Consider a monopolist which sells batteries. Each battery works for h hours, and so if a consumer buys q batteries she obtains H = qh hours of use. Each consumer has
utility u(H) - pq if she obtains H hours of use and buys q batteries each at price p. (Here, u(
) is increasing and concave.) The marginal cost of producing a battery which lasts h hours is c(h), where c(
) is increasing and convex.
(a) For a given h, derive the inverse demand function for batteries.
(b) Find the monopolist's most profitable choice of h and q.
(c) Calculate the socially optimal choice of h and q, and show that h is the same as in part (b).