The inverse demand curve for widgets is P = 130−2Q. There are two firms, A and B, who produce wid-gets. Each firm has a constant marginal and average cost of producing the good that equals 10. Firms compete in quantities and they make their quantity choices simultaneously (Cournot competition). Firms can choose any quantity to produce.
(a) Write down the profit function of each firm.
(b) Find the best-response function of each firm. Remember that to compute this, you need to differentiate firm i’s profit function with respect to qi, assuming that q−i is held constant, where q−i is the quantity produced by firm i’s rivals. Once you have the first-order condition, solve for qi as a function of q−i, marginal costs and demand parameters (i.e., the 130 and the 2 of the demand function).
(c) Plot the best response functions of both firms. In your graph, put qA in the horizontal axis and qB on the vertical axis.
(d) What is the Cournot-Nash equilibrium? You need to solve the system of equation characterized by the two first-order conditions. Show the equilibrium graphically.
(e) Suppose there are three firms (instead of two). Each firm still has constant marginal and average cost equals 10 and they compete in quantities. What is the Cournot-Nash equilibrium in this case? To solve for the NE you need to re-do the first two questions of this problem, now with three firms.
(f) Suppose again that there are only two firms, but now firms have different costs. In particular, firm A has constant marginal and average cost equals 10, whereas firm B has constant marginal and average cost equals 34. What is the Cournot-Nash equilibrium in this case? Note that, as we did in class, when firms have different marginal costs, the only way to solve for the NE is to replace one best-response in the other.