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Question: Two firms, one and two, produce a homogeneous good at zero marginal cost. Demand in each period is D = 10A - 5p. Firms meet in the market repeatedly and expect to do so into the infinite future. They maximize present discounted profits. The discount factor is δ < 1. Study an equilibrium in which the firms use a grim-trigger strategy.
a) Suppose first that firms compete in prices. Derive the critical discount factor where collusion is an equilibrium.
b) How does the critical discount factor varies if deviations are only detected with probability Derive the new critical discount factor where collusion is an equilibrium.
c) Suppose then that firms compete in quantities. The probability of detection of a deviation is back to one. Derive the critical discount factor where collusion is an equilibrium. Is collusion more or less likely than in a)?
d) There are now 71 firms in the market that compete in quantities. Generalise the critical discount factor to be a function of 71 firms. Known that the critical discount factor in case of price competition ie δcv = 1 - 1/n. Plot the critical values against the number of firms, n, and determine whether, for all n, the critical value of one mode of competition exceeds the other.
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