Question

Consider a monopoly facing inverse demand function ?(?) = 12 − ?, where ? = ?1 + ?2 denotes the monopolist’s production across two plants, 1 and 2. Assume that total cost in plant 1 is given by ??1 (?1 ) = (5 + 4?1)?1, while that of plant 2 is ??2 (?2 ) = [5 + (4 + ?)?2]?2, where parameter ? ≥ 0 represents plant 2’s inefficiency to plant 1. When ? = 0, the total (and marginal) cost of both plants coincide; but when ? > 0, plant 2 has a higher total and marginal cost than plant 1.

a) Write down the monopolist’s joint profit maximization problem ? = ?1 + ?2.

b) Find the optimal production in each plant.

c) How does total optimal production change in the inefficiency of plant 2, ?? Answer this question by finding ?? ??. What is the optimal production in each plant if ? = 0?

Answer 1