On Pure-Strategy Nash Equilibria in Price-Quantity Games

Transcription

1 On Pure-Strategy Nash Equilibria in Price-Quantity Games Iwan Bos Dries Vermeulen April 5, 2016 Abstract This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence is proved for a broad and natural class of price-quantity games. With differentiated products, the equilibrium outcome is similar to that of a price-only model. With undifferentiated products and limited spillover demand, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a wide variety of market conditions. Keywords: Bertrand Oligopoly; Bertrand-Edgeworth Competition; Cournot Oligopoly; Oligopoly Theory; Price-Quantity Competition. JEL Codes: D4; L1. We appreciate the comments of Attila Tasnádi, participants at the 2015 Oligo Workshop (Carlos III University of Madrid), the 2015 Society for Economic Design Conference (Istanbul Bilgi University), the 2015 European Association for Research in Industrial Economics Conference (Ludwig-Maximilians University Munich) and seminar participants at the University of East Anglia (Norwich, UK) and the University of Exeter (Exeter, UK). All opinions and errors are ours alone. Department of Organization & Strategy, Maastricht University. Corresponding author at: P.O. Box 616, 6200 MD Maastricht, The Netherlands. i.bos@maastrichtuniversity.nl. Department of Quantitative Economics, Maastricht University. 1

2 1 Introduction Imagine a remote village with two bakeries baking breads that are close substitutes in the eyes of the villagers. If this was a Cournot market, then both bakers would decide how much bread to bake and bring to the marketplace. An auctioneer would be waiting there to set the price in such a way that market demand equals the total amount offered. If, instead, this was a Bertrand market, then each bakery would set its selling price and wait for customers. After the realization of demand, both would be obliged to bake exactly the amount of bread their clients are asking for. A glance at reality shows, however, that such a bread market is neither a Cournot nor a Bertrand market. 1 It is not a Cournot market as bakeries commonly choose their own prices, but it is not a Bertrand market either as it is usually up to the bakers to decide how much bread to make. More generally, in free-market societies there typically is no law telling firms how much to supply and what price to charge. 2 A natural assumption in theories of oligopoly would therefore be that firms choose both their price and quantity of output. Edgeworth (1922, 1925). 3 The foundation for this type of analysis is due to Francis He recognized that price-setting oligopolists may be unable or unwilling to meet their demand. For example, an undertaking might not be able to generate sufficient supplies when there is a shortage and it is already producing at capacity. Similarly, it can be de facto capacity-constrained when production precedes sales and cannot be boosted instantly. Also, even when a firm has the ability, it will lack the incentive to satisfy all its customers when demand exceeds its competitive supply. 4 As is well-known, the disturbing conclusion of Edgeworth s explorations was that oligopoly prices are essentially indeterminate. In more modern language, we would say that no pure-strategy Nash equilibrium exists. This non-existence problem received much attention ever since and has been demonstrated in a variety of settings where firms compete in prices and quantities. It is often conjectured that the non-existence problem is, at least in part, driven by the frequently used homogeneous goods assumption as it gives rise to discontinuities in demand. In fact, Edgeworth himself hypothesized the degree of indeterminateness to diminish with an increase in product differentiation. 5 generally yield a solution. Assuming imperfect substitutes does, however, not Price-quantity games with heterogeneous products have been studied by Shapley and Shubik (1969), Alger (1979), Friedman (1988), Benassy (1989) and 1 For a detailed discussion of classical models of oligopoly, see Vives (1999). 2 Evidently, there may be regulated markets, but these are the exception rather than the rule. 3 The original version of his 1925-paper dates back to This may occur when firms cost functions exhibit decreasing returns to scale. Edgeworth (1922) studies the case of quadratic costs, whereas capacity constraints are considered in Edgeworth (1925). 5 See Edgeworth (1925, p. 121). 2

3 Canoy (1996). Under a variety of cost and demand conditions, all establish that the problem of non-existence remains for industries with a few firms selling sufficiently close substitutes. In this paper, we re-examine price-quantity competition in oligopolies. 6 Specifically, our purpose is to shed light on the non-existence problem by presenting conditions under which a pure-strategy equilibrium does and does not exist. We consider price-quantity games with continuous as well as discontinuous demand in terms of the own prices. The first is characteristic for differentiated goods, whereas the second applies to perfect substitutes. For both cases, we establish that there is a broad and natural class of price-quantity games with a pure-strategy equilibrium. Additionally, we are able to characterize these equilibria. With differentiated products, the equilibrium is shown to coincide with that of a basic Bertrand model. Moreover, and contrary to previous findings, its existence need not directly depend on the number of firms or the degree of differentiation. With discontinuous demand, equilibria come in two types. First, there is a range of rationing equilibria in which combined production falls short of market demand. This result is surprising as it has long been argued that the competitive outcome would be the only equilibrium candidate. 7 Although we do find that firms in equilibrium price at marginal costs, these prices are typically too low to ensure saturation of the market. Second, there may again be an equilibrium reflecting the outcome of a price-only model. Therefore, our analysis additionally provides a foundation for Bertrand price models. The non-existence problem pertains to pure-strategy Nash equilibria. Mixed-strategy equilibria have been repeatedly shown to exist under a variety of assumptions. 8 Mixed equilibria are, however, problematic within this context for at least two reasons. Perhaps the most prominent problem is simply that it is at odds with reality. Indeed, butchers, brewers or bakers typically do not roll dice to determine their prices or quantities of output. This is especially cumbersome when considering repeated interaction among firms. 9 Another issue 6 Regarding price-quantity models, it is not uncommon in the literature to distinguish between, on the one hand, production to order or Bertrand-Edgeworth competition and, on the other hand, production in advance (see, e.g., Shubik (1955, 1960), Maskin (1986) or Tasnádi (2004)). In case of the former, production takes place after the realization of demand, whereas in case of the latter production precedes sales. All results derived in this paper apply equally well to both cases. 7 See the pioneering works of Shubik (1955, 1960) and, in particular, Theorem 2 in Shubik (1960, p. 100). 8 See, among others, Levitan and Shubik (1972), Dixon (1984), Gertner (1986), Maskin (1986) and Tasnádi (2004). 9 In this regard, Friedman (1988, p. 608) stated that:... a deep understanding of oligopolistic behavior cannot be achieved by means of singleperiod models with results that are unreasonable in a multiperiod setting. This article is intended as a prelude to the study of dynamic models. Mixed strategies, under which prices and outputs are randomly selected in each period, seem bizarre in an infinite-horizon oligopoly. 3

4 is that models of oligopoly become significantly more complicated when allowing for random strategies. In particular, it is often difficult to provide an explicit characterization of mixedstrategy equilibria. As such, this concept is of limited use in applied research on oligopolistic industries. The next section gives a general outline of the model and offers some preliminary insights that serve as a basis for the ensuing study. The main analysis and results are presented in sections 3 and 4. Section 3 considers price-quantity competition with continuous demand, whereas the discontinuous demand case is described in Section 4. A few illustrative examples are presented in Section 5. Section 6 concludes. 2 A Model of Price-Quantity Competition In this section, we lay out the basics of our price-quantity oligopoly model. It is assumed there are n 2 profit-maximizing firms where the set of firms is N = {1,..., n}. Each firm i N simultaneously picks a price-quantity pair (p i, q i ), with p i 0 and q i 0. Let the vector of prices and quantities be respectively given by p = (p 1,..., p n ) and q = (q 1,..., q n ). Moreover, p i = (p j ) j i and q i = (q j ) j i. Given the price-quantity choices of competitors, demand for the products of firm i is denoted D i (p i, p i, q i ). Firm i s cost of production is given by C i (q i ), which is a strictly increasing continuous function of its own output q i. Both the demand and cost structure will be further specified below. The firm s objective is then to pick a profit-maximizing price-quantity pair: max Π i (p, q) = p i s i (p, q) C i (q i ), i N, (1) p i,q i where s i (p, q) denotes firm i s sales. It is clear that a firm cannot offer more than it produces and cannot sell more than is demanded. Therefore, sales equals the minimum of production and demand: 10 s i (p, q) = min {q i, D i (p i, p i, q i )}. A price-quantity pair (p i, q i ) is firm i s best response to (p i, q i ) when for each choice ( p i, q i ) it holds that Π i (p i, q i, p i, q i ) Π i ( p i, q i, p i, q i ). 10 Note that, due to the profit-maximizing assumption, firms bring all produced quantities to the market. Additionally, they will not produce more than the profit-maximizing output at the price set. This condition, albeit more general, therefore effectively coincides with the voluntary-trading constraint in standard Bertrand- Edgeworth settings. 4

5 The above specifications are sufficient to show the following basic and useful result stating that none of the firms find it optimal to produce in excess of demand. Lemma 1. For all i N, a strategy (p i, q i ) for which q i > D i (p i, p i, q i ) is not a best response to (p i, q i ). Proof Suppose that (p i, q i ) is firm i s best response to (p i, q i ) and suppose further that q i > D i (p i, p i, q i ). Since s i = min{q i, D i (p i, p i, q i )}, it holds that s i = D i (p i, p i, q i ). Now define q i = q i + D i (p i, p i, q i ). 2 As C i is continuous and strictly increasing, it follows that C i (q i ) > C i ( q i ). Combining with s i = D i (p i, p i, q i ), this implies Π i (p i, q i, p i, q i ) < Π i (p i, q i, p i, q i ), which contradicts the assumption of (p i, q i ) being a best response. In light of the purpose of this study, we restrict ourselves in the following to pure-strategy Nash equilibria. A price-quantity vector (p, q) is an equilibrium when for all i N and each choice ( p i, q i ) it holds that Π i (p i, q i, p i, q i ) Π i ( p i, q i, p i, q i ). Lemma 1 excludes the possibility that in equilibrium a firm operates above its demand curve. As will become clear in the ensuing analysis, and depending on the specifics of the model, the two other types of outcome might occur in equilibrium. Akin to the Bertrand model, all firms may choose to meet their demand at the price set. Alternatively, one or more firms may find it optimal to satisfy only a subset of their consumers. To indicate these possibilities, we will use the following definitions: An equilibrium (p, q ) is a Bertrand equilibrium when (p i, q i ) is a solution to: subject to max p i Π i (p i, q i, p i, q i) = p i s i (p i, q i, p i, q i) C i (q i ), q i = D i (p i, p i, q i ), i N. An equilibrium (p, q ) is a rationing equilibrium when q i D i (p i, p i, q i ) for each firm i and qj < D j(p j, p j, q j ) for at least one firm j. 5

6 3 Price-Quantity Competition with Continuous Demand Let us begin by considering price-quantity competition with differentiated products. If goods are less than perfect substitutes, firms typically do not lose their entire clientele when setting their price just above that of the competition. In a similar vein, slightly undercutting rivals prices is unlikely to yield the market. To capture this idea that small changes should have small effects, it is natural to think of demand as a continuous function of prices. Hotelling (1929, p. 44) formulated this as follows:...a discontinuity, like a vacuum, is abhorred by nature. More typical of real situations is the case in which the quantity sold by each merchant is a continuous function of two variables, his own price and his competitor s. Quite commonly, a tiny increase in price by one seller will send only a few customers to the other. To formalize, let Z i (p i, q i )(p i ) = D i (p i, p i, q i ) be the (residual) demand of firm i and let p i be the firm s marginal costs at zero output. We start by making the following assumptions on demand and cost. For all (p i, q i ) and i N: A1 Z i (p i, q i )(p i ) is continuous and non-increasing in p i. A2 Z i (p i, q i )(p i ) > 0. A3 C i (q i ) is differentiable and C i (0) = 0. Moreover, C i (q i) is continuous and non-decreasing with C i (q i) > 0 at all q i > 0. A1 reflects the law of demand for firm i s product. A2 and A3 together ensure that firms have an incentive to be productive. In particular, note that p i C i (0) and that cost functions are assumed to exhibit constant or decreasing returns to scale. Because both price and quantity are decision variables, it is in principle possible that production and demand do not coincide. Recall, however, that overproduction will not occur in equilibrium (Lemma 1). As the next result shows, the properties A1-A3 are sufficient to show that, in equilibrium, shortages will not occur either. Moreover, equilibrium profits are strictly positive. Lemma 2. Under A1-A3, all equilibria are Bertrand equilibria and equilibrium profits are strictly positive. 6

7 Proof Suppose that (p i, q i ) is a best reply of firm i to (p i, q i ). For an equilibrium to be a Bertrand equilibrium it must hold that q i = D i (p, q i ) for all i N. By Lemma 1, we know that q i > D i (p, q i ) cannot occur in equilibrium. Therefore, it remains to be shown that q i < D i (p, q i ) cannot occur either. Towards that end, we assume q i < D i (p, q i ) and derive a contradiction. To begin, we show that equilibrium profits are strictly positive. By A2, there is a price p i > p i with D i ( p i, p i, q i ) > 0. Because marginal cost functions are continuous and nondecreasing by A3, there exists a q i > 0 such that C i ( q i) < p i and D i ( p i, p i, q i ) > q i. As C i (q i) < p i for all q i < q i and C i (0) = 0 by A3, taking integrals on both sides of the inequality over the range [0, q i ] gives C i ( q i ) < p i q i. Since s i ( p i, q i, p i, q i ) = q i, we conclude that Π i ( p i, q i, p i, q i ) > 0. Thus, equilibrium profits are strictly positive. By the preceding argument, we can exclude the possibility that a firm s best response is to produce nothing, i.e., q i = 0 cannot occur in equilibrium. Therefore, suppose that 0 < q i < D i (p i, p i, q i ). By A1, there exists a ε > 0 such that for p i = p i + ε it still holds that q i < D i ( p i, p i, q i ). However, this implies Π i ( p i, q i, p i, q i ) = p i q i C i (q i ) > p i q i C i (q i ) = Π i (p i, q i, p i, q i ), which contradicts (p i, q i ) being a best reply of firm i to (p i, q i ). We conclude that in equilibrium it must hold that q i = D i (p, q i ), for all i N. A best reply (p i, q i ) of firm i to (p i, q i ) thus satisfies q i = D i (p i, p i, q i ). Consequently, (p i, q i ) is a best reply to (p i, q i ) precisely when p i maximizes R i (p i ) := p i D i (p i, p i, q i ) C i (D i (p i, p i, q i )) and q i = D i(p i, p i, q i ). Hence, all equilibria are Bertrand equilibria. Firms thus have a strict preference for price-quantity pairs on their demand curve; a finding that appears to be very robust. Indeed, it has repeatedly been found to hold under a variety of mild conditions. 11 This result greatly reduces the number of equilibrium candidates and is commonly taken as a basis for establishing non-existence of an equilibrium. By contrast, we will now use it as a foundation for proving existence. Towards that end, we introduce two assumptions on demand. In stating these assumptions, let E i (p i, p i, q i ) be a function such that D i (p i, p i, q i ) = max{e i (p i, p i, q i ), 0}. 11 See, for instance, Alger (1979, Theorem 3.1), Friedman (1988, Lemma 3), Benassy (1989, Theorem 1) and Canoy (1996, Lemma 1). In the context of an evolutionary model, Khan and Peeters (2015) have recently shown that this outcome may also result from firms imitating the most profitable producer. 7

8 For all (p i, q i ) and i N: B1 E i (p i, p i, q i ) is continuously differentiable and non-increasing in p i. Moreover, ξ > 0 such that E i p i ξ < 0. B2 K > 0 such that D i (p i, p i, q i ) K. i N B1 is similar to A1, but adds the requirement that a firm s demand is differentiable with respect to its own price and that this derivative is bounded away from zero. B2 states that market demand is bounded. Together with assumptions A2 and A3, these two conditions are sufficient for showing existence. Theorem 3. Under A2-A3 and B1-B2, there exists at least one equilibrium. Proof By Lemma 2, we know that all equilibria are Bertrand equilibria. In particular, (p i, q i ) is a best reply to (p i, q i ) precisely when p i maximizes R i (p i ) := p i D i (p i, p i, q i ) C i (D i (p i, p i, q i )) and q i = D i (p i, p i, q i ). Using this fact, we now proceed in three steps to prove existence of a pure-strategy Nash equilibrium. Step 1. We show that there is a unique price p i > 0 that maximizes R i. To begin, we know by Lemma 2 that equilibrium profits are strictly positive. Consequently, a price that maximizes R i is strictly positive and D i = E i. Specifically, using A3 and B1, this profit-maximizing price is obtained by equating marginal revenue with marginal cost: Rearranging gives E i = E i p i [ dci p i E i + p i E i p i dc i E i p i = 0. ]. (2) We now compare the LHS of (2) with the RHS. Define A i (p i ) = E i (p i, p i, q i ). By B1 and A2, we know that A i is non-increasing and A i (0) > 0. We now analyze the RHS of (2). Define F i (p i ) = E i p i [ dci (E i (p i, p i, q i )) p i Let us first have a closer look at the part between square brackets. Define B i (p i ) = dc i (E i (p i, p i, q i )) p i. 8 ].

9 Since E i is non-increasing in p i by B1 and C i (q i) is strictly positive and non-decreasing by A3, it follows that G i (p i ) = dc i (E i (p i, p i, q i )) is strictly positive and non-increasing. As all functions involved are continuous, there is a unique p m i > 0 with G i (p m i ) = p m i. Thus, B i (p m i ) = 0 and B i(p i ) p i. Because E i p i < ξ by B1, it follows that F i (p m i ) = 0 and F i (p i ) ξ (p i p m i ), for all p i > p m i. Finally, combining the previous observations on A i and F i, there is a unique p i > p m i > 0 with A i (p i ) = F i(p i ). Since dr i dp i = A i F i, dr i dp i (p i ) = 0. Moreover, dr i dp i (p i ) > 0 for p i < p i and dr i dp i (p i ) < 0 for p i > p i. We conclude that p i is the unique maximizer of R i. For each firm i N and a given choice vector (p, q), we can now define f i (p, q) := p i. Before proceeding, let us define a function H : R 2n + R 2n + for each choice vector (p, q) R 2n + : H(p, q) = (f 1 (p, q),..., f n (p, q), D 1 (p, q 1 ),..., D n (p, q n )). Step 2. We show that there is a point (p, q ) with H(p, q ) = (p, q ). By B2, we know that E i (0, p i, q i ) K and by B1 we further know that E i (p i, p i, q i ) K + ξ p i. Therefore, 0 p i K ξ and, following B2, 0 D i(p i, p i, q i ) K. Define M = max{k, K ξ }. Then, for any choice vector (p, q), H(p, q) 2n M. Next, we argue that H is continuous. Recall that f i (p, q) := p i is the unique maximizer of R i (p i ) := p i D i (p i, p i, q i ) C i (D i (p i, p i, q i )). Because both D i and C i are continuous, the maximum principle implies continuity of f i. As the demand of each rival is also continuous, H is continuous. By the Brouwer Fixed Point Theorem, we then know there is a point (p, q ) with H(p, q ) = (p, q ). Step 3. We show that (p, q ) is an equilibrium. Since p i = f i(p, q ), p i maximizes the function R i (p i ) := p i D i (p i, p i, q i) C i (D i (p i, p i, q i)). 9

10 Moreover, qi = D i(p i, p i, q i ). Thus, (p i, q i ) is a best reply to (p i, q i ). As this holds for each firm i N, (p, q ) is an equilibrium. This finding complements the results of Benassy (1989). Under the assumption of symmetry, he established that (i) for a given number of firms, there does not exist an equilibrium when the degree of substitutability between heterogeneous goods is sufficiently high, and (ii) for a given level of imperfect substitutability, there does exist an equilibrium provided that the number of undertakings is sufficiently large. 12 Theorem 3 not only allows for asymmetry, but also admits demand curves to be very flat independent of market size. In particular, an equilibrium may also exist when a small number of sellers produce close but imperfect substitutes. Let us conclude this part of our analysis by showing conditions under which an equilibrium does not exists. The next demand property is, in conjunction with A1-A3, sufficient. For all (p i, q i ) and i N: C1 If (p, q) is such that q j = D j (p j, p j, q j ), for all j, then Z i (p i, q i )(p i ) has a left derivative L and a right derivative R at p i. Moreover, both derivatives are strictly negative and the right derivative is larger than the left derivative (i.e., L < R < 0). C1 states that residual demand Z i (p i, q i )(p i ) has a (convex) kink when firms meet their demand. Therefore, and in contrast to assumption B1 above, firms demand functions are not differentiable when C1 holds. Theorem 4. Under A1-A3 and C1, there exists no equilibrium. Proof Suppose that (p, q ) is an equilibrium. In the following, we derive a contradiction. To begin, note that by A1-A3 Lemma 2 applies so that each equilibrium is a Bertrand equilibrium, i.e., qi = D i(p i, p i, q i ) for each firm i N. Next, define Zi = Z i (p i, q i )(p i ) and consider firm i. By C1, there exists an ε > 0 and numbers X and Y with X < L < Y < R < 0 such that for all p i with p i ε < p i p i it holds that Z i (p i, q i )(p i) Zi X (p i p i ) and for all p i with p i p i < p i + ε it holds that Z i (p i, q i )(p i) Z i Y (p i p i ). 12 The first finding has been confirmed by Canoy (1996), whereas the second conclusion can also be found in Shapley and Shubik (1969). 10

11 Since X < Y, Z i (p i, q i )(p i) Zi X (p i p i ) for all p i with p i ε < p i < p i + ε. Thus, (p i, q i ) is a solution to the following optimization problem: max p i q i C i (q i ) subject to q i = Z i X (p i p i ) and p i ε < p i < p i + ε. Using these constraints, this implies that p i of the function: is a solution to the problem of finding a maximum h X (p i ) = p i (Z i X (p i p i )) C i (Z i X (p i p i )) with domain p i ε < p i < p i + ε. Taking the derivative with respect to p i gives dh X dp i (p i ) = Z i + X p i 2X p i + X dc i (Z i X (p i p i )). Following A3, it is easy to see that dh X dp i (p i ) is strictly decreasing. Consequently, the equation dh X dp i (p i ) = 0 has at most one solution, which is the unique maximum. As we already established that p i is a solution to the above optimization problem, it holds that dh X dp i (p i ) = 0 and this equation can be rewritten to p i = dc i (Zi ) + Z i X. In a similar fashion, it can be deduced that p i = dc i (Zi ) + Z i Y. Yet, as X < Y, both cannot hold simultaneously. We believe it is instructive to clarify how this finding compares to Friedman (1988), because he also established non-existence in a simultaneous-move price-quantity setting. 13 Following Friedman s Assumption 3, it is easy to verify that our cost conditions (A3) are satisfied with the exception of C i (0) = 0. Regarding the demand side, both A1 and C1 follow from his demand specifications. 14 Therefore, the only two conditions that are in some sense more restrictive are C i (0) = 0 and Z i (p i, q i )(p i ) > 0, for all i N (see assumptions A2 and A3). The key difference with Friedman (1988) is that he establishes non-existence of interior equilibria, but allows oligopoly models that may have equilibria in which some or all firms have zero production levels. Theorem 4 above also excludes equilibria on the boundary. That is, as we impose C i (0) = 0 and Z i (p i, q i )(p i ) > 0, for all i N, our analysis precludes any type of pure-strategy equilibrium; also ones where one or more firms produce nothing. In sum, this section presents sufficient conditions for the (non-)existence of a pure-strategy equilibrium in price-quantity games with continuous demand. Specifically, we have shown 13 See Friedman (1988, p. 611), Theorem Specifically, condition C1 follows from the observation that the left derivative in Friedman (1988) equals fi i (p) < 0, whereas the right derivative equals fi i (p) + j i f j i (p) < 0 and f j i (p) > 0, for all j i. 11

12 there is a broad and natural class of differentiated goods games with an equilibrium. This class is broad and natural in the sense that many well-known oligopoly models meet the required assumptions. A few standard textbook examples are presented in Section 5 of this paper. Moreover, all equilibria exhibit Bertrand behavior. Indeed, even though sellers have no obligation to satisfy the demand forthcoming to them, they are happy to do so. 4 Price-Quantity Competition with Discontinuous Demand Let us now direct our attention to price-quantity games with demand functions that are discontinuous in price. Discontinuous demand is characteristic of industries with undifferentiated products. Though arguably not as realistic as price-quantity competition with continuous demand, this case is of interest because it plays a central role in classical theories of oligopoly and remains popular today. To begin, let us list some assumptions on cost and demand that will hold throughout this section. For all (p i, q i ) and i N: D1 C i (q i ) is differentiable and C i (0) = 0. Moreover, C i (q i) is continuous and non-decreasing with C i (q i) > 0 at all q i > 0 and p i C i (0) = 0. D2 Z i (p i, q i )(p i ) is non-increasing in p i. D3 δ, κ > 0 such that for all p i δ with p i p j, j i, it holds that Z i (p i, q i )(p i ) > κ. Our assumptions on cost are comparable to the continuous demand analysis. Specifically, D1 is identical to A3, but the requirement that marginal costs are zero at zero output. Regarding residual demand, we impose D2, which is standard and a weaker version of A1. Finally, D3 ensures that each firm can guarantee itself demand by charging a sufficiently low price. 15 In general, the demand of each firm comprises direct and indirect or spillover demand. Note that in case goods are homogeneous, a firm faces direct demand only when there is no cheaper supplier in the market. Indeed, when consumers are only concerned with price, those that are not among the lowest-priced firms may only receive demand from unserved customers who still plan to purchase the product or service even at a higher price. In the following, the degree of spillover demand is captured by the parameter α [0, 1], where α = 1 allows for 15 This is ensured when demand is continuous and strictly positive at price zero. Yet, as this section deals with discontinuous demand, we need such a slightly more technical assumption. 12

13 complete spillover and α = 0 indicates the absence of spillover demand. Specifically, we make two additional assumptions on firm demand. For all (p i, q i ) and i N: D4 α > 0 such that for every (p, q) and i with p i = p j, for at least one j i, and for every p i with p i < p i it holds that Z i (p i, q i )(p i) α Z i (p i, q i )(p i ). D5 r i > 0 for which D i (r i, p i, q i ) = 0. D4 captures a certain type of consumer behavior. Specifically, it states that there is a fraction of at least 1 α of customers that base their decisions only on price and visit at most one firm. Furthermore, note that any demand function that satisfies D4 for α < 1 is automatically discontinuous. D5 in some sense mirrors D3 by imposing a choke price. That is, each firm will find itself without customers when it charges a sufficiently high price. These assumptions are sufficient to show the existence of a pure-strategy equilibrium in which at least one firm does not meet its demand, i.e., there are rationing equilibria. Theorem 5. Under D1-D5, ᾱ (0, 1) such that α < ᾱ rationing equilibria exist. Proof Consider a fixed quantity ˆQ > 0 and let ˆP = min i { dc i ( ˆQ)}. By D1, we know that dc i ( ˆQ) > 0 for each firm i N and therefore ˆP > 0. Next, consider a value P with 0 < P < ˆP. By D1 and the definition of ˆP, we know there is a quantity q i (P ) with dc i (q i (P )) = P. Now take δ > 0 and κ > 0 as in D3. By D1, there exists a P > 0 with (i) P < ˆP, (ii) P δ and (iii) q j (P ) < κ, j. Next, we write p i = P and q i = q i(p ) and argue that is an equilibrium. (p, q ) = (P,..., P, q 1,..., q n) Consider firm i and let the strategies of the other firms be as given above. We will argue in three steps that (P, qi ) is a best response. Step 1. We show that q i < D i(p, q i ). Note that 0 < p i = P δ. So, following D3, D i (p, q i ) > κ and therefore q i = q j (P ) < κ < D i (p, q i ). Step 2. Let (p i, q i ) be any best response. We show that p i = P. 13

14 Suppose p i < P. We derive a contradiction. We first argue that q i < q i. Since p i < P = p i, D i (p, q i) = Z i (p i, q i)(p i ) Z i (p i, q i)(p i ) = D i (p i, p i, q i), by D2. By Step 1, therefore, q i < D i(p, q i ) D i(p i, p i, q i ). Consider the function f i (q i ) = p i q i C i (q i ). This function is concave in q i, with derivative f i (q i) = p i dc i (q i ). Notice that f i(q i ) = p i dc i (q i ) = p i dc i (q i (P )) = p i P < 0. Thus, since q i is a best choice given p i, it holds that p i = dc i (q i ). Then dc i (q i ) = p i < P = dc i (q i ), or dc i (q i ) < dc i (q i ). So, since dc i is non-decreasing, q i < q i. We argue that (p i, q i) is a strictly better choice than (p i, q i ). Since q i < qi, we know by Step 1 that q i < D i (p, q i ). Hence, Π i (p i, q i, p i, q i) = p i q i C i (q i ) < p i q i C i (q i ) = Π(p i, q i, p i, q i). We show that p i = P. By the analysis above, we know that p i P. Suppose (p i, q i ) is such that p i > P. Write Π i = p i q i C i (q i ). By Step 1, Π i is the profit for firm i corresponding to (p, q ). Let r i be as in D5 and suppose Π that α < i r i D i (p,q i ). We show that (p i, q i ) yields a strictly higher payoff than (p i, q i ). Since p i > P, using D4 to get the second inequality, the resulting profit of (p i, q i ) is Π i (p i, q i, p i, q i) = p i s i (p i, q i, p i, q i) C i (q i ) We conclude that p i = P. Step 3. p i D i (p i, p i, q i) r i α D i (p, q i) Π i < r i r i D i (p, q i ) D i(p, q i) = Π i. Given p i = P, we show that the quantity q i maximizes profit. Consider the function g i (q i ) = p i q i C i (q i ). This function is concave in q i, with derivative g i (q i) = p i dc i (q i ). Then, by the definition of q i, g i (q i ) = 0. So, since q i < D i(p, q i ) by 14

15 Step 1, q i indeed maximizes the profit function of firm i given price p i. This completes the proof that (p, q ) is an equilibrium. Finally, note that, for any choice of P as above, the corresponding quantity q i is strictly less than the demand for firm i s products. Thus, there exists rationing equilibria for α sufficiently small. The above result shows there are pure-strategy solutions when spillover demand is not too strong. The logic behind this result is as follows. On the one hand, none of the firms should have an incentive to cut price. Clearly, a price cut will be profitable only when it generates sufficient additional sales. Since firms are operating on their competitive supply curve, however, none of them would be willing to serve more demand. Indeed, when firms are already unwilling to meet demand at the going price, shortages would be even larger at lower prices. On the other hand, none of the firms should have an incentive to elevate prices. This is the case when a price increase would result in a sufficient drop of sales. In the context of our model, this implies that spillover demand as reflected by α must be sufficiently small. It is noteworthy that this significant drop in demand requirement is at the heart of various known existence solutions to the Edgeworth paradox. 16 What is more surprising is the type of equilibrium. In particular, there is an interesting contrast with the pioneering work of Shubik (1955, 1960). Even though he did not solve the Edgeworth paradox, he did establish the efficient point as the sole equilibrium candidate. 17 In the efficient point, firms price at marginal cost and saturate the market. The above result confirms the first requirement (i.e., firms indeed produce at a level where price equals marginal cost), but not the second. That is, sellers may not meet their demand at the price set and satisfy only part of their customers in equilibrium. In particular, and as reflected by equilibrium rationing, market prices might be too low to be efficient. To be clear, this is not to say that the efficient point is never an equilibrium. In fact, with identical technologies and absent spillover demand, there does exist a Bertrand equilibrium. In accordance with Theorem 5, however, there also still is a range of rationing equilibria. Theorem 6. Assume α = 0 and C i (q i ) = C(q i ), i N. There exists a set of symmetric equilibria. Precisely one of these equilibria is a Bertrand equilibrium. The remaining equilibria are rationing equilibria. 16 See, among others, Levitan and Shubik (1972) and Dixon (1990, 1992). The underlying logic can already be found in Shubik (1955, 1960). More recently, Tasnádi (1999) proves the existence of a pure-strategy equilibrium when demand is sufficiently elastic. 17 See Theorem 2 in Shubik (1960, p. 100). 15

16 Proof Using the symmetry conditions, all equilibria (p, q ) = (P,..., P, q 1,..., q n) from Theorem 5 reduce to (P,..., P, Q,..., Q ). In this case, Q D i (p, q i ). All equilibria with Q < D i (p, q i ) are rationing equilibria. The equilibrium with Q = D i (p, q i ) is a Bertrand equilibrium. Similar to the previous section, let us conclude this part by presenting a non-existence result. Towards that end, we introduce the following two assumptions. For all i N: E1 If (p, q) is such that q j D j (p, q j ), j, then there is an ε > 0 and K > 0 such that for each p i with p i < p i < p i + ε it holds that Z i (p i, q i )(p i) Z i (p i, q i )(p i ) K (p i p i ). E2 If q j D j (p, q j ), j, and p i < p i, then D i (p i, p i, q i ) D i (p, q i ) + D j (p, q j ) for each p j p i, j i. Property E1 captures a spillover effect. It states that a seller who raises his price slightly will not lose too much demand when his competitors are capacity-constrained. One can imagine consumers to accept a somewhat higher price when they cannot get the product or service somewhere else. Notice that as Z i (p i, q i )(p i ) is non-increasing by D2, Z i (p i, q i )(p i ) is right-continuous at p i. E2 ensures that buyers visit the lowest-priced firms first. They may visit a higher-priced firm, but only when the cheapest suppliers produce strictly less than their demand. E2 thus forces the demand function to be discontinuous. Together with some of the previous assumptions, these conditions are sufficient to undermine the existence of an equilibrium. Theorem 7. Under D1-D3 and E1-E2, there exists no equilibrium. Proof Assume that (p, q ) is an equilibrium. We proceed in five steps to derive a contradiction. Step 1. We show that Π i (p, q ) > 0, for all i. 16

17 Consider a firm i. Given (p i, q i ) it is sufficient to show that there exists a strategy (p i, q i ) for which firm i s profits are strictly positive. Suppose p i = 0 so that firm i has at most zero profits. By D2 and E1, we know that Z i (p i, q i )(p i) is right-continuous at p i = 0. As Z i(p i, q i )(0) > 0 by D3, there is a p i > 0 with Z i (p i, q i )(p i) > 0. Moreover, C i (0) = 0 by D1 and dc i (0) = 0 by D1. Consequently, firm i s profits are strictly positive for q i sufficiently small; a contradiction. Suppose now that p j > 0, for all j, and assume that δ and κ are as in D3. Furthermore, define ε = min{δ, p j j i}. Observe that ε > 0. Hence, and following D1, there is an η > 0 with C i (q i ) < ε q i for all q i η. Next, define p i = ε and q i = min{η, κ}. Thus, q i κ < D i ( p i, p i, q i ) and therefore s i = q i. Moreover, 0 < q i η so that C i ( q i ) < ε q i. In turn, this implies Π i ( p i, q i, p i, q i ) = p i q i C i ( q i ) = ε q i C i ( q i ) > 0. We conclude that equilibrium profits of all firms are strictly positive. Step 2. We show that qi = D i(p, q i ), for all i. By Lemma 1, q j D j(p, q j ), for all j. Suppose now that q i < D i (p, q i ). Since q j D j (p, q j ), for all j, Z i(p i, q i )(p i) is right-continuous at p i by assumptions D2 and E1. Thus, firm i can increase its payoff by raising its price slightly, while still selling the same amount q i as at p i. Step 3. We show that p i C i (q i ), for all i. Consider a firm i and notice that, for a given price p i, q i solves: max p i q i C i (q i ) subject to 0 q i D i (p, q i ). By D1, the objective function is concave in q i. At q i, we then either have that q i < D i(p, q i ) and p i = dc i (qi ) or q i = D i (p, q i ) and p i dc i (qi ). By Step 2, we know that q i = D i (p, q i ) and therefore p i C i (q i ). 17

18 Step 4. We show that p i C i (q i ), for all i. Consider a firm i and suppose that p i = dc i (qi ). Following Step 2, we can choose ε > 0 and K > 0 such that E1 holds. Let L > K and consider the following optimization problem: max p i q i C i (q i ) Using Lagrange, it must hold that subject to q i = Z i (p, q )(p i ) L (p i p i ) and p i p i. p i = dc i (q i ) + q i L. Notice that by D1, if q i > q i, then p i > p i. Furthermore, by picking q i > q i sufficiently close to q i and by choosing L sufficiently large, we can ensure that p i = dc i (q i ) + q i L < dc i (q i ) + ε 2 + q i L < p i + ε. For this choice, therefore, the profit at (p i, q i ) is strictly larger than at (p i, q i ) provided that firm i will sell its entire production, q i. Note that by our choice of ε and K it holds that: Z i (p i, q i)(p i ) Z i (p i, q i)(p i ) K (p i p i ) > Z i (p, q )(p i ) L (p i p i ) = q i. Thus, firm i will indeed sell all its output. Step 5. Suppose that p i p j, j. We construct p i and q i such that Π i(p i, p i, q i, q i ) > Π i(p i, p i, q i, q i ). Let us first define q i. As there are at least two firms, there is at least one firm j with p i p j. By Step 1, we know that p i > 0 and by Step 2 we know that q j D j (p, q j ), for all j. Now consider a price p i < p i, which by E2 means that: D i (p i, p i, q i) D i (p, q i) + D j (p, q j). Define A = D j (p, q j ). Note that by Step 1, A > 0. Moreover, we know by Step 2 that qi = D i(p, q i ) and by Step 3 and 4 that p i > dc i (qi ). Thus, because dc i is continuous in q i by D1, we can take q i > q i such that dc i (q i ) < p i and q i < D i(p, q i ) + A. Next, let us define p i. Given p i, define the function F i (q i ) = p i q i C i (q i ). 18

19 Following D1, F i is concave and F i (0) = 0. Furthermore, F i (q i ) is strictly increasing as long as p i > dc i (q i ). Since q i > q i and dc i (q i ) < p i, it follows that F i (q i) > F i (q i ). Notice that when p i converges to p i, the expression p i q i C i (q i). converges to p i q i C i(q i ) = F i(q i ). We can therefore choose p i < p i such that p i q i C i (q i) > F i (q i ). Finally, suppose that firm i charges price p i and produces quantity q i. Observe that D i (p i, p i, q i) = D i (p i, p i, q i) D i (p, q i) + A > q i. As q i < D i(p i, p i, q i ), it follows that Π i (p i, p i, q i, q i) = p i q i C i (q i) > F i (q i ) = Π i (p, q ). Thus, we conclude that (p i, q i ) is not a best response against (p i, q i ), which contradicts the assumption that (p, q ) is an equilibrium. The intuition behind this result is similar to the one underlying the original analyses of Edgeworth. Moreover, and unlike the continuous demand case, Theorems 5 and 6 show that with discontinuous demand it is the degree of spillover that plays a critical role in establishing existence. Specifically, firms have no incentive to put their focus on indirect rather than direct demand when spillover is sufficiently small. The willingness to fight for direct demand will be strong when, for example, a substantial part of unlucky customers prefers to leave the market after having failed to obtain the product or service directly. Also, it might be suspected that spillover demand will not endure in fairly stable markets. Under the assumption that time is valuable and transportation costly, it seems evidently suboptimal for customers to repeatedly receive their product indirectly. In other words, one would expect indirect demand to diminish in cases where the same customers are served each period. 18 Let us conclude this section with a note on costs. Recall that our assumptions on production technologies are the same as in the continuous demand analysis with the exception of p i 18 This suggests that spillover demand is more likely to last when products are sold on a first-come-first-served basis rather than under efficient rationing or any other situation where the same customers are (un)served. 19

20 C i (0) = 0, i.e., marginal costs are zero at zero output. The latter ensures that all firms are productive in equilibrium, but also excludes the possibility that marginal costs are constant at all output levels. In that particular case, our findings would be different, although the underlying logic remains. One notable difference is that, under the assumption that firms choose to meet their demand in case of zero economic profits, rationing equilibria will no longer exist leaving Bertrand-like outcomes as the only equilibrium candidates. Results, in this case, have more of an all-or-nothing flavor. With heterogeneous costs, it will be the lowest-cost firm that serves the entire market. With identical cost functions, by contrast, an equilibrium exists when there is no spillover demand (i.e., when α = 0). Note that this latter outcome is equivalent to the traditional Bertrand-Nash equilibrium. 5 Examples Let us now illustrate our main findings with a few representative examples. In this section, we present five cases; the first three of which deal with a differentiated duopoly. Example 1 serves to show how equilibrium prices and quantities can be derived in a well-known textbook model. Example 2 illustrates how an equilibrium may fail to exist when demand is nondifferentiable (a violation of assumption B1), whereas Example 3 shows that an equilibrium may still exist when demand functions exhibit a kink. Example 2 and 3 specifically highlight how (non-)existence of equilibrium depends critically on whether a firm s spillover demand is a function of its own selling price. The final two examples, Example 4 and 5, consider a homogeneous good oligopoly with and without spillover demand. 5.1 Heterogeneous Products EXAMPLE 1. (Bowley (1924)) In this first example, we examine a well-known representative consumer model as introduced by Bowley (1924). The (quasi-linear) utility of consuming a quantity s 1 at a price p 1 and a quantity s 2 at price p 2 is given by U = α (s 1 + s 2 ) 1 2 β (s θs 1 s 2 + s 2 2) p 1 s 1 p 2 s 2, which is the gain from consumption V = α (s 1 + s 2 ) 1 2 β (s θs 1 s 2 + s 2 2) minus the total expenditure of p 1 s 1 + p 2 s 2. Notice that utility depends on sales (s i ) and not on demand (D i ) or production (q i ). The degree of differentiation is captured by the differentiation (or distance ) parameter θ [0, 1]. Both goods are perfect substitutes when 20

21 θ = 1 and independent in demand when θ = 0. For simplicity, it is assumed in the following that firms have common, constant marginal cost c > 0. To determine the utility-maximizing allocation of sales, take the first-order conditions with respect to s 1 and s 2 : α βs 1 βθs 2 p 1 = 0, α βs 2 βθs 1 p 2 = 0. Rearranging gives the inverse demand function for both firms: p 1 = α β (s 1 + θs 2 ), p 2 = α β (s 2 + θs 1 ). Thus, given prices p 1 and p 2, the representative consumer aims to acquire [ 1 s 1 = β (1 + θ) α p 1 1 θ + θp ] 2, 1 θ s 2 = 1 β (1 + θ) [ α p 2 1 θ + θp 1 1 θ ]. Note that this model meets the requirements of A1-A3 and B1-B2 provided that production is not too expensive. Thus, by Theorem 3, we know there is a pure-strategy equilibrium. Moreover, by Lemma 2, we know that this equilibrium is a Bertrand equilibrium. Thus, firm 1 maximizes Π 1 = (p 1 c) [ 1 β (1 + θ) α p 1 1 θ + θp ] 2, 1 θ which yields the following corresponding best reply function: p 1 = 1 2 [c + (1 θ) α + θ p 2]. Using symmetry and solving the resulting system of best response functions gives the equilibrium (p, q ): p 1 = p 2 = c + α (1 θ) 2 θ and D 1 = D 2 = q 1 = q 2 = α c β (1 + θ) (2 θ). For instance, suppose that α = 18, β = 2 3, θ = 1 2 and C i(q i ) = 3q i for i = 1, 2. This gives demand functions D 1 (p 1, p 2 ) = [18 2p 1 + p 2 ] + and D 2 (p 1, p 2 ) = [18 2p 2 + p 1 ] + and the following equilibrium prices and quantities: p 1 = p 2 = c + α (1 θ) 2 θ = 8 and D 1 = D 2 = q 1 = q 2 = α c β (1 + θ) (2 θ) =

22 EXAMPLE 2. ((Almost) Friedman (1988)) Let us now use the previous specifications (i.e., α = 18, β = 2 3, θ = 1 2 and c = 3) and extend the model by adding the possibility of complete spillover. Thus, a firm s demand includes all its competitors unserved customers. Demand for the products of firm 1 is then given by { [18 2p 1 + p 2 ] + if 18 + p 1 2p 2 q 2, D 1 (p 1, p 2, q 2 ) = [36 p 1 p 2 q 2 ] + if 18 + p 1 2p 2 > q 2. Firm 2 s demand function is symmetric. Such a setting closely mimics the framework of Friedman (1988) and the underlying logic for non-existence of equilibrium is comparable. 19 To see that this demand specification does indeed not admit an equilibrium, first observe that assumptions A1-A3 still hold. Hence, Lemma 2 applies so that in equilibrium q i = D i, for i = 1, 2. This implies D 1 = [18 2p 1 + p 2 ] + and D 2 = [18 2p 2 + p 1 ] +. Note that this is precisely the demand structure of Example 1 leaving p 1 = p 2 = 8 and q 1 = q 2 = 10 as the only equilibrium candidate. Given that p 2 = 8 and q 2 = 10, firm 1 s demand function reduces to D 1 (p 1 ) = { [26 2p 1 ] + if 2 + p 1 10, [18 p 1 ] + if 2 + p 1 > 10. It can now be easily verified that (p 1, q 1 ) = (8, 10) is no longer a best reply. In fact, given (p 2, q 2 ) = (8, 10), firm 1 has an incentive to generate spillover demand by hiking its price. Specifically, it maximizes Π 1 = (p 1 3) [18 p 1 ] +, which reaches its maximum at p 1 = Thus, in this case, a best reply to (p 2, q 2 ) = (8, 10) is no longer (p 1, q 1 ) = (8, 10), but (p 1, q 1 ) = (10.5, 7.5). EXAMPLE 3. (Kinked Demand with a Solution) Following Friedman (1988) and the previous example, it may be tempting to conclude that no equilibrium can exist in models with complete spillover. We will now present an example showing that such a conclusion would be premature. We again use the above Bowley model and assume that c = 3, β = 1 and θ = 0 (i.e., both firms are (local) monopolists). Absent spillover, demand functions are D 1 (p 1 ) = α p 1, D 2 (p 2 ) = α p Formally, this setting does not fall within the class of models considered by Friedman (1988) as it allows for strictly positive demand even when direct demand is zero. That is, for a sufficiently high price, demand of a firm may consist exclusively of customers that visited the rival first. Unfortunately, we have been unable so far to construct meaningful or tractable demand specifications that fully fit the Friedman (1988) context. 22

23 In this case, equilibrium prices and quantities are respectively given by p 1 = p 2 = α+3 2 and q 1 = q 2 = α 3 2. In contrast, the demand functions with complete spillover are D 1 (p 1, p 2, q 2 ) = [α p 1 ] + + [α p 2 q 2 ] +, D 2 (p 2, p 1, q 1 ) = [α p 2 ] + + [α p 1 q 1 ] +. By Lemma 2, we know that in equilibrium q i = D i, for i = 1, 2. Thus, actual spillover is zero and therefore q1 = α p 1 and q 2 = α p 2. We conclude that there is an equilibrium with p 1 = p 2 = α+3 2 and q1 = q 2 = α 3 2, which is identical to the equilibrium absent spillover demand. 20 It is worth highlighting the difference with the previous example as in both cases the demand function exhibits a kink. The fundamental distinction lies in the nature of the kink. In this example, it is due to the production level of the rival. In particular, the indirect demand is independent of the own selling price. In Example 2, by contrast, the indirect demand does depend on the own price. That is, the demand functions exhibit a kink in p i and it is precisely this property that destroys the equilibrium in Example Homogeneous Products EXAMPLE 4. (No Spillover) Let total demand be given by D(P ) = A P, A > 0, and suppose that demand is divided equally among equally-priced firms. Moreover, let C i (q i ) = q 2 i, for all i N. This is a symmetric model and a special case of the model in Section 4 with α = 0 (see Theorem 6). Let P be a price associated with a symmetric equilibrium. As C i (q i) = 2 q i, it must be true that q i = 1 2 P. In equilibrium, it holds that qi D(P ) n or 1 2 P A P n, which can be rearranged to P 2A n+2. If P < 2A n+2, then the equilibrium is a rationing equilibrium. The equilibrium is a Bertrand equilibrium when P = 2A n+2. EXAMPLE 5. (Spillover) Consider the model of the previous example, but now with spillover demand (i.e., α > 0). Let P 1 := P and define recursively P k+1 = min{p i p i > P k }. For each k, write N k = {i p i = P k } and let Q k = i N k q i. Demand is defined as follows. Let A 1 = A and D(P ) = A 1 P. Suppose that D k is defined recursively and let A k+1 = D k (P k ) Q k. Furthermore, D k+1 (P ) = α A k+1 (P P k ) for 20 A similar result can be derived in the spatial model of Hotelling (1929). See Alger (1979, Theorem 3.1). 21 It is furthermore noteworthy that assumption B1 may also be satisfied when customers take into account other factors than price. For instance, when there are costs associated with queuing, buyers might approach another supplier well before their most preferred choice (absent queuing costs) is sold out. 23