business-Problem Set 5 Fall 2014

| August 14, 2017

1. in a market where inverse demand function comes from the equation: p = 6 2q, two rms are playing the standard version of the Bertrand game for 3 periods. They have the same marginal cost which is equal to 2. Discount factor for the rst and second rm is .8 and .6, respectively.a. What would be the price in the market if one of them was the monopolist in the market.b. Using backward induction, nd the unique SPNE. of this game. Remember an equilibrium of this games includes one action for each player at each period. So it would look like: (p1 , p1 , p1 ), (p2 , p2 , p2 ) , where superscripts represents the rm and 1 2 31 2 3subscripts represents the period.c. Find each rms present value of the prot they make in this equilibrium.Now suppose they play this game for innitely many times.d. (Tacit Collusion): If pm is the monopoly price you found in part a, can you nd a strategy prole under which p1 = p2 = pm is a SPNE of this game. If yes give that strategy and show that it is a SPNE. Find the present value of the prot for both rms in this equilibrium.Hint 1: Consider grim trigger strategies.Hint 2: 1 = + 2 + 3 … = 1 for both of discount factors.2. Demand function in a market is q = 8 p.a. What would be the price if the market structure was monopoly with the marginal cost equal to 4?b. What would be the price if market structure was competitive and full of identical rms with marginal costs equal to 4?Now consider the ex-ante production model in order to explain the eect of capacity constraint on the outcome of a duopoly. There are two rms which have ex-ante marginal cost of production c0 = 4.c. What is the highest level of production (investment) that each rm would have in this market? In particular, assume that one of the rms is the monopolist in the market, so she maximizes:max p(8 p)pFind the optimal level of prot ( ) and then nd the highest level of production,q , beyond which prot (net of investment cost) is negative. That is nd q such that for any q > q :c0 q < 0d. Argue that for q1 , q2 [0, 8/3] dumping the whole capacity is always a N.E. Note that if they dump their capacities the price in the market will be equal to p = 8(q1 +q2 ).e. Among the set of equilibria you found in part d does any of them coincide with the pricing behavior in competitive equilibrium (part b)? How about pricing in monopoly (part a)?3. Demand function in a market is p = 6 2q. There are ve rms in the market which are playing the Cornuot game. Each rms marginal cost is a constant and equal to 2. Find the N.E. for this games.Hint: Use the strategy we used in class to nd the equilibrium of the game with n rms.

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