Strategic complements and substitutes in bilateral oligopolies

Economics Letters 70 (2001) 83–87 www.elsevier.com / locate / econbase

Strategic complements and substitutes in bilateral oligopolies ´ ` Ferrer b Francis Bloch a , *, Helene a

IRES and CORE, Department of Economics, Universite´ Catholique de Louvain, Louvain-La-Neuve, Belgium b CORE, Universite´ Catholique de Louvain, Louvain-La-Neuve, Belgium Received 4 January 2000; accepted 15 June 2000

Abstract This paper characterizes the equilibrium of a bilateral oligopoly where traders have CES utility functions. We show that the offers of traders on the two sides of the market are strategic substitutes if and only if the goods are complements.  2001 Elsevier Science B.V. All rights reserved. Keywords: Bilateral oligopoly; Strategic substitutes and complements; CES utility functions JEL classification: D43

1. Introduction The strategic market games introduced by Shapley and Shubik (Shapley, 1976; Shubik, 1973) offer a simple description of an exchange economy where all traders behave strategically. A specific example of Shapley–Shubik games, recently introduced by Gabszewicz and Michel (1997), can be viewed as an extension of the Cournot oligopoly model, where both sides of the market (buyers and sellers) are strategic players. In this ‘bilateral oligopoly’, there are two types of traders and two commodities, and each type of trader is endowed with one unit of the two commodities and wants to consume both. General conditions for existence and uniqueness of an interior equilibrium have recently been studied by Bloch and Ghosal (1997) and Bloch and Ferrer (2000). In this note, we investigate further the structure of bilateral oligopolies by analyzing the relation between product complementarity and the Nash equilibrium of the strategic market game, in the simple setting of a CES (constant elasticity of substitution) utility function. We show that the offers of traders on the two sides of the market are strategic complements (substitutes) if and only if the two

*Corresponding author. Tel.: 132-10-473-963; fax: 132-10-473-945. E-mail address: [email protected] (F. Bloch). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 00 )00343-8

 2001 Elsevier Science B.V. All rights reserved.

F. Bloch, H. Ferrer / Economics Letters 70 (2001) 83 – 87

84

goods are substitutes (complements).1 Furthermore, we prove that the outcome of the game converges monotonically to the competitive equilibrium, as the elasticity of substitution between the goods decreases to zero.

2. The model We consider an economy with two goods, labeled x and y, and two types of traders. There are n $ 2 traders of type I (who are endowed with one unit of good x) and n traders of type II (who are endowed with one unit of good y). Preferences for the two types of traders are identical, and given by the CES utility function U(x,y) 5 (x r 1 y r )1 / r , with r # 1. This utility function encompasses both situations where the goods are substitutes ( r $ 0) and complements ( r # 0). When r 5 1, the goods are perfect substitutes, when r 5 0, the utility function is Cobb–Douglas, and when r → 2 `, the goods are perfect complements. This exchange economy has a unique competitive equilibrium given by p * 5 (1,1) and (x * ,y * ) 5 (1 / 2,1 / 2) for traders of types I and II. We analyze a strategic market game where each trader offers a fraction of the commodity he owns on the market. Hence, the strategy spaces for the two types of traders are SI 5 hqi [ Ru0 # qi # 1j, SII 5 hb j [ Ru0 # b j # 1j. The final allocations obtained by the traders are given by

O O2

bj (x i ,y i ) 5 1 2 qi ,qi ]] , for traders of type I, qi

1

O 1O

qi (x j ,y j ) 5 b j ]],1 2 b j , for traders of type II, bj

2

with corresponding utility levels

Ob U (x ,y ) 51(1 2 q ) 11 q ]]2 2 Oq r

j

r

i

i

i

i

i

1 /r

,

i

1

See Bulow et al. (1985) for the original definition of strategic substitutes and complements in traditional oligopoly models.

F. Bloch, H. Ferrer / Economics Letters 70 (2001) 83 – 87

Oq U (x ,y ) 5 b ]] 1 (1 2 b ) 11 Ob 2 2 r

j

j

j

1 /r

r

i

j

85

.

j

j

3. The equilibrium To compute the reaction functions of the two types of traders, we solve the maximization problem faced by agents of type I: Max Ui (qi ,b j ).

0#q i #1

We obtain ≠U ]]i 5 A(qi ) ? B(qi ), ≠qi where

O qq Ob ) ]]] (O q ) Ob B(q ) 53(1 2 q ) 11 q ]]2 4 . 0. Oq A(qi ) 5 2 (1 2 qi ) r 21 1 (

r

k±i

j

k r 11

r 21 i

,

i

r

i

i

(1 / r )21

j

r

i

i

It is easy to check that the second-order conditions are satisfied, so that the maximization problem has a unique interior solution given by

3

A(qi ) 5 2 (1 2 qi ) r 21 1 (

O q q 5 0. Ob ) ]]] 4 (O q ) r

k±i

j

k r 11

r 21 i

i

We use this reaction function to study whether offers of traders on the two sides of the market and on the same side of the market are strategic substitutes or complements. Lemma 1. The offers of traders on the two sides of the market are strategic complements (substitutes) if and only if the goods are substitutes (complements). If the goods are complements, offers of traders on the same side of the market are strategic complements. If the goods are substitutes, offers of traders on the same side of the market are neither strategic complements nor strategic substitutes. Proof. By implicit differentiation, we get

O

O

O

r ( b j ) r 21 [ k ±i qk /( qi ) r 11 ]q ri 21 ≠qi ] 5 2 ]]]]]]]]]]]. ≠b j ≠A / ≠qi Hence, ≠qi / ≠b j $ 0 iff r $ 0. Similarly, we compute

F. Bloch, H. Ferrer / Economics Letters 70 (2001) 83 – 87

86

O

O

O

( b j ) r [(qi 2 r k±i qk ) /( qi ) r 12 ]q ir 21 ≠qi ] 5 2 ]]]]]]]]]]]]]. ≠q j ≠A / ≠qi The sign of ≠qi / ≠q j thus depends on the sign of (qi 2 r o k ±i qk ). We now turn to the computation of the equilibrium. It is easy to see that the equilibrium has to be symmetric, i.e. all traders of the same type adopt the same strategy. Hence, we may denote by q and b the offers of traders of type I and type II on the market. The reaction functions are then given by ≠U (n 2 1) ]]i 5 2 q(1 2 q) r 21 1 ]]]b r 5 0, ≠qi n

(1)

≠U (n 2 1) ]]i 5 2 b(1 2 b) r 21 1 ]]] q r 5 0. ≠b j n We can show that the system of equations (1) characterizes a unique symmetric Nash equilibrium, where all traders adopt the same strategy.2 Proposition 1. The strategic market game has a unique interior Nash equilibrium. All traders adopt the same strategy: 1 q * 5 b * 5 ]]]]]]] . [n /(n 2 1)] 1 / (12 r ) 1 1 Lemma 2. As the degree of substitution of the two goods increases the equilibrium offers q * and b * decrease. Proof. We compute log[n /(n 2 1)][n /(n 2 1)] 1 / (12 r ) ≠b * 1 ]] 5 2 ]]]2 ]]]]]]]]]] , 0. ≠r (1 2 r ) h1 1 [n /(n 2 1)] 1 / (12 r ) j 2 We obtain for a linear utility function ( r → 1) q * 5 b * 5 0, for a Cobb–Douglas utility function ( r 5 0), q * 5 b * 5 (n 2 1) /(2n 2 1). As the goods become perfect complements ( r → 2 `), q * 5 b * 5 1 / 2, the equilibrium outcome converges to the competitive equilibrium.

4. Discussion In this note, we analyze the Nash equilibrium of a bilateral oligopoly when the degree of substitution between the two goods is parametrized by the constant elasticity of substitution 1 /(1 2 r ). We first show that the offers of traders on the two sides of the market are strategic substitutes (or complements) if and only if the goods are complements (substitutes). To understand this result, consider the behavior of an agent i of type I. If the offer b j of agents of type II increases, the amount 2

The proof is available from the authors.

F. Bloch, H. Ferrer / Economics Letters 70 (2001) 83 – 87

87

of good y in agent i’s allocation, y i , increases. If the two goods are substitutes, this decreases the marginal utility of x i and induces trader i to increase her offer qi . If, on the other hand, the two goods are complements, this increases the marginal utility of x i and induces trader i to reduce her offer. Note that this effect is not related to the traditional analysis of strategic substitutes and complements in oligopoly. We also show that, as the complementary between the two goods increases, the equilibrium offers on the two sides of the market increase. The intuition underlying this result is easily grasped. When the complementarity increases, the marginal utility of good y to traders of type I increases. Hence, for any fixed offer b j of traders of type II, the offer qi increases. In equilibrium, both offers b j and qi are increasing with the degree of complementarity between goods.

Acknowledgements We thank Jean J. Gabszewicz and Nicolas Boccard for discussions which led to improvements in the paper. The second author is grateful to the European Community for support through the Cooperation and Information Network, contract ERB FMRX CT 96 0055. This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

References Bloch, F., Ghosal, S., 1997. Stable trading structures in bilateral oligopolies. Journal of Economic Theory 74, 368–384. Bloch, F., Ferrer, H., 2000. Trade fragmentation and coordination in strategic market games. Journal of Economic Theory (forthcoming). Bulow, J., Geanakoplos, J., Klemperer, P., 1985. Multimarket oligopoly: strategic substitutes and complements. Journal of Political Economy 93, 488–511. Gabszewicz, J.J., Michel, P., 1997. Oligopoly equilibrium in exchange economies. In: Eaton, B.C., Harris, R.G. (Eds.), Trade, Technology and Economics: Essays in Honor of Richard G. Lipsey. Edward Elgar, Cheltenham. Shapley, L., 1976. Noncooperative general exchange. In: Lin, S. (Ed.), Theory and Measurement of Economic Externalities. Academic Press, New York. Shubik, M., 1973. Commodity money, oligopoly, credit and bankruptcy in a general equilibrium model. Western Economic Journal 11, 24–38.