### Demonstrate collusion at monopoly price is not sustainable

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Suppose that there are three firms, who produce homogeneous products, and whom have the same marginal cost which is constant over output. These firms play an infinitely repeated Bertrand pricing game. Each period they simultaneously set prices. One period monopoly profits are given by ?M. Firms 1 and 2 have discount factors of 0.8, while Firm 3 has a discount factor of 0.6.

(a) Supposing that when the three firms set the same price they share the market evenly, demonstrate that collusion at the monopoly price is not sustainable, using the Bertrand Nash Equilibrium of marginal cost pricing forever as the punishment strategy. Explain and show your work.

(b)Suppose now that when all three firms set the same price, they do not share the market equally. Rather, when all three firms set the same price, Firms 1 and 2 each get a share of the market equal to ?<1/3; Firm 3's market share is 1 - 2 ?, which is bigger than 1/3. It is still the case that when a firm undercuts the others, it serves the entire market. Are there any values of ? for which collusion is now sustainable using the Bertrand Nash Equilibrium of marginal cost pricing forever as the punishment strategy?

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