Estimation of quantity games in the presence of indivisibilities and heterogeneous firms

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Journal of Econometrics 134 (2006) 187–214

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Estimation of quantity games in the presence ofindivisibilities and heterogeneous firms

Peter Davisa,b,c,�

aLondon School of Economics, STICERD, UKbCEPR, UK

cApplied Economics Ltd., UK

Available online 31 August 2005

Abstract

This paper presents a theoretical framework that allows estimation of game theoretic

models of quantity competition, including a non-trivial class of differentiated product quantity

games. The simplest examples of quantity games are entry games, where the strategy each firm

i makes is discrete, si 2 Si ¼ f0; 1g (do not enter/enter). I consider a general class of games

where strategy sets are ‘‘chains’’, which includes the situation where they are a finite set of

integers, Si ¼ f0; 1; . . . ;Mg. In addition, I assume that profits of each firm can be written as a

function of the firm’s own strategy si and a possibly parametric index of market output,

Qðsi; s�i; y1Þ, so that piðsi; s�i; yÞ ¼ Ciðsi;Qðsi; s�i; y1Þ; y2Þ where y ¼ ðy1; y2Þ: The main theore-

tical result in the paper establishes easily verifiable conditions under which the index of market

output Qðsi; s�i; y1Þ is uniquely determined within the set of Nash equilibria of

the game. The model’s parameters may then be estimated by comparing the predicted index

of market output in a cross section of markets. The paper provides both a generalization and

an extension of the theoretical results developed by Bresnahan and Reiss (1991. Empirical

models of discrete games. Journal of Econometrics 48, 57–82.) and Berry (1992. Estimation

of a model of entry in the airline industry. Econometrica 60 (4), 887–917.), which

allowed estimation of the homogeneous products entry game and where the index of market

output was the number of active firms, Qðsi; s�iÞ ¼PN

i¼1si, with si 2 f0; 1g: I illustrate one

member of the class of models that these results allow us to estimate by developing a model of

discrete quantity competition using count data from the supermarket industry. The discrete

see front matter r 2005 Elsevier B.V. All rights reserved.

.jeconom.2005.06.020

nding author at: R518 STICERD, London School of Economics, Houghton Street, London

UK.

dress: [email protected].

www.elsevier.com/locate/jeconom

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P. Davis / Journal of Econometrics 134 (2006) 187–214188

quantity game is the game theoretic analogue to ordered LDV models such as the ordered

probit model.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: D0; C1; C7; L0; L1

Keywords: Game theoretic empirical models; Multiple equilibria; Entry games; Quantity games; Limited

dependent variable (LDV) models; Simulation estimators; Retail markets; Leveraged buy-outs

1. Introduction

In this paper I develop an estimation methodology for a class of multivariatelimited dependent variable (LDV) models. I build on Bresnahan and Reiss (1991)and Berry (1992) who study an entry game where firm i chooses to enter a market ornot, si 2 Si ¼ f0; 1g, according to whether it is profitable to do so. Firm profitabilityis treated as a latent variable guiding the observed choices, so that si ¼ 1 only ifpið1; s�iÞ40, where s�i denotes the actions of rivals. Since s�i are choices generated byanalogous latent variable models, game theoretic models with discrete strategyspaces correspond to multivariate LDV models.

The generic existence of multiple Nash equilibria in such settings introducesdifficulties for conventional estimation methods since a single set of unobservables ina model can produce multiple predicted outcomes so the sum of predictedprobabilities of all feasible outcomes will add to more than 1 (see Blundell andSmith, 1994). Such econometric models are known as ‘‘incoherent’’ and theassumptions required to ensure ‘‘coherency’’ essentially rule out all simultaneousdecision making (see Schmidt, 1981). More optimistically, Tamer (2003) showed thatgames with multiple equilibria can be estimated without imposing coherencyconditions if the model is used only to inform upper and lower bounds on theprobabilities of outcomes rather than assigning exact probabilities to each potentialoutcome. While an attractive approach, it is currently impractical for games withlarge outcome spaces and so Ciliberto and Tamer (2004) for example, in their studyof airline entry, must reduce the number of potential entrants to just three firms forcomputational reasons. For both intellectual and practical reasons therefore, itseems worthwhile to continue to consider potential alternative methodologies.

The approach I take is to provide a class of models that do have a uniqueprediction about the world, even in the presence of multiple equilibria. Thus I followthe approach of Bresnahan and Reiss (1990) and Berry (1992) but study a far wider,though still restrictive, class of games. Specifically, I consider a class of games whereprofits of each firm can be written as a function of the firms own strategy si and apossibly parametric index of market output, Qðsi; s�i; y1Þ, so that piðsi; s�i; yÞ ¼Ciðsi;Qðsi; s�i; y1Þ; y2Þ where y ¼ ðy1; y2Þ and s�i denotes the vector of strategies ofrivals. I provide a set of sufficient conditions which ensures existence of at least onepure strategy Nash equilibrium and, crucially, that the index of market outputQðsi; s�i; y1Þ is uniquely determined within the set of pure strategy Nash equilibria.

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P. Davis / Journal of Econometrics 134 (2006) 187–214 189

The required sufficient conditions are far weaker than (i) those necessary to ensurethat the equilibrium is unique (see, for example, Moulin, 1984) and (ii) the conditionsrequired by the previous literature for the homogeneous products entry game(see Berry, 1992).

Examples of games within the class I consider include (i) the homogeneousproducts entry game where si 2 Si ¼ f0; 1g and Qðsi; s�i; y1Þ ¼

PN

i¼1 si is the numberof active firms, (ii) the discrete quantity or Cournot game wherein each firm choosesa number of a homogeneous product to produce Si ¼ f0; 1; . . . ;Mg for some Mo1and the index of market activity is the total quantity produced (the game theoreticanalogue to ordered LDV models such as the ordered probit model) and (iii) a classof games with differentiated products. One might consider the move from (i) to (ii)important if multiple entry is possible, for example doctors’ and dentists’ practises intruth have different numbers of partners (cf., Bresnahan and Reiss, 1991), whileairlines can enter a market with multiple scheduled flights on a given route(cf., Berry, 1992). Naturally, product differentiation will often also be vitallyimportant, and I extend the results to apply to an important, but by no meansuniversal, class of differentiated product games.

In homogeneous product entry games, the assumptions I require to establishuniqueness of the index of market activity are weak (even weaker than inBerry, 1992), but for many more general quantity and differentiated product gamesI show they are restrictive in particular directions. A recent paper in the spiritof this one is provided by Mazzeo (2002). Mazzeo’s model is not within the classof single index (of market activity) models that I explore; his is a two-indexmodel. On the other hand, my results allow for heterogeneity across firms whileMazzeo’s is a homogeneous firm model in the sense that profit functions are notindexed by i.

An alternative approach is suggested by Seim (2002) who considers a class ofBayesian Nash equilibrium (BNE) models wherein the firm’s optimal strategybecomes a mapping from its private information about its ‘‘type’’ ci to its choices si

and therefore does not depend directly on the actions chosen by rival firms s�i. Thisclass of models makes different, but not obviously less stringent, assumptions aboutthe determinants of firm profitability and the nature of equilibrium in an effort tosolve the multiplicity problem. While undoubtedly useful, it is easy to constructexamples of models with multiple BNEs (see, for example, p. 159 in Gibbons, 1992).Similar assumptions regarding information structures are made by a series ofinfluential recent papers by Aguirregabiria and Mira (2004), Bajari et al. (2004),Berry et al. (2002) and Pesendorfer and Schmidt-Dengler (2003). However,Doraszelski and Satterthwaite (2004) argue that their class of models can also havemultiple equilibria when firms make entry/exit decisions, even if firms are assumedsymmetric. The very different sets of assumptions regarding information structureand equilibrium concepts mean that the approach followed by these authors isprobably best considered complementary to the conventional pure strategy Nashequilibrium approach that I take here.

I illustrate the use of my theoretical results by estimating one member of thisclass of models, the discrete Cournot (quantity) game, using count data from the

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US supermarket industry. As with any econometric model, it is certainly best seenas an approximation to the true data generating process. Estimating the structuralmodel allows (i) the researcher to control for the endogeneity of decisions madeby other firms and (ii) the determinants of firms’ reaction functions to beuncovered rather than reporting correlations between observed equilibrium activitylevels. Here I supplement the reduced form work of Chevalier (1995), by uncoveringthe effect of changes in a supermarket chain’s financial structure on its reactionfunction.

The paper is organized as follows. Section 2 introduces the model. Section 3contains the main theoretical results in the paper, establishing sufficient conditions toensure that market output is uniquely determined within the set of Nash equilibria.Section 4 presents an application and empirical results, proposes an equilibriumcomputation procedure, and sketches the generalized method of moments (GMM)estimation routine. Section 5 concludes.

2. The model

Let i ¼ 1; . . . ;N index firms. I assume a firm’s strategy is chosen from a strategyspace Si which is a completely ordered set, technically a ‘‘chain’’.1 For example, itcould be a bounded set of integers Si ¼ f0; 1; 2; 3; . . . ;Mg or a closed and boundedsubset of the real line, Si ¼ ½0;M� where Mo1. (An example of a strategy setwhich is not completely ordered occurs if each firm could simultaneously enterby producing two or more qualities of goods so that si ¼ ðs1i; s2iÞ 2

Si ¼ fð0; 0Þ; ð0; 1Þ; ð1; 0Þ; ð1; 1Þg:) As usual, each player considers her rival’s strategiess�i ¼ ðs1; . . . ; si�1; siþ1; . . . ; sNÞ fixed when choosing her own strategy si to maximizeher profits, piðsi; s�i; yÞ. Let the best response correspondence, riðs�i; yÞ, denote theset of solutions to this problem for a given s�i, riðs�i; yÞ ¼ fsi 2 Si j

pðsi; s�i; yÞXpðti; s�i; yÞ for all ti 2 Sig. Finally, define the Cartesian product,S ¼ X N

i¼1 Si, and let F ¼ fs 2 S j si 2 riðs�i; yÞ for all ig be the set of all purestrategy Nash equilibria of the game. I follow the literature and focus only on purestrategy Nash equilibria. I also (mostly) drop the explicit dependence of functionalforms on the parameters y for brevity.

1I use some basic results and terminology from supermodular game theory. Since the basic concepts are

all I require, I refer the reader to Fudenberg and Tirole (1991) or Vives (1990) and references therein for a

full discussion, while including some required definitions here for completeness. Let X be the binary

relation on a non-empty set S. The pair ðS;XÞ is a partially ordered set (POSET) if X is reflexive,

transitive and antisymmetric, whereX is antisymmetric if for x; y 2 S, xXy and yXx implies that x ¼ y. A

lattice is a POSET ðS;XÞ in which any two elements, x; y 2 S have a least upper bound (sup) and a greatest

lower bound (inf ) in the set S. The strategy set Si ¼ f0; 1; . . . ;Mg is a lattice with respect to the standard

orderings,X orp. A lattice is complete if every non-empty subset of S has a supremum and an infimum in

S. When ðS;XÞ is a POSET, a real-valued function f with domain S is increasing if for all x; y 2 S, xXy

implies that f ðxÞXf ðyÞ.

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3. Characterizing the set of equilibria

3.1. The firm’s decision problem

In this Section 1 characterize the solution to the firm’s decision problem. Threeweak assumptions suffice to ensure that best response functions are well-defineddecreasing functions. I discuss each assumption in turn before stating the main resultof this subsection.

First, I assume that the effect of competition on profitability can be summarizedthrough its effect on a single index function Qðsi; s�iÞ which is additively separable inthe components of s and assumed not to vary across firms. While restrictive, I showbelow that this class of models does incorporate many important models includingthe homogeneous products Entry and Cournot games as well as a non-trivial class ofmore general models including some differentiated product models.

Assumption 1 (A1). For all i and for all s�i, piðsi; s�iÞ ¼ Ciðsi;Qðsi; s�iÞÞ, whereQðsi; s�iÞ is a real-valued increasing function (the market index function) that isadditively separable in each firm’s strategy so that Qðsi; s�iÞ ¼

PN

i¼1QiðsiÞ with eachsub-function QiðsiÞ increasing in si.

This assumption does rule out many models of differentiated product entry,though not all (see below). Mazzeo (2002) for example provides a two-index modelwhere the two indexes are the number of low- and high-quality products beingproduced respectively. By assuming that entry decisions are taken prior to theproduct quality decision, he retains a completely ordered strategy set Si ¼ f0; 1gassociated with a single entry decision, but he considers a two-index profitfunction pðsi; s�iÞ ¼ eCðsi;Q1ðsi; s�iÞ;Q2ðsi; s�iÞÞ ¼ maxfFLðQ1ð0; s�iÞ þ si;Q2ð0; s�iÞÞ;FHðQ1ð0; s�iÞ; Q2ð0; s�iÞ þ siÞg where the maximization is over the type of productproduced, given the decision to enter. Our results are non-nested since he assumesthat firms are homogeneous, i.e. that the reduced-form profit function pðsi; s�iÞ iscommon across i; while I consider only single-index models in this paper.

Second I assume that best response functions are single valued. If not, a firm couldobtain the same profits from different output choices and aggregate output may notbe uniquely determined. Assumption 2 is attractive for empirical work since theaddition of an action-specific random component to profits, drawn from acontinuous distribution of possible values, would make indifference between anytwo actions a zero probability event. A sufficient condition for A2 is that marginalprofitability is strictly monotonically decreasing in si.

Assumption 2 (A2). For all i and s�i 2 S�i, riðs�iÞ is a singleton.

Third, I make an assumption on the bivariate function Ciðsi;QÞ, requiring that if itis not profitable for firm i to increase output from ti to the higher level of output si

when the index of market activity is low QL, then it is also not profitable for thefirm to produce at the higher level of output si when the index of market activityis higher QH . This assumption essentially captures the idea that firms producestrategic substitutes.

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Assumption 3 (A3). The function Ciðsi;QÞ is decreasing in Q and satisfies the dualsingle crossing (DSC) condition in ðsi;QÞ i.e., for all ðsi; tiÞ 2 S2

i and ðQH ;QLÞ 2 Z2

where Z is a subset of R such that siXti and QHXQL; Ciðsi;Q

LÞ �Ciðti;Q

LÞp0)

Ciðsi;QHÞ �Ciðti;Q

HÞp0:

Lemma 1 states that these conditions are sufficient to ensure that each firm’s bestresponse function can be written as the composition of a decreasing function and thecontribution to the index of total market activity provided by the other players,Q�iðs�iÞ: The latter is by assumption an increasing and additively separable functionof the contribution of each player to the index. Thus

Lemma 1. A1– A3 imply riðs�iÞ ¼ s�i ðQ�iðs�iÞÞ; where

s�i ðQ�iÞ ¼ argmaxsi

Ciðsi;Q�i þQiðsiÞÞ

is decreasing in Q�i; so that riðs�iÞ is decreasing in s�i.

Proof. See Appendix. &

Lemma 1 makes clear that A1–A3 restrict any behavior that would result in firmshaving increasing best response functions (e.g., positive externalities or complemen-tarities between firms). To ensure that best response functions are decreasing onecould just assume that piðsi; s�iÞ has the DSC property in ðsi; s�iÞ instead

2 of A1–A3,but the latter have some advantages. First they are easier to check since they involvethe properties of the bivariate function Ciðsi;QÞ instead of the N-variate functionpiðsÞ. Second, they are less restrictive than any sufficient conditions I could developto establish that piðsi; s�iÞ satisfies the DSC property directly.3 Proposition 1 providesthe weakest set of conditions I could uncover, which requires that CiðQ; siÞ beconcave in Q while no such restriction is included in Assumptions 1–3.

2The proof is a special case of Theorem 4 in Milgrom and Shannon (1994). piðsi ; s�iÞ has the dual single

crossing property in ðsi; s�iÞ if for all ðsi; tiÞ 2 S2i and ðs�i; t�iÞ 2 S2

�i such that siXti and s�iXt�i

piðsi ; t�iÞ � piðti; t�iÞp0) piðsi; s�iÞ � piðti; s�iÞp0. To see why this is sufficient to generate downward

sloping reaction functions, suppose that piðsi; s�iÞ has the dual single crossing property in ðsi; s�iÞ,

si 2 riðs�iÞ, ti 2 riðt�iÞ and s�iXt�i. Since ti is a best response to t�i, piðx; t�iÞppiðti; t�iÞ for all x 2 Si. By the

dual single crossing condition, for any xXti piðx; s�iÞppiðti; s�iÞ and hence sipti.3Consider smooth quantity games with continuous strategy sets and differentiable profit functions

where piðsi; s�iÞ ¼ Ciðsi;Qðsi ; s�iÞÞ. A sufficient condition forCiðsi;QÞ to satisfy A3 is that it has decreasing

differences, q2i Ciðsi;QÞ=qsiqQp0. (See for example Milgrom and Shannon, 1994, p. 164.) In contrast, the

equivalent sufficient condition for piðsi; s�iÞ to satisfy dual single crossing in ðsi ; s�iÞ would be that it satisfies

decreasing differences—all its pairwise cross derivatives are negative q2piðsi; s�iÞ=qsiqsjp0. If QðsÞ is

additively separable and increasing then decreasing differences in ðsi ; s�iÞ requires the condition that

q2piðsi; s�iÞ

qsiqsj

¼q2Ciðsi;QÞ

qsiqQþq2Ciðsi;QÞ

qQ2

qQðsÞ

qsi

� �qQðsÞ

qsj

p0,

so that sufficient conditions are to require q2Ciðsi;QÞ=qsiqQp0 and q2Ciðsi;QÞ=qQ2p0: As I have already

noted, A3 only requires the first of these two conditions and not the second. This relaxation is important,

for instance, when studying the differentiated product entry model considered by Sutton (1998) where the

function Ciðsi;QÞ is not concave in Q. I provide greater detail for this example below.

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Proposition 1. Let S be a lattice and Z a convex subset of R containing the range of

QðsÞ, where QðsÞ is increasing and supermodular in s 2 S ¼ Si � S�i: If Ciðsi;QÞ is

submodular on ðsi;QÞ 2 Si � Z;Ciðsi;QÞ decreasing and concave in Q for any fixed si,then piðsi; s�iÞ ¼ Cðsi;QðsÞÞ satisfies the dual single crossing property in ðsi; s�iÞ.

Proof. See Appendix A which follows Lemma 2.6.4 in Topkis (1998).4 &

3.2. Existence of equilibrium for the two-player game

First I establish an equilibrium in pure strategies exists in the N ¼ 2 player game.This result provides a useful foundation for subsequent results since existence in theN-player game is proven by induction. Milgrom and Roberts (1990) observed that asuitable ordering on the strategy space places any game with two players anddecreasing best response functions within the class of supermodular games. The idea isto turn the original best response mappings (each decreasing in rival’s output) into bestresponse functions which are increasing with respect to the new ordering. The existenceresult provided in Tarski (1955) for increasing functions then applies directly.

Theorem 1 (Existence of equilibrium; Tarski, 1955). If S is a non-empty, compact

sublattice of RK and f : S! S is an increasing function, f has a fixed point in S.

Proposition 2 (Milgrom and Roberts, 1990). Under assumptions A1–A3, the two-

player game has a pure strategy Nash equilibrium.

Proof. By Lemma 1, each firm’s best response function riðs�iÞ is non-increasing in theother’s strategy with the conventional ordering. Transform the problem from oneanalyzed on the partially ordered set (POSET) ðS1 � S2;XÞ, where X is the usualvector ordering ða; bÞXðc; dÞ iff aXc and bXd, to a problem analyzed on the POSETðS1 � S2;XrÞ where ða; bÞXrðc; dÞ iff apc and bXd. With respect to this new order,the vector function, f ðsÞ ¼ ðr1ðs2Þ; r2ðs1ÞÞ, is an increasing function defined on asublattice of R2 and so Tarski’s theorem applies. &

3.3. The N-player game

In this section I make two further assumptions and then state my main theoreticalresults. First I demonstrate that provided a Nash equilibrium exists, then A1together with these two additional assumptions A4 and A5 are collectively sufficientto ensure that the index of market activity is uniquely determined within the set ofNash equilibria. Second, I show existence of Nash equilibria for the N-player gameunder assumptions A1–A5.

4Note that if Ciðsi;QÞ is submodular in ðsi;QÞ then it also has dual single crossing property (A3 holds).

Note also that if A1 holds so that QðsÞ ¼PN

i¼1QiðsiÞ is increasing and additively separable then it is

immediately supermodular in the vector s since the sum of any two supermodular functions is

supermodular and every univariate function is supermodular—provided only that its domain is completely

ordered (see, for example, Topkis, 1998, p. 46).

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Assumption 4 (A4). For all i and any s; t 2 S such that si4ti and QðsÞ4QðtÞ; thereexists an a40 (sufficiently small) such that si � aXti with si � a 2 Si andQðsi � a; s�iÞXQðti; t�iÞ.

In the homogeneous products entry game where Si ¼ f0; 1g and QðsÞ ¼PN

i¼1si,notice that the function QðsÞ can only increase in steps of size 1. Thus QðsÞ4QðtÞ

implies that QðsÞXQðtÞ þ 1 and A4 holds with a ¼ 1 since Qðsi � a; s�iÞ ¼

QðsÞ � 1XQðtÞ. Later I show that both the discrete Cournot game and also theentry then symmetrically differentiated quantity game studied by Sutton (1998) alsoeach satisfy this condition. Moreover, if Si ¼ ½0;M� � R and QðsÞ is a continuousfunction of si then A4 will hold, since if si4ti it is always possible to find asufficiently small a so that (i) si � aXti; and (ii) Qðsi � a; s�iÞXQðti; t�iÞ; where thelatter follows by continuity of QðsÞ and the assumption that QðsÞ4QðtÞ. ConditionA4 is therefore not at all restrictive for continuous quantity choice games withcontinuous market index functions, whether they are homogenous or differentiatedproduct quantity games. Unfortunately, A4 is not satisfied for a more general integerdifferentiated product entry game where Si ¼ f0; 1g, a ¼ 1 and Qðs; y1Þ ¼

PN

i¼1y1isi

with y1iX0 since the minimum increment in this function is minj y1j so thatQðs; y1Þ4Qðt; y1Þ impliesQðs; y1Þ �minj y1j XQðt; y1Þ, but no more, which is clearlynot sufficient to ensure what we would require—namely that Qðsi � a; s�i; y1Þ ¼

Qðs; y1Þ � y1i XQðt; y1Þ for arbitrary i.

Assumption 5 (A5). For any pair of strategy profiles, ðs; tÞ 2 S2 with siXti andQðsÞXQðtÞ, if ti þ a; si þ a 2 Si then pð:; :Þ satisfies piðti þ a; t�iÞ � piðti; t�iÞp0)piðsi þ a; s�iÞ � piðsi; s�iÞp0 for some a40.

To make clear the requirement embodied in A5, consider the game represented inFig. 1 where I assume that Si ¼ f0; 1; 2g. Arrows on the vertices indicate thepreferences of the firms. For example, when firm 2 plays the strategy s2 ¼ 2, thehorizontal arrows indicate that firm 1 prefers playing 0 to 1. Similarly at ð1; 1Þ,the vertical arrow (below) indicates that firm 2 would prefer to play 0 over 1. Nashequilibria are nodes at which neither firm would prefer to change her strategy, giventhe strategy of her rival, here F ¼ fð1; 0Þ; ð0; 2Þg.

Notice that total output differs across equilibria. To see why A5 rules out thiscase, compare the actions of firm 2 at the nodes ð1; 0Þ and ð0; 1Þ. A5 requires that if itis unprofitable for firm 2 to produce a first unit of output when s1 ¼ 1, then it mustbe unprofitable for firm 2 to produce a second unit when s1 ¼ 0. Thus if movingfrom ð1; 0Þ to ð1; 1Þ is unprofitable, then it must also be unprofitable to move fromð0; 1Þ to ð0; 2Þ, since total output is at least as large (it is the same) and firm 2’s ownoutput is higher. Clearly, this strong assumption rules out fixed costs in all cases butthe entry game.

Next I state the first of our main results, which establishes uniqueness of the indexof market activity, given the existence of at least one equilibrium. As we will see, thisresult relies only on assumptions A1, A4 and A5. Having established uniqueness ofthe index of market activity conditional on existence of equilibrium, I progress toestablish existence of equilibrium under A1–A5. The intuition for the next result is

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s2

s1

(0.2)

(0.1)

(0.0) (1.0)

(1.1)

(2.0)

Fig. 1. A two-player game illustrating the requirements embodied in A5.


provided by considering any two equilibria where total market output is strictlyhigher in one of the two equilibria. At least one firm must be producing more in thehigh output equilibrium. Since she does not want to produce more in the low marketoutput equilibrium, her marginal profitability of expanding output from these lowlevels of own- and market-output must be negative but A5 requires it must then alsobe negative at higher levels of own and market output. One set of sufficientconditions for A5 is thus that marginal profitability is declining in own and marketoutput, as it would be if marginal revenue is declining in ðs;QÞ and costs are convexso marginal cost is increasing in own output.

Proposition 3 (Uniqueness of the index of market activity in the N-player game). If

the set of Nash equilibria F in the N-player game is non-empty then under assumptions

A1, A4 and A5, the index of market activity is uniquely determined within the set of

Nash equilibria.

Proof. Suppose there exists es; t 2F such that QðesÞaQðtÞ. Without loss of generality,suppose QðesÞ4QðtÞ. Since market output is higher under es than under t, there mustexist at least one player—who we will call player i—that produces more under esthan t so that esi4ti. By A4, there exists an a40 such that esi � aXti 2 Si andQðesi � a;es�iÞXQðti; t�iÞ: Now we can apply A5 with the strategy profiles s �

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ðesi � a;es�iÞ and t � ðti; t�iÞ giving

piðti þ a; t�iÞppiðti; t�iÞ ) piðesi;es�iÞppiðesi � a;es�iÞ for some a40.

The first inequality is guaranteed to hold since ti 2 riðt�iÞ thus the second inequalitymust also hold but it directly contradicts esi ¼ riðes�iÞ. &

Proposition 4 (Existence of equilibrium). Under assumptions A1– A5, the N-player

game has a pure strategy Nash equilibrium.

Proof. See Appendix A. &

3.3.1. Examples

Homogeneous product entry games. Bresnahan and Reiss (1990) and Berry (1992)consider the homogeneous product entry game where Si ¼ f0; 1g and QðsÞ ¼PN

j¼1si ¼ Ns

A; the number of active players. In this case, A3 is trivially always satisfiedwhen si ¼ ti and we need only consider the case with si4ti; i.e., si ¼ 1 and ti ¼ 0:Assuming pið0; s�iÞ ¼ Cið0;Qð0; s�iÞÞ ¼ 0 8i; s�i, A3 simplifies to the requirement thatif NH

XNL then Cið1;NHÞp0) Cið1;N

LÞp0, i.e., that profitability of entry is

declining in the number of active players. As I noted above, A4 does not imposerestrictions in this entry game. Next consider A5. Since we require ti þ a; si þ a 2Si ¼ f0; 1g with a40; for the entry game we need only consider ti ¼ si ¼ 0 and it isonly possible to allow a ¼ 1. Thus A5 collapses to the conditionpið1; t�iÞp0) pið1; s�iÞp0, or Cið1; 1þN

t�iA Þp0) Cið1; 1þN

s�iA Þp0 whenever

QðsÞ ¼ Ns�iA þ 1XN

t�iA þ 1 ¼ QðtÞ or equivalently N

s�iA XN

t�iA . That is, we require

only that each firm’s profitability of entry declines weakly in the number ofactive rivals. One important implication of this result is that in the homogeneousproducts entry game, fixed cost and variable profit heterogeneity is only restrictedin a very limited way. The results for the entry game presented here thereforestrictly generalize those provided in Berry (1992) where only heterogeneity in fixedcosts is allowed.

Smooth Cournot games. Consider the Cournot game where si 2 Si ¼ ½0;M�; Q ¼PN

i¼1si; PðQÞ is an inverse demand function, and CiðsiÞ is the firm’s cost function.Then profits are piðsi; s�iÞ ¼ Pð

PN

j¼1sjÞsi � CiðsiÞ ¼ Ciðsi;QðsÞÞ where Ciðsi;QÞ ¼PðQÞsi � CiðsiÞ: A1 is satisfied by construction while strict quasi-concavity in si issufficient to ensure A2 and q2Ciðsi;QÞ=qsiqQ ¼ P0ðQÞp0 is sufficient to ensure A3. Ihave already noted that A4 is satisfied for any continuous market index functiondefined over a compact set, while A5 requires that qpiðti; t�iÞ=qsip0¼)qpiðsi; s�iÞ=qsip0 whenever siXti and QðsÞXQðtÞ; so that a sufficient condition isthat qpiðsi; s�iÞ=qsipqpiðti; t�iÞ=qsip0 whenever siXti andQðsÞXQðtÞ: Here

qpiðsi; s�iÞ

qsi

¼qCiðsi;QÞ

qsi jQ¼QðsÞ

þqCiðsi;QÞ

qQ jQ¼QðsÞ

qQðsÞ

qsi

¼ PðQðsÞÞ þ P0ðQðsÞÞsi � C 0iðsiÞ.

Sufficient conditions for A5 are therefore that (i) the firm’s marginal revenueMRiðsi;QÞ ¼ PðQÞ þ P0ðQÞsi is decreasing in own output and the index of market

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output,5 i.e., ðsi;QÞ, and (ii) that marginal cost C 0iðsiÞ is increasing in own output, si.The latter is evidently, and unfortunately, a rather restrictive assumption since iteffectively rules out fixed costs in this case.

Discrete Cournot games. With discrete Cournot games, Si ¼ f0; 1; . . . ;Mg,Ciðsi;QÞ ¼ PðQÞsi � CiðsiÞ and QðsÞ ¼

PN

j¼1sj : All increments of the function QðsÞ

are at least 1, so that for any s; t with QðsÞ4QðtÞ implies QðsÞ � 1XQðtÞ and henceA4 is satisfied with a ¼ 1. Given A1, a sufficient condition for A5 is that the firm’smarginal profitability DCiðsi;QÞ � Ciðsi þ 1;Qþ 1Þ �Ciðsi;QÞ is decreasing in ownoutput and the index of market output ðsi;QÞ, so that ðsi;QsÞXðti;QtÞ

¼)DCiðsi;QsÞpDCiðti;QtÞ.Smooth differentiated product games. Consider the profit function piðsi; s�iÞ ¼

siPiðs1; . . . ; sNÞ � CiðsiÞ; where CiðsiÞ is the firm specific cost function and pi ¼

Piðs1; . . . ; sNÞ is the standard differentiated product inverse demand function withsi 2 Si ¼ ½0;M�. A1 requires that we can write Piðs1; . . . ; sNÞ ¼ ePiðsi;QðsÞÞ and sorules out lots of interesting cases, but not all of them. Consider, for example, thelinear in parameters inverse demand function where Piðs1; . . . ; sNÞ ¼ ai þPN

k¼1 bikQkðskÞ: In general, the index of market activity in this class of modelsdepends directly on i through the parameters bik, so this class of models is larger thanthe class I study here. However, there is some substantial intersection. For example,suppose bik ¼ libk for all iak and bkX0. The inverse demand function may bewritten as

Piðs1; . . . ; sNÞ ¼ ai þ ½bii � libi�QiðsiÞ þ li

XJ

k¼1

bkQkðsiÞ

!� ePiðsi;QðsÞÞ,

which satisfies A1 with the increasing and additively separable market index functionQðsÞ ¼ ð

PJ

k¼1 bkQkðsiÞÞ. In this case, the parameters which will control the own priceelasticity of demand bii are entirely unrestricted, while the parameters which willcontrol the cross elasticities can take on only the more restricted product form.While this product form is restrictive, it will nonetheless evidently allow a fairly largeset of potential substitution behavior to be modelled. As before, A4 is satisfied forSi ¼ ½0;M� and continuous market index function. As in the smooth Cournot game,A5 requires (i) that the constructed marginal revenue functions

MRiðsi;QÞ ¼ ePiðsi;QðsÞÞ þqePiðsi;QÞ

qsi jQ¼QðsÞ

þqePiðsi;QÞ

qQ jQ¼QðsÞ

qQðsÞ

qsi

" #si

are each decreasing in ðsi;QÞ; and (ii) that marginal cost is increasing in ownoutput CiðsiÞ.

Symmetrically differentiated entry then quantity competition games. Sutton (1998)considers two-stage games where the first stage involves entry decisions by N

potential entrants and the second involves symmetrically differentiated product

5Note that the condition P00ðQÞsi þ P0ðQÞp0 is the assumption required by Novshek (1985) to establish

existence of Cournot–Nash equilibrium which clearly is sufficient to ensure that marginal revenue is

decreasing in total output, while P0ðQÞp0 ensures that marginal revenue is decreasing in own output si:

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quantity game among the set of firms that entered at the first stage. At the first stage,each product introduced incurs a cost of entry F i which is sunk at the second stage.Sutton (1998) shows (see particularly Section 2.4) that with linear inverse demand atthe second stage of Piðq1; . . . ; qNÞ ¼ 1� 2qi � 2s

Plaiql where the summation is over

other firms active firms output, zero marginal costs and differentiation parameters 2 ½0; 1�; that game has the stage 1 ‘‘reduced form’’ profit function:

piðsi; s�iÞ ¼

ð1�sÞþssi

2½2ð1�sÞþssi �2 si

f1þQðsÞg2� F isi: ¼ Ciðsi;QðsÞÞ,

where QðsÞ ¼PN

j¼1 QjðsjÞ, with sub-functions QjðsjÞ ¼ ð2ð1� sÞ=ssj þ 1Þ�1 that areeach strictly increasing in their argument sj for any s 2 ½0; 1�. This reduced-formprofit function is obtained by solving for the Nash equilibrium in quantities atstage 2, given the entry decisions si 2 Si ¼ f0; 1g made at stage 1.

First notice that the function Ciðsi;QÞ is not concave in Q, which as I mentionedearlier provides one important justification for making assumptions A1–A3 insteadof assuming dual single crossing on the profit function piðsÞ directly.

Next notice that with Si ¼ f0; 1g the function QðsÞ can only increase in steps of sizeð2ð1� sÞ=sþ 1Þ�1, so that QðsÞ4QðtÞ implies that QðsÞXQðtÞ þ ð2ð1� sÞ=sþ 1Þ�1.Hence with si ¼ 1;

Qð0; s�iÞXQðtÞ þ2ð1� sÞ

sþ 1

� ��1�Qið1Þ

" #¼ QðtÞ

as required for A4 to hold with a ¼ 1 for all s 2 ½0; 1� since the term in squarebrackets is exactly zero.

Finally, notice that since this is an entry game, ti þ a; si þ a 2 Si with a40,we need only consider the incentives for expansion at si ¼ ti ¼ 0 and a ¼ 1.Since pið0; s�iÞ ¼ 0, A5 reduces to the requirement that if Qð0; s�iÞXQð0; t�iÞ,then pið1; t�iÞp0) pið1; s�iÞp0, or equivalently that Cið1;Qð1; t�iÞÞp0)Cið1;Qð1; s�iÞÞp0 which will be true provided only that Ciðsi;QÞ is decreasing inQ, since Qð0; s�iÞXQð0; t�iÞ implies Qð1; s�iÞXQð1; t�iÞ by assumption. Thus A5 holdsfor this single product firm version of Sutton’s game.

4. Modelling competition in the supermarket industry

I illustrate the use of the theoretical results by building a structural econometricmodel of competition in the supermarket industry. The discrete nature of thedecision to build a supermarket, combined with the fact that count data is the bestreadily available measure of quantity choice, motivates the decision to modelcompetition among supermarkets as a discrete Cournot game with chain storeschoosing the number of stores to operate in a market. This is a modelling relaxationrelative to the existing structural homogeneous products entry literature, butcertainly assumes away potentially very important aspects of competition including

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(i) product differentiation across stores within the market and (ii) all dynamics ofcompetition.

I illustrate the use of the model by examining the effect of financial structure onproduct market competition following a number of large changes in the financialstructure of supermarkets, including the $5 billion Safeway LBO, $4.1 billion Krogerleveraged recapitalization, the $1.8 billion Supermarkets General leveraged buyout(LBO), and the $1.2 billion Stop and Shop LBO. Chevalier (1995) uses the same datato estimate a reduced-form model to test for significant interaction between thefirm’s financial structure and the intensity of product market competition. Thereduced-form results, while suggestive, include variables such as the firm’s ownmarket share and the total number of stores in the market. An advantage of thestructural model is that each of these variables are appropriately treated asendogenous, while consistency of the reduced-form estimates requires that they beexogenous. A second advantage is that we can uncover movements in underlyingreaction functions, rather than examining only movements between equilibria. HereI quantify the impact of the firm’s LBO on its optimal quantity decision, as well asreport the equilibrium effects on LBO firms, their responses by their rivals and thenet effect of such a change on the equilibrium number of supermarkets.6

4.1. Specification of the profit function

Firm i’s profit function is assumed to have the form Cikðsik;QkÞ ¼ PkðQkÞsik�

CikðsikÞ, where k ¼ 1; . . . ;K indexes markets and Qk denotes aggregate quantity inmarket k. For simplicity, I assume that the market inverse demand curve is linear inmarket demand, so that PðQkÞ ¼ b1

1 � b2Qk þ X 0kg1 where X k represents a vector ofmarket level variables that shift demand exogenously. Examples include market size,median income, the percentage of households with income below $10,000 andregional dummy variables. The firm’s cost functions are assumed quadratic in ownquantity with one source of heterogeneity across firms that is unobserved to theeconometrician, but known to the firms and its rivals, CiðsiÞ ¼ ðb

2

1 þ b3shLBOik þ

b4LBOi þ b5si þ suuikÞsi:Heterogeneity in marginal costs, uik, is assumed independent and identically

distributed (iid) across players and markets and is drawn from a standard Gaussiandistribution. Heterogeneity in variable costs is important in an empirical specifica-tion because it allows observationally identical firms to own a different number of

6Theoretical analyses of the interaction between financial structure and product market competition

delineate circumstances under which debt results in either a softening or a toughening of competition.

However, the theoretical predictions depend on precise timing of the firm’s output and financial structure

decisions, about which there is little reason to hold a strong a priori position. For example, Brander and

Lewis (1986) argue that increases in debt make firms more competitive. Making one small change to the

model, allowing firms to change their level of output twice after the increase in debt levels, Glazer (1994)

argues the opposite is true, that increases in debt soften competition. Maksimovic (1988) and Rotemberg

and Scharfstein (1990) also argue that increases in debt toughen product market competition while

Fudenberg and Tirole (1986), Bolton and Scharfstein (1990) and Phillips (1991) each argue the contrary

position, that increases in debt soften competition.

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stores within a market. Other variables hypothesized to affect marginal costs are alsoincluded. For example, shLBOik denotes the share of rivals in market k who haveundergone leveraged buyouts, while the dummy variable LBOi indicates whether ornot firm i has undergone an LBO by 1991.

In a Cournot game, a firm’s best response function is a decreasing function of thefirm’s own marginal cost of production. Thus, a variable which increases a firm’smarginal cost will result in lower own output for every level of output of her rivals.For example, if b440, then undergoing a leveraged buyout ‘‘softens’’ productmarket competition since the firm’s reaction function shifts inward. Similarly, ifb340, a larger share of rival’s output provided by LBO stores leads to a lower levelof firm i’s output for every level of rival’s output. As is usual in discrete choicemodels, the units of profits are not identified in the estimation procedure developedbelow. I impose the traditional constraint that the variance of the unobservable isunity, su � 1: Exogenous movements in rivals’ reaction functions, such as thatcaused by a change in status of LBOj for some firm j in the market, will identify firmi’s reaction function in much the same way that movement in demand identifiesmovements in supply in a simple demand and supply diagram.

The model satisfies A1 and A4 without further assumptions, with Cikðsik;QkÞ ¼

aiksi � b2Qksi � b5s2i , where aik � b1 þ X 0kg1 � b3shLBOik � b4LBOi þ suuik with b1 ¼

b1

1 � b2

1 and a ¼ 1, respectively. I now show that provided b2X0 and b5X0 theconditions in A2, A3 and A5 are also satisfied. These parameter restrictions embodythe assumptions that demand is decreasing and marginal costs are non-decreasingrespectively. A sufficient condition for A2 (unique best responses) is that i’s marginalprofitability is always weakly decreasing in si, and never exactly zero for multiplevalues of si. Note that marginal profitability piðsi þ 1; s�iÞ � piðsi; s�iÞ may be writtenas

DCiðsik;QkÞ � Cikðsik þ 1;Qk þ 1Þ �Cikðsik;QkÞ

¼ aik � b2½sik þQk þ 1� � b5ððsik þ 1Þ2 � s2ikÞ,

which is strictly declining in sik if b240 or b540. If b2 ¼ 0 and b5 ¼ 0 then marginalprofitability is either strictly positive or strictly negative almost everywhere,making s�ik ¼ riðs�ikÞ equal to M or 0, respectively. (Only on a set of measure zerowill aik ¼ 0 and hence marginal profitability would also be.) For A3, note thatCikðsik;QkÞ �Cikðtik;QkÞ ¼ aikðsik � tikÞ � b2Qkðsik � tikÞ � b5ðs

2ik � t2ikÞ, which is de-

creasing in Qk provided b2X0 for sikXtik: Finally, to see that A5 is satisfied witha ¼ 1, note that DCiðsik;QkÞ is decreasing in the pair ðsik;QkÞ provided b2X0 andb5X0. The fact that the theoretical restrictions amount only to parameter restrictionsin this linear model makes it a particularly attractive baseline model for estimation.

Identification of the parameters in the discrete Cournot model will follow theanalysis provided for the continuous choice Cournot model provided by Bresnahan(1982) and Lau (1982) if there is price data. If not, then exogenous variation in firmj’s reaction functions will act to identify firm i’s reaction function (or rather the profitfunction that generated it). While I motivate the form of the profit functionestimated in terms of familiar demand and cost function specifications, it is more

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appropriate to think of the pair implying a specification for variable profitabilitysince, for example, the model’s prediction about the world will only depend upon thevalue of b1 � b1

1 � b2

1, not each individual component. Similarly, the same profitfunction would result whether uik entered because it represents heterogeneity inmarginal cost or in the firm’s demand intercept. Only parameters on variables thatprovide independent variation in the variable profit function can be identified, andhere that amounts to the restriction that we can only estimate b1 � b1

1 � b2

1 and asingle unobservable in the variable profit function which interacts with si.

4.2. Estimation

In this section, I outline the procedure used to estimate the parameters of themodel. For clarity of exposition, I ignore several aspects of the actual estimationprocedure in this section, with full details provided in Appendix B. For a given valueof the parameters, the model uniquely identifies the total market output providedA1–A5 are satisfied. Define the prediction error, nk, as the difference between theobserved index of market activity and the model’s conditional expectation of

equilibrium market activity nkðQ�

k;Z�k; yÞ ¼ Q�k � E½Q�kjZ�k; y�; where Z�k �

ðfX ikgN

i¼1;X kÞ is the exogenous data for market k (i.e., the market and firm

characteristics) and y is the vector of model parameters ðb; gÞ0. By construction, theprediction error nk is mean independent of the exogenous data when evaluated at thetrue parameter value y0;E½nkð�;Z�k; y0ÞjZ�k� ¼ 0 and this condition, together withsome standard regularity conditions, provides the basis for a generalized method ofmoments estimator of y0 (see Hansen, 1982). While E½Q�kjZ�k; y� is difficult to

compute exactly, as it requires computation of a high dimensional integral, over theunobserved (by the econometrician) heterogeneity, following Berry (1992) asimulation estimator is easy to construct. Given a fixed set of R random draws on

each market, ðffur

ikgN

i¼1gR

r¼1Þ, I calculate an unbiased estimate, Qk, of the expected

market output, E½Q�kjZ�k; y� as QR

k ¼ ð1=RÞPR

r¼1QkðZ�k; fur

ikgN

i¼1; yÞ; and calculate the

simulated prediction errors and the corresponding simulated sample moment

condition nk ¼ Q�k � QR

k ð�; yÞ; GR

KðyÞ ¼ ð1=KÞPK

k¼1nkðyÞf ðZ�kÞ where f ðZ�kÞ is an

arbitrary ðp� 1Þ function of the exogenous data. The estimate of the parameter

vector y is then obtained as the value of y 2 Y that solves the minimization problem

kGR

KðyÞk2A where k � kA is the Euclidean norm with respect to the weight matrix A.

Pakes and Pollard (1989) provide weak regularity conditions that are satisfied hereunder which the resulting estimator is consistent and asymptotically normal. Theirresults are far more general than we require here because the moment conditions areactually linear in the simulation error, ensuring that simulation error will average outacross observations in the moment conditions. As a result, consistency is achievedfor any fixed number of simulation draws per observation, R. Their results establishthat

ffiffiffiffiKpðyns

K � y0Þ�Nð0; ð1þ 1=RÞðG0AGÞ�1G0AVA0GðG0AGÞ�1Þ where V ¼

Evkðy0Þ2f ðZ�kÞf ðZ�kÞ

0; and G ¼ Eqvkðy0Þf ðZ�kÞ=qy. The Nelder Meade method is usedto minimize the objective function because it is not smooth in finite samples and I

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refer the reader to a very interesting recent working paper (Ackerberg, 2001), wherealternative estimators for this and other economic models that must be estimated bysimulation are considered.

Finally, I discuss estimation models where the index function Qðs; y1Þ depends onsome parameters, y1. Examples include the symmetrically differentiated entry thenquantity game where y1 ¼ s and the differentiated product quantity game, wherey1 ¼ ðy11; . . . ; y1JÞ: In each case, for a fixed known y1 the approach described abovewill work and so it will also work if consistent estimates of by1 are available. (Itremains only to correct the standard errors of by2 for the uncertainty in by1 arisingfrom sampling variation.) For example, in the differentiated product game,consistent estimates of by1 may be obtained from a first-stage estimation of adifferentiated product demand system, and the identical approach could be used. Analternative is to use additional moment style conditions. For example, Sutton (1998)shows that in entry then differentiated product quantity games, the one firmconcentration ratio will be bounded below by a term that depends on the productdifferentiation parameter s. Requiring all markets to have observed concentrationratios that are greater than those computed from the model would then beinformative about at least the range of possible s’s that are consistent with the data. Ileave pursuit of this idea to future research.

4.3. Equilibrium computation: Cournot tattonnement

The model developed in this paper has a large strategy space. Each of N firmschooses a level of output from the set Si ¼ f0; 1; 2; 3; . . . ;Mg. Consequently, there areMN points in the strategy space and finding an equilibrium is not a trivialcomputational task. Evidently, complete enumeration is not a feasible option exceptfor very small M and N. In the supermarket data, M ¼ 163 and N ¼ 48. We mustconsider more practical solutions than complete enumeration.

To estimate the parameters of the model, the predicted equilibrium must becalculated for every simulation draw in every market at each stage of the estimationprocedure. Thus, if the non-linear optimization over the parameter space takes 1000iterations to converge, and we use 100 simulation draws in each of 85 markets,estimation involves finding 8.5 million equilibria. Unfortunately, Cournot gamessuch as this are not generally globally stable except under special conditions (seeMoulin (1984). Nonetheless, severe difficulty in finding an equilibrium is unlikelysince (i) we know at least one equilibrium exists, usually many, and (ii) our task isonly to find one: any equilibrium will suffice, since we know that market output is thesame at every point in the equilibrium set.

In practice, player by player iterated Cournot tattonnement always convergedto an equilibrium without problems. That is, starting at the zero point, s0 ¼ 0,player 1 plays her best response to 0, moving to her monopoly output,s1 ¼ ðr1ðs

0Þ; 0; 0; 0; . . . ; 0Þ. Next, player 2 picked his best response to s1, and so on.Cycling through the players generally provided convergence within a reasonablysmall number of rounds.

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4.4. Estimation results

Table 1 presents the parameter estimates for a variety of specifications. In eachcase, the estimates are computed using data from 1985 and 1991, before and after theleveraged buyout wave in the supermarket industry. Cotterill and Haller (1992) findthat new supermarkets are effectively exclusively built by supermarket chains alreadyactive in a market, that de-novo entry is rare, so I assume that the set of potentialentrants into a market in each year include only the set of chains already active inthat market. (Table 3 in Appendix B provides robustness checks for thisassumption.) Columns (1)–(4) in Table 1 provide a variety of specifications of themodel. Column (5) reports a reduced-form OLS regression of the total number ofactive stores in each market during 1991 on market characteristics and the fractionof stores operated by firms which had undergone a leveraged buyout by 1991.Column (5) illustrates that the estimation results are not a quirk of either thestructure imposed by the model or the over-identifying restrictions imposed toestimate the full model.

First, notice that in all four specifications the estimates of b2 and b5 are non-negative as required. b2 is estimated to be robustly positive, which implies that theestimated inverse demand function slopes down in the total number of operatedstores, a necessary condition for marginal revenue to decline in total output. The

Table 1

Parameter estimates using data from 1985 and 1991

(1) (2) (3) (4) OLS: 1991

y ðjtyjÞ y ðjtyjÞ y ðjtyjÞ y ðjtyjÞ bOLS ðjtbOLSjÞ

Median income ð�1e4Þ 1.30 (0.45) 0.83 (0.45) 3.39 (0.16)

Percent income o10k 0.04 (0.38) 0.04 (0.44) 0.89 (0.64)

Percent income 450 k �0.12 (0.52) �0.13 (0.85) 0.25 (0.14)

Median age 0.04 (0.38) 0.12 (0.77) �1.14 (1.12)

Households ð�1e5Þ 5.78 (2.78) 5.73 (4.74) 5.32 (3.89) 3.91 (3.26) 28.4 (19.6)

Landarea ð�1e5 sq. miles) 1.68 (2.05) 1.57 (3.05) 1.27 (4.70) 1.39 (4.71) 6.34 (3.08)

Households2 �0.04 (2.42) �0.03 (2.53) �0.04 (3.23) �0.03 (3.39) �0.24 (5.41)

Land area2 �0.05 (1.73) �0.05 (2.71) �0.04 (5.18) �0.05 (5.64) �0.20 (2.53)

b1 2.94 (0.54) 1.66 (0.44) 5.84 (3.59) 3.91 (3.25)

b2 0.21 (2.59) 0.21 (5.06) 0.19 (4.02) 0.14 (6.34)

b3 4.03 (0.83) 5.80 (2.47)

b4 5.07 (1.64) 4.91 (3.13)

b5 0.03 (1.11) 0.07 (2.36) 0.03 (0.95) 0.04 (1.90)

Fraction LBO �50.1 (3.17)

R2 0.90 0.89 0.90 0.89 0.97

Columns (1)–(4) report results in which the firms that were actually present in 1985 and 1991, respectively

are considered to be the set of potential entrants in those years. The final column reports a simple ordinary

least squares regression of the number of stores operating in each market area in 1991 on the market

characteristics, together with a variable reporting the fraction of stores operated by firms who have

undergone an LBO by 1991.

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estimates of b5 are less robustly strictly positive, but are never estimated to benegative. Since demand is decreasing and costs are non-concave, we may concludethat market output is uniquely determined in all equilibria and, consequently, thatthe estimation method is consistent with the estimated parameters.

Land area and the total number of households play an important role indetermining the intercept of the inverse demand curve. These variables may beinterpreted as determinants of market size and explain the observed positivecross-sectional relationship between market size and the number of active stores.7

The remaining coefficients on market level variables in columns (1) and (2) areestimated imprecisely. In particular, no statistically significant role was found forthe variables capturing cross market variation in demographics such as ageand income. For this reason, columns (3) and (4) report estimates of the modelsin columns (1) and (2), deleting these insignificant market demographic variables. Ifocus on the results in columns (3) and (4) for the remainder this section.

Following Chevalier (1995) I consider the impact of undergoing a leveragedbuyout on incumbent firms. Columns (1) and (3) allow the marginal costs of LBOand non-LBO firms to differ systematically by introducing a dummy variable LBOi

in the cost function indicating whether firm i undertook an LBO by 1991. Thisdifference might perhaps arise from differences in the cost of capital to LBO andnon-LBO firms or simply that higher leverage provides stronger incentives forminimizing costs. The estimated effect is a direct strategic effect of debt since thefirm’s own debt structure is allowed to change its own reaction function. Since bb440,the results suggest that a firm’s own reaction function shifts inwards when the firmundertakes an LBO and as a result equilibrium own output falls and equilibriumrivals output will increase. If the increase in rivals’ output is less than the reduction inown output, equilibrium total output decreases.

Fig. 2 shows the effect of undertaking an LBO on the firm’s reaction functionwhen all market characteristics are held at their median values. Undertaking an LBOsoftens competition, reducing a firm’s optimal number of stores in median market byapproximately eleven stores. The ability to recover the firm’s reaction function fromobserved equilibrium movements is one great advantage of having estimated astructural model (see also below).

Finally, column (4) reports estimates of the model allowing for an indirectstrategic effect of debt. I find that as the fraction of rival stores operated by firmswho have undergone an LBO by 1991 increases, the firm’s reaction function shiftsinward, softening competition (bb340). In principle, the model could distinguishbetween the direct and indirect strategic effects of debt by estimating a model withboth b3 and b4 included. Unfortunately, in practise the data did not allow such adistinction to be drawn.

7Although the estimated relationship between these measures of market size and variable profitability is

non-linear, the exact degree of the non-linearity is somewhat sensitive to the inclusion or exclusion of

specific observations since there are very few markets as large as New York or LA.

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70

60

50

40

30

20

20

10

40 60 80 120100 140 16000

Rivals Combined Output

Fir

m i'

s O

utp

ut

Fig. 2. Estimated best response functions using the point estimates reported in column (3) of Table 1.

These results are reported using all explanatory variables held at their median values. Undergoing a

leveraged buyout shifts the best response function inward for every level of output produced by rivals,

making competition softer.


4.5. Quantifying the effect of LBOs on competition

Next I use the estimated model to quantify the effect of an LBO on equilibriumoutcomes. I consider the game played by LBO and non-LBO firms using theirrespective estimated reaction functions for the median market, the reaction functionsdepicted in Fig. 2 above. For the purposes of this section, I make an additionalassumption—namely that LBO and non-LBO firms are symmetric within their typebut asymmetric across types, LBO and non-LBO. Doing so allows us to work withjust two reaction functions, one for each type of firm:

sLBOi ¼ 50� 0:42 ðNLBO

� 1ÞsLBOi þ snonLBO

i NnonLBO� �

,

snonLBOi ¼ 61� 0:42 NLBOsLBO

i þ snonLBOi ðNnonLBO

� 1Þ� �

,

where the terms in square brackets are the total number of stores operated by rivalfirms in each case.

Since de-novo entry is unusual in the industry (see Cotterill and Haller, 1992), Iexogenously fix the total number of active firms in the market and consider the effectof incumbent chains undergoing LBOs on (i) the aggregate equilibrium number ofstores operating and (ii) the equilibrium number of stores operated by each type offirm. Table 2 reports the results.

Table 2 demonstrates the three main effects that occur when a firm undertakes anLBO which reduces product market rivalry. First, the left-hand panel shows that the

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Table 2

The left-hand panel reports the aggregate equilibrium number of stores in a market, Q� with rows varying

the number of LBO firms NLBO and columns the total number of chains ðNLBOþNnonLBO

Þ. The right-hand

panel shows the resultant equilibrium number of stores operated by each type of firm ðsLBOi ; snonLBO

i Þ

No. LBO

chains ðNLBOÞ

Total number of chains ðNLBOþNnonLBO

Þ

1 2 3 4 5 1 2 3 4 5

0 61 86 99 108 114 (—,61) (—,43) (—,33) (—,27) (—,23)

1 50 78 93 103 110 (50,—) (30,49) (19,37) (12,31) (7,26)

2 70 88 98 106 (35,—) (23,42) (15,34) (10,29)

3 82 93 101 (27,—) (19,38) (13,32)

4 88 97 (22,—) (16,35)

5 93 (19,—)


equilibrium number of stores in the market, Q�; declines as incumbent chainsundertake LBOs (we move down a column). Second, moving to the right-hand panel,the table shows that the size of the LBO firms sLBO

i increases when there are fewer ofthe (aggressive) non-LBO firms in the market (move down a column looking at thefirst entry in each pair, sLBO

i Þ. Third, that non-LBO firms expand in response to arival undertaking an LBO, so that snonLBO

i also increases as we go down a column. Thelatter is the effect that Chevalier emphasizes in her reduced-form model; she findsthat in this dataset when a supermarket chain with a 10% market share undertakesan LBO, the probability that a given non-LBO firm will add stores to the marketincreases by approximately 6.5%.

5. Conclusions

In this paper I develop and implement a method to estimate structuraleconometric models of quantity setting oligopolistic competition allowing forindivisibilities and firm level heterogeneity. I focus on a class of models sufficientlygeneral to allow them to be used with count data. The results provided by Bresnahanand Reiss (1991) and Berry (1992) are important, but very special, cases of this classof models.

The main contribution of this paper is to provide conditions on firms profitfunctions that are sufficient to ensure that an index of market activity is the same inall the equilibria of a quantity setting game. In those circumstances, the index ofmarket activity may be used to identify the parameters of the model by matchingobserved and predicted index of market activity in a cross section of markets.Broadly, I show that if (i) marginal revenue is non-increasing in own output and anappropriate index of market activity and (ii) marginal cost is non-decreasing in ownoutput, then the index of market activity will be uniquely determined within the setof equilibria. I provide a variety of examples of games which satisfy these conditionsand outline the important and in most cases rather restrictive assumptions required.

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Unfortunately, in all but entry games the required assumptions rule out non-convexities in costs which can introduce the potential for wide variation in marketoutput across equilibria.

Using data from the $300 billion per annum supermarket industry, I estimate themodel with data from 1985 and 1991, before and after a wave of leveraged buyoutschanged the financial structure of many firms in the industry. The empirical resultsare consistent with those provided by Chevalier (1995), who found that LBOs softenproduct market competition, but the richer nature of the structural model allows adirectly game theoretic consideration of the equilibrium consequences of firmsundertaking LBOs. In contrast to the reduced-form results, the results I present arenot subject to econometric concerns regarding the endogeneity of rival or indeedtotal market output; they allow for unobserved heterogeneity in variable profitabilityacross firms and they enable presentation of the results in terms of familiar gametheoretic objects such as reaction functions.

Acknowledgements

My thanks to Steve Berry, Moshe Buchinsky, Ariel Pakes, David Pearce, JohnRust, Chris Shannon and Nadia Soboleva for helpful conversations, advice andcomments. Thanks are also due to the Associate Editor of this journal and twoanonymous referees who provided numerous helpful comments. I also thank theparticipants of the Game Theory workshop at Yale, the International Conference onGame Theory at Stony Brook and the Harvard-MIT Industrial Organizationseminar. Judith Chevalier kindly provided the data. Financial support from YaleUniversity and the Robert M. Leylan Doctoral Fellowship Fund is gratefullyacknowledged. This paper is a substantially extended and revised version of chapter1 of my PhD thesis.

Appendix A. Proofs

Lemma 1. A1– A3 imply riðs�iÞ ¼ s�i ðQ�iðs�iÞÞ, where s�i ðQ�iÞ ¼ argmaxsiCiðsi;Q�i þ

QiðsiÞÞ is decreasing in Q�i, so that riðs�iÞ is decreasing in s�i.

Proof. Define Q�i � Qð0; s�iÞ. Suppose s�i ðQ�iÞ is not decreasing in Q�i, then thereexists a QH

�iXQL

�i such that sHi ¼ s�i ðQ

H

�iÞ4s�i ðQL

�iÞ ¼ sLi . Since i’s contribution to the

market index, QiðsiÞ; is strictly increasing in si; QiðsHi Þ4Qiðs

Li Þ and hence the index of

market output must be strictly higher, QH� QH

�i þQiðs�

i ðQH

�iÞÞ4QL

�i þQiðs�

i ðQL

�iÞÞ � QL. Now since sHi XsL

i and QHXQL, by the dual single crossing

condition in A3 we know that

CiðsHi ;Q

LÞpCiðs

Li ;Q

LÞ ) Ciðs

Hi ;Q

HÞpCiðs

Li ;Q

HÞ.

Hence CiðsHi ;Q

HÞpCiðs

Li ;Q

HÞpCiðs

Li ;Q

H

�i þQiðsLi ÞÞ where the first inequality follows

from increasing differences and the second inequality follows since Cið�;QÞ is

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decreasing in Q while QH¼ QH

�i þQiðsH�iÞXQH

�i þQiðsL�iÞ. Clearly Ciðs

Hi ;Q

H

�i þ

QiðsHi ÞÞpCiðs

Li ;Q

H

�i þQiðsLi ÞÞ contradicts the definition of sH

i as the maximizer ofCiðsi;Q

H

�i þQiðsiÞÞ since i would be better off playing sLi : Finally it remains to show

that riðs�i; yÞ ¼ s�i ðQ�iðs�iÞÞ is decreasing, which follows directly from the fact thats�i ðQ�iÞ is decreasing, Q�iðs�iÞ is increasing and the composition of a decreasingfunction and an increasing one is decreasing. &

Lemma 2. For any univariate concave function gðxÞ and for any xpxpx with x; x;x 2X ; the domain of g, (i) ðgðxÞ � gðxÞÞ=ðx� xÞXðgðxÞ � gðxÞÞ=ðx� xÞ and (ii)ðgðxÞ � gðxÞÞ=ðx� xÞXðgðxÞ � gðxÞÞ=ðx� xÞ.

Proof. For any x satisfying xpxpx there exists l 2 ð0; 1Þ such thatx ¼ l xþð1� lÞx. In particular set l ¼ ðx� xÞ=ðx� xÞ. By concavity, for anyl 2 ð0; 1Þ, and hence this one in particular,

gðxÞXlgðxÞ þ ð1� lÞgðxÞ ¼x� x

x� x

� �gðxÞ þ

x� x

x� x

� �gðxÞ.

Adding ð� xþxÞ to the numerator of the first term and rearranging establishes therequired result in (i). Adding ð�xþ xÞ provides the result in (ii). &

Proposition 1. Let S be a lattice and Z a convex subset of R containing the range of

QðsÞ, where QðsÞ as increasing and supermodular in s 2 S ¼ Si � S�i. If CiðQ; siÞ

submodular on ðQ; siÞ 2 Z � Si;CiðQ; siÞ decreasing and concave in Q for any fixed si,then piðsi; s�iÞ ¼ Cðsi;QðsÞÞ satisfies the dual single crossing property in ðsi; s�iÞ.

Proof (Closely related to Lemma 2.6.4 in Topkis (1998)). By cancelling terms, it iseasy to show that the following identity holds:

CiðQðs ^ tÞ; si ^ tiÞ þCiðQðs _ tÞ; si _ tiÞ �CiðQðtÞ; tiÞ �CiðQðsÞ; siÞ

� ½CiðQðs _ tÞ; si _ tiÞ �CiðQðs _ tÞ þQðs ^ tÞ �QðtÞ; si _ tiÞ

�CiðQðtÞ; si _ tiÞ þCiðQðs ^ tÞÞ; si _ tiÞ�

þ ½CiðQðs _ tÞ þQðs ^ tÞ �QðtÞ; si _ tiÞ �CiðQðsÞ; si _ tiÞ�

þ ½CiðQðtÞ; si _ tiÞ þCiðQðs ^ tÞ; tiÞ �CiðQðs ^ tÞ; si _ tiÞ �CiðQðtÞ; tiÞ�

þ ½CiðQðsÞ; si _ tiÞ þCiðQðs ^ tÞ; si ^ tiÞ �CiðQðs ^ tÞ; tiÞ �CiðQðsÞ; siÞ�.

The right-hand side of the identity has been grouped into four terms, each insquare brackets. We progress by establishing that each term in square brackets isnon-positive and hence so is the sum of all four which establishes piðsi; s�iÞ ¼

CiðQðsÞ; siÞ is submodular in s and hence satisfies the DSC in ðsi; s�iÞ:Step 1. First consider the four terms in the first bracket; we wish to show

CiðQðs _ tÞ; si _ tiÞ �CiðQðs _ tÞ þQðs ^ tÞ �QðtÞ; si _ tiÞ

�CiðQðtÞ; si _ tiÞ þCiðQðs ^ tÞÞ; si _ tiÞp0.

For brevity, define Q1 ¼ Qðs ^ tÞ;Q2 ¼ QðtÞ;Q3 ¼ Q4 þQ1 �Q2; andQ4 ¼ Qðs _ tÞ where Q1p Q2;Q3pQ4. Notice that each term in the inequality ofthe second argument takes on the same value, si _ ti, in this inequality, so we may

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suppress it defining gðQÞ � CiðQ; si _ tiÞ: The inequality to be shown can then berewritten compactly as

gðQ4Þ � gðQ3Þ

Q4 �Q3

pgðQ2Þ � gðQ1Þ

Q2 �Q1

since Q4 �Q3 ¼ Q2 �Q1. Intuitively, this inequality is about the magnitude ofslopes of the function as we move along it. Since CiðQ; sÞ is concave in Q, so byconstruction is gðQÞ and we know that by part (i) of Lemma 2,

gðQ2Þ � gðQ1Þ

Q2 �Q1

XgðQ4Þ � gðQ1Þ

Q4 �Q1

since Q1p Q2pQ4 and moreover by part (ii) of Lemma 2,

gðQ4Þ � gðQ1Þ

Q4 �Q1

XgðQ4Þ � gðQ3Þ

Q4 �Q3

since Q1p Q3pQ4: Thus,

gðQ4Þ � gðQ3Þ

Q4 �Q3


Q4 �Q1


Q2 �Q1

as desired.Step 2. For the second bracket, note that CiðQðs _ tÞ þQðs ^ tÞ �QðtÞ; si _ tiÞ �

CiðQðsÞ; si _ tiÞp0 since Qðs _ tÞ þQðs ^ tÞ �QðtÞXQðsÞ by supermodularity of QðsÞ

in s and CiðQ; si _ tiÞ is decreasing in Q.Step 3. For the penultimate bracket to be non-positive we want to show

CiðQðtÞ; si _ tiÞ þCiðQðs ^ tÞ; tiÞ �CiðQðs ^ tÞ; si _ tiÞ �CiðQðtÞ; tiÞp0.

Set u ¼ ðQðtÞ; tiÞ; v ¼ ðQðs ^ tÞ; si _ tiÞ. Then u _ v ¼ ðQðtÞ _Qðs ^ tÞ; ðsi _ tiÞ _ tiÞ ¼

ðQðtÞ; si _ tiÞ and u ^ v ¼ ðQðtÞ ^Qðs ^ tÞ; ti ^ ðsi _ tiÞÞ ¼ ðQðs ^ tÞ; si ^ tiÞ: Thus thepenultimate inequality can be re-written as Ciðu _ vÞ þCiðu ^ vÞ �CiðvÞ �CiðuÞp0which is true by submodularity of CiðQ; sÞ in ðQ; sÞ:

Step 4. For the last bracketed terms to be non-positive we want to show

CiðQðsÞ; si _ tiÞ þCiðQðs ^ tÞ; si ^ tiÞ �CiðQðs ^ tÞ; tiÞ �CiðQðsÞ; siÞp0.

Set u ¼ ðQðsÞ; siÞ; v ¼ ðQðs ^ tÞ; tiÞ. Then u _ v ¼ ðQðsÞ _Qðs ^ tÞ; si _ tiÞ ¼

ðQðsÞ; si _ tiÞ and u ^ v ¼ ðQðsÞ ^Qðs ^ tÞ; si ^ tiÞ ¼ ðQðs ^ tÞ; si ^ tiÞ. Thus the lastinequality can be re-written as Ciðu _ vÞ þCiðu ^ vÞ �CiðvÞ �CiðuÞp0 by sub-modularity of CiðQ; sÞ in ðQ; sÞ.

Step 5. Finally (for completeness) I show that if piðsÞ is submodular in s thenpiðsi; s�iÞ satisfies the DSC property. To do so let x ¼ ðsi; t�iÞ and y ¼ ðti; s�iÞ wheresiXti and s�iXt�i so that x ^ y ¼ t and x _ y ¼ s. Then by submodularity piðx _

yÞ � piðyÞppiðxÞ � piðx ^ yÞ or piðsÞ � piðti; s�iÞppiðsi; t�iÞ � piðtÞ, so that ifpiðsi; t�iÞ � piðtÞp0 then piðsÞ � piðti; s�iÞ p0. &

Proposition 4 (Results for the N-player game). Under assumptions A1– A5, the N-

player game has a pure strategy Nash equilibrium.

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Proof (By induction). The result is proven by induction, noting first thatProposition 2 has established existence for the two-player game and that Proposition3 shows that for the N-player game, the output index is uniquely defined within theset of equilibria provided one or more equilibria exist. Throughout, denote thevector of strategies chosen by players in group A ¼ f1; . . . ;N � 1g as sA ¼ ðsi; i 2 AÞ

and note that by additive separability we can always write the market index functionas QðsÞ ¼ QðsA; 0Þ þQð0; sNÞ; where QðsA; 0Þ ¼

Pi2AQiðsiÞ and Qð0; sNÞ ¼ QNðsNÞ.

The induction hypothesis is to suppose there exists an equilibrium in the N � 1player game; we must then prove that there exists at least one equilibrium in theN-player game.

Step 1. Assumption A1 means we can reparameterize the profit function of playerN to depend directly on the contribution to the aggregate output index from playersin group A rather than their full strategy profile sA, viz. CNðsN ;QA þQNðsNÞÞ. Doingso facilitates construction of a reparameterized best response function for player N,sN ¼ s�NðQAÞ, which in turn means we can construct the function

f NðQAÞ ¼ Qð0; s�NðQAÞÞ,

which describes player N’s contribution to the market index when the group Aplayers are producing QA. Note that f NðQAÞ ¼ Qð0; s�NðQAÞÞ is a decreasing functionof QA since Qð0; sNÞ is increasing in sN by assumption, and s�NðQAÞ is decreasingin QA by Lemma 1.

Step 2. Similarly, we can reparameterize the profit function of any player i 2 A todepend directly on the contribution to the aggregate output index from player N, viz.Ciðsi;QAðsA; 0Þ þQNÞ: The induction hypothesis states that an equilibrium exists forthe N � 1 player subgame for any fixed value of sN (or reparametrized QN) so thatFAðQNÞa;. Since any equilibrium of the N � 1 player subgame s�AðQNÞ 2FAðQNÞ

generates the same output index Qðs�AðQNÞ; 0Þþ QN we can define the function (in thetechnical sense)

f AðQNÞ ¼ Qðs�AðQNÞ; 0Þ

using an arbitrary selection from the set of Nash equilibria, s�AðQNÞ 2FAðQNÞ: Thefunction f AðQNÞ captures the equilibrium contribution to the aggregate index byplayers in group A when the Nth player is contributing QN . Next we would like toshow that f AðQNÞ is decreasing in QN . To do so, suppose not. Then there exists aQH

NXQL

N (generated by an underlying sHNXsL

N) such that f AðQH

N Þ4f AðQL

NÞ. Notice firstthat the index of market output must be higher in the equilibrium indexed by QH

N

than in that indexed by QL

N , since QH¼ f AðQ

H

N Þ þQH

N4f AðQL

NÞ þQL

N ¼ QL: Sincef AðQ

H

N Þ4f AðQL

NÞ; there must be a player i 2 A which is producing strictly more in theequilibrium to the N player game parameterized by QH

N than in the oneparameterized by QL

N : For that player sHi 4sL

i while equilibrium total output is alsohigher, QH4QL. Thus by A3, Ciðs

Hi ;Q

LðsL

i ; sL�iÞÞpCiðs

Li ;Q

LðsL

i ; sL�iÞÞ )

CiðsHi ;Q

HðsH

i ; sH�iÞÞpCiðs

Li ;Q

HðsH

i ; sH�iÞÞ.Since QH

ðsHi ; s

H�iÞXQH

ðsLi ; s

H�iÞ and Ciðsi;QÞ is

decreasing in Q, we know that CiðsHi ;Q

HðsH

i ; sH�iÞÞpCiðs

Li ;Q

HðsH

i ; sH�iÞÞp

CiðsLi ;Q

HðsL

i ; sH�iÞÞ; which contradicts sH

i is a unique best response to the equilibriumunderlying sH

�i.

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Step 3. Now we have constructed two decreasing functions, the first of whichdescribes the equilibrium contribution to the aggregate market index of group A forany given strategy of player N and the second describes the contribution to theaggregate market index of player N for any given contribution to the index by theN � 1 players in group A.

QA ¼ f AðQNÞ ð¼ Qðs�AðQNÞ; 0ÞÞ,

QN � f NðQAÞ ð¼ Qð0; s�NðQAÞÞÞ.

Notice that f Að�Þ; f Nð�Þ are each decreasing functions of the other group’s totaloutput. Therefore, f ð�Þ is increasing on a sublattice of R2 with respect to the neworder, Xr. Hence, by Tarski’s theorem, there exists a fixed point. By construction ofthese output functions from group A and player N, each player is playing a bestresponse against the other players’ actions; hence a fixed point of f ðQA;QNÞ ¼

ðf AðQNÞ; f NðQAÞÞ corresponds to at least one strategy profile ðs�1; . . . ; s�

NÞ0 which is a

Nash equilibrium of the N-player game. &

Appendix B. Further estimation details

A two-stage estimation procedure was found to be most effective. At the firststage, a consistent estimate of the parameters y were obtained using the momentconditions

GR

KðyÞ ¼1

K

XK

k¼1

n1kZ1k

n2kZ2k

!,

where nt

k denotes the simulated prediction error in market k from the data in period t.Since the data are observed for two periods, a more efficient estimator for y isobtained by using data from both periods.

At the second stage, I found that the best results were obtained using the simulatedanalogue of non-linear least squares (see Laffont et al., 1995; Gourieroux andMonfort, 1990, 1993) Specifically, I replaced the instruments Zt

k constructed fromthe data by a simulated estimate of qnt

kðyÞ=qy evaluated at the consistent estimateobtained from the first stage. Since the estimate of y is consistent, this function needsto be approximated only once before the second-stage estimator is computed and,consequently, a very large number of simulation draws can be used. Naturally, toobtain a consistent second-stage estimator, each of the simulation draws usedto calculate the instruments must be independent of the R simulation draws used tocalculate nt

kðyÞ.

GR

KðyÞ ¼1

K

XK

k¼1

n1k^qn1kðy

1Þ

qy

n2k^qn2kðy

1Þ

qy

0BBBB@1CCCCA.

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Table 3

Results verifying that the conclusions are robust to alternative assumptions regarding the set of potential

entrants into a market

Regional chains All chains Regional chains All chains

y ðjtyjÞ y ðjtyjÞ y ðjtyjÞ y ðjtyjÞ

Median income ð�1e4Þ �0.76 (0.71) 0.04 (0.14)

Percent income o10k �0.03 (0.30) �0.01 (0.11)

Percent income 450 k 0.05 (0.43) �0.02 (0.37)

Median age �0.01 (0.09) �0.07 (0.39)

Households ð�1e5Þ 5.69 (2.40) 5.44 (2.74) 5.51 (2.64) 5.61 (1.52)

Landarea ð�1e5 sq. miles) 1.22 (2.67) 1.34 (1.56) 1.32 (2.53) 1.18 (2.46)

Households2 �0.05 (2.00) �0.03 (2.74) �0.03 (2.01) �0.04 (1.95)

Land area2 �0.04 (2.70) �0.04 (1.63) �0.04 (3.57) �0.04 (2.34)

b1 2.87 (1.38) 3.97 (0.56) 2.61 (1.54) 2.26 (0.41)

b2 0.20 (2.50) 0.20 (2.58) 0.21 (2.68) 0.21 (1.44)

b3 8.79 (2.09) 3.46 (1.27) 5.37 (1.89) 5.56 (0.93)

b5 0.02 (0.11) 0.03 (0.12) 0.03 (0.25) 0.04 (0.16)

R2 0.90 0.89 0.86 0.85

I report supplementary estimates computed under the assumptions that (i) all chains which operate stores

in the region are potential entrants (in the columns headed ‘‘Regional chains’’), and (ii) all 48 chains are

potential entrants (in the columns headed ‘‘All chains’’). In the former case, regions are defined by dividing

the country into four broad zones: Northeast, Center, Southeast and West. jtj-statistics are reported in

parentheses. The parameters that are estimated with precision remain remarkably similar in magnitude

across the specifications, but different assumptions about the set of potential entrants do lead to different

estimates of the precision of some parameters. Generally, the smaller the set of potential entrants is

assumed to be, the higher the reported precision of the estimates.


Since data from both cross sections are utilized, the model is over-identifiedat both stages of the estimation procedure. At the first stage, the includedmarket-specific variables were treated as instruments. In addition, the averageacross active firms of the fraction of stores operated by rival LBO supermarkets in1991 is included as an instrument in Z2

k. Thus in columns (1) and (2) of Table 1, 19moment conditions are used to identify the 12 parameters, nine instruments arein Z1

k and ten instruments are in Z2k at the first stage.8 At the second stage,

the number of over-identifying restrictions is equal to twice the number ofparameters, i.e., 2� 12 ¼ 24. For the results reported in columns (3) and (4)of Table 1, five instruments are included in Z1

k and six in Z2k at stage 1, making 11

moment conditions used to estimate the eight parameters at the first stage(Table 3).

8In fact, 18 instruments are used since median income and the percentage of households with incomes

above $50,000 have a raw correlation of 0.98 in the second cross section.

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Estimation of quantity games in the presence of indivisibilities and heterogeneous firms

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