Consider a monopolist who sells batteries. Each battery works for h hours and then needs to be replaced. Therefore, if a consumer buys q batteries, he gets H = qh hours of operation. Assume that the demand for batteries can be derived from the preferences of a representative consumer whose indirect utility function is v = u (H) - pq, where p is the price of a battery. Suppose that u is strictly increasing and strictly concave. The cost of producing batteries is C (q) = qc (h), where c is strictly increasing and strictly convex.

1. Derive the inverse demand function for batteries and denote it by P(q).

2. Suppose that the monopolist chooses q and h to maximize his profit. Write down the first-order conditions for profit maximization assuming that the problem has an interior solution, and explain the meaning of these conditions.

3. Write down the total surplus in the market for batteries (i.e., the sum of consumer surplus and profits) as a function of H and h. Derive the first-order conditions for the socially optimal q and h assuming that there is an interior solution. Explain in words the economic meaning of these conditions.

4. Compare the solution that the monopolists arrives at with the social optimum. Prove that the monopolist provides the socially optimal level of h. Give an intuition for this result.