1. (Bertrand's Model of Oligopoly.) Consider three ready-mix concrete manufacturers, Joe, Louie and Rebecca, operating in a small town with the same marginal cost. The manufacturers produce homogeneous products and set their prices simultaneously.
Show that in any Nash equilibrium, all sales must occur at the marginal cost. Extend your argument to show that this statement will be true as long as two or more firms are competing in the market.
2. (Price Competition with Differentiated Products.) Suppose the demand for Coke and Pepsi in a small city are given by:
QCoke = 45 - 50PCoke + D(PPepsi - PCoke)
QPepsi = 45 - 50PPepsi + D(PCoke - PPepsi)
(a) Assuming D = 50 and MC = $0:30 per can for both firms, find the Nash equilibrium prices.
(b) How does a change in D affect competition?
(c) Assume that Coke's marginal cost is now $0:24 (Pepsi's is still $0:30) and D = 50. What are the equilibrium prices (to the nearest penny) and sales quantities when the firms set their prices simultaneously?
(d) Now suppose a single monopolist controls the market for Coke and Pepsi. If the monopolist sets the same price for Coke and Pepsi, what price would maximize its pro t? How does that price compare to the equilibrium prices in part (a)?

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Let us assume that the marginal cost of all the firms is MC=$30 for ready-mix concrete manufacturers, Joe, Louie and Rebecca. If any one of the three firms say Joe, is charging at a higher price than its marginal cost, then it must be that the other two firms of Louie and Rebecca should also be charging either at marginal cost at $30 (and split the market between themselves) or charging higher than MC and sell nothing. In this case it is not a nash equilibrium as either of the two firms can cheat and increase their profits by reducing the price just below the price of Joe. Hence there can be no nash equilibrium that would involve any of the firms selling a positive quantity at a higher price than its MC. Similarly, no firm would be willing to charge a price less than its MC as it would be losing its money....

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