1 Oligopoly: Bertrand Model

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1 1 Oligopoly: Bertrand Model Bertrand model: There are two rms and no entry is possible. Homogeneity of product. Single period. Consumers always purchase from the cheapest seller. If the two selllers charge the same price then half of the consumers purchase from rm 1 and the other half from rm. Firms chose price. The demand function faced by the rm i is 8 0 if p i > a >< 0 if p q i = i > p j a p >: b if p i = p j = p < a a if p i < minfa; p j g p i b Let the rms 1 and have constant marginal cost c 1 and c respectively. Formally: Pro t function of rm 1 assuming that the second rm has chosen p = p b 1 (p 1 ; p b ) = p 1 q 1 T C 1 = p 1 q 1 c 1 q 1 The imization problem is p1 1 (p 1 ; p b ) Similarly for the second rm the pro t function is p (p b 1; p ) = p q c q q 1 and q are determined from the demand function described above. Result: 1. If c 1 = c = c, then a Bertrand equilibrium is p b 1 = p b = c and q1 b = q b = a c b :. If c 1 < c ; and let be very small, then Bertrand equilibrium is p b = c ; p b 1 = c and q b = 0; q1 b = a c+ b Bertrand equilibrium under capacity constraints. Let the rms, with constant zero marginal costs face a demand where there are 4 consumers consuming a imum of an unit each. Consumer 1 has a reservation price of $3, consumer has reservation price of $, consumer 3 has reservation price of $1, consumer 4 has reservation price of $0. 1

2 If the rms could produce as much as they wanted then the Bertrand Nash equilibrium would be p b = p b 1 = 0: Let us assume that the rms cannot produce more than units. If the rms were at p = p 1 = 0 (whre each rm makes 0 pro ts), then this cannot be equilbrium outcome. For rm can deviate and set price equal to $3 and make a pro t of $3. Edgeworth cycles: an equilibrium is never reached. Stackelberg (leader-follower) model: Let us make the following assumptions. Let rm 1 be the leader and rm be the follower. There are two rms, 1 and. Homogeneity of product. The rms chose quantity. The rms move sequentially. Firm 1 knows the demand function and can predict how rm will react. Let the demand functions be p = a bq The marginal cost of both rms is constant and is c:there is no xed cost. Timeline. Stage 1: Firm 1 choses quantity. Stage : Firm choses quantity. Solution: In order to solve this game we need to use backward induction. Stage : Firm behavior q = (a bq 1 bq )q cq d dq = (a bq 1 bq ) c = 0 q = a bq 1 c b = R(q 1 )

3 Stage 1: Firms 1 behavior q 1 1 = (a bq 1 bq )q 1 cq 1 = (a bq 1 br(q 1 ))q 1 cq 1 = (a bq 1 + c)q 1 = a + c dq 1 q 1 = a c b and q = a c 4b ; Q = q 1 + q = 3(a c) 4b 3 Di erentiated products: bq 1 c = 0 : p = a+3c 4 Drop the assumption of identical products: sellers sell di erentiated products. There are two rms, rm 1 and rm. The two products have an inverse demand function: cq 1 p 1 = q 1 q ; p = q 1 q > 0; > And the direct demand functions are Measure of di erentiation is q 1 = a bp 1 + cp ; q = a + cp 1 bp The quantity demand game: The players play a Cournot game. Let us assume that there is zero cost. For rm 1 this gives the reaction function Therefore equilibrium is q 1 = q = q 1 1 = ( q 1 q )q 1 + ; p 1 = p = q 1 = q + ; 1 = = ( + ) 3

4 Claim 1 In a quantity game the pro ts increase as the products get more di erentiated. Price game: The market structure is similar to that of the Bertrand game. Given the direct demand funtion we can write the behavior function of one of the players as 1 p 1 = (a bp 1 + cp )p 1 dp 1 = a bp 1 + cp p 1 = a + cp b Therefore the equilibrium outcome is p 1 = p = a b c ; q 1 = q = ab b c ; a b 1 = = (b + c) Strategic complements and strategic substitutes Claim In Bertrand game with di erentiated products the pro t of the rms increase with di erentiation. Cournot vs Bertrand outcome Sequential move price game: Let Firm 1 choose price rst and then frim follows by selecting p :Let cost be zero and the demand function be q 1 = (168 p 1 + p ); q = (168 + p 1 p ) Therfore the imization problem is 1 (p 1 ; R (p 1 )) = (168 p p 1 )p 1 p = 10 dp 1 p 1 = 0 p 1 = 60; p = 57 Claim 3 1. Both rms make more pro t if they play sequentially than if they play the game simultaneously. The leader makes smaller pro t than the follower. 4

5 4 Monoploistic Competition Di erentiated products. Let there be n rms. The residual demand function is p i = D(q 1 ; ::::; q N ) Cournot/quantity game is being played but with free entry. Characteristic space. Represenative model: Let the consumer have a utility function U(q 1 ; ::::; q N ) = And budget function p i q i I Therefore the optimization Lagrangian is p qi p qi + [I p i q i ] For rms: all are identical with cost functions T C i = F + cq i Equlibrium: 1. Consumers imize utility.. Firms imize pro ts. 3. Free entry till the pro ts are zero. 4. All resources are utilized. Too much or too little: 1. Fixed costs.. Externality e ects. 5