Question: Cournot Theory of Duopoly. Duopoly refers to a market structure in which two firms compete to serve the industry demand. Since price varies with total production, it is clear that each firm must account for the actions of the other when determining its own best production level. Various dynamic processes result, depending on the particular strategies employed by the two firms.

Suppose the total industry demand in any period is

d(k) = 200 - 2p(k)

where p(k) is current price. Let q1(k), q2(k) denote the output levels of the two firms. Assuming that the price adjusts so as to sell all current output, it follows that

p(k) = 100 - 1/2[q₁(k) + q₂(k)]

Suppose further that the total cost of production for the two firms are

C₁(k) = 5q₁(k)

C₂(k) = 1/2q₂(k)²

Both firms know the demand curve, but each knows only its own cost curve. The current profits for the two firms are revenue minus cost; that is, p(k)q1(k) - C1(k) and p(k)q2(k)- C­2(k), respectively.

The Cournot theory is based on the assumption that each firm selects its output level to maximize its own profit, using some estimate of its competitor's output. The corresponding function, for each firm, expressing the best output level as a function of the estimate of the competitor's output, is called a reaction function.

(a) Assuming each firm estimates Its competitor's output by using its competitor's actual output of the previous period, derive two reaction functions In the form of two first-order linear difference equations. (b) Find the equilibrium outputs of each firm.

(c) Derive a general solution for even periods and verify your answer to part (b).

(d) Suppose that both firms estimate each others' output as a simple average of the previous two periods in an effort to smooth out the oscillatory response. Show that for arbitrary initial conditions, the convergence to equilibrium need not be more rapid. (Hint: You need not factor, but you must analyze, the new characteristic polynomial.)