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Nash Equilibrium Nash Equilibrium u A game consists of – a set of players – a set of strategies for each player – A mapping from set of strategies to a set of payoffs, one for each player N.E.: A Set of strategies form a NE if, for player i, the strategy chosen by i maximises i’s payoff, given the strategies chosen by all other players u NE is the set of strategies from which no player has an incentive to unilaterally deviate u NE is the central concept of non- cooperative game theory I.e. situtations in which binding agreements are not possible Example Player 2 C D C (10,10) (0,20) This is the Player 1 game’s D (20,0) (1,1) payoff matrix. Player A’s payoff is shown first. Player B’s payoff is shown second. NE: (DD) = (1,1) Another Example…. Player B L R U (3,9) (1,8) Player A D (0,0) (2,1) Two Nash equilibria: (U,L) = (3,9) (D,R) = (2,1) Applying the NE Concept Modelling Short Run ‘Conduct’ Bertrand Competition Cournot Competition [Building blocks in modeling the intensity of competition in an industry in the short run] p pmonop P(N))? C N Bertrand Price Competition u What if firms compete using only price-setting strategies,? u Games in which firms use only price strategies and play simultaneously are Bertrand games. Bertrand Games (1883) 1. 2 players, firms i and j 2. Bertrand Strategy - All firms simultaneously set their prices. 3. Homogenous product 4. Perfect Information 5. Each firm’s marginal production cost is constant at c. Bertrand Games pi = 0 if pi >pj pi = ½ (pi – c)Q if pi = pj pi = (pi – c)Q if pi < pj u Q: Is there a Nash equilibrium? u A: Yes. Exactly one. All firms set their prices equal to the marginal cost c. Why? Bertrand Games Proof by Contradiction u Suppose one firm sets its price higher than another firm’s price. u Then the higher-priced firm would have no customers. u Hence, at an equilibrium, all firms must set the same price. Bertrand Games u Suppose the common price set by all firm is higher than marginal cost c. u Then one firm can just slightly lower its price and sell to all the buyers, thereby increasing its profit. u The only common price which prevents undercutting is c. Hence this is the only Nash equilibrium. Illustration p1=p2 p2 E B C F A D C C p1 NE: at point A where p1=p2=Cost Bertrand Paradox u For n>=2 with firms simultaneously setting prices, prices = marginal cost and profits are zero……. Perfectly competitive outcome is replicated u Intuitive assumption …..surprising result! u This result holds where firms have identical costs u If firms have different costs, then there may or may not be a pure strategy equilibrium If firms are capacity constrained, then a mixed strategy equilibrium results u Edgeworth (1897) - Capacity Constraints Neither firm can meet the entire market demand, but can meet half market demand. Constant MC to a point, then decreasing returns Under these conditions, Edgeworth cycle: prices fluctuate between high and low Kreps & Scheinkman (1983) u If there is a two stage game, u in which firms set capacity in stage 1 u And in stage 2, given their capacity, set price u Then the Cournot result is observed Differentiated Products resolve the Bertrand Paradox u Differentiated Products allow price competing oligopolists to mark up Cournot Competition (1838) 1. 2 Players (identical) 2. Cournot strategy - All firms simultaneously set their output 3. Homogenous product 4. Perfect Information 5. Linear demand 6. Constant MC 2 identical firms, linear demand, constant marginal cost P = a - Q = a - q1 - q2 TCi = c( qi ) Choose q to max p , given q P1 = ( a - q1 - q2 ).q1 - c( q1 ) 1 1 2 ¶P1 = a - 2q1 - q2 - c = 0 ¶q1 Higher q , lower level q to max p a - q - c 2 1 1 q = 2 q = R( q ) 1 2 1 2 Similarily,identical firms Þ a - q - c q = 1 q = R( q ) 2 2 2 1 u An equilibrium is when each firm’s output level is a best response to the other firm’s output level - then neither wants to deviate from its output level. u A pair of output levels (q 1 *,q 2 *) is a Cournot-Nash equilibrium if * * * * q 1 =R1(q2 ) and q2 =R1(q1 ) Firm 1’s “reaction curve” q *=R1(q *) q2 1 2 Cournot-Nash equilibrium q * = R (q *) and q * = R (q *) * 1 1 2 2 2 1 q2 q * 1 q1 * * Firm 2’s “reaction curve” q2 =R1(q1 ) Solve reaction curves to find cournot equilibrium… R 1 (q 2 ) = R 2 (q 1 ) = a - q - c a - ( 1 ) - c a - q - c q = 2 Þ q = 2 1 2 1 2 a - c solving : q * = 1 3 identical firms : in equilibriu m a - c q * = q * = cournot nash equil 1 2 3 2 (a - c ) Total Cournot quantity : 3 (a + 2c ) Solve for price : p = a - Q = 3 Solve for profit p = p q = p u Q: Are the Cournot-Nash equilibrium profits the largest that the firms can earn in total? u A: Firms could earn higher profits if both agreed to set half the monopoly output (and thus earn half monopoly profit each) HOWEVER Collusive\Joint profit max output levels q m1q m2 not sustainable – incentives to unilaterally deviate – not a NE m , if firm 1 continues to produce q 1 firm 2’s profit-maximizing response is q 2 m = R2(q 1 ) P monop > Pcournot > Pperfcomp=bertrand Q monop < Qcournot < Qperfcomp=bertrand p monop > p cournot > p perfcomp=bertrand Example: P = 140 – Q; Ci = 60(qi); 2 firms play Cournot. What are equilibrium outcomes? P = 140 - q 1 - q 2 P 1 = (140 - q 1 - q 2 ). q 1 - 60 q 1 ¶P 1 = 140 - 2 q 1 - q 2 - 60 = 0 ¶q 1 1 q 1 = 40 - 2 q 2 and solving profit max q 2 , given q 1 1 q 2 = 40 - 2 q 1 solve reaction functions : 1 1 q1 = 40 - 2 q 2 and q 2 = 40 - 2 q1 1 1 * 80 2 q1 = 40 - 2 (40 - 2 q1 ) Þ q1 = 3 = 26 3 * * 2 identical firms : q1 = q 2 = 26 3 160 1 Total Cournot quantity : 3 = 53 3 260 2 Solve for price : p =140 - Q = = 86 3 3 Solveforpr ofit :p1 = p 2 = Monopoly….1 firm P = 140 - q m P m = (140 - q m ).q m - 60 q m ¶P m = 140 - 2q m - 60 = 0 ¶q m q m = 40 p m =140 - Q = 140 - 40 = 100 Perfect Competition (& Bertrand) P = MC = 60 P = a-Q so Q = a-c = 140-60 = 80 Profit = p.q – c.q = 0 since p = c Shows Q m < Qc < Qpc And P m > Pc > Ppc Thus, Bertrand ® for N> or = 2, get perfectly competitive outcomes Can show that as • N, cournot outcome ® perfectly competitive N player Cournot 1. N Players (identical) 2. Cournot strategy - All firms simultaneously set their output 3. Homogenous product 4. Perfect Information 5. Linear demand 6. Zero Cost N Q = q P = a - bQ ; å i i = 1 since N firms are identical Q = Nq i TCi = 0+0q1 so c = 0 Firm : i P i = ( a - bQ ). q i Note: if y = bQ.qi & ¶P i N Q = q + q + ….+ q Q = q i j n = a - bQ - bq =å i0 i i =1 ¶ q i (so dQ/dqi =1) a - b ( Nq i ) - bq i = 0 dy/dqi= bQ.1+ bqi.dQ/dqi a - b (1 + N ) q i = 0 = bQ+ bqi * a q i = b ( N + 1) * a p i = a - b .N .q i = ( N + 1) P = a / N+1 p N®¥ p ®c =0 N=1: a/2 N=2: a/3 N=3: a/4 D Duo olig M Q Q Q Q P(N) function links price cost margins to a given N p p monop Joint Max. Cournot MC Bertrand 1 2 N P for any given N depends on the ‘intensity of competition’ (Bertrand: Most intense) Entry? Entry s P(N) Stage 1 Stage 2 (Long Run) (Short Run) The entry decision is a backward induction procedure We return to modelling entry, where N is endogenous and depends on P(N) in the latter part of the course…….
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