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1. (25) In class we briefly discussed the Bertrand oligopoly model, and showed that the existence of only two firms competing on price can produce the same outcome as under perfect competition, where price equals marginal cost and the total surplus is maximized. In this question we will consider the other widely used oligopoly model, that of French mathematician Antoine Augustin Cournot. Unlike in the Bertrand model, where firms choose prices and then produce to meet demand, in the Cournot model firms instead choose the quantities of the good they will produce, and the entire industry output is then sold at the same market clearing price.
Suppose that in the market for gears there are two firms, Spacely Sprockets and Cogswell Cogs. Suppose that each firm has constant marginal cost c = 1. The market price depends only on the total output of the industry Q = qs + qc, and is given by the inverse demand function p = P(Q) = 3 Q.
(a) (3) Find the perfectly competitive equilibrium price p and quantity q.
(b) (3) Suppose that Spacely was the only firm in the industry. Find the monopoly price pm, quantity
qm, and profit πm.
(c) (4) Now consider the optimal strategies of the two firms playing a quantity game. Just like a perfectly competitive firm or a monopolist, a duopolist will also choose a quantity that sets marginal revenue equal to marginal cost. However, as this depends on the current market price, Spacely’s marginal revenue will depend not only on its own quantity qs but also the quantity of its rival qc, and likewise for Cogswell. Find Spacely’s marginal revenue curve rs(qs,qc).
(d) (4) Using your answer to the last question, find Spacely’s profit maximizing quantity as a function of Cogswell’s output qs(qc). This is also known as a best response function, since it prescribes for Spacely the optimal level of output in response to whatever it observes Cogswell doing.
(e) (3) Recalling that the firms are identical, write Cogswell’s best response function qc(qs).
(f) (4) In a Nash equilibrium, neither player has any incentive to deviate from their chosen strategy, or stated otherwise, both players are optimally responding to what they observe their opponent doing. Using the best response functions, find the equilibrium quantities for Spacely and Cogswell, qs and qc, as well as the duopoly price pd. How do the Cournot equilibrium price and total quantity compare with the equilibrium in the Bertrand model?
(g) (4) Find the profits to each firm πs and πc. How do they compare to the results from the Bertrand model?
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