[2009] find thatR exactly compensates amonopolist direct purchaser (their Proposition 1). Thedifference in our result is explained by the fact that they consider only amarginal price increase,while we deal with discrete price increases.



VI(ii). Recovering both Direct and Downstream Overcharges

The second question asks, if direct purchasers are allowed to recover basedupon the full overcharge they paid, and indirect purchasers are allowed torecover based on the downstream overcharge, will the combined overchargesprovide a closer estimate to the true harmcreatedby the price fixing, or could itin fact overestimate the total true harm? To answer this question, we first needto derive the ‘pass-through rate’ (‘PTR’), that is, the change in downstreamprice divided by the change in the upstream (collusive) price. For our example,this is easy to obtain. The price for the intermediate output as a function of theinput price is given by (13) and from there we can directly obtain:

PTR ¼ p1 � p0

w1 � w0¼ ðbþ eðn� 1ÞÞ

2bþ eðvþ n� 1Þ � 1

The PTR in our numerical examples are presented in Tables V.A and V.B.36

Next, if we let R0 represent the combined direct and indirect purchaser over-charges,wecanwrite:R0 ¼ R�ð1þ PTRÞ.Fromouranalysisaboveweknowthatthe total true harm (TH) is given by: TH ¼ x�R, with x given by Tables II.Therefore, we have R0 ¼ ðTH=xÞ�ð1þ PTRÞ ! TH ¼ x=ð1þ PRTÞ�R0.

The values of x/(1þPRT) for our numerical examples are given in Tables

Table IV.B

Rvs. Harm toDirect Purchasers: Bertrand

e/b

0.3 0.6 1

n 1 1.50 1.50 1.505 0.80 0.37 0.00

10 0.51 0.19 0.00

Harm to direct purchasers is ‘z’ times R, where z are cell entries.

Table IV.A

Rvs. Harm toDirectPurchasers: Cournot

e/b

0.3 0.6 1

n 1 1.50 1.50 1.505 0.95 0.68 0.51

10 0.64 0.41 0.28

Harm to direct purchasers is ‘z’ times R, where z are cell entries.

36 These values clearly depend on the assumptions we have made about demand and costcurves. Kosicki and Cahill [2006] provide a detailed analysis of the pass-through rate for avariety of shapes of demand and cost curves.



VI.A and VI.B. Hence, as expected, R0 will not underestimate total harm asmuch as R does. However, for lower levels of competition downstream, itwill still underestimate total harm. It may however, overestimate totaldamages in some cases (i.e., some cell entries are less than 1), somethingneither R nor RþT would ever do. In such cases, damage awards equal tothe sum of the overcharges paid by direct and indirect purchasers would belarger than the true harm created by the price-fixing.

TableV.A

Pass-ThroughRates: Cournot

e/b

0.3 0.6 1

n 1 0.50 0.50 0.505 0.69 0.77 0.83

10 0.79 0.86 0.91

Pass-through rates (PTR).

TableV.B

Pass-ThroughRates: Bertrand

e/b

0.3 0.6 1

n 1 0.50 0.50 0.505 0.73 0.88 1.00

10 0.83 0.94 1.00

Pass-through rates (PTR).

TableVI.A

R0vs.TrueHarm: Cournot

e/b

0.3 0.6 1

n 1 1.50 1.50 1.505 1.17 1.04 0.95

10 1.02 0.91 0.86

True harm is ‘r’ times R0 where r are cell entries.

TableVI.B

R0vs.TrueHarm: Bertrand

e/b

0.3 0.6 1

n 1 1.50 1.50 1.505 1.10 0.90 0.75

10 0.96 0.83 0.75

True harm is ‘r’ times R0 where r are cell entries.



The final question asks how the allocation of damages to direct andindirect purchasers compares to the allocation of true harms suffered. Toanalyze this, one needs to compare numbers in Table III (which shows thefraction of true harm borne by direct purchasers) to R/(R(1þPTR))5 1/(1þPTR), which is the fraction of total damages that are awarded to directpurchasers. The latter can easily be calculated from Tables V, and resultsclearly show that it is not only the case that under this regime damages donotcorrespond to true harm, but also that damages are not proportionallyawarded. For example, for n5 10, e/b5 1 and Cournot competition, theshare of harm suffered by direct purchasers is 17%, while with R andR�PTR, the share of damages direct purchasers obtain is 52%.

VII. DISCUSSION AND CONCLUSIONS

The primary purpose of this paper has been to demonstrate just how large anerror can be made when one measures the harm due to price-fixing usingsimple overcharge measures – particularly when the price-fixing activitieswere undertaken by input producers – as has often been the case. Our resultsshow that, as downstream markets get less competitive (measured in anumber of ways), the error associated with simple overcharge methods getslarger. We also use our model to provide some guidance as to how the trueharm as a result of price-fixing will be allocated between direct and indirectpurchasers. We show that this is also highly dependent on the degree ofcompetition downstream, making the portion of harm that resides withdirect purchasers highly variable.As damage awards are likely to depend on measures of actual harm, we

think it is important to get these measures right. While we fully understandthat further work needs to be done to consider alternative market structuresand functional forms not evaluated here, we believe our resultsmaybe usefulin a number of ways. At a most basic level, our results point to the fact thatsimple overcharge estimates may be gross understatements of the true harmsuffered by others as a result of cartel pricing. These are errors of the firstorder such that simple overcharges cannot in general even be thought of as‘rough approximations’ of the true harm. Just understanding how far fromthe true harm the simple overcharge can be, as illustrated by the examples inour tables, could be helpful to enforcement agencies and courts (and settlingparties) in determining appropriate fines and damage awards.Our results also show that legal regimes that allow direct purchasers to

recover the direct overcharge and then allow indirect purchasers to recoverthe ‘downstreamovercharge’may actually be allowingdamagesmuch closerto the actual harm.37

37However, the allocation of those damages between direct and indirect purchasers will notgenerally correspond to the actual allocation of the economic harm suffered by those groups.



Secondly, our results provide some guidance as to how harm is sharedbetween direct and indirect purchasers and so might be helpful in legalregimes in which both types of purchasers have rights to recover damages(which seems likely to become more common). Courts or settling partiescould, for the purposes of allocating an award between indirect and directclasses of buyers, workwith the techniques described here to create estimatesof how the harm was shared.Conventional approaches tend to use incorrect measures of harm – such

as the overcharge – and then apply pass-through share estimates (possiblycoming from reduced form regressions) in order to exclude from the directpurchasers’ damage awards, any harm that was allegedly passed ontoindirect purchasers through higher output prices.We show that this practicemay be incorrect. For example, we show that it is incorrect to think thatbuyers which are monopolists in their output markets pass on some of, andtherefore suffer less than, the overcharge on the units they purchase. In factthe simple overcharge measure still understates the harm suffered bymonopoly purchasers (and does not consider the harm to indirectpurchasers at all).Under some circumstances, our method may be a useful alternative to

actually trying to estimate the degree of pass-through and the subsequentharm further downstream from the direct purchasers. If a court wants toobtain an accuratemeasure of the degree of pass-through (using for exampleformulas such as the ones neatly obtained by Verboven and Van Dijk[2007]), key parameters for the market – such as number of firms, type ofcompetition, and degree of product substitutability – would need to bechosen. But, in such a case, the true measure of harm could be obtained byadjusting the traditional overcharge measure, and from there the actualdivisionof harm could be obtained aswell. Therewould be noneed touse theextent of pass-through or even to know it.Clearly, this paper is hardly the last word on these topics – there is a great

deal more that could be done to refine and expand on these results. Forexample, most of the analysis above was done for linear cost and demandfunctions. We did this to help us focus on the important first-order effects,but certainly future work should consider the consequences of changing thefunctional forms.38 Other important features that were not captured inour model, but which we believe are worthy of further work include:

38As indicated above, we have begun work in this direction by considering the case of anupward sloping, linear marginal cost curve. These results are reported in the appendix. Whilethe qualitative results are very similar, there certainly are some differences, including: (i) theerror associated with using RþT will be somewhat smaller with rising marginal costs; (ii) theerror shrinks as the marginal cost curve becomes steeper; and (iii) the fraction of the harmfalling on direct purchasers will be somewhat greater with risingmarginal costs andwill only bezero if the number of firms goes to infinity. Also, there is a nice discussion of the effects ofdifferent types of demand curves in Kosicki and Cahill [2006].



(i) downstream prices that adjust to increasing input prices only with a lag;(ii) more than two stages in the vertical chain with the price-fixing occurringat any of the stages, and; (iii) direct purchasers’ being able to employ someform of non-linear pricing.

APPENDIX A

This appendix provides the technical background, drawing on the results of Basso

[2006] that will take us from equation (6) in the text to equation (7). To begin, we note

that Basso shows that, if there exists a differentiable function H(q,w) such that:

ðA1Þ @Hðq�;wÞ=@qi ¼ @piðq�;wÞ=@qi; 8i

ðA2Þ @Hðq�;wÞ=@wz ¼ �Xi

@ciðq�;wÞ=@wz; 8z

then

ðA3ÞZ w1

w0

Xz2Z XzðwÞdwz ¼ � H q�ðwÞ;wð Þj jw1

w0

where q�ðwÞ denotes an interior Nash-equilibrium of the downstream game as a

function of w. Thus, this result provides a useful way to describe what is captured

by the inputmarket surplus, that is, areaRþT. It suffices to find a functionH fulfilling

(A1) and (A2) above, which essentially means that we look for a function with

critical values that coincide with the equilibria of the non-cooperative game we are

studying.

As Basso [2006] shows, there is a subset of the H functions above that fulfill the

required conditions and that are particularly well-behaved: the exact potential

functions of potential games (Monderer and Shapley [1996]). These functions fulfill a

stronger condition than (A3); as the equality holds for every value of q and not only at

equilibrium points, there is a well-defined way to calculate them, and they are defined

up to an additive constant (and the constant is unimportant in (A3)). As it happens, the

gamewehavehere is a potential gameand, hence, an exact potential function exists. It is

given by:39

ðA4Þ Hðq;wÞ ¼Xnj¼1

a� bþ ev

2

� �qj

� �qj � cjðqj ;wÞ

h i� e

Xn�1k¼1

Xnj¼kþ1

qkqj

It is straightforward to verify that:

@H

@qi¼ a� qið2bþ evÞ � e

Xj 6¼i

qj � cii ¼@pi

@qi

��Conj:Var:

39 See Basso [2006], Lemmas 1 and 2, for explanations onhow to determinewhether a game ispotential or not, how to obtain the potential function when it exists, and for other properties ofpotential functions.



So condition (A1) is fulfilled (and for every value of q). It is also easy to see that,

@H

@wz¼ �

Xj

@cjðqj ;wÞ@wz

So that condition (A2) is fulfilled.Hence, from (A3) and Proposition (1) in Basso [2006]

we obtain that:

ðA5Þ Rþ T ¼ � Hðq�;wÞj jw1

w0

Tomake (A5) useful, we need to relateH to downstreamprofits and (final) consumer

surplus. For this, we first rewrite H as:

ðA6Þ Hðq;wÞ ¼Xnj¼1

pjðq;wÞ þ eXnk¼2

Xk�1j¼1

qkqj �Xnj¼1

ev

2q2j

Next, consumer surplus is given by the line integralR ap

Pi qiðpÞdpi, which is

straightforward to compute since the solution is path independent. A linear integration

path leads to

ðA7Þ CS ¼ ðb=2ÞXk

q2k þ eXnk¼2

Xk�1j¼1

qkqj

Replacing (A6) and (A7) in (A5) we finally obtain:

ðA8Þ Rþ T ¼ �Xnj¼1

pjðq�;wÞ þ CSðq�;wÞ� bþ ev

2

Xnk¼1ðq�kÞ

2

��w1

w0

which is Equation (7) in the text.

With cost symmetry, consumer surplus in (A7) is given by:

ðA9Þ CS ¼ Q2ðbþ eðn� 1ÞÞ2n

And the last term on the RHS of (A8) is now

bþ ev

2

Xnk¼1ðq�kÞ

2 ¼Q2ðbþ evÞ2n

:

Hence, (A8) can be re-written as:

ðA10Þ Rþ T ¼ �Xnj¼1

pj þ CS

e

bðn� 1� vÞ

1þ e

bðn� 1Þ

0B@

1CA

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}D

��

��

w1

w0

Which is equation (8) in the text.



APPENDIX B

In this appendix we repeat much of the analysis of Section 4 using a quadratic cost

function that gives us a rising marginal cost curve. We number equations here

according to the related equations in Section 4. Specifically, we use the following:

ðB11Þ ciðqi;wÞ ¼ qiðaþ wÞ þ b2q2i

Therefore, marginal costs for firm i will be given by: ciiðqi;wÞ ¼ aþ wþ bqi. With this

cost function, the first-order conditions for profit-maximization become:

@pi

@qi¼ a� qið2bþ evÞ � e

Xj 6¼i

qj � ðaþ wÞ � bqi ¼ 0

Solving for qi and imposing qi 5Q/n, we get total output and equilibrium price as:

ðB12Þ Q ¼ nða� a� wÞ2bþ bþ eðvþ n� 1Þ

ðB13Þ p ¼ aðbþ evþ bÞ þ ðbþ eðn� 1ÞÞðaþ wÞ2bþ bþ eðvþ n� 1Þ

Total downstream profits are then given by:

ðB14ÞXnj¼1

pj ¼ nð2bþ 2evþ bÞða� a� wÞ2

2 2bþ bþ eðvþ n� 1Þ½ �2

Consumer surplus, solved for as before but using these altered functions becomes:

ðB15Þ CS ¼ Q2ðbþ eðn� 1ÞÞ2n

¼ nðbþ eðn� 1ÞÞða� a� wÞ2

2 2bþ bþ eðvþ n� 1Þ½ �2

And therefore, the percentage error term, F, now becomes:

ðB16Þ F ¼1þ e

bv

3þ e

bð2vþ n� 1Þ þ b

b

If the harmwas correctly measured, the percentage,C, of that harm that will be fall on

direct purchasers would be:

ðB19Þ C ¼2 1þ e

bv

� �þ b

b

3þ e

bð2vþ n� 1Þ þ b

b

Notice the following about these new formulae:

(i) It remains true thatF andCdo not dependon the level at which pricewas fixed,w1,

or the ‘but-for’ price, w0. They depend only on model parameter values.

(ii) It is no longer true that C � 2F when b4 0. Indeed, in general, C � 2F.



(iii) The values of C and F associated with particular values of n, v and e/b will

change, of course. The signs of the comparative statics on F and C are the same,

however. There are new comparative statics, which are straightforward to derive:

@F/@bo 0 and @C@b ¼

bþeðn�1Þ½3bþeð2vþn�1Þþb�2>0. That is, increasing the steepness of the

marginal cost curve will, ceteris paribus, reduce the error associated with using

RþT tomeasure total harm, and it will increase the share of harm falling on direct

purchasers.

(iv) With quadratic cost functions, the percentage error in usingRþT, F, reaches 1/3only formonopolies (e5 0, or v5 0 and n5 1) or downstream collusion (v5 n� 1)

andwhen b5 0. In ‘perfect competition’ cases (i.e., homogenous product Bertrand

competition or when n ! 1), the error, F, is still zero. Note from (B14) that

profits are not zero in this case, however, unless n ! 1.

(v) With quadratic cost functions, the upper limit percentage on the harm that falls on

direct purchasers is no longer 2/3 – it will be higher. The share is no longer zero

under homogeneous product Bertrand competition. It is still zero when n ! 1.

To illustrate the potential impact of rising marginal costs in this model, we present the

tables fromSection 5here amended to allowb5 b, giving usTables B.I.A,B.I.B, B.II.A

and B.II.B.

In Tables B.I.A, B.I.B, B.II.A and B.II.B we see that the errors in using RþT or R,

thoughoften still large, are both somewhat lesswhenwehave risingmarginal costs. The

differences are smaller when n is large or e/b is close to 1.

When compared with their counterparts in Section 5, Tables B.III.A and B.III.B

show that the fraction of harm that remains with direct purchasers will be higher with

rising marginal costs.

Table B.I.A

RþT vs.TrueHarm: Cournot (RisingMC)

e/b

0.3 0.6 1

n 1 1.33 1.33 1.335 1.24 1.19 1.14

10 1.18 1.12 1.08

True Harm is ‘x’ times RþT where x are cell entries.

Table B.I.B

RþT vs.TrueHarm: Bertrand (RisingMC)

e/b

0.3 0.6 1

n 1 1.33 1.33 1.335 1.20 1.10 1.00

10 1.14 1.06 1.00

True Harm is ‘x’ times RþT where x are cell entries.



REFERENCES

Antitrust Modernization Commission (United States), 2007, Report and Recommenda-tions, available at http://www.amc.gov/report_recommendation/amc_final_report.pdf.

Baker, J. B. and Rubinfeld, D. L., 1999, ‘Empirical Methods in Antitrust Litigation:Review and Critique,’ American Law and Economics Review, 1, pp. 386–435.

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e/b

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n 1 2.00 2.00 2.005 1.86 1.78 1.71

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Table B.II.B

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n 1 75.00 75.00 75.005 57.69 46.88 37.50

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