Price Competition with Decreasing Returns-to-Scale: A General Model of Bertrand-Edgeworth Duopoly Blake A. Allison and Jason J. Lepore
December 10, 2014
Abstract 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 existing literature on BE games. In particular, our model allow for convex costs of production and general demand rationing schemes including but not limited to e¢ cient and proportional rationing. Within this framework, we provide bounds on equilibrium pricing and expected payo¤s, necessary and su¢ cient condition for the existence of a pure strategy equilibrium, and a classi…cation of all pure strategy pricing as either “Cournot” or “Bertrand.” We provide some uniqueness results with additional restrictions. 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 that 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 formulation 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 questions in a model with general (and potentially asymmetric) production technologies and consumer rationing. 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 is1
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 a 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 rationing rule speci…es that the highest value consumers are served by the low price …rm while low value consumers are rationed. The proportional rule speci…es that a uniform distribution of the consumers are served by the low price …rm, resulting in a larger residual demand than the e¢ cient rule. 5 Almost all of 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 the interesting case in which …rms have constant marginal costs that are asymmetric. Additionally, 2
3 sues in price determination, including 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)], and uncertainty [Reynolds and Wilson (2000), Lepore (2008, 2011), De Frutos and Fabra (2011)]. A consequence of the assumptions made thus far in 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 has an optimal supply function that is continuous and nondecreasing in price. This formulation allows for both endogenous and exogenous quantity restriction. The second major generalization we allow for is in the process by which consumers are rationed between …rms when the low priced …rm faces excess demand. Rather than considering a particular rule, we allow for a general class of (possibly asymmetric) rationing schemes which need only satisfy the the bulk of this literature further restricts demand to be such that a …rm’s monopoly pro…t is concave in its price. Our analysis is based on the considerably weaker assumption that a …rm’s monopoly pro…t is single peaked in its price. 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.
4 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. 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. Unlike the all-pay contests studied by Siegel, the bounds on a player’s bids in our model do not depend on that player’s payo¤s, but rather on the payo¤s of the other player.7 Thus, our analysis can also be viewed as analysis of a new type of contest. We begin the analysis by establishing bounds on the …rms’expected pro…ts in all equilibria. In order to establish these bounds, we de…ne the following preliminary objects. First de…ne the critical judo price as the highest price either …rm can set to guarantee that 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).8 The second important price we de…ned is a …rm’s safe price, which is the maximum price that a …rm may set such that the other …rm would earn its max-min pro…t if it were to undercut. The …rst primary result of the paper is that the expected pro…ts of each …rm in all equilibria are bounded between its monopoly pro…t at the maximum of each …rm’s safe price and the critical judo price. 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 necessary and su¢ cient conditions under which a 7
Speci…cally, a player’s ‘reach’in our model corresponds most closely to what we will de…ne to be the safe price, the price at which the other …rm’s monopoly pro…t is equal to its max-min pro…t. Another distinction from the traditional all-pay contest is that each player’s bids need not approach the ‘threshold’in our model. 8 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 concept 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).
5 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 conditions for such an outcome. We take the analysis further by providing conditions that are both necessary and su¢ cient for existence of pure strategy equilibrium, as well as presenting conditions that guarantee that any pure strategy equilibrium must be the unique Nash equilibrium of the BE game. 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 each …rm’s residual pro…t is independent of the other …rm’s price, allows for the computation of a closed form of the unique equilibrium strategies. This case is a generalization of the BE game with e¢ cient demand rationing and constant marginal cost up to capacity. The rest of the paper proceeds as follows. In Section 2, we formally de…ne model and our terminology. Section 3 presents our main result. Section 4 is results particular to pure strategy equilibrium. Results on 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. We denote by pi the price
6 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 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.
Each …rm i has a capacity ki that 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 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. We assume that si (pi ) is a continuous, nondecreasing function. Further, we assume that if x < si (pi ), then
i
(pi ; y) >
i
(pi ; x) for all y 2 (x; si (pi )). This is necessarily
true if ci is strictly convex.10 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
: f(pi ; pj ) 2 R2+ : pi
pj g ! [0; 1] 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, allowing for the possibility of asymmetric rationing. The residual demand served by …rm j is Qrj (p) = min fmax fD (pj ) 9
Qi (pi )
ij
(p) ; 0g ; sj (pi )g ;
This speci…cation of si (pi ) follows from Dixon (1984), Maskin (1986), Bagh (2010) and Allison and Lepore (2014). 10 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 .
7 where Qi (pi )
(D(pi ) Qi (pi )
D(pj )) ij
(pi )
D(pj ) : Qi (pi )
This general framework is consistent with the assumption that consumers prefer lower prices, and so the high price …rm may sell a positive quantity only if the low price …rm exhausts its supply. To highlight the role that sider two choice for the functions The rationing rule under under
p ij
e ij
ij
plays in determining the rationing rule, con-
ij
given by
e ij
(p) = D(pj )=Qi (pi ) and
p ij
= D(pj )=D(pi ).
is the well known e¢ cient, or parallel rule, whereas the rule
is the proportional rationing rule.
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)
Based on our speci…cations above, 'i and
ci (Qri (p)) :
i
are continuous. Further, there exists an
a
0 such that for each …rm i, si (pi ) = 0 for all pi 2 [0; a]. We restrict prices so that
pi
a.12 We make the following assumptions about the pro…t functions 'i and
i.
Assumption 1 'i has a unique maximizer pbi .13 'i is positive and strictly increasing at all pi 2 (a; pbi ]. 12
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 as analyzed by Deneckere and Kovenock (1996). 13 Although we have omitted the capacities from our notation, it should be apparent that pbi can vary based on …rm i’s capacity.
8 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 i
(p0i ; pj ) for all p0i The continuity of
pbi . i
0,
i (pi ; pj )
is nonincreasing in pj . For any pj
0,
i
(b p i ; pj )
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
pj by Pei (pj ).
Based on the construction of Qri (p), 'i (pi )
i (pi ; pj )
for all pj
i
at any
a. The following
property is important for our characterization. Lemma 1 There exists a
a such that for each …rms i;
for all x 2
'i (x) >
i
(x; x)
'i (x0 ) =
i
(x0 ; x0 ) for all x0
The existence of such a price
;x ; :
follows from the structure of the demand rationing as-
sumptions. If the front-side pro…t is equal to the residual pro…t 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
ui (p) =
where
i (p)
2 [0; 1] and
8 > > > < > > > :
1 (p)
'i (pi ) i (p)'i (pi )
+ (1 i (p)
+
2 (p)
pi < p j i (p))
i (p)
pi = pj ; pi > p j
2 (0; 2). In our formulation, we put the sharing rule
on the pro…t instead of directly on the demand. In particular, permitting the sum of the
9 shares to be greater than one allows this speci…cation to generalize a model where rationing is put directly on the demand shares. De…ning the sharing rule in this general way on pro…ts does not impact the results and actually provides notational parsimony to the equilibrium characterization. Its useful to note that for each …rm i, any price pi > pbi is always strictly dominated by
p0i = pbi . It will be useful in what follows to de…ne the notation equilibrium prices must be less than .
max pbi and note that all
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, rj = maxfpj 2 [ ; pbj ]j'i (pj ) = ui g;
where ui = suppi inf pj ui (pi ; pj ) = maxx2[
;b pj ]
i (x; x).
De…ne the larger of the two …rms’ judo prices to be critical judo price, denoted by r = max ri . Similarly, de…ne the larger of the two …rms’safe prices to be the critical safe price, denoted by r = max ri . Based on their de…nitions, the judo price is always weakly greater than the safe price, that is, r
r.
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
10 j at the critical safe price, 'j For equilibrium strategies
'j (r). = ( 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 …rm i’s mixed strategies on [x; x], with 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 (2014).
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 > .
11 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))
the continuity of 'i , for > 0 small enough 'i (x Suppose that there is an equilibrium deviation by …rm i to ei de…ned as ei (fx ei (E) = Z
i
(E) for all sets E with x
ui (p)dei
j
= > >
This violates
Z
Z Z
ui (p)d +
)
< m < 'i (x) i (x; x)
ui (p)d + m
Based on
> m. Choose such an :
that has mass at p1 = p2 = x. Consider a g) =
i (fxg)
+
i (fx
;x 2 = E. Then, i
i (x; x).
(x) ('i (x i
('i (x)
)
g), ei (fxg) = 0, and
[ i 'i (x) + (1
i)
i (x; x)])
i (x; x))
ui (p)d :
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, de…ne Mi (x) to be 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.
12 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
so it must be that uj Z
Z
x
uj (x0 ; pi )d
i
= 'j (x) ,
x
'j (x). However, note that
x
uj d
=
x
=
lim
k!1
Z
x
uj (xk ; pi )d
i
x k
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
Proof of Lemma 4. First, we argue that x means that 'i (x) > e i (r) mixed pricing x
x
r.
r. Suppose to the contrary that x > r. This
e i (x) for both …rms. This implies that for any upper bound on
x, the expected pro…t for some …rm i (since at most one …rm may have
13 an atom at x) of playing x is
lim x"x
Z
x
ui (x; pj )d
x
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
that rj = r. By de…nition of rj , it must be that ui
'i (r). Further, since 'i is strictly
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 .
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 an equilibrium being in non-degenerate mixed strategies. The second level of indeterminacy present in this framework is 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
14 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, all equilibria will have prices bounded below the maximum of the two …rms monopoly price and it is straightforward to show that the least upper bound on pricing must be weakly greater than the minimum of all residual maximizers across the …rms. This provides abstract bounds on the range of pricing for 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 demonstrates that this is both necessary and su¢ cient 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
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
15 …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 equi-
;
librium. Let (x ; x ) be an equilibrium and suppose …rst that x < , yielding a pro…t of 'i (x ) = of
i
(x ; x ) for each …rm i. Then note that by playing pi = , …rm i earns a pro…t
i
(pi ; x )
'i (x ) >
i (x
i
. Suppose next that x > . Then
(pi ; pi ) = 'i (pi ) > 'i (x ). Thus x
; x ) for each …rm i and
ui =
i (x
i (x
; x ) < 1 for at least …rm i. Thus,
; 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)
< 'i (x0 ) = ui (x0 ; x ); a contradiction to x as an equilibrium. It remains to be shown that i. Note that and so if
;
;
2 Pei ( ) for each …rm
is an equilibrium if and only if
2 = Pei ( ), then there exists a pi > is an equilibrium, then
such that
i
pi ;
>
i
;
2 Pei ( ) for each …rm i. Further, if
each …rm i, then for each …rm i and all prices pi > ,
i
pi ;
<
i
;
= ui
;
,
2 Pei ( ) for
, 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 ;
is an equilibrium.
2 Pei ( ) for each …rm i, then
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
16 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 …rm i, then both …rms pricing at
= max Pei ( ) for each
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
'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
does not have an atom at xi . Then note that when choosing a price at or near xi , …rm i R earns a pro…t of approximately i (xi ; pj ) d j . By assumption, Z
If
i
(xi ; pj ) d
= max Pei ( ) for each …rm i, then
i
xi ;
j
<
i
xi ;
i
;
:
= 'i
. This contradicts prices
at or near xi as equilibrium strategies.
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. Thus, in this environment, the necessary and su¢ cient condition for existence of pure strategy equilibrium also guarantees its uniqueness. In getting back to Edgeworth’s theme of price indeterminacy, Proposition 1 provides the conditions for determinant prices in this class of BE game.
17 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 price. 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 )
min si (x ) or D(x ) =
s1 (x ) + s2 (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.
(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 ) 1
s1 (x ). By assumption,
1
(x0 ; s1 (x )) <
1
(x0 ; s1 (x0 )), and since
1
(x0 ; s1 (x )) <
(x ; s1 (x )), it must be that x0 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.
18 Consider the special case that each …rm has a strictly increasing cost function ci . The following lemma demonstrates that in this case, there is no possibility of Bertrand pricing in equilibrium.
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 ) is an equilibrium. For at least one …rm i, ui < x D(x )
i (x
min si (x )
) < 1 and consequently …rm i’s pro…t is
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 strategy 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, then 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. front-side pro…t is D(x )(x
Suppose there is x > a such that D(x )
min ki . At x the
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.
Proposition 4 is signi…cant in that it establishes that, regardless of the rationing scheme, all pure strategy equilibria are either classical Bertrand marginal cost pricing or market clearing Cournot pricing in the constant marginal cost setting.
19
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 for all equilibria to have the same expected pro…ts. The next two subsections o¤er di¤erent restrictions on the …rms’residual pro…ts to achieve di¤erent levels of uniqueness. 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; pei ],
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
20 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 equilibrium 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.
Proof of Lemma 5. Suppose to the contrary that xi < min fe p1 ; pe2 g for some …rm i. We 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, must be that
i
i
(e p i ; pj ) >
i
(x; pj ) for all x 2 (xj ; xi ] and all prices pj , and so it
((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
ej )) j ([xi ; p
= 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 pj g) 'i (xi ) + j (fe Since
j
Z
xi i
(xi ; pj ) d j .
x
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
j
(fe pj g) 'i (min fxj ; pei g
Z ")+
x
xi i
(min fxj ; pei g
"; pj ) d
j
Z > j (fe pj g) 'i (xi )+
x
xi i
(xi ; pj ) d j .
21 Thus, a deviation to min fxj ; pei g
xi cannot be in the support of
i,
" would yield a strictly higher pro…t for …rm i, and so 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 conclude that each xi
min fe p1 ; pe2 g.
Suppose that pe1 = pe2 = pe. Then note that
pj 2 [r; pei ], and 'i (e pi ) >
i
i
i
(e p i ; pj ) >
as an equilibrium strategy. We
i
(x; pj ) for all x > xj and all
(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). Proof of Lemma 6.
Part 1. We begin by showing that each Mi is strictly increasing
p1 ; pe2 g). It is su¢ cient to demonstrate that for each …rm i there is no interval on (x; min fe (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.
22 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
") 'j (a
Mi (a)) 'j (a) +
= (1
Mi (a)) 'j (b) +
< (1
") + lim Z
Z
#0
Z
x
(a
j
"; pi ) d
i
a "
x 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 that any interval (a; b)
p1 ; pe2 g) must be such that (x; min fe
i
((a; b)) > 0 for each …rm i.
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
p1 ; pe2 g). Suppose are atomless on (x; min fe
to the contrary that there is a …rm i with an atom of arbitrary size (x; min fe p1 ; pe2 g). Based on Lemma 4, we know that xi
> 0 at some x 2
. 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
Z
x
uj (x + ; pi ) d
= lim Mj (x + ") 'j (x + ") + lim "#0 "#0 [x;x+") Z = (1 Mi (x)) 'j (x) + j (x; pi ) d
i
x
j
(x + "; pi ) d
i
i
[x;x)
Alternatively, picking a price arbitrarily below x would yield an expected pro…t of
lim #0
Z
x
x
uj (x
; pi ) d
i
= lim Mj (x #0
= (1
) 'j (x
) + lim #0 Z Mi (x) + ) 'j (x) +
Z
[x;x)
x j
(x
x j
(x; pi ) d i :
; pi ) d i :
23 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
and so it must be the case that
j
"; x + "),
"; x + ")) = 0. This contradicts part 1 of this proof.
((x
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
24 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
Based on the fact that
i (pi ; pj )
pe
Y i (x; pj )d j :
is non-increasing in pj , then Proposition 6.D.1 of Mas-Colell MiY (x) for all x 2 [x; x]. Thus, for some x0 < pe,
et al. (1995) it cannot be that MiX (x)
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
X j
>
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).
25 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 as discussed in Remark 1.
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 constant marginal cost up to capacity, we can re…ne our results further in order to obtain a unique equilibrium that can be explicitly computed. The condition required is as follows.
De…nition 2 A BE game has independent residual pro…t if for both …rms i, i
i
(pi ; pj ) =
pi ; p0j for all pj ; p0j .
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. Moreover, it is possible to construct a game with convex costs in which the rationing rule is not e¢ cient and the
ij ’s
are chosen to be decreasing in such a
way that the residual quantities are constant, thereby satisfying 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 that pei
pej whenever ri < rj . Then there is a unique equilibria
26 such that ui = 'i for all i. The unique equilibrium is the (possibly degenerate) cumulative distribution strategies
M(x) = on [r; min fe p1 ; pe2 g)
'2 (x) '2 (x)
'1 (x) '2 ; 2 (x) '1 (x)
'1 1 (x)
,
[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 p1 ; pe2 g). Therefore the equilibrium must take continuous and strictly increasing on (x; min fe the form
'2 (x) '2 (x)
'1 (x) d2 ; 2 (x) '1 (x)
d1 1 (x)
,
on [x; min fe p1 ; pe2 g). Since both …rms must get the front-side pro…t for sure at x; for both …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 = so
i
(pj ) <
j
pj ), j (e
and
(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 min fe p1 ; pe2 g. Otherwise, ri < rj , so 'i =
i
pej , and thus would never price higher than
(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
27 pi
pei , and so will never choose a price higher than min fe p1 ; pe2 g.
The uniqueness of the residual maximizers combined with the restriction that ri < rj implies pei
pej is necessary for using this method to verify uniqueness of equilibrium. The proof
of Lemma 6 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 guarantee continuous and strictly increasing strategies on (e pj ; pei ), nor can we pin down the upper bound on the strategies. The issue is
driven by the possibility of gaps in each of the …rm’s strategies. Both …rms can have a gap in their support above pej : …rm j will not increase prices from below the gap because its residual
pro…t is decreasing in its price, and …rm i will not increase its prices from above because that would strictly decrease its expected payo¤. Since there can be gaps, it is possible that there are multiple equilibria with di¤erently placed atoms above pej . We resolve this issue by
assuming that the residual pro…t maximizers are unique and ri < rj implies pei
pej . These
condition are not as general as we would like, but still cover the case of constant marginal cost up to capacity and e¢ cient rationing.
Remark 4 It is worth highlighting the signi…cance of the fact that the critical safe price is equal to the critical judo price. This implies that at least one of the two …rms earns at most its max-min payo¤ in equilibrium (both if …rms have the same safe prices), implying that rents are maximally dissipated in equilibrium. This corresponds exactly with the equilibria of the contests studied by Siegel (2009). This correspondence is to be expected, as the most important di¤erence between our model and the traditional all-pay contest is that the players’ payo¤s conditional on losing the contest is a function of the winner’s strategy, and this is no longer the case under the assumption of independent residual pro…t. The remaining distinction is that …rms in our model may be able to obtain a positive payo¤ even if they are unable to guarantee that they will have the lowest price.
28
6
Conclusion
We have provided a characterization of equilibrium payo¤s in a large class of BertrandEdgeworth games that have not been previously studied. 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 a minor restriction is guarantees that 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 pricing greater than marginal cost (if marginal cost is well de…ned). The methodology we have used to provide this characterization involves 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 price 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 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), and (iii) the analysis of tari¤s between foreign …rms as in Fisher and Wilson (1995). The setting also permits the possibility of considering topics not studied by previous BE models, such as endogenous technology choice or strategic rationing.
29
7
Appendix
Proof of Lemma 1. Let D be the total demand available at pi = p i . Clearly, Q1 +Qr2 and Q1 + Qr2
D
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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