4) Suppose there are two firms A and B that sell an identical product (consumers view them as perfect substitutes). Total demand for the product is QD = 100 - P, where Q is measured in 1000s. The firms have the same costs: CA = 500 + 20QA + QA 2 (CB has the same form with QB). Both firms know all of this information.

a) Calculate the marginal revenue of firm A, and set MRA=MCA to find firm A's profit maximizing output. Remember, QS = QA+QB. This will be a partial derivative: you're taking the derivative of total revenue with respect to QA, so treat QB as a constant. The profit maximizing level of QA will be a function of QB - this is A's "best response function."

b) Repeat these calculations for firm B. Use these two best response functions to find the Cournot equilibrium in this market: the quantity each firm will produce, the price of the good, and each firm's profit. Again, decimal values for Q are OK since Q is in 1000s. Now suppose that the two firms collude: they form a cartel and cooperatively set their price and quantity to maximize joint profits, splitting production and profit equally.

c) First, calculate the marginal revenue and marginal cost curves for the cartel together. MR is based on total market demand for the good, and is with respect to the total quantity produced by the two firms. Similarly, the cartel's costs are the sum of each firm's cost, assuming each firm produces half of the cartel's output: C = CA + CB, but QA = QB = Q*. MC will be different than in 2a.

d) Use these to calculate the total level of production, the amount produced by each firm, the price of the good, and the profits of the firm. Compare these results to the Cournot values from 4b.

e) Take the quantity firm B produces when the firms collude, and plug it into firm A's best response function from 4b. What value do you get? Explain the economics of this value: what does it imply about the stability of the cartel?