Judo Economics: Capacity Limitation and Coupon Competition Author(s): Judith R. Gelman and Steven C. Salop Source: The Bell Journal of Economics, Vol. 14, No. 2 (Autumn, 1983), pp. 315-325 Published by: RAND Corporation Stable URL: http://www.jstor.org/stable/3003635 Accessed: 22-09-2017 09:30 UTC 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 support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms RAND Corporation is collaborating with JSTOR to digitize, preserve and extend access to The Bell Journal of Economics This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms Judo economics: capacity limitation and coupon competition Judith R. Gelman* and Steven C. Salop** This article analyzes the effect of credible capacity limitation on the interaction between an incumbent and an entrant. The prospect of price competition deters entry by a less efficient firm with unlimited capacity. If, however, the entrant practices 'Judo economics" and limits its capacity sufficiently, it is in the incumbent's interest to accommodate entry. The entrant can increase its profits by selling rights (coupons) to its scarce discount output. If the entrant is less efficient, the incumbent will choose to serve coupon holders at the discount price. When this occurs, the entrant sells coupons, but no output, and industry production is rationalized. 1. Introduction * This article explores the role of precommitment in the strategic interaction between a dominant firm and a fringe competitor. In the model examined, a fringe competitor partially offsets its demand disadvantage by engaging in capacity limitation and discount pricing. This strategy credibly reduces the threat posed to the dominant firm and makes retaliation more expensive. The rational incumbent is thus induced to accommodate the entrant. To capture the image of a small firm using its rival's large size to its own advantage, we call this a strategy of judo economics. The article is organized as follows. Section 2 sets out a simple capacity-limitation model in which a less efficient entrant with unlimited capacity is deterred by the rationally based prospect that the incumbent will not accommodate it. In contrast, following Schelling (1960), when the entrant judiciously limits its capacity, it is in the incumbent's interest to accommodate entry. Section 3 examines a more sophisticated entry strategy. When the entrant both limits its capacity and sets a low price, its scarce output must be rationed. The entrant can increase its profits by selling coupons-transferable rights to its output. If the entrant's coupons are transferable, the incumbent may choose to honor the coupons, serving coupon holders at the discount price. When this occurs, the entrant produces no output and earns its profits solely from the sale of its coupons. Section 4 reviews the results of the model and suggests a number of extensions. * Federal Trade Commission. ** Georgetown University Law Center. The opinions in this article do not necessarily represent the views of the Federal Trade Commission. We would like to especially thank E. Bailey, A. Dixit, W. Dudley, J. Farrell, R. Frank, J.J. Gabszewicz, D.K. Levine, S. Salant, R. Schmalensee, D. Scheffman, W. Schwartz, C. Shapiro, and A.M. Spence for helpful conversations and the editors and referees for useful suggestions. The authors gratefully acknowledge research support from the FTC's Bureau of Economics. 315 This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms 316 / THE BELL JOURNAL OF ECONOMICS 2. Capacity and competition * Consider the following market. Consumer demand is given by the demand function D(p). Initially, the industry consists of an incumbent monopolist with unlimited capacity to produce output at a constant marginal cost cl. We denote the incumbent's monopoly price as Pim. Suppose a single potential entrant appears. The entrant can produce up to k units of output at a constant marginal cost c2 . We assume that the entrant has no cost advantage so that c2 > cl. The entrant freely selects a capacity level k.' Upon entry, it incurs a nominal sunk cost which is invariant with respect to k.2 The entrant's prospects depend on consumers' relative demand for the two brands and the strategic interaction that follows entry. In this article we focus on the effect of the entrant's ability to limit capacity on postentry pricing. To highlight this issue, we make a number of unrealistic simplifying assumptions. First, we assume consumers lexicographically prefer the incumbent's brand at equal prices; with any price differential, all consumers prefer the less expensive brand.3 Formally, let demands xi and x2 for incumbent and entr respectively be given as follows: xl =I T D(pI) if P -P2 Xi O otherwise (1 00 if PI --P2 = D(P2) otherwise. Second, we assume the following strategic interaction. The entrant plays first, irre- vocably choosing a price/capacity pair (P2, k).4 The incumbent then must react by choosing a price pi. In standard parlance, the entrant is the von Stackelberg price "leader" and the incumbent is the price "follower." For now, we assume that the incumbent may not price discriminate. O Competition with unlimited capacity. Suppose the entrant's capacity is unlimited (i.e., k = oo). Because entry costs must be paid, entry is only profitable if the entrant's price exceeds its marginal cost. But if the entrant has unlimited capacity and no cost advantage, the incumbent surely maximizes its profits by meeting the entrant's price.5 This is a result of consumers' lexicographic preferences: at any price pi > P2, the incumbent obtains no customers; at equal or lower prices (p1 < P2) the incumbent obtains the entire market demand D(p1).6 Thus, entry does not occur unless the entrant is strictly more efficient. What may not be obvious is the importance of the entrant's unlimited capacity and the incumbent's inability to price discriminate in generating these results. ' Here the entrant's choice is restricted to supply curves with marginal costs that are constant at c2 up to a maximum capacity level k. More generally, the entrant's strategy space could include a family of cost functions. 2 Any sunk cost prevents entry unless price strictly exceeds the entrant's marginal cost. If sunk costs are large, entry may be unprofitable even if accommodated. Presumably, the incumbent has already sunk this cost. 3Lexicographic preferences are a limiting case of differentiated products (Gelman and Salop, 1982). See Schmalensee (1982) and Dudley (1982) for alternative specifications of the advantage of the first entrant's "pioneer" brand. 4 The qualitative results of our model do not depend on the entrant's ability to make a credible price commitment, only on its ability to credibly limit capacity. 5A postentry competition based on the Bertrand price equilibrium concept is more complicated, but the results are similar. Without capacity limitations, the Bertrand equilibrium also implies zero entry. 6 Assuming a concave profit function, if the entrant sets P2 > pl,,,, the incumbent earns greater profits by undercutting (i.e., p, = Pi,,). If the entrant sets cl < P2 < Pl,?, the incumbent earns greater profits by matching (i.e., p, = P2). This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms GELMAN AND SALOP / 317 o Competition with capacity limitations. By credibly committing itself to a limited ca- pacity, the entrant makes itself less threatening to the incumbent and therefore improves its strategic position. Instead of matching the entrant's price, the incumbent may allow the entrant to sell out its limited capacity and serve the remainder of the market itself at a higher price. Formally, suppose the entrant irrevocably chooses a price/capacity pair (P2, k) to which the incumbent responds by choosing a price pl. Since price discrimination is impossible, the incumbent has two basic choices-pi > P2 (accommodating) or pi < P2 (undercutting or matching).7 The entrant obtains no customers if the incumbent matches or undercuts its price. Hence, a profit-maximizing entrant chooses among the (P2, k) pairs that induce an accommodation response. To derive this set of price/capacity pairs (the accommodation set), we first examine the incumbent's problem. If the entrant chooses P2 > pi, (the initial monopoly price), the incumbent will surely undercut by setting pi = pim. Hence, no entrant will pick P2 > Pim. For any price P2 < Plm, if the incumbent matches, it earns profits HM(P2) given by HM(P2) = (P2 - cl)D(p2) for Pi = P2. (2) Alternatively, the incumbent can accommodate the entrant by choosing p, > P2 and allowing the entrant to sell its k units. At P2 < pl, the entrant must ration the rights to its scarce output on some nonprice basis for k < D(p2). Two simple rationing schemes are random rationing and reservation-price rationing. Under random rationing, the rights are distributed randomly among all consumers willing to pay at least P2.' This more realistic random selection scheme is presented in the Appendix. Under reservation-price rationing, the k customers with the highest reservation prices are given the rights to the entrant's product.9 This rationing scheme is quite simple to analyze and, as shown in Section 3, selling the rights by means of coupons induces exactly this latter type of rationing. For now, we assume that the rights are rationed by reservation price but are not transferable, so that no aftermarket exists. When the entrant distributes its output to the k customers with the highest willingnessto-pay, the incumbent's residual demand when accommodating the entrant is given by xi = D(pl) - k. The incumbent's maximized profits are therefore given by 11A(k) = max (pi - c,)[D(p,) - k]. (3) Pi The profit-maximizing entry-accommodating price p,(k) can be derived by differ- entiating equation (3) and solving the following first-order condition: (pi - c1)D'(p1) + D(p,) - k = 0. (4) Totally differentiating equation (4), it is clear that p'1(k) < 0 as long as the profit function is strictly concave. 7 Under certain circumstances, the incumbent may selectively match the entrant's price, offering discounts to only those customers approached by the entrant. Such a price-discrimination strategy requires that the entrant's actual or potential customers be distinguishable, perhaps by use of a screening device. Unless, however, customers whom the incumbent serves have exclusive "rights" to the entrant's capacity, the entrant can offer to sell its discount output to other customers. As discussed in Section 3 below, selective matching is facilitated if the entrant issues coupons. 8 Of course, if the rights were transferable, then an aftermarket for these rights could exist and rights could have resale value. If so, even consumers with reservation prices below P2 would desire them. 9 These customers must be selected on a nonprice basis. If the high-reservation-price customers are identified by using price (i.e., if the entrant offers its units at a price p,), the incumbent will match. This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms 318 / THE BELL JOURNAL OF ECONOMICS We now return to the entrant's choice of (P2, k), which temporally precedes the incumbent's choice of p,(k). Because the entrant only obtains sales when the incumbent accommodates it, the profit-maximizing entrant chooses a (P2, k) pair such that'" HIA(k) > HIM(P2). (5) We denote the set of (P2, k) pairs that satisfy equation (5) with equality as X(k) so that the accommodation set satisfies the constraint P2 < X(k)." The frontier 4(k) represents the entrant's demand curve. If the entrant chooses a price/capacity pair along ?(k), the incumbent accommodates its entry and the entrant sells its entire capacity k. The p,(k) function and the (P2, k) accommodation set are illustrated in Figure 1 below. For the entrant's profit-maximization problem, we now have H2 = max (P2 - c2)k (6) p2ik subject to P2 X(k). At the optimum, the constraint clearly holds with equality. Substituting the constraint into the entrant's profit function and differentiating, we have the first-order condition that defines optimal capacity k*: 1'(k*)k* + o(k*) - C2 = 0. (7) FIGURE 1 ACCOMMODATION SET P\ D(p) k '1 Equation (5) reflects the convention that if the incumbent is indifferent between the two strategies, it chooses to accommodate. " It must also be shown that all (P2, k) satisfying equation (5) have the entrant charging a lower price than the incumbent (i.e., P2 < p,(k)). We prove that P2 < p,(k) for all k in the relevant open interval (0, D(c,)) as follows. Consider a (P2, k) pair that satisfies equation (5). Substituting equations (2) and (3) into (5), we have (p,(k) - cl)D(p1(k)) - (P2 - cl)D(P2) > k(p,(k) - cl) > 0. Since the profit function 11(p) is concave, P2 < P,tn and p,(k) < p,n, the inequality implies P2 <p,(k). This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms GELMAN AND SALOP / 319 This, in turn, defines the entrant's optimal price p' = k(k*) and the incumbent's best response p' = pl(k*). When the incumbent accommodates the entrant, it reduces its price below the initial monopoly level. The incumbent does not, however, match the entrant's price. Rather, it maintains an "umbrella" under which the entrant can prosper, as long as it remains satisfied with its modest market share. Even given our one-entrant assumption, this alternative formalization captures some features of dominant-firm/fringe-competitor interaction that are absent in the usual model. These features have been discussed informally in the literature and are prominent in the antitrust oral tradition. For example, Scherer (1980, p. 233) observes that in many industries the fringe firms choose to remain small rather than risk retaliation by the dominant firm.'2 Our model explicitly incorporates these concepts of punishment and industry discipline. If the entrant were to increase capacity slightly beyond the ?(k) fron- tier, the incumbent would cut its price discreetly from pi(k*) to P2, thereby driving the entrant's sales to zero. Our model applies to only those situations in which the entrant can make credible capacity-limitation commitments. Credibility is enhanced by the use of contracts, strategic product design, and other strategies.'3 Irreversibly adopting a technology with sharply increasing marginal costs can represent a credible commitment to limit capacity. If prod- ucts are differentiated, an entrant can effectively commit to a low capacity strategy by designing a specialty product or one of substantially inferior (or superior) quality. Producing a product with limited consumer appeal is analogous to capacity limitation.'4 On the other hand, if the entrant can secretly produce and sell additional units at low cost, the judo economics model does not apply. 3. Coupon competition * Coupons-transferable rights to discount output-provide a twist on standard price competition. Traditionally used to compete for price-sensitive consumers of "convenience" goods, coupons have recently been introduced into other markets. Airline coupons were first introduced by United in May 1979, after a long strike. United distributed transferable coupons that entitled recipients to a 50% discount on a future flight. Within days, American also began distributing coupons. An aftermarket for the coupons developed and considerable speculation ensued.'5 Each airline subsequently chose to accept the others' coupons. The demand for flights on which coupons were issued increased substantially. In the Summer of 1980, Eastern Airlines used coupons to deter entry as well as to induce accommodation. Eastern distributed coupons on its Air Shuttle routes which New 2 That incumbents only respond fully to low-priced entrants when they grow large is nicely illustrated by experience in the airline industry. For details, see Gelman and Salop (1982). " A credible commitment can also sometimes serve as an incumbent's counterstrategy. For example, a meeting-competition contractual clause can commit a firm to match its rivals' bids and thus prevent the incumbent from accommodating entry, even if accommodation is the most profitable response. Delta Airlines may have attempted such a strategy with its "We will not be undersold" advertising campaign in the Spring of 1982. See Salop (1982) for application to cartel stabilization. " See Gelman and Salop (1982) and d'Aspremont et al. (1979). See Yip (1982) for a practical discussion of how product differentiation and other structural characteristics traditionally thought to be entry barriers may be used to facilitate entry. 5 This aftermarket was far more complicated than the simple one we analyze below. First, the coupons had an expiration date, introducing dynamic considerations into the market. Second, carriers who accepted rivals' coupons had the option of reselling them. Third, coupon values depended on the price of restrictive discount fares as well as on the coach fare to which they were pegged. Finally, the airlines distributed coupons on other flights rather than selling them directly. This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms 320 / THE BELL JOURNAL OF ECONOMICS York Air had recently entered. The coupons were good for a 50% discount on Eastern's new transcontinental service. After United and American decided to accept these coupons, Eastern cut back its transcontinental flights. By distributing coupons, Eastern effectively cut its Shuttle fare without reducing its revenue. Coupons can be analyzed in our incumbent/entrant framework. So far, we have assumed that the entrant rations its scarce output on a nonprice basis. We now analyze the market for transferable rights and its natural extension-coupon competition. o The market for rights. Suppose that the entrant randomly distributes transferable rights, each of which represents an option to purchase one unit of the entrant's output at the price P2. Assume that the incumbent accommodates entry by choosing Pi > P2. Consider a resale market for these rights, where q denotes the transfer price of a right. Consumers compare the "full price" of the entrant's output (q + P2) to the incumbent's price pl. Given their lexicographic preference for the incumbent's product, consumers will demand rights only if q < P, - P2. This leads to an equilibrium price in the rights market just below Pi - P2, which can be approximated by'6 q = p- P2* (8) When Pi > P2, all the rights to the entrant's k units are ultimately purchased by consumers with reservation prices no less than Pi. These high-reservation-price consumers obtain the entrant's output, thereby leaving the incumbent with a residual demand given by xi = D(pl) - k. Not surprisingly, this is identical to the incumbent's residual demand in the case of reservation-price rationing. If transferable rights are allocated on a nonprice basis, the k individuals who initially obtain the rights gain a windfall profit of q on each right. If, instead, the entrant sells the rights, it obtains the additional revenue.'7 Finally, the incumbent may choose to honor the rights, serving bearers at the P2 and all others at Pi > P2. We refer to this price discrimination strategy as selective matching.'8 If the entrant sells the rights to its output, its optimal strategy may change for two reasons. First, it receives additional revenue for every unit sold. Second, if the incumbent honors the coupons, the entrant's profit function is altered. o The incumbent's problem. If the entrant issues coupons, the incumbent may selectively match the entrant's price by honoring the k coupons at the price P2 and setting a higher price p, for sales to its remaining D(pl) - k customers. Under this selective matching strategy (denoted by the subscript S), the incumbent maximizes profits as follows: 11s = max (pi - cl)[D(pi) - k] + (P2 - ci)k. (9) Pi 16 If q > pi - P2, consumers strictly prefer the incumbent's product. At any q < pi with reservation prices above p, prefer the entrant's brand. If there is excess demand for the rights when q < pi - P2 holds (i.e., if D(p,) > k), the equilibrium price of the k coupons must be infinitesimally less than the price differential. Although there is no true equilibrium, by using the approximation in equation (8), we derive an epsilon equilibrium. 17 The timing of this sale is crucial. For the market for these rights to equilibrate as assumed, the output prices pi and P2 must already be known when the sale occurs. Given this timing, selling coupons represents a credible commitment by the entrant slightly to undercut the incumbent's price as long as pi > P2. Thus, coupons act as a "beating competition" clause. See Salop (1982). 18 The entrant may prevent selective matching by keeping a secret registry of customers entitled to the low-priced output or by replacing lost (or honored) coupons, thereby maintaining its output sales at k. We assume throughout that the entrant precommits itself to limiting its coupon sales to the original k units. This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms GELMAN AND SALOP / 321 Differentiating equation (9) with respect to pi yields a first-order cond to that of the accommodation strategy, as given by equation (4). Hence, for any k, the incumbent chooses the same pi whether honoring the coupons or accommodating entry The incumbent's optimal response function is derived by comparing profits from selective matching, accommodating, and matching. Selective matching is more profitable than accommodating only if P2> Cl. (10) If equation (10) is satisfied, the incumbent compares its profits from matching and selective matching. Under either strategy, the incumbent serves the entrant's k potential customers at the price P2 and earns profit Hlc = (P2 - cl)k from these sales. Subtracting LIc from equations (2) and (9), and recalling that p,(k) maximizes Hl(pl) = (pi - cl)[D(p) -k], it follows that Ils > rIM if and only if (p,(k) - cl)(D(pl(k)) - k) > (P2 - cl)(D(p2) - k). (11) Hence, a (P2, k) pair induces the incumbent to selectively match the entrant if and only if p,(k) _ P2-'9 Figure 2 illustrates the (P2, k) pairs for which the incumbent's optimal response is undercutting (U), matching (M), selectively matching to coupon holders (S), or accommodating (A). 3 The entrant's problem. Taking into account the incumbent's reaction function, the entrant chooses the (P2, k) pair that maximizes its profits. If the entrant issues coupons, it has two potential revenue sources: the sale of coupons and the sale of output. FIGURE 2 STRATEGY REGIONS D(p) Pi m XP1 (k) 19 The incumbent cannot di cannot be induced to reveal th This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms 322 / THE BELL JOURNAL OF ECONOMICS If the entrant picks P2 < cl, the incumbent's best response is to accommodate its entry, allowing the entrant to sell coupons and output. Given this accommodation re- sponse, the entrant maximizes its profits as follows: JI2A = max (P2 - c2)k + qk (12) P2,k subject to q= pi -P2; pi = p,(k); P2 < cl; P2 <P, Substituting for q and pi, we have max (p,(k) - c2)k (13) P2,k subject to P2 < Cl. Differentiating the maximand in (13) with respect to k and rearranging the first-order condition, we have p,(ka) - c2 + kapIl(ka) = 0, (14) where ka denotes the entrant's optimal capacity choice, given that it issues coupons and induces an accommodation response. The optimal choice of P2 is indeterminate as long as P2 < Cl . If the entrant picks P2 such that p,(k) > P2 > cl, the incumbent selectively matches by honoring the coupons. This implies that the entrant sells only coupons and no output. The entrant maximizes its revenue from coupon sales as follows: I12S P2,k = max qk (15) subject to q =pi - P2; pi = p,(k); P2 > Cl; P2 <P,. Substituting for q and pi, we have max (pi(k) - p2)k (16) p2,k subject to P2 > Cl. By inspection, it is clear that the entrant maximizes its profits by setting P2 = c Substituting this equation into equation (16) and differentiating, we have the first-order condition p,(ks) - cl + kspIi(ks) = 0, (17) where kS denotes the entrant's optimal capacity choice, given that it issues coupons and induces selective matching. Comparing equations ( 13) and ( 16), it is clear that if C2 > cl, the entrant's profits are higher when the incumbent honors its coupons.2' By comparing equations (14) and (17), 20 For c2 > cl, this implies P2 < C2. This introduces the possibility that the incumbent could destroy the entrant by subsequently refusing to honor its coupons. Such a strategy violates the leader/follower equilibrium concept used in this article. 21 If the more efficient entrant issues coupons, it sets P2 < cl, thus inducing accommodation by the incumbent. This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms GELMAN AND SALOP / 323 it is easy to show that in inducing the incumbent to honor coupons, the less efficient entrant selects a larger k (i.e., kS > ka). Finally, by comparing the coupon strategy with pure capacity limitation, we show that the less efficient entrant earns additional profits from selling coupons.22 Coupons can yield a peculiar equilibrium in which the entrant sells no output and earns revenue only from coupon sales. Although the entrant's output price (when ac- companied by a coupon) is P2, consumers only realize the benefit of the reduction in the incumbent's price p1(k) because they must purchase coupons at price q = pi - P2. The entrant obtains the windfall profit. But coupons do result in a higher level of industry output than would occur under pure capacity limitation. The sale of coupons also facilitates the rationalization of industry production. The entrant sets its price so that the incumbent accepts coupons and produces the k units of output only if the incumbent is the more efficient producer. If the entrant is less efficient, this results in an industry cost savings of (C2 - cl)k-a savings which accrues entirely to the entrant as profits. Finally, selling coupons represents a counterstrategy by which the entrant can par- tially overcome the incumbent's lexicographic demand advantage. In effect, the entrant extorts some of the incumbent's profits by credibly threatening to produce k units unless it is bought off. Faced with this threat, the incumbent purchases the rights to the output. Although the market mechanism by which this deal is carried out is complex in its details, it is quite simple in its essence: the entrant blackmails the incumbent into sharing its profits by threatening to spoil the market. 4. Conclusions * This article has presented a new model of incumbent/entrant interaction. By setting a low price and limiting its capacity, the entrant makes accommodation more profitable than retaliation in two ways. The capacity limitation restricts the incumbent's loss of market share from accommodation. The entrant's low price increases the incumbent's loss from matching. The entrant can further increase its own profits by coupling the capacity limitation with sales of transferable coupons. If the incumbent has a cost advantage, optimal pricing by the entrant leads to an industry equilibrium in which the entrant sells coupons and no output. All output is produced and sold by the incumbent. Many questions are left unanswered. The structure of the model is special, and many properties of the equilibrium remain unanalyzed. Extension of the model to consider multiple entrants in a free-entry equilibrium is needed. Even within the context of the present model, we have not explored the relative profits nor the share gained by an equally efficient entrant. In addition, we have not fully analyzed alternative commitment strategies such as meeting-competition, beating-competition, and most-favored-nation provisions. Appendix Random rationing * In Section 2 we assumed that the consumers with the highest reservation prices obtain the entrant's scarce output. In the absence of transferable rights and an aftermarket, this 22 By comparing equations ( 13) and (6), it is easy to show that a more efficient entrant also prefers selling coupons to limiting capacity. A more efficient entrant has, however, an alternative strategy: it can undercut the incumbent and supply the entire market. For some range of c2 < cl, the entrant prefers undercutting the incumbent to limiting capacity, but prefers selling coupons to undercutting. For such an entrant, the ability to sell coupons results in lower output by the entrant, a higher market price pi, and more production by the highcost incumbent. See Gelman and Salop (1982) for details. This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms 324 / THE BELL JOURNAL OF ECONOMICS class of consumers would not necessarily obtain the product. We now consider the perhaps more realistic case in which nontransferable rights to the scarce output are distributed randomly among the willing purchasers. Let the density of consumers' reservation prices be given by h(w). Total demand is thus the integral of this density, or D(p)= f h(w)dw. (A1) Suppose the incumbent accommodates the entrant's price/capacity choice (P2, k) by setting pi > P2. Each of the D(p2) consumers with reservation prices greater than or equal to P2 prefers to buy from the entrant. Under random rationing, the rights to the k units are apportioned as follows. Let r be the proportion of consumers served at each reservation price, where r is defined according to k = f rh(w)dw. (A2) 2 Substituting equation (A1) into (A2) and rewriting, we have the rationing function r(p2, k), k r = r(p2, k) = D(P) (A3) When the entrant's units are rationed to k lucky customers in this way, the incumbent is left with a residual demand of disappointed customers xi who constitute a fraction (1 - r) of the potential market, or xi = (1 -r) h(w)dw. (A4) Substituting equation (Al) into (A4), we have X= (1 - r)D(p1). (A5) Under the optimal accommodation strategy, the incumbent's profits are given by HA = max (pi - cl)(1 - r)D(pl) (A6) Pi subject to Pi > P2. Differentiating (A6), we have the first-order condition (1 - r)[(p1 - cl)D'(pl) + D(pl)] = 0. (A7) Inspecting equation (A7), it is obvious that for the case of constant marginal cost, the accommodation price is invariant with respect to the entrant's (P2, k) choice. In fact, the accommodation price equals the (preentry) monopoly price (i.e., Pi = pim). Defining the incumbent's maximum accommodation profits as IIA(P2, k), we have 11A(P2, k) = [1 - r(p2, k)](pim - cl)D(plm). (A8) Of course, the incumbent still has the choice between accommodating and matching. Following equation (5) in the text, we define the accommodation set as the (P2, k) pairs such that HA > HM. Upon rewriting this constraint, we have (1 - r)H1(pim) > H(P2), (A9) This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms GELMAN AND SALOP / 325 where 11(p) = (p - c)D(p). Equation (A9) can be inverted to derive the demand curve P2 1< kr(k), and the entrant's optimal strategy can be calculated as in equation (7) in the text. This uniform rationing case captures an often-asserted feature of dominant-firm behavior. As long as the fringe (entrant) stays within the accommodation set, the incumbent does not react at all, but continues to price at the monopoly level. Only if the fringe violates the industry "discipline" does the incumbent respond. When the punishment comes, it is discontinuous and extreme-the incumbent matches the entrant's price and captures all its customers. References d'ASPREMONT, C., GABSZEWICZ, J.J., AND THISSE, J.F. "On Hotelling's 'Stability in Competition.'" Econometrica, Vol. 47 (1979), pp. 1145-1150. DUDLEY, W. "Limit Pricing Revisited: A Consumer Inertia Model of Entry Deterrence." Ph.D. Dissertation, University of California at Berkeley, June 1982. GELMAN, J.R. AND SALOP, S.C. "Judo Economics, Entrant Advantages and the Great Airline Coupon Wars." Federal Trade Commission Bureau of Economics Working Paper No. 58, June 1982. SALOP, S. C. "Practices That (Credibly) Facilitate Oligopoly Coordination" in J.E. Stiglitz et al., eds., New De- velopmnents in Market Structure, Proceedings of an IEA Symposium, forthcoming. SCHELLING, T.C. The Strategy oJ Conflict. Cambridge: 1960. SCHERER, F.M. Industrial Market Structure and Economic PerJbrmance, 2d ed. Chicago: 1980. SCHMALENSEE, R. "A Model of Pioneer Brands." American Economic Review, July 72 (1982), pp. 349-365. YIP, G.S. "Gateways to Entry." Harvard Business Review (September-October 1982), pp. 85-92. This content downloaded from 128.252.67.66 on Fri, 22 Sep 2017 09:30:06 UTC All use subject to http://about.jstor.org/terms