a theory of bilateral oligopoly - sscc -...

A THEORY OF BILATERAL OLIGOPOLY

KENNETH HENDRICKS and R. PRESTON MCAFEE∗

In horizontal mergers, concentration is often measured with the Hirschman–Herfindahl Index (HHI). This index yields the price–cost margins in Cournot com-petition. In many modern merger cases, both buyers and sellers have market power,and indeed, the buyers and sellers may be the same set of firms. In such cases, the HHIis inapplicable. We develop an alternative theory that has similar data requirementsas the HHI, applies to intermediate good industries with arbitrary numbers of firmson both sides, and specializes to the HHI when buyers have no market power. Themore inelastic is the downstream demand, the more captive production and consump-tion (not traded in the intermediate market) affects price–cost margins. The analysisis applied to the merger of the gasoline refining and retail assets of Exxon and Mobilin the western United States. (JEL L13, L41)

I. INTRODUCTION

The seven largest refiners of gasoline onthe west coast of the United States accountfor over 95% of the production of CaliforniaAir Resources Board (CARB) certified gasolinesold in the region. The seven largest brands ofgasoline also account for over 97% of retail salesof gasoline. Thus, the wholesale gasoline marketon the west coast is composed of a numberof large sellers and large buyers who competeagainst each other in the downstream retailmarket. What will be the effect of a merger ofvertically integrated firms on the wholesale andretail markets? This question has relevance withthe mergers of Chevron and Texaco, Conoco andPhillips, Exxon and Mobil, and BP/Amoco andArco, all of which have been completed in thepast decade.

When monopsony or oligopsony faces anoligopoly, most analysts consider that the needfor protecting buyers from the exercise of mar-ket power is mitigated by the market power ofthe buyers and vice versa. Thus, even when thebuyers and sellers are separate firms, an analysisbased on dispersed buyers or dispersed sellersis likely to err. How should antitrust authori-ties account for the power of buyers and sellers

*We thank Jeremy Bulow and Paul Klemperer fortheir useful remarks and for encouraging us to exploredownstream concentration.Hendricks: Department of Economics, University of Texas

at Austin, Austin TX 78712.McAfee: Yahoo! Research, 3333 Empire Blvd., Burbank, CA

91504. Phone 818-524-3290, Fax 818-524-3102, [email protected]

in a bilateral oligopoly market in evaluating thecompetitiveness of the market? Merging a netbuyer with a net seller produces a more bal-anced firm, bringing what was formerly tradedin the intermediate good market inside the firm.Will this vertical integration reduce the exerciseof market power and produce a more compet-itive upstream market? Or will the verticallyintegrated firm restrict supply to other nonin-tegrated buyers, particularly if they are rivals inthe downstream market?

There is a voluminous theoretical literaturethat addresses these questions. Most of theliterature considers situations in which one ortwo sellers supply one or two buyers whocompete in a downstream market and modelstheir interactions as a bargaining game.1 Sellersnegotiate secret contracts with buyers specifyinga quantity to be purchased and transfers tobe paid by the buyer. The bilateral bargainingin these models is efficient, so there is nodistortion in the wholesale market. Gans (2007)uses the model of vertical contracting to derive

1. See Rey and Tirole (2008) for a survey of thisliterature. Several of the main papers in this literatureare Hart and Tirole (1990), McAfee and Schwartz (1994),O’Brien and Shaffer (1992), Segal (1999), and de Fontenayand Gans (2005).

ABBREVIATIONS

CARB: California Air Resources BoardHHI: Hirschman–Herfindahl IndexMHI: Modified Hirschman–Herfindahl Index

391

Economic Inquiry doi:10.1111/j.1465-7295.2009.00241.x(ISSN 0095-2583) Online Early publication November 12, 2009Vol. 48, No. 2, April 2010, 391–414 © 2009 Western Economic Association International

392 ECONOMIC INQUIRY

a concentration index that measures the amountof distortion in the vertical chain as a result ofboth horizontal concentration among buyers andsellers and the degree of vertical integration.However, the vertical contracting models donot describe intermediate good markets like thewholesale gasoline market in the western UnitedStates. The market consists of more than twosellers and two buyers, and trades occur at afixed, and observable, price. Other papers studyvertical mergers by assigning the market powereither to buyers or to sellers, but not both.2

These models are excellent for assessing someeconomic questions, including the incentive toraise rival’s cost, the effects of contact in severalmarkets, or the consequences of refusals-to-deal.But they do not address the implications ofbilateral market power that we wish to studyin this paper.

Traditional antitrust analysis presumes dis-persed buyers. Given such an environment, theCournot model (quantity competition) suggeststhat the Hirschman–Herfindahl Index (HHI),which is the sum of the squared market shares ofthe firms, is proportional to the price–cost mar-gin, which is the proportion of the price thatis a markup over marginal cost. Specifically,the HHI divided by the elasticity of demandequals the price–cost margin. The HHI is zerofor perfect competition and one for monopoly.The HHI has the major advantage of simplicityand low data requirements. In spite of well-publicized flaws, the HHI continues to be theworkhorse of concentration analysis and is usedby both the U.S. Department of Justice and theFederal Trade Commission. The HHI is inap-plicable, however, to markets where the buyersare concentrated, particularly if they compete ina downstream market.

Our objective in this paper is to offer analternative to the HHI analysis that applies tohomogenous good markets with linear pricingwhere buyers are concentrated and with (i) sim-ilar informational requirements, (ii) the Cournotmodel as a special case, and (iii) an underly-ing game as plausible as the Cournot model.The model we offer suffers from the sameflaws as the Cournot model. It is highly styl-ized and static. It uses a “black box” pricingmechanism motivated by the Cournot analysis.

2. See, for example, Hart and Tirole (1990), Ordover,Saloner, and Salop (1990), Salinger (1988), Salop andScheffman (1987), Bernheim and Whinston (1990). Analternative to assigning the market power to one side ofthe market is Salinger’s sequential model.

Moreover, our model will suffer from the sameflaws as the Cournot model in its applicationto antitrust analysis. Elasticities are treated asconstants when they are not, and the relevantelasticities are taken as known. However, theanalysis can be applied to markets with arbitrarynumbers of sellers and buyers, who individuallyhave the power to influence price, and buyerswho may compete against each other in a down-stream market. The analysis is simple to apply,and permits the calculation of antitrust effects ina practical way.

Our approach is based on the Klemperer andMeyer (1989) market game. In their model, sell-ers submit supply functions and behave strate-gically, buyers are passive and report their truedemand curves, and price is set to clear the mar-ket. We allow the buyers to behave strategicallyin submitting their demand functions, and applya similar concept of equilibrium as Klempererand Meyer. As is well known, supply functionmodels have multiple equilibria. Klemperer andMeyer (1989) reduce the multiplicity by intro-ducing stochastic demand, and they show that,if the support is unbounded, then the equilib-rium is unique. More recently, Holmberg (2004,2005) has shown that capacity constraints anda price cap can lead to uniqueness. A numberof authors (e.g., Turnbull 1981; Green 1996,1999; and Akgun 2005) obtain uniqueness byrestricting the supply schedules to be linear. Ourapproach is similar but we do not require lin-earity. In our model, sellers can select froma one-parameter family of nonlinear schedulesindexed by production capacity, and buyers canselect from a one-parameter family of nonlinearschedules indexed by consumption or retailingcapacity. Thus, sellers can exaggerate their costsby reporting a capacity that is less than it in factis, and buyers can understate demand. The mainadvantage of restricting the selection of sched-ules is that it allows us to study the strategicinteraction between sellers and buyers.

In a traditional assessment of concentrationaccording to the U.S. Department of JusticeMerger guidelines, the firms’ market sharesare intended, where possible, to be shares ofcapacity. This is surprising in light of the factthat the Cournot model does not suggest the useof capacity shares in the HHI, but rather theshare of sales in quantity units (not revenue).Like the Cournot model, the present studysuggests using the sales data, rather than thecapacity data, as the measure of market share.Capacity plays a role in our theory, and indeed

HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 393

a potential test of the theory is to check thatactual capacities, where observed, are close tothe capacities consistent with the theory.

The merger guidelines assess the effect of themerger by summing the market shares of themerging parties.3 Such a procedure provides auseful approximation, but is inconsistent withthe theory (either Cournot or our theory), sincethe theory suggests that, if the merging parties’shares do not change, then the prices are unlikelyto change as well. We advocate a more com-putationally intensive approach, which involvesestimating the capacities of the merging partiesfrom the pre-merger market share data. Giventhose capacities, we then estimate the effect ofthe merger on the industry, taking into accountthe incentive of the merged firm to restrict out-put (or demand, in the case of buyers). Horizon-tal mergers among sellers in intermediate inputmarket, where buyers are manufacturing firms,are more likely to raise price and be profitablein our model than in the Cournot model becausecapacity reports of sellers are typically strategiccomplements, not strategic substitutes. In whole-sale markets, the buyers are retailers, and theytypically respond to enhanced seller power byreducing their reported demand, thereby miti-gating the effects of the merger and complicat-ing its impact on prices and profits. The modeltreats horizontal mergers by buyers symmetri-cally. Vertical mergers in our model generatelarge efficiency gains because they eliminatetwo “wedges,” the markup by the seller and themarkdown by the buyer. Foreclosure effects areimportant when the merging firms are large.

Structural models of homogenous good mar-kets with dispersed buyers use an ad hoc mod-ification of the Cournot first-order conditions toestimate seller markups. They find that markupsare typically much lower than Cournot markups4

and attribute this finding to the Cournot model’sfailure to account for a firm’s expectations ofhow rivals will respond to its output choices. Inour model, each firm understands that reductionsin its supply will be partially offset by increasesin the outputs of its rivals. Their responsemeans that the elasticity of each firm’s resid-ual demand function exceeds the elasticity of

3. Farrell and Shapiro (1990) and McAfee and Williams(1992) independently criticize the Cournot model whileusing a Cournot model to address the issue.

4. See Bresnahan (1989) for a description of the method-ology and a survey of a number of empirical studies. Morerecent studies include Genesove and Mullin (1998) and Clayand Troesken (2003).

demand, so markups are lower in our model thanin the Cournot model. Furthermore, the rivals’responses are determined by their marginalcosts, so markups depend upon cost elasticitiesas well as the demand elasticity. Larger firmshave higher markups, and markups are higher inmarkets where marginal costs are steep. Struc-tural models of vertically related markets typ-ically estimate markups under the assumptionthat sellers post prices that buyers take as fixedin a sequential vertical-pricing game.5 In ourmodel, sellers and buyers move simultaneouslyand the division of rents from market powerdepends upon cost and demand functions. Weinvestigate the conditions under which the first-order conditions of our model can be used toestimate demand and cost parameters.

Our model also provides an interesting alter-native for studying spot electricity markets. Theoperation of these markets closely resemblesour market game: generating firms submit sup-ply schedules, buyers report the demands oftheir retail customers who face regulated prices,and an independent system operator chooses thespot price to equate reported supply to marketdemand. Green and Newberry (1992) was thefirst study of wholesale markets for electricityas a game in which firms’ strategies are theirchoices of supply functions.6 Empirical stud-ies of these markets have applied supply func-tion models or Cournot models to predict thepotential for generating firms to exercise marketpower in spot electricity markets and to estimatetheir markups.7 Our model generates a markupequation that combines the advantages of thesupply function model with the simplicity of theCournot model.

The second section presents a model of inter-mediate good markets and derives the equilib-rium price/cost margins and the value/price mar-gins, which is the equivalent for buyers, forvertically separated markets and for vertically

5. Several recent studies include Goldberg and Verboven(2001), Villas-Boas and Zhao (2005), and Villas-Boas andHellerstein (2006).

6. Other studies include Anderson and Philpott (2002),Anderson and Xu (2002, 2005), and Baldick et al. (2004).

7. The studies of markups include Borenstein andBushnell (1999), Borenstein, Bushnell, and Wolak (2003),and Wolak (2003) on California; Hortacsu and Puller (2008)on Texas; Mansur (2007), Bushnell, Mansur, and Saravia(2008) on California, New England, and Pennsylvania, NewJersey, Maryland (PJM); Green and Newbery (1992), Green(1996), Brunekreeft (2001), Sweeting (2007), and Wolfram(1999) on England and New Wales; and Wolak (2000, 2003,2007) on Australia.


integrated markets. The third section extendsthe model to spot markets in electricity andwholesale markets in which buyers compete in adownstream market. The fourth section analyzeshorizontal and vertical mergers. The fifth sectionexamines identification issues that would arise intrying to apply our model to market data. Thesixth section applies the model to the mergerof the west coast assets of Exxon and Mobil toillustrate the plausibility and applicability of thetheory. The final section concludes.

II. INTERMEDIATE GOOD MARKETS

We begin with a standard model of a marketfor a homogenous intermediate good Q. Thereare n firms, indexed by i from 1 to n. Each selleri produces output xi using a constant returns toscale production function with fixed capacity γi .Thus, seller i’s production costs take the form

C(xi, γi ) = γic

(xi

γi

),(1)

where c(·) is convex and strictly increasing.8

Each buyer j consumes intermediate outputqj and values that consumption according toa function V (qj , kj ) where kj is buyer j ’scapacity for processing the intermediate output.We assume that V is homogenous of degree oneso that it can be expressed as

V (qj , kj ) = kjv

(qj

kj

),(2)

where v(·) is concave and strictly increasing.9

A firm may be both a seller and a buyer, thatis, it may produce the intermediate good andalso consume it. Such firms are called verticallyintegrated, although they may be net sellers ornet buyers. Letting p denote the market-clearingprice in the intermediate good market, the profitsto a vertically integrated firm i are given by

πi = p (xi − qi) + kiv

(qi

ki

)− γic

(xi

γi

).

(3)

The profits to a firm if it is either a pure seller ora pure buyer can be obtained by setting qi = 0or xi = 0, respectively.

Markets in which both sellers and buy-ers exercise market power are called bilateral

8. In addition, we assume that c′(z) −→ ∞ as z −→ ∞.9. In addition, we assume v′(z) −→ 0 as z −→ ∞.

oligopoly. A market with no vertically integratedfirms is called a vertically separated market.These markets can be further decomposed intooligopoly markets, in which sellers have marketpower and buyers do not, and oligopsony mar-kets, in which buyers have market power butsellers do not. A market with one or more ver-tically integrated firms is a vertically integratedmarket.

In what follows, we will need to distinguishbetween two kinds of intermediate good mar-kets based on the type of buyer. Markets inwhich buyers are manufacturing firms are calledintermediate input markets. In these markets,a buyer j combines the intermediate input qj

with capacity kj using a constant returns to scaletechnology F(qj , kj ) to produce a good yj thatit sells at price r .10 Thus, its revenue functioncan be expressed as

V (qj , kj ) = rkif

(qi

ki

).

A manufacturing firm that has twice the capacityof another firm can produce twice as much at thesame average productivity.

Markets in which buyers are retail firms arecalled wholesale markets. In these markets, abuyer j purchases qj to resell to final consumersat price r . Here yj = qj and firm j ’s valuationof qj is given by

V (qj , kj ) = rqj − kjw

(qj

kj

)= kj

[r

(qj

kj

)− w

(qj

kj

)]where w represents unit selling costs. A retailerwith twice as much selling capacity (e.g., num-ber of stores) can sell twice as much at the sameunit cost.

In the Cournot model of oligopoly markets,sellers submit quantities and the market choosesprice to equate reported supply to demand.The equilibrium price is equal to each buyer’smarginal willingness to pay, but exceeds eachseller’s marginal cost of supply. In the standardoligopsony model, buyers submit quantities andthe market chooses price to equate reported sup-ply and demand. The equilibrium price is equalto each seller’s marginal cost, but exceeds eachbuyer’s true willingness to pay. In each of these

10. The production function could include other inputsprovided their quantities are proportional to qi .


models, one side of the market is passive andthe other side behaves strategically, anticipat-ing the market-clearing mechanism in order tomanipulate prices. Our interest, however, is in amarket where both buyers and sellers recognizetheir ability to unilaterally influence the priceand behave strategically. In order to model thistype of market, we extend the Klemperer andMeyer model, in which sellers behave strategi-cally in submitting supply schedules, to allowbuyers to behave strategically in submitting theirdemand schedules.

In adopting this model, however, we imposesome restrictions on the schedules that the firmscan report. Sellers have to submit cost scheduleswhich come in the form γc(x/γ), and buyershave to submit valuation functions which comein the form kv(q/k). In a mechanism designframework, agents can lie about their type, butthey cannot invent an impossible type. Theadmissible types in our model satisfy (1) and(2), and agents are assumed to be bound bythis message space. For sellers, the messagespace is a one-parameter family of schedulesindexed by production capacity, and for buyers,the message space is a one-parameter familyof schedules indexed by consumption capacity.Therefore, a seller’s type is a capacity γ, abuyer’s type is a capacity k, and if the firmis vertically integrated, its type is a pair ofcapacities (γ, k). Agents (simultaneously) reporttheir types to the market mechanism, but indoing so, they do not have to tell the truth.A seller can exaggerate its costs by reporting acapacity γ̂ that is less than what it is in reality,and a buyer can understate its willingness topay by reporting a capacity k̂ that is less thanwhat it is in reality.11 Values and costs are notwell specified at zero capacity. However, thesolution can be calculated for arbitrarily smallbut positive capacities and zero capacity handledas a limit. Firms with zero capacity would thenreport zero capacity.

Given the agents’ reports, the market mecha-nism chooses price p to equate reported supplyand reported demand, and allocates the outputefficiently. The solution is characterized by the

11. A buyer’s marginal willingness to pay for anotherunit of input at q units of input is measured by v′ ( q

k

). Since

v is concave, this derivative is increasing in k. Therefore,a buyer understates its willingness to pay by underreportingits capacity.

balance equation,

Q =n∑

i=1

qi =n∑

i=1

xi = X(4)

and the marginal conditions,

v′(

qj

k̂j

)= p = c′

(xi

γ̂i

), i, j = 1, .., n.

(5)

Note that, if everyone tells the truth, then theequilibrium outcome is efficient. Our model canbe viewed as turning the market into a blackbox, as in fact happens in the Cournot model,where the price formation process is not mod-eled explicitly. Given this black box approach,it seems appropriate to permit the market to beefficient when agents do not, in fact, exerciseunilateral power. Such considerations dictate thecompetitive solution, given the reported types.Any other assumption would impose inefficien-cies in the market mechanism, rather than havinginefficiencies arise as the consequence of therational exercise of market power by firms withsignificant market presence.

Each firm anticipates the market mecha-nism’s decision rule in submitting its reports.From Equation (4), it follows that

qi = k̂iQ

K, xi = γ̂iQ

�,(6)

where

K =n∑

i=1

k̂i , � =n∑

i=1

γ̂i .(7)

Thus, given the firms’ reports, market outputQ(�, K) solves the equation

v′(

Q

K

)= c′

(Q

�

),(8)

which depends only on the aggregate productionand consumption capacity reports. Market pricep(�, K) is obtained by substituting Q(�, K)into the marginal conditions of Equation (5), andthe output is allocated to sellers and buyers usingthe market share Equation (6).

The firms’ actual types are common knowl-edge to the firms. Thus, in choosing their reports,firms know the true types of other firms. Thenthe payoff to a vertically integrated firm i from


submitting reports (̂γi , k̂i) is

πi = kiv

(k̂iQ(�, K)

kiK

)(9)

−γic

(γ̂iQ(�, K)

γi�

)

−p(�, K)Q(�, K)

(k̂i

K− γ̂i

�

).

If firm i is a seller with no consumption capac-ity, then ki = k̂i ≡ 0; similarly, if firm i is abuyer with no production capacity, then γi =γ̂i ≡ 0. The Nash equilibrium to the marketgame consists of a profile of reports with theproperty that (i) each firm correctly guesses thereports of other firms and (ii) no firm has anincentive to submit a different, feasible report.

A. Equilibrium

We first derive and discuss equilibriummarkups in vertically separated markets. Weconsider several special cases including theCournot model. We then derive and discussthe equilibrium markups in vertically integratedmarkets.

Before stating the theorems, we require someadditional notation. The market demand functionQd is given by v′(Qd/K) = p, so the marketelasticity of demand is

ε = −(

p

Q

) (dQd

dp

)= −v′(Q/K)

(Q/K)v′′(Q/K).

(10)

Similarly, the market supply function Qs isgiven by c′(Qs/�) = p, so the market elasticityof supply is

η =(

p

Q

) (dQs

dp

)= c′(Q/�)

(Q/�)c′′(Q/�).(11)

Let σi and si denote firm i’s market sharein production and consumption, respectively.Given any profile of reports, the market sharesare equal to reported capacity shares, that is,

σi = γ̂i

�, si = k̂i

K.(12)

Finally, define

c′i ≡ c′

(σiQ

γi

), v′

i ≡ v′(

siQ

ki

).(13)

as firm i’s equilibrium marginal cost andmarginal valuation.

THEOREM 1. Suppose markets are verticallyseparated. Then

p − c′i

p= σi

ε + η(1 − σi ),(14)

and

v′i − p

p= si

ε(1 − si) + η.(15)

COROLLARY 1. (i) γ̂/γ is less than 1 and de-creasing in γ; (ii) k̂/k is less than 1 and decreas-ing in k.

The exercise of market power by sellers andbuyers creates a double markup problem. Sellersreport less than their true capacity, therebyoverstating their marginal cost. Since the marketmechanism equates price to reported marginalcosts, it exceeds each seller’s actual marginalcost. Buyers report less than their true capacity,thereby understating their true willingness topay. As a result, price is less than each buyer’sactual marginal willingness to pay. The corollaryestablishes that, on both sides of the market,the distortion is larger for firms with largercapacities.

As in the standard Cournot model, sellermarkups are constrained by the elasticity ofdemand. If demand is elastic, then ε is largeand sellers’ profit margins are small. However,the seller’s margins also depend upon the elas-ticity of supply. In the Cournot model, if aseller restricts output, market supply falls by thesame amount, and the price response dependsonly upon the elasticity of demand. In a sup-ply function model such as ours, if a seller triesto restrict output by reporting a higher marginalcost schedule, the reported market supply shiftsto the left causing price to rise, but other sellersmove up their reported supply curves, expand-ing their output. Thus, the fall in market supplyis less than the reduction in the seller’s output.The price response depends upon the slope ofthe reported supply curves. If marginal costs areroughly constant (i.e., c′′ � 0), then η is verylarge, individual sellers cannot raise price sig-nificantly by constricting their supply schedule,and the Bertrand outcome arises. On the otherhand, if marginal cost curves are steeply sloped(i.e., c′′/c′ −→ ∞), then η approaches 0, andthe Cournot outcome arises. Since our model


treats buyers and sellers symmetrically, the samereasoning applies to buyer markups.

We turn next to vertically integrated markets.

THEOREM 2. Suppose markets are verticallyintegrated. Then, in any interior equilibrium,v′

i = c′i and

v′i − p

p= c′

i − p

p= si − σi

ε(1 − si) + η(1 − σi ).

(16)

There are two immediate observations. First,each vertically integrated firm is technically effi-cient about its production; that is, its marginalcost is equal to its marginal value. Thus, thefirm cannot, in the equilibrium allocation, gainfrom secretly producing more and consumingthat output. This is not to say that the firm couldnot gain from the ability to secretly produceand consume, for the firm might gain from thisability by altering its reports appropriately. Forexample, if the firm is a net seller, it will tryto raise price by restricting supply and overstat-ing demand. It accomplishes the first by report-ing a production capacity γ̂ that is less than itsactual capacity γ, and the second by reporting aconsumption capacity k̂ that exceeds its actualcapacity of k. A net buyer does the opposite,reporting higher production capacity and lowerconsumption capacity than its actual capacities.Second, net buyers value the good more thanthe price, and net sellers value the good lessthan price. Thus, net buyers restrict their demandbelow that which would arise in perfect competi-tion, and net sellers restrict their supply. In bothcases, the gain arises because of price effects.

THEOREM 3. The (quantity weighted) aver-age difference between marginal valuations andmarginal costs satisfies:

1

p

(n∑

i=1

siv′i −

n∑i=1

σic′i

)(17)

=n∑

i=1

((si − σi )

2

ε(1 − si) + η(1 − σi )

).

In evaluating proposed horizontal mergers invertically separated markets, antitrust agencies(and courts) focus primarily on demand elas-ticity, the concentration levels in the industryprior to the merger and the predicted change inconcentration levels due to the merger, whereconcentration is measured using the HHI. This

analysis is motivated by the Cournot model.Theorem 3 gives the equivalent of the HHI forthe present model. We will refer to it as themodified Hirschman–Herfindahl index (MHI).It has the same useful features—it depends onlyon market shares and elasticities—but there aretwo important differences. First, it suggests thatanalysts also need to consider the elasticity ofsupply in evaluating the competitiveness of themarket. Second, it generalizes the analysis tovertically integrated markets and suggests thatanalysts use the firms’ net positions to measurethe effects of market power. As noted above,zero net demand causes no inefficiency. Thus,an intermediate good market in which each firmis vertically integrated and supplies only itself isperfectly efficient. However, with even a smallbut nonzero net demand or supply, size exacer-bates the inefficiency.

In this framework, the shares are of pro-duction or consumption, and not capacity. TheU.S. Department of Justice Merger Guidelinesgenerally calls for evaluation shares of capac-ity. While our analysis begins with capacities,the shares are actual shares of production (σi)or consumption (si), rather than the capacityfor production and consumption, respectively.Firms may have the same capacity in produc-tion and consumption but nevertheless chooseto be a net seller or a net buyer depending uponmarket conditions. The use of actual consump-tion and production is an advantageous featureof the theory, since these values tend to be read-ily observed, while capacities are not. Finally,the shares are shares of the total quantity andnot revenue shares. However, like the Cournotmodel, our model is not designed to handleindustries with differentiated products, which isthe situation where a debate about revenue ver-sus quantity shares arises.

B. Intermediate Input Markets with ConstantElasticities

An important special case of our model is onein which value and cost elasticities are constantand buyers are manufacturing firms. Supposemarginal cost is given by

c′(z) = z1/η,

where η > 0, and marginal willingness to pay isgiven by

v′(z) = z−1/ε,


where ε > 1.12 Given any vector of capacityreports, the market clearing conditions yieldclosed form solutions for output

Q(�, K) = �ε/(ε+η)Kη/(ε+η)

and price

p(�, K) = �−1/(ε+η)K1/(ε+η),

which facilitates a quantitative assessment offirm misrepresentations and the cost of thosemisrepresentations. If elasticities vary, the for-mulae derived from the constant elasticity caseapply approximately, with the error determinedby the amount of variation in the elasticities.

Let Qf represent the first best quantity, whicharises when all firms are sincere in their behav-ior, and pf be the associated price. Then:

THEOREM 4. With constant elasticities, thesize of the firms’ misrepresentations is given by

k̂i

ki

=(

1 + si − σi

ε(1 − si) + η(1 − σi )

)−ε

(18)

γ̂i

γi

=(

1 + si − σi

ε(1 − si) + η(1 − σi )

)η

.

Moreover,

(19)

Qf

Q=

[n∑

i=1

si

(1 + si − σi

ε(1 − si) + η(1 − σi )

)ε]η/(ε+η)

×[

n∑i=1

σi

(1 + si − σi

ε(1 − si) + η(1 − σi )

)−η]ε/(ε+η)

and

pf

p=

⎡⎢⎢⎢⎣∑n

i=1σi

(1 + si − σi

ε(1 − si ) + η(1 − σi )

)−η

∑n

i=1si

(1 + si − σi

ε(1 − si ) + η(1 − σi )

)ε

⎤⎥⎥⎥⎦1/(ε+η)

.

(20)

Equation (18) confirms the intuition that themisrepresentation is largest for the largest nettraders, and small for those not participating

12. The associated cost and valuation functions are

c(z) =(

η

η + 1

)z(η+1)/η, v(z) =

(ε

ε − 1

)z(ε−1)/ε.

significantly in the intermediate good market.Indeed, the size of the misrepresentation is pro-portional to the discrepancy between price andmarginal value or cost, as given by Theorem 2,adjusted for the demand elasticity. This is hardlysurprising, since the constant demand and sup-ply elasticities insure that marginal values can beconverted to misrepresentations in a log-linearfashion.

Equation (19) provides the formula for losttrades. Here, there are two effects. Net buyersunderrepresent their demand, but overrepresenttheir supply. On balance, net buyers under-represent their net demands, which is whythe quantity-weighted average marginal valueexceeds the quantity-weighted average marginalcost. Equation (19) provides a straightforwardmeans of calculating the extent to which amarket is functioning inefficiently, both beforeand after a merger, at least in the case where theelasticities are approximately constant.

Equation (20) gives the effect of strategicbehavior in the model on price. Note that theprice can be larger, or smaller, than the efficientfull-information price. Market power on thebuyer’s side (high values of si) tends to decreasethe price, with buyers exercising market power.Similarly, as σi increases, the price tends torise.

III. EXTENSIONS

In this section we consider two extensionsof the model. The first is to markets in whichthe buyers compete in a downstream market.The second is to spot markets such as electricitymarkets in which firms are net traders.

A. Downstream Competition

In many, perhaps, even most, applications,the assumption that a buyer in the intermedi-ate good market can safely ignore the behaviorof other firms in calculating the value of con-sumption is unfounded. This is particularly truewhen the buyers are retail firms. In this section,we extend the model to wholesale markets inwhich retail firms compete in quantities in thedownstream market.

Recall that the value of consumption to aretail firm is given by

V (qi, ki) = r(Q)qi − kiw

(qi

ki

).


where r(Q) is the downstream inverse demandand w represents unit selling costs. Firm prof-its are:

πi (γ, k) = r(Q)qi − kiw

(qi

ki

)(21)

−γic

(xi

γi

)− p(qi − xi).

As before, we calculate the efficient solution,which satisfies:

p = c′(Q/�) = c′(xi/γi )(22)

and

r(Q) = p + w′(Q/K) = p + w′(qi/ki).(23)

Let α be the elasticity of downstream demand,and β be the elasticity of the selling cost w. Letθ be ratio of the intermediate good price p tothe final good price r . The observables of theanalysis will be the market shares (both produc-tion, σi , and retail, si), the elasticity of final gooddemand, α, of selling cost, β, of production cost,η, and the price ratio θ = p/r . It will turn outthat the elasticities enter in a particular way, andthus it is useful to define:

A = α−1, B = (1 − θ)β−1,(24)

and C = θη−1.

We replicate the analysis of Section 3 inthe Appendix for this more general model. Thestructure is to use the efficiency equations toconstruct the value to firm i of reports ofk̂i and γ̂i . The first-order conditions providenecessary conditions for a Nash equilibrium tothe reporting game. These first-order conditionsare used to compute the price/cost margin,weighted by the firm shares. In particular, welook for an MHI given by:

MHI =n∑

i−1

1

r[(r(Q)−p−w′

i )si + (p−c′i )σi],

where wi = w(qi/γi ).The main theorem characterizes the MHI for

an interior solution.

THEOREM 5. In an interior equilibrium,

MHI =n∑

i=1

[BC(si − σi )

2 + ABs2i (1 − σi ) + ACσ2

i (1 − si )

A(1 − si )(1 − σi ) + B(1 − σi ) + C(1 − si )

].

(25)

While complex in general, this formulahas several important special cases. If A = 0,

the downstream market has perfectly elasticdemand. As a result, r is a constant, and (25)readily reduces to Theorem 3. Note, however,that Theorem 4 does not apply since, in thewholesale market, the market-clearing condi-tions fail to yield closed form solutions for out-put and prices.13 As we shall see in the nextsection, this distinction between the intermedi-ate input and wholesale market also matters formerger analysis.

When B = 0, there is a constant retailing costw. This case is analogous to Cournot, in that allfirms are equally efficient at selling, althoughthe firms vary in their efficiency at producing.In this case, (25) reduces to

MHI|B=0 =n∑

i=1

[ACσ2

i

A(1 − σi ) + C

]

=n∑

i=1

[θσ2

i

η(1 − σi ) + θα

].

The Herfindahl index reflects the effect of thewholesale market through the elasticity of sup-ply η. If η = 0, the Cournot HHI arises. Forpositive η, the possibility of resale increases theprice/cost margin. This increase arises becausea firm with a large capacity now has an alter-native to selling that capacity on the market.A firm with a large capacity can sell some ofits Cournot level of capacity to firms with asmaller capacity. The advantage of such salesto the large firm is the reduction in desire ofthe smaller firms to produce more, which helpsincrease the retail price. In essence, the largerfirms buy off the smaller firms via sales in theintermediate good market, thereby reducing theincentive of the smaller firms to increase theirproduction.

Equation (25) can be decomposed intoHerfindahl-type indices for three separate mar-kets: transactions, production, and consumption.Note,

13. In wholesale markets, Q(�,K) is defined implicitlyby the equilibrium condition(

Q

�

)1/η

= r −(

Q

K

)1/β

,

whereas in intermediate good markets, market clearingimplies (

Q

�

)1/η

=(

Q

K

)−1/ε

,

which yields an explicit solubion for Q(�,K).


MHI =n∑

i=1

[B(1 − σi ) + C(1 − si)

A(1 − si)(1 − σi ) + B(1 − σi ) + C(1 − si)

] [(si − σi )

2

C−1(1 − σi ) + B−1(1 − si)

]

+n∑

i=1

[A(1 − σi )(1 − si)

A(1 − si)(1 − σi ) + B(1 − σi ) + C(1 − si)

] [Bs2

i

(1 − si)+ Cσ2

i

(1 − σi )

].

The MHI is an average of three separate indices.The first index corresponds to the transactionsin the intermediate good market. In form, thisterm looks like the expression in Theorem 3,adjusted to express the elasticities in terms ofthe final output prices. The second expressionis an average of the indices associated withproduction and consumption of the intermediategood. These two indices ignore the fact thatfirms consume some of their own production.

When the downstream market is very elas-tic, as we have already noted, A is near zero. Inthis case, the MHI reduces to that of Theorem 3,because elastic demand in the downstream mar-ket eliminates downstream effects, so that theonly effects arise in the intermediate good mar-ket. In contrast, when the downstream marketis relatively inelastic, downstream effects dom-inate, and the MHI is approximately an averageof the Herfindahl indices for the upstream anddownstream markets, viewed as separate mar-kets.

In a sense, these limiting cases provide a res-olution of the question of how to treat captiveconsumption. When demand is very inelastic,as with gasoline in California, then the issueof captive consumption can be ignored withoutmajor loss; it is gross production and consump-tion that matters. In this case, it is appropriateto view the upstream and downstream marketsas separate markets and ignore the fact thatthe same firms may be involved in both. Inparticular, a merger of a pure producer and apure retailer should raise minimal concerns. Onthe other hand, when demand is very elastic(A near zero), gross consumption and gross pro-duction can be safely ignored, and the markettreated as if the producers and consumers of theintermediate good were separate firms, with nettrades in the intermediate good the only issuethat arises.14 Few real-world cases are likely to

14. However, the denominator still depends on grossproduction and consumption, rather than net production andconsumption. This can matter when mergers dramaticallychange market shares, and even the merger of a pureproducer and pure consumer can have an effect.

approximate the description of very elastic mar-ket demand.15 However, the case of A = 0 alsocorresponds to the case where the buyers do notcompete in a downstream market, and may thushave alternative applications.

In the Appendix, we provide the formulaegoverning the special case of constant elas-ticities. It is straightforward to compute thereduction in quantity that arises from a con-centrated market, as a proportion of the fullyefficient, first-best quantity. Moreover, we pro-vide programs which take market shares asinputs and compute the capital shares of thefirms, the quantity reduction, and the effects of amerger.16

B. Electricity Markets

Theoretical and empirical studies of whole-sale electricity markets have applied supplyfunction models or Cournot models to predictthe potential for generating firms to exercisemarket power in spot electricity markets andto estimate markups. Our model generates amarkup equation that combines the advantagesof the supply function models with the simplic-ity of the Cournot model.

In day-ahead or real-time balancing mar-kets, generating firms submit supply schedules,buyers report the amounts that they need fortheir retail customers, and an independent sys-tem operator chooses price to equate reportedsupply to market demand. An important fac-tor determining the generating firms biddingbehavior in these markets is their contract posi-tions. With the exception of firms in the Cal-ifornia market, generating firms typically signforward contracts with buyers in which theyagree to deliver a fixed amount of electricityat a predetermined price. Generating firms that

15. When market demand is very elastic, it is likely thatthere are substitutes that have been ignored. It would usuallybe preferable to account for such substitutes in the market,rather than ignoring them.

16. This program is available on McAfee’s Web site.


have signed such vertical contracts are essen-tially vertically integrated, and they may benet sellers or net buyers in the spot market.Green (1999) and Wolak (2000) discuss the the-oretical implications of forward contracts andshow that they make the spot market morecompetitive.

Let qi denote firm i’s forward contract quan-tity and let r denote the contract price. Gener-ating firms typically take short positions in theforward market, in which case qi is positive.Firm i’s profit from supplying xi at price p isgiven by

πi = p(xi − qi) − vic

(xi

vi

)+ rqi.

The firm’s revenues consist of two components:the amount it earns from its contract positionand the payment it receives from either reducingits supply below qi or from increasing its supplyabove qi . When it reduces its supply, it is in a netbuyer position, buying the reduction in supplyat price p from the spot market and sellingthis amount to its customers at the contractprice of r . When it increases its supply, firmi is in a net seller position, selling the increaseat price p to retailers. The profit function canalso be interpreted as the profits of a verticallyintegrated firm that sells electricity to its retailcustomers at a regulated price of r.

We assume that the firms face a downward-sloping inverse demand function p(X).17 Theoperator knows the marginal cost schedules butdoes not know the capacity that firms have avail-able or at least cannot force them to make allof their capacity available. Firms are asked toreport their available generating capacities. Con-tract positions are common knowledge amongthe firms. On the basis of these reports, theoperator equates demand to supply and allo-cates output across firms by equating reportedmarginal costs so in equilibrium,

p(X) = c′(xi

ν̂

)and for i = 1, .., n. Note that the allocationrule does not depend upon the firms’ contractpositions, so firms report only their productioncapacity. Let α(p) be the elasticity of demandat price p and define si = qi/X.

17. Market demand is downward-sloping in some elec-tricity markets because it is equal to the fixed retail demandless a competitive import supply.

THEOREM 6. In an interior equilibrium

p − c′i

p= σi − si

η(1 − σi ) + α.

The theorem states that the firm’s reportedcapacity exceeds its true capacity (i.e., marginalcost exceeds reported marginal cost) when thefirm produces less than its contract quantity, andthe opposite is true when it produces more thanits contract quantity. It reports truthfully whenit is balanced. (If demand is perfectly inelastic,then α is equal to zero in the above formula.)The intuition is that, in former case, the firm isin a net buy position and wants to lower price,whereas in the latter case, it is in a net sell posi-tion and wants to raise price. Markups in ourmodel will vary across firms depending upontheir contract positions, and with demand con-ditions if the elasticity of costs is not constant.Since marginal cost functions in the electricitymarkets are approximately L-shaped, our modelpredicts that markups are essentially zero in lowdemand periods and higher during high demandperiods, particularly for large net sellers. This isconsistent with the evidence presented in Bush-nell, Mansur, and Saravia (BMS) (2008).18 Theyfind that prices are very close to marginal costsduring off-peak hours and higher during peakhours.

Our markup equation is closely related to themarkup equations that have been estimated inthe literature. Wolak (2000, 2003) assumes thateach firm i faces a stochastic residual demand,RDi (p, ε), and submits a bid schedule that is expost optimal. That is, for each realization of therandom variable ε, xi(p) is a best reply to the expost residual demand and satisfies the optimalitycondition

p − c′i

p= σi − si

αi (p, ε).

where αi (p, ε) is the elasticity of the residualdemand facing firm i. It incorporates both theelasticity of demand and the aggregate elasticityof supply bid by firm i’s rivals and can be esti-mated from data on the bid schedules of firm

18. Bushnell, Mansur, and Saravia (2008) study markupsin California, New England, and the Pennsylvania, New Jer-sey, Maryland (PJM) electricity markets using the Cournotmodel to predict the potential for generating firms toexercise market power. Borenstein and Bushnell (1999) andBorenstein, Bushnell, and Wolak (2002) also use the Cournotmodel to study markups in the California electricity market.


i’s rivals. In his study of Australian electric-ity markets, Wolak has data on a firm’s con-tract positions, and he develops a procedure forrecovering the firm’s cost function from the expost optimality condition. Hortacsu and Puller(2008) show that ex post optimal bid functionsare a Bayesian equilibrium when the firms’ con-tract positions are private information and bidstrategies are additively separable in the pri-vate information. The additivity assumption alsoimplies that the elasticity of the residual demandfunction does not depend upon ε. They exploitthe availability of data on the firms’ marginalcost and bid schedules in the Texas electric-ity market to infer the firms’ contract positionsand then use the markup equations to test theex post optimality conditions. Sweeting (2007)also uses the ex post optimality condition in hisstudy of the Wales electricity market to test thehypothesis of optimal bidding behavior undervarious assumptions about the firms’ contractpositions.

IV. MERGERS

In this section we examine the equilibriumeffects of horizontal and vertical mergers. Theconstant returns to scale assumption facilitatesthe study of mergers. It implies that the mergerof two firms i and j produces a firm with con-sumption capacity ki + kj and production capac-ity γi + γj , and thereby is subject to the sameanalysis. In what follows, we focus primarily onmergers in vertically separated markets for tworeasons. First, previous merger studies typicallymake this assumption and we wish to comparethe results of our analysis to their results. Sec-ond, the vertically separated market provides apolar case in which qualitative results can beobtained under the assumption of constant elas-ticities. The analysis illustrates the economicforces at work in vertically integrated markets,where the impact of mergers depends upon thevalues of the elasticity parameters and hencerequires a more quantitative analysis. We willassume throughout this section that elasticities,including that of downstream demand, are con-stant.

A. Horizontal Mergers

The DOJ’s Merger Guidelines estimate theimpact of a horizontal merger in oligopolymarkets under the assumption that the merged

firms do not change their capacity reports.19

However, as Farrell and Shapiro (1990) haveobserved, this rule ignores the fact that post-merger behavior is likely to be different frompre-merger behavior since the merging firmswill internalize the negative externality that theirpre-merger actions imposed on each other’sprofits. An equilibrium analysis is necessary andFarrell and Shapiro provide such an analysisfor Cournot oligopoly markets. They investigatethe relationship between HHI and consumerand social welfare, and provide necessary andsufficient conditions under which a merger raisesprice. They also provide sufficient conditionsunder which profitable mergers raise welfare.Mergers without cost synergies generally raiseprice but are often not profitable in the Cournotmodel, since the merging firms reduce outputand rival firms respond by expanding theiroutput. McAfee and Williams (1992) provideconditions under which mergers are profitablefor the special case of quadratic costs and lineardemand.

We begin our equilibrium analysis of hori-zontal merger by examining firms’ best replies.Substituting equilibrium output and price intothe profit function of seller i and taking logs20,we obtain

log πi =(

η + 1

η

) [log Q(�, K) − log �

]+ log

[̂νi −

(η

η + 1

)ν−1/η

i ν̂(η+1)/η

i

].

Differentiating yields firm i’s best reply, whichsolves

v̂i

�

[1 − ∂Q

∂�

�

Q

]= 1 − (̂vi/v)1/η[

((η + 1)/η) − (̂vi/vi)1/η

] .

(26)

The right-hand side of Equation (26) is decreas-ing in v̂i . The two terms on the left-hand side

19. Applying this rule to a merger of firms 1 and 2 usingour index of market power yields

�MHI = (1 − ρ1)σ21 + (1 − ρ2)σ

22 + 2σ1σ2,

where

ρi = ε + (1 − σ1 − σ2)

ε + (1 − σi )η< 1.

Therefore, �MHI exceeds �HHI (although MHI < HHI).20. Note that maximizing log πi is the same as maxi-

mizing πi .


of the equation capture the impact of capacityreports of other firms. The first term is firmi’s market share, which falls with reports byother sellers, and the second involves the pro-duction capacity output elasticity. An analogousequation determines the buyer’s best reply. Thekey issue that determines whether reports ofother firms are strategic substitutes or comple-ments is their impact on the production capacityoutput elasticity (or consumption capacity out-put elasticity in the case of buyers). This impactwill depend upon the type of market.

In intermediate input markets in which buy-ers face a constant downstream price, it iseasily verified that the output elasticities areconstants.21

LEMMA 1. Consider a vertically separated,intermediate input market with constant cost andvalue elasticities and a fixed downstream price.Then (i) the capacity reports by a seller and abuyer are independent of each other and (ii) thecapacity reports of any pair of sellers or buyersare strategic complements.

Lemma 1 implies that intermediate input mar-kets with no vertically integrated firms are notonly structurally separate, they are also strategi-cally separate. It also implies that the reportinggame is a log supermodular game.22

THEOREM 7. Consider a vertically sepa-rated, intermediate input market with constantcost and value elasticities and a fixed down-stream price. A horizontal merger among sellersreduces reported production capacity, increasesprice, and decreases output. A horizontal mergeramong buyers reduces reported consumptioncapacity and decreases price and output. Hor-izontal mergers are always profitable for themerging firms.

We show in the Appendix that the best replyof merging firms to pre-merger reports of otherfirms is always to report a capacity that is lessthan the sum of their pre-merger reports. Itthen follows from Lemma 1 that only firms

21. Recall that

Q(�,K) = �ε/(ε+η)Kη/(ε+η).

22. Substituting the expression for Q(�,K) given inthe previous footnote, it is easily verified that each seller’s(buyer’s) profit function is log supermodular in the capacityreports of other sellers (buyers) and independent of thecapacity reports of buyers (sellers).

on the same side of the market will respond,and their responses are mutually reinforcing.This leads to the following predictions aboutthe equilibrium impact of horizontal mergers.A merger among sellers reduces reported supplybut does not affect reported demand. Henceprice increases and output falls. Similarly, amerger among buyers reduces reported demandbut does not affect reported supply, so both priceand output fall. The merging firms profit froma merger in two ways. First, it gives them moremarket power to reduce capacity and raise price,and second, other firms on the same side ofthe market will do the same. The latter effectreflects the key difference between our modeland the Cournot model. In our model, bestreplies are strategic complements, whereas inthe Cournot model, they are strategic substitutes.Akgun (2005) obtains similar results in a supplyfunction model of oligopoly markets in whichthe sellers are restricted to reporting linearsupply schedules.

In wholesale markets, the equilibrium outputelasticities are functions of the aggregate pro-duction and retailing capacity.23 This introducestwo new effects into the analysis of a horizon-tal merger, which complicates the analysis ofa merger. First, the market is no longer strate-gically separate. If two sellers merge, buyerswill respond. Tedious calculations reveal that thecapacity reports of a buyer and a seller are strate-gic complements as long as downstream demandis not too inelastic.24 Thus, when the merg-ing sellers try to reduce their reported capac-ity, buyers will reduce their capacity reports,thereby lowering reported demand. In this way,the enhanced market power of sellers is miti-gated by buyers exercising buyer power. Pricesare lower, mergers are less profitable, and inef-ficiency costs increase. In fact, the strategiccomplementarity between the buyer and sellerreports can lead to nonexistence of an interiorsolution. For example, if the merger among sell-ers creates a monopoly, and there is only onebuyer who sells at a fixed price of r , it canbe shown that the only intersection point ofthe best replies is (0,0). Clearly, in this case,

23. In this case, Q(�,K) is defined implicitly by(Q

�

)1/η

+(

Q

K

)1/ε

− Q1/α = 0.

24. A sufficient condition for strategic complementarityis ε ≥ α. However, if α is sufficiently small, the second-orderconditions will be violated.


a merger would not be profitable. Second, theseller reports may no longer be strategic com-plements. An increase in the reports of othersellers reduces the production capacity outputelasticity (assuming downstream demand is nottoo inelastic). This effect dominates the marketshare effect when sellers have a lot of marketpower (i.e., η is small).

B. Vertical Mergers

The voluminous literature on vertical merg-ers is primarily concerned with two issues: effi-ciency and foreclosure. Vertical mergers cangenerate substantial efficiency gains by eliminat-ing the double markup problem that arises fromsellers exercising market power in the upstreammarket and buyers exercising market power indownstream markets. This is the reason whyantitrust agencies are less likely to contest avertical merger than a horizontal merger. Theprimary concern that the agencies have about avertical merger is the risk that the vertically inte-grated firm will foreclose the intermediate inputmarket to other buyers with whom it may becompeting or, more generally, raise their costsby increasing input prices. Analogous effectsarise when the vertically integrated firm preventsother sellers from selling input into the interme-diate market or forcing them to accept a lowerprice.

The standard models assign market powereither to buyers or to sellers, but not both.In contrast, in our model, sellers misrepresenttheir costs and buyers misrepresent their willing-ness to pay. Thus, in equilibrium, the “wedge”between marginal cost and marginal value isthe sum of the seller markup and buyer mark-down. A vertical merger eliminates this “wedge”between the merging seller and buyer. Con-sequently, vertical mergers in our model havestronger efficiency effects than in the standardmodels.

What about foreclosure effects? To obtainsome insight into this issue, we consider inter-mediate input markets with a single seller (orbuyer) and a constant downstream price. In thiscase, it can be shown that a vertical mergerdoes not change the seller’s production capacityreport, but it does cause the merged firm to over-state its consumption capacity. It then followsfrom Lemma 1 that reported demand increasessubstantially as other buyers respond by increas-ing their reported consumption capacity. Hence,both output and price increase. Similarly, when

only the buyer in the market merges with aseller, reported demand does not change, butreported supply increases, lowering price andincreasing output. Thus, a vertical merger inmonopoly or monopsony markets always leadsto foreclosure, with the magnitude of the effectdepending upon the elasticities of supply anddemand and upon market shares. In marketswith multiple buyers and sellers, the verticallyintegrated firm increases both capacity reports,which causes rivals on both sides of the mar-ket to increase their capacity reports. Hence,reported supply and demand increase, outputincreases, but the impact on price is ambiguous.Intuitively, the foreclosure effect is importantwhen the vertically integrated firm is either alarge net seller or a large net buyer.

The assumption that the market is verti-cally separated is crucial to the merger analysis.A vertical merger in a vertically integrated mar-ket can lead to a decrease in reported demandor supply. The reason is that reported pro-duction capacity and consumption capacity arestrategic substitutes for a vertically integratedfirm. A similar issue arises with vertical merg-ers in wholesale markets. Even if the market isvertically separated, increases in reported con-sumption capacity reduce the reported capacityof sellers, and increases in reported productioncapacity reduce the reported capacity of buyers.Thus, in these cases, the impact of a verticalmerger on price and output needs to be com-puted on a case–case basis.

V. IDENTIFICATION

Suppose a researcher has data on prices andquantities in a market for t = 1, .., T periodsand wants to use our model to estimate cost anddemand parameters, under what conditions is themodel identified?

In addressing this question, it is useful tobegin with the standard model that has been esti-mated in numerous empirical studies of marketpower (e.g., Porter 1983; Genesove and Mullin1998; Clay and Troesken 2003). The modelassumes that the market is vertically separatedand that buyers are price-takers. Market demandis given by

pt = P (Qt, Kt , Zt , udt ; δ),

where Zt are observed demand shifters, udt rep-resents unobserved (to the researcher) demandshocks that are independent over time, and δ are


the unknown demand parameters. The pricingequation for sellers is specified as

pt = c′(Qt , Wt , ust ; φ)

−λQt

∂P (Qt, Kt , Zt ; δ)

∂Q,

where Wt are observed factors that shift marginalcost, ust represents unobserved supply shocks,and φ are the unknown cost parameters. Herewe have assumed that only the aggregate quan-tity data are available, so c′ is the marginal costof the average firm. The parameter λ is knownas the “conduct” parameter and interpreted asthe average of the firms’ conjectures on howaggregate supply will change with an increase intheir output. In the Cournot model, rivals cannotreact so λ is equal to 1 but, in empirical work,it is often useful to allow markups to vary fromthe Cournot markups. As is well known (seeBresnahan 1989), the above model is identifiedif instruments are available for the endogenousvariables in the two equations. Shifts in marginalcost can be used to identify the demand param-eters, and shifts in demand and in slope ofdemand can be used to identify the cost andconduct parameters. In the various empiricalstudies surveyed by Bresnahan (1989), estimatesof λ range from 0.05 to 0.65. More recently,Genesove and Mullin report estimates of λ forthe sugar industry at the turn of the century rang-ing from 0.038 to 0.10, with the latter computeddirectly from the data on prices and marginalcosts. Clay and Troesken report similarly lowestimates for λ in the whiskey industry at theturn of the century. These estimates suggest thatdynamic considerations do matter and lead tolower markups.

In our model,

λ = ε

ε + (1 − σ)η,

where σ denotes the market share of the aver-age firm. It is not a free parameter but in generaldepends upon the elasticities of reported demandand supply evaluated at the equilibrium mar-ket quantity. Note that λ is bounded between0 and 1, with the upper bound achieved whenη = 0 (i.e., the Cournot case). Hence, our modelprovides a potential explanation for why esti-mated markups are typically lower than Cournotmarkups.

Our model is identified in vertically sepa-rated, intermediate input markets with constant

cost and value elasticities. More precisely, sup-pose

c′(Qt , Wt , ust ) = Wφt Q

1/ηt �

−1/ηt ust

and

v′(Qt , Zt , udt ) = Zδt Q

−1/εt K

1/εt udt ,

where (ust , udt ) are distributed multivariate log-normal with mean zero and covariance �.This is the model that Porter estimates underthe assumption that buyers are price-takers,which, in terms of our model, means that theyare reporting their true capacity. As we haveobserved previously, the elasticities of reporteddemand and supply are constant in this model,independent of the capacity reports. Further-more, the argument given for Lemma 1 alsoimplies that capacity reports of sellers andbuyers are independent of the observed andunobserved factors shifting demand and sup-ply. Thus, as long as actual capacities are con-stant over time, � and K are constants, as arethe firms’ market shares. This in turn impliesthat λ is a constant and that the derivative ofthe reported demand schedule does not varywith reported buyer capacity. The variation inmarkups over time is coming from exogenousvariation in the observed and unobserved factorsbut not from the endogenous variables, � and K .As a result, changes in W shift the reported sup-ply but not the reported demand, thereby identi-fying the demand parameters; changes in Z shiftthe reported demand but not the reported supply,thereby identifying the cost parameters.25

Our model is not identified if elasticities(and/or slopes of reported demand and supply)depend upon reported capacities and these differfrom actual capacities due to the exercise ofmarket power. This will typically be the case in

25. Solving for equilibrium and taking logs, the structuralmodel is given by

log Qt = εδη

ε + ηlog Zt − φεη

ε + ηlog Wt + ε

ε + ηlog �

+ η

ε + ηlog K + εη

ε + ηlog

udt

ust

log Pt = εδ

ε + ηlog Zt + φη

ε + ηlog Wt − ε

ε + ηlog �

+ 1

ε + ηlog K + ε

ε + ηlog

udt

ust

It is straightforward to show that the structural parameters(ε,η, δ, φ) can be recovered from parameter estimates of thereduced form regressions of log Qt and log Pt on a constant,log Zt and log Wt .


TABLE 1Approximate Market Shares

Company iRefining Market

Share (σi )Refining Capital

ShareRetail Market

Share (si )Retail Capital

Share

Chevron 1 26.4 (26.6) 29.5 (29.5) 19.2 (19.5) 19.0 (19.0)

Tosco 2 21.5 (21.7) 21.7 (21.7) 17.8 (18.0) 17.8 (17.8)

Equilon 3 16.6 (16.7) 16.1 (16.1) 16.0 (16.2) 16.0 (16.0)

Arco 4 13.8 (13.9) 13.0 (13.0) 20.4 (20.7) 22.0 (22.0)

Mobil 5 7.0 (13.3) 6.2 (12.4) 9.7 (17.5) 9.3 (17.8)

Exxon 6 7.0 (0.0) 6.2 (0.0) 8.9 (0.0) 8.5 (0.0)

Ultramar 7 5.4 (5.4) 4.7 (4.7) 6.8 (6.9) 6.4 (6.4)

Paramount 8 2.3 (2.3) 2.0 (2.0) 0.0 (0.0) 0.0 (0.0)

Kern 9 0.0 (0.0) 0.0 (0.0) 0.3 (0.3) 0.27 (0.27)

Koch 10 0.0 (0.0) 0.0 (0.0) 0.2 (0.2) 0.18 (0.18)

Vitol 11 0.0 (0.0) 0.0 (0.0) 0.2 (0.2) 0.18 (0.18)

Tesoro 12 0.0 (0.0) 0.0 (0.0) 0.2 (0.2) 0.18 (0.18)

PetroDiamond 13 0.0 (0.0) 0.0 (0.0) 0.1 (0.1) 0.09 (0.09)

Time 14 0.0 (0.0) 0.0 (0.0) 0.1 (0.1) 0.09 (0.09)

Glencoe 15 0.0 (0.0) 0.0 (0.0) 0.1 (0.1) 0.09 (0.09)

Note: West coast CARB gasoline post-merger numbers are given in parentheses.

wholesale markets and in vertically integratedmarkets. In these markets, markups will be afunction of K and �, which are likely to dependupon the unobserved shocks affecting demandand supply. As a result, λ and P ′ will vary overtime and the variation will be correlated withthe unobserved shocks. One could try to findinstruments for K and � but data on reportedcapacities are typically not available.

VI. APPLICATION: THE EXXON–MOBIL MERGER

Our second application is to a merger ofExxon and Mobil’s gasoline refining and retail-ing assets in the western United States. The westcoast gasoline market is relatively isolated fromthe rest of the nation, both because of transporta-tion costs26 and because of the requirement ofgasoline reformulated for lower emission, a typeof gasoline known as CARB.

Available market share data are generallyimperfect, because of variations due to shut-downs and measurement error, and the presentanalysis should be viewed as an illustrationof the theory rather than a formal analysis

26. There is currently no pipeline permitting transfer ofTexas or Louisiana refined gasoline to California, and thePanama Canal cannot handle large tankers, and in any caseis expensive. Nevertheless, when prices are high enough,CARB gasoline has been brought from the Hess refinery inthe Caribbean.

of the Exxon–Mobil merger. Nevertheless, wehave tried to use the best available data forthe analysis. In Table 1, we provide a listof market shares, along with our estimatesof the underlying capital shares and the post-merger market shares, which will be discussedbelow. The data come from Leffler and Pulliam(1999).

From Table 1, it is clear that there is asignificant market in the intermediate good ofbulk (unbranded) gasoline, prior to branding andthe addition of proprietary additives. However,the actual size of the intermediate good marketis larger than one might conclude from Table 1,because firms engage in swaps. Swaps tradegasoline in one region for gasoline in another.Since swaps are balanced, they will not affectthe numbers in Table 1.

It is well known that the demand for gasolineis very inelastic. We consider a base case of anelasticity of demand, α, of 1/3. We estimate θ tobe 0.7, an estimate derived from an average of60.1 cents spot price for refined CARB gasoline,out of an average of 85.5 (net of taxes) at thepump in the year 2000.27 We believe the selling

27. We will use all prices net of taxes. As a consequence,the elasticity of demand builds in the effect of taxes, sothat a 10% retail price increase (before taxes) correspondsapproximately to a 17% increase in the after tax price. Thus,the elasticity of 1/3 corresponds to an actual elasticity ofcloser to 0.2.


TABLE 2Analysis of Exxon–Mobil Merger

Cases Markup as a Percent of Retail Price

α β η SymmetricBalanced

AsymmetricPre-merger

MarkupPost-merger

Markup Refinery Sale Retail Sale

1/3 5 1/2 6.9 (98.4) 18.4 (95.3) 20.0 (94.6) 21.3 (94.3) 20.1 (94.6) 21.2 (94.3)1/5 5 1/2 7.9 (98.7) 21.6 (96.0) 23.6 (95.4) 25.2 (95.2) 23.7 (95.4) 25.2 (95.2)1/3 3 1/2 7.0 (98.4) 18.7 (95.3) 20.3 (94.6) 21.7 (94.3) 20.5 (94.6) 21.6 (94.4)1/3 5 1/3 8.7 (98.2) 23.0 (94.6) 25.1 (93.8) 26.7 (93.5) 25.2 (93.8) 26.7 (93.5)

Note: Quantity as a percent of fully efficient quantity is given in parentheses.

cost to be fairly elastic, with a best estimateof β = 5. Similarly, by all accounts refiningcosts are quite inelastic; we use η = 1/2 asthe base case. We will consider the robustnessto parameters below, with α = 1/5, β = 3, andη = 1/3.

Table 2 presents our summary of the Exxon/Mobil merger. The first three columns providethe assumptions on elasticities that define thefour rows of calculations. The fourth columnprovides the markups that would prevail undera fully symmetric and balanced industry, that is,one comprised of 15 equal sized firms. This isthe best outcome that can arise in the model,given the constraint of 15 firms, and can beused as a benchmark. The fifth column consid-ers a world without refined gasoline exchange,in which all 15 companies are balanced, andis created by averaging production and con-sumption shares for each firm. This calculationprovides an alternative benchmark for compari-son, to assess the inefficiency of the intermediategood exchange. The next four columns use theexisting market shares, reported in Table 1, as aninput, and then compute the price–cost marginand quantity reduction, pre-merger, post-merger,with a refinery sale, or with a sale of retail out-lets, respectively.

Table 2 does not use the naive approach ofcombining Exxon and Mobil’s market shares, anapproach employed in the Department of JusticeMerger Guidelines. In contrast to the mergerguidelines approach, we first estimate the capitalheld by the firms, then combine this capitalin the merger, then compute the equilibriumgiven the post-merger allocation of capital. Theestimates are not dramatically different fromthose that arise using the naive approach of themerger guidelines. To model the divestiture ofrefining capacity, we combine only the retailingcapital of Exxon and Mobil; similarly, to model

the sale of retailing, we combine the refiningcapacity.

The estimated shares of capital are presentedin Table 1. These capital shares reflect theincentives of large net sellers in the intermediatemarket to reduce their sales in order to increasethe price, and the incentive of large net buyersto decrease their demand to reduce the price.Equilon, the firm resulting from the Shell-Texaco merger, is almost exactly balanced andthus its capital shares are relatively close toits market shares. In contrast, a net seller inthe intermediate market like Chevron refinessignificantly less than its capital share, but retailsclose to its retail capital share. Arco, a net buyerof unbranded gasoline, sells less than its sharefrom its retail stores, but refines more to itsshare of refinery capacity. The estimates alsoreflect the incentives of all parties to reducetheir downstream sales to increase the price,that is, the larger an incentive the larger is theretailer.

The sixth column of Table 2 provides thepre-merger markup, or MHI, and is a directcalculation from Equation (24) using the mar-ket shares of Table 1. The seventh, eighth, andninth columns combine Exxon and Mobil’s capi-tal assets in various ways. The seventh combinesboth retail and refining capital. The eighth com-bines retail capital, but leaves Exxon’s Beniciarefinery at the hands of an alternative suppliernot listed in the table. This corresponds to a saleof the Exxon refinery. The ninth and last columnconsiders the alternative of a sale of Exxon’sretail outlets. (It has been announced that Exxonwill sell both its refining and retailing operationsin California.)

Our analysis suggests that without divestiturethe merger will, under the hypotheses of thetheory, have a small effect on the retail price.In the base case, the markup increases from


20% to 21%, and the retail price increases 1%.28

Moreover, a sale of a refinery eliminates most ofthe price increase; the predicted price increaseis less than a mil. Unless retailing costs aremuch less elastic than we believe, a sale of retailoutlets accomplishes very little. The predictedchanges in prices, as a percent of the pretaxretail price, are summarized in Table 2. Theunimportance of retailing is not supported byHastings (2004).

The predicted quantity, as a percentage of thefully efficient quantity, is presented in Table 3, inparentheses. The first three columns present theprevailing parameters. The next three columnscorrespond to the conceptual experiments dis-cussed above. The symmetric column considersfifteen equal-sized firms. The balanced asym-metric column uses the data of Table 1, but aver-ages the refining and retail market shares to yielda no-trade initial solution. The pre-merger col-umn corresponds to Table 1; post-merger com-bines Exxon and Mobil. Finally, the last twocolumns consider a divestiture of a refinery andretail assets, respectively. We see the effects ofthe merger through a small quantity reduction.Again, we see that a refinery sale eliminatesnearly all of the quantity reduction.

The analysis used the computed marketshares rather than the approach espoused by theU.S. Department of Justice Merger Guidelines.Our approach is completely consistent with thetheory, unlike the merger guidelines approach,which sets the post-merger share of the merg-ing firms to the sum of their pre-merger shares.This is inconsistent with the theory becausethe merger will have an impact on all firms’shares. In Table 1, we provide our estimateof the post-merger shares alongside the pre-merger shares. Exxon and Mobil were respon-sible for 18.6% of the refining, and we esti-mate that the merger will cause them to contractto 17.4%. The other firms increase their share,though not enough to offset the combined firm’scontraction.

There is little to be gained by using the naivemerger guidelines market shares, because theanalysis is sufficiently complicated to requiremachine-based computation. However, we repli-cated the analysis using the naive market shares,and the outcomes are virtually identical. Thus, itappears that the naive approach gives the rightanswer in this application.

28. The percentage increase in the retail price can becomputed by noting that p = q−A.

TABLE 3Analysis of Exxon–Mobil Merger

Expected Percentage QuantityDecrease

Cases

α β η

FullMerger

RefinerySale

RetailSale

1/3 5 1/2 0.31 (0.94) 0.03 (0.09) 0.30 (0.90)1/5 5 1/2 0.27 (1.36) 0.02 (0.11) 0.25 (1.29)1/3 3 1/2 0.32 (0.97) 0.05 (0.15) 0.30 (0.89)1/3 5 1/3 0.35 (1.06) 0.03 (0.08) 0.34 (1.03)

Note: Percentage price increase is given in parentheses.

VII. CONCLUSION

This paper presents a method for measur-ing industry concentration in intermediate goodmarkets. It is especially relevant when firmshave captive consumption, that is, some of theproducers of the intermediate good use someor all of their own production for downstreamsales.

The major advantages of the theory are itsapplicability to a wide variety of industry struc-tures, its low informational requirements, andits relatively simple formulae. The major dis-advantages are the special structure assumed inthe theory and the static nature of the analysis.The special structure mirrors Cournot, and thusis subject to the same criticisms as the Cournotmodel. For all its defects, the Cournot modelremains the standard model for antitrust anal-ysis; the present theory extends Cournot-typeanalysis to a new realm.

We considered the application of the the-ory to wholesale electricity markets and to themerger of Exxon and Mobil assets in the west-ern United States. Several reasonable predic-tions emerge. In wholesale electricity markets,firm markups are approximately zero during lowdemand periods, high during high demand peri-ods, and vary depending upon the firm’s netposition and market share. In west coast gaso-line markets, the industry produces around 95%of the efficient quantity and the merger reducesquantity by a small amount, around 0.3%. Theprice–cost margin is on the order of 20% andrises by a percentage point or two. A sale ofExxon’s refinery eliminates nearly all of the pre-dicted price increase. This last prediction arisesbecause retailing costs are relatively elastic, sothat firms are fairly competitive downstream.Thus, the effects of industry concentration ariseprimarily from refining, rather than retail. Hence


the sale of a refinery (Exxon and Mobil have onerefinery each) cures most of the problem associ-ated with the merger. The naive approach basedpurely on market shares gives answers similarto the more sophisticated approach of first com-puting the capital levels, combining the capitalof the merging parties, then computing the newequilibrium market shares. Finally, it is worthnoting that the computations associated with thepresent analysis are straightforward, and run ina few seconds on a modern PC.

As with Cournot analysis, the static natureof the theory is problematic. In some industries,entry of new capacity is sufficiently easy thatentry would undercut any exercise of marketpower. The present theory does not accommo-date entry, and thus any analysis would needa separate consideration of the feasibility andlikelihood of timely entry.29 When entry isan important consideration, the present analy-sis provides an upper bound on the ill-effectsmerger.

Another limitation of the theory is the restric-tion to homogenous good markets. An extensionof the theoretical approach to markets in whichsellers offer differentiated good would be chal-lenging but worth exploring.

APPENDIX

Proof of Theorems 1 and 2

Before proceeding, note that differentiating the equilib-rium condition in Equation (8) implies that(

K

Q

)∂Q

∂K= ε−1

ε−1 + η−1,

(�

Q

)∂Q

∂�= η−1

ε−1 + η−1

and

(K

P

)∂P

∂K= −

(�

P

)∂P

∂�= (ηε)−1

ε−1 + η−1.

Differentiating the firm’s profit function in Equation (9)and substituting the relations given above yields the follow-ing first-order conditions:

∂πi

∂k̂i

=(

v′(

siQ

ki

)− p

) (Q

(1

K− k̂i

K2

)+ si

∂Q

∂K

)

+(

p − c′(

σiQ

γi

)) (σi

∂Q

∂K

)− Q(si − σi )

∂p

∂K

29. McAfee, Simons and Williams (1992) present aCournot-based merger evaluation theory that explicitlyaccommodates entry in the analysis. Ghosh and Morita(2007) study entry in Salinger-type model (with downtreamfirms as price-takers).

= Q

K

[(v′

i − p)

((1 − si) + si

ε−1

ε−1 + η−1

)

+(p−c′i )

(σi

ε−1

ε−1 + η−1

)−p(si −σi )

(ηε)−1

ε−1 + η−1

].

Similarly,

∂πi

∂ γ̂i

=(

v′(

siQ

ki

)− p

) (si

∂Q

∂�

)+

(p − c′

(σiQ

γi

)) (Q

(1

�− γ̂i

�2

)+ σi

∂Q

∂�

)

−Q(si − σi )∂p

∂�= Q

�

[(v′

i − p)

(si

η−1

ε−1 + η−1

)

+(p − c′i )

(1 − σi + σi

η−1

ε−1 + η−1

)

+p(si − σi )(ηε)−1

ε−1 + η−1

].

Thus,

K

Q

∂πi

∂k̂i

+ �

Q

∂πi

∂ γ̂i

= v′i − c′

i .

In an interior equilibrium, then, v′i − c′

i = 0. Either ofthe first-order conditions yields (16).

The Case of si = 0, σi> 0.If σi> 0, the first-order condition on γ̂i holds with

equality. Consequently, using si = 0,

0 = Q

�

[(p − c′

i )

(1 − σi + σi

η−1

ε−1 + η−1

)

+p(0 − σi )(ηε)−1

ε−1 + η−1

].

This yields

p − c′i

p= σi

ε + η(1 − σi ),

a formula that respects (15) (for si = 0). In addition, wehave

0 � ∂πi

∂k̂i

∣∣∣∣si=0

,

which gives

v′(0) − p

p� −σi

ε + η(1 − σi ).

Summarizing,

v′i − p

p� si − σi

ε(1 − si) + η(1 − σi ), with equality if si > 0.


and

p − c′i

p� si − σi

ε(1 − si) + η(1 − σi ), with equality if σi > 0.

Suppose ki = 0, then kv(q/k) = [(v(q/k))/(1/k)] ≈qv′(q/k) −−→

k→00. Thus, an agent with ki = 0 will report

k̂i = 0. Similarly, an agent with γi = 0 produces zero. Thisyields (14) and (15).

Proof of Corollary 1

Suppose γ̂i > γ̂j . Then Equation (6) implies that xi >

xj and hence that σi > σj . It then follows from the first-order conditions of firms i and j that

c′(

xi

γi

)< c′

(xj

γj

)⇒ xi

γi

<xj

γj

⇒ γ̂i

γi

<γ̂j

γj

and γi > γj .

The reasoning for buyers is similar.Proof of Theorem 3 : It is readily checked that the

following substitutions hold, even when a share is zero.

n∑i=1

v′i si −

n∑i=1

c′iσi =

n∑i=1

(v′i − p)si

+n∑

i=1

(p − c′i )σi =

n∑i=1

(v′i − p)si +

n∑i=1

(p − v′i )σi

=n∑

i=1

(v′i − p)si −

n∑i=1

(v′i − p)σi

=n∑

i=1

(v′i − p)(si − σi )

=n∑

i=1

p(si − σi )2

ε(1 − si) + η(1 − σi ).

Proof of Theorem 4

Applying (8), (16) and (18):

(siQ

ki

)−1/ε

= v′(

siQ

ki

)= p

(1 + si − σi

ε(1 − si ) + η(1 − σi )

)

= v′(

Q∑ni=1 k̂i

)(1 + si − σi

ε(1 − si ) + η(1 − σi )

).

This readily gives the first part of (18); the second halfis symmetric.

Rewrite (18) to obtain

ki = k̂i

(1 + si − σi

ε(1 − si) + η(1 − σi )

)ε

.

Thus

ki∑nj=1 k̂j

= si

(1 + si − σi

ε(1 − si) + η(1 − σi )

)ε

.

Thus

∑ni=1 ki∑ni=1 k̂i

=n∑

i=1

si

(1 + si − σi

ε(1 − si ) + η(1 − σi )

)ε

.

A similar calculation gives

∑ni=1 γi∑ni=1 γ̂i

=n∑

i=1

σi

(1 + si − σi

ε(1 − si) + η(1 − σi )

)−η

.

Note that, with constant elasticity, actual quantity is

Q =(

n∑i=1

k̂i

)η/(ε+η) (n∑

i=1

γ̂i

)ε/(ε+η)

,

and

Qf =(

n∑i=1

ki

)η/(ε+η) (n∑

i=1

γi

)ε/(ε+η)

.

Substitution gives (19). One obtains (20) from

p

pf= v′ (Q/

∑ni=1 k̂i

)v′ (Qf/

∑ni=1 ki

)=

(Q

Qf

∑ni=1 ki∑ni=1 k̂i

)−1/ε

=(

Qf

Q

∑ni=1 k̂i∑ni=1 ki

)1/ε

=⎡⎣(∑n

i=1 ki∑ni=1 k̂i

)η/(ε+η) (∑ni=1 γi∑ni=1 γ̂i

)ε/(ε+η) ∑ni=1 k̂i∑ni=1 ki

⎤⎦1/ε

=[(∑n

i=1 γi∑ni=1 γ̂i

) (∑ni=1 k̂i∑ni=1 ki

)]1/(ε+η)

=

⎡⎢⎣∑n

i=1 σi

(1 + si−σi

ε(1−si )+η(1−σi )

)−η

∑ni=1 si

(1 + si−σi

ε(1−si )+η(1−σi )

)ε

⎤⎥⎦1

ε+η

.


Proof of Theorem 5

Using the market calculations (22) and (23), rewriteprofits to obtain

πi = r(Q)qi − kiw(qi/ki )

−γi c(xi/γi ) − p(qi − xi )

= (r(Q) − p)qi − kiw(qi/ki )

+pxi − γi c(xi/γi )

= w′(Q/K)qi − kiw(qi/ki )

+c′(Q/�)xi − c(xi/γi )

= w′(Q/K)k̂i

KQ − kiw

(k̂i

K

Q

ki

)

+c′(Q/�)γ̂i

�Q − γi c

(γ̂i

�

Q

γi

).

The equilibrium quantity is given by

r(Q) − w′(Q/K) − c′(Q/�) = 0.

From this equation, and applying (24), it is a routinecomputation to show:

K

Q

dQ

dK= K

Q

−(w′′)Q/K2

r ′ − w′′/K − c′′/�

= (r − p)β−1

rα−1 + (r − p)β−1 + pη−1

= B

A + B + C.

Similarly,

�

Q

dQ

d�= C

A + B + C.

Differentiating πi , and using the analogous notation for

0 = K

Q

∂πi

∂k̂i

= (w′ − w′i )

(1 − si + si

K

Q

dQ

dK

)−w′′ siQ

K+ (c′ − c′

i )σi

K

Q

dQ

dK

+(

w′′si

Q

K+ c′′σi

Q

�

)K

Q

dQ

dK

= (w′ − w′i )

(1 − si + si

K

Q

dQ

dK

)+(c′ − c′

i )σi

K

Q

dQ

dK− β−1w′si

+(β−1w′si + η−1c′σi )K

Q

dQ

dK

= (w′ − w′i )

(1 − si + si

K

Q

dQ

dK

)+(c′ − c′

i )σi

K

Q

dQ

dK− r

(Bsi − (Bsi + Cσi )

K

Q

dQ

dK

)= (w′ − w′

i )

(1 − si + si

K

Q

dQ

dK

)+(c′ − c′

i )σi

K

Q

dQ

dK

−r

(Bsi

(1 − K

Q

dQ

dK

)− Cσi

K

Q

dQ

dK

).

Similarly, and symmetrically,

0 = �

Q

∂πi

∂ γ̂i

= (w′ − w′i )si

�

Q

dQ

d�

+(c′ − c′i )

(1 − σi + σi

�

Q

dQ

d�

)−r

(Cσi

(1 − �

Q

dQ

d�

)− Bsi

�

Q

dQ

d�

).

These equations can be expressed, substituting the elas-ticities with respect to capacity, as

[A + B + C − si (A + C) Bσi

Csi A + B + C − σi (A + B)

]

×(

(w′ − w′i )/r

(c′ − c′i )/r

)=

(Bsi(A + C) − BCσi

cσi (A + B) − BCsi

).

The determinant of the left-hand-side matrix is given by

DET = [A + B + C − si (A + C)]

×[A + B + C − σi (A + B)] − BCsiσi

= (A + B + C)[(A + B + C)(1 − si )(1 − σi )

+Bsi(1 − σi ) + C(1 − si)σi ]

= (A + B + C)[A(1 − si )(1 − σi )

+B(1 − σi ) + C(1 − si )].


Thus,

((w′ − w′

i )/r

(c′ − c′i )/r

)= 1

DET

[A + B + C − σi (A + B) −Bσi

−Csi A + B + C − si (A + C)

] (Bsi(A + C) − BCσi

cσi (A + B) − BCsi

)

= 1

DET

((A + B + C − σi (A + B))(Bsi(A + C) − BCσi ) − Bσi (cσi (A + B) − BCsi)

−Csi(Bsi (A + C) − BCσi ) + (A + B + C − si(A + C))(cσi (A + B) − BCsi)

)

= A + B + C

DET

((Bsi (A + C) − BCσi ) − ABsiσi

(Cσi (A + B) − BCsi) − ACsiσi

)= A + B + C

DET

(B[C(si − σi ) + Asi(1 − σi )]C[B(σi − si) + Aσi (1 − si)]

).

Thus,

MHI =n∑

i=1

((r(Q) − p − w′

i )si + (p − c′i )σi

r

)

=n∑

i=1

((w′ − w′

i )si + (c′ − c′i )σi

r

)

=n∑

i=1

[(A + B + C

DET

)(siB[C(si − σi )

+Asi(1 − σi )] + σiC[B(σi − si)

+Aσi (1 − si)])

]

=n∑

i=1

[(A + B + C

DET

)[BC(si − σi )

2

+ABs2i (1 − σi ) + ACσ2

i (1 − si)]

]

=n∑

i=1

BC(si−σi )2+ABs2

i(1−σi )+ACσ2

i(1−si )

A(1−si )(1−σi )+B(1−σi )+C(1−si ).

The Constant Elasticity Case

(w′ − w′

i

c′ − c′i

)= r

⎛⎝ B[C(si−σi )+Asi (1−σi )]A(1−si )(1−σi )+B(1−σi )+C(1−si )

C[B(σi−si )+Aσi (1−si )]A(1−si )(1−σi )+B(1−σi )+C(1−si )

⎞⎠≡ r

(ψi

χi

).

Then(siQ

ki

)1/β

= w′i = w′ − rψi = r(1 − θ − ψi ),

or

ki = siQ[r(1 − θ − ψi )]−β.

Similarly,

γi = σiQ[r(θ − χi )]−η.

The efficient solution satisfies r − w′(

Qf∑ki

)−

c′(

Qf∑γi

)= 0, or

0 = Q−1/α

f −(

Qf

Q

)1/β

r

(n∑

i=1

si [1 − θ − ψi ]−β

)−1/β

−(

Qf

Q

)1/η

r

(n∑

i=1

σi [θ − χi ]−η

)−1/η

.

Thus, substituting and dividing by r = Q−1/α

0 =(

Qf

Q

)−1/α

−(

Qf

Q

)1/β(

n∑i=1

si [1 − θ − ψi ]−β

)−1/β

−(

Qf

Q

)1/η(

n∑i=1


)−1/η

,

or

1 =(

Qf

Q

)A+(1/β)(

n∑i=1

si [1 − θ − ψi ]−β

)−1/β

+(

Qf

Q

)A+(1/η)(

n∑i=1


)−1/η

,

or

1 =(

Q

Qf

)−(A+(1/β))(

n∑i=1

si [1 − θ − ψi ]−β

)−1/β

+(

Q

Qf

)−(A+(1/η))(

n∑i=1


)−1/η

.

This equation can be solved for the ratio Qf/Q, whichyields the underproduction.

Proof of Theorem 6

First, we need to define a couple of elasticities. Differ-entiating the equilibrium condition,

Q1/α =(

Q

�

)1/η

,


with respect to � yields the supply elasticity

�

Q

∂Q

∂�= α

η + α.

Similarly, define the equilibrium price elasticity

�

p

∂p

∂�= −1

η + α.

Differentiating firm i’s profits with respect to its capacityreport and substituting the above elasticities into the first-order condition, we obtain

0 = ∂p

∂�[Qσi − qi ] + p

∂Q

∂�σi + pQ

[1

�− v̂i

�

]−c′

i

[(1

�− v̂i

�

)X + σi

∂Q

∂�

]

= ∂p

∂�

�

p[σi − si ] + p − c′

i

p

[∂Q

∂�

�

Qσi + (1 − σi )

]

= −(σi − si)

η + α. + (p − c′

i )

p

[η(1 − σi ) + α

η + α

].

Proof of Theorem 7

Let v̂∗i denote the equilibrium pre-merger reports and

let �∗−12 =

n∑i=2

v̂∗i denote the aggregate capacity reported by

sellers 2 through n. Define z12 = v̂12/v1 and assume, withoutloss of generality, that v1 ≥ v2. Fixing its rival reports totheir pre-merger levels, the merged firm’s equilibrium bestreply solves

z12

(ν1 + v2)−1�∗−12 + z12

(εη + αη

εη + αη + αε

)

= 1 − z1/η

12[((η + 1)/η) − z

1/η

12

] .

The fact that

(ν1 + v2)−1�∗

−12 < v−11

(�∗

−12 + v̂∗2

)implies that z12 < ((̂v∗

1)/v1)). Hence,

v̂12

ν1 + v2<

v̂∗1

v1�⇒ v̂12 <

v̂∗1

v1(v1 + v2) < v̂∗

1 + v̂∗2 ,

where the last inequality follows from Lemma 1. Thus, themerging firms best reply to its rival’s pre-merger capacityreports is to reduce its reported total capacity. The resultthen follows from Lemma 1.

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