Question: (Duopoly) Each of two firms A and B produces its own brand of a commodity such as mineral water in amounts denoted by x and y, which are sold at prices p and q per unit, respectively. Each firm determines its own price and produces exactly as much as is demanded. The demands for the two brands are given by
x = 29 - 5p + 4q, y = 16 + 4p - 6q
Firm A has total costs 5 + x, whereas firm B has total costs 3 + 2y. (Assume that the functions to be maximized have maxima, and at positive prices.)
(a) Initially, the two firms collude in order to maximize their combined profit, as one monopolist would. Find the prices (p, q), the production levels (x, y), and the profits of firms A and B.
(b) Then an anti-trust authority prohibits collusion, so each producer maximizes its own profit, taking the other's price as given. If q is fixed, how will A choose p? (Find p as a function p = pA(q) of q.) If p is fixed, how will B choose q? (Find q as a function q = qB(p) of p.)
(c) Under the assumptions in part (b), what constant equilibrium prices are possible? What are the production levels and profits in this case?
(d) Draw a diagram with p along the horizontal axis and q along the vertical axis, and draw the "reaction" curves pA(q) and qB(p). Show on the diagram how the two firms' prices change over time if A breaks the cooperation first by maximizing its profit, taking B's initial price as fixed, then B answers by maximizing its profit with A's price fixed, then A responds, and so on.