Problem: Consider a firm that has a domestic monopoly and that also exports goods for sale in a competitive world market. The firm chooses two quantities: y for the domestic market and z for the world market. Its total cost is C(Q) = F + Q2/4, where Q = y + z and F is a known constant.
Suppose that the price in the local market is given by p(y) = 80 - y, while the price in the world market is equal to 30. (Since the world market is competitive, this firm's choice of z will not affect the world price.)
Assume that the firm's profit-maximizing choice (y*, z*) has both y* > 0 and z* > 0.
Use the first-order conditions for profit-maximization to identify y* and z*.