Loyal Consumers and Bargain Hunters: Price Competition with Asymmetric Information

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1 Loyal Consumers and Bargain Hunters: Price Competition with Asymmetric Information Lester M.K. Kwong Created: February 12, 2002 This Version: April 15, 2003 COMMENTS WELCOME Abstract This paper considers a homogeneous good Bertrand market with asymmetric information. Consumers differ by a unidimensional type space as well as grouped exogenously as being either a loyal consumer or a bargain hunter. We find that under such an asymmetric information environment, the equilibrium will be supported by mixed strategies. Furthermore, we find that firms will compete via nonlinear price schedules bearing similarities to a monopolist to the extent that a flattening out effect occurs. More importantly, competition occurs in the reservation utility of the marginal consumer. Ph.D. program in economics, University of British Columbia East Mall, Vancouver, British Columbia, Canada V6T 1Z1. mkit@interchange.ubc.ca. I would like to thank Sukanta Bhattacharya, Patrick Francois, Thomas Ross, Kazutaka Takechi, Markus von Wartburg, Okan Yilankaya, participants in the IO study group and in the microeconomics lunch seminar, at the University of British Columbia, and especially Gorkem Celik and Guofu Tan for helpful comments and suggestions. I am also grateful to the A.D. Scott Fellowship in Economics for financial support.

2 1 Introduction The theory of monopoly pricing and the practice of price discrimination has received a lot of attention from economists because of their welfare implications. While inefficiencies arise due to conflicts of interest between the firm and consumers, there are methods of minimizing such inefficiencies. For example, if the willingness to pay by each consumer in the market is observable, then a monopolist may implement a perfect price discriminating policy to extract all consumer surplus. Under such a pricing scheme, all inefficiencies are eliminated. However, if the willingness to pay is unobservable, nonlinear pricing schemes may still be implemented for the same purpose of consumer surplus extraction. One way of modelling such environments is to allow consumers to differ in their willingness to pay usually represented by a unidimensional type space. 1 Then, a firm, given some prior belief over the probability distribution of this taste parameter may construct a nonlinear pricing scheme to discriminate between different types of consumers. Such pricing strategies are, therefore, means for extracting information from consumers regarding their tastes. Furthermore, the implementation of such nonlinear pricing schemes require little more than the usual assumptions of microeconomic theory. The growth of informational economics and mechanism design in the past couple of decades provide us with insights into the construction of such pricing mechanisms. In particular, with the aid of the revelation principal, the focus of the modeler on the space of strategies, in which the optimal mechanism lies within, is reduced. This allows for much simplifications from a theoretical point of view. The study of nonlinear pricing to monopoly markets has been studied extensively by Mussa and Rosen (1978), Goldman, Leland, Sibley (1984), and by Maskin and Riley (1984), to name a few. And the extension into oligopolistic markets have been studied by Oren, Smith and Wilson (1983, 1984), Ivaldi and Martimort (1994), Stole (1995), Hamilton and Thisse (1997), and Rochet and Stole (2002). A conclusion one may draw from such extensions is that a theory for competing mechanisms brings about complications absent in monopoly markets. For example, a pricing mechanism for one firm may include messages, from the consumer, regarding the mechanism its opponent is offering them, in addition to the revelation of their private information. As a result, the structure of such mechanisms will 1 It is for theoretical simplicity that a unidimensional type space is considered. Extensions beyond the unidimensional case has been studied by Armstrong (1996), Armstrong and Rochet (1999), Laffont, Maskin and Rochet (1987), McAfee and McMillan (1988), and Rochet and Choné (1998). 1

3 have an inherent recursive nature. One way to over come such issues, as the authors above have noted, is by introducing differentiated products into the market. 2 Therefore, by introducing a spatially competitive oligopolistic setting, a simplification of the problem is attained. On the other hand, Mandy (1992) considered a pure homogeneous good Bertrand game where firms compete via nonlinear price schedules. The equilibrium characterized in his study suggests that when firms may enter freely into the market, then firms profits are driven to zero. However, the characterization of the equilibrium is incomplete which leads to the possibility that consumers may be segmented, in terms of which firm they purchase from, by types. Alternatively, when entry is restricted, firms may earn positive profits. The profitability of firms is solely determined by the structure of costs. However, much like the previous case, the exact price schedules offered by each firm may not be characterized. Consequently, some general questions that may be of interest in environments of price competition with asymmetric information is whether the introduction of a second firm into the market disrupts the pricing behavior of a monopolist. From the mechanism design literature, it is known that there exists an incentive compatible direct mechanism if there is a single principal. However, when the number of principals are increased, the existence of such a mechanism is, generally, unknown. 3 In other words, will there be incentives for the second firm to charge, for example, a uniform price so as to disrupt the incentive compatible direct mechanism a monopolist may offer? Heuristically, such a strategy is feasible so long as there exists a large number of consumers on the low end compensated well enough by the uniform price. But then, one may predict the incumbent s mechanism to be suboptimal if such segmentation of the market is to be expected. Alternatively, one may propose the question regarding the distribution of consumers, in terms of types, among the firms in the market. For example, will one firm specialize in serving all high end consumers while another firm serves the low end consumers? More specifically, will consumers be partitioned according to their types in equilibrium? Such questions, in our opinion, are of particular interest as competing mechanisms are often an observed phenomenon. In attempts to provide a partial answer to the questions above, we ad- 2 This is with the exception of Oren, Smith and Wilson (1983, 1984) who took a different route and modelled nonlinear pricing in terms of a Cournot model. 3 This issue was addressed in Epstein and Peters (1999) in which they examined the revelation principal with a language capable of describing mechanisms. In view of their findings, the model developed in this paper will be one in which the mechanisms a seller may offer is restricted to those absent of this language. 2

4 dress the issue of competing mechanisms from a different approach. We adopt a structure, similar to the framework from previous studies such as Rosenthal (1980) and Varian (1980), where each firm has access into two different markets; a captive market and a competitive market. On the assumption that firms cannot distinguish between consumers from the two markets, the benefits from this assumed structure is obvious; the existence of the captive markets will restrict the amount of competition in the competitive market. In other words, a zero profit equilibrium, from undercutting, will not result. This paper purports to address the question of nonlinear price competition in the framework described above. The remainder of the paper is as follows. Section 2 will set up the basic framework. We then analyze this framework under a duopoly setting in Section 3. In Section 4, we analyze some extensions to the model. More specifically, we consider the case when the size of the captive market goes to zero and when we allow for more firms in the market by considering a generalized n-firm oligopoly market. Finally, some concluding remarks will follow in Section 5. 2 The Basic Framework Assume a continuum of consumers in the unit interval [0, 1] with unit density. Associated with each consumer is a taste parameter θ [ θ, θ ] Θ R + which is drawn from a continuously differentiable distribution function F (θ) with density f (θ). Nature draws a θ independently for each individual according to F (θ) and the realization of θ is private information for each consumer. Furthermore, consumers are assumed to fall into one of the two groups; loyal consumers and bargain hunters. A loyal consumer is here defined as one whose objective is to consume the product from their current firm, not necessarily the one with the lowest price. We assume loyal consumers are distributed evenly among all the firms in the market. In contrast, a bargain hunter is one who seeks to consume the product at the lowest available price. Consequently, a bargain hunter is well informed regarding the prices of all firms in the market. 4 At the outset, this assumption may seem restrictive, but we may interpret a loyal consumer to have very high search costs whereas a bargain hunter does not. Consequently, the distribution of prices among all firms is 4 Alternatively, one may think of loyal consumers as uninformed consumers while bargain hunters as informed ones. Under this interpretation, an informed consumer is well informed about the prices in the market while uninformed ones are not. 3

5 known to bargain hunters but not to loyal consumers. We further assume consumers only purchase from one firm. As a result, consumers are unable to split their total consumption between the two firms in the market. This, again, although restrictive, greatly simplifies the analysis to follow. 5 Let α [0, 1] denote the proportion of consumers who are bargain hunters and 1 α denote the proportion of consumers who are loyal. We assume that the distribution of θ, F (θ), as well as the proportion of loyal consumers and bargain hunters, given by α, is common knowledge to all firms in the market. However, the actual type of each consumer remains unknown to the individual firms. Due to the nature of the asymmetric information, firms in the market will have incentives to price nonlinearly. Firms in the market are homogeneous and produce with a constant marginal cost of c. Denote T i (q) as the price schedule offered by firm i where q denotes quantity. Firms choose T i (q) simultaneously after which consumers make their purchase decision. Utility for the consumer depends on his taste parameter, θ, as well as his total consumption of the good q. 6 Let u : Θ R + R be the utility function for a consumer. We assume that the utility function satisfies the properties: u (θ, q) θ > 0; for all θ Θ and q R +. u (θ, q) q 3 A Duopoly Market > 0; 2 u(θ, q) q 2 < 0; 2 u (θ, q) > 0 θ q Given the assumption of a homogeneous good market and price competition, this implies that we essentially have a Bertrand price competition game. As in the case of the monopoly, a consumer maximizes utility by choice of her consumption level. However, because there are two firms in the market, the distinction between bargain hunters and loyal consumers is not trivial. 7 5 An alternative approach is to interpret each firm as offering a class of groups differing in quality and each consumer simply demands one good at most. Under this interpretation, the assumption seems justified. 6 Note that the utility of a consumer only depends on the taste parameter but not on being a bargain hunter or a loyal consumer. The additional utility from being a bargain hunter is the cost savings from consumption which we assume to be equal to search costs. As a result, consumers are loyal or bargain hunters by nature and have no real strategic advantages over loyal consumers. 7 The reason for this distinction to be trivial in the case of the monopoly is that a monopolist has no incentives to differentiate between the two groups of consumers. In 4

6 With the assumption that α < 1, each firm in the market is guaranteed (1 α)/2 number of consumers each period. 8 The remaining α share is where competition between the two firms takes place. Given the objective of a loyal consumer, we may obtain a consumption rule as a function of θ by her optimization problem. We define qi L (θ) as: qi L (θ) arg max u (θ, q) T i (q) (1) q where L denotes a loyal consumer and i represents the firm the loyal consumer is loyal to. A bargain hunter s consumption decision, on the other hand, involves two stages. Aside from the decision on how much to consume, he must also choose from which firm to consume. Therefore, a typical θ-type bargain hunter s optimization problem may be expressed as: { } max max q u (θ, q) T 1 (q), max u (θ, q) T 2 (q) q This optimization problem may be solved by a two stage process. More specifically, for any firm i {1, 2} in the market, a bargain hunter will have an optimal consumption rule. We denote this as q i (θ) and define it as: (2) q i (θ) = arg max u (θ, q) T i (q) (3) q Clearly then, the reduced form optimization problem, with Eq. 3, may be written as: max {u (θ, q i (θ)) T i (q i (θ))} (4) i {1,2} The solution to this optimization problem, Eq. 4, may not be unique but a solution necessarily exists, for all θ Θ. So, given the two price schedules offered by the two firms, we may define each firm s respective segment of bargain hunters by Θ i. More formally: { } Θ i = θ Θ max {u (θ, q j (θ)) T j (q j (θ))} = i (5) j {1,2} fact, by definition, a bargain hunter is a loyal consumer if there only exists one firm in the market. 8 While each firm is guaranteed this fraction of consumers each period, identifying the consumer as being loyal or as a bargain hunter is not possible. As a result, the same price must be offered to all consumers who purchase from the same firm. 5

7 u u(θ, q 2(θ)) T 2(q 2(θ)) u(θ, q 1(θ)) T 1(q 1(θ)) }{{}}{{} Θ 1 θ Θ 1,2 Θ 2 Figure 1: Θ 1 and Θ 2 given T 1 and T 2 A graphical representation of the partition of Θ into Θ 1 and Θ 2 is given in Figure 1. While the figure shows a nice simple partition of Θ into Θ 1 and Θ 2, ignoring Θ 1,2 with zero measure, in general this need not be the case. Furthermore, the utility level in terms of θ is shown to be smooth, which, again, may not necessarily be the case. This may especially be true if the optimal price schedules by the two firms are discontinuous. Furthermore, Θ 1 Θ 2 Θ 1,2 may have positive measure but Θ 1 Θ 2 = Θ. We may interpret Θ 1,2 as the θ-type bargain hunters who are indifferent to consuming from either firm. Let Θ i, j = Θ \ Θ j be the θ-type bargain hunters who strictly prefers consuming from firm i. Note that by construction, the subsets of Θ into each firm s respective market segment is solely determined by the price schedules offered by each firm. Consequently, an equilibrium may be defined by this characterization. Therefore, given two price schedules T 1 and T 2, we may define a bargain hunter s optimal consumption rule as: { q B q1 (θ), if θ Θ (θ) = 1 (6) q 2 (θ), if θ Θ 2 Claim 1 Given T 1 and T 2 and the induced partition of Θ: 1. for all θ Θ i, q B (θ) = q L i (θ). Furthermore, if for some i {1, 2}; 2. Θ i =, then for all q B (θ) > 0, T j < T i with j i. 6

8 3. Θ i / {, Θ} for all i {1, 2}, then there exists some q, q R + such that T 1 (q) T 2 (q) and T 1 (q ) T 2 (q ) with at least one inequality strict. Proof: See appendix. The above claim states that for all θ Θ i, all θ-type consumers, regardless of being a bargain hunter or a loyal consumer, will choose to consume the same quantity level from firm i. Furthermore, we note that the price schedule offered by firm i must be uniformly higher than that of firm j s over the set of quantity levels the bargain hunters are consuming at if Θ i =. This further drives the result that if both Θ i and Θ j are nonempty and not equal to Θ, then there must exist at least two quantity levels for which T 1 (q) T 2 (q) and T 1 (q ) T 2 (q ) with at least one inequality strict. This idea that the choice set for a bargain hunter is larger than that of the loyal consumer is important for the analysis to follows. This implies that if a firm offering T i as their price schedule does not capture the θ Θ j bargain hunters, then simply replicating T j over the quantity levels such θ-type bargain hunters are consuming will not change q B (θ) for all θ Θ i. Consequently, Claim 1 allows us to examine deviations when analyzing an equilibrium in a simple and systematic way. Define C P as the set of all possible piecewise continuous price schedules. More formally: C P = {T : R + R + T is piecewise continuous} (7) Since C P may be interpreted as the set of all possible tariffs offered by each firm, given our restrictions, it is essentially the set of pure strategies for each firm. Alternatively, a mixed strategy is given by a measurable function i : C P [0, 1] with the property that T C P i (x) dx = 1 and i (x) 0 for all x C P. We seek a Nash equilibrium in the price schedules offered by each firm. 9 Given T 1 and T 2, each firm s market segment of the bargain hunters is given by the partition of Θ, expected profits for each firm may be expressed as: π i = α (T i (θ) cqi L (θ))f(θ)dθ + α (T i (θ) cqi L (θ))f(θ)dθ Θ i, j 2 Θ i,j + 1 α (T i (θ) cqi L (θ))f(θ)dθ (8) 2 Θ 9 Note that we have not restricted our attention to purely nonlinear price schedules as the class of functions C P does not eliminate the uniform price as a possibility. 7

9 where T i (θ) T i (qi L(θ)). In equilibrium, each firm s choice of T i maximizes Eq. 8. Since for all T i C P, qi L(θ) is well defined, we therefore, let CP to be the set of all possible price schedules a firm may offer evaluated at the induced demand of its loyal consumers, qi L (θ). More specifically: C P = {T q L : Θ R + T C P and q L = arg max u(θ, q) T (q)} (9) q It is clear that there exists a one to one mapping between C P and C P. Define T to be the set of monetary transfer rules of all incentive compatible direct mechanisms. Then: T = { T : Θ R + for all θ, θ Θ, U (θ, θ) U ( θ, θ )} (10) where U(θ, θ ) u(θ, q(θ )) T (θ ). It is clear that T C P. Note that all incentive compatible direct mechanisms are well defined. 10 In fact, for all T T, they must differ only in one dimension. More importantly, they differ by a degree of a constant given by the constant of integration. 11 This, may be interpreted as the reservation utility for the marginal consumer which we define to be û. We characterize our first result in the following proposition. Proposition 1 Given α (0, 1), there exists no pure strategy equilibrium. Proof: See appendix. In establishing this result, we first considered the idea that the price schedules offered by the two firms will not, roughly speaking, intersect, in equilibrium. In particular, we rely on two specific deviations in our proof of this proposition. More specifically, we consider a deviation for firm i by replicating firm j s tariff over the quantity levels for which bargain hunters purchase from firm j. From Claim 1, we know that if θ Θ j, i then such θ-type bargain hunters purchase from firm j. Furthermore, T i (q L j (θ)) > T j (q L j (θ)) for all θ Θ j, i. So in equilibrium, the profits obtained from the proportion of loyal consumers who are consuming such quantity levels must be strictly greater than those obtained by replicating such portions of the price schedule. In addition, we consider a deviation by firm i, and firm j by replicating T j and T i, respectively, over all q R +. If both deviations are jointly unfeasible then it cannot be the case that profits for a firm from selling only 10 See, for example, Maskin and Riley (1984). 11 See, for example, Laffont (1989). 8

10 to θ-type loyal consumers is strictly greater than replicating its competitors tariff over such quantity levels. This, therefore, suggests that it must be the case Θ i {, Θ} in any pure strategy equilibrium. In other words, no equilibrium where bargain hunters are segmented between the two firms, in pure strategies, may be supported. This further allows us to consider equilibria where such segmentation does not occur. We may further deduce that with the existence of a group loyal consumers, profits must be bounded above zero. Therefore, motives to undercut an opponent s price schedule, uniformly, by some arbitrary ɛ always exist. As a result, an equilibrium where Θ i = Θ j = Θ occurs is not rationalizable. This result, in essence, allows us to reduce the set of pure strategies from C P to T since equilibrium behavior necessarily implies an incentive compatible direct mechanism. In other words, an equilibrium in pure strategies necessarily implies that T i qi L, T j qj L T and that competition is for all of the θ-type individuals. As a result, the individual firm must offer a contract as prescribed by the monopolist. 12 This further implies that in the duopoly case, all competition must occur in û since any functional changes in the optimal tariff will not be incentive compatible and will, therefore, be suboptimal. As a result, the proof for the nonexistence of a pure strategy equilibrium follows nicely from the discontinuities in the profit function. Because undercutting an opponent s tariff, uniformly, by an arbitrarily small amount will cause jumps in the profit function due to monopolization of the bargain hunting group of consumers, profits will be driven to zero. However, in such a process of undercutting, the inability to differentiate between bargain hunters and loyal consumers essentially eliminates all monopoly power. Consequently, the loss of all monopoly power is not an equilibrium in this model, since an outside option with strictly positive profits exists. This corresponds to the case when a firm forgoes competition for bargain hunters and simply monopolizes loyal consumers. Due to Proposition 1, equilibrium, in so far as it exists, will be in mixed strategies. In specifying a mixed strategy equilibrium, one essentially has to consider all strategies within the set C P, the set of all piecewise continuous price schedules, which makes characterization cumbersome. Therefore, we provide the following lemma to simplify this process. 12 This is by the assumption that a direct incentive compatible mechanism is the optimal mechanism for a monopolist in this environment. 9

11 Lemma 1 In any mixed strategy equilibrium, if S(G 1 ), S(G 2 ) C P are the equilibrium supports for firms 1 and 2, then no two pure strategies T i s, T j k {S(G 1 ) S(G 2 )} are such that Θ s,k is a countable set, for s, k = 1, 2 except when Θ s,k =. Furthermore, Θ s,k is a convex set. Proof: See Appendix. The implications of Lemma 1 is that firms will not randomize with price schedules such that the resulting distribution of bargain hunters is segmented. This follows from the proof that Θ s,k is convex. In other words, whenever two price schedules, T s and T k, intersect, then Θ s Θ k or vice versa. In addition, we may establish the following corollary. Corollary 1 In any mixed strategy equilibrium, T T restricted to the domain Θ Θ such that for all θ Θ, T (θ) cq(θ) 0. Furthermore, Θ s,k {Θ \ Θ}, and T (Θ \ Θ) = c q(θ \ Θ). Proof: To establish this corollary, consider, first, the strategies such that for all T k, T s S(G i ), i = 1, 2, with Θ k,s =, they lie within the space C P. Define the set of such strategies as S and assume T S but T / T. Since such strategies are uniformly above or below one another, sup{s} = T is well defined. 13 Let T M be the optimal price schedule offered by the monopolist. Then, it is clear that for all T S, T T M for all θ Θ. Then, if T / T, expected profits can be improved by simply charging T M so T T. It is then clear that for every T / T but in S a deviation to some T T exists by the idea that Θ k,s = for all T S. 14 Now consider a strategy T = T M k such that θ Θ for which T (θ) cq(θ) < 0. It is evident that a deviation to offering T where T ( Θ) = T ( Θ) and T (Θ \ Θ) = cq(θ \ Θ) is profitable since incentive compatibility for all θ Θ is preserved and zero profits are earned by all θ {Θ \ Θ} types. 15 Given the structure of the mixed strategy equilibrium, we may, therefore, simply restrict our attention to strategies that satisfy Corollary 1. In other words, we may simply consider price schedules T T with the flattening 13 By this we mean the highest price schedule offered in equilibrium. 14 This follows since for all T, T T, T T = k, for all θ Θ, since maps within T differs only by a constant of integration. 15 Note that incentive compatibility for all θ Θ is preserved since the price schedule is flattened out for lower θ-types at a higher price than along the unrestricted map T T. Furthermore, we do not claim that incentive compatibility is preserved over θ {Θ \ Θ} types. However, their behavior is irrelevant as zero profits are earned from such θ-types anyways. 10

12 out property along θ-types where T (θ) cq(θ) 0. Figure 2. T (q(θ)) T (q(θ)) û c q( θ) q(θ) q(θ) Figure 2: The flattening out property This is depicted in Let G i (û i ) be firm i s equilibrium distribution function over û i with density g i (û i ) and S (G i ) denote the support of this distribution. 16 Let πi M (û i ) be monopoly profits from offering û i to the marginal consumer. Then we may establish S (G i ) in the following lemma. Lemma 2 In equilibrium, S (G i ) = [0, u] for i {1, 2} where u satisfies the equality: ( ) ( ) 1 + α 1 α πi M (u) = πi M (0) (11) 2 2 where: πi M (û) = Θ and I( ) is the indicator function. Proof: See Appendix. (T (θ) cq L (θ) û)i(t (θ) cq(θ) û 0)f(θ)dθ (12) The idea behind lemma 2 is that a firm will not wish to exhaust all profits since there is always an outside option of not pursuing bargain hunters because they may exercise monopoly power over loyal consumers. Therefore, 16 Essentially, a firm s mixed strategy is i(t i) where T i q L i T. But given that for all T, T T, they only differ by a constant, we abuse notation and simply refer to a mixed strategy as G i(û i) to imply the randomization given by i. 11

13 the region over which firms will randomize their price schedule must not yield profits lower than this outside option. The objective of the firm, given that it offers an incentive compatible price schedule, faces the following maximization problem: 17 subject to: max g i (û i ) u 0 u 0 ( ) πi s G j (û i ) + π f i (1 G j (û i )) g i (û i ) du i (13) g i (û i ) du i = 1; g i (û i ) 0 for all û i S (G i ) (14) π s i (û i ) G j (û i ) + π f i (û i) (1 G j (û i )) π for all û i S (G i ) (15) where πi s and π f i represent firm i s profits when it successfully offers the lowest nonlinear tariff and when it does not, respectively. More specifically, πi s and πf i as: πi s (û i ) = 1 + α πi M (û i ) (16) 2 π f i (û i) = 1 α πi M (û i ) (17) 2 The constraints in Eq. 14 are the boundary conditions of the equilibrium density function and those in Eq. 15 may be interpreted as the firm s profit maximizing constraint. Note that π is the minimum profit the firm is guaranteed, which is equal to the expected profits from serving only the loyal group of consumers. In other words, π π f i (0). The solution to the optimization problem, Eq. 13, is obtained trivially since we may exploit the indifference condition of any two pure strategies within S (G i ). By construction, equilibrium strategies will be symmetric, and in particular, in mixed strategies. Proposition 2 In equilibrium, both firms will randomize their price schedules according to the distribution function: ) ( π M i (0) πi M (û i ) ) for i {1, 2}. G i (û i ) = ( 1 α 2 απ M i (û i ) (18) 17 We may express the optimization problem in this way since we have already restricted our attention to incentive compatible price schedules. Therefore, a more formal way to express this is with the condition that T i q L i T over θ Θ such that T (θ) cq(θ) 0 and T i = c q i otherwise. 12

14 Proof: See Appendix. The interpretation of the equilibrium distribution, Eq. 18, is quite simple. In equilibrium, the frequency at which competition for the bargain hunters, with changes in û i, is determined by the gains of successfully capturing the α share of the market and by the losses due to increases in û i in the remaining (1 α)/2 share of the market in which the firm has monopoly power. Corollary 2 The equilibrium in Proposition 2 is the unique equilibrium of the game. Proof: This is immediate given Corollary 1. From the analysis above, we derived a symmetric mixed strategy equilibrium in nonlinear price schedules. Some interesting properties of this equilibrium suggests that the price schedules offered by both firms will be pseudo-incentive compatible, in terms of a direct mechanism, over the whole of Θ. 18 Consequently, the existence of the loyal group of consumers is sufficient in disciplining the equilibrium from segmentation. Furthermore, due to the degree of freedom in the price schedules given by û, this is where competition for consumers exists. In essence, the problem considered here reduces down to a single dimension. 4 Extensions In this section of the paper, we consider some basic extensions that may be of interest in this context. More specifically, we will focus on a generalization to an arbitrary n number of firms as well as examining the equilibrium of the limiting game when α approaches one. This corresponds to a game where there exists no loyal consumers. We begin by analyzing the case when α approaches one. 4.1 When α Approaches One The question regarding the equilibrium derived in Proposition 2 when α approaches one is one of particular interest. This corresponds to a game where a captive market, or no loyal consumers, exist. Correspondingly, this 18 By this, we mean that incentive compatibility holds, in terms of a direct mechanism, over θ-type consumers in which positive rent is extracted. 13

15 will be a game of pure Bertrand competition without a positive outside option. Note that by taking the limit of the definition of the equilibrium support, given by Lemma 2, one finds that: π M (u) = (T (θ) cq L (θ) û)i(t (θ) cq(θ) û 0)f(θ)dθ = 0 Θ Furthermore, taking the limit of the equilibrium distribution function given by Eq. 18, we find that: lim G i(û i ) = 0 α 1 This implies that as α approaches one, all the probabilities are transferred to the upper-bound of the equilibrium support. This is intuitively plausible. As the number of loyal consumers decreases, the gains from competition for the bargain hunters increase. Put in a different way, the loss from competition due to the loss of monopolization over loyal consumers, decreases. As a result, firms compete more rigorously by placing higher probabilities on the upper-bound of the equilibrium support of the distribution function. In the limit, the flattening out process, as suggested by the mixed strategy equilibrium developed in the preceding section implies that price is flattened out to marginal cost for all θ Θ. In fact, this is also the unique equilibrium of the limiting game. We state this formally in the following proposition. Proposition 3 If α = 1, then the unique equilibrium is that for all i {1, 2}, T i = c q. Proof: See Appendix. We now turn our attention by examining a more general case of a n-firm oligopoly type market. 4.2 n Number of Firms In extending the model to include n number of firms, note that each firm captures (1 α)/n number of loyal consumers. The objective of the loyal consumer remains unchanged but the choice set of a bargain hunter increases. We maintain our notation and denote q i (θ) as the induced demand for a θ-type bargain hunter facing the price schedule T i (q) from firm i. Similarly, we may define a partition of Θ given the set of price schedules offered by the n firms, {T i } n i=1. Let N = {1, 2,..., n} be the set of firms in the market. Then: { } Θ i = θ Θ max {u(θ, q j(θ)) T j (q j (θ))} = i (19) j N 14

16 Similarly, a bargain hunter s optimal consumption rule is, therefore: q B (θ) = q i (θ), if θ Θ i (20) We now state the equivalence of Claim 1 under this more general environment. Claim 2 Given {T i } n i=1 and the induced partition of Θ: 1. for all θ Θ i, q B (θ) = q L i (θ). Furthermore, if for some i N; 2. Θ i =, then for all q B (θ) > 0, T i > min{t j } n j=1. 3. Θ i / {, Θ} for all i N, then there exists a vector {q i } i N R n +, such that T i (q) min{t j } j i for all i N, with at least one inequality strict. Proof : The proof is analogous to that of Claim 1 and is thus omitted here for brevities. Define Θ j i to be the θ-type bargain hunters who are indifferent to purchasing from exactly j number of firms with firm i being one of them. More formally: Θ j i = {θ Θ i!n N with #{N } = j and θ s N Θ s } (21) Then profits for firm i may be expressed as: π i = n j=1 α j Θ j i (T i (θ) cq L i (θ))f(θ)dθ+ 1 α n Θ (T i (θ) cq L i (θ))f(θ)dθ (22) The remaining analysis in the case of n firms is similar to that of the duopoly and we summarize the results in the following proposition. Proposition 4 For n <, the equilibrium may be characterized by the following four properties. 1. No firm will employ a pure strategy. 2. Firms will randomize over strategies as prescribed by Corollary 1. 15

17 3. The support of the mixed strategy is given by S(G i ) = [0, u], for all i N, where u solves: ( ) ( ) 1 (n 1)α 1 α πi M (u) = πi M (0) n n where π M i (u) as in Eq Each firm randomizes according to the distribution function: G i (û i ) = (( 1 α n ) ( π M i (0) π M i (û i ) ) απ M i (û i ) ) 1 n 1 Proof: See Appendix. As can be seen, one may construct a similar equilibrium, to the case of the duopoly, taking on a n-firm oligopoly type market. As the number of firms increases, the proportion of loyal consumers each firm will capture will decrease. This is evident if one fixes the market demand to the unit interval and distributes (1 α)/n number of loyal consumers to each firm. Consequently, since the support of the equilibrium distribution, S (G i ), is determined by this allocation of loyal consumers to each firm, this will inevitably change the equilibrium support for each firm s strategies. An interesting question that one may ask, at this conjuncture, is what will happen when n approaches infinity. It is intuitively clear that as n approaches infinity, the number of loyal consumers each firm has approaches zero. Similar to the case when α approaches one, by taking the limit of the equilibrium support, we find that the upper bound implies: π M i (u) = 0 This is consistent to the case when α approaches one as both cases suggest that each firm has no loyal consumers. Similarly, by taking the limit of the equilibrium distribution function, we find that: lim G i(û i ) = 1 n This implies that as n increases, the probabilities each firm place on the lower-bound of their equilibrium support increases. This contrasts the case when α approaches one since the probabilities, there, are transferred to the upper-bound. While the interpretation of α approaching one and n approaching infinity is similar, (i.e., the number of loyal consumers each 16

18 firm has approaches zero,) the difference lies in the gains and the losses due to competition. As mentioned above, as α approaches one, The gains from competition for bargain hunters increases. On the other hand, as n approaches infinity, the number of loyal consumers each firm has decreases. However, the probability of being able to offer the lowest price schedule also decreases. Consequently, the probability of offering something in the upperbound of the equilibrium support goes down and the probability of offering something in the lower-bound increases. 5 Conclusion This paper has derived an equilibrium in a homogeneous good duopoly market with asymmetric information. Consumers are assumed to have private information regarding their type, which corresponds to the realization of some θ Θ, as well as being classified into two groups: loyal consumers and bargain hunters. The equilibrium derived is a symmetric mixed strategy equilibrium where firms randomize their price schedules in hopes of capturing all of the bargain hunters. An important factor in deriving this equilibrium is the assumption that firms must offer the same schedule to both groups of consumers. 19 This may be translated into the scenario in which the two groups are indistinguishable when transactions take place in the market. Furthermore, it has been shown that both firms will offer a price schedule which partially coincides to that of an incentive compatible direct mechanism through a flattening process. Clearly then, introduction of entrants into an environment considered in this paper will not highly disrupt the incentive compatibility of mechanisms offered. While we have only analyzed the equilibrium of a static game, one may interpret the mixed strategy as the randomization of the price schedules each firm will offer in each of an infinitely, or finitely, repeated game. However, one explicit assumption must be made, namely, firms will not collude in the extended game. Another important point to note is that the assumption of the exogenous grouping of consumers given by α, suggests an equilibrium where only positive profits are sustained. As shown above, without such an assumption, 19 See, for example, Rosenthal (1980) and Varian (1980). Alternatively, if the two groups of consumers are distinguishable by the firm, then the equilibrium may be trivially determined as the optimal nonlinear price schedule, in terms of a monopolist s problem, is offered to loyal consumers while pure Bertrand competition for the bargain hunters will occur (i.e., T = c q for all θ Θ). 17

19 or if one takes α to be equal to 1, then we essentially derive the outcome of a Bertrand game where a zero profit, with price equalling marginal cost, equilibrium occurs. To extend this model to capture the notion of an arbitrary n number of firms, we derived a similar equilibrium. Previous studies such as Varian (1980) and Rosenthal (1980) have shown a similar equilibrium in markets with n number of firms in the absence of asymmetric information with consumers differing by types. The difference in the results lie in the pricing for the goods. In our case, equilibrium was in a price schedule which is incentive compatible, in the direct mechanism sense, where in the others, only a uniform price is considered. The general results of this model seem to be consistent with the studies cited here. The crucial assumption behind such results though, lies in the inability of firms to differentiate between the captive market and the competitive market. This, in essence forces them to offer the same price between the two markets. 18

20 A Appendix We provide the proofs of our results presented in the paper. Proof of Claim 1: 1. Suppose that for some θ Θ i, q B (θ) qi L (θ). This implies that there exists some other bundle q = q B (θ) or q = qi L (θ) with the associated transfer T i (q ) such that it yields higher utility for either the loyal consumer or the bargain hunter. Note that the associated transfer must be from firm i since θ Θ i. Clearly then, such a choice of q is available for both loyal consumers and bargain hunters so if one finds it optimal to deviate so must the other type which contradicts the assumption that there exists some θ Θ i such that q B (θ) qi L(θ). 2. Suppose Θ i = but that for some θ Θ j with q B (θ) > 0, T j > T i. Then a deviation to consuming from firm i exists which contradicts the assumption that Θ i =. 3. If Θ i, Θ j / {, Θ} then some bargain hunters consume from firm 1 and some from firm 2. If there does not exist some q, q R + such that T 1 (q) T 2 (q) and T 1 (q ) T 2 (q ) with at least one inequality strict, then from part 2, this implies that for some i {1, 2}, Θ i = or that Θ i = Θ j, which contradicts the assumption that Θ i / {, Θ}. Proof of Proposition 1: To establish this proposition, we begin by first showing that the two firms will not, in equilibrium, offer two price schedules T 1 and T 2 such that Θ i / {Θ, } for i = 1, 2. Suppose given T i and T j, Θ i, Θ j / {, Θ}. Then firm i s profits may be written as: π i = α (T i (θ) cqi L (θ))f(θ)dθ + α (T i (θ) cqi L (θ))f(θ)dθ Θ i, j 2 Θ i,j + 1 α (T i (θ) cqi L (θ))f(θ)dθ (23) 2 Θ Clearly then, if the pair (T i, T j ) is an equilibrium, then firm i will not find it profitable to deviate by replicating T j over the set qj B (θ) > 0. Therefore, we derive the inequalities that for all i {1, 2}: 1 α (T i (θ) cqi L (θ))f(θ)dθ 1 (T j (θ) cqj L (θ))f(θ)dθ (24) 2 Θ j, i 2 Θ j, i Alternatively, we may consider a deviation by firm i by replicating T j over all q R +. Such a deviation is unprofitable if and only if: π i 1 (T j (θ) cqj L (θ))f(θ)dθ (25) 2 Θ 19

21 Eq. 25 must also hold for firm j i if the pair (T i, T j ) is, indeed, an equilibrium. Therefore, we have: (T i (θ) cqi L (θ))f(θ)dθ + (T j (θ) cqj L (θ))f(θ)dθ Θ i, j Θ j, i (T i (θ) cqi L (θ))f(θ)dθ (T j (θ) cqj L (θ))f(θ)dθ 0 (26) Θ j, i Θ i, j Clearly, this cannot be true given Eq. 24 and thus, such a partition of Θ is not possible. Therefore, if an equilibrium exists, for all i {1, 2}, Θ i {, Θ}. Therefore, if a pure strategy equilibrium exists, then Θ i {Θ, } for i = 1, 2. Suppose that Θ i = Θ j = Θ. Then T i = T j and profits are such that π i = π j. As a result, if such an equilibrium exists, it must be unprofitable for one firm to undercut by ɛ > 0. This amounts to the condition: ɛ α (T i (qi B (θ)) cqi B (θ))f(θ)dθ > 0 (27) 2 Θ which, by assumption, is not possible since α > 0 and ɛ maybe arbitrarily chosen. Therefore, suppose that given T i and T j, Θ i = and Θ j = Θ. Then all bargain hunters purchase from firm j. Clearly then profit maximization requires that T i qi L T. Similarly, profit maximization requires that T j qj L = T i ɛ qi L T for some ɛ > 0. The above implies that all competition between firms 1 and 2 must occur in û 1 and û 2. Suppose these are equilibrium values and, without loss of generality, assume that û 1 < û By continuity, ɛ > 0 such that û 1 < û 2 ɛ and that profits for firm 2 increase. Clearly then, a unilateral deviation for firm 2 exists, violating the notion of an equilibrium. Therefore, suppose that û 1 = û 2. Then each firm captures exactly half of the market. Again, by continuity, ɛ > 0 such that for firm i that increases û i to u i = û i + ɛ will capture the whole of the bargain hunters. This gain in profits is clearly greater than the loss due to the ɛ change so long as profits, given û i, are not zero. In the case that π i (û i ) = 0, this contradicts profit maximization since offering a contract with û i = 0 yields strictly positive profits. This may be accomplished by forgoing competition for bargain hunters and by monopolizing loyal consumers. Therefore, unilateral 20 Roughly speaking, one could imagine that û i is so low such that for some θ Θ, negative profits are earned. Consequently, we consider price schedules which flatten out over θ Θ whenever T (θ) c q(θ) < 0 along the total cost curve. 20

22 deviations exists for all values of û 1 and û 2 proving the nonexistence of a pure strategy equilibrium. Alternatively, one may think of a Bertrand price competition game with a strictly positive outside option. Proof of Lemma 1: First note that if Θ s,k is countable, then Ts i S(G s ) and T j k S(G k) intersects at least once at unique points over the relevant domain. 21 The proof of this, then, is analogous to that of Proposition 1. By definition of a mixed strategy equilibrium, all pure strategies in the support will yield the same level of expected profits. Therefore, consider any two price schedules offered by firms s, k = 1, 2 such that Θ s Θ k / {Θ, }. Define Θ s (Ts, i T j k ), with s k T s i S(G s ) and T j k S(G k), to be the θ Θ type bargain hunters that will purchase from firm s given Ts i and T j k. Θ s, k (Ts, i T j k ) and Θ s,k(ts, i T j k ) are similarly defined. Therefore, consider the strategy Ts i for firm s. Expected profits, given G k is: E(π s Ts, i G k ) = π s (Ts, i T j k )g k(t j k )dt j k (28) T j k S(G k) A similar expression may be derived for another pure strategy Ts l S(G s ). Then, first consider the case when Ts i and Ts l are such that Θ i (Ts, i Ts) l / {Θ, }. 22 We can define an ordering over a partition, indexed by Λ, of Θ such that either Θ λ s (Ts, i T j k ) {Θλ s (Ts, l T j k ) } or {Θλ s (Ts, i T j k ) } Θλ s (Ts, l T j k ) for all T j k S(G k) and λ Λ Θλ s = Θ. Then consider the pure strategy T s where: T s = { T l s if Θ λ s (T i s, T j k ) {Θλ s (T l s, T j k ) } T i s if {Θ λ s (T i s, T j k ) } Θλ s (T l s, T j k ) (29) It is clear that T s takes the lower envelope of the two functions Ts i and Ts. l Then, by the logic of the proof of Proposition 1, E(π s Ts, i G k ) = E(π s Ts, l G k ) E(π s T s, G k ) provides a contradiction, unless Θ λ s (Ts, i T j k ) Θλ s (Ts, l T j k ), or vice versa, for all λ Λ and for all T j k S(G k) which further implies that Ts i and Ts l are uniformly above or below, or partially overlaps, T j k for all T j k S(G k). For the remainder of the first part of the Lemma, we are left to show that if one firm randomizes price schedules, Ts i S(G s ), then the other firm will not randomize with any schedules, T j k S(G k) such that both Θ s, Θ k / {Θ, }, for all T j k S(G k) and for all Ts i S(G s ). Again, this 21 By this, we mean over R + such that q j z(θ) > 0 for z = 1, Since both T i s and T l s are offered by the same firm, we define Θ i(t i s, T l s) as the set of θ Θ who will prefer T i s over T l s if they were both available. 21

23 may be established in a manner similar to that of Proposition 1. Without loss of generality, suppose T j k T k l over the relevant domain. Then for any Ts, i Θ k (T j k, T s) i Θ k (Tk l, T s). i Consider π s (T j k, T s) i and π s (Tk l, T s) i and assume they are not pointwise equal in Ts. i If there exists a Ts i S(G s ) such that π s (T j k, T s) i = π s (Tk l, T s), i then consider a deviation to T k such that T k = Tk l over Θ k(tk l, T s) i and T k = Tk s elsewhere.23 It follows that π s ( T k, Ts) i > π s (T j k, T s) i = π s (Tk l, T s) i at the point Ts. i Furthermore, for all Ts v S(G s ) such that Θ k (Tk l, T s) i Θ k (Tk l, T s v ), π s ( T k, Ts) i π s (Tk l, T s) i in Ts. i Conversely, for all Ts v S(G s ) such that Θ k (Tk l, T s) i Θ k (Tk l, T s v ), the inequality π s ( T k, Ts) i π s (T j k, T s) i holds in Ts. i 24 Therefore, a deviation in pure strategy exists. In the case where there does not exist any Ts i S(G s ) such that π s (T j k, T s) i = π s (Tk l, T s), i then choose Ts i S(G s ) such that πs (T j k, T s) i π s (Tk l, T s) dx, i over the relevant domain, is minimized. A similar argument now completes the proof. The contradiction derived in the above proof requires both Θ l, Θ s / {Θ, }. Therefore, convexity of Θ s,k immediately follows if we allow for either Θ l / {Θ, } or Θ s / {Θ, } since non-convex sets will generate Θ k and Θ s otherwise. Proof of Lemma 2: Suppose û i < 0, then the individual rationality constraint will not hold. Therefore, in equilibrium, it must be the case that û i 0. Suppose û i > u. Then the maximum profits a firm receives will be lower than if it simply monopolizes the loyal group of consumers and does not compete for bargain hunters. 25 This is clearly not profit maximizing behavior, thus establishing an upper-bound on the support of G i in equilibrium for all i {1, 2}. Proof of Proposition 2: A mixed strategy must yield the same profits for all pure strategies in its support in order for it to be an equilibrium,. Therefore, it is without loss of generality that we restrict our attention to [0, u]. The expected profits for any û i [0, u] are given by: E (π i û i ) = π s i G j (û i ) + π f i (1 G j (û i )) (30) By the firm s individual profit maximizing constraint, Eq. 15, expected 23 Similarly, if π s(t j k, T s) i and π s(tk, l Ts) i are pointwise equal in Ts, i then consider any Ts i S(G s). 24 These inequalities follow since if Θ k (Tk, l Ts) i is an increasing set in Ts, i then Θ k ( T k, Ts) i is nondecreasing in Ts i and vice versa. Consequently, π s( T k, Ts) i is nondecreasing in all directions of Ts i if, in fact, Tk, l T j k S(G k) is, indeed, a mixed strategy equilibrium. 25 Maximum, here, is interpreted as in the case that the firm does offer the lowest price schedule and captures all of the bargain hunters. 22

24 profits must be at least equal to π. Therefore: E (π i û i ) π (31) In equilibrium, this condition must bind with equality since if E (π i û i ) > π, then firm j may simply reduce û j until this condition binds with equality. Therefore, rearranging Eq. 30 yields the equilibrium distribution for firm j. Conversely, firm i s equilibrium distribution function may be determined by the indifference condition of firm j. The symmetry of this problem makes the solution trivial. Note that the nonexistence of an optimal deviation is guaranteed by Lemma 1. Consequently, off equilibrium strategies may be disregarded. Proof of Proposition 3: For α = 1, the proof for the nonexistence of two pure strategies which yield Θ 1 Θ 2, both with positive measure, in Proposition 1 still holds. Thus, for all i {1, 2}, Θ i {, Θ}. So if Θ i =, then from Condition 2 of Lemma 1, we know that for all θ Θ j such that qj B(θ) > 0, T j < T i. Furthermore, π i = 0, and π j 0. In fact, π j = 0 since if π j > 0, then there exists some ɛ > 0 such that firm i may deviate by setting T i = T j ɛ and earn positive profits. Now, note that if for some θ Θ, qj B(θ ) > 0 and that T j (qj B(θ )) cqj B(θ ) > 0, then there must exist some θ θ such that T j (qj B(θ)) cqj B (θ) < 0. Since T j(qj B(θ)) < T i(qi B (θ)), by definition, firm j will not deviate by setting T j (qj B(θ)) = T i(qi B (θ)) if the profits from the permutated price schedule, Tj, is lower. This is only possible if T j ( q j B(θ)) c qb j (θ) T j (qj B(θ)) cqb j (θ) < 0 where qb j is the induced demand from T j. Note that T j ( q j B(θ)) < T i( q j B (θ)) since otherwise, a type θ individual will switch and firm j makes positive profits. So given any increases on portions of the tariff by firm j where negative profits are earned, such θ-types individuals will always maintain on such portions, a uniform increase over that curve is always feasible. But that implies that setting T j = c q is also possible over all q such that for all θ Θ, T j (qj B(θ)) cqb j (θ) < 0. But then, this is a contradiction since zero profits are earned from such θ-type consumers. Consequently, there cannot exist any θ-type for which firm j is making positive profits from and thus, profits must be pointwise equal to zero over all q for which qj B (θ) > 0. A simple Bertrand argument now completes the proof and is therefore, omitted. Proof of Proposition 4: 1. We prove this first by establishing that no firms will employ pure strategies such that, in equilibrium, Θ i / {Θ, }. We proceed by induction. 23