review questions_ak.pdf - Review Questions 1 Yuya Takahashi...

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Review Questions 1 Yuya Takahashi 7 = 26 = 2018 1. Consider an industry where there is only one °rm. Demand is given by Q = 10 ° p: Assume the °rm±s cost function is c ( Q ) = 2 Q: Find a competitive equilibrium. Since we are °nding a competitive equilibrium, the °rm maximizes its pro°t taking the price as given. Thus, p = MC gives p = 2 : From the demand function, we know Q = 8 . 2. Consider a market of homogenous products with two active °rms (the number of °rms is °xed). Demand is given by Q ( p ) = 40 ° p: Two °rms are identical and the total cost function is given by TC ( q ) = 10 q: (a) Find a competitive equilibrium (i.e., °nd p ° and Q ° ). p = MC implies p ° = 10 : Using the demand function, we have Q ° = 30 : (b) Find a Cournot-Nash equilibrium (i.e., °nd p C ; q C 1 ; and q C 2 ). Firm 1 maximizes ° 1 = pq 1 ° 10 q 1 = (30 ° q 1 ° q 2 ) q 1 : The °rst order condition is 30 ° q 2 ° 2 q 1 = 0 : In the same way, we also obtain the best response function of °rm 2: 30 ° q 1 ° 2 q 2 = 0 : Solving this system gives q C 1 = 10 ; q C 2 = 10 ; and p C = 40 ° q C 1 ° q C 2 = 20 : (c) Find an equilibrium for a sequential game where °rm 1 moves °rst and °rm 2 moves second. (i.e., °nd p S ; q S 1 ; and q S 2 ). We solve this problem backward. Firm 2±s best response function for any given q 1 is q 2 ( q 1 ) = 30 ± q 1 2 : This is the same best response function for °rm 2 as in question (b). Then, °rm 1 maximizes ° 1 = pq 1 ° 10 q 1 = (30 ° q 1 ° q 2 ( q 1 )) q 1 = ° 30 ° q 1 2 ± q 1 : The °rst order condition implies q S 1 = 15 : Using q 2 ( q 1 ) ; we have q S 2 = 15 2 : Finally, p S = 35 2 : (d) Find a Bertrand-Nash equilibrium (i.e., °nd p B ; q B 1 ; and q B 2 ).
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