Price Discrimination Through Offers to Match Price Author(s) - CiteSeerX

Price Discrimination Through Offers to Match Price Author(s): I. P. L. Png and D. Hirshleifer Source: The Journal of Business, Vol. 60, No. 3 (Jul., 1987), pp. 365-383 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2352876 . Accessed: 18/02/2011 17:41 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ucpress. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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1. P. L. Png D. Hirshleifer University of California, Los Angeles

Price Discrimination through Offers to Match Price*

I.

Introduction

Offersby one seller to match the prices of others are a feature of both consumer and industrial marketing. In industrial marketing, meet-orrelease provisions are common in long-termcontractsof sale: these providethe buyerwith assurance that, shouldhe be offered a lower price, the original seller will either match that price or release the buyer from the contract. Such provisions assure the buyer that he will benefit from subsequent drops in the price of the item to be purchased. They protect the buyer from subsequent ex post opportunism on the part of the seller (Klein and Kenney 1985).Such provisions, however, have the effect of deterringprice competition, as arguedby Salop (1986)and Holt and Scheffman (1985).

In retail markets, sellers frequently offer to match any other advertised price-we have noticed such offers by majordepartmentstores, a renter of furniture, and several electronics goods stores. It is not obvious why such practices should arise outside a long-term sale contact. If a retailer wishes to lower his price, he could do so directly by cutting his price. A * We are grateful to the University of California,Los Angeles, ResearchProgramin CompetitionandPublicPolicy for financialsupportand to S. Bikhchandani,D. Levine, J. Mamer, D. Schwartz, participants at the University of California,Los Angeles, Sloan IndustrialOrganizationSeminar, and the refereefor their advice. (Journal of Business, 1987,vol. 60, no. 3) ? 1987by The Universityof Chicago.All rightsreserved. 0021-9398/87/6003-0003$01.50

365

In this paper, a firm discriminatesbetween two classes of customer who have different cost of information by couplinga list price with an offer to match the price of any other shop. If the list price elsewhere is lower, the firmwill be successful in discrimination.The list price of each firmis increasingin the number of sellers, and the total sales are decreasing in the numberof sellers. Furthermore,if sellers coordinate,they discriminatemore efficaciouslyand increase their profitsby increasingtheir total sales.

366

Journal of Business

distinguishingfeature of retail marketsis that the cost of information about price may be large relative to the value of the item to be purchased. Moreover, Consumersmay well differin their cost of information. In these circumstances, an offer to match any other advertised price has the effect of screeningcustomersby theircost of information. There have been several recent papers on the subject of price discriminationin retail markets. Shilony (1977), Rosenthal (1980), and Varian(1980)present models in which identicalcompetingsellers of a homogeneous good select strategies in price to discriminatebetween two classes of customer with differentelasticities of demand. In their analyses, the uniquesymmetricequilibriumwas a mixed-strategyequilibrium.This was interpretedas a model of sales or discounts. Sobel (1984) introducedtime explicitly into the model to study the timingof the discounts. Jeulandand Narasimhan(1985)gave a monopoly analysis of a similarproblem.Narasimhan(1984)presenteda theoretical and empirical analysis of the use of cents-off coupons as a method of price discrimination.'Lazear (1986) solved for the profitmaximizingprice path where a monopolistic seller learns about customer demandthroughthe price that he sets. In this paper, we consider a largerstrategyspace in which firmsmay not only choose price but also choose to offer to matchthe lowest price listed by a competitor.It is shown that, in symmetricequilibriumwith more than one firm,each firmwill attemptto discriminateby randomization of its list price and, in addition,by offeringto matchthe price of its competitors. Unlike other methods of discriminationsuch as periodic sales and coupons, price matchingis a device that depends on the existence of competition:it cannotbe implementedby a singlefirm. In equilibrium,the list price of each firmis increasingin the number of firms, and the total sales are decreasing in the number of firms. Furthermore,for a given number of firms, if firms coordinate their pricing, they will be able to discriminatemore efficaciously and increase their profit by selling more to the customers with low cost of information.Thus increased competition through the entry of more sellers reduces social welfare, while coordinationincreases social welfare. The remainderof the paper is organized as follows. The setting is introducedin Section II, and the equilibriumin the case of two firmsis presented in Section III. In Section IV, this is compared with the equilibriumwith correlated strategies. In Section V, the model is extended to the case of more than two firms, and the numberof firmsis made endogenous. Some comparativestatics are presented in Section VI. The paper concludes with some remarkson directions for future work. 1. For a generalreviewof the literatureon coupons, see VilcassimandWittink(1985).

Price Discrimination

II.

367

Basic Model

Assume, for simplicity, that all firms produce under identical conditions of zero fixed cost and constant marginalcost, c. The strategyof firmi consists of a list pricepi and, possibly, a qualificationin the form of an offer to match the competitor's list price. The equilibriumconcept is Nash in the firms' strategies. The firms announcetheir strategies simultaneously. Once the strategies are announced, price may alteronly throughthe effect of offers to match.For instance, if one firm announces a higher list price but has offered to match, in equilibrium the customers of that firmwho take advantageof its offer to matchwill be charged a price equal to the lowest list price announced. There are two classes of customer.One class, tourists,has an infinite cost of search and an inelastic demandfor 2Q, units of the good at any price at or below a reservation level, r. The tourists choose between the firmsat random:each firmreceives an equal fractionof the tourist demand. The other class, locals, can collect informationabout the firms'pricingstrategiesat zero cost and has an elastic demand,2Qe(p), for the good. If all firmsoffer the same price (eitherby the same list or throughmatching),the demandof the locals is dividedequallybetween the firms. Let e(P)

def =

Qe(P) * (p - c)

(1)

and HAP)

otherwise.

(2)

Assume that the tourist demandand the local demandare such that the following two assumptionshold. ASSUMPTION 1. arg max [FHe(p)] < r. ASSUMPTION 2. He(H)is twice differentiableand Hli(p)< 0. Assumption 2 is a sufficient condition for there to exist a unique profit-maximizingprice for a monopolistthat caters to a populationof locals only. For purposesof comparison,it will be helpfulfirstto studythe profitmaximizingstrategyin the case in which there is only one seller in the market. In these circumstances, if the seller could discriminate,he would chargea price r to the tourists and arg max[2Qe(p) * (p - c)] = def = Pe to the locals. The snag is how to distinguisha arg max[IFIe(p)] touristfrom a local. Suppose that the monopolistis unable to discriminate.Then he will choose price to maximize 2Qe(p) * (p - c) + 2Qt (p - c) = 2[H-e(p) + [It(p)].

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Let the profit-maximizingprice for the (nondiscriminating) monopolist be denoted pm, that is,

[1f(pm)+ ;t(Pm)= 0.

(3)

This will be a compromisebetween the two prices that a discriminating monopolist would charge. Since, by assumption 1, Pe < r, it follows that Pe < Pm < r. The total sales will be 2Qe(pm) to the local customers and, since Pm < r,

a quantityof 2Qt to the tourists. III.

DuopolyEquilibriumwith Price Matching

Now let there be two firms in the industry.2A naturalmechanismby which to screen the two classes of customerthat arises when there are several sellers is the offer to match. The tourists do not know the list prices of the various firms; only the locals do. Hence only the locals can take advantageof offers to match prices. Let HI(pi:(D)denote the profitof firmi when the list price of firmi is pi and the strategyof firm j is given by the distribution F( ). In the particularcase in which the strategyof firmj is a single pricepj, the functionwill be writtenas (Pi: pi).

In the following lemma, it will be shown that both firmswill couple their list prices with offers to match the price of the other. The reason is very simple:a firmloses nothingby doing so, but in the event thatits list price is higher than that of the other firm, an offer to match will prevent the defection of its local customers.3 LEMMA 1. For each firm, it is a weakly dominantstrategyto couple its list price with an offer to match. In the following lemma, it is proved that, in the model, all strategies that have supportoutside the interval [pm, r] are dominated. LEMMA

2.

Any strategy with support in [0, pm) or in (r, + oo) is

dominated. It remainsto analyze the equilibriaof the game. In the remainderof this section, it will be shown that there are two pure-strategyequilibria, both of which are asymmetric. Next, it will be shown that there is a unique symmetricdifferentiablemixed-strategyequilibrium. A.

Pure-Strategy Equilibrium

The characterof the pure-strategyequilibriais quite simple. Each equilibriumconsists of one firmlisting price Pmand offeringto match and the other firmlisting price r and offeringto match. 2. In a subsequentsection, we considerthe case of more than two firms. 3. The proofs of all results will be presentedin the App.


369

PROPOSITION 1. There are two pure-strategyequilibria:(a) one in which firm 1 sets list price pm and offers to match and firm2 sets list price r and offers to match; (b) another with the pricing strategies interchangedbetween the firms. The intuition of the equilibriumis simple. Suppose that one seller prices at pm. Then the other surely will price at r, for by doing so he gains more profitfrom his tourists while his offer to matchensures that he loses no customersto the lower-list-pricefirm.On the other hand, if one firmprices at r, the other is in the position of being a price leader. His decision is exactly like that of the nondiscriminatingmonopolist; hence he will choose Pm The local customers utilize the offer to match of the higher-priced firm.In equilibrium,all locals pay pricePm, and each firmproducesan identicaloutput Qe(pm)+ Qt. The combinedoutputof the two firmsis 2[Qe(pm) + Qt], which is exactly equal to the output of a nondiscriminatingmonopoly. By takingadvantageof the low-pricefirm,the high-pricefirm(which sets list price r) effects price discriminationand realizes profitof

Qe(Pm)* (pm - c) + Qt (r - c) = HIe(pm)+ HIt(r). The low-price firmrealizes a smallerprofitof Qe(Pm) (P.

-

c) + Qt (P.

-

c) = fef(pm) + Ht(Pm).

Because some discrimination occurs, the combined profit of the duopoly exceeds the profit of a nondiscriminatingmonopoly. B.

Mixed-Strategy Equilibrium

In the previous subsection, it was shown that there are only two purestrategyequilibriaand that both are asymmetric.In pure-strategyequilibrium, the firm that sets the higher list price will earn largerprofit. Immediatelythe question arises, In a situationin which the two firms are identical, which will be the one to price higher? Intuitively, a theory in which identicalfirmspursue identical strategies may seem more plausible. The next propositionpresents the main result of the paper. There is a unique symmetricdifferentiablemixedstrategy equilibrium.Under the equilibriumstrategy, each firm sets price on the range [pi, r) with continuousdensity - QtfI (p)/[Hle(p)]2 and at r with probability mass - QtlfIH(r).

This strategy may be interpretedas one in which each firm has a standardlist price, r, but may offer discounts from that accordingto the distributiongiven. Such discountsin themselves have a discriminatory effect.4 4. The discriminatoryeffect of such discountsis the focus of Shilony(1977),Rosenthal (1980),and Varian(1980).

Journal of Business

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2. The unique symmetric differentiableequilibrium PROPOSITION strategy is given by the distributionfunction F

1 + QtIHt(p) 11 if p = r.

if p E [pm, r),

It was shown earlier that there are no symmetric pure-strategy equilibria.Proposition2 shows that there is a uniquesymmetricdifferentiable mixed-strategyequilibrium.The supportof this equilibriumis the interval [pm, r]. This has significantwelfare implications. Under nondiscriminatingmonopoly, the price to locals is Pmalways. Proposition 2 implies that, under duopoly, the (effective) price to locals will exceed pm with positive probability. This implies the striking result that, under duopoly, the expected combined sales will be less than under nondiscriminatingmonopoly. The presence of competitionallows each firman additionalmethod of discrimination,namely, the offer to match price. This leads each firm to try to extract more from its tourists by raisingits list price with the result that total expected sales are reduced. In contrast, an increase in the numberof sellers alone would lead to largerexpected sales.5 To summarize, in the mixed-strategyequilibrium,the total sales to tourists are 2Qt, while the total expected sales to locals are 2 fp Qe(p)dF(p) < 2Qe(pm). The expected profitof each firm is rr

=

I [Qe(p)

+ Qt](p - c)dF(p) = HI(Pm)+ It(Pr)

(5)

~Pm

where %(.)denotes the expectation operator. IV. CorrelatedStrategies The problemthat confrontsthe two sellers is inherentlyone of coordination. Profits may be increased by discriminationbetween the two classes of customer. There is, however, only one means of discrimination-the offer to matchprice. But this will workfor a seller only if the other seller sets a lower list price. In the mixed-strategyequilibrium,both sellers randomizeover the prices in the interval [pm, r]. But the profitsthat result are dominated by the profitsin the pure-strategyequilibrium,in which one seller sets price at Pmand the other at r. The essential reason is that, in the mixedstrategyequilibrium,each firmmust be indifferentbetween settinglist price at some p > pmand setting it at pm. Thereforethe expected profit must be He(pm) + Ht(Pm), which is exactly the profit of the low-price firm in pure-strategy equilibrium. 5. For a detailedanalysis of the effect on outputof the introductionof (third-degree) price discrimination,see Schmalensee(1981)and Varian(1985).

371


It is likely that the firmsin such a situationwill appreciatethat there are gains from coordination.In a static model, they may exploit external devices to achieve a correlated equilibrium.6That is, they may adopt strategiesthat specify moves contingenton externalsignals. For instance, one strategymight be, Set list price at pmif the closing Dow Jones the previous day is even and at r otherwise. The best response to such a strategy is, Set a list price at r if the closing Dow Jones the previous day is even and at Pm otherwise.7 There is a multiplicityof possible correlatedequilibria.First, there are many possible external signals on which the equilibriummay be based. Second, for any given signal, there are multipleequilibriathat differin the division of profitsbetween the two sellers. We shall focus attention on a symmetric equilibrium,in which the external signal is such that, with probabilityone-half, one firmsets list price pmand the other list price r and, with probabilityone-half, the roles are reversed between the two firms. In this equilibrium,the expected profitof a firm will be ?/2[e(pm) + flt(Pr)] =

HIe(pm)

+

+ ?/2[fIe(pm)+ HIt(r)]

?/2Flt(pm)

+

(6)

?/2FIt(r).

This is largerthan the profitunder the mixed-strategyequilibrium. A fundamentaldifficulty for a conspiracy to support prices in an oligopoly is that each seller has an incentive to shade his price to draw more business. In coordinationto price discriminatethroughoffers to match price, however, neither seller has an incentive to deviate from the (agreed)pattern of pricing. The effect of the offers to match is to make deviations unprofitableand hence supportthe coordinatedprice discrimination.

It is worth notingthat, in the equilibriumof correlatedstrategies,the realized list prices always are r for one firm and Pm for the other. Hence, in all instances, the effective price to local customers is Pm. Thereforeit is unambiguousthat, relative to the mixed-strategyequilibrium,the industry output in correlated equilibriumis larger-it is equal to the output of a nondiscriminatingmonopoly. One possible measureof welfare is the expectation of combined supplierand buyer surplus. Since price is above marginalcost in both the independent 6. In a settingof repeatedplays, a sellermay playa strategyof the form,Priceatpmon odd dates and at r on even dates. The best responseto this is the strategy,Price at r on odd dates and at Pm on even dates. By alternatingtheirprices, the two firmscan achieve an equaldivisionof profit.A similarpracticeoccurredin the GreatElectricalEquipment Conspiracyof 1955-56: General Electric, Westinghouse,I-T-E, and Allis-Chalmers coordinatedtheirbids to supplylargeturbinegeneratorsby the phases of the moon (see Sultan 1974,p. 39). 7. For the original reference on the subject of correlatedstrategies, see Aumann (1974).

Journal of Business

372

mixed strategy and the correlated equilibria, the increase in output leads to an increase in expected total surplus. In this sense it may be argued that welfare is higher in the correlatedequilibrium;however, there are distributivedifferences between the two equilibriaas well. V. EquilibriumNumberof Firms Up until this point, it has been assumed that the numberof firmsin the marketis fixed at two. In this section, we allow the numberof firms,n, to be an endogenousfunction of a fixed cost of production,K. Assume that this cost satisfies the following assumption. 3. K < Ile(pm) + [lt(Pm). ASSUMPTION To recall, the total demandof touristsis 2Qtif the price is no greater than r, and the total demand of locals is 2Qe(p). Each firm will sell a quantity 2Qtln to tourists if its price is no greater than r. If all firms chargethe same effective pricep to locals, the locals also divide themselves equally, so that each firmwill sell quantity2Qe(p)In to locals. The unique symmetricdifferentiableequilibriumstrategyis for each firmto set price on the range [pm, r) with continuous density _0 n-

1

ll(-

) II "(P)

He(P)

fHe(P)

and at r with mass probability [- QtIfJ(r)]"/n '). PROPOSITION 3. There are n pure-strategyequilibria:one firm lists price Pm and the remainingfirms list price r. The unique symmetric

differentiableequilibriumstrategyis given by the distributionfunction Fn(P) =

-

[_Qt/He(p)j1(

)

if

p E [pm,r),

The effective price to the locals is the minimumprice quoted by the n firms. The distributionof the minimumprice is pr[min(p1,

.

* * pP) ?p]

= 1 - pr[pi > p, =

1

-

[1

for all i = 1, . . ., n]

_ Fn(p)]n.

(8) The total sales to the locals are 2Qe[min(pi)]. Notice that the distributionFn is decreasingin n. Hence the distribution of the minimumprice is decreasingin n. Thatis, the effective price to locals is higher and the total sales to locals smaller in the sense of first-orderstochastic dominancethe largeris the numberof firms. The intuitiveexplanationof this result is that setting a low price is a public good to all the other firms;once one firm sets a low price, the others are free to set list price at r to maximizeearningsfrom discrimi-


373

nation. The largerthe numberof firms, the smalleris the pressure on any firm to set a low price.8 In equilibrium,the profitfor a firmif it sets list price p E [Pm,r] is given by f hPmP)Qe(x)(x - c) * dGn(x) + [1 - GQ(p)] * Qjp)(p - c)

-(p

= F,) = H~~p F~f)

+ HA(P) if P E [Pm, r), [pmr) Qe(x)(x

+

1-

-

(9)

c) * dG'(x)

QtlfH(r)] * Qe(p)(p

+ HIt(r) ifp = r,

- c)

where G,(*) is the distributionof the minimumof the list prices of the other n - 1 firms. In the proof of proposition3, it is shown that the expected profitfrom the strategyF(-), ,=

rr

| H(p *Fn)dF"(p) =

-lIfe(Pm)

+ Ht(Pm)].

(10)

This is decreasing in n, the number of sellers.

In equilibrium,it must not be profitablefor anotherfirmto enter the industry, that is, 2

2 + ['t(Pr)] 2 K 2 + [lIe(pm) + n n ~~~~~+ 1 Ite(Pm) Ht(P)].

(11)

Thus the equilibriumnumberof firms,n, in the industryis boundedby K2[e(Pm) + Ht(Pm)] - 1 ?

n

s4 2HTe(Pm) + HAPm)]

It is interestingto compare these results with the equilibriumwith correlated strategies. Consider a symmetric correlated equilibrium with N firmsand with an externalsignalthat takes values i = 1, ... , N with equal probability:if the value of the signalis i, firmi sets list price Pm, and all the other firms set list price r. The expected profitof each firmwill be I

+ A~t(PM)

Hft(Pm)1+ N

2 [He(pm) + k1t(Pm)

-[k fe(Pm)

+

+ NN- 1t(r)].

kf (12)

8. Rosenthal(1980)presenteda modelin whichan increasein the numberof sellersled to a higherprice in the sense of first-orderstochastic dominance.In this model, the strategyspace of a seller was simply price. It was assumed that each additionalseller broughtalong with him a captive customer base ("tourists") but competed with the existing sellers for a fixed groupof price-sensitivecustomers("locals"). In our paper, each additionalseller competes for a share of fixed touristand local markets.

374

Journal of Business

For a given number of firms, the expected profit is larger in correlatedequilibriumthanin the mixed-strategyequilibrium.An increase in the number of firms has two effects on the expected profit in correlated equilibrium.With more sellers, the fixed market must be divided into smaller shares; hence the sales of each firmwill be smaller. On the other hand, with more sellers, the probabilitythatany one seller must set the low pricePm is reduced. On balance, the net effect is that the expected profit is decreasing in the numberof firms. New firms will enter the industryuntil 2

f~e(P.) + HIt(r)+ I[HIt(p.)

'K 2

N + Ie\PmI ffle(p)

+ HT(r)

+

- fHt(r)]j

+ 11

] It(fl(\M

This provides bounds on the equilibriumnumberof firms, N, in the industry. VI. ComparativeStatics In this section, we study the effect of shifts in demandand cost on the characterof the mixed-strategyequilibriumof the industry. To recall from (7), the equilibriumstrategyis F (p)

=

{1

-

1(n

[-Q'(p)]

-)

if p E [pm, r),

(7)

Suppose that the total local demandis 2aQe(p), where a. is a multiplicativefactor. The distributionof the equilibriumstrategyis Fn(p) = 1 - [- QtlhxHe(p)]1n- 1), which is increasing in a. Thus it is unambiguous that the larger is the total local demand, the lower will be the list price of each firmin the sense of first-orderstochastic dominance.By (8), it follows that the effective price to locals will be lower and the total sales to locals larger. From (10), the expected profit of a firmfrom the strategyFn(-) is %II = 2 [HA(PM) + Mt(Pm)]. n

Hence d

TOII= 2

I[aH-1(Pm)+ It(P )]dPm +

[le(Pm)4

Now by definition of pm, otf7I(pm) + [1(pm) = 0; hence

daT dot ~

-

n211(pM)>

O.


375

Therefore'CHis increasingin a. The larger is the local demand, the largerwill be the expected profit and hence the largerthe equilibrium numberof firms in the industry. Next we consider the effect of an increase in the total tourist demand, 2Q,. Now

r Qt'

dFn,(p) =__1 n

dQt

I1 l

H(P)]

2-n-

0

Hence -

-F

le

(p)

nj_1

1l

[

p)J

dce 1(p)] Therefore it is unambiguousthat the list price of each firm will be higher in the sense of first-orderstochastic dominance. Furthermore, the expected profitof each firmis decreasingin c. Therefore,as might be expected, when marginalcost is higher,list prices are higher,profits lower, and the numberof firms smaller. dc

=

-

_

_

_

Ht(x) since pm > x. Therefore Jle(x) + flt(pm) > is positive.

le(p) + fLt(p), and hence the second term in (Al)

The thirdterm in (Al) is [lt(pm) - flt(p)]do(x)

=

which is positive. Thus H(pm F) > fl(p

Qt *(pm - p)dD(x), )

b) For any p > r, the demand from tourists is zero; hence flt(p) = 0, Hl(p: D)

=

[I - 4D(p)]- [Ele(p)] +

fP

He(x)dD(x)

f

while IL(r: 4D) = [1

-

D(r)] * [r-e(r) + Ht(r)] +

[Hle(x) + Jlt(r)]dID(x).

Thus Hl(r: D) - HI(p PD)= t[l - I(p)] + +

f d((x)r}

[He(r)+ H#(r)]

[Hle(x) + Ht(r)]d'I(x) - [1 - cI(p)]IIe(p)

[j~He(x~d'(x)+ f H~e(x)d(p)]*

+ +

[lle(r) + Ht(r) - Ie(P)]

f[le(r) +

Ot(r)- He(x)]d(D(x)

sr [Ie(x)

+ 171(r)- fle(x)]dF(x).

(A2) The function He(-)is concave and attainsa maximumat Pe < r < p. Hence, for x E [r, p], fHe(r)- le(x). Thus the first and second terms in (A2) are positive. The thirdterm is fp IIt(r)dI(x) > 0; thereforeH(r: ')> H(p: (D). Proof of proposition1. In lemma 1, it was shown that each firmwill couple its list price with an offer to match. To determinethe equilibriumstrategies,it remainsto solve for the list prices. a) We firstprove that, in pure-strategyequilibrium,the list price of each firm will be eitherpmor r. From lemma 2, we know that any strategywith support outside [Pm, r] is weakly dominatedand hence not an equilibriumstrategy. Therefore,in pure-strategyequilibrium,pi E [pm, r], a = 1, 2. Let firm 2 price at P2 E [pm, r], and study the pricingdecision of firm 1. Considerthose p I > P2. Notice that, for such values of p1, the locals will take advantageof firml's offerto match;hence its effective priceto local customers will be P2. Since both firmschargean identicalpriceto locals, each will receive half the local demand, that is, Qe(P2)For all PI E [P2, r], the profitof firm 1 is fli(P I

P2) = Qt

(PI - c) + Qe(P2)

(P2 - c,

which is less than Qt (r - c) + Qe(P2) (P2 - c). Moreover, for allpI > r, the

profitof firm 1 is tI(PI : P2) = 0 + Qe(P2)

(P2 - c,

which is less than Qt (r - c) + Qe(P2) *(P2

- c). Thereforethe best response that satisfies the constraint p1 > P2 is PI = r. Considerthose Pi < P2. For such values of p1, the effect of firm2's offer to

matchwill be thatthe price to locals will be p I at both firms.Hence the profitto firm 1 will be Ili(Pi ; P2) = Qt (PI

-

c) + Qe(Pi)

(Pi - c).

The profit-maximizingvalue of Pi is the solution to max 1ll(Pi: P2)

subject to Pi s P2.

p1

This is the same programas thatfor the nondiscriminating monopolistwith the addition of the constraintPi ' P2. Now the unconstrainedsolution for the nondiscriminating monopolistis pm But P2 E [Pm, r], and hence min (P2, Pm) = pm. Thus the best response for firm 1 is to set list price Pi = Pm. By a similarargumentbeginningwith some price Pi for firm 1, it may be demonstratedthat firm 2 will choose a list price of eitherPm or r. This com-

Journal of Business

380

pletes the proof that, in pure-strategyequilibrium,the list price of each firm will be eitherpm or r. b) Now suppose that firm2 sets P2 = r. By the definitionof pm, Qt (P.

c) + Qe(pr)

-

(P. - c) 2 Qt (P - c) + Qe(P) (p - C)

for all p. This implies that Hi(pn

r) 21H1(r: r) and hence that firm 1 will set

Pi = Pm.

Suppose that seller 1 sets p I = pm. Now, profitof seller 2 Pm) = Qt

12(Pm

(Pm

-

c) + Qe(Pm)

(Pm -c),

and pm) = Qt * (r - c) + Qe(Pm)

12(r

(pm

-

c).

Since pm < r, it follows that 112(r:pm) > 12(fnm:fm). Thereforeseller 2 will set = r. Thus strategiesof a list pricePmcoupledwith an offerto matchfor firm 1 and a list price r coupledwith an offer to matchfor firm2 are an equilibrium. Similarly, it may be shown that the same strategies, reversed between the firms, also are an equilibrium.

P2

Suppose that the distribution F is a symmetric

Proof of proposition 2.

differentiableequilibriumstrategy. Let S be its supportandf the density function. a) The first step is to prove that pm E S and r E S. Suppose that min (p PE( S) p

def = P

> PM.

Then F) = Hle(p) + Ht(p).

H(p Since pm < p,

+ HJlt(pm)> Hl(pm F) = HJIe(pm)

He(p)+

[It(p),

by the definition of pm. Thus H(pm: F) > H(p: F), which is a contradiction. defSuppose that max (p p E S) d jp < r. Then H(j

F)

=

H(r: F)

=

[He(x) +

Hft(I)]dF(x),

and, since p-< r, {

rr [He(X)+

Ht(r)]dF(x)

[He(x) + Ht(r)]dF(x).

=

But Ht(r) > flt(O); hence H(r F) > H(ji F), which is a contradiction. b) Since F is an equilibrium strategy, it must be that H(p: F) = H(pm :F)

(A3)

for all p E [pm, r]. Now for any p E [Pm, r], H(p:

F) =

Hex) ~Pm

.

dF(x)

+ [1 - F(p)]He(p)

+

,t(p).

381


Since the function Hl(p: F) is differentiablein p, (A3) implies that dIl

=

0

dp

for all p E (Pm, r), or He(p)dF(p) + [1 - F(p)] dfe dp

dF(p) *He(P) + Qt = 0,

-

that is, F(p) = 1 + QtIiIe(p).

It remainsto show that the strategy F(p) =

I

+ Qt17J;(P) p E [Pm, r),

is a symmetricequilibrium.First, note that, by the definitionof Pm,Hj(pm) + Hl;(pm) = 0, and hence Hl;(pm) = -l;(pm) = - Qt. Now, by assumption 2, fl'(P) < 0. Hence, for all p E [pm, r], 11H(p) - Qt. Since F(-) is differentiable on (pm, r), the density is f

)= -

Q~ll(p)]2

At any p E [pm, r), H(p

F)

He(x)f(x)dx + [1 - F(p)]171e(p) + HIt(p)

=

11(x) dx + [Pm [H fl(x) e(x) - ljdx ) ] ~~~[H-1(X)]21(p {|le(P

PH(x) =

-

[11M(x)

=

_

jPHm(p

= _Qt

=

-

Qt(p)

-

F11e(x)

Now sj(p,)

]I(p) + II(p) QIIe()P) +II,(p)

Q

-

(P)

+flt(t -

-

c)

I

I

Ql; hence, for all p E [p,, r),

H(p F)

=

'e(Pm)+ H,(Pm)= I(pt

F).

Likewise, 11(r: F)

+ I1t(r) H HIe(x)f(x)dx + pr(r) Ile(r)

= Pm

_____

Ho L lfe(X)

=

=

-

-

x] IPM

Qtl7I(r) + It(r) Hj(r) fif(r)

Qtfl7e(pm) + Qt(pm - c) =

T1,(pm)

Thus, for all p Ez[pm, r], H-(p :F)

=

Hl(pm F).

1(pm:F).

Journal of Business

382

In lemma 2 it was proved that any strategywith supportoutside [pm, r] is weaklydominated.ThereforeF(p) is the uniquesymmetricdifferentiableequilibriumstrategy. Proof of proposition 3. The proof of the pure-strategyequilibriumis a directextension of the proof of proposition1 andwill be omitted.Likewise, the proof of the mixed-strategyequilibriumis an extension of the proofof proposition 2, and only a sketch will be given here. Considerthe equilibriumstrategy of some firm i. As in the proof of proposition 2, it may be shown that the support of a symmetricdifferentiableequilibriumstrategyis the interval [Pm, r].

Let Gnbe the distributionof the minimumof the prices listed by the other

(n - 1) firms. Then the expected profit of firmi for list price p E [Pm, r] is

H(p : GO)=

+

+ I1t(P)}. + [1 - Gn(P)]IIe(P) IHe(x)dGn(X)

{fPP

In mixed-strategyequilibrium,the firmmust be indifferentamongall the list prices in the interval [Pm,r], which implies that He(x)dGn(x)+ [1 - Gn(p)]HIe(p) + I

dp nf

dp-

0.

-

Notice thatthis is identicalto the equationfor the symmetricequilibriumin the case of two firms. Hence the solution must be + QtlfIlH(P) I1 1 if p = r

=

Gn(p)

if P

E [Pm, r),

and thereforeHl(p: GO) = (2ln)[He(pm) + Ht(pm)]. Now Gn(p) is the distributionof the minimumof the list prices of the other (n - 1) firms. Thus Gn(p) = pr[minimumof (n - 1) prices is no greaterthanp] = 1 - pr[all (n - 1) prices are greater than p] =

1

-

Hence 1 - Fn(p) = [1 Fn(p)

_

[1 -

Fn(p)]n-1.

Gn(p)]"11(n 1),

=

1

-

=

1

-

=

1-

[1

{1

and thus 1(n-

1)

-

Gn(P)]

-

[1 + Qtl1le(p)]}

10

1)

[-Qt~/If(p)]lI(n-1)

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383

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