Capacity Choice and List Pricing in a Duopoly* 1)

Sung‐Hwan Kim

**

(접수일 2006.6.27 / 게재확정일 2006.11.27)

When we include a list pricing stage before capacity‐constrained price competition, a pure‐ strategy equilibrium always exists. This paper investigates this equilibrium with different assumptions on capacity levels. First, if the capacities are at Cournot level, the equilibrium in prices coincides with the Cournot outcome if and only if there is parallel rationing. Second, if we allow the firms to choose their capacity levels, the equilibrium coincides again with the Cournot outcome under the same rationing rule. The results suggest that Kreps and Scheinkman (1983)’s result is robust to this change in the structure of price competition.

Keywords: List Pricing, Capacity Choice, Price Competition JEL Codes: D43, L13, C72

I. Introduction In oligopoly price competition subject to limited capacity, which is known as

Bertrand‐Edgeworth model, there is a problem of non‐existence of pure strategy

equilibrium. Osborne and Pitchik (1986) showed that equilibrium must involve

randomization in case of intermediate level of capacities and characterized the set of Nash equilibria. However, the notion of mixed strategy in price

competition remains doubtful since it is hard to imagine that the decision‐makers * I began to work on this paper while I was in Johns Hopkins University as a graduate student. I am grateful to my advisor, Joseph Harrington, for his kind guidance and comments. I am also indebted to two anonymous referees for their careful reading and comments. ** Fair Competition Policy Devision, Korea Information Society Development Institute, [email protected].

126 産業組織硏究 第14集 第4號 ▶

in firms shoot dice as an aid to selecting price (Friedman, 1988). Recently, Diaz

and Kujal (2003) introduced a two‐stage duopoly competition model that consists

of a list pricing stage and a price discounting stage and showed there exists a specific type of pure strategy subgame perfect Nash equilibrium. We will call this game and its equilibrium list pricing game and Diaz and Kujal’s SPNE

respectively in this paper. The intuition behind the equilibrium is that the firms

could make credible commitments to price using list pricing mechanism, which enables a pure strategy equilibrium to obtain.

The primary objective of this paper is to examine Diaz and Kujal’s pure

strategy subgame perfect equilibrium when endogenizing capacity levels. In their

work, they supposed that capacities are predetermined. I will pursue two basic

questions. First, if the predetermined capacities correspond to the Cournot output

levels, what will the pure strategy equilibrium of list pricing game be like? Second, when we extend the list pricing game by allowing the simultaneous and independent choice of capacities, what kind of equilibrium will result?

These questions are simply in the same spirit as of the well known Bertrand

versus Cournot issue1) that Kreps and Scheinkman (1983), Davidson and

Deneckere (1986) and others addressed. Kreps and Scheinkman showed that, given a concave inverse demand curve2) and the parallel rationing rule,

simultaneous capacity choice prior to price competition leads to a unique equilibrium and the Cournot outcome. In contrast, Davidson and Deneckere argued that, for other rationing rules, equilibrium does not coincide with the Cournot outcome and actually leads to a more competitive result.

I investigate this old issue with a new price game Diaz and Kujal’s list

pricing model offers. It is meaningful to examine the robustness of the old

results since the new list pricing model is more sophisticated than the old 1) A comprehensive discussion of the issue especially along with some useful cautions can be found in Chapter 5 of Tirole (1988).

2) Osborne and Pitchik (1986) used a weaker assumption on demand curve and showed that the result of Kreps and Scheinkman still obtains. In their model, however, the set of pure-strategy subgame perfect equlibria was only a proper subset of the set of pure-strategy Cournot equilibria.

◀ Capacity Choice and List Pricing in a Duopoly 127

Bertrand‐Edgeworth model and may be a better description of certain oligopoly situations. Most significantly, the model ensures the existence of a pure‐strategy

equilibrium.

The subsequent analyses will show the following main results. First, if the

capacities are at Cournot level, the equilibrium in prices coincides with the

Cournot outcome if and only if there is parallel rationing. Second, if we allow the firms to choose their capacity levels, the equilibrium coincides with the Cournot outcome under the same rationing rule.

In Section 2, we briefly review Diaz and Kujal’s model and results along

with some discussion. Section 3 examines our first question, what the pure

strategy equilibrium will be like given Cournot capacities. In section 4, we endogenize the choice of capacity and examine the equilibrium to answer our primary question. We conclude in Section 5.

2. List Pricing Game – Model, Equilibrium, and Discussion Diaz and Kujal (2003)’s model of list pricing game consists of two stages. At

L the first stage, firm i ∈ {1,2} sets a list price pi , and at the second

stage the firms have the opportunity to offer a discount on the list

d price. In particular, each firm’s discounted price pi is subject to the

d L condition pi ≤ pi . The profits are realized according to the second‐

stage discounted prices, given market demand D( p ) and the rationing

rule. But, first‐stage list pricing choices are also important since they could limit the second‐stage choices.

The following is the basic assumptions they used. (And, I will follow the

same notations and assumptions although I will have to introduce additional ones later as they are necessary for the analysis.) Assumption 1

128 産業組織硏究 第14集 第4號 ▶

(a) The market has two firms that produce a homogeneous good. (b) Unit cost of production is zero up to capacity.3)

(c) Each firm’s capacity k i is restricted, 0 < k i ≤ D(0) .

0 (d) D( p ) is downward sloping and twice differentiable on (0, p ) and

0 zero for p ≥ p > 0 .

(e) Revenue curve pD( p ) is strictly concave.

(f) Residual demand for firm i when it is the high‐priced firm is, Rλ ( p i , p j , k j ) = max(λ ( D( pi ) − k j ) + (1 − λ ) D( pi )(1 −

kj D( p j )

),0)

where i ≠ j , 0 ≤ λ ≤ 1 .

,

With the formulation of residual demand as above, two special cases R0

and R1 will indicate the proportional and the parallel (or efficient)

residual demand respectively, which are familiar in the literature on B

e

r

t

r

a

n

d

‐

E

d

g

e

w

o

r

t

h

competition.4) As max( D ( p i ) − k j ,0 ) represents parallel residual demand a

n

max( D( pi )(1 −

kj D( p j )

d

),0) proportional residual demand, the above residual

demand function allows for such generalized case as c

a

p

a

c

i

λ 1

of the firm’s t

y

being allocated by parallel rationing rule and the rest by proportional rationing rule.

3) This zero cost assumption is no stronger than the constant unit cost assumption since we can interpret the price p as the excess of price over the unit cost.

4) For a discussion of the two rationing rules, see Section 2 of Davidson and Deneckere (1986). They suggest that the proportional rule is most appropriate since it treats the consumers symmetrically and also it is expected to be chosen as equilibrium when the firms are allowed to select a rationing rule after the quantity game. Interestingly, Osborne and Pitchik (1986) supports the use of parallel rule. They think that the low-priced firm will choose the parallel rule to minimize the profit of its opponent.

◀ Capacity Choice and List Pricing in a Duopoly 129 L d E Under the above assumptions, Diaz and Kujal showed [ p1 = p1 = p1 ,

p 2L ≥ p 2d = p M ( p1E , k1 )]

is

a

pure

strategy

subgame

perfect

Nash

equilibrium of the list pricing game. ( p1 represents Edgeworth price5) E

M and p monopoly price given the residual demand.) We will call this

type of equilibrium Diaz and Kujal’s SPNE. In words, in this equilibrium

E firm 1 sets its list price equal to its Edgeworth price p1 and does not

discount at the discounting stage while firm 2 sets a high enough list

price and sets the discount price such that it maximizes its profit given the residual demand. Note that by definition of Edgeworth price p1E ≤ p M ( p1E , k1 ) and hence p1L = p1d ≤ p 2d ≤ p 2L in this equilibrium. They

also argued that their equilibrium is a focal point by showing that any

other kind of subgame perfect equilibrium of the list pricing game leads to lower expected profits for both firms.

Their two‐stage model and the resulting equilibrium overcome the

conventional problem of non‐existence of pure strategy equilibrium in Bertrand‐

Edgeworth models. The secret of their success is in providing the list prices as instruments for commitment. Without any commitment as to the price as it is in

usual price game, the low‐priced firm with its binding capacity would always be

tempted to raise its price and it would eventually lead to the undercutting of the high‐priced firm and hence the breakdown of the equilibrium. The commitment

based on list prices could ensure the two firms peace in which the high‐priced firm serves the residual demand and the low‐priced firm focuses on its own market without being tempted to raise its price or worried about its opponent’s

undercutting it. Besides being technically successful, the consideration of list

pricing also seems quite reasonable as we do see the prevalence of list pricing 5) The Edgeworth price is defined as the highest price by which you could let your opponent choose to feed on the residual demand rather than undercut your price. The Edgeworth price and the relatively general form of residual demand are the two key elements in Diaz and Kujal’s analysis. They provide a quite detailed discussion of the two in Section 2 and 3 of their paper.

130 産業組織硏究 第14集 第4號 ▶

in reality.

3. Equilibrium of List Pricing Game given Cournot Capacities It is worth noting that the equilibrium is determined by firm 1’s Edgeworth

price and its capacity except for the firm 2’s list price that could be set at any level greater than its equilibrium discount price. In fact, it turns out that given a demand function the equilibrium is completely determined only by the two firm’s capacities because firm 1’s Edgeworth price in turn is defined as a function of the capacities k1 and k 2 .

As the Edgeworth price, as a function of the capacity pair, will take a critical

role in our analysis, we need to make clear how it is defined. Diaz and Kujal (2001) showed the Edgeworth price is characterized by the following implicit equation that has the unique solution of p1E .

min(k 2 , D( p1 )) p1 = R( p 2R ( p1 , k1 ), p1 , k1 ) p 2R ( p1 , k1 )

(1)

where R ( p 2 , p1 , k1 ) represents the residual demand function of firm 2,

and p 2R ( p1 , k1 ) represents firm 2’s optimal price based on the given residual demand function, i.e.

p 2R ( p1 , k1 ) = arg max p ∈[ p , p 0 ] R ( p 2 , p1 , k1 ) p 2 . 2

1

Recall that the residual demand function Diaz and Kujal supposed is

Rλ ( p 2 , p1 , k1 ) = max(λ ( D ( p 2 ) − k1 ) + (1 − λ ) D( p 2 )(1 − (2)

k1 ),0) D ( p1 )

To understand the equation (1) intuitively, note that the maximum profit that

◀ Capacity Choice and List Pricing in a Duopoly 131

firm 2 can obtain for p 2 < p1 is arbitrarily close to min(k 2 , D( p1 )) p1

while it is R ( p 2R ( p1 , k1 ), p1 , k1 ) p 2R ( p1 , k1 ) for p 2 > p1 . Hence, firm 2 would want to undercut if min(k 2 , D( p1 )) p1 > R ( p 2R ( p1 , k1 ), p1 , k1 ) p 2R ( p1 , k1 ) . For

the opposite inequality, however, firm 1 can always raise p1 a bit

without inducing the undercut by firm. Therefore, the solution of the

equation gives the Edgeworth price, p1E , that is the highest price firm

can set without inducing firm2’s undercut. Refer to Theorem 3.1 of Diaz

and Kujal for a rigorous proof.

From this point on, we summarize the above characterization of firm 1’s

Edgeworth price by the function p1E (k1 , k 2 ) . Now that we know each pair of k1 and k 2 are mapped to some pair of equilibrium discount prices in

Diaz and Kujal’s SPNE, we can ask what the equilibrium would be like when capacities from the Cournot equilibrium are given. We

denote

Cournot

capacity

and

price

by

k c = D( p c ) / 2

and

p c = P( 2k c ) . And by the term Cournot outcome, we will indicate the

outcome with Cournot capacities and prices in this paper. Given the capacity cost function c (k i ) , Cournot capacities k1 = k 2 = k c are defined

by the following first order conditions.

∂P( k1 + k 2 ) k1 + P ( k1 + k 2 ) − c ' ( k1 ) = 0 ∂k1 ∂P( k1 + k 2 ) k 2 + P ( k1 + k 2 ) − c ' ( k 2 ) = 0 ∂k 2 Proposition 1

In two‐firm list pricing game, if parallel residual demand ( λ = 1 ) is

supposed, then Cournot capacities lead to equilibrium discount prices

132 産業組織硏究 第14集 第4號 ▶

equal to the Cournot prices, but otherwise ( λ ≠ 1 ), equilibrium discount prices are higher than the Cournot prices.

Proof. [case of λ = 1 ] We know from Kreps and Sheinkman (1983) that,

in a two‐stage oligopoly game, if λ = 1 , we have a Cournot outcome, which c involves the pure strategy equilibrium p1 = p 2 = p in the second‐stage

price game (i.e. Bertrand‐Edgeworth model) given Cournot capacities.

Since Corollary 2 of Diaz and Kujal (2003) says the Bertrand‐Edgeworth

E model has a pure strategy equilibrium if and only if p1 = P (k1 + k 2 ) , we can

conclude

p1E (k c , k c ) = P( k c + k c ) = p c .

Given

p1E = p c ,

p M ( p1E , k c )

c degenerates6) and firm 2’s equilibrium choice is also p by Theorem 5 of L d c L d c Diaz and Kujal. Therefore, the SPNE is [ p1 = p1 = p , p 2 ≥ p 2 = p ] .

[case of λ ≠ 1 ] At Cournot state ( p1 = p 2 = p c , k1 = k 2 = k c ), note that limc

p2 ↓ p

∂Rλ ( p 2 , p c , k c ) ∂R1 ( p 2 , p c , k c ) kc c = [λ + (1 − λ )(1 − )] D ' ( p ) > lim = D' ( p c ) p2 ↓ p c ∂p 2 ∂p 2 D( p c )

because

k c < D( p c ) .

Since

Rλ ( p c , p c , k c ) = R1 ( p c , p c , k c ) = k c ,

following inequality will also be true. limc [

p2 ↓ p

the

∂Rλ ( p 2 , p c , k c ) ∂R ( p , p c , k c ) p 2 + Rλ ( p 2 , p c , k c )] > limc[ 1 2 p 2 + R1 ( p 2 , p c , k c )] p2 ↓ p ∂p 2 ∂p 2

From Kreps and Sheinkman (1983) we know that the right‐hand‐side term is equal to 0. Therefore, we have limc [ p2 ↓ p

∂Rλ ( p 2 , p c , k c ) p 2 + Rλ ( p 2 , p c , k c )] > 0 . ∂p 2

(The proof for λ ≠ 1 case so far just followed the same steps that were

used for proof of Theorem 1 by Davidson and Deneckere (1986).) That

is, firm 2 can increase its profit by raising p 2 above p c , which means

6) By the term “degenerate”, I mean the firm 2 does not set strictly higher discount price than firm 1 in list pricing game.

◀ Capacity Choice and List Pricing in a Duopoly 133

in

our

framework

k c p c < R ( p 2R ( p c , k c ), p c , k c ) p 2R ( p c , k c ) .

and

p 2R ( p c , k c ) > p c

Considering

min(k c , D( p c )) = k c ,

equation (1) doesn’t hold for Cournot capacities and prices. Hence,

p1E (k c , k c ) ≠ p c . Furthermore, we can see p1E (k c , k c ) > p c by showing

that the right hand side of (1) is decreasing in p1 as follows. ∂R ( p 2R ( p1 , k1 ), p1 , k1 ) p 2R ( p1 , k1 ) ∂R( p 2R , p1 , k1 ) R = p2 < 0 ∂p1 ∂p1

theorem and by

by

the

envelope

∂R ( p 2 , p1 , k1 ) k1 = (1 − λ ) D( p 2 ) D' ( p1 ) < 0 . ∂p1 ( D( p1 )) 2

L d c Therefore, the SPNE of the list pricing game satisfies [ p1 = p1 > p ,

p 2L ≥ p 2d > p c ] .■

The above proposition says, given Cournot capacities and parallel residual

demand, the Diaz and Kujal’s SPNE degenerates to be equivalent to Cournot equilibrium where firm 2 does not monopolize on the residual demand with higher price. But, if we suppose a residual demand other than the parallel one,

the equilibrium discount prices will be maintained higher than the Cournot

prices. Intuitively speaking, parallel residual demand is the worst possible rationing rule for the firm 2 that is supposed to call higher price and when this worst rule is combined with Cournot capacities the structure of list pricing game

does not make any difference from the conventional price game. (In fact, we

will see further that the parallel residual demand leads the list pricing game to be this trivial also when we endogenize capacity choice.) However, any other

type of rationing rule we are considering here would be better for firm 2 than the parallel rule and it was shown to be enough for guaranteeing that the two firms can find a pure strategy equilibrium with higher prices than the Cournot level by exploiting the structure of list pricing game.

134 産業組織硏究 第14集 第4號 ▶

4. Capacity Choice Prior to List Pricing Game Instead of examining the equilibrium with some predetermined capacities like

Cournot capacities, we can move on to endogenize the choice of capacities. That

is, it would be natural to extend the model to include capacity competition stage

prior to the list pricing game. From this point on, our analysis will concern this three‐stage game. The reduced form payoff functions for capacity choice under Diaz and Kujal’s SPNE will be

Π 1 (k1 , k 2 ) = p1E ( k1 , k 2 )k1 − c1 (k1 )

Π 2 (k1 , k 2 ) = p2R ( p1E (k1 , k 2 ), k1 ) R( p2R ( p1E (k1 , k 2 ), k1 ), p1E (k1 , k 2 ), k1 ) − c2 (k 2 ) where ci (k i ) represents each firm’s cost of installing capacity level

k i . In firm 1’s payoff function above, we can notice that firm 1’s

quantity is set to be exactly same as its capacity choice, which reflects the condition from the Diaz and Kujal’s SPNE that firm 1 cannot meet

all its demand ( D( p1 ) > k1 ) so that firm 2 could feed on the residual demand in the equilibrium of list pricing game.7) Firm 2’s payoff

function indicates that firm 2’s equilibrium quantity is not constrained

by the chosen capacity. It is not hard to see the reason. Suppose k 2 is

effectively binding (i.e. R ( p 2R , p1E , k1 ) > k 2 ) in equilibrium. Then, there is

no threat of undercut from firm 2 since lower price will always result

in lower profit with binding capacity. To be aware of this, firm 1 can 7) Of course, the payoff function for firm 1 will not be correct if k1 > D ( 0) . But, we

already assumed away the case in Assumption 1 (c). Even without the assumption, such case can never be considered for firm 1’s best response as long as marginal cost of capacity is positive, since capacity beyond maximal demand can do nothing good. Thus, there would be no problem for working with the given specification of payoff function of firm 1 although we may need to check if the equilibrium capacity solved is relevant in order to make sure at the end.

◀ Capacity Choice and List Pricing in a Duopoly 135

increase

p1 a little bit without inducing firm 2’s undercut, which

contradicts the definition of p1E .

The question we intend to address now is whether this extended game will

result in Cournot outcome. Our focus will have to be on the case of parallel residual demand ( λ = 1 )8) that looks promising because we already

know Cournot capacities bring us Cournot prices in list pricing game by

Proposition 1. Indeed, we will see in the following analysis that capacity competition that precedes the list pricing game will result in Cournot

capacities given parallel residual demand and some additional conditions. From now on, we will assume the following. Assumption 2

(a) Residual demand is subject to the parallel rationing rule. ( λ = 1 )

(b) Each firm’s marginal capacity cost is strictly positive. ( ci ' (k i ) > 0 ) (c) Firm 1’s payoff function Π 1 (k1 , k 2 ) is concave in k1 .

Assumption (c) can be satisfied, if ci (k i ) is convex and p1E (k1 , k 2 )k1

is concave, or if ci (k i ) is sufficiently convex relative to p1E (k1 , k 2 )k1 , or if p1E (k1 , k 2 )k1 is sufficiently concave relative to ci (k i ) .

Before we look at our main result as to the question of Counot outcome, it

will be useful to note one interesting feature of capacity choice equilibrium in our three‐stage game. Lemma 1 and Proposition 2 show this. Lemma 1

Firm 2’s best response choice of k 2 given k1 satisfies the condition 8) As already shown in Proposition 1, in case of λ ≠1 , even if Cournot capacities are chosen in the first stage, the following two stages of the list pricing game will lead to higher prices than in Cournot equilibrium. Hence, our equilibrium will be different from Cournot outcome in case of λ ≠ 1 .

136 産業組織硏究 第14集 第4號 ▶ * p1E (k1 , k 2BR ) ≥ p 2* ( k1 ) where p 2 ( k1 ) = arg pmax p 2 ( D( p 2 ) − k1 ) . 2

Proof. Suppose p1E (k1 , k 2 ) < p 2* (k1 ) . Then, p 2R ( p1E , k1 ) = p 2* ( k1 ) . (One

might suspect that

p 2R ( p1E , k1 ) > p 2* ( k1 ) can also happen when k 2 is

binding. But, R ( p 2R , p1E , k1 ) > k 2 is not possible as explained before for

the specification of payoff function.) It implies that as long as

p1E (k1 , k 2 ) < p 2* (k1 ) , firm 2 has a chance to be better off by decreasing

k 2 as it saves capacity cost with the revenue fixed. Therefore, best

response satisfies p1E (k1 , k 2 ) ≥ p 2* (k1 ) . ■

Proposition 2

Firm 2’s best response choice of k 2 given k1 will lead to the result

that the two firms set the same discount prices and there is no extra capacity for firm 2.

Proof. Using the condition p1E (k1 , k 2BR ) ≥ p 2* ( k1 ) from Lemma 1 and the

definition

of

Edgeworth

price,

we

know

p ( p (k1 , k ), k1 ) ≥ p (k1 , k ) ≥ p (k1 ) . Now note that the Assumption 1 R 2

E 1

BR 2

E 1

(e) of strictly concave

BR 2

* 2

pD( p ) implies

p ( D( p ) − k i ) is also strictly

concave. Since firm 2 maximize p 2 ( D( p 2 ) − k1 ) and its unconstrained solution

is

p 2* ( k1 ) ,

the

solution

under

the

constraint

p 2R ≥ p1E (k1 , k 2BR ) ≥ p 2* ( k1 ) will be p 2R = p1E (k1 , k 2BR ) considering the strict

concavity. This is a necessary condition for the equilibrium and we showed the two firms set the same equilibrium discount prices given

◀ Capacity Choice and List Pricing in a Duopoly 137

k1 , k 2BR . To see if there could be extra capacity for firm 2 with this

equilibrium, suppose there exists some positive amount of extra capacity for firm 2. Then, when the two firms are setting the same

price, firm 2 could be better off by slight undercut. This contradicts what we already proved. Therefore, there is no extra capacity for firm 2 in equilibrium induced by k1 , k 2BR . ■

Since the best response of firm 2 is a necessary condition for capacity choice

equilibrium, Proposition 2 tells us that we will always observe the same discount prices and no extra capacity in our three‐stage game equilibrium. (Recall that

there is no extra capacity for firm 1 either in Diaz and Kujal SPNE.) It is

indeed a little surprising to find that, in case of parallel residual demand, Diaz

and Kujal’s SPNE will always degenerate and market sharing by different prices

will not actually happen when we endogenize capacity choices. Furthermore, it is

encouragingly consistent with the property of Cournot outcome we have been

considering from Proposition 1. But, we will still have to check whether the equilibrium really is of Cournot or not. For proof of this, Proposition 2 and the following Lemma 2 will be necessary. Lemma 2

c c Suppose k1 < k and k 2 = k . Then, p1E (k1 , k 2 ) = P (k1 + k 2 ) if and only if

p 2* ( k1 ) ≤ P( k1 + k 2 ) .

Proof. Consider the Bertrand‐Edgeworth price game between firm 1 and firm

2 with k1 and k 2 . First, we will show that there is a pure strategy equilibrium

in

the

Bertrand‐Edgeworth

model

if

and

only

if

p ( k1 ) ≤ P( k1 + k 2 ) , which will result in the proof of the statement using * 2

the Corollary 2 of Diaz and Kujal. Suppose

p 2* ( k1 ) ≤ P( k1 + k 2 ) .

Then,

we

can

show

that

p1 = p 2 = P (k1 + k 2 ) is a pure strategy equilibrium as follows. As there

138 産業組織硏究 第14集 第4號 ▶

is no incentive for either firm to undercut at P (k1 + k 2 ) , we need only

to check that either firm will not want to raise its price. Given p1 = P (k1 + k 2 ) , the condition p 2* ( k1 ) ≤ P( k1 + k 2 ) and strict concavity of

revenue make firm 2 not raise its price above p 2 = P (k1 + k 2 ) .

We need a similar but slightly different reasoning for firm 1’s choice. Note

that firm 1 will face parallel residual demand, which is assumed, if it

raises p1 from P (k1 + k 2 ) given p 2 = P (k1 + k 2 ) . We already know that, c c if k1 = k and k 2 = k , then firm 1 will choose p1 = P (2k c ) . When we c c consider k1 < k instead of k1 = k , we have P (k1 + k 2 ) > P( 2k c ) . But

c since k 2 = k is maintained, the residual demand as a function of p1 it

c faces remains the same. Therefore, considering its choice k1 = k and

k 2 = k c and the strict concavity of revenue, we conclude it will not have

any

incentive

p 2 = P ( k1 + k 2 ) .

to

raise

Now consider the case

its

price

from

p1 = P (k1 + k 2 )

given

p 2* ( k1 ) > P( k1 + k 2 ) . Firm 2 will choose

p 2 = p 2* ( k1 ) provided that p1 < p 2* (k1 ) . However, firm 1’s best response

to p 2 = p 2* ( k1 ) is p1 ≥ p 2* (k1 ) if it exists since otherwise it will be able

to gain by raising its price. And if we suppose p1 ≥ p 2* (k1 ) instead, then firm 2 will always try to slightly undercut with extra capacity since p1 > P (k1 + k 2 ) . Therefore, it is implied by p 2* ( k1 ) > P( k1 + k 2 ) that there

is no pure strategy equilibrium.

c c So far, we showed for k1 < k and k 2 = k that there exists a pure

strategy equilibrium in Bertrand‐Edgeworth model if and only if

p 2* ( k1 ) ≤ P( k1 + k 2 ) . Finally, using Corollary 2 of Diaz and Kujal, we can

conclude that p1E (k1 , k 2 ) = P (k1 + k 2 ) if and only if p 2* ( k1 ) ≤ P( k1 + k 2 ) . ■

◀ Capacity Choice and List Pricing in a Duopoly 139

Proposition 3

In our three‐stage game with the Assumption 1 and 2, if the demand curve is

concave, then the Cournot outcome is a subgame perfect Nash equilibrium outcome.

Proof. As a preliminary step, we first show that the concave demand curve

c c will guarantee p 2* ( k1 ) ≤ P( k1 + k 2 ) for k1 < k and k 2 = k so that we

could make use of Lemma 2. First notice that

p 2* ( k c ) ≤ P( k c + k c )

because the Cournot pure strategy equilibrium would not hold otherwise. Hence, if the inequality

dp 2* (k1 ) ∂P (k1 + k 2 ) ≥ holds, p 2* ( k1 ) ≤ P( k1 + k 2 ) dk1 ∂k1

c c will be guaranteed for k1 < k and k 2 = k . In order to see whether this

is the case, consider

∂P (k1 + k 2 ) dp 2* (k1 ) and respectively. ∂k1 dk1

The first order condition of p 2* ( k1 ) is given by D' ( p 2* ) p 2* + D( p 2* ) − k1 = 0 .

Taking derivatives of this condition by k1 , we get [ D' ' ( p ) p + 2 D' ( p )]

other hand, t

dp 2* dp 2* 1 − 1 = 0 . Therefore, = . On the dk1 dk1 D' ' ( p ) p + 2 D' ( p )

∂P (k1 + k 2 ) 1 = P' ( k1 + k 2 ) = . ∂k1 D' ( p)

Since D' ' ( p) ≤ 0 by the concavity assumption and D' ( p ) < 0 , we have h

inequality D' ' ( p) p + 2 D' ( p ) < D' ( p ) , which implies

e

dp (k1 ) ∂P (k1 + k 2 ) > . dk1 ∂k1 * 2

c c Therefore, we proved p 2* ( k1 ) ≤ P( k1 + k 2 ) holds for k1 < k and k 2 = k .

Given this result, we will show both of the firms will produce according to the Cournot equilibrium.

140 産業組織硏究 第14集 第4號 ▶

[Firm 1] The first order condition for firm 1 is

∂p1E (k1 , k2 ) k1 + p1E (k1, k2 ) − c'1 (k1 ) = 0 ∂k1

if p1E (k1 , k 2 ) is differentiable. Unfortunately, it is not differentiable at

Cournot

capacities due to the discontinuity of the residual demand curve. So, we will c c consider the cases of k1 < k and k1 > k separately. (a) Consider

k1 < k c and k 2 = k c . According to the concavity assumption and Lemma

2, we know that p1E (k1 , k 2 ) = P (k1 + k 2 ) holds. So, firm 1’s problem is

exactly the same as the Cournot problem in this case, and hence firm 1 c c will choose to increase k1 up until k1 = k . (b) Consider k1 > k and

k 2 = k c . When we increase k1 from k1 = k c , lower residual demand for

firm 2 will give firm 2 lower profit given p1 , which means the value of the right hand side of equation (1) will be smaller and hence the solution

p1E (k1 , k 2 ) becomes lower. Note that this change does not

involve a discontinuity. With some abuse of notation, therefore, we can

c take derivatives of the equation (1) at the neighborhood of k1 = k such

c that k1 > k . (Since we are considering the neighborhood of Cournot

capacities the left hand side of the equation (1) should be k 2 p1 ) k2

∂p1 ∂R1 ( p 2R ( p1 , k1 ), p1 , k1 ) p 2R ( p1 , k1 ) ∂R1 ( p 2R ( p1 , k1 ), p1 , k1 ) p 2R ( p1 , k1 ) ∂p1 = + ∂k1 ∂k1 ∂p1 ∂k1

=

∂R1 ( p 2R , p1 , k1 ) R ∂R1 ( p 2R , p1 , k1 ) R ∂p1 p2 + p2 by the envelope theorem. ∂k1 ∂p1 ∂k1

Since R1 ( p 2 , p1 , k1 ) = D( p 2 ) − k1 , k 2

∂p1 ∂p1 pR =− 2 . = − p 2R and hence ∂k1 ∂k1 k2

◀ Capacity Choice and List Pricing in a Duopoly 141

Evaluated at Cournot capacities, Therefore,

∂p1E (k c , k c ) pR ( pc ,k c ) pc =− =− 2 . ∂k1 kc kc

∂p1E (k c , k c ) c k + p1E (k c , k c ) − c '1 (k c ) = −c '1 (k c ) < 0 . ∂k1

Combining the results from (a) and (b), and the Assumption 2(b) (i.e.,

concavity of the payoff function Π 1 (k1 , k 2 ) ), we conclude firm 1 will

choose the Cournot capacity when the other firm chooses Cournot capacity.

[Firm 2] According to Proposition 2, firm 2’s best response choice always

satisfies p 2R = p1E (k1 , k 2 ) = P (k1 + k 2 ) in equilibrium and quantity equalizes

c BR c to k 2 . Suppose firm 2’s best response to k1 = k is k 2 ≠ k and it is the

equilibrium. The equilibrium payoff with k 2BR will be P (k c + k 2BR )k 2BR by

c Proposition 2. By Proposition 1, we know that k1 = k 2 = k will give firm

2

the

payoff

P ( 2k c ) k c .

Since

we

supposed

k 2BR ≠ k c ,

P (k c + k 2BR )k 2BR > P (2k c ) k c . However, this contradicts the definition of

Cournot equilibrium. Therefore, firm 2 will also choose the Cournot

capacity when the other firm chooses the Cournot capacity in equilibrium. ■

In Proposition 3, we have seen that, besides Assumptions 1 and 2, the

concavity of the demand curve is used to ensure the Cournot outcome as a

subgame perfect Nash equilibrium in out three‐stage game.9) As is apparent from

the proof, however, the sufficient condition may be weakened. In particular, if we assume D' ' ( p) p + D' ( p ) < 0

around the neighborhood of Cournot

outcome, it will be enough for the same result as in Proposition 3. In

other words, we could even allow a convex demand curve as long as its 9) Kreps and Scheinkman (1983) also used the assumption of concave demand function to prove the Cournot outcome is the unique equilibrium in their two‐stage game.

142 産業組織硏究 第14集 第4號 ▶

curvature is not too large.

The reason why the curvature matters here is related to the potential market

size for firm 2 as a higher‐priced firm. The more convex the demand curve is,

the larger the market that firm 2 can expect by setting a higher discount price.

In firm 1’s view, reducing its capacity from the Cournot level could induce firm

2 to shift to a higher discount price when the size of the potential market for

firm 2 is relatively large. And in turn this might benefit firm 1 by enabling it to have higher Edgeworth price (i.e. higher list and discount price), which will let firm 1 deviate from Cournot outcome if the new potential Edgeworth price is sufficiently high.

5. Conclusion As in the existing literature that studied Bertrand versus Cournot competition,

our results depended critically on the rationing rule. In the case of the parallel rationing rule, we showed that the predetermined Cournot capacities lead exactly to the Cournot outcome in Diaz and Kujal’s list pricing game and furthermore

an extended game that allows capacity choice has Cournot outcome as an

equilibrium provided that the demand curve is concave. It implies that the Cournot equilibrium is still an important reference point even after introducing a more sophisticated price competition structure. And, Kreps and Scheinkman

(1983)’s well‐known result proved to be robust to this model change. In case of any other rationing rule we considered (i.e., λ ≠ 1 ), however, the

predetermined Cournot capacities brought forth a more collusive list

pricing equilibrium than with parallel rationing. From this we could see that the Cournot outcome cannot hold in our three‐stage game without

parallel rationing rule. However, we did not directly investigate the

nature of the equilibria of this case and it will have to await the future research. If we get some hint from Davidson and Deneckere (1986)’s

insight, the reasonable conjecture would be that more collusion in the

◀ Capacity Choice and List Pricing in a Duopoly 143

price game leads to more competition in capacity choice, i.e. higher capacity levels than Cournot.

Finally, I would like to mention a potentially interesting topic we could

pursue in the future extending the list pricing model. It would be a natural extension to suppose random demand at the list pricing stage and realized

demand at discounting stage. In that case, the merit of the pure strategy SPNE

to firm 1 will definitely be decreased since in case of high demand it should bear a low‐price commitment and in case of low demand it should face a higher

possibility of competition caused by undercut of firm 2. (The more risk averse it is, the lower will be its list price.) To the contrary, firm 2 would have nothing

to lose by setting a high enough list price. So it seems to be interesting to try investigating the equilibrium with this kind of intuition. Moreover, I think we

could also check out with certain data whether market sharing by list pricing break down at low demand period.

REFERENCES 1. Davidson, C. and R. Deneckere (1986) Long‐Run Competition in Capacity, Short‐Run Competition in Price and the Cournot Model,

Rand Journal of Economics, Vol.17, 404‐415.

2. Diaz, A. and P. Kujal (2003) List Pricing and Pure Strategy

Outcomes in a Bertrand‐Edgeworth Duopoly, Working Paper 03‐49, Universidad CarlosⅢ de Madrid.

3. Friedman, J. (1988) On the Strategic Importance of Prices versus Quantities, Rand Journal of Economics, Vol.19, 607‐622.

4. Kreps, D. and J. Scheinkman (1983) Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes, Bell Journal of

Economics, Vol.14, 326‐337.

5. Osborne, M. and C. Pitchik (1986) Price Competition in a Capacity‐

144 産業組織硏究 第14集 第4號 ▶

Constrained Duopoly, Journal of Economic Theory, Vol.38, 238‐260.

6. Tirole, J. (1988) The Theory of Industrial Organization. Cambridge: MIT Press.

◀ Capacity Choice and List Pricing in a Duopoly 145

복점에서의 설비량 선택과 표시요금제 김 성 환 정보통신정책연구원 공정경쟁정책연구실

설비량이 제한된 복점 가격경쟁 모형에 사전적인 표시요금 설정 단계가

추가되면 순수전략(pure strategy)균형이 존재한다는 사실이 알려져 있다. 이 논문은 설비량 가정에 초점을 맞추어 이러한 균형의 특성에 대해 분석하였 는데 그 주요결과는 다음과 같다. 첫째, 만약 설비량이 쿠르노 균형에서와

같다고 가정하는 경우에, 잔여 수요(residual demand)가 평행적 배분(parallel rationing)을 따르면 그리고 그럴 경우에만 균형 가격은 쿠르노 균형과 일치 한다. 둘째, 만약 설비량이 기업에 의해 사전적으로 (표시요금 설정 이전에)

자유롭게 선택가능한 경우에도, 평행적 배분 규칙하에서 그 균형은 쿠르노

균형과 일치한다. 이는 Kreps and Scheinkman (1983)의 결과가 가격경쟁의 구 조가 이처럼 변화하는 경우에도 여전히 유효하다는 것을 보여준다. 주제어: 표시요금제, 설비량 선택, 가격경쟁 JEL Classifications: D43, L13, C72