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Electronic copy available at: http://ssrn.com/abstract=1387096Electronic copy available at: http://ssrn.com/abstract=1387096
The Overcharge as a Measure for Antitrust Damages
Martijn A. Han, Maarten Pieter Schinkel and Jan Tuinstra
Amsterdam Center for Law & Economics Working Paper No. 2008-08
The complete Amsterdam Center for Law & Economics Working Paper Series is online at: http://ssrn.acle.nl
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Electronic copy available at: http://ssrn.com/abstract=1387096Electronic copy available at: http://ssrn.com/abstract=1387096
The Overcharge as a Measure for AntitrustDamages∗
Martijn A. Han†, Maarten Pieter Schinkel‡ and Jan Tuinstra§
July 2009
Abstract
Victims of antitrust violations can recover damages in court. Yet, the quan-tification of antitrust damages and to whom they accrue is often complex. Anillegal price increase somewhere in the chain of production percolates throughto the other layers in a ripple of partial pass-ons. The resulting reductions insales and input demands lead to additional harm to both downstream (in)directpurchasers and upstream suppliers. Nevertheless, U.S. civil antitrust litigationis almost exclusively concerned with direct purchaser claims for (treble) dam-ages calculated on the basis of the overcharge. Similar best practice rules areemerging in Europe. In this paper, we show that there is no structural rela-tionship between the direct purchaser overcharge and the true harm inflictedby an antitrust violation on all of the direct and indirect purchasers and sellersin the chain of production.
JEL-codes: C13, D43, L41.Keywords: Antitrust, damages, pass-on, overcharge.
1 Introduction
Anticompetitive acts to eliminate competition and prevent new entry can cause severeand widespread harm. In the U.S., under Section 4 of the Clayton Act: “Any personwho shall be injured in his business or property by reason of anything forbidden in theantitrust laws [...] shall recover threefold the damages by him sustained.” The vast
∗We are grateful to Simon Anderson, Leonardo Basso, Patrick van Cayseele, John Connor, An-drew I. Gavil, Franklin M. Fisher, Martin Hellwig, Pierre Larouche, William Page, Martin Peitz andJeroen van de Ven for constructive comments to earlier versions of this paper. We also thank semi-nar participants of the 2007 workshop on The Law and Economics of Competition Policy in Bonn,the 6th IIOC meeting in Washington, DC, the 7th IIOC meeting in Boston, the 2008 workshop onPrivate Enforcement of Competition Law at TILEC, University of Antwerp, ESMT in Berlin, ZEWin Mannheim and The Netherlands Competition Authority NMa. Opinions and errors remain ours.
†Faculty of Economics and Business and ACLE, University of Amsterdam.‡Faculty of Economics and Business and ACLE, University of Amsterdam. Corresponding author
at: Roetersstraat 11, 1018WB Amsterdam, The Netherlands. E-mail: [email protected].§Faculty of Economics and Business and CeNDEF, University of Amsterdam.
1
majority of civil actions for antitrust damages concerns cartels. A similar litigationpractice is currently developing in Europe.1
The identification of antitrust harm can be complicated. In longer supply chains,in which one product is an input in the production of the next, an illegal priceincrease somewhere in the chain can percolate through to the other layers in a rippleof partial pass-ons. The resulting reductions in sales cause additional harm to directand indirect customers and suppliers of the wrongdoer(s).In order to determine who is affected by an antitrust violation and to what extent,
in principle all actual trades need to be compared to what would have been the mar-ket allocation without the anticompetitive behavior–the so-called ‘but-for’ world.2
In practice this is often difficult. At a minimum, it requires information about con-sumer demand and the structure of the market, such as the number of layers in theproduction chain, the type and level of competition amongst firms in each layer, theirproduction technologies and costs.In the U.S., some of these complexities have been reduced by case law. At least
since Chattanooga Foundry (1906) have direct purchasers been entitled to recoverdamages on the basis of the overcharge they paid as a result of the antitrust infringe-ment.3 According to this prevailing method, basic damages–before trebling andinterest, if applicable–are calculated as the difference between the anticompetitiveprice and the competitive but-for price, multiplied by the amount actually purchased.The overcharge ignores lost profits on transactions that could have been made at lowerprices, which courts have been reluctant to award.Indirect purchasers often do not have standing to sue. In Hanover Shoe (1968),
the Supreme Court ruled against the use of the pass-on defense in Federal antitrustdamage actions.4 In a pass-on defense, the defendant attempts to show that theplaintiff did not in fact suffer the amount of damages claimed on the argument thatit was able to pass on all or part of the overcharges on downstream to it customers.In addition, in Illinois Brick (1977) the Supreme Court established that only thedirect purchasers have legal standing in Federal court to sue for antitrust damages.5
Hanover Shoe and Illinois Brick together cemented the use of the overcharge, whichindeed disregards pass through.Direct suppliers to a buyers cartel that colluded to depress input prices can in
principle maintain a treble-damages action after Mandeville Island Farms (1948).6
Yet, standing was denied by the Supreme Court to suppliers damaged by anticom-petitively restricted demand for their produce in Associated Contractors (1983).7 As
1See, e.g., Commission of the European Communities (2008).2See Fisher (2006) for a survey of some of the methods that can be applied in the determination
of but-for prices.3Chattanooga Foundry & Pipe Works v. Atlanta, 203 U.S. 396 (1906).4Hanover Shoe Inc. v. United Shoe Machinery Corp. 392 U.S. 481 (1968).5Illinois Brick Co. v. Illinois 431 U.S. 720 (1977). Later, in California v. ARC America Corp.
490 U.S. 93 (1989) the Supreme Court left it to the discretion of individual states whether or notto allow indirect purchaser suits. As a result, the rules on antitrust standing vary across the states,with presently a small majority allowing indirect purchaser suits under state law. See also Schinkelet al. (2008).
6Mandeville Island Farms, Inc. v. American Crystal Sugar Co., 334 U.S. 219 (1948).7Associated Gen. Contractors of Cal., Inc. v. Carpenters, 459 U.S. 519 (1983). Suppliers
2
a result of these legal constraints, in the vast majority of U.S. antitrust damages ac-tions, the plaintiffs are direct purchasers and their claim is based on the overcharge.In this paper, we consider the effects of anticompetitively raised prices somewhere
in a chain of production with an arbitrary number of layers. We will mostly refer tocartels, but our results have wider antitrust application. Competition in each layer isspecified between perfect competition and monopoly. This allows us to exactly char-acterize the effects of the cartel’s direct and indirect purchasers, as well as its directand indirect suppliers. We assess the bias introduced by relying on the overchargeon the direct purchasers–which we refer to as the ‘direct purchaser overcharge’–forthe estimation of the actual antitrust harm in the chain.We find that even in the most basic of settings–with unit pricing and input price
taking–the direct purchaser overcharge is a poor measure of the true antitrust harm.The overcharge can grossly underestimate the actual antitrust harm, depending onsuch characteristics of the market as the shape of demand, the number of producers,the type of competition, and the location of the cartel in the chain of production. Inparticular, we show that lost profit harm ignored by the direct purchaser overchargemay increase without bound with the length of the production chain. Moreover,the method misses harm sustained upstream from the cartel, which can be substan-tial. The ratio of antitrust harm to the direct purchaser overcharge can be anythingbetween one and infinity.The existing literature on cartel pass-on effects uses a model with only three lay-
ers: a top layer of input producers that form a cartel upstream, a layer of directpurchasers downstream who sell to a third layer of final consumers.8 Hellwig (2006)shows that the deadweight-loss of a direct purchaser monopolist from discrete cartelprice increases is equivalent to the part of the overcharge it was able to pass on toconsumers. On this basis, Hellwig argues that the direct purchaser overcharge is agood measure for the actual antitrust harm sustained by this group. Verboven andvan Dijk (2007) use the mainstream model to analyze an infinitessimal cartel priceincrease to determine ‘discounts’ to be given on the direct purchaser overcharge tocorrect for pass-on to consumers and output effects locally. Basso and Ross (2007)extend the approach to differentiated products, so that there can be input substi-tution, to produce a numerical table of correction factors for a discrete cartel priceincrease. For all practical purposes, Boone and Mueller (2008) express the share of(otherwise unspecified) total antitrust harm borne by consumers for an infinitessimalprice increase as a function of common measures such as HHI and PCM.This paper is organized as follows. In Section 2, we decompose the various welfare
effects caused by a cartel anywhere in a chain of production and relate aggregateand individual effects to the overcharge on the direct purchasers. In Section 3, weevaluate the direct purchaser overcharge as an estimator for antitrust harm. Section 4concludes. Appendix A illustrates our decomposition of harm. Appendix B presentsan example of an upstream ‘undercharge’ without collusive buyer power. AppendixC contains the proofs.
standing is discussed more extensively in Section 4.8See Harris and Sullivan (1979) and Kosicki and Cahill (2006).
3
2 Cartel Effects in a Chain of Production
2.1 A Vertical Model of Production
Consider a vertical chain with several layers of intermediaries, each adding value toproduce a homogenous consumer product. Let consumer demand for the final productbe represented by an inverse demand function P : R+ → R+, which is nonincreasing,twice differentiable and continuous in aggregate production Q. Let there be K layersof production, with nk firms active in layer k. Except for layer 1, where the rawmaterials originate, the firms in any layer k each transform a homogeneous input theypurchase from firms in layer k−1, using a one-to-one technology, into a homogeneousnew output, which they sell on to the firms in layer k + 1. Eventually, the firms inlayer K sell the final product to consumers. Figure 1 illustrates.
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Consumers P (Q)
1 j n1 layer 1
1 j nK layer K
1 2 nk − 1 nk layer k
. . . . . .
. . . . . .
. . .
Figure 1: A longer vertical chain of production.
We assume that the number of firms in each layer is exogenously given and fixed.The cost function for firm j in layer k is given by pk−1q + cjk (q), where pk−1 isthe unit price for the input from layer k − 1 (with p0 = 0), and cjk (q) are thecosts for transforming q units of the input into q units of the output. We abstractfrom nonlinear pricing or more general types of vertical relations between firms fromdifferent layers.Firms move simultaneously within the same horizontal layer, and sequentially
following the layer above–so layer 1 moves first, layer 2 second, and so on. That is,we assume that all firms in each layer purchase from the producers in the layer aboveat going prices, without bargaining. As a result, all upstream antitrust effects in ourmodel are reduced input demand effects.The market equilibrium is found by backward induction. First consider layer
K. For any possible input price pK−1–and given final consumer demand P (Q)–we can determine the resulting equilibrium output in layer K. Let the relationshipbetween this equilibrium quantity and pK−1 be represented by a uniquely defined,nonincreasing, continuous and differentiable function pK−1 (Q). This function serves
4
as the inverse demand function for the firms in layerK−1.9 Firms in layerK−1 thendetermine, for any pK−2–and given their inverse demand function pK−1 (Q)–theiroptimal production quantity, in turn leading to an inverse demand function pK−2 (Q)for firms in layer K − 2, and so on.For analytical convenience, we analyze a model with conjectural variations to
simulate various types of competition in each layer.10 That is, in each layer k firm j’sconjecture about the reaction of the other firms in that layer to its quantity decisionis ϑk =
∂Q∂qjk. We assume that ϑk is the same for all firms in a horizontal layer, but
may be different for firms from different layers vertically. Hence, given input pricepk−1, the first-order condition for a symmetric equilibrium in layer k, with conjecturalvariations parameter ϑk is
pk (Q) +ϑknk
Qp0k (Q)− ck − pk−1 = 0. (1)
Note that the classic Cournot conjecture corresponds to ϑk = 1. If ϑk = 0, all firmsin layer k are price takers, so that in equilibrium prices will equal marginal costs, i.e.,pk = pk−1 + ck. The specification ϑk = nk is analytically equivalent to full horizontalcollusion in layer k. Other values of ϑk close to nk can be interpreted as formsof imperfect collusion, in which joint-profit maximization is further constrained, forexample when the cartel members understand that the risk of discovery and the sizeof a consequential damage claim are likely to depend upon the cartel’s pricing andproduction strategy.11
We denote the ultimate equilibrium quantity on the inverse demand functionp1 (Q) faced by the firms in layer 1 by Q∗. Equilibrium prices then clear as p∗1 =p1 (Q
∗), . . . , p∗K−1 = pK−1 (Q∗) and p∗K = P ∗ = P (Q∗). The individual level of
production of firm j in layer k equals q∗jk, withPnk
j=1 q∗jk = Q∗. Equilibrium profits
and consumer surplus follow straightforwardly from these quantities and prices.Now suppose that the firms in some layer g ∈ {1, . . . ,K} form a cartel, while
competitive conditions in all other layers remain the same–note that these mayinclude pre-existing cartels elsewhere in the chain. Our setup implies that the carteluses its obtained market power to raise unit prices vis-a-vis its customers, but remainsa price-taker on the market for its inputs. We further assume that there are no cartel-specific efficiency gains that would somehow allow the cartel to produce at lower coststhan its members could in competition. Let the resulting equilibrium quantity andequilibrium prices under the cartel regime be denoted byQg and pgk for k = 1, . . . ,K.
12
Firm j in layer k produces qgjk withPnk
j=1 qgjk = Qg, for all k.
9This is an assumption in so far that a priori it can be that for a certain value of pk there existmultiple equilibria in the quantity-setting subgame in layer k. For all specifications considered inthis paper, however, an explicit, continuous and differentiable relationship between pk and Q existsfor every k.10Basso and Ross (2007) take the same approach. For a conceptual critique of conjectural varia-
tions, see Hahn (1989).11Salant (1987) and Harrington (2004) offer explicit analyses of some of the effects of such addi-
tional incentive constraints.12In our model it is optimal for the cartel to increase prices symmetrically, so that the cartel
price pgg is the same for each direct purchaser. Verboven and van Dijk (2007) also consider variousasymmetric cartel price mark-ups in their reduced form model. This could be relevant for example
5
2.2 Decomposition of Cartel Effects
The presence of the cartel causes harm to welfare in the form of high unit prices,resulting in lost profits throughout the chain of production, lost consumers surplus,and deadweight-losses, while the cartel members raise their profits. That is, Qg < Q∗
and pgg > p∗g. Typically also pgk > p∗k for k > g and downstream intermediariesand consumers are harmed by the price conspiracy. Under certain specifications,the profits of some downstream intermediaries–in particular direct purchasers–mayactually increase in response to the upstream price increases.13 Also, prices higherup in the chain may either in- or decrease, depending upon the shape of demand andcost functions. In toto, however, collusion on unit prices is always bad for welfare.The impact of the g-level cartel’s unit price increase on one particular layer k of
production that is downstream from layer g can be decomposed into three distincteffects. The overcharge effect on layer k is the amount by which the firms in this layerare overcharged by the previous layer k−1. Part of the burden of this overcharge maybe passed on by the firms in layer k to the next layer of production, layer k+1. Thisis the pass-on effect. Finally, the output effect results from the decrease in productiondue to the cartel. It amounts to the losses in profits from the reduction in sales.14
We consider each of these effects separately, as they are borne out in lost profits.Consider the aggregate profits of firms in layer k. In the competitive benchmark,
these are π∗k =¡p∗k − p∗k−1
¢Q∗−
Pnkj=1 cjk
¡q∗jk¢. Under the cartel regime, they are πgk =¡
pgk − pgk−1¢Qg −
Pnkj=1 cjk
¡qgjk¢. The difference 4πk = π∗k − πgk can be decomposed
as follows
4πk = Qg¡pgk−1 − p∗k−1
¢−Qg (pgk − p∗k)
+
"(Q∗ −Qg)
¡p∗k − p∗k−1
¢+
nkXj=1
cjk¡qgjk¢−
nkXj=1
cjk¡q∗jk¢#
(2)
= ξk − ωk + σk.
The first factor is the overcharge-effect on firms in layer k, thus defined as
ξk = Qg¡pgk−1 − p∗k−1
¢, (3)
or the price increase of the product of the previous layer k − 1, multiplied by thequantity purchased under the cartel regime.The second factor,
if some of the direct purchasers are integrated with a colluding firm, however Verboven and van Dijk(2007) do not derive mark-up differentiation as an optimal pricing strategy.13For an analysis of comparative statics effects in Cournot models, see Dixit (1986) and Quirmbach
(1988).14Our decomposition follows Hellwig (2006), but we use a slightly different terminology. Where
he distinguishes between a direct cost effect and a business loss effect, we use overcharge effect andoutput effect, respectively. Verboven and van Dijk (2007) also analyse both effects, but for exogenousinfinitessimal changes in the input prices, rather than endogenous discrete equilibrium effects. Theyspeak of direct cost effect and output effect , respectively. All three papers share the definition of thepass-on effect.
6
ωk = Qg (pgk − p∗k) , (4)
corresponds to the pass-on effect, which is the amount of the price increase that layerk passes on to layer k + 1. It is equal to the price increase of layer k multiplied bythe quantity produced under the cartel regime. Note that ωk, the pass-on effect oflayer k, equals the overcharge effect suffered by layer k + 1, that is ξk+1 = ωk.The last factor in equation (2) is the output effect,
σk = (Q∗ −Qg)
¡p∗k − p∗k−1
¢+
nkXj=1
¡cjk¡qgjk¢− cjk
¡q∗jk¢¢
. (5)
This part represents the loss of profits that could have been made on the largervolume in the competitive benchmark. It can be rewritten as the sum of individualfirm output effect, σk =
Pkj=1 σjk, with
σjk =¡q∗jk − qgjk
¢ ¡p∗k − p∗k−1 − cjk
¡q∗jk¢¢+ qgjk
¡cjk¡qgjk¢− cjk
¡q∗jk¢¢
.
Here cjk (qjk) = cjk (qjk) /qjk are the average costs for firm j in layer k, evaluated atqjk. The first part of the individual output effect,
¡q∗jk − qgjk
¢ ¡p∗k − p∗k−1 − cjk
¡q∗jk¢¢,
equals the lost sales times the average profit margin and is always positive. The signof the second part, qgjk
¡cjk¡qgjk¢− cjk
¡q∗jk¢¢, is ambiguous. It is positive (negative)
if average costs for firm j in layer k are decreasing (increasing).The effects on layers upstream from cartel layer g–i.e., layers k < g–can be
decomposed in much the same way. These layers also each face an overcharge, apass-on and an output effect. The effect of the cartel on upstream prices resultsfrom reduced derived demand and are ambiguous. As a result, so are the signs of theupstream overcharge and pass-on effects. If all upstream prices increase, the upstreamovercharge and pass-on effect are positive. If all upstream prices decrease, boththe overcharge effect and the pass-on effect of the upstream layers will be negative,corresponding to a decrease in input costs and a decrease in revenues, respectively.In certain specifications, it may also be that all upstream prices remain the same sothat there are no upstream overcharge and pass-on effects. Generally, some of theupstream prices may increase and others decrease.All the way at the end of the supply chain, the loss in consumer surplus of the
final consumer is given by
4CS = CS∗ − CSg = ξC + σC ,
with
ξC = Qg (pgK − p∗K) and σC =
Q∗ZQg
[P (Q)− P (Q∗)] dQ,
where pgK = P (Qg) and p∗K = P (Q∗). Note that, since these are the final consumers,there is no pass-on effect. Also note that, because Qg < Q∗, both ξC and σC arestrictly positive and final consumers unambiguously suffer from the cartel.Our decomposition of cartel effects straightforwardly allows us to formulate the
following basic insight.
7
Proposition 1 The direct purchaser overcharge is equal to the sum of all dowstreamovercharges, net of pass-ons,
PKk=1 (ξk − ωk) + ξC = ξg+1.
Note that in an overcharge conception of compensatory damages, this result justifiesobtaining the direct purchaser overcharge proceeds first, and then redistributing themamongst indirect purchasers later.
2.3 Measure of Antitrust Harm
We are primarily interested in the relationship between the direct purchaser over-charge and the net actual antitrust harm to total welfare. The latter is equal to thechange in total profits in the chain,
PKk=14πk, plus the change in consumer surplus,
4CS. That is,
4W =
g−1Xk=1
4πk +4πg +
"KX
k=g+1
4πk +4CS
#= dU +4πg + dD.
The cartel gains are 4πg. The downstream damages dD =PK
k=g+14πk + 4CScorrespond to losses in profits and consumer surplus by all direct and indirect pur-chasers. In addition, there are upstream damages, dU =
Pg−1k=14πk, equal to profit
losses incurred by direct and indirect suppliers to the cartel.We can use our decomposition of harm in equation (2) to evaluate each of these
terms separately. We find
4W =KXk=1
(ξk − ωk + σk) + ξC + σC =KXk=1
σk + σC ,
where we used ξ1 = 0 and the fact that the overcharge on layer k + 1 equals thepass-on of layer k, ξk+1 = ωk for k = 1, . . . ,K and ξC = ωK . The total welfare effecttherefore coincides with the sum of the output effects.Cartel profits are 4πg = ξg − ωg + σg. Downstream harm can be represented as
dD =KX
k=g+1
(ξk − ωk + σk) + ξC + σC = ξg+1 +KX
k=g+1
σk + σC ,
or the sum of all output effects of direct and indirect purchasers plus the directpurchaser overcharge. Upstream harm is equal to
dU =
g−1Xk=1
(ξk − ωk + σk) = −ωg−1 +
g−1Xk=1
σk = −ξg +g−1Xk=1
σk.
We study the direct purchaser overcharge, ξg+1, in relation to these actual welfareeffects. That is, we evaluate the ratio
λW =4W
ξg+1= λg + λD + λU ,
8
in which
λg =4πgξg+1
=ξg + σg
ωg− 1, (6)
are the cartel gains expressed in the direct purchaser overcharge,
λD =dDξg+1
= 1 +KX
k=g+1
σkξg+1
+σCξg+1
, (7)
is the downstream harm to the direct purchaser overcharge ratio, and
λU =dUξg+1
= −ξgωg+
g−1Xk=1
σkξg+1
, (8)
is that ratio upstream.In addition to these aggregate measures, we specify the individual harm to direct
purchasers and to consumers as:
λg+1 =ξg+1 − ωg+1 + σg+1
ξg+1= 1− ωg+1 − σg+1
ξg+1and λC =
ξC + σCξg+1
. (9)
In the next section, we evaluate how these various ratios vary with the intensityof competition, the number of firms in each layer, the number of layers, and theposition of the layer in which the anticompetitive behavior emerges in a specifiedvertical production model.
3 Quantifying Antitrust Damages using the DirectPurchaser Overcharge
In order to explicitly characterize the ratios introduced above, we need to furtherspecify our model. Suppose that marginal costs are constant and identical for everyfirm in the same layer. That is, for layer k we have cjk (q) = ckq, for each j ∈{1, . . . , nk}.15 Let the inverse demand function be
P (Q) = a− bQγ, (10)
with a, b and γ > 0.16 Inverse demand is a convex function of quantity for 0 < γ < 1,a concave function for γ > 1 and a linear function for γ = 1.
15In order to have gains from trade in this market, we naturally require a >PK
j=1 cj . That is,the consumer’s willingness to pay for the first unit (a) must exceed total costs to produce that unit(PK
j=1 cj).16Note that demand is nonnegative and nonincreasing as well for a ≥ 0, b < 0 and γ < 0–see
Genesove and Mullin (1998). We restrict attention to b, γ > 0, since in that case second-orderconditions are always satisfied. Corbett and Karmarkar (2001) develop a multi-layered Cournotmodel with linear demand.
9
In this setup, the equilibrium quantity and prices can be expressed as follows:17
Q∗ =
"1
b
ÃKYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!# 1γ
, (11)
p∗k =
Ã1−
kYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!+
kXl=1
cl ∀ k ∈ {1, ...,K} . (12)
The competitive benchmark is characterized by a vector of conjectural variationsparameters (ϑ1, ϑ2, . . . , ϑK) ∈ ×K
k=1 [0, nk].Collusion amongst the ng firms in layer g (with ng > 1) results in an increase
in ϑg to ϑcg ∈ (ϑg, ng]. It follows straightforwardly from equation (11) that such anincrease in any ϑk decreases the equilibrium quantity. For ng > 1 and ϑcg ∈ (ϑg, ng],we can write the ratio of collusive output to total competitive output r as
r =Qg
Q∗=
µng + γϑgng + γϑcg
¶ 1γ
.
Note that, although both Q∗ and Qg depend on market characteristics of every layer,apart from γ, their ratio is a function only of characteristics of the colluding layer.If γ = 1, r ∈
£12, 1¢, with the lower bound corresponding to perfect competition in
layer g in the but-for world. An increase (decrease) in γ above (below) 1 decreases(increases) this lower bound value.18
Taken together, we can now characterize cartel profits in layer g in terms of thedirect purchaser overcharge as
λg =ξg + σg
ωg− 1 = γϑg
ng
(1− r)
r (1− rγ)− 1.
Note that λg = −1 if prior to collusion layer g was in perfect competition (ϑg = 0or ng → ∞). In that case, σg = 0 and the total cartel profit equals the overchargeon the direct purchasers. In all other benchmarks, (positive) cartel profits are alwayssmaller than the direct purchaser overcharge.
3.1 Downstream Damages
Downstream from cartel layer g, it follows from equation (12) that equilibrium pricesp∗k (weakly) increase in all layers k ≥ g, as each subsequent layer passes on part of theprice increase it receives from its suppliers to its customers.19 A full characterization
17See Proposition 6 in Appendix A.1.18In particular, limγ→∞ r = 1 for all parameter values and limγ→0 r = e−1 for ϑcg = ng and ϑg = 0.19For k ≥ g + 1, this pass-on rate Rk can be expressed as
Rk =ωkξk=
pgk − p∗kpgk−1 − p∗k−1
=nk
nk + γϑk.
Note that unless in perfect competition (ϑk = 0), each layer will absorb some of the price overchargeit receives. The less competitive a layer is, the lower is the pass-on fraction.
10
of which player faces what passed-on price increase allows us to consider the directpurchaser overcharge ratios derived above. To begin with, consider
λg+1 =γϑg+1
ng+1 + γϑg+1
1− rγ+1
r (1− rγ).
This expression immediately reveals that the direct purchaser overcharge must gen-erally be a poor estimator for actual direct purchaser harm. In the case of linear de-mand, for example, we obtainQ∗ ≤ 2Qg from equation (11), resulting in λg+1 ∈
£0, 3
2
¤.
The upperbound is reached when the pre-cartel equilibrium in layer g was perfectlycompetitive, so that Q∗ = 2Qg.20 The region only slightly changes when demand isnon-linear.The actual antitrust harm of direct purchasers will typically be small when there
is strong competition between them, and/or competition in layer g was weak to beginwith. In these cases, the direct purchaser overcharge will significantly overestimatethe actual harm that direct purchasers suffer. In particular, if ng+1 ≥ 2 and ϑg+1 ≤ 1,in all cases λg+1 ≤ 1.21 If on the other hand layer g+1 is governed by a monopolist or acartel itself (e.g., ϑg+1 = ng+1), the direct purchaser overcharge always underestimatesthe actual harm sustained by indirect purchasers. Naturally, it is possible to constructmarket structures in which the direct purchaser overcharge turns out to be exact. Thisis so, for example, if direct purchasers have sufficient market power and pre-cartelcompetition in the colluding layer was strong. Such examples are specific, however.22
Next consider normalized harm to final consumers23
λC =γ
γ + 1
1− rγ+1
r (1− rγ)
KYi=g+1
nini + γϑi
.
Again we find for γ = 1 that λC ∈£0, 3
2
¤with a slightly changed upperbound for non-
linear demand. If all intermediate layers downstream from the cartel are sufficientlycompetitive, the direct purchaser overcharge underestimates actual final consumerharm and λC > 1. If instead there is substantial market power in enough of theselayers, the direct purchaser overcharge will overestimate consumer harm and λC < 1.Aggregate downstream welfare effects relate to the direct purchaser overcharge
20This is the principal result in Basso and Ross (2007).21Verboven and van Dijk (2007) propose their ‘discounts’ on the direct purchaser overcharge when
awarding damages in direct purchaser suits on the claim that λg+1 ≤ 1, i.e., that the pass-on effectwould always outweigh the output effect, ωg+1 ≥ σg+1 and therefore the direct purchaser overchargeoverestimates the actual harm. Note that this need not be true in our more general setting.22Hellwig (2006) bases his argument for restricting standing to sue to direct purchasers to the
claim that the direct purchaser overcharge exactly coincides with the actual harm if the directpurchaser layer is monopolized–and thus overestimates the actual harm in all other cases. Verbovenand van Dijk (2007) reproduce this result as a special case of their analysis. The marginal priceincreases in both papers correspond to a marginal increase in ϑg in our setting, for which we obtainλg+1 =
2ϑg+1ng+1+ϑg+1
, which indeed equals 1 for ϑg+1 = ng+1–and is smaller otherwise.23Note that λc can be written as λc = 1
2Q∗+Qg
Qg RC , where RC =QK
i=g+1Ri =pgK−p
∗K
pγg−p∗gis the part
of the price increase due to the cartel that ends up being paid by the final consumers.
11
as24
λD =1− rγ+1
r (1− rγ)
Ã1− 1
γ + 1
KYi=g+1
nini + γϑi
!. (13)
Clearly, λD decreases with r, a decrease in competition in one of the downstreamlayers, and an increase in the number of downstream layers. Note also that λD isunaffected by changes in the number of upstream layers and their competitiveness.In the case of linear demand, 1
2Q∗+Qg
Qg ≤ λD < Q∗+Qg
Qg . So we find quite intuitivelythat downstream harm is greater when the cartel reduces output more, which is thecase for example when the pre-cartel equilibrium is more competitive. When all in-termediate layers are perfectly competitive, every layer fully passes on the overchargewhich then is eventually borne by the end users, i.e., λD = λC =
ξC+σCξC
= 12Q∗+Qg
Qg > 1.This provides the lower bound of the actual harm. Monopolistic competition in theintermediate downstream layers increases actual downstream antitrust harm, with astrict upper bound of λD < Q∗+Qg
Qg .25 Interestingly enough, we here find a rationalfor awarding treble damages according to Section 4 of the Clayton Act.
Proposition 2 For linear demand, three times the direct purchaser overcharge is theexact upper bound of total downstream harm, i.e., 1 < λD < 3.
The two top-panels of Figure 2 plot the value of absolute and relative aggregatedownstream antitrust harm against the number of cartel members (upper-left panel)and the number of direct purchasers (upper-right panel) in an example with lineardemand.26 The value of λD increases monotonically with the number of firms in thecartel layer, since more competing firms implies a lower but-for price p∗g and highercompetitive output Q∗. These changes increase the direct purchaser overcharge ξg+1,as well as the output effects,
PKk=g+1 σk + σC. The net effect on the downstream
damages measure dD is positive, because output effects grow faster than the directpurchaser overcharge. The downstream multiplier decreases if competition in anydownstream layer rises. Both the downstream harm dD and the direct purchaserovercharge ξg+1 increase because of increased output, but the latter increases at afaster rate. As a result, λD decreases in the number of direct purchasers.27
In the lower-left panel of Figure 2, the three parameters are plotted for the samespecifications and a cartel in layer 1, varying the total number of downstream layersK.Increasing the number of similarly imperfectly competing downstream layers steeplydecreases λD. Each additional downstream layer of production introduces an extra
24See Proposition 7 and Corollary 1 in Appendix A.1 for details of the derivation.25This upper bound is reached with an infinite number of imperfectly competitive layers down-
stream. This limit case also implies zero equilibrium quantities, limK→∞Q∗ = limK→∞Qg = 0–seeequation (11).26Although the multiplier λD does not depend upon the values of a, b and
PKk=1 ck, the direct
purchaser overcharge ξg+1 and the total downstream harm dD do. We have chosen these parametersin such a way that the three functions can be conveniently presented in the same graph, but theyare otherwise non-specific.27For this parameter configuration, the downstream damage multiplier is equal to 329
216 ≈ 1.52 forng = 2 and equal to 47
24 ≈ 1.96 when ng → ∞, as can be easily checked from (13). Moreover, it isequal to 52
27 ≈ 1. 93 for ng+1 = 2 and equal to149 ≈ 1. 56 when ng+1 →∞.
12
2 4 6 8 100
0.5
1
1.5
2
ng
λD
ξg+1
dD
2 4 6 8 100
0.5
1
1.5
2
ng+1
λD
ξg+1
dD
2 4 6 8 100
0.5
1
1.5
2
2.5
K (g=1)
λD
ξg+1
dD
1 2 3 4 50
0.5
1
1.5
2
2.5
γ
λD
ξg+1
dD
Figure 2: Downstream cartel effects under different numbers of firms per layer for
γ = 1, K = 5, nk = 5, ϑk = 1, g = 3, and 1b
³a−
PKk=1 ck
´2= 10. For the upper two
panels, ϑc3 = 5. The upper-left panel shows the effect of an increase in the numberof cartel members, the upper-right panel in the number of direct purchasers. Thelower-left panel displays downstream harm under different numbers of downstreamlayers of production when there is a cartel in the first layer for ϑc1 = 5. The lower-rightpanel plots the effects of non-linearity of consumer demand.
mark-up, which reduces implied demand for the colluding layer, thereby decreasingthe direct purchaser overcharge. As a result, less demand is affected by the cartel.28
The sum of output effects is the resultant of two opposite effects: the extra markup increases downstream harm and the decrease in production decreases downstreamharm. In the example in the figure, the first effect outweighs the second. However, theeffect on the direct purchaser overcharge dominates, leading to a monotonic increasein λD.The lower-right panel of Figure 2 shows the effect of variations in the shape
of demand. Both dD and ξg+1 decrease both sides from linearity, resulting in amonotonically decreasing value of λD over the spectrum.It appears in these examples that the ratio between actual downstream harm and
28Note that an increase in the number of downstream layers decreases the equilibrium quantity–ascan be seen from equation (11)–but leaves the equilibrium price in layer g unaffected–see equation(12).
13
the direct purchaser overcharge may be high. The following result establishes thatthere is no upperbound on the downstream damages multiplier.
Proposition 3 For any finite number M > 0, there exists a market structure suchthat λD ≥M .
This result obtains as long as there is some degree of market power in suffi-ciently many downstream layers.29 The intuition is that in a two-layered model withcartelized suppliers directly supplying final consumers, the deadweight loss trianglebecomes a smaller fraction of the overcharge quicker when demand is more concavethan when demand is increasingly convex. In both limit demand curves, the outputeffect goes to zero, but a decrease in γ increases the output effect relative to theovercharge. The proof of Proposition 3 uses the fact that when demand becomes rel-atively elastic in the relevant region (γ → 0), the output effect becomes large relativeto the overcharge.Finally, we compare total downstream harm with the sum total of direct purchaser
overcharges and lost profits (dead-weight loss). That is, we define
eλD = dDξg+1 + σg+1
.
For some parameter specifications, adding the direct purchasers’ dead-weight loss tothe denominator changes the damage multiplicator somewhat. The total direct effectremains a poor estimator of total true antitrust harm, however. To see this, first notethat if layer g + 1 is perfectly competitive–that is, ϑg+1 → 0 and/or ng+1 → ∞–there is no output effect for this layer of direct purchasers, i.e., σg+1 = 0. Thisimplies eλD = λD. Obviously, eλD can therefore also take on any value. On the otherhand, the output effect of the direct purchaser layer is large when this layer is fullymonopolized, i.e., ϑg+1 = ng+1. Yet even in that case there exists no upper boundon eλD.30 One way of interpreting this finding in contract to Proposition 1 is as the‘direct purchaser dead-weight loss’ being a poor estimator of ‘total chain deadweightlosses’. Hence, even if lost profits would be recognized as (part of) antitrust harm,damage assessment at the direct purchaser level only in general provides no propersense of the sum of actual downstream harm sustained in the chain.
3.2 Upstream Damages
Upstream from the cartel, the derived demand for inputs is reduced. Depending onmarket conditions, suppliers may optimally respond to this shift in demand by in-or decreasing output, resulting in an ambiguous effect on upstream prices. Underconstant marginal costs of production, in our chosen demand specification (10) theprice decreasing effect from reduced demand and the price increasing effect from
29In fact, λD is bounded from above by (1 +K − g) (e− 1), with e being the natural number.This bound is exact for γ → 0, ϑg = 0 and ϑcg = ng, and ϑk = nk for all downstream layersk = g + 1, . . . ,K. See Appendix C.3.30See Lemma 3 in Appendix A.2.
14
reduced production exactly offset. To see that upstream prices are indeed not affectedby downstream overcharges, note that in equation (12) the equilibrium price in layerk does not depend on ϑl or nl for any l > k. This implies that collusion in layer gwill not have any effect on prices upstream: pgk = p∗k for all k < g and all upstreamovercharges and pass-on effects vanish, ξk = 0 for k = 1, . . . , g and ωk = 0 fork = 1, . . . , g − 1.31 This setup allows us to focus on upstream output effects.32
Moreover, the case of linear demand suffices to establish some limit properties ofλU . Aggregate upstream antitrust harm relates to the direct purchaser overcharge as
λU =(1− r)
r (1− rγ)
ng + γϑgng
Ãg−1Yi=1
ni + γϑini
− 1!, (14)
so that we immediately have the following result.
Proposition 4 For any finite number M > 0, there exists a market structure suchthat λU ≥M .
Note also that λU increases with a decrease in competition in one of the upstreamlayers, as well as with an increase in the number of imperfectly competitive upstreamlayers. It is furthermore invariant to changes in the number of downstream layersand their competitiveness.The upper-left panel of Figure 3 shows numerically that both upstream antitrust
harm dU and the direct purchaser overcharge ξg+1 grow with the number of firmsin the colluding layer–increasing competition in the ‘but-for’ world–but that theupstream damage multiplier λU remains constant. The upper-right panel shows theeffect of an increase in the number of firms in one of the upstream layers. Thisdecreases upstream harm dU , since an increase in the number of firms in one layerdecreases the mark up, and thereby the output effect, in that layer. This effect isonly partially outweighed by an increase in the output effect that occurs due to theincrease in production. At the same time, the direct purchaser overcharge decreaseswith an increase in the number of firms in an upstream layer and the net effect is adecrease in the upstream damage multiplier.33
The lower-left panel shows how the upstream damage multiplier varies with thenumber of upstream layers. If the number of (imperfectly competitive) upstream lay-ers increases, production decreases, which decreases the direct purchaser overcharge.The relationship between downstream damages dU and the number of upstream layersis nonmonotonic in this linear example. An increase in the number of layers introducesadditional mark-ups but decreases production. The first effect has a positive, and the
31Greenhut and Ohta (1976) and Haring and Kaserman (1978) find analogously for the case oflinear demand and constant marginal costs that output changes but upstream input prices remainthe same under vertical integration. In the next subsection we discuss variations in demand andcost functions.32In Appendix B we offer an example of an upstream input price decrease (an ‘undercharge’) by
which damages are passed up, still without the cartel exercising collusive buying power.33For the parameter values here considered, the upstream damage multiplier is equal to 8
5 forng−1 = 2 and equal to 2
5 when ng−1 →∞, as can be easily checked from equation (14).
15
2 4 6 8 100
0.2
0.4
0.6
0.8
1
ng
λU
dU
ξg+1
2 4 6 8 100
0.5
1
1.5
2
ng−1
λU
dUξ
g+1
2 4 6 8 100
0.5
1
1.5
2
2.5
K (g=K)
λU
ξg+1
dU
1 2 3 4 50
0.5
1
1.5
2
2.5
γ
λU
ξg+1
dU
Figure 3: Upstream cartel effects under different numbers of firms per layer for γ = 1,
K = 5, nk = 5, ϑk = 1, g = 3 and 1b
³a−
PKk=1 ck
´2= 10. For the upper two panels,
ϑc3 = 5. The upper-left panel shows the effect of an increase in the number of cartelmembers, the upper-right panel in the number of direct suppliers. The lower-left paneldisplays upstream harm under different numbers of downstream layers of productionwhen there is a cartel in the final layer K for ϑcK = 5. The lower-right panel displaysthe effects of non-linearity of consumer demand.
second a negative impact upon upstream output effects. From Figure 3 it follows thatthe first (second) effect dominates when the number of layers is small (large). Thenet effect of an increase of the number of upstream layers on the upstream damagemultiplier is always positive.
3.3 Cartel Location and the Distribution of Harm
Equations (13) and (14), as well as Figure 3 suggest that the upstream damagesmultiplier λU is substantially larger than the downstream damage multiplier λD.This raises the question whether it is possible for total antitrust harm to be higherwhen the location of the colluding layer is closer to the final consumers. A priorisuch proximity might increase the harm done upstream more than it decreases theharm done downstream. While indeed the multipliers do not trade off perfectly, theabsolute level of total cartel harm is unaffected by the location of the cartel.
16
Proposition 5 Consider two distinct layers, g and h > g that have the same char-acteristics: nh = ng, ϑh = ϑg, and, if one of these layers would collude, ϑch = ϑcg. Let4Wk be the change in welfare if layer k colludes. Then 4Wh = 4Wg, ξh+1 < ξg+1and λW,h > λW,g.
The location of the cartel affects its ability to impose and overcharge. All otherthings equal, the lower the cartel is in the chain, the smaller is the direct purchaserovercharge, while total cartel harm remains the same. Therefore, the upstream dam-ages multiplier increases when the cartel is lower in the chain. The reason for thisis that if the number of layers upstream from where the cartel forms is large, eachof these layers will have put some mark-up on the price. This effectively reduces thescope for abuse of market power by the colluding layer. At the same time, more lay-ers upstream and less downstream from the cartel implies that the direct purchaserovercharge misses more of the total of effects.Formally, the price overcharge is given as
pgg − p∗g = Ψ
gYi=1
nini + γϑi
,
where Ψ is a function of the parameters of the model and independent of the lo-cation of the colluding layer. Clearly, when the number of imperfectly competitivelayers–or for that matter market power of existing upstream layers–increases, thisdifference decreases. As a consequence, the total welfare measure λW increases whenthe colluding layer is further downstream in the chain of production, not becauseabsolute welfare effects are higher, but because the direct purchaser overcharge goesdown. Figure 4 illustrates.Given that final output is independent of the location of the cartel, so is the
increase of final consumer prices. Therefore if the colluding layer is higher up in thechain of production, only a larger increase in prices would result in the same increasein final consumer prices. This is because part of the price increase is absorbed byintermediate downstream layers. For this reason, the direct purchaser overchargefor a cartel higher up the chain is higher. Hence, the closer the cartel is to thefinal customers, the more the direct purchaser overcharge underestimates the actualharm, the larger share of which is borne by the producers upstream in the chain ofproduction.Figure 4 also plots the ‘consumer harm share’ (CHS), a concept introduced in
Boone and Mueller (2008) as the share of final consumers in the sum total of down-stream harm, that is CHS = ∆CS
dD. So defined, the value of CHS increases, the
closer the cartel is to the consumer market. In our setup, the CHS is more naturallydefined as the share in the total welfare effects. Since the sum of all welfare effects inthe chain is independent of the location of the cartel, in our definition the consumerharm share is constant.
17
2 4 6 8 100
1
2
3
4
5
λU
λD
ξg+1
dD
dU
CHS
Location of colluding layer
Figure 4: Down- and upstream damages and the CHS as a function of the loca-tion of the cartel in the chain of production for nk = 5 and ϑk = 1 for all k,1b
³a−
PKk=1 ck
´2= 50 and ϑcg = 5.
4 Concluding Remarks
We have assessed the pass-on of antitrust welfare effects in longer vertical supplychains, in which an antitrust violation may occur in any layer of production. Wefind that the direct purchaser overcharge generally grossly underestimates the totalantitrust harm. While the direct purchaser overcharge is equal to the sum of allpassed on overcharges downstream, it misses the output effects in every layer. Thisproblem is not generally remedied by including the direct purchaser output effectin the damage assessment. The share in total harm sustained by suppliers to thewrongdoer(s) may be large. Relying on the direct purchaser overcharge becomesincreasingly problematic, the longer the vertical chain of production and the closerthe illegal price increases occur to the final consumers.We conclude that there exist no simple multiplication factors to correct the es-
tablished direct purchaser overcharge for actual markets that rely on basic marketstructure characteristics–not even in the case of unit price overcharges. Nor are thelost profits of the direct purchasers a good measure for total chain deadweight losses.This implies that it will not generally suffice to collect the direct purchaser overchargefirst, and then redistribute this money over all antitrust victims later. The flip sideis that duplicative recovery of uncorrected overcharges at several levels in the chainof production can easily create a problem of multiple liability for the defendant.Even though we offer a more general picture of antitrust effects than the existing
the literature, ours also remains a partial model. We have abstracted from nonlinear
18
cartel price elements, for example, which can reduce deadweight-losses, but also fromadditional negative effects that many cartels have on product quality and innovation.We have also not considered the effects of partial vertical integration in the chain ofproduction. The cost structures we use are special–albeit that our results remainqualitatively unchanged in various variations with smoothly in- and decreasing mar-ginal costs of production. In addition, we have not specified how cartels could usetheir market power vis-a-vis suppliers.34
Our assumptions of homogeneity of products within each layer and one-to-oneproduction between layers are not entirely innocuous either. Our fixed-proportionproduction technology implies that products of different layers are perfect comple-ments. When there would be substitution possibilities between inputs instead, a priceincrease in one layer might induce firms in the next layer to substitute away fromthat input and towards an input produced in another chain all together. We ignorethis competitive constraint on pricing in our analysis. The industry supplying theother input is furthermore likely to benefit from the cartel, since its demand may goup and it can raise prices as a result of the reduced competition–this is sometimescalled the ‘umbrella effect’ of antitrust violations.35
In light of our results, it appears to be an unbalanced spending of resources todevote considerable effort and econometric expertise to the characterization of but-for worlds–as is often the case in U.S. antitrust damage cases–only to subsequentlyuse the so found but-for price to calculate the direct purchaser overcharge. With theinformation obtained in such detailed but-for economic analyses, structural estima-tions are often possible and courts would do well to consider them. The same is truefor claims of antitrust injury sustained by direct and indirect suppliers to antitrustviolators.We warn in this context also against the European Commission’s call for “simpli-
fied rules on estimating the loss” from infringements of the competition rules in its2008 White Paper 36 The Commission’s stated primary objective and guiding princi-ple in this document is full compensation of victims of a breach of the EC antitrustrules–which is in line with relevant decisions of the European Court of Justice. Itfurthermore recognizes “actual damages, lost profits, and interest” for direct as wellas indirect purchasers. The Commission should realize that there is no hope for verysimple rules on antitrust damage estimation.37
34One way to represent upstream bargaining power in our model is as a reduction of ϑg−1 inresponse to a cartel forming in layer g–to ϑg−1 = 0 if the cartel has complete bargaining power.Lower costs of inputs will typically induce the cartel to produce more, thus reducing some of itsdetrimental effects. This approach is not entirely free from problems, however. For example, itleaves pre-cartel bargaining power unspecified.35See Areeda and Hovenkamp (1992), para. 337.3 on the rules of standing for “umbrella” plaintiffs.
The authors argue that purchasers from non-cartel members that were overcharged because of theumbrella effect should generally be granted standing to sue the cartel for damages, in particularwhen products are homogeneous. When products are differentiated, the authors recognize that itmay be difficult to draw the line where standing ends.36Commission of the European Communities (2008), p.3 and p.7.37In addition, the Commission means to assist indirect purchasers by introducing a peculiar “re-
buttable presumption that the illegal overcharge was passed on to them in its entirety.” Commissionof the European Communities (2008), p. 8. The use of “them” in this quote suggests that the
19
Finally, our analysis has clarified the antitrust damages caused to direct andindirect suppliers. These upstream effects of a downstream cartel appear not to beproperly viewed as antitrust damages in U.S. antitrust law. In Mandeville IslandFarms, the Supreme Court had held that growers of sugar beets could maintain atreble-damages action against refiners who had alledgedly conspired to fix the pricethat they would pay for the beets. In Radovic (1957) the Court further recognized awide “zone of harm” caused by an antitrust violation that could potentially includecustomers as well as suppliers.38 Indeed in Wilson v. Ringsby Truck Lines (1970),truck drivers and warehousemen of Ringsby, a common carrier transporting betweenColorado and Wyoming, were recognized by a district court as antitrust victims whentheir services were no longer needed after Ringsby joined a cartel that divided theirmarkets geographically.39
However, in the 1970s a number of supplier suits for antitrust damages failed whencircuit courts gave no standing to classes of employees and suppliers of cartels.40 InIllinois Brick, the Supreme Court had concluded that: “An antitrust violation maybe expected to cause ripples of harm to flow through the Nation’s economy; butdespite the broad wording of 4 there is a point beyond which the wrongdoer shouldnot be held liable.” In Associated Contractors (1983), the Court then denied standingto a class of carpenters who sought antitrust damages for business loss resulting fromthe contractors association using anticompetitive means to work around their union.It ruled that the carpenters union was not a person injured by reason of a violationof the antitrust laws within the meaning of Section 4, because of the “tenuous andspeculative character of the causal relationship”, and the existence of “more directvictims” of the conspiracy, meaning consumers and competitors.Clearly antitrust violations cause direct and indirect supplier effects sound and
sure. Analogous to customers of a cartel facing higher prices and reduced quantitiesof output, suppliers to a cartel may obtain lower prices for a reduced volume of inputsales, even when the cartel does not execute collusive buyer power. Whilst it remainsan open question of law, therefore, where exactly are the points beyond which thewrongdoers should no longer be held liable for the harm caused by their illegal actions,there appears to be no good reason to define the zone of harm symmetrically in bothdirections, that is up- and downstream.
References
[1] Areeda, P. E., and H. Hovenkamp (1992), Antitrust Law: An Analysis of An-titrust Principles and Their Application, Aspen Publishers, New York;
drafters of the White Paper indeed had the mainstream two-layer upstream cartel model in mind.38Radovich v. National Football League 352 U.S. 445 (1957). See Comment (1972), pp.403-4.39See Wilson v. Ringsby Truck Lines, Inc., 320 F. Supp. 699 (D. Colo. 1970).40In Contreras v. Grower Shipper Veg. Ass’n of Cent. Cal., 484 F.2d 1346 (9th Cir. 1973) the
9th Circuit Court denied a class of employees laid off by an alleged price-fixer standing to sue forantitrust damages. In Comet Mechanical Contractors, Inc v. E.A. Cowen Constr. Inc, 609 F.2d 404(10th Cir. 1980) a subcontractor could not seek compensation for reduced demand for its servicesresulting from alleged bidrigging by general contractors. See Page (1985), p.1492.
20
[2] Anderson, S.P. and R. Renault (2003), “Efficiency and Surplus Bounds inCournot Competition,” Journal of Economic Theory, 113, 253-264;
[3] Basso, L.J. and T.W. Ross (2007), “Measuring the True Harm from Price-Fixingto Both Direct and Indirect Purchasers,” working paper, University of BritishColumbia;
[4] Blair, R.D. andW.H. Page (1996), “The Role of Economics in Defining AntitrustInjury and Standing,” Managerial and Decision Economics, 17, 127-142;
[5] Boone, J and W. Müller (2008), “The Distribution of Harm in Price-FixingCases,” working paper, University of Tilburg;
[6] Comment (1972), ““Mangano” and Ultimate-Consumer Standing: The Misuseof the Hanover Doctrine,” Columbia Law Review, 72(2), 394-414;
[7] Commission of the European Communities (2008), White Paper on DamagesActions for Breach of the EC Antitrust Rules, Brussels;
[8] Corbett, C.J. and U.S. Karmarkar (2001), “Competition and Structure in SerialSupply Chains with Deterministic Demand,” Management Science, 47, 2001,966-978;
[9] Dixit, A. (1986), “Comparative Statics for Oligopoly,” International EconomicReview, 27, 107-122;
[10] Fisher, F.M. (2006), “Economic Analysis and Antitrust Damages,”World Com-petition, 29, 383-394;
[11] Genesove, D. and W. Mullin (1998), “Testing Static Oligopoly Models: Conductand Cost in the Sugar Industry: 1890-1914,” RAND Journal of Economics, 29,355-377;
[12] Greenhut, M.L. and H. Ohta (1976), “RelatedMarket Conditions and Interindus-trial Mergers,” American Economic Review, 66, 267-277;
[13] Hahn. F.H. (1989), “Conjectural Equilibria,” in: J. Eatwell, M. Milgate and P.Newman, The New Palgrave: General Equilibrium, Norton, New York, 98-107;
[14] Haring, J.R. and D.L. Kaserman (1978), “Related Market Conditions and In-terindustrial Mergers: A Comment,” American Economic Review, 68, 225-227;
[15] Harrington, J.E., Jr. (2004), “Post-Cartel Pricing during Litigation,” Journal ofIndustrial Economics, 52, 517-533;
[16] Harris, R.G. and L.A. Sullivan (1979), “Passing on the Monopoly Overcharge: AComprehensive Policy Analysis,” University of Pennsylvania Law Review, 128,269-360;
21
[17] Hellwig, M. (2006), “Private Damage Claims and the Passing-On Defense in Hor-izontal Price-Fixing Cases: An Economist’s Perspective,”in: J. Basedow (ed.),Private Enforcement of EC Competition Law, Kluwer Law International, DenHaag, 2007, 121-159;
[18] Kosicki, G. and M.B. Cahill (2006), “Economics of Cost Pass Through andDamages in Indirect Purchaser Antitrust Cases.” Antitrust Bulletin, 51, 599-630;
[19] Lande, R.H. (1993), “Are Antitrust ‘Treble’ Damages Really Single Damages?,”Ohio State Law Journal, 54, 115-174;
[20] Page, W.H. (1985), “The Scope of Liability for Antitrust Violations,” StanfordLaw Review, 37, 1445-1512;.
[21] Quirmbach, H.C. (1988), “Comparative Statics for Oligopoly: Demand ShiftEffects,” International Economic Review, 29, 451-459;
[22] Salant, S.W. (1987), “Treble Damage Awards in Private Lawsuits for Price Fix-ing,” Journal of Political Economy, 95, 1326-1336;
[23] Schinkel, M.P., J. Tuinstra and J. Rüggeberg (2008), “Illinois Walls: How Bar-ring Indirect Purchaser Suits Facilitates Collusion,” RAND Journal of Eco-nomics, 39, 683-698;
[24] Verboven, F. and T. van Dijk (2007), “Cartel Damages Claims and the Passing-on Defence,” forthcoming in Journal of Industrial Economics.
22
A Decomposition of Harm Illustrated
Figure 5 illustrates the various typical effects decomposed in Section 2.2 in a simplemodel with three production layers (K = 3) and final consumers.
-
6
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
@@@@@@@@@@@@@@@@@@@@@
JJJJJJJJJJJJJJJJJJJJJ
BBBBBBBBBBBBBBBBBBBBB
G
H
I
J
A
B
C
D
E
F
P (Q)p2(Q)
p1(Q)p01(Q)
p∗1
p21
p∗2
P ∗p22
P 2
Q2 Q∗
Figure 5: Decomposition of antitrust harm in a three layer model.
We assume that marginal own production costs are constant and equal to zerofor all firms, i.e., cj1 (q) = cj2 (q) = cj3 (q) = 0, for all j and every q. Given inverseconsumer demand P (Q), the implied inverse demand for the product of firms in layer2 is given by p2 (Q). Competition in layer 2 results in inverse demand function p1 (Q)for the firms in layer 1. Their competitive benchmark is given by the quantity Q∗
and prices P ∗, p∗2 and p∗1, respectively. Now suppose firms in the second layer collude.
This leads to a reduction in the quantity they supply. That is, for every price p1,layer 2 demands less of the input supplied by layer 1, resulting in an inwards shift ofp1 (Q) to p01 (Q). Under the cartel regime, output decreases to Q2, and equilibriumprices become P 2, p22 and p21, respectively. Note that in this illustration we assumethat p1 decreases under the cartel, which need not be the case.It is insightful to identify some of the areas in the graph. The loss in profits of
the direct purchasers of the cartel (layer 3) equal
4π3 = π∗3 − π23 = C +H −A = (B + C)− (A+B) +H = ξ3 − ω3 + σ3,
with ξ3 = (p22 − p∗2)Q2 = B + C being the amount by which the firms in layer 3
are overcharged, ω3 = (P 2 − P ∗)Q2 = A + B the amount passed-on to the finalconsumers, and σ3 = H the output effect. The loss in consumer surplus is
4CS = CS∗ − CS2 = (A+B) +G = ξC + σC ,
23
where ξC = A + B = ω3 is the overcharge imposed by layer 3, and σC = G is theoutput effect for consumers. Profits of the (direct) suppliers to the cartel change by
4π1 = π∗1 − π21 = E + F + J − F = E + J = −ω1 + σ1,
where ω1 = −E is the pass-on from layer 1 to layer 2, which is negative since p1 hasincreased. Note that there is no overcharge for layer 1. Its output effect is given byσ1 = J . Finally, consider the colluding layer 2. Profits of the cartel members changeby
4π2 = π∗2 − π22 = D + I − (B + C +D +E) = −E − (B + C) + I = ξ2 − ω2 + σ2,
where ξ2 = −E = ω1 is the overcharge from their direct suppliers, ω2 = B + C isthe pass-on to the next layer and σ2 is the output effect. Obviously, the sum total ofthese effects is negative or otherwise it would not be profitable for the cartel to form.Notice also that
4π1 +4π2 +4π3 +4CS = G+H + J + I = σ1 + σ2 + σ3 + σC ,
that is, the sum of all effects combined reduces to the sum of output effects.
24
B An Example of Upstream ‘Undercharges’
Following Mandeville Island Farms, direct suppliers to a buyers cartel that colludedto decrease input prices can maintain a treble-damages action for the ‘undercharge’.41
In this paper, we abstract from buyer power effects. Yet, as argued in Section 3.2,variations in demand or the cost of production can generate upstream price effects aswell. Input prices may increase when the cartel reduces demand. Direct suppliers mayalso obtain a lower price for their inputs by the cartel members than they would undercompetition, however, in which case they suffer a straightforward ‘undercharge’.Figure 6 illustrates an example of input prices in layer g−1 decreasing as a result
of a cartel forming in layer g when the marginal upstream costs of production increasein production.
-
6
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
@@@@@@@@@@@@@@@@@@@@@
SSSSSSSSSSSSSSSSSSSSS
TTTTTTTTTTTTTTTTTTTTT
pg−1(q)©©©©
©©©©
©©©©
©©©©
©©©©©
cg−1(q)
pg−1
qg−1
pgg−1(q)
pgg−1
qgg−1
U
Figure 6: Direct sellers undercharged by a purchasers cartel.
The derived demand for inputs under downstream competition, pg−1(q), turnsinwards to pgg−1(q). Profit maximization given cg−1 (q) results in lower input pricesto the purchaser cartel, pgg−1 < pg−1. As a result, the upstream industry sustains anundercharge of size
¡pg−1 − pgg−1
¢qgg−1 on its actual sales–or area U in the figure.
Given linear demand, upstream prices increase after downstream collusion whencosts are concave, and decrease when costs are convex.42 The intuition for the latterbecomes clear from the case of perfect competition in the upstream market, in whichprices are equal to marginal costs. Decreasing returns to scale result in lower marginalcosts of production in equilibrium when the quantity of inputs demanded is reduced.
41See Blair and Page (1996), pp.131-3.42Analytical results quickly become intractable in longer supply chains. From tedious but straight-
forward computations it follows that in a chain with two layers of production the upstream Cournot
25
These results carry over to the other forms of imperfect competition upstream thatour model allows.
equilibrium price decreases with an increase in ϑ2, whenever
(ΨP +Ψc)P0 (Q) >µ
(n2 + ϑ2)P0 (Q) + ϑ2QP
00 (Q)− c002
µQ
n2
¶¶((n1 + 1)P
0 (Q) +QP 00 (Q)) .
where ΨP = (n1 + 1) (n2 + ϑ2)P0 (Q) + (n2 + (n1 + 3)ϑ2)QP
00 (Q) + ϑ2Q2P 000 (Q) and Ψc =
−n2c001³Qn1
´− (n1 + 1) c002
³Qn2
´− Q
n2c0002
³Qn2
´. If consumer demand is linear and marginal costs
downstream are constant, this condition reduces to c001³Qn1
´> 0. Furthermore, the class of demand
specifications for which there are no upstream price effects when the marginal costs of productionare constant both up- and downstream is characterized by
P 00 (Q)P 0 (Q) +QnP 000 (Q)P 0 (Q)− [P 00 (Q)]2
o= 0.
Obviously, whereas (10) does, many nonlinear demand functions do not satisfy this condition.
26
C Characterizations and Proofs
In this appendix we present the analysis that underlies the text.
C.1 Proofs of the Introductory Results
Proof of Proposition 1. This follows directly from ξk+1 = ωk for all k andξC = ωK .
Proof of Proposition 2. For γ = 1 we have
λD =Q∗ +Qg
Qg
Ã1− 1
2
KYi=g+1
nini + ϑi
!.
It follows that we have 12Q∗+Qg
Qg ≤ λD < Q∗+Qg
Qg . The lower bound is reached if alldownstream layers are perfectly competitive and the upper bound is reached withan infinite number of imperfectly competitive layers downstream. Since the cartelreduces production, we have Qg < Q∗ and hence λD > 1. In addition, it follows fromQ∗
Qg =ng+ϑ
cg
ng+ϑgthat Q∗ ≤ 2Qg, with Q∗ = 2Qg if the pre-cartel industry was perfectly
competitive (ng →∞ or ϑg = 0) and the cartel sets the full cartel quantity (ϑcg = ng).
In this case Q∗+Qg
Qg = 3 and therefore λD < 3.
C.2 Intermediate Results
Our first intermediate result characterizes the implied inverse demand function foreach layer k.
Lemma 1 Given final consumer demand P (Q) = a− bQγ, constant marginal costsck in layer k and conjectural variations parameter ϑk in layer k, the implied inversedemand function faced by firms in layer k is
pk (Q) = a−Ãb
KYi=k+1
ni + γϑini
Qγ +KX
l=k+1
cl
!, (15)
for k = 1, . . . ,K − 1. Furthermore, pK (Q) = P (Q).
Proof. We will show that for a given K, (15) holds for all 1 ≤ k ≤ K − 1. Considerfirm i in layer k. It maximizes profits pk (Q) qi − (ck − pk−1) qi, where pk (Q) is theimplied inverse demand function that the industry in layer k faces. Using symmetry(qi = Q/nk) and conjectural variations (ϑk = dQ/dqi) the first-order condition equals(1). Solving for pk−1 then gives
pk−1 (Q) = pk (Q) +ϑknk
Qp0k (Q)− ck. (16)
27
Given pk (Q), the implied inverse demand for layer k − 1, pk−1 (Q) can therefore bedetermined recursively. We need to show that (15) satisfies the recursive relation (16)for all k = 2, . . . ,K. First consider k = K. Using pK (Q) = P (Q) = a − bQγ (16)reduces to pK−1 (Q) = a−bnK+γϑK
nKQγ−cK , which is equivalent to (15) for k = K−1.
Now we proceed by induction. Assuming that (15) holds for k we want to show thatit also holds for k − 1. We have
pk−1 (Q) = pk (Q) + p0k (Q)ϑknk
Q− ck
= a−Ãb
KYi=k+1
ni + γϑini
Qγ +KX
l=k+1
cl
!−Ãγb
KYi=k+1
ni + γϑini
!Qγ ϑk
nk− ck
= a−Ãbnk + γϑk
nk
KYi=k+1
ni + γϑini
Qγ + ck +KX
l=k+1
cl
!
= a−Ãb
KYi=k
ni + γϑini
Qγ +KXl=k
cl
!.
Therefore equation (15) holds for all k with 1 ≤ k ≤ K − 1. Finally, it is easilychecked that γ > 0 is a sufficient condition for the second-order condition for anoptimum to be satisfied for every individual firm at the equilibrium.
Knowing the implied inverse demand functions (15), we can now determine equilib-rium quantities and prices.
Proposition 6 In this model, equilibrium prices and quantities are given as
Q∗ =
"1
b
ÃKYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!# 1γ
, (17)
p∗k =
Ã1−
kYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!+
kXl=1
cl ∀ k ∈ {1, . . . ,K} . (18)
Aggregate profits of firms in layer k and consumer surplus are then respectively givenby
π∗k =
µ1
b
¶ 1γ γϑknk + γϑk
k−1Yi=1
nini + γϑi
ÃKYi=1
nini + γϑi
! 1γÃa−
KXj=1
cj
!γ+1γ
(19)
CS =γ
γ + 1
µ1
b
¶ 1γ
ÃKYi=1
nini + γϑi
!γ+1γÃa−
KXj=1
cj
!γ+1γ
. (20)
Proof. Using (15) the first-order condition (1) for k = 1 reduces to
a−Ãb
KYi=1
ni + γϑini
Qγ +KXl=1
cl
!= 0.
28
Solving for Q gives (17). Substituting Q∗ into (15) gives (18). Profits and consumersurplus follow from substituting (17) and (18) in π∗k =
¡p∗k − p∗k−1 − ck
¢Q∗ and CS =
Q∗Z0
P (Q) dQ− p∗KQ∗ = γ
γ+1b (Q∗)γ+1 = γ
γ+1(a− p∗K)Q
∗, respectively.
We are interested in how the equilibrium quantity Q∗ and equilibrium prices p∗kdepend upon the underlying model parameters. It is easily checked that Q∗ increaseswith an increase in a or ni and with a decrease in ϑi, b or the number of (imperfectlycompetitive) layersK. The equilibrium price for layer k, p∗k, increases with an increasein a, γ, ci or ϑi, for i ≤ k, and with a decrease in ni for i ≤ k, with a decrease in cifor l > k.The dependence of Q∗ on γ, however, is ambiguous and Q∗ could either decrease
or increase with γ.43 Moreover, we have
limγ→∞
p∗k = a−KX
j=k+1
cj and limγ→0
p∗k =kX
j=1
cj.
Firms in layer 1 have all the market power, that is, they extract the entire surplusby setting price p∗1 equal to a minus the marginal costs of the other layers. The otherlayers price competitively, in the sense that they only recover their marginal costs.
Using limγ→∞
³1 + ϑi
niγ´ 1
γ= 1 and limγ→0
³1 + ϑi
niγ´ 1
γ= exp
hϑini
i, respectively,
we find
limγ→∞
Q∗ = 1 and limγ→0
Q∗ =
⎧⎪⎨⎪⎩0,
exph−PK
i=1ϑini
i,
∞,
if b > a−PK
j=1 cjif b = a−
PKj=1 cj
if b < a−PK
j=1 cj
.
Hence, when demand becomes infinitely concave (γ →∞), the equilibrium quantity is1, independent of all other parameters. Therefore, when demand becomes infinitelyconcave (γ → 0) , the equilibrium price is 0 for all layers, independent of all otherparameters. The equilibrium quantity is 0 (∞) when b > (<) a−
PKj=1 cj; only when
this condition holds with equality equilibrium quantity is positive and finite.
Collusion in layer g is modelled as an increase in the conjectural variations parameterfrom ϑg to ϑ
cg ∈ (ϑg, ng]. Cartel quantity and prices are
Qg =
"1
b
ng + γϑgng + γϑcg
ÃKYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!# 1γ
, (21)
pgk =
( ³1− ng+γϑg
ng+γϑcg
Qki=1
nini+γϑi
´³a−
PKj=1 cj
´+Pk
l=1 cl
p∗k
k ≥ gk < g
. (22)
The next lemma provides expressions for pass-on and output effects (recall that ξk+1 =ωk).43Take for example the case with ϑi = 0 for all i. Then Q∗ increases (decreases) with γ if
a−PK
k=1 ck is larger (smaller) than b.
29
Lemma 2 The pass-on effect ωk for layer k ≥ g is given by
ωk =
µ1
b
¶ 1γµng + γϑgng + γϑcg
¶ 1γµγϑcg − γϑg
ng + γϑcg
¶Ã kYi=1
nini + γϑi
!(23)
×Ã
KYi=1
nini + γϑi
! 1γÃa−
KXj=1
cj
!γ+1γ
,
and ωk = 0 for every k ≤ g − 1. The output effect for layer k and the output effectfor consumers are given as
σk =
µ1
b
¶ 1γ γϑknk
Ã1−
µng + γϑgng + γϑcg
¶ 1γ
!ÃkYi=1
nini + γϑi
!(24)
×Ã
KYi=1
nini + γϑi
! 1γÃa−
KXj=1
cj
!γ+1γ
,
σC =
µ1
b
¶ 1γ
⎛⎜⎝γ +³ng+γϑgng+γϑcg
´γ+1γ
γ + 1−µng + γϑgng + γϑcg
¶ 1γ
⎞⎟⎠ (25)
×Ã
KYi=1
nini + γϑi
!γ+1γÃa−
KXj=1
cj
!γ+1γ
.
Proof. Straightforward computations show that (for k ≥ g) we have
pgk − p∗k =γϑcg − γϑg
ng + γϑcg
ÃkYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!,
p∗k − p∗k−1 − ck =γϑknk
ÃkYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!,
Q∗ −Qg =
Ã1−
µng + γϑgng + γϑcg
¶ 1γ
!"1
b
ÃKYi=1
nini + γϑi
!Ãa−
KXj=1
cj
!# 1γ
.
Equations (23)—(24) follow immediately from substituting the above expressions intoωk = Qg (pgk − p∗k) and σk = (Q
∗ −Qg)¡p∗k − p∗k−1 − ck
¢. Furthermore, we have
σC =
Q∗ZQg
[P (Q)− P (Q∗)] dQ =1
γ + 1b£(Qg)γ+1 − (Q∗)γ+1
¤+ b (Q∗)γ (Q∗ −Qg) .
Equation (25) then follows from substituting Qg = rQ∗, with r =³ng+γϑgng+γϑ
cg
´ 1γ.
Using Lemma 2, we can express the measures of harm discussed in Section 2.3 interms of the parameters of the model.
30
Proposition 7 Denote by r = Qg
Q∗ =³ng+γϑgng+γϑ
cg
´ 1γthe fraction by which the cartel
reduces output. The damages measures (7)—(9) are equal to
λD =1− rγ+1
r (1− rγ)
Ã1− 1
γ + 1
KYi=g+1
nini + γϑi
!, (26)
λU =(1− r)
r (1− rγ)
ng + γϑgng
Ãg−1Yi=1
ni + γϑini
− 1!, (27)
λg =γϑgng
(1− r)
r (1− rγ)− 1, (28)
λg+1 =γϑg+1
ng+1 + γϑg+1
1− rγ+1
r (1− rγ), and (29)
λC =γ (1− rγ+1)
(γ + 1) r (1− rγ)
KYi=g+1
nini + γϑi
. (30)
λW =
µng + γϑcgγϑcg − γϑg
¶"rγ − r−1
γ + 1
ÃKY
i=g+1
nini + γϑi
!+¡r−1 − 1
¢Ã gYi=1
ni + γϑini
!#(31)
Proof. Substituting equations (23)—(25) into (7) and using³ng+γϑgng+γϑ
cg
´ 1γ= r and
γϑcg−γϑgng+γϑ
cg= 1− rγ we obtain
λD = 1 +1− r
r (1− rγ)
KXk=g+1
γϑknk
kYi=g+1
nini + γϑi
+γ + rγ+1 − (γ + 1) r(γ + 1) r (1− rγ)
KYi=g+1
nini + γϑi
.
Equation (26) then follows from using
1−KX
k=l+1
γϑknk
ÃkY
i=l+1
nini + γϑi
!=
KYi=l+1
nini + γϑi
, (32)
which can straightforwardly be shown to hold by induction.Similarly, (8) reduces to
λU =(1− r)
r (1− rγ)
g−1Xk=1
γϑknk
gYi=k+1
ni + γϑini
.
Equation (27) then follows from using
g−1Xk=1
γϑknk
gYi=k+1
ni + γϑini
=ng + γϑg
ng
Ãg−1Yi=1
ni + γϑini
− 1!.
Equations (28)—(30) are derived analogously.
31
Corollary 1 For linear inverse demand (γ = 1), the measures of harm are given as
λD =Q∗ +Qg
Qg
Ã1− 1
2
KYi=g+1
nini + ϑi
!, λU =
ng + ϑcgng
Ãg−1Yi=1
ni + ϑini
− 1!
λg =ϑgng
Q∗
Qg− 1, λg+1 =
ϑg+1ng+1 + ϑg+1
Q∗ +Qg
Qg, and λC =
1
2
Q∗ +Qg
Qg
KYi=g+1
nini + ϑi
.
Proof. This follows immediately from substituting γ = 1 and 1−r2r(1−r) =
1+rr= Q∗+Qg
Qg
and 1−rr(1−r) =
1r= Q∗
Qg into equations (26)—(30).
C.3 Proofs of the Main Results
First we establish that for small γ, λD can increase without bound.
Proof of Proposition 3. First note that we have limγ→∞ r = 1 and, using
limγ→0 (1 + αγ)1γ = exp [α], we have limγ→0 r = exp
hϑg−ϑcgng
i. Moreover, the following
results are useful
limγ→∞
rγ+1 = limγ→∞
⎛⎝1 + γ ϑgng
1 + γϑcgng
⎞⎠ limγ→∞
r =ϑgϑcg,
limγ→0
rγ+1 = limγ→0
⎛⎝1 + γ ϑgng
1 + γϑcgng
⎞⎠ limγ→0
r = exp
∙ϑg − ϑcg
ng
¸.
Now it follows immediately that
limγ→∞
λD = limγ→∞
1− rγ+1
r − rγ+1limγ→∞
Ã1− 1
γ + 1
KYi=g+1
nini + γϑi
!= 1.
In order to evaluate limγ→0 λD first notice that λD can be rewritten as
λD =1− rγ+1
r(1− rγ)
Ã1− 1
γ + 1
KYi=g+1
nini + γϑi
!
=1− rγ+1
r
ng + γϑcgγ (γ + 1)
¡ϑcg − ϑg
¢ Ãγ + 1− KYi=g+1
nini + γϑi
!
=1− rγ+1
r
ng + γϑcgγ (γ + 1)
¡ϑcg − ϑg
¢ Ãγ + QKi=g+1 (ni + γϑi)−
QKi=g+1 niQK
i=g+1 (ni + γϑi)
!
=1− rγ+1
r
ng + γϑcgγ (γ + 1)
¡ϑcg − ϑg
¢⎛⎝γ +
γhPK
i=g+1
³ϑiQK
j=g+1, j 6=i nj
´+ f (γ)
iQK
i=g+1 (ni + γϑi)
⎞⎠ ,
32
where f (γ) is a function with the property that limγ→0 f (γ) = 0. Taking the limitnow gives
limγ→0
λD =
⎛⎝1− exphϑg−ϑcgng
iexp
hϑg−ϑcgng
i⎞⎠ ng
ϑcg − ϑg
⎛⎝1 +PKi=g+1
³ϑiQK
j=g+1 nj/ni´
QKi=g+1 ni
⎞⎠= h
µϑcg − ϑg
ng
¶Ã1 +
KXi=g+1
ϑini
!,
with h (x) = exp(x)−1x
. Note that limx→0 h (x) = 1, h0 (x) > 0 and h (1) = e−1. Clearly,the second part of the above expression,
³1 +
PKi=g+1
ϑini
´, is bounded from above by
K − g + 1. Taken together we find that limγ→0 λD ≤ (1 +K − g) (e− 1), with theupperbound being exact when all downstream layers are monopolized (implying ϑi =ni for all i), and there is full collusion in the colluding layer and perfect competitionotherwise (ϑ
cg−ϑgng
= 1).
Alternatively, one can consider the relation between total downstream harm andtotal profit loss (i.e. overcharge plus output effect) of the direct purchasers. That is,
eλD = dDξg+1 + σg+1
=λDΞ,
where Ξ = ξg+1+σg+1ξg+1
. Obviously, Ξ ≥ 1. Moreover, if layer g + 1 is perfectly com-petitive (that is, ϑg+1 → 0 and/or ng+1 →∞) there is no output effect for this layerof direct purchasers, σg+1 = 0. This implies Ξ = 1 and eλD = λD. Obviously, thedownstream damage multiplier eλD can therefore also take on any value. However, thisholds even if the indirect purchaser layer is imperfectly competitive as the followinglemma shows.
Lemma 3 For any M > 0 and any value of ϑg+1/ng+1 ≤ 1 there exists a marketstructure such that eλD > M .
Proof. First, using Lemma 2 we find that
Ξ = 1 +1− r
r
ϑg+1ng+1 + γϑg+1
ng + γϑcgϑcg − ϑg
.
From the proof of Proposition 3 we know that limγ→0 r = exphϑg−ϑcgng
i. Using h (x) =
exp(x)−1x
again we can write
limγ→0
eλD = limγ→0
λDΞ=
h³ϑcg−ϑgng
´1 + ϑg+1
ng+1h³ϑcg−ϑgng
´ Ã1 + KXi=g+1
ϑini
!.
33
Since h (x) is maximized at x = 1 and h (1) = e− 1 we find an upper bound for eλDby taking γ → 0, ϑg = 0, ϑcg = ng and ϑk = nk for k = g + 1, . . . ,K. This upperbound is given by
e− 1e
(1 +K − g) .
Clearly, any level of eλD can be reached by choosing K − g appropriately.
Moreover, it is easily verified that for the case with ϑg+1 = ng+1, and a perfectlycompetitive benchmark in the colluding layer, ϑg = 0, we have that Ξ equals 2 forγ = 1, and Ξ goes to e (1) for γ → 0 (γ →∞)Next we show that λU can increase without bound.
Proof of Proposition 4. It is sufficient to prove the proposition for the linear casewith γ = 1. Then we have
λU =ng + ϑcg
ng
Ãg−1Yi=1
ni + ϑini
− 1!.
In the extreme case where all upstream layers are monopolized (ϑi = ni for i =1, . . . g − 1) and there is full collusion in the colluding layer (ϑcg = ng) we obtain
λU = 2¡2g−1 − 1
¢,
which obviously can reach any finite level as the number of upstream layers g − 1increases.
Finally, we show the effects of the location of the colluding layer on total welfare andthe different measures of harm.
Proof of Proposition 5. The change in welfare is
4Wg =KXk=1
σk + σC = (Q∗ −Qg)
ÃP ∗ −
KXk=1
ck
!+ σC
= Φ
"Ã1−
µng + γϑgng + γϑcg
¶ 1γ
!− 1
γ + 1
Ã1−
µng + γϑgng + γϑcg
¶γ+1γ
!KYi=1
nini + γϑi
#,
where Φ =¡1b
¢ 1γ
³QKi=1
nini+γϑi
´ 1γ³a−
PKj=1 cj
´γ+1γ. The change in welfare is inde-
pendent of the location of the cartel. The direct purchaser overcharge is given by
ξg+1 = ωg = Φ
µng + γϑgng + γϑcg
¶ 1γµγϑcg − γϑg
ng + γϑcg
¶ gYi=1
nini + γϑi
,
which does decrease in g.
34
