Market power & product differentiation

Transcription

1 & product Applications of game theory 1 Department of Economics, University of Oslo ECON5200 Fall 2009 Introduction 3 lectures where game theory is applied: 13 October & product. 20 October Adverse selection, signaling and screening. 27 October Principal-agent problems. 4 November Seminar.

2 To be covered under & product by a monopoly firm by two (or more) firm with identical and constant unit costs (Bertrand oligopoly model) Are two firms are sufficient for a perfectly competitive outcome? Competition can be relaxed by constraints (Cournot oligopoly model) Another topic: to affect future by a profit-maximizing monopolist Monopoly Duopoly x( ): continuous and strictly decreasing demand function for all p with x(p) ; x(p) =0forp p max p px(p) c(x(p)) p( ) =x 1 ( ): inverse demand function; withp(0) = p. First-order condition: max q p(q)q c(q) p (q m )q m + p(q m ) c (q m ) with equality if q m If q m, then p(q m ) > c (q m ): Price under monopoly exceeds marginal cost; this leads to quantity distortion and welfare loss. Other distortions; investment in cost reductions & quality choice.

3 by profit-maximizing duopolists Bertrand model with homogeneous products (1) Monopoly Duopoly Structure: (1) Prices set (2) Demand determines quantities x( ): continuous and strictly decreasing demand function for all p with x(p) ; x(p) =0forp p. Firms1and2have constant unit cost c and set prices p 1 and p 2 simultaneously. Sales are given by: x j (p j, p k )= x(p j ) if p j < p k 1 2 x(p j) if p j = p k 0 if p j > p k Proposition There is a unique Nash equilibrium (p1, p 2 ) in the Bertrand duopoly model. In this equilibrium, both firms set their prices equal to cost: p1 = p 2 = c. by profit-maximizing duopolists Bertrand model with homogeneous products (2) Monopoly Duopoly Proof. Part 1: (p1, p 2 )=(c, c) is a Nash equilibrium. Assume that p k = c. Iffirmj chooses p j = c, then profit equals 0. If firm j chooses p j < c, then profit is negative. If firm j chooses p j > c, then profit is 0. Therefore, p j = c is best response to p k = c. Part 2: (p 1, p 2 ) is not a Nash equil. if (p 1, p 2 ) (c, c). Case 1: min{p 1, p 2 } < c. At least one firm j has negative profit. Zero profit by setting p j 0. Case 2: p k > p j = c. Firm j can increase its profit by setting p j = 1 2 (p k + c). Case 3: p k p j > c. Firm k can increase its profit by setting p k = p j ε, forε>0 sufficiently small. Conclusion: If J firms have constant unit cost c, then all firms set price equal to c, providedj 2.

4 Firms set prices. But the conclusion of the Bertrand model that two firms are sufficient for perfectly competitive outcomes is unrealistic. Competition can be relaxed by Not no capacity contraints, but capacity constraints (Cournot oligopoly model) Not homogeneous products, but product Not a static game, but repeated Quantity choices by profit-maximizing duopolists Cournot duopoly model (1) Structure: (1) Quantities set (2) Demand determines price Assume firms first choose quantities q 1 and q 2 and that these are sold at price p(q 1 + q 2 ) in the market. Each firm j s problem: First-order condition: max q j p(q j + q k )q j cq j p (q j + q k )q j + p(q j + q k ) c with equality if q j For each q k,letb j ( q k ) denote j s best response.

5 Quantity choices by profit-maximizing duopolists Cournot duopoly model (2) (q1, q 2 ) is a Nash equilibrium if and only if for each j, qj b j (qk ). Hence, if (q 1, q 2 ) is a Nash equilibrium, then p (q1 + q2)q 1 + p(q1 + q2) c with equality if q1 p (q1 + q2)q 2 + p(q1 + q2) c with equality if q2 If p(0) > c, then(q1, q 2 ) 0, implying that ( ) p (q1 + q2) q 1 +q2 2 + p(q1 + q2)=c Proposition In a Nash equilibrium of the Cournot duopoly model with p(0) > c and p (q) for all q, the market price is greater than c (the competitive price) and smaller than p(q m ). Quantity choices by profit-maximizing duopolists Cournot duopoly model (3) Proof. Part 1: p(q1 + q 2 ) > c in a Nash equilibrium. Follows from the FOC and the assumption that p (q). Part 2: p(q1 + q 2 ) < p(qm ) in a Nash equilibrium. Since p (q) forallq, this is equivalent to q1 + q 2 > qm. Subpart (i): q1 + q 2 qm. Suppose q1 + q 2 < qm.ifonefirmjincreases quantity to q m qk, then total profit is increased and the profit of the other firm is decreased. Hence, firm j can do better. Subpart (ii): q1 + q 2 qm. Suppose q1 + q 2 = qm. The FOCs for monopoly and Cournot duopoly cannot both be satisfied.

6 choices followed by price Kreps & Scheinkman (1983) Structure: (1) Capacities set (2) Prices set (3) Demand determines quantities In the subgames starting at stage (3), all possible profiles of capacities and prices must be considered. How are consumers rationed? 1 Proportional rationing: Any consumer willing to pay a good offered at p has an equal change of buying at p. 2 Efficient rationing: Aconsumerwiththehighest willingness-to-pay is served first. Proposition With efficient rationing, the game considered by Kreps & Scheinkman (1983) leads to the Cournot outcome. Two issues How does product affect price? Bertrand model where demanded quantity is a continuous function of the price profile Horizontal How do firms product differentiate to relax price? Not covered

7 by profit-maximizing duopolists Bertrand model with differentiated products (1) Structure: (1) Prices set (2) Demand determines quantities Firms 1 and 2 have constant unit cost c and set prices p 1 and p 2 simultaneously. Sales are given by x j (p j, p k ), which is the continuous demand function for firm j. Proposition max p j (p j c)x(p j, p k ) If x 1 (p 1, p 2 ) when p 1 min{c, p 2 } and x 2 (p 2, p 1 ) when p 2 min{c, p 1 }, then in any Nash equilibrium (p1, p 2 ) we have that (p1, p 2 ) (c, c). by profit-maximizing duopolists Bertrand model with differentiated products (2) Proof. To be shown: (p 1, p 2 ) is not a Nash equilibrium if min{p 1, p 2 } c. Case 1: min{p 1, p 2 } < c. At least one firm j has negative profit. Non-negative profit by setting p j 0. Case 2: min{p 1, p 2 } = c. If p j = c and p k c, thenby assumption x j (p j, p k ), and firm j s profit is zero. Since x j (, ) is continuous, firm j can attain positive profit by setting p j = c + ε for ε>0 sufficiently small.

8 Horizontal product Linear city with linear transportation costs (1) Assume that M consumers are distributed uniformly along the unit inverval: [0, 1]. Also, assume that firm 1 is located at 0 and firm 2 is located at 1. For a consumer located at z ( [0, 1]) the cost to buy from firm 1 is p 1 + tz and the cost to buy from firm 2isp 2 + t(1 z). Finally, assume that every consumer buys one (and only one) unit from a firm with lowest cost. Indifferent consumer ẑ: p 1 + tẑ = p 2 + t(1 ẑ), which implies ẑ = t + p 2 p 1 2t x j (p j, p k )= and 1 ẑ = t + p 1 p 2 2t M if p j p k t (t+p k p j )M 2t if p j (p k t, p k + t) 0 if p j p k + t Horizontal product Linear city with linear transportation costs (2) max (p j c) (t + p k p j )M p j [p k t,p k +t] 2t FOC if p j (p k t, p k + t): t + p k + c 2p j =0 b j ( p k )= p k t if p k c +3t t+p k +c 2 if p k (c t, c +3t) p k + t if p k c t Unique Nash equilibrium, (p1, p 2 )satisfies: p1 = p 2 = c + t. Hence, price exceeds c. Do firms maximize product differentiate to relax price? Problematic to discuss choice of product with linear transportation costs. Why? What product maximizes welfare?

9 Bertrand duopoly (1) Firms 1 and 2 have constant unit cost c. Ineachstage,firms choose prices simultaneously. The chosen prices determine quantities in this stage and becomes observable for both firms before the next stage. Infinite number of stages. Payoff is profits discounted at constant discount factor δ (0, 1). Consider the following paths, where p (c, p m ]: (p(0) t )=(p, p), (p, p),... (p(1) t )=(c, c), (c, c),... (p(2) t )=(c, c), (c, c),... Proposition Thesimplestrategyprofileσ((p(0) t ), (p(1) t ), (p(2) t )) is a subgame-perfect equilibrium if and only if δ 1 2. Bertrand duopoly (2) Proof. Enough to consider unilateral one-period deviations. Why? Is a unilateral one-period deviation from (p, p) profitable? (1 δ)(p c)x(p) }{{} Supremum of payoff with deviation 1 2 (p c)x(p) }{{} Payoff without deviation Not profitable if 1 δ 1 2 or, equivalently, δ 1 2. Is a unilateral one-period deviation from (c, c) profitable? Payoff with deviation is non-positive. Payoff without deviation equals zero. What if there are more than 2 firms? What if prices are not perfectly observable?

10 to affect future (1) Structure: (1) Firm 1 makes strategic investment k. Observable. (2) Firms 1 and 2 play some oligopoly game, choosing s 1 and s 2 Payoffs: π 1 (s 1, s 2, k) andπ 2 (s 1, s 2 ) Actions are aggressive : π 1 (s 1, s 2, k)/ s 2 and π 2 (s 1, s 2 )/ s 1. Best response functions: b 1 (s 2, k) andb 2 (s 1 ). Assume unique Nash equilibrium: (s 1 (k), s 2 (k)). Assume sufficient differentiability. to affect future (2) Identity: s 2 = b 2(b 1 (s 2, k)) ds 2 = db 2 ( b1 ds2 + b ) 1 s 2 ( 1 db ) 2 b 1 ds2 = db 2 b 1 s 2 ds 2 = db 2 ( b 1 ) 1 db 2 b 1 s 2

11 to affect future (3) ( ds 2 = db b1 2 1 db 2 b 1 s 2 b 1 ( ) ) substitutes: db 2 ( ) ds 2 (k) complements: db 2 ( ) ds 2 (k) b 1 ( ) ds 2 (k) ds 2 (k) to affect future (4) To Deter Entry b 1 ( ) b 1 ( ) substitutes: db 2 ( ) Δk Top Dog big & strong to look tough & aggressive Δk Mean & Hungry Look small & firm to look tough & aggressive complements: db 2 ( ) Δk Top Dog big & strong to look tough & aggressive Δk Mean & Hungry Look small & firm to look tough & aggressive

12 to affect future (5) To Accommodate Entry b 1 ( ) b 1 ( ) substitutes: db 2 ( ) Δk Top Dog big & strong to look tough & aggressive Δk Mean & Hungry Look small & firm to look tough & aggressive complements: db 2 ( ) Δk Puppy Dog small & weak to look soft & inoffensive Δk Fat Cat fat & mellow to look soft & inoffensive