Price Equilibria in Pure Strategies for Homogeneous Oligopoly

PRICE EQUILIBRIA

1N P U R E STRATEGIES

FOR HOMOGENEOUS OLIGOPOLY

BETH ALLEN Deyartment of Economics

University of Pennsylvania 3718 Locust Walk

Philadelphia, PA 191 04-6297

JACQUES-FRAN~OIS THISSE CORE, 34 voie d u Roman Pays,

7348 Lnuvain-la-Neuve, BELGlUM and

University of Puris I-Sorbonne 12 Place du Pantheon

75231 Paris. Cedex 0.5 FRANCE

For a homogeneous product oligopoly market, possibilities for pure strategy Nash equilibria in prices are studied. Consumers, who each nunstrategically purchase one unif up to a common resevvation price, are hypothesized to be more concerned with large price differcnces (and therefore buy from the cheapest firm) than slightly different prices. For the duopoly case, exisfence, ulziqueness, and characterization results are provided. Liriear examples are given with 2 and n firms.

1. INTRODUCTION

The hypothesis that firms in an oligopoly market pick prices strategi- cally appeals to us for several reasons. First, it provides a direct game theoretic explanation of price formation; market equilibrium prices arise as the endogenous profit-maximizing decisions of firms. More- over, for many markets, the price-setting Bertrand model seems more realistic than Cournot quantity competition. Last, the use of noncoop- erative pricing models for the analysis of homogeneous product oli-

This research was supported by C.I.M. (Belgium), CORE, the U.S. National Science Foundation, the Fishman-Davidson Center for the Study of the Service Sector at the Wharton School of the University of Pennsylvania, and the Deutsche Forschungsge- meinschaft through the Gottfried-Wilhelm-Leibniz-F6rderpreis. Paul Milgrom, Dan Spulber, and an anonymous referee provided very helpful comments. We retain respon- sibility for any remaining errors.

Q 1992 The Massachusetts Jnstitute of Technology lournal of Economics & Management Strategy, Volume 1, Number I , Spnng 1YY2

64 Journal of Ecorionzics & Management Strategy

gopolies facilitates their comparison with markets for differentiated products, where price competition has been examined extensively in the recent literature.

However, the price approach suffers from a well-known draw- back. As Bertrand (1883) observed, price-setting firms undercut each other. As a consequence, firms’ profits (or the payoff functions of the noncooperative game among price-setting firms) exhibit discontinu- ities. If a single firm can serve the entire market at constant marginal cost, firms undercut until profits are zero (or price equals marginal cost). This prediction of perfectly competitive outcomes even with only two firms strikes us as implausible.

In this paper, we examine the possibilities for pure strategy equi- libria in prices that need not be perfectly competitive for a modified Bertrand game. In particular, we derive existence and uniqueness results, as well as a number of features of these equilibria. We analyze a noncooperative game played among identical firms producing a single homogeneous commodity at constant (possibly zero) marginal cost. The capacity of each firm is sufficient to serve the entire market, which is composed of many consumers who each demand one unit at any price up to a (uniform) reservation price.

Our games possess pure strategy Nash equilibria because we envision consumers who do not necessarily react to small price differ- ences, so that firms’ demands and, hence, profits are continuous functions of prices. Discontinuous market behavior is difficult to imag- ine and impossible to verify empirically from any finite data set. The continuity hypothesis yields continuous payoff functions. Then we derive sufficient conditions for the existence of a (symmetric) Nash pricing equilibrium in pure strategies. In this way, a more accurate description of consumer decision-making permits characterization of a more realistic market equilibrium.

If firms’ prices are very close, many customers do not necessarily purchase from the lowest-priced firm; instead, a large market segment is divided among similarly priced firms. When firms offer very distinct prices, more consumers will appreciate the difference, thereby choos- ing to purchase from the lowest-priced firm. Thus, in the duopoly case, consumers are divided into a concerned group, which necessarily pa- tronizes the cheaper firm, and an unconcerned group, which is split between the firms. The relative sizes of these groups depend continu- ously on the magnitude of price differences. This continuity is based on a distribution of just-noticeable thresholds of price differences. See Luce (1956) for an axiomatic treatment of individuals with just- noticeable thresholds.

Surprisingly, pure strategy Nash price equilibria in static noncoop- erative games played among firms selling a homogeneous commodity

Price Equilibria for Homogeiieous Oligopoly 65

have not been analyzed in the literature.’ The only exception of which we are aware is a very recent paper by Kamien, Li, and Samet (1989), who tackle the problem of subcontracting by identical firms with strictly convex costs to obtain subgame perfect Nash equilibrium in a two-stage game involving first prices and then division of production.

The remainder of this paper is organized as follows: Section 2 presents our model and suggests some alternative interpretations. A linear example is examined in Section 3 . Sections 4-6 then proceed to characterize, for our duopoly games, the possible pure strategy equi- libria, to show that they are unique, and to derive sufficient condi- tions for the existence of such equilibria. Returning to the previous linear example, we extend the model to the n firm case in which consumers compare a given firm’s price to the average market price. Section 8 concludes. An appendix contains proofs.

2. A MODEL IN WHICH P R I C E D I F F E R E N C E S M A T T E R

Consider two firms (denoted 1 and 2) selling a single homogeneous commodity. Either production has already occurred (so that product specific production costs are sunk) or production takes place at con- stant and equal marginal costs. In the latter case, we write all prices net of these marginal (or average) costs. Let p , 2 0 and pl z 0 denote the prices chosen by firms i and j , respectively.

A continuum of individually negligible identical consumers par- ticipate in this market. Each consumer buys one unit of the product, provided that it can be purchased at a price not exceeding the com- mon reservation price p . These consumers do not behave strategically.

2.1 CONSUMER BEHAVIOR

For an individual consumer, let 8 E 0 denote the customer’s thresh- old above which price differences matter. Thus, if p f < p I and ( p , - pi)@ > 8, then the consumer with parameter 8 necessarily purchases one unit (assuming that p I 5 p ) from the lower-priced firm i, while if (p , -

1. Some work has addressed equilibrium prices in models focusing on other issues, such as limited ability to visit firms (Peters, 1984), demand uncertainty, costless search, and potential entrants (Bryant, 1980), alternating price choices in a dynamic context (Maskin and Tirole, 1988), heterogeneous search costs and asymmetric equilibrium prices (Salop and Stiglitz, 1977), and endogenous price dispersion with costly advertis- ing and search (Butters, 1977). Recent work of E. Dierker (1991), Caplin and Nalebuff (1991), and H. Dierker (1989) demonstrates that certain distributional assumptions suffice to guarantee the existence of pure strategy equilibria in prices for models with product differentiation. Note that the paper by E. Dierker and that of Caplin and Nalebuff use very similar aggregation techniques, whereas H. Dierker employs quite different methods.

66 Journal of Economics & Managemelit Strategy

pJ/P 5 8, the customer may buy from either store. Suppose that 0 =

[0,1], and let g be the density function defined on 0. While at this point we do not assume that g is continuous or smooth, the existence of a density for the distribution of 8 implies that its distribution is atomless.

PRoPosiTioN 1 : I f the dzstributzon of 0 has a density function g: f O , Z l + IX,, then the function f: [-Z,Z] + [-1,Zl defined b y f ( x ) = I” g(B)d8for x E f0,Zl and f (x ) = Jo g(8)dO = -J-” g(8)dO for x E IvZ,Ol satisfies (a) continuity, (b) increising (weak) mtnotonicity, and (c) asymmetry CfC-x)

Continuity (a) is used in our demonstration that there is a nonco- operative equilibrium in pure strategies. Conditions (b) and (c) to- gether imply that f is sign preserving: f(0) = 0, f(x) > 0 for x > 0 and f(x) < 0 for x < 0. Let G: [0,1] + [0,1] be the cumulative distribution function for 8, where G ( x ) = Pg(8)dO. Thenf(x) = G ( x ) if x 2 0 andf(x) = - G ( - x ) if x < 0. Consequehtly, any assumptions to be imposed onf have natural counterparts in terms of the cumulative distribution func- tion G, while assumptions on the derivative off correspond to condi- tions on the density function g.

The previous model can hardly be interpreted in terms of prod- uct differentiation in which a nonnegligible subset of consumers do not consider the two products to be equivalent. Indeed, if a con- sumer with parameter 8 prefers firm 1’s product to that of firm j, then he will purchase from firm i whenever p , 5 p, + 8. Conversely, if firm J‘S product is preferred, purchase occurs whenever p, 5 pz + 8. These conditions are not equivalent to lpl - pll I 8 unless, for each value of 8, identical masses of consumers belong to the i-preferring group as to the j-preferring group. In other words, exactly half of the consumers are in a vertically differentiated world with i as the supe- rior product, while the remaining half have exactly diametrically opposed views. Moreover, the preference parameters must follow exactly the same distribution law for each group. Each consumer must have an ”antimatter clone.” Admittedly, this structure imposes a weird combination of aspects of both horizontal and vertical prod- uct differentiation. However, with homogeneous goods, our specifi- cation is natural.

Remark. The model can be extended to incorporate distributions of reservation prices so that f~ is replaced by a parameter y distributed over some compact interval centered at p, where a consumer with parameter y is willing to buy one unit at a price not exceeding y. In this case, the compact strategy set [O,p] would be replaced by [O,;],

= -f(x)).

Price Equilibria for Homogeneous Oligopoly 67

where support of the distribution of y}.

is defined by 7 = max { p E IR, 1 p belongs to the (compact)

2.2 THE GAME BETWEEN FIRMS

In the market, each firm i chooses a price p I L 0 at which to sell its product. Thus, we consider a noncooperative game played by the two firms in which prices are the strategic variables. Payoff functions are defined by the firms’ total revenues from sales of the product. In view of Section 2.1, we envision a scenario in which some consumers are concerned with the price difference and buy from the lower-priced firm, while others do not. Thus, we have two groups of variable size, where the latter group is shared equally by the firms. If p , = p,, then exactly half of all consumers purchase from each firm. By definition, f: [-1,1] + [-1,1] describes the ”size” of the group of concerned con- sumers. In particular, if p , < pi, then the (positive) fraction f [ ( p , - p l ) l p ] of consumers necessarily purchase from firm i, while the remainder 1 - f [ ( p j - p , ) / p ] are divided equally between the two firms. Hence, in this case, firm i sells (assuming pt < pl)

while the higher-priced firm j has sales given by

On the other hand, if p1 = pz = p , then

1 S h P ) = S,(P,P) = 2

Under Proposition 1, we can simplify the firms’ sales functions to

for all pi and p,. Note that this formulation avoids the necessity of examining different formulae for the cases p , > pl and p1 > p,.

This then defines symmetric payoff functions

T(Pl .P , ) = P1 ‘ SI(Pi.P,) for the noncooperative game played by the firms. We are interested in the Nash equilibria of this game, that is, a pair p ; and p z of prices such that

~,( , (p: ,pr ) 2 T,(pi,pr) for all p , 2 0, i, j = 1, 2, and i # j .

68 Journal of Economics & Manageiizeizt Strategy

However, observe that prices greater than p need never be chosen by firms,2 because they yield zero profits as no consumer will ever pur- chase the product at prices exceeding the (uniform) reservation price p . Consequently, we may restrict firms’ strategies to the compact inter- vals [O,p].

2.3 SOME REMARKS ABOUT THE MODEL

Our research strategy is to focus on distributions of consumers’ thresh- olds of just-noticeable price differences. However, our results are de- rived in terms of the function f, which can be considered the reduced form of alternative microeconomic models. The logit model provides an example of admissible specifications. The same is true more gener- ally when firms believe that consumers make their choices according to some probabilistic rule based on price differences. Anderson et al. (1989) provide microeconomic underpinnings for these models. Such an approach is taken by Perloff and Salop (1985) in the context of differentiated products, but the same kind of idea can be applied to seemingly homogeneous products (i.e., identical products sold by distinctive sellers). For a detailed survey of this family of models, see McFadden (1981).3

An additional interpretation is based on the idea that individuals differ in their tendencies to notice and remember price differences. If some consumers are aware of small price differences while others are not, and if these informed consumers necessarily purchase from the cheaper firm while exactly half of the group of uninformed house- holds visit each firm, then the aggregate demand functions facing each firm are equivalent to the sales functions defined in Section 2 in terms of our f function. Continuity results from endogenizing the sizes of the groups of informed and uninformed consumers. This continuity could also be rationalized by dispersed incomes or diverse search costs.4

2. Technically, we claim that prices above p never belong to a firm’s best reply, except for some inessential multiple best replies in which both firms sell zero. In this situation, zero and all prices strictly above p lead to the same (zero) profits.

3. See also the recent article by E. Dierker (1991), which deals explicitly with price competition in discrete choice models with differentiated products. Also in a model with differentiated products, Bester (1990) develops conditions similar to (1) and (3) of Proposition 5 on the distributions of consumers’ preferences-in terms of the addi- tional willingness to pay for one product rather than the other-which imply existence and uniqueness of pure strategy pricing equilibria.

4. As suggested by Paul Klemperer, another possible formal interpretation of the aggregate demand in our duopoly model is to view the firms as being located at the endpoints of an interval while symmetrically distributed consumers face a continuous

Price Equilibria for Homogeneous Oligopoly 69

3. AN EXAMPLE

To illustrate our model, we take the function f describing consumers' concern for price differences to be piecewise linear. Specifically, let f : [-1,1] .+ [-1,1] be defined, for a > 0, by

f(x) = f( - T'i

P ) = ax if lP/ -

In this case, firms' sales functions become

if pt I p, - pIa

S,(P,JJ,) = 0 if p, + pla 5 p,.

This is depicted in Figure 1. Note that the parameter a indicates the tendency for consumers

to notice price differences. Moreover, in this example, the fraction of concerned consumers is a linear function of price differences, provid- ing that some but not all consumers are concerned.

We can easily compute the Nash equilibrium in closed form. Notice first that if only one firm serves the entire market, the other firm necessarily earns zero profits. If this were to occur, the firm selling zero could do strictly better by selling, say, to half of the mar- ket at its competitor's (strictly positive) price. Hence, we only need to consider prices where sales are a linear and increasing function of price differences. In this region, first-order conditions for a Nash equi- librium are as follows, for i, j = 1,2 and i # j :

Setting FJ: = p; = p" and solving for a symmetric equilibrium yields p* = p/a, where p" must not exceed p , that is,

p" = min {P , pic.}.

transportation cost function. (Note that, for some specifications of the transportation cost functions, firms choose to locate in equilibrium at the endpoints of the market in order to relax price competition.)

70 Journal of Economics & Management Strategy

1

0,5

0

ts i

P J

I! +p/cx J

FIGURE 1 . F I R M i’s SALES AS A FUNCTION OF p , FOR p , GIVEN.

(Recall that we have restricted firms’ strategies to [O,?] because con- sumers’ total demand would be zero at all prices above p . ) In fact, these necessary first-order conditions are also sufficient because firms‘ profit functions are strictly concave over the region where both firms receive positive profits.

4. CHARACTERIZATION OF EQUILIBRIA

Here we consider the issue of whether equilibrium prices are charac- terized by competitive or monopoly features. In particular, we find necessary and sufficient conditions for p; = p;i‘ = p and for p; = p; = 0. We assume that payoff functions are continuously differentiable.

PROPOSITION 2: If there exists a symmetric equilibrium, then p: = p: = min {p/f’(O), p } . In particular, p: = p; = p whenever f ’ (0) 5 1. On the other hand, as f’(0) + m, p: -+ 0 and p; + 0.

We have shown that firms set their prices equal to the reserva- tion price of consumers when consumers do not react strongly to small price differences. In this case, each firm serves exactly half of the market, and this sharing would be unaffected by small price changes-in particular, by an attempt to undercut slightly the other firm. Thus, each firm essentially acts as a monopolist. One may argue that our model is most convincing for commodity industries with f’(0) > 1, where our model may be viewed as a slight perturbation of the standard Bertrand model. However, we include the f ’(0) I: 1 case in

Price Equilibria for Homogeneous Oligopoly 71

order to state necessary and sufficient conditions for monopoly pric- ing to arise in our model.

At the other extreme, competitive pricing approximately pre- vails when consumers react strongly to small price differences. In this case, at equal positive prices, undercutting is extremely effective; a tiny price reduction would capture a substantial portion of the rival firm’s customers. The solution thus almost resembles the classical Bertrand equilibrium.

Between these two polar cases, our game displays a genuinely intermediate solution. Firms choose prices so as to equilibrate the marginal gain caused by the ”revenue per customer” factor and the margnal loss caused by the ”number of customers served” factor. A lower-priced firm will increase its price only if the gain in revenue per customer outweighs the loss of customers who are less concerned by the (smaller) price difference. A higher-priced firm will decrease its price only if, at some lower price, losses on the sale of each unit can be balanced by additional consumers attracted away from the other firm. When firms pick equal prices in equilibrium, this means that not only does each sell to exactly half of the consumer population but also that no other price is advantageous on balance, given consumers’ concern for price deviations. The magnitude of f ’ (0) quantifies this concern. Its inverse can be interpreted as a measure of local monopoly power, that is, the responsiveness of an individual firm’s demand to small price variations.

5. U N I Q U E N E S S OF E Q U I L I B R I A

Next we state that our duopoly game cannot possess multiple Nash equilibria in pure strategies. As our argument uses results from the preceding section, we must continue to impose continuous differ- entiability. Moreover, smoothness plays an additional direct role in that our proof relies on the examination of first-order conditions. The manipulations in our proof are based on the characterizations of the f function provided by Proposition 1. Recall that Proposition 2 asserts the uniqueness of symmetric equilibria. Properties of the f function intuitively relate to symmetry of the resulting games, which thus can frequently be expected to have only symmetric equilibria.

PROPOSITION 3: The noncooperative duopoly game has at most one Nash equilibrium in pure strategies.

Having established the uniqueness of Nash equilibrium in pure strategies, we may now consider comparative statics. By Proposition 2, as the reservation price p increases, Nash equilibrium prices p“ and

72 @trnal of Economics & Management Strategy

firms’ profits increase. The intuition that increases in consumers’ res- ervation prices increase equilibrium prices and profits is clear, given that each firm serves exactly one-half of the market, so that n-,(py,p:) = 7r2(p:,p:) = p“/2. Furthermore, as f(0) decreases, equilibrium prices p* and equilibrium profits do not decrease, and, in fact, they strictly increase unless p“ = p (i.e., whenf’(0) 5 1). Our result forf’(0) is more interesting because it implies that only the behavior of the f function near zero matters. However, the fact that increasing concern for price differences by consumers causes equilibrium prices and profits to be closer to competitive levels is not surprising. Of course, these claims make sense only if the parameter changes do not destroy the exis- tence of a pure strategy equilibrium.

Next we consider the possibility of mixed strategy equilibria, even though we recognize that they are less tractable than equilibria restricted to pure strategies. A fairly strong condition on f guarantees that there does not exist any equilibrium in mixed strategies. Observe that this result depends on the shape of the demand function rather than the underlying model that generates the demand. In particular, threshold price differences need not be hypothesized for our results.

PROPOSITION 4: Iffur all x E [-1,Zl, we have

f f ( x P 2 (1 + f(M))f”fx),

then the duopoly game has no equilibrium involving genuinely mixed strafe- gies .

Note that the linear example described in Section 3, as well as the logit specification, satisfies the above condition.

A further remark is that our general model admits no symmetric equilibria in genuinely mixed strategies. This follows from the fact that firms’ demands necessarily sum to one.

6. EXISTENCE OF P U R E STRATEGY E Q U I L I B R I A

In this section, we provide three different sufficient conditions, each of which guarantees that our duopoly game possesses a Nash equilibrium in pure strategies. For convenience, we hypothesize sufficient differen- tiability to utilize first- and second-order conditions for maximization.

PROPOSITION 5: Assume that the function f is twice continuously differ- entiable and satisfies one of the fulluwing conditions:

(1) f”(x) 5 2f’(x) fur all x E [ - 1 , I I (2) f ’ ( x ) 5 1 fur all x E f-1,ZI (3) ff’Cx,P 2 f” (x ) for all x E f-1,ZI.

Price Equilibria for Homogeneous Oligopoly 73

Then the duopoly pricing game has a Nash equilibrium in pure strategies.

Notice that these conditions are genuinely distinct. None of them implies or is implied by any of the others.

An interpretation of the first hypothesis is that the functions f do not depart too much from linearity. Hence, we have given a general- ization of the existence result derived by calculation for the example in Section 3. Condition (1) can be restated in terms of the requirement that the cumulative distribution function G must be approximately linear-more precisely, IG"(x)l I 2G'(x). In the proof, we appeal to the standard condition that payoff functions be quasiconcave by showing that they are concave. This guarantees convex-valuedness of the best reply correspondence, so that a well-known fixed point argument applies.

Our proof under condition (2) shows that when consumers are not too sensitive to price differences [recall that f'(x) 2 0 for all x by Proposition 1, so that the hypothesis of Proposition 5 (2) requires 0 5

f'(x) 5 1 for all x E [-1,1]], then there is always an equilibrium with monopoly pricing. Each firm charges p . Intuitively, firms do not re- duce prices because doing so would not capture sufficiently many new consumers to compensate for the price decrease on every unit sold. Note that condition (2) is equivalent to the hypothesis that 8 is uniformly distributed. In fact, (2) requires G'(x) 5 1, and because G' equals the density g defined on [0,1], we must have g(x) = 1 every- where for the total mass to integrate to one.

The third condition resembles the first in that both are clearly satisfied by linear f functions. However, whether the less demanding upper bound is given by (1) or (3) at some x E [ - 1,l J depends on the relation between 2f'(x) and Cf'(x)]'. Because we impose no hypotheses other than nonnegativity on the first derivative off, there are clearly examples that show that (1) and (3) are not directly comparable.

The previous three conditions are similar in that each gives a sufficient condition to prevent a firm from jumping down to a price lower than that charged by its competitor. Hence, best reply corre- spondences cannot jump down at all or cannot jump down over the diagonal. Recall that lack of downward jumps was the key ingredient in the McManus (1962, 1964) and Roberts and Sonnenschein (1976) proof that a symmetric Cournot oligopoly game has an equilibrium in pure strategies. In contrast to quantity-setting games, such down- ward jumps cannot be ruled out, in general, for price-setting games. Indeed, all selections from the Bertrand best reply correspondences (even for our games with continuous payoff functions) may jump down but not up. Absence of downward jump suffices (with compact-

74 Journal of Economics & Management Strategy

ness) to establish a fixed point theorem, but simple examples show that a downward jump can negate the possibility for a fixed point. Thus, the fundamental nature of the jumps constitutes an inherent major difference between the pricing-setting and quantity-setting oli- gopoly models. With price strategies, jumps down over the diagonal occur when a firm suddenly finds it profitable to undercut the other firm’s price. No such phenomenon occurs in quantity-setting games, thus facilitating the existence of pure strategy Cournot equilibria. A related difference between the two models is based on the observa- tion that total market quantity matters for Cournot payoffs, while price differences determine Bertrand payoffs. Note that these funda- mental differences between the models do not depend on the pres- ence of discontinuities in Bertrand payoffs, but rather on the nature of the competition itself. In other words, Bertrand discontinuities are not the sole reason for the nonexistence problem in pure strategy pricing games.

Yet another sufficient condition for the existence of Nash equilib- rium in pure strategies is to require the density g of 0 to be continu- ously differentiable, decreasing, and log-concave5 over [0,1]. We then know from Theorem 1 of Caplin and Nalebuff (1991) that each firm’s profit function is quasiconcave wherever the first-order conditions are satisfied. For example, this sufficient condition holds for any normal density centered at 0 that has been truncated over [0,1] and rescaled. Different values of the variance correspond to various dispersion pat- terns of distributions of consumers’ sensitivities to price differences.

Reniark. Iff’(0) > 0, then there exists a unique local Nash equilibrium in pure strategies that is symmetric and given by min { F / f ’ ( O ) , p } . In other words, even if none of the conditions stated in Proposition 5 hold, our games necessarily have local Nash equilibrium under only very mild conditions.6

7. AN AVERAGE PRICE COMPARISON EXAMPLE WITH It FIRMS

Perhaps the simplest way to incorporate dependence on n prices si- multaneously while retaining symmetry among firms is to model con- sumers as focusing on the difference between a given firm’s price and

5. In fact, (-$)-concavity would suffice, as either condition guarantees that f has an inflection point at x = 0 and that firms’ sales function? are (-1)-concave in their own prices.

6 . For details, see Allen and Thisse (1990).

Price Equilibria for Homogeneotis Oligopoly 75

the average price offered in the market.: For instance, if consumers know only a single statistic-the mean-of the price distribution, they obviously have no other data on which to base their decision. Let f: [-1,1] + [-1/1] be the linear function of Section 3 except that for firm j , the argument off is now given by (paz, - pl ) /p , where

Observe that linearity guarantees that total sales always exactly ex- haust the market when the slope a 5 1. If a > 1, for some price configurations, total sales may not equal one.8 However, the problem occurs only when some store either sells nothing or sells to all consum- ers by the above formula. Such a configuration cannot be an equilib- rium. To economize on notation, we avoid giving the formulae for adjusted sales in this situation.

Then firms' revenues (or payoffs) become, forj = 1, . . . ,n,

Assuming an interior solution, we can characterize Nash equilibria by their fi rst-ord er conditions

where P:~, = 2 p:/n and f ( x ) = ax with IpaZ, - p,l 5 ?/a. Once more, we solve for a sy&netric equilibrium and calculate

provided that p* 5 p . Notice that p* is a strictly decreasing function of n. This agrees

with the common economic intuition that increasing the number of competitors in a market should reduce equilibrium prices. As n be- comes large, p" approaches ?Icy, the duopoly equilibrium derived in Section 3. However, note that for n = 2, p* is twice as great as our earlier

7. For differentiated products with linear demands, Shubik with Levitan (1980, pp. 89-92) has calculated Bertrand and Cournot equilibria in a market in which consumers compare a firm's price to the average price.

8. The problem was pointed out by Suzanne Scotchmcr when this article was pre- sented a t the Northwestern Suininer Workshop on Theoretical Industrical Organiza- tion. See also her note (Scotchmer, 1986).

76 Iournal of Economics & Management Strategy

duopoly price. This illustrates that different underlying choices for the function f can give rise to vastly different equilibrium outcomes.

If in the formulae for firm jrs,~ales and profits, we replace the average pav = 5 p , / n by the average &j(n-l) of all other firms' prices

we would obtain a symmetric equilibrium p" = pia (p* c p) that does not depend on n.

1=1 I t 1

8. CONCLUDING REMARKS

If consumers do not react perfectly to small price differences, then the market equilibrium departs from the classical Bertrand or perfectly competitive outcome. This grants oligopolistic firms some market power and can be expected to lead to higher profits. Such behavior may be more likely to arise if consumers pay less attention to slight price differences as their incomes increase. Our model predicts that firms would take advantage of slight deviations from perfect rationality. As shown by Luce (1956), this departure from the standard model of ra- tional economic behavior also can be based on psychological princi- ples. In this respect, our model can be viewed as an attempt to integrate these approaches within a model of strategic behavior by firms. Alterna- tively, the economic concepts of switching costs or transactions costs possibly could serve as an intuitive foundation for our model.

We believe that in many markets, firms strategically set prices that generate positive profit levels. Such strategic behavior arises from managers' realizations that small price changes need not result in the capture or loss of the entire market. Implicitly, prices are set by firms in consideration of the tradeoff between the size of their clienteles (or market shares) and their price levels (because average revenue per customer depends on the prices chosen). Survival pressures encour- age managers to adopt such policies, which can substitute for the more costly alternatives of product differentiation, advertising, etc. Such strategic decision making leads to positive profits and pricing above cost even in markets that otherwise appear perfectly competi- tive. In other words, potential undercutting by rival firms is less attrac- tive. Its extent is limited by the unconcern of at least some consumers about small price differences.

9. APPENDIX

Proof of Proposition 1. Observe that pg(O)dO and -J-"g(O)d@ are con- tinuous and nondecreasing in x. Asyhmetry follow: from the defini- tion. 0

Price Equilibria for Hornogeizeous Oligopoly 77

Proof of Proposition 2. Examine the first-order conditions for an inte- rior Nash equilibrium:

for i, j = 1,2 and i # j . Setting p: = p ; = p* yields

1 P* 3 + f(0) - -=- f’(O)] = 0. P Recalling that f(0) = 0 (by Proposition l), we obtain p* = p/f’(O), which gives the desired result.

If the symmetric Nash equilibrium (p*,p*) occurs at the boundary (i.e., p* = 0 or p“ = B ) , then the above first-order conditions must be written as inequalities. However, we only need to consider strict in- equalities because analysis of the equality case follows our earlier reasoning. Accordingly, we have either

1 P“ - [I + f(0) - : f’(O)] > 0 for p* = P 2 P

or

However, as f(0) = 0 and f’(0) is bounded, the second inequality can never hold at p* = 0. Hence the only possible boundary equilibrium is p* = P , which requires f’(0) < 1.

As f ’(0) + m, the first part of the above argument applies and p* + 0. 0

Proof of Proposition 3 . By Proposition 2, there is at most one symmet- ric equilibrium. Hence, it suffices to demonstrate that asymmetric Nash equilibria are impossible in our (symmetric) duopoly game.

Once again, consider the first-order (necessary) conditions for an interior Nash equilibrium:

d r , 1 P“ - P* r j : I P; - P: - (p:,p;) = - [1 + f( dp, 2 Y P P ) - -- f ( 7- )] = 0

and

Use property (c) of Proposition 1 [asymmetry: f ( -x) = -f(x), which im- plies?(-~) = f’(x)] to rearrange these equations to the following form:

78 Journal of Economics & Management Strategy

f ( P: - - p; ) = - l + : r ( m ) P P P

Upon adding the preceding two equations, we obtain

which is equivalent to

- 2 = ( - P: - P: ) f ' ( P: - P1' )if( P: - P: ). P P F

The first and third terms on the right-hand side are of the same sign (because Proposition 1 implies that f is sign preserving), while the middle term is necessarily positive as f is increasing (again by Proposi- tion 1). This then yields a contradiction unless p ; = pz , that is, unless the equilibrium is symmetric9

It remains yet to consider the possibility of an asymmetric equi- librium in which at least one firm charges 0 or p. We first examine the case in which exactly one firm, say firm 2, charges p ; = P while the other firm charges a (lower) strictly positive price p: E (0,p). In this case, either the previous argument applies if firm 2's first-order condi- tion is satisfied with equality, or firm 2's first-order condition becomes

(Recall that if p; = p, the opposite inequality cannot hold because a decrease in firm 2's price would thus lead to increased profits for firm 2 . ) Upon rearrangement this becomes

Adding this to firm 1's rearranged first-order condition yields

2f( W ) < ( &Sy( P: - p: ), P P P

which is again (providing that f [ ( & - p:)/P] # 0) equivalent to

-2 > ( F )f'( P:+: )/f( p: - P p1' ).

9. If p: = pz , f [ ( p ; ~ p; ) /p] = 0 because f is a sign preserving function. In this case, division by zero in the last rearrangement was invalid as the equation reduced to 0 = 0.

Price Equilibria for Honiogetieous Oligopoly 79

As before, the right-hand side of this inequality is necessarily posi- tive, which gives the desired contradiction.

Finally, we claim that there exists no asymmetric equilibrium in which (exactly) one firm charges 0. Suppose p ; = 0 and p ; > 0. Then firm 1’s profits n1 must equal p: S,(p;,p:) = 0. This cannot be a Nash equilibrium, because whenever firm 1 charges E (0,p3, firm 1 serves at least half of the market at a strictly positive price, thus earning strictly positive profits T~ = . S,(@,,p;) 2 f412 > 0. Hence, such (O,p**) or (p**,0) with p** > 0 cannot form a Nash equilibrium in our game.

Proof of Proposition 4. This condition is equivalent to concavity of the logarithm of firms’ sales functions. Hence, we can appeal to the result of Milgrom and Roberts (1990, Section 4.2) to conclude that there exists a unique equilibrium and that it uses only pure strategies. 0

Proof of Propositio~z 5(1). First note that continuity of payoff func- tions implies, by the Maximum Theorem, that firms’ best reply corre- spondences are nonempty and upper hemicontinuous; recall that we have already argued that best replies can be restricted to the compact interval [O,p]. To obtain convex valuedness, compute the second de- rivative of

0

with respect to pi:

P, - F’l 1 P , - P 1 s - [ - f ( - ) + T y ( 4 ) ] P P P 5 0

whenever the hypothesis of the proposition is satisfied. Hence, there is a fixed point (pT,p;), which is a Nash equilibrium in pure strate- gies. 0

Proof of Proposition 5f2). We want to show that, for all T J ! , T, is a nondecreasing function of p l on [O,p,). Calculating the first derivative of rI with respect to pi yields

If pt E [O,pl), then (pl - p , Y p < 0, so that f [ ( p , - p,)ipl E [-1,0) as a consequence of Proposition 1. Hence, 1 - f[(pi - p , ) /p ] E (1,2] and

by hypothesis, so that d ~ , / + ~ 2 0 as desired. This shows that, for any p , E [O,p], either firm i’s best response set R,(pJ = { p , E [O,p]l~,(p~,p.) 2

80 Journal of Economics & Management Strategy

nL(p:, pJ for all p : E [O,p]> is contained in [p,,pJ, or it is the union of an interval containing p, and a nonempty subset of [ p , , p ] .

Define altered best reply correspondences I?,, R2: [O,p] - [O,p] by RL(p1) = R,(p,) n [p,,p] # 0. The graphs of these altered best reply correspondences stay above the diagonal in [O,p] x [ O , p ] . To obtain a pure strategy equilibrium, it suffices to find a fixed point for this altered game or, alternatively because our games are symmetric, to show that the graph of I?, = R2 intersects the diagonal in [O,p] x [ O , p ] . This follows from the following lemma. In particular, note that ( p , p ) is always a fixed point because R,(p) = R,(p) = { p } by definition.

LEMMA: Let F : fa,bl+ la,bl be a nonempty valued correspondence, where --x < a < b < fw, suck that for all c E fa,b], F(c} [c,bf. Then there is x* E fa,bl suck that x* E Ffx*) .

Proof. By hypothesis F(b) = {b}, and, hence, b E F(b) is a fixed point, as desired. 0

Proof of Proposition 5(3). Condition (3) guarantees the quasi- concavity of each firm’s profit function in its own price at all prices (if any) satisfying the first-order conditions. If no prices satisfy the first- order conditions, the quasiconcavity follows from monotonicity of the profit function; then the best reply correspondence contains only the reservation price p . Our condition is derived from convexity [ ( f ’ (x))* 2

(1 + f(x)) f”(x)/2] of the reciprocal of the demand function. See Ander- son and de Palma (1988).

0

0

REFERENCES

Allen, B. and J.-F. Thisse, 1990, ”Price Equilibria in Pure Strategies for Homogeneous Oligopoly,” CORE Discussion Paper No. 9034, Universiti. Catholique de Louvain, Louvain-la-Neuve, Belgium.

Anderson, S. and A. de Palma, 1988, “Spatial Price Discrimination with Heterogeneous Products,” Rwieio of Economic Studies, 55, 573-592.

~ _ _ and J.-F. Thisse, 1989, ”Demand for Differentiated Products, Discrete Choice Models and the Characteristics Approach,” Rwiew of Economic Studies, 56, 21-35.

Bertrand, J . , 1883, “Review of ’Theorie Mathematique de la Richesse Sociale’ and ‘Re- cherches sur les Principes Mathematiques de la Theorie des Richesses,‘ ” Joimal des Savants, 499-508.

Bester, H., 1990, ”Bertrand Equilibrium in a Differentiated Duopoly,“ Discussion Paper No. A-209, Sonderforschungsbereich 303, University of Bonn, Germany.

Bryant, J., 1980, “Competitive Equilibrium with Price Setting Firms and Stochastic Demand,” International Economic Rez~iezu, 21, 619-626.

Butters, G.R., 1977, “Equilibrium Distributions of Sales and Advertising Prices,” Reuiezu of Economic Studies, 44, 465-491.

Price Equilibria for Homogeneous Oligopoly 81

Caplin, A. and B. Nalebuff, 1991, "Aggregation and Imperfect Competition: On the Existence of Equilibrium," Econometrica, 59, 25-59.

Dierker, E., 1991, "Competition for Customers," in W.A. Barnett et al., eds., Equilibrium Theory and Applications: Proceedings of the Sixth Intermtioizal Symposium in Economic Theory and Econometrics, Cambridge: Cambridge University Press, 383-402.

Dierker, H., 1989, "Existence of Nash Equilibrium in Pure Strategies in an Oligopoly with Price Setting Firms," Department of Economics Working Paper No. 8902, Uni- versity of Vienna.

Kamien, M.I., L. Li, and D. Samet, 1989, "Bertrand Competition with Subcontracting," Rand Journal of Economics, 20, 553-567.

Luce, R.D., 1956, "Semiorders and a Theory of Utility Discrimination," Econoiizetrica, 24, 178-191.

Maskin, E. and J. Tirole, 1988, "A Theory o f Dynamic Oligopoly 11: Price Competition, Kinked Demand Curves, and Edgeworth Cycles," Econometrica, 56, 571-599.

McFadden, D., 1981, "Econometric Models of Probabalistic Choice," in C.F. Manski and D. McFadden, eds., Structural Analysis of Discrete Data with Economefric Applications, Cambridge, MA: MIT Press, 198-272.

McManus, M., 1962, "Numbers and Size in Cournot Oligopoly," Yorkshire Bulletin of Social a i d Economic Research, 14, 14-22.

-, 1964, "Equilibrium, Number and Size in Cournot Oligopoly," Yorkshire Bulletin of Social a i d Ecuizuinic Resenrch, 16, 68-75.

Milgrom, l? and J. Roberts, 1990, "Rationalizability, Learning and Equilibrium in Games with Strategic Complementarities," Econometricn, 58, 1255-1277.

Perloff, J.M. and S.C. Salop, 1985, "Equilibrium with Product Differentiation," Review of Economic Studies, 52, 107-120.

Peters, M., 1984, "Bcrtrand Equilibrium with Capacity Constraints and Restricted Mobil- ity," Econornetrica, 52, 1117-1127.

Roberts, J. and H. Sonnenschein, 1976, "On the Existence of Cournot Equilibrium without Concave Profit Functions," Journal of Economic Theory, 13, 112-117.

Salop, S. and J. Stiglitz, 1977, "Bargains and Ripoffs: A Model of Monopolistically Competitive Price Dispersion," Review of Economic Studies, 44, 493-510.

Scotchmer, S., 1986, "Market Share Inertia with More than Two Firms: An Existence Problem," Economics Letters, 21, 77-79; also erratum, 1986, 22, 105.

Shubik, R., with R. Levitan, 1980, Market Structure and Behavior, Cambridge, MA: Har- vard University Press.