There are two players, a seller and a buyer, and two dates. At date I, the seller chooses his investment level I ≥ 0 at cost I. At date 2, the seller may sell one unit of a good and the seller has cost c(I) of supplying it, where c'(0) = —∞, c' < 0,="" c"=""> 0, and c(0) is less than the buyer's valuation. There is no discounting, so the socially optimal level of investment, /*, is given by 1 + c'(/*) = 0.
(a) Suppose that at date 2 the buyer observes the investment I and makes a take-it-or-leave-it offer to the seller. What is this offer? What is the perfect equilibrium of the game?
(b) Can you think of a contractual way of avoiding the inefficient out-come of (a)? (Assume that contracts cannot be written on the level of I.)