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THE FOERDER INSTITUTE FOR ECONOMIC RESEARC_IfI.3 TEL-AVIV UNIVERSITY

RAMAT AVIV ISRAEL

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111P071 nutTr13t31m ley ASYMMETRIC DYNAMIC PRICE COMPETITION WITH PRICE INERTIA

by

Arthur Fishman

Working Paper No.6-89

Revised February, 1 9 8 9

This research was supported by funds granted to the Foerder Institute of Economic Research by the JOHN RAUCH FUND

, FOERDER INSTITUTE FOR ECONOMIC RESEARCH Faculty of Social Sciences, Tel-Aviv University, Ramat Aviv, Israel. ABSTRACT

This paper presents a model of dynamic price duopoly with short-term price inertia. A perturbation of consumer behavior, specifying that a consumer may purchase from the higher priced seller with positive probability, endows each duopolist with an arbitrarily small degree of monopoly power. It is shown that the slightest degree of asymmetry between the firms with respect to this monopoly power unambiguously identifies a firm which appropriates almost all the surplus from those consumers over whom the firms compete. This outcome is the unique Pareto dominant equilibrium if firms are restricted to state dependent strategies. Introduction

Price inertia plays an important role in many economic models.

Thus, for example, the assumption that prices and wages may only adjust slowly to aggregate disturbances is a cornerstore of Keynesian analysis.

Indeed, as Anderson (1985) has argued persuasively, the notion that prices are "sticky" is inherent in the Bertrand model of price competition. Recall how the Bertrand argument eliminates any but the zero profit price: If any firm charges a higher price a competitor may profitably undercut that price and capture its rival's. customers. If the firm could respond to a price cuts instantaneously, however, it could match the defector's price cut without losing sales, a response which if anticipated removes the incentive to undercut. Price inertia also lies at the heart of the recent theory of contestable markets (Baumol, Panzar and Willig, 1982). There it is argued that a monopoly may be prevented from charging any but the zero profit price for fear that potential entrants are ready to capture the incumbent's market by undercutting its price at the first opportunity. Obviously this argument has no bite if the incumbent is able to quickly match a price challenge.

Recently, Maskin and Tirole, henceforth M-T, (1985, 1988 (ii); see also Maskin and Tirole, 1988 (i)) have presented a formal analysis of price inertia in the context of a dynamic oligopoly. The firms' post prices in alternate periods and each firm is committed to its price for price two periods. Thus, whenever a firm posts a price it observes the a to which its competitor is already bound. Maskin and Tirole introduce state set of strategies, termed Markov, which depend only on the current are presently of the game defined as the prices to which all firms actions which committed. Thus a Markov strategy does not depend on past do not directly affect the current payoffs. This modelling approach serves to dramatically reduce the set of (subgame perfect) equilibria for this game. In particular, M-T show that equilibria are either of the kinked demand or of the "Edgeworth" cycle variety. In either case, the firms' profits are bounded away from zero.

M-T argue that their alternating move structure emerges (in finite time) as the (unique) perfect equilibrium of the following simultaneous move game. The firms move simultaneously and in each period that it is uncommitted a firm may either charge a price, in which case it is committed for two periods, or refrain from charging any price. In the latter event the firm earns zero current profit but remains unconstrained in the following period)

This note presents an analysis of price inertia which is similiar in structure to that of Maskin and Tirole but which diverges from their analysis by introducing the following perturbation of consumer behavior.

Consumers generally buy from the cheapest priced seller but are subject to an arbitrarily small probability of buying from the higher priced 2 seller. This probability may be interpreted as an occasional "urgent" need for the product which precludes selective buying. Buyers of the former type are referred to as "discriminate", of the second type as

1 M-T refer to this option as the "null action". 2 This interpretation is also found in Sobel (1984). -3-

"indiscriminate". There are two infinitely lived sellers of a

homogeneous product who compete in prices in continuous time and there

is a constant flow of customers into the market. Observe that the

presence of indiscriminate consumers endows the firms with a limited

(perhaps arbitrarily small) degree of monopoly power.

The commitment structure is structured along the lines of M-T.

Specifically, whenever a firm "announces" a price it is committed to

that price for a period of positive duration, A > 0, which may be

arbitrarily short. This assumption is likely to hold for firms

competing in prices by mailing price lists or catalogues or who are

pressured by their retailers not to change prices too quickly (M-T, 1988

(i), p.565, footnote 12). An uncommitted firm may post a new price at

any time. Furthermore, whenever it is uncommitted, the firm may 'stay out of the market' in which case it makes no sales until the time a new price is announced. We also follow Maskin and Tirole in restricting the players to Markov strategies. However, it must be emphasized that an equilibrium in Markov strategies is required to be immune to deviation to any strategy, not necessarily Markov. See M-T, 1988(i), for a more detailed motivation and discussion of the Markov assumption.

In this framework, a striking result emerges: The slightest asymmetry between the firms with respect to their limited monopoly power

(i.e., their respective "shares" of indiscriminate consumers) ensures a unique Pareto dominating (subgame perfect) equilibrium in (pure) Markov strategies. In this equilibrium the firm whose share of indiscriminate customers is smaller consistently "captures" the entire market of -4

discriminate customers and earns profits in the neighborhood of the monopoly level.

The intuition behind this result is the following. First, observe that the first firm to post a price is at a disadvantage; its uncommitted competitor may immediately undercut its price and retain this advantage for the duration of the "commitment" period, A. Thus the strategic interaction between the firms is interpreted as concession game (see for example Nalebuff and Riley (1985)) in which discriminate buyers are the prize. In fact, it is a repeated game of concession which recommences whenever commitments expire. Also observe that a firm is always guaranteed a minimum, strictly positive level of profits by selling to its share of indiscriminate consumers at the monopoly price, a profit which is foregone as long as it remains uncommitted to any price. In a natural sense, then, the firm with less monopoly power enjoys a strategic advantage, deriving from the fact that its opportunity cost of postponing a price commitment is smaller than its rival's. Let us therefore refer to this firm as the "strong" firm and to its rival as the "weak" firm.

It is this strategic advantage which enables the strong firm to consistently underprice its rival and enjoy a monopoly status vis-a-vis indiscriminate consumers. No corresponding equilibrium exists in which the weak firm makes sales to discriminate consumers. There does, however, exist a second equilibrium in pure strategies in which the stronger seller consistently concedes to its opponent. In this equilibrium, however, the strong firm continues to underprice its rival , 5

so that the former's "concession" serves only to diminish its own profit without enhancing that of the weaker seller. Thus this equilibrium is

Pareto dominated by the preceding one.

In contrast to the preceding result, it is well known, e.g.,

Nalebuff and Riley, 1985) that the classical concession game has two equilibria in pure strategies: either player concedes at once and its rival wins the entire surplus. This is the case even if the players are asymmetric, e.g. one player is more impatient or has a greater opportunity cost than the other. This distinction between the classical concession game and the outcome of our price commitment game is due to the following fact. In the price commitment game, the sellers simultaneously engage in a concession game (who commits to a price first?) and a Bertrand price setting game (who charges the lowest price?). Due to the asymmetric monopoly power of the firms, the weak firm is unable to enjoy the fruits of victory upon "winning" the concession game because it is consistently underpriced by its rival in any perfect equilibrium as will be shown. The fact that the weaker player's payoff is not increased by causing the stronger player to concede provides the basis for eliminating equilibria of the second type.

It should be noted that the equilibrium we derive is only an equilibrium: The payoff from unilateral deviation can be made arbitrarily small, but not eliminated entirely. This is because the strong firm strategically postpones announcing its price in order to -6

observe its rival's action and this postponement is associated with some avoidable loss.

The following section presents our formal model and a brief summary concludes the paper.

THE MODEL

Time, denoted by t 0, is contnuous and the horizon is infinite.

There are two sellers, indexed i — 1,2, each of whom costlessly supplies an unlimited quantity of a homogeneous product whenever it is operative in the market. There is a constant flow of consumers into the market at each date t. Each consumer has unit demand up to a reservation price z > 0. The constant instantaneous flow of consumers is denoted /3 > O. A consumer may be of two types. With probability

1 - r, 0 < r < 1, she buys from the lowest priced seller (provided the price does not exceed z) only. Such a consumer is called discriminate. With probability r, she buys from a particular seller only, even if its competitor charges less. A consumer of this type is referred as indiscriminate. We denote by r the proportion of i 3 indiscriminate from firm i, i — 1,2, r1 + r. T consumers who buy r2

3 Alternatively, r may be interpreted as an arbitrarily small measure of irrationality to which buyers are subject. One may also think of r as representing a proportion of buyers who have high search costs and thus stick with one particular seller. See, for example Salop and Stiglitz (1987), Varian (1980). 7

represents the probability that a buyer faces an urgent need for the product and must buy immediately from the most conveniently located firm. We allow r to be arbitrarily small. Furthermore, we assume

firms' monopoly power is asymmetric so that r > r but we that the 1 2' allow r - r to be arbitrarily close to zero; i.e., the asymmetry 1 2 may be arbitrarily small. For example, firm 1 may be slightly more conveniently located.

Whenever a firm posts a price it is committed to that price for a period of duration A > 0. Following the elapse of A, the commitment expires and it must post a new price. At any date at which it is uncommitted, the firm may recommit to any price or remain out of the market. In the latter event, no sales are made until a new price has been posted.

It is assumed that the set of admissible prices includes discrete multiples of an indivisible unit of currency, say a penny, denoted a so that the second highest possible price is z - a, the third highest is z 2a, etc. In what follows it shall be assumed that a is 4 sufficiently small that integer problems may be ignored.

Strategies: Following M-T we restrict the firms to Markav strategies which depend only on data which is directly pertinent to the current payoff. This includes only the present commitment status of both firms and is independent of any other past actions taken by the players. More precisely, a Markov strategy for firm i at date t depends only on the answers to the following questions:

4 This setup follows M-T (1985(ii)), and is necessary to ensure that an optimal reaction to a rival's commitment exists. See M-T, 1988(ii) p.573, fn.4. (a) is i currently committed?

(b) is j 0 i currently committed and if so for how long and to

what price?

If the answer to (a) is in the affirmative, is constrained to

the price it has previously posted. Only pure strategies are

considered. i Let p i's price at t, i — 1,2, and, abusing notation let i p = 0 be the "null action", indicating that i is out of the market t at t. If 0 0 pt < p! or if p! — (k, i monopolizes discriminate

buyers in the market at t, provided that pt :5_z. If p t _— pt 0 46,

the firms share discriminate consumers equally. Denoting the 1 instantaneous profit function of i as H(pi, pi), we have

[(1.-r)/3 4' riPiP if z > pi < pi

or

if z pi and pj =

(ripPi if z pi > pi 0 0 i i j H (P P ) = 1 i j 1-r)fi + rifflp if z >p p 0 0

i 0 if P

1 The subscript is omitted since strategies are assumed to be Markov. Let r be the common discount rate. Then the intertemporal

discounted profit of i is:

i -rt i e H(pi, pj)dt. 0 Let

i -rt N = e zr.fl dt 0

i f°3 -rt dt. 0

N is i's intertemporal profit from selling to its share of

indiscriminate customers only. Mi is i's intertemporal profit if it

enjoys monopoly status with respect to all discriminate consumers in

addition to its share of the indiscriminate.

Equilibrium: The equilibrium concept is epsilon subgame perfection.

pair of Markov strategies forms an epsilon subgame perfect equilibrium

if for any e > 0, however, small, unilateral defection to any strategy,

not necessarily Markov, cannot increase the defector's payoff by more

than E at any subgame. 0

Define: pi (min) = zri/(1-ri). -10-

Firm i is just indifferent between selling to only its indiscriminate customers at the monopoly price or selling to. both its share of i indiscriminate consumers and all discriminate consumers at p (min).

2 Proposition In any equilibrium, II is arbitrarily near i.e., within e of) N2.

2 2 2 Proof: Obviously H >_ N - e. To show that H N + e, it

suffices to show that 2 almost never makes any sales to

discriminate consumers.

Define:

T = ft 0 I Neither firm is committed to any price).

By the Markov assumption, each firm's pure strategy at any t' E T must be constant. It is apparent that no pure strategy equilibrium exists in which both firms announce a price at t' e T. Thus at least 2 one firm takes the "null action" 0 at t' E T. Suppose p at t' 2 2 t' E T. Then, since obviously p p (min), firm 1 charges a price t' 1 2 p', p (min) < p' < p at t E [t', t' + 61(] so as to monopolize t' discriminate consumers for almost the entire interval A. Here Ikl is chosen to be arbitrarily large so as to make this interval arbitrarily -11-

short. This being the case, pt = z so as to maximize profits on

indiscriminate consumers. Now consider the interval [t' + A, t'+A+c ]. then • If 2 does not commit to a new price at any date in ti-O_s interval, k t' + A + e e T so that the preceding argument obtains once more. If 2

does commit to a new price during this interval then for sufficiently k small e, 1 profitably undercuts 2's price at t' + A + e which again 2 1 that p — > p Thus in either case an implies ek Ic. ti+A+ t'+A+6 infinitely repeating cycle is initiated in which firm 2 sells only to

its indiscriminate customers at almost every date.

Thus if the proposition is untrue, it must be the case that at t' 2 1 1 2 E T, t, =0 and p 0 0. If pt, > p (min), 2 profitably undercuts p t' its price, capturing (almost) all discriminate consumers during this 1 1 p2 p (min) < p (min) to which 2 optimally interval. Therefore t'

responds with a greater price immediately after Again an

infinitely repeating cycle is initiated during which only 1 sells to

discriminate consumers. This completes the proof. 0

1 2 2 M Theorem 1: There exists an equilibrium in which H — N and 11

as a -4 O.

Proof: We construct the equilibrium strategies: -12-

.I if neither firm is committed at t

2 2 1 . 2 if p (mln) and p — Pa- t' k for •t' E [t - e ,t] where k is an 6 arbitrarily large positive number

2 1 if p < p (min) t

2 pt = z for all t O.

In words, firm l's strategy is to stay out of the market whenever 2 is uncommitted and to underprice 2 "almost immediately" following a p1 commitment on the part of firm 2 to a price exceeding (min). This strategy profile induces the following outcome. 1 is out of the market at t = 0 and perpetually charges a price as close as possible to z. 2 charges z at each date. 1 monopolizes all discriminate consumers, while 2 sells to its share of indiscriminate consumers only.

The above specification of seller 2's strategy is incomplete because it does not specify its actions if, off the equilibrium path, k seller 1 is committed to z - a for more than e at the time that 2's commitment expires. In this case 2 might optimally underprice 1 in order to capture discriminate consumers at least until l's commitment

6 Thus is small if e < 1. If e > 1, k is chosen to be negative. -13-

expires. Note, however, that regardless of 2's planned actions at this node, 1 can do no better than follow its equilibrium strategy (i.e., timing its commitment to expire "immediately" following the expiration of 2's commitment). Thus, as long as firm 1 adheres to its equilibrium strategy, 2's strategy is an optimal response.

Observe that along the equilibrium path, 1 incurs an arbitrarily small loss - in the order of e - by deferring its commitment at t = 0.

By choosing k sufficiently large, this loss tends to vanish.

As specified in theorem l's proof, firm ,1 plays an unyielding strategy, never committing itself first. As soon as it observes firm

2's price, it undercuts that price. In equilibrium, firm 2 perpetually charges the monopoly price while firm I perpetually charges the second highest possible price, i.e. z-a. By Proposition 1, no parallel equilibrium exists in which firm 2, by playing an unyielding strategy, 2 earns more than N . There does, however, exist an equilibrium in which firm 2, by playing an unyielding strategy, forces firm 1 to "accept" a 2 low profit of p (min); if 1 "believes" 2's "threat" to never commit 2 itself first, its optimal response is to commit itself to p (min) at once, following which 2 commits to z. It is easy to verify that the above sketched (Markov) equilibrium is (epsilon) subgame perfect.

However, this outcome is Pareto dominated by that described in theorem pareto 1. The latter is, therefore, by proposition 1, the unique dominant outcome. 4-

Conclusion

In the Bertrand model of price competition with a homogeneous product, absent any explicit price inertia, the equilibrium price is the competitive one and each firm earns zero profit. In this paper, a diametrically opposed reult has been obtained by explicitly introducing price inertia and allowing for some "irrational" behavior on the part of consumers: One firm appropriates (almost) all the surplus.

Furthermore, the identity of the latter is unambiguously determined in

the presence of even an arbitrarily small degree of asymmetry between

the firms' opportunity costs. Though highly counterintuitive, this

outcome is, like Bertrand's, to a large extent a consequence of assuming

that (discriminate) consumers perceive the firms' products as perfect

substitutes and are endowed with perfect . information. Obviously,

neither of these assumptions are realistic.

Perhaps more fundamentally, the formulation of the model in

continuous time allows for an instantaneous response to a rival's

observed commitment. Since an instantaneous response is not possible in

a discrete time framework, the surplus the unyielding firm is able to

appropriate is likely to be sensitive to the length of the term of

commitment. -15-

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