Monopoly profit maximization with linear demand: Bertrand vs. Cournot approaches Consider a monopolist that has a constant returns to scale production function and can produce any (continuous) amount of a good q = 0 at a constant marginal c = 0. Suppose the monopolist faces a linear demand function for its product, qd = a - b p, where qd is the quantity of the monopolist's good that customers demand when the price is p and a > 0 and b > 0 are positive constants. However we can also compute the inverse demand function as p = 1 (a - q) and interpret p as the per unit price the monopolist could b receive if the monopolist produced an amount q and put the entire amount q up for auction in the market under the requirement that all consumers pay the same per unit price p (i.e. no price discrimination).
a. Compute the monopolist's optimal price and quantity under the assumption that the monopolist is a price setter, i.e. the monopolist chooses the price that maximizes its profits. We refer to this as the Bertrand model of the monopolist's behavior.
b. Compute the monopolist's optimal price and quantity under the assumption that the monopolist is a quantity setter, i.e. the monopolist chooses the quantity that maximizes its profits. We refer to this as the Cournot model of the monopolist's behavior.
c. Show that the Bertrand and Cournot solutions are the same in this case.
d. Now suppose that instead of a linear demand function the monopolist faces a general demand function q = D(p) where D(p) is a differentiable function of p satisfying D'(p) < 0. Will the Cournot and Bertrand solutions be the same in the case of a general demand function? (If you say yes, then provide a proof that they result in a same profits, price and quantity produced, otherwise if you say no, then provide an example of a demand function q = D(p) where the Bertrand and Cournot solutions are different.)
e. How do your answers to parts a to d above change if the monopolist faces a fixed capacity con- straint K? That is, the monopolist can produce at constant returns to scale with marginal cost of c for any q = K, but at least in the short run (i.e. assuming that we are considering a period of time too short for the monopolist to have time to invest and increase its production capacity K) the monopolist cannot produce any more than K.