101A_finalsp_sol

101A_finalsp_sol - Econ 101A Solutions to Final Exam Th 15...

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Econ 101A — Solutions to Final Exam Th 15 December. Please solve Problem 1, 2 and 3 in the f rst blue book and Problems 4 and 5 in the second Blue Book. Good luck! Problem 1. Shorter problems. (50 points) Solve the following shorter problems. 1. Compute the pure-strategy and mixed strategy equilibria of the following coordination game. Call u the probability that player 1 plays Up, 1 u the probability that player 1 plays Down, l the probability that Player 2 plays Left, and 1 l the probability that Player 2 plays Right. (20 points) 1 \ 2 Left Right Up 3 , 21 , 1 Down 1 , 12 , 3 2. For each of these cost functions, plot the marginal cost function and the supply function, and write out the supply function S ( p ) , with quantity as a function of price p (30 points): (a) C ( q )=2 q (8 points) (b) C ( q q 2 q +2 (12 points) (c) C ( q )= q 3 +10 q (10 points) Solution to Problem 1. 1. The pure strategy Nash equilibria can be found in the matrix once we underline the best responses for each player: 1 \ 2 Left Right Up 3 , 2 1 , 1 Down 1 , , 3 The equilibria therefore are ( s 1 ,s 2 )=( U, L ) and ( s 1 2 D,R ) .T o f nd the mixed strategy equilibria, we compute for each player the expected utility as a function of what the other player does. We start with player 1. Player 1 prefers Up to Down if lu 1 ( U, L )+(1 l ) u 1 ( U, R ) lu 1 ( D,L l ) u 1 ( ) or 3 l +(1 l ) l +2(1 l ) or l 1 / 3 . Therefore, the Best Response correspondence for player 1 is BR 1 ( l u =1 if l> 1 / 3; any u [0 , 1] if l / 3; u =0 if l< 1 / 3 . 1

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We then compute the Best Response correspondence for player 2. Player 2 prefers Left to Right if u u 2 ( U, L )+(1 u ) u 2 ( D,L ) u u 2 ( U, R u ) u 2 ( D,R ) or 2 u +(1 u ) u +3(1 u ) or u 2 / 3 . Therefore, the Best Response correspondence for player 2 is BR 2 ( u )= l =1 if u> 2 / 3; any l [0 , 1] if u =2 / 3; l =0 if u< 2 / 3 . Plotting the two Best Response correspondences, we see that the three points that are on the Best Re- sponse correspondences of both players are ( σ 1 2 )=( u ,l =1) , ( u =0) , and ( u / 3 / 3) . The f rst two are the pure-strategy equilibria we had identi f ed before, the other one is the additional equilibrium in mixed strategies. 2. We proceed case-by-case: (a) C 0 ( q C ( q ) /q . The marginal cost function is always (weakly) above the average cost function). Supply function: S ( p q + if p> 2 any q [0 , ) if p q if p< 2 (b) C 0 ( q )=4 q 1 ,C ( q ) /q q 1+2 /q. Marginal cost is higher than average cost whenever 4 q 1 2 q /q, or 2 q 2 2 0 , or q 2 1 , or q 1 . (we do not care about solutions with negative q )P lugin q in the marginal cost curve to f nd the lowest price level such that the marginal cost function lies above the average cost function: p =4 (1) 1 , or p =3 . We invert the marginal cost function C 0 ( q q 1= p to get q = p/ 4+1 / 4 . The supply function therefore is S ( p ½ q = p/ / 4 if p 3 q if 3 3. C 0 ( q )=3 q 2 +10 ( q ) /q = q 2 . Marginal cost is higher than average cost whenever 3 q 2 q 2 , or 2 q 2 0 , which is always true. We invert the marginal cost function C 0 ( q q 2 +10= p to get q = q ( p
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This note was uploaded on 09/05/2009 for the course ECON 101a taught by Professor Staff during the Spring '08 term at Berkeley.

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101A_finalsp_sol - Econ 101A Solutions to Final Exam Th 15...

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