Price Competition with Decreasing Returns-to-Scale: A General Model of Bertrand-Edgeworth Duopoly Blake A. Allison and Jason J. Lepore

November 5, 2014

Abstract This paper studies the equilibria of a price setting Bertrand-Edgeworth duopoly with general cost and demand structure. The generality of this formulation allows a broad range of speci…cations of costs of production, capacities/prior investment, and residual demand rationing. We bound the range of pricing in all equilibria and establish closed-form bounds on …rms’equilibrium expected payo¤s. The payo¤ bounds are used to provide a necessary and su¢ cient condition for the existence pure strategy equilibrium, and classify all pure strategy pricing as either “Cournot” or “Bertrand.” Further, we show that a moderate restriction guarantee that the existence of pure strategy equilibrium also implies uniqueness of equilibrium. Precise closed-form formula for expected payo¤s are shown for a class games that encompasses e¢ cient or proportional demand rationing with constant marginal cost up to capacity. Key words: Price competition, Demand rationing, Convex costs, Capacity constraints.

Allison: Department of Economics, University of California, Irvine; Lepore: Department of Economics, California Polytechnic State University, San Luis Obispo, CA 93407;

1

2

1

Introduction

In this paper, we present a novel approach to analyzing models of Bertrand-Edgeworth (BE) price competition. This approach allows us to greatly generalize the models in which pricing behavior can be characterized, accommodating a broader range of industries, including a large class of industries do not …t within the assumptions made in the literature on BE games. Since the inception of mathematical economics, the determination of prices in markets with very few sellers has been a central subject of inquiry. Edgeworth (1925) moved the understanding of this subject forward by appreciating the impact of consumer rationing and the prominence of price indeterminacy, or pricing cycles, in duopoly with decreasing returns to scale.1 Edgeworth’s basic insights were formalized into a game theoretic model by Shubik (1959), the …rst formulations of what is now called the Bertrand-Edgeworth oligopoly model.2;3 Shubik focused on understanding the range of pricing in mixed strategy equilibrium and the character of pure strategy equilibria when they exist. Motivated by Shubik’s work we aim to …nally answer the seminal questions he posed in a general setting. In particular, we answer these question in a model with general (and potential asymmetric) production technologies and consumer rationing in order to have broad implications. There is an extensive literature studying BE games in which …rms possess, constant marginal costs and demand is rationed according to either the e¢ cient or proportional rationing rule.4;5 This class of Bertrand-Edgeworth games has been used to understand fundamental issues in price determination. (Duopoly pricing and capacity investment: Levitan and Shubik (1972), Kreps and Scheinkman (1983), Osborne and Pitchik (1986), Davidson and Deneckere (1986), Allen and Hellwig (1993), Deneckere and Kovenock (1996), Lepore (2009); Sequential pricing: Deneckere and Kovenock (1992), Allen (1993), Allen et al. (2000); Large markets: Allen and Hellwig (1986), Vives (1986), Dixon (1992); Oligopoly: Hirata (2009), De Francesco and Salvadori (2010);Uncertainty: Reynolds and Wilson (2000), Lepore (2008, 2011), De Frutos and Fabra (2011)). A consequence of the assumptions made thus far in 1

Vives (1989, 1993) both provide excellent context for Edgeworth’s contribution to oligopoly. Before Shubik (1959), Shapley (1957) published an abstract with a description of results derived from game theoretic model of pricing. 3 Other early seminal contributions to BE competition were made by Beckman (1965), Shapley and Shubik (1969), and Levitan and Shubik (1972). 4 The e¢ cient rule for residual demand rationing speci…es that the highest value consumers are served by the lower price …rm and low value consumers are rationed. On the other hand, the proportional rule for residual demand rationing speci…es that a uniform distribution of the consumers can buy at the low price …rm, resulting in a larger residual demand than the e¢ cient rule. 5 Almost all the BE literature assumes that the …rms have symmetric constant marginal cost up to capacity. Deneckere and Kovenock (1996) and Allen et al. (2000) are the notable exceptions. These papers focus on interesting case of constant marginal costs of capacity that are asymmetric. Additionally, the bulk of this literature also restricts demand to be such that a …rm’s pro…t is concave in their price. Our analysis is based on the considerably weaker assumption that a …rm’s pro…t is single peaked in their price. 2

3 the literature is that the models fail to accommodate industries in which sales occur prior to production, as the argument that having an inventory of goods leads to approximately constant marginal cost does not apply. Despite the pertinence of factors such as general production technologies and demand rationing, only incremental progress has been made including these features in a theoretical model. Dixon (1992) is alone in characterizing a model of BE oligopoly with strictly convex costs, deriving conditions for the existence of pure strategy equilibrium such a setting.6 Advancement along these dimensions has been hindered by theoretical problems with existence of equilibrium (pure or mixed) in this setting. However, recent advances in the literature on existence of equilibrium in discontinuous games by Bagh (2010) and Allison and Lepore (2014) allow for the straightforward veri…cation of existence of equilibrium in vast generalizations of BE oligopoly. Our model allows for two large generalizations of the existing literature. First, we allow for production technology to vary greatly across …rms, assuming only that each …rm’s cost function induces an optimal supply function that is continuous and nondecreasing in price. The supply function inherently re‡ects capacity restrictions, serving a similar role in the analysis. An important distinction between the two is that unlike the capacity constraint, the supply function allows for an endogenous restriction of quantity that depends on the price. The second major generalization we allow for is in the process by which consumers are rationing between …rms when the demand facing the low priced …rm exceeds its supply. Rather than considering a particular rule, we allow for a general class of (possibly asymmetric) rationing schemes which need only satisfy the basic conditions that the residual pro…t be continuous and weakly decreasing in the other …rm’s price. This class includes both the e¢ cient and proportional rules in particular. We begin the analysis by establishing bounds on the expected pro…ts of all equilibria. In order to establish these bounds, we de…ne the following preliminary objects. First de…ne the judo price as the highest price a …rm can use to guarantee the other …rm would rather maximize its residual pro…t than undercut. This terminology is based on the sequential pricing model of Judo Economics in Gelman and Salop (1983).7 The maximum of the two …rm’s judo prices is de…ned as the critical judo price, and the monopoly pro…t that each …rm earns when setting this price is de…ned to be its critical judo pro…t. The second important price is de…ned as the safe price. A …rm’s safe price is the lowest price that a …rm may set such that the other …rm earns its max-min pro…t if it were to undercut. The maximum of the two …rm’s safe prices is de…ned as the critical safe price, and the monopoly pro…t of each …rm at this price is de…ned to be its critical safe pro…t. The …rst primary result of the paper is that the expected pro…ts of each …rm in all equilibria are bounded between its critical safe 6

Hoernig (2007) provides a thorough treatment of price competition with general cost structure and sharing rules for the classical Bertrand model with no consumer rationing. 7 Gelman and Salop (1983) show that, in a two period sequential game, a single potential entrant can use capacity restriction and judo pricing to induce an unconstrained monopolist to allow entry. The basic idea of judo economics has been used as a basis to understand the equilibrium of Bertrand-Edgeworth duopoly by Deneckere and Kovenock (1992) and Lepore (2009).

4 pro…t and the critical judo pro…t. Notice that this payo¤ characterization does not rely on uniqueness of equilibrium and applies to both pure and mixed strategy equilibria. In the process of establishing the payo¤ bounds we provide abstract bounds for the range of pricing in all equilibria in the spirit of Shubik (1959). The second primary result provides the necessary and su¢ cient conditions under which a BE game has a pure strategy equilibrium. Dixon (1992) a¢ rmed the postulate of Shubik (1959) that su¢ ciently convex costs would admit a pure strategy equilibrium by providing necessary conditions for such an outcome. We take the analysis further by proving necessary and su¢ cient conditions for existence of pure strategy equilibrium, as well as showing that any pure strategy equilibrium must be the unique Nash equilibrium of the BE game. In the proof of our …rst result, it is established that a relevant tie cannot occur with positive probability in equilibrium. Here, a relevant tie is one in which a …rm’s front-side pro…t (pro…t earned if the …rm has the lower price) is strictly greater than the residual pro…t. This is used to show that all pure strategy equilibria are characterized by symmetric pricing at the largest non-relevant tie. Moreover, this price corresponds to both …rms’judo prices and is a lower bound for equilibrium pricing. Consequently, no non-degenerate mixed strategy equilibrium may exist as long as the judo price is the largest residual maximizer given that the other …rm sets the judo price. This characterization allows us to classify all pure strategy equilibria as either Bertrand pricing (marginal cost pricing) or Cournot pricing (market clearing price). The third set of results pertain to the case in which the aforementioned bounds on equilibrium expected pro…ts can be reduced to precise expected payo¤s. We begin by addressing the case in which each …rm has what we label an independent residual pro…t maximizer. In this case, we show that all equilibria yield the same expected payo¤s if both …rms have the same residual maximizing price. These conditions encompass the cases of e¢ cient and proportional rationing with (symmetric) constant marginal cost up to capacity. Further restriction to the case of independent residual pro…t, in which a …rm’s residual pro…t is independent of the other …rm’s price, allows for a closed form computation of the unique equilibrium strategies. This case is a generalization of the BE game with e¢ cient residual demand rationing and constant marginal cost up to capacity. Our model of BE duopoly can also be viewed as a generalization of a two player single prize all-pay contest as in Siegel (2009, 2010), with a lower price corresponding to a higher score (bid). From this perspective, our model is an all-pay contest in which the payo¤ of the losing player is a function of the bid of the winning player and such that each player’s ‘reach’ corresponds to his safe price. The ‘reach’functions di¤erently in our model, as prices need not approach the ‘threshold’in all equilibria.8 This can never be the case in Siegel’s game. Thus, our analysis can also be viewed as analysis of a new type of contest. The rest of the paper proceeds as follows. In Section 2, we formally de…ne model and 8

This terminology is taken from Siegel (2009, 2010), The reach is the bid such that a player makes as much winning as they can guarantee themselves by playing the best losing strategy. In a contest in which the top M bidders are awarded a prize, the threshold is the M + 1-th highest reach.

5 our terminology. Section 3 presents our main result. In Section 4 we present the results particular to pure strategy equilibrium. Results for uniqueness of equilibrium expected payo¤s are shown in Section 5. In Section 6, we conclude.

2

The Model

Consider a homogeneous product industry with two …rms i = 1; 2. We will use j to refer to the …rm other than i. The …rms simultaneously announce prices. Each …rm i has a continuous, nondecreasing cost of production ci : R+ 7! R+ with ci (0) = 0. The market demand D : R 7! R is continuous and nonincreasing. Further, assume that there exists a price x > 0 such that D (x) = 0 for all x x. We denote by pi the price of …rm i and by p the vector of both …rms’prices. Since p is the vector of prices (p1 ; p2 ), we will unambiguously use x to denote a single price when it is not associated with a particular …rm. Each …rm i has a capacity ki , which serves as an upper bound on the quantity that can be produced. Thus, the production problem faced by …rm i at a price pi is max

z2[0;ki ]

i

(pi ; z) = pi z

ci (z) .

Let si (pi ) denote the quantity that solves this optimization problem.9 We assume that si (pi ) is a continuous nondecreasing function. Further, we assume that if x < si (pi ), then 10 i (pi ; y) > i (pi ; x) for all y 2 (x; si (pi )). This is necessarily true if ci is strictly convex. The quantity si (pi ) may be referred to as …rm i’s supply, the maximum quantity that it is willing to produce at any given price. Inherently, si ki , so the supply functions account for the capacity constraints. If pi < pj , the demand served by …rm i is Qi (pi ) = min fD (pi ) ; si (pi )g. We make minimal assumptions as to which portion of the demand is served by …rm i when pi < pj , only that there is a continuous function ij (p) which denotes the share of i’s quantity that satiates j’s demand.11 Note that the function ij depends on which …rm i is the low price …rm. This allows for asymmetric rationing. The residual demand served by …rm j is Qrj (p) = min fD (pj )

Qi (pi )

ij

(p) ; sj (pi )g ;

where ij (p) = 1 if pi pj . We also add the intuitive restriction that if pi < pj and Qi = D (pi ), then ij = 1. 9

The speci…cation of si (pi ) follows from Dixon (1984), Maskin (1986) and Bagh (2010). Notice that for symmetric constant marginal cost c 0, we can restrict the strategy space to X = [c; x]2 and the supply functions satisfy our assumptions. 11 A simple way to understand the purpose of i is to consider the case in which a continuum of consumers have unit demand. In this case, i speci…es the fraction of consumers served by …rm i that have willingness to pay of at least pj . 10

6 This general framework is consistent with the notion that consumers shop …rst at …rms with lower prices. To highlight the role that ij plays in determining the rationing rule, consider two choice for the functions ij given by eij (p) = 1 and pij de…ned as p ij (p)

= min

D(pi ) ;1 . D (pj )

The rationing rule under eij is the well known e¢ cient, or parallel rule, whereas the rule under pij is the proportional rationing rule.12 We de…ne two di¤erent pro…t functions for a …rm based on whether the …rm has the lower price or higher price. The front-side pro…t of a …rm i with a lower price than …rm j is 'i (pi ) = pi Qi (pi )

ci (Qi (pi )) :

On the other hand, the residual pro…t of …rm i with a higher price than …rm j is i (p)

= pi Qri (p)

ci (Qri (p)) :

Notice that based on our speci…cations above, 'i and i are continuous. Further, there exists an a 0 such that for each …rm i, 'i (pi ) = 0 for all pi 2 [0; a]. We restrict prices so that pi a.13 We make the following assumptions about the pro…t functions 'i and

i.

Assumption 1 'i has a unique maximizer pbi .14 'i is positive and strictly increasing at all pi 2 (a; pbi ].

This assumption on the front-side pro…t is very close to assuming that the 'i strictly quasiconcave except for the fact that behavior at price pi > pbi is less restrictive. Assumption 2 For any pi 0 0 pbi . i (pi ; pj ) for all pi

0,

i (pi ; pj )

is nonincreasing in pj . For any pj

0,

i

(b p i ; pj )

The continuity of i and the compactness of its domain imply that the residual pro…t function has a largest maximizer pei (pj ) pbi . We denote the set of maximizers of i at any pj by Pei (pj ). 12

Our speci…cation of residual demand rationing is general enough to include the endogenous rationing scheme that is found to be the equilibrium with strategic consumers in Allison (2014). 13 This is intended to allow for constant marginal cost up to capacity and still …t our supply function assumptions. These assumptions rule out the possibility of asymmetric constant marginal cost analyzed by Deneckere and Kovenock (1996). 14 Although we have omitted the capacities from our notation, it should be apparent that pbi can vary based on …rm i’s capacity.

7 Notice that based on the construction of Qri (p), 'i (pi ) following property is important for our characterization. Lemma 1 There exists a

i (pi ; pj )

for all pj

a. The

a such that for each …rms i;

'i (x) > 0

'i (x ) =

i

(x; x)

i

0

0

for all x 2

(x ; x ) for all x

0

;x ; :

The existence of such a price follows from the structure of the demand rationing assumptions. If the front-side pro…t is equal to the residual demand for player i, then the demand is not fully exhausted at that tie. As such, it must be that the demand is not fully exhausted if the tie went the other way, so …rm j’s front-side and residual pro…t must be equal as well. We provide a formal proof of Lemma 1 in the appendix. Each …rm i’s pro…t is speci…ed as follows 8 'i (pi ) < ui (p) = (p)'i (pi ) + (1 : i i (p)

pi < p j i (p)) i (p) pi = pj ; pi > p j

where i (p) 2 [0; 1] and 1 (p) + 2 (p) 2 (0; 2). In our formulation, we put the sharing rule on the pro…t instead of directly on the demand. By allowing the sharing rule to be very general, this does not impact the results and provides parsimony to the characterization. In particular, allowing the sum of the shares to be greater than one allows this speci…cation to generalize a model where rationing is put directly on the demand shares. Its useful to note that for each …rm i, any price pi > pbi is always strictly dominated by = pbi . It will be useful in what follows to de…ne the notation max pbi and note that all equilibrium prices must be less than .

p0i

Denote the maximized residual pro…t by e i (pj ), that is, e i (pj ) = max pi pj

i

(pi ; pj ) :

De…ne rj to be …rm j’s judo price, the highest price that …rm j can set to guarantee that …rm i would rather maximize its residual pro…t than undercut. Formally, rj = maxfpj 2 [ ; pbj ]j'i (pj ) = e i (pj )g:

De…ne rj to be …rm j’s safe price, the highest price such that the front-side pro…t of …rm i equals the highest pro…t that …rm i can guarantee itself. Formally de…ned as rj = maxfpj 2 [ ; pbj ]j'i (pj ) = ui g;

8 where ui = suppi inf pj ui (pi ; pj ) = maxx2[

;b pj ]

i (x; x).

De…ne the larger of the two …rm’s judo prices to be critical judo price, denoted by r = max ri , and de…ne the larger of the two …rm’s safe prices to be the critical safe price, denoted by r = max ri . Notice that based on their de…nitions, the judo price is always weakly greater than the safe price. De…ne …rm j’s judo pro…t to be the front-side pro…t of …rm j at the critical judo price, denoted by 'j 'j (r). Similarly, de…ne …rm j’s safe pro…t to be the front-side pro…t of …rm j at the critical safe price, 'j 'j (r). For equilibrium strategies = ( 1 ; 2 ), we use xi and xi to denote the in…mum and supremum of the support of …rm i’s strategy, respectively. We will occasionally use x to denote the minimum of x1 and x2 , and x to denote the maximum of x1 and x2 . Further we de…ne Mi as the distribution (CDF ) of the …rm i’s mixed strategies on [x; x], where M = (M1 ; M2 ).

3

Equilibrium Expected Payo¤s

In this section we present our …rst substantial result, the characterization of the bounds on equilibrium expected pro…ts. Because of the general structure of our model, we cannot rule out the existence of multiple non-payo¤ equivalent equilibria and, consequently, we are unable to provide an exact characterization of equilibrium payo¤s. Existence of equilibrium for the BE duopoly follows directly from Proposition 2 in Allison and Lepore (2013). Theorem 1 All equilibria are such that ui 2 ['i ; 'i ]. The proof of the Theorem 1 is based on a serious of lemmas that provide character to all equilibria in order to establish the bounds on equilibrium expected payo¤s. We begin by establishing that relevant ties (p1 = p2 > ) occur with probability zero in all equilibria of this game. That is, all equilibria are atomless at pricing ties above the price such that front-side pro…t equal its residual pro…t. This result is used as a basis for many of the results that follow. Lemma 2 All equilibria are atomless at any pricing p1 = p2 > . Proof of Lemma 2. For all , at every tie p1 = p2 = x 2 ( ; ] there is some player i such that i < 1. De…ne m > 0 such that i ('i (x) i (x; x)) < m < 'i (x) i (x; x). Based on the continuity of 'i , for > 0 small enough 'i (x ) i (x; x) > m. Choose such an :

9 Suppose that there is an equilibrium that has mass at p1 = p2 = x. Consider a deviation by …rm i to ei de…ned as ei (fx g) = i (fxg) + i (fx g), ei (fxg) = 0, and ei (E) = i (E) for all sets E with x ;x 2 = E. Then, Z Z ui (p)dei ui (p)d + i (x) ('i (x ) [ i 'i (x) + (1 i ) i (x; x)]) j = Z > ui (p)d + m i ('i (x) i (x; x)) Z > ui (p)d : This violates

as an equilibrium. We conclude that

does not have mass at x.

The next result establishes a key property of any equilibrium, that the in…mum of the support of each …rm’s strategy is identical. For any …xed equilibrium, denote Mi (x) as the probability that …rm i gets the front-side pro…t when playing the price x. Lemma 3 In any equilibrium, the in…mum of the support of each …rm’s strategy is identical. That is, x1 = x2 = x. If x > , then neither …rm’s equilibrium strategy may have an atom at x. Proof of Lemma 3. We begin by proving that in any equilibrium, both …rms must play prices that approach or are equal to x. That is for both …rms i, Mi (x) < 1 for all x > x. Suppose to the contrary that xj > xi for some …rm i. We will show that this contradicts xi as the greatest lower bound of the support of …rm i’s equilibrium pricing. By de…nition, it must be that there are is a sequence of prices xk in the support of …rm i’s strategy such that xk ! xi . It follows that Mi xk ! 1, and so E ui xk ! 'i (xi ) < 'j xj , violating the prices xk as equilibrium prices. Thus, it must be that x is the in…mum of the support of both players’strategies. Now suppose that …rm i has atom at x > with mass > 0. We have just shown that xi = xj = x, so Lemma 1 implies that there is a sequence xk in the support of …rm j’s strategy such that with xk > x for all k and xk ! x. Note that lim 0

x "x

Z

x

x

uj (x0 ; pi )d

i

= 'j (x) ,

10 so it must be that uj Z

'j (x). However, note that

x

uj d

=

lim

k!1

x

Z

x

uj (xk ; pi )d k

=

i

x k

lim Mj (x )'j (x ) + lim

k!1

= (1

k!1

) 'j (x) +

j (x; x)

Z

xk j (x

k

; pi )d

i

x

< 'j (x) . The …nal inequality follows form the fact that x > . This contradicts xk as a sequence of equilibrium actions, and so we conclude that neither …rm’s strategy may have an atom at x. We are now able to prove that the lower bound of any equilibrium must lie between the critical safe price and the critical judo price. Lemma 4 The lower bound of equilibrium pricing x is such that r

x

r.

Proof of Lemma 4. First, we argue that x r. Suppose to the contrary that x > r. This means that 'i (x) > e i (r) e i (x) for both …rms. This implies that for any upper bound on mixed pricing x x, the expected pro…t for some …rm i (since at most one …rm may have an atom at x) of playing x is lim x"x

Z

x

x

ui (x; pj )d

j

=

Z

x i (x; pj )d j

x

Z

x

x

e i (pj )d

j

e (x) i

< 'i (x) Z x = ui (x; pj )d j : x

This contradicts x > r as a lower bound of equilibrium pricing. Second, we argue that x r. Suppose to the contrary that x < r. Let player j be such 'i (r). Further, since 'i is strictly that rj = r. By de…nition of rj , it must be that ui increasing at all pi < pbi , it follows that 'i (x) < 'i (r). The previous lemma implies that ui = 'i (x), yielding the contradiction that ui < ui . We conclude that x r.

Proof of Theorem 1. We have demonstrated in the proofs of Lemmas 2 and 3 that each …rm i’s equilibrium expected pro…t is ui = 'i (x). The statement of the theorem thus follows immediately from Lemma 4 and the fact that 'i is strictly increasing for all pi < pbi .

11 These payo¤ bounds provide a solid foundation to understand the properties of all BE equilibria in this general setting. While the literature on BE games has in some cases been able to provided precise payo¤s, the bounds presented here apply to a much larger class of games than have previously been studied. Remark 1 In contrast to the seminal work of Edgeworth (1925) and Shubik (1959), the generality of our speci…cation introduces an additional level of pricing indeterminacy. The …rst level of indeterminacy, which Edgeworth and Shubik focus on, is based the equilibrium being in non-degenerate mixed strategies. The second level of indeterminacy driven by the fact that there can be multiple non-payo¤ equivalent equilibria. The preceding analysis of this section provides abstract bounds on range of pricing for all equilibria, which contains the range of total indeterminacy. Lemma 4 directly states that the lower bound of all equilibria must lie between the critical safe price and critical judo price. As we noted previously in the text, it is immediate based on the structure of the model that all equilibria will have prices bounded below the maximum of the two …rms monopoly price. Further, it is straightforward to show that the least upper bound on pricing for all equilibria must be bounded below by the minimum of all residual maximizers across the …rms. Thus, in addition to providing expected payo¤ bounds, our analysis has provided abstract bounds on the range of pricing of all equilibria.

4

Pure Strategy Equilibria

We now turn to addressing the circumstances in which a pure strategy equilibrium exists in the BE pricing game, with the goal being to classify all such equilibria as one of two types, corresponding to the classic understandings of Bertrand and Cournot equilibria. The key aspect that permits a pure strategy equilibrium is that the highest price that makes the …rms indi¤erent between receiving the front-side and residual pro…ts is also a maximizer of the residual pro…t for both …rms. The following theorem provides necessary and su¢ cient conditions for the existence of a pure strategy Nash equilibrium. Theorem 2 Any pure strategy equilibrium must be symmetric. There is a unique pure strategy equilibrium price candidate x . Both …rms pricing at x is an equilibrium if and only if x = 2 Pei ( ) for each …rm i.

Proof of Theorem 2. To begin the proof we argue that any pure strategy equilibrium must be symmetric. Suppose to the contrary that there is an asymmetric equilibrium with pi < pj . This means …rm i gets 'i (pi ) with certainty. There are two cases to consider: (i) pi < pbi and (ii) pi = pbi . We may ignore the case in which pi > pbi since …rm i would trivially be better o¤ with a price of pbi . In case (i), playing p0i 2 (pi ; pj ) is strictly better for …rm i

12 since it would earn a pro…t of 'i (pi ) > 'i (pi ) with certainty. In case (ii), there must be no residual demand for …rm j at pj . If there were any residual demand remaining at pbi , then …rm i could increase its price slightly, sell the same quantity and make strictly greater pro…t than at pbi . Thus, …rm j will want undercut and get positive pro…t.

Next we show that any symmetric strategy pro…le (x ; x ) 6= ; cannot be an equilibrium. Let (x ; x ) be an equilibrium and suppose …rst that x < , yielding a pro…t of 'i (x ) = i (x ; x ) for each …rm i. Then note that by playing pi = , …rm i earns a pro…t . Suppose next that x > . Then of i (pi ; x ) i (pi ; pi ) = 'i (pi ) > 'i (x ). Thus x 'i (x ) > i (x ; x ) for each …rm i and i (x ; x ) < 1 for at least …rm i. Thus, ui =

i (x

; x )'i (x ) + (1

i (x

; x ))

i (x

;x )

< 'i (x ) . We have already demonstrated that ui = ' (x), thus we have a contradiction. We conclude that x = , and by the continuity of 'i there is a x0 < x close enough to x such that ui =

i (x

; x )'i (x ) + (1

i (x

; x ))

i (x

; x)

0

< 'i (x ) = ui (x0 ; x ); a contradiction to x as an equilibrium. It remains to be shown that ; is an equilibrium if and only if 2 Pei ( ) for each …rm i. Note that 2 = Pei ( ), then there exists a pi > such that i pi ; > i ; = ui ; , and so if ; is an equilibrium, then 2 Pei ( ) for each …rm i. Further, if 2 Pei ( ) for each …rm i, then for each …rm i and all prices pi > , i pi ; < i ; , and so neither …rm can increase its pro…ts by increasing it’s price. Since 'i is strictly increasing, neither …rm can increase it’s pro…ts by reducing it’s price. Thus, if 2 Pei ( ) for each …rm i, then ; is an equilibrium. Theorem 2 is important in that it speci…es the exact circumstances that Edgeworth’s concerns about price determinacy can be alleviated. But in this general setting, existence of a pure strategy equilibrium does not guarantee uniqueness of this equilibrium. Since is uniquely de…ned, it is the only pure strategy equilibrium candidate, however, it may be that another mixed strategy equilibrium concurrently exists. The following proposition demonstrates that no other equilibrium in pure or mixed strategies may exist as long as is the largest residual maximizer.

Proposition 1 Suppose that a pure strategy equilibrium exists. If = max Pei ( ) for each …rm i, then both …rms pricing at is the unique equilibrium of the BE game. Proof of Proposition 1. Assume that a pure strategy equilibrium exists. Then note that any price x < is strictly dominated by x0 = , since ui (x; pj ) = 'i (x) for any x < and

13 'i is strictly increasing below the monopoly price. Moreover, each …rm i’s judo price ri = . Thus, from Lemma 4, it must be that in any equilibrium, xi = . Suppose that there exists an equilibrium in which the support of some …rm i’s strategy is such that xi > . Without loss of generality, assume that xi xj and that …rm j’s strategy note that when choosing a price at or near xi , …rm i does not have an atom at xi . Then R earns a pro…t of approximately i (xi ; pj ) d j . By assumption, Z : i (xi ; pj ) d j i xi ; If = max Pei ( ) for each …rm i, then at or near xi as equilibrium strategies.

i

xi ;

<

i

;

= 'i

. This contradicts prices

Remark 2 Based on the Proposition 1, it is evident that in a model such that both …rms have unique residual pro…t maximizers, non-degenerate mixed strategy and pure strategy equilibrium cannot coexist for the same parameters; the BE game either has a unique pure strategy equilibrium or non-degenerate mixed strategy equilibrium. That is, in this environment the necessary and su¢ cient condition for existence of pure strategy equilibrium also guarantees uniqueness. In getting back to Edgeworth (1925) them of indeterminacy of pricing, our results for this class of BE game corresponds to the conclusions of Shapley (1959). We turn to strengthening this characterization by classifying all pure strategy equilibria of this game as one of two distinct types. The …rst type of pricing requires price equals marginal cost a la Bertrand pricing. The second type of equilibrium requires supply to equal demand with prices above marginal cost a la Cournot pricing. These types are de…ned formally as follows. Again we use x to denote a pure strategy equilibrium. Type B: D(x )

min si (x ),

Type C : D(x ) = s1 (x ) + s2 (x ). In the following proposition we show that all pure strategy equilibria must be of Type B or C. Proposition 2 All pure strategy equilibria x are such that D(x ) s1 (x ) + s2 (x ).

min si (x ) or D(x ) =

Proof of Proposition 2. Let (x ; x ) be a pure strategy equilibrium. Suppose to the contrary that either (i) D(x ) > s1 (x ) + s2 (x ), or (ii) D(x ) < s1 (x ) + s2 (x ) and D(x ) > si (x ). We will use the result from Theorem 2 that all pure strategy equilibrium must be such that x = as the basis of our arguments.

14 (i) Suppose …rst that D(x ) > s1 (x ) + s2 (x ). By continuity of D and each si , it follows that D (x0 ) > s1 (x0 ) + s2 (x ) for some x0 > x . Thus, it must be that Qr1 (x0 ; x ) = s1 (x0 ) s1 (x ). By assumption, 1 (x0 ; s1 (x )) < 1 (x0 ; s1 (x0 )), and since 1 (x0 ; s1 (x )) < 0 1 (x ; s1 (x )), it must be that x is a pro…table deviation from x for …rm 1. This contradicts (x ; x ) as an equilibrium. (ii) Suppose next that D(x ) < s1 (x )+s2 (x ) and D(x ) > si (x ). In this case, Qi (x ) = si (x ) > D(x ) sj (x ) Qri (x ; x ). Thus, 'i (x ) > i (x ; x ), which implies that x > , a contradiction. As mentioned previously, Type B pricing has the ‡avor of Bertrand pricing and Type C has the ‡avor of Cournot. Now we address two special cases where this pricing is exactly that of Bertrand or Cournot. Consider the special case that each …rm has a strictly increasing cost function ci . In this case, there is no possibility of Bertrand pricing. We prove this in the following lemma. Proposition 3 If each …rm has strictly convex cost ci , then there can only be pure strategy pricing of Type C. Proof of Proposition 3. Suppose to the contrary that x such that D(x ) min si (x ) is an equilibrium. For at least one …rm i, i (x ) < 1 and consequently …rm i’s pro…t is ui < x D(x ) ci (D(x )). Firm i can undercut by picking x0 < x and get arbitrarily close to the pro…t x D(x ) ci (D(x )). Therefore, there exists x0 such that ui (x0 ; x ) > ui . Consider a speci…cation of the game with constant marginal cost a 0. The following lemma shows that in this case all pure equilibria of Type B must be the classical Bertrand marginal cost pricing x = a and all Type C equilibria must be market clearing pricing up to capacity (quantity) as in Cournot D(x ) = k1 + k2 . Proposition 4 If each …rm has a constant marginal cost a 0, all Type B equilibria are such that x = a and all Type C equilibria are such that D(x ) = k1 + k2 . Proof of Proposition 4. Suppose there is x > a such that D(x ) min ki . At x the front-side pro…t is D(x )(x a) > 0 and the residual pro…t is zero, a contradiction to x as a pure strategy equilibrium. The second part of the lemma is immediate, since si (pi ) = ki for all pi a. The signi…cance of Proposition 4 is that it establishes that, regardless of the rationing scheme, all pure strategy equilibrium are either classical Bertrand marginal cost pricing or market clearing Cournot pricing in the constant marginal cost setting.

15

5

Unique Equilibrium Expected Payo¤s

Although it is not possible to rule out many equilibria with di¤erent expected payo¤s within the bounds established by Theorem 1, we are able to derive su¢ cient conditions such that all equilibria have the same expected pro…ts. The next two subsections o¤er di¤erent restrictions on the …rms’residual pro…ts. In 5.1, we restrict each …rm to have a residual pro…t maximizers that is not a¤ected by the other …rm’s price and show conditions for all equilibria to have the same expected pro…ts. In 5.2, we consider the much stronger restriction that each …rm’s residual pro…t is not a¤ected by the other …rm’s price and characterize the strategies of the unique equilibrium.

5.1

Independent Residual Pro…t Maximizers

The complexity of the analysis of the general model is driven by the unrestricted nature of each …rm’s residual pro…t. To guarantee that all equilibria have the same expected pro…t for each …rm, we impose more structure on the residual pro…t function. In particular, we consider the case in which each …rm’s residual pro…t is monotonic in its own price up to the same unique maximizer which is independent of the other …rm’s price. This is formalized in the de…nition below. De…nition 1 A BE game has independent residual pro…t maximizers if for all i, j 6= i 1. Pei (pj ) = fe pi g for all pj 2 [r; pei ] and all pj such that e i (pj ) > 0, where pei = arg maxpi 2. 'i (e pi ) >

i

i

(pi ; a),

(pi ; pj ) for all pi > pj > pei , and

3. For any pj 2 [r; pb],

i (pi ; pj )

is nondecreasing in pi 2 [r; pei ]:

Its worth noting that any BE game with symmetric constant marginal cost up to capacity and e¢ cient or proportional rationing satis…es this condition. Consequently, our results pertaining to the uniqueness of equilibrium expected payo¤s generalizes the prominent cases in the literature on BE duopoly. We begin by establishing properties of any equilibria of a BE game with independent residual pro…t maximizers. The next two lemmas apply to all non-degenerate mixed strategy equilibria (i.e., x < x). Lemma 5 In any mixed strategy equilibrium, the support of each …rm i’s strategy must be such that xi min fe p1 ; pe2 g. If pe1 = pe2 = pe, then x1 = x2 = pe.

16 p1 ; pe2 g for some …rm i. We Proof of Lemma 5. Suppose to the contrary that xi < min fe consider the three cases (i) xi > xj , (ii) xi < xj , and (iii) xi = xj . Case (i): xi > xj .

Note that when setting any price x 2 (xj ; xi ], …rm i receives the residual pro…t with certainty. By de…nition, i (e pi ; pj ) > i (x; pj ) for all x 2 (xj ; xi ] and all prices pj , and so it must be that i ((xj ; xi ]) = 0. This contradicts xi as the supremum of the support of i . Case (ii): xi < xj . Using the same logic as in Case (i), we may conclude that j ([xi ; pej )) = 0 and that xj = pej . We will show that …rm i possesses a pro…table deviation to a price near min fxj ; pei g. When …rm i chooses a price of xi , it receives an expected pro…t of j

(fe pj g) 'i (xi ) +

Z

xi

i

(xi ; pj ) d j .

x

Since j does not have an atom at xi , it must be that this pro…t is left continuous in pi at xi . Since 'i is strictly increasing and i is weakly increasing in pi for all pj , there exists an " > 0 su¢ ciently small that pj g) 'i (min fxj ; pei g j (fe

Z ")+

x

xi

ei g i (min fxj ; p

"; pj ) d

j

Z > j (fe pj g) 'i (xi )+

xi i

(xi ; pj ) d j .

x

Thus, a deviation to min fxj ; pei g " would yield a strictly higher pro…t for …rm i, and so xi cannot be in the support of i , a contradiction. Case (iii): xi = xj = x.

From lemma 1, at most one …rm’s strategy may have an atom at x. Without loss of generality, assume that …rm j does not have an atom. Then by the same logic as Case (ii), …rm i possesses a pro…table deviation to pei , contradicting i as an equilibrium strategy. We conclude that each xi min fe p1 ; pe2 g.

Suppose that pe1 = pe2 = pe. Then note that i (e pi ; pj ) > i (x; pj ) for all x > xj and all pj 2 [r; pei ], and 'i (e pi ) > i (x; pj ) for all x > xj and all pj > pei . Thus, it must be that ui (e pi ; pj ) > ui (x; pj ) for all x > xj , and so x1 = x2 = x. Suppose further that x > pe. Since only one …rm’s strategy may have an atom at x, there must be a sequence of prices pki in the support of some …rm i such that Mi pki ! 0. Repeating the previous argument, it follows that ui (e pi ; pj ) > limk ui pki ; pj , and thus, pei would be a pro…table deviation from pki for some k. We conclude that x = pe. Lemma 6 In any non-degenerate mixed strategy equilibrium of a BE game with constant residual maximizers, Mi is continuous and strictly increasing on (x; min fe p1 ; pe2 g).

17 Proof of Lemma 6. Part 1. We begin by showing that each Mi is strictly increasing on (x; min fe p1 ; pe2 g). It is su¢ cient to demonstrate that for each …rm i there is no interval (a; b) (x; min fe p1 ; pe2 g) such that i ((a; b)) = 0. Suppose to the contrary that there exists such an interval. Without loss of generality, let (a; b) be the largest interval such that i ((a; b)) = 0. That is, let a and b be such that they are in the support of i . Note that for any x; x0 2 (a; b) with x < x0 , Mj (x) = Mj (x0 ). Moreover, since 'j is strictly increasing and j is weakly increasing in pj , it follows that x yields a strictly lower expected pro…t than does x0 . Thus, j ((a; b)) = 0, and (a; b) must also be the largest such interval with zero mass under …rm j’s strategy. From Lemma 1, we know that at most one …rm’s strategy may have an atom at a. Without loss of generality, assume that …rm i’s strategy does not have an atom at a. Then note that the payo¤ that …rm j earns when setting a price arbitrarily below a is lim #0

Z

x

uj (a

"; pi ) d

i

x

= lim Mj (a #0

= (1 < (1

") 'j (a

Mi (a)) 'j (a) + Mi (a)) 'j (b) +

") + lim Z

Z

#0

x

Z

x

(a

j

"; pi ) d

i

a "

j

(a; pi ) d

j

(b; pi ) d i .

i

b x

b

Thus, for " > 0 su¢ ciently small, a price of x0 = b " yields a higher expected pro…t than any price x 2 [a "; a]. This implies that j ((a "; b)) = 0, contradicting the assumption that (a; b) is the largest interval with no mass under either of the …rms’strategies. We conclude p1 ; pe2 g) must be such that i ((a; b)) > 0 for each …rm i. that any interval (a; b) (x; min fe Part 2. We will now show that each strategy Mi is continuous on (x; min fe p1 ; pe2 g). It is su¢ cient to demonstrate the strategies i are atomless on (x; min fe p1 ; pe2 g). Suppose to the contrary that there is a …rm i with an atom of arbitrary size > 0 at some x 2 (x; min fe p1 ; pe2 g). Based on Lemma 4, we know that xi . Together with Lemma 1, this implies that …rm j cannot also have an atom at x. For …rm j, picking a price arbitrarily higher than x gives the expected pro…t lim #0

Z

x

Z

x

uj (x + ; pi ) d

i

= lim Mj (x + ") 'j (x + ") + lim "#0 "#0 [x;x+") Z = (1 Mi (x)) 'j (x) + j (x; pi ) d [x;x)

j

i

(x + "; pi ) d

i

18 Alternatively, picking a price arbitrarily below x would yield an expected pro…t of lim #0

Z

x

uj (x

; pi ) d

= lim Mj (x

i

#0

x

= (1

) 'j (x

) + lim #0 Z Mi (x) + ) 'j (x) +

Z

x j

(x

; pi ) d i :

x j

(x; pi ) d i :

[x;x)

It follows from the continuity of the front-side and residual pro…t functions that for small enough ", a price of x " yields a higher pro…t for …rm j than any price pj 2 (x "; x + "), and so it must be the case that j ((x "; x + ")) = 0. This contradicts part 1 of this proof. We conclude that each Mi is continuous on (x; min fe p1 ; pe2 g).

Lemma 7 Suppose that if pe1 = pe2 = pe. Then all non-degenerate mixed strategy equilibria have the same support. Proof of Lemma 7. From Lemmas 6 and 5, we know that the support of any equilibrium strategy i must be an interval [x; pe] for some x < pe and the corresponding distribution functions must be continuous. We will show via contradiction that there cannot exist two equilibria with di¤erent supports.

Suppose to the contrary that there are two equilibria whose supports have in…ma of x and y, respectively, where without loss of generality x > y. We denote the equilibrium with support [x; pe] as X and the equilibrium with support y; pe as Y . As we demonstrated in the proof of Lemma 3, the expected pro…t for a …rm in equilibrium is equal to its front-side pro…t when setting a price equal at the in…mum of the support of the equilibrium strategies. Since the front-side pro…t function is strictly increasing, the expected payo¤ of the two equilibria Y is ordered for both players i with uX i = 'i (x) > 'i (y) = ui . From Lemma 1, at most one …rm j’s strategy may have an atom at pe. Thus, for some …rm i, it must be that uX i

=

Z

x

pe

X i (x; pj )d j .

It follows from the payo¤ ordering that for …rm i, (1)

Z

x

pe

p; pj )d i (e

X j

>

Z

y

pe

p; pj )d Yj : i (e

In the equilibrium X, when …rm i sets a price of x, it receives its front-side pro…t in equilibrium X with certainty. Alternatively, in the equilibrium Y , when …rm i sets a price of x, it Y receives its front-side pro…t with some probability MiY (x) < 1. Since uX i > ui , 'i (x) >

MiY (x)'i (x)

+

Z

y

pe

Y i (x; pj )d j :

19 Based on the fact that i (pi ; pj ) is non-increasing in pj , then Proposition 6.D.1 of Mas-Colell et al. (1995) it cannot be that MiX (x) MiY (x) for all x 2 [x; x]. Thus, for some x0 < pe, it must be that MiX (x0 ) < MiY (x0 ). Since the distributions are continuous on (x; pe), there must be a smallest x00 2 (x; pe) such that MiX (x00 ) = MiY (x00 ). To simplify the notation denote by the probability that …rm i receives the front-side pro…t when choosing a price of x00 . That is, MiX (x00 ) = MiY (x00 ) = . Consider this x00 , and note that uX i

00

=

'i (x ) +

Z

x00 i (x

00

; pj )d

X j

i (x

00

; pj )d

Y j :

x

uYi

00

=

'i (x ) +

Z

x00

y

Y Since uX i > ui ,

Z

x00 i (x

00

; pj )d

X j

>

x

Z

x00 i (x

00

; pj )d

Y j :

y

Again based on Proposition 6.D.1 of Mas-Colell et al. (1995), this inequality implies that there exists an xo < x00 such that MiX (xo ) = MiY (xo ), which contradicts the assumption that x00 is the smallest such price. We conclude that x = y: Now we present the main result of this section. Proposition 5 All equilibria have the same expected payo¤ if the game has independent residual pro…t maximizers and pe1 = pe2 :

Proof of Proposition 5. As we have previously argued, the equilibrium payo¤s are such that ui = 'i (x). If pe1 = pe2 , then Lemma 7 implies that all equilibria must have the same support. As such, all equilibrium payo¤s are exactly 'i (x). Remark 3 In terms of equilibrium pricing, Proposition 5 shows that with independent residual pro…t maximizers the range of pricing of all equilibria must be the same. Thus, this condition eliminates the second level of indeterminacy introduced by our general speci…cation. Next we turn to su¢ cient conditions to guarantee the BE game has a unique equilibrium.

5.2

Independent Residual Pro…t

By restricting the game to a very special class of rationing schemes and cost functions that includes the popular e¢ cient rationing rule with symmetric constant marginal cost up to capacity, we can re…ne our results further in order to obtain uniqueness of equilibrium. The condition required is as follows.

20 De…nition 2 A BE game has independent residual pro…t if for both …rms i, 0 0 i pi ; pj for all pj ; pj .

i

(pi ; pj ) =

In this case, we will abuse notation and write the residual pro…t as i (pi ). Independent residual pro…t and e¢ cient rationing are not equivalent. In fact, neither concept implies the other. A game with e¢ cient rationing and strictly convex cost up to capacity does not have independent residual pro…t, The following proposition is a characterization of the unique equilibrium of a BE game with independent residual pro…t. Proposition 6 Suppose that the game has independent residual pro…t, each …rm has a unique maximizer pei and ri < rj , then pei pej . Then there is a unique equilibria such that ui = 'i for all i. The unique equilibrium is the (possibly degenerate) cumulative distribution strategies '2 (x) '2 '1 (x) '1 M(x) = ; , '2 (x) 2 (x) '1 (x) 1 (x) p1 ; pe2 g) on [r; min fe

[r; min fe p1 ; pe2 g) and M(min fe p1 ; pe2 g ; min fe p1 ; pe2 g) = (1; 1).

Proof of Proposition 6. The proof that ui = 'i is an obvious corollary to Theorem 1 since, in this case r = r. It remains to be shown that the equilibrium is unique. Based on the proof of Proposition 5, we know that if the equilibrium is in pure strategies, then it must be unique. It only remains to rule out the case of multiple mixed strategy equilibria. The remainder of the proof is constructive and follows a similar argument to the proof of Theorem 3 in Siegel (2010). We know from Lemma 2 that no player has an atom at x and from Lemma 6 that Mi is continuous and strictly increasing on (x; min fe p1 ; pe2 g). Therefore the equilibrium must take the form '1 (x) d1 '2 (x) d2 ; , '2 (x) 2 (x) '1 (x) 1 (x) p1 ; pe2 g). Since both …rms must get the front-side pro…t for sure at x; for both on [x; min fe …rms i at pi = x the probability that the other …rm plays lower must be zero. This is only true if di = 'i for both i. It remains to be shown that no player j will ever choose a price pj > min fe p1 ; pe2 g. Without loss of generality, suppose that pej = min fe p1 ; pe2 g. If ri = r, then 'j = j (e pj ), and so i (pj ) < j (min fe p1 ; pe2 g) for all pj > min fe p1 ; pe2 g. In this case, limx!epj Mi (x) = 1, so …rm i does not play prices higher than min fe p1 ; pe2 g. It follows that …rm j receives its residual pro…t with certainty at any price pj pej , and thus would never price higher than min fe p1 ; pe2 g. Otherwise, ri < rj , so 'i = i (e pi ). Since pei pej , then pei = pej . This further implies that limx!epj Mj (x) = 1, and so …rm j does not play prices higher than min fe p1 ; pe2 g. It follows that …rm i receives its residual pro…t with certainty when choosing any price pi pei , and so will never choose a price higher than min fe p1 ; pe2 g.

21 Remark 4 The uniqueness of the residual maximizers and restrictions that ri < rj implies pei pej are necessary for using this method verifying uniqueness. The proof of the lemma only gives us continuous and strictly increasing strategies on [r; min fe p1 ; pe2 g). If (1) ri < rj and (2) pei > pej , then we can’t get continuous and strictly increasing on (e pj ; pei ), nor can we pin down the upper bound on the strategies. The issue is as follows. Both …rms can have a gap in his support above pej : …rm j will not increase prices from below the gap because his residual pro…t is decreasing in his price, and …rm i will not increase his prices from above because that would strictly decrease his expected payo¤. Since there can be gaps, it is possible that there could be multiple equilibria with di¤erently placed atoms above pej . We resolve this pej . issue by assuming the residual pro…t maximizers are unique and ri < rj implies pei These condition are not as general as we would like, but still covers the case of constant marginal cost and e¢ cient rationing.

6

Conclusion

We have provided a characterization of equilibrium payo¤s in a large class of BertrandEdgeworth games which have previously been unstudied. This characterization bounds expected pro…ts between …rms’safe pro…ts and judo pro…ts. Further, these bounds have been shown to be tight, as we have examined a special subclass of these games in which these bounds coincide given any exogenous capacity constraints. Additionally, we have provided necessary and su¢ cient conditions for the existence of a pure strategy pricing equilibrium. There is a unique candidate for a pure strategy equilibrium, and it is symmetric. When such an equilibrium exists, then it is the unique Nash equilibrium of the game. Pure strategy equilibria in this model have been classi…ed as being either a Bertrand type of equilibrium, exhibiting characteristics akin to marginal cost pricing (exactly that when marginal cost is well de…ned), or a Cournot type, exhibiting market clearing and price greater than marginal cost (if marginal cost is well de…ned). The methodology we have used to provide this characterization involved a realization of the payo¤s abstractly as front-end and residual pro…ts. This abstraction allows for simpler analysis, and more importantly demonstrates the connection between the literature on pricing competition and contests. These classes of games exhibit similar characteristics, and the methodology used here should be applicable to more general game structures that encompass both of these classes. Finally, this analysis provides a starting point for a number of other research topics. There is the possibility of …nding new insights from applying our BE model to prominent topics studied considered in the previous literature, which include: (i) the introduction of a capacity choice stage as in Kreps and Scheinkman (1983), (ii) sequential entry as in Allen et al. (2000), (iii) the analysis of tari¤s between foreign …rms as in Fisher and Wilson (1995).

22 The setting also permits the possibility of considering topics outside previous BE models, such as endogenous technology choice or strategic rationing.

7

Appendix

Proof of Lemma 1. Let D be the total demand available at pi = p i . Clearly, Q1 +Qr2 D and Q1 + Qr2 D, otherwise quantity sold exceeds demand. A problem point requires that Q1 > Qr1 and Q2 = Qr2 . Suppose to the contrary that Q1 > Qr1 and Q2 = Qr2 . Note that Q1 + Qr2 = D, otherwise there would be more consumers to satiate …rm 1. Then D = Q1 +Qr2 < Q1 +Q2 = Q1 +Qr2 D, a contradiction. Therefore, if Q1 > Qr1 , Q2 > Qr2 . Equivalently, if Q1 = Qr1 , Q2 = Qr2 .

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