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. 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 ° ). (b) Find a Cournot-Nash equilibrium (i.e., °nd p C ; q C 1 ; and q C 2 ). (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 ). (d) Find a Bertrand-Nash equilibrium (i.e., °nd p B ; q B 1 ; and q B 2 ). (e) Compare four di/erent equilibria. Order the price levels and explain why you obtain that result. 3. Consider a market with demand given by Q ( p ) = 100 ° p: The total cost function is given by TC ( q ) = 50 + 2 q 2 + 5 q: Firms are identical. Let n denote the number of °rms. Note that P n i =1 q i = Q: (a) Derive the average cost function, AC ( q ) and marginal cost function, MC ( q ) . (b) Assume there are now 25 °rms. Consider the short run such that the number of °rms is °xed. Find a competitive equilibrium (i.e., °nd p ° ; q ° ; Q ° ). Are °rms making a positive, zero, or negative pro°t? (c) Find a competitive equilibrium in the long run (i.e., °nd p ° ; q ° ; Q ° ; n ° ). Hint: °rms enter and exit until each °rm earns zero pro°t. 4. Consider a market with inverse demand function p = 14 ° Q . Firms have constant marginal cost 2 and °xed cost 2. Firms compete by simultaneously choosing quantities. (a) Suppose there are n °rms in this market. Derive the Nash equilibrium prices, quantities and pro°ts.
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