2. Quality provision. A seller owns a single unit of an invisible good. It is worth $v(s)$ to a buyer, where $s \in \mathbb{R}_+$ is the quality of the good. The function $v(.)$ is strictly increasing and strictly concave with $v(0) = 0$. Quality is chosen by the seller, and the cost of providing a good of quality $s$ is given by a function $c(s)$, which is strictly increasing and strictly convex with $c(0) = c'(0) = 0$ and $c'(0) < v'(0)$. (Implicit in this description is the assumption that the buyer's preferences are quasilinear is some good other than the one being traded.)
(a) Let $s^*$ denote the Pareto optimal quality level (i.e. the level which maximizes total surplus, $v(s) - c(s)$). Write down the first-order condition which determines $s^*$ and confirm that $s^* > 0$. Illustrate your conclusion in a graph with quality on the horizontal axis and net surplus, $v - c$, on the vertical axis.
(b) Consider the following game. Simultaneously and independently the seller chooses a quality $s \ge 0$ and the buyer chooses an offer $p \ge 0$ (Note that a quality choice of $s = 0$ is both costless and worthless, so it is equivalent to not supplying the good.) The buyer's surplus is $v(s) - p$ and the seller's surplus is $p - c(s)$ from the strategy profile $(p, s)$. Identify any Nash equilibria and comment on their Pareto optimality and whether any of them are also dominant strategy equilibria.
(c) Suppose now that the buyer and seller play the game from part (b) over an infinite horizon. They discount period surplus at rate $\delta \in (0, 1)$. After each period, the buyer observes the quality level chosen by the seller. Describe a Nash equilibrium that supports the seller choosing $s^*$ at each date, provided that the players are patient enough. Is it possible to do this with the buyer offering $p = c(s^*)$ at each date? What is the smallest discount factor that supports the seller choosing $s^*$ at each date? (This last question takes some thinking and calculating; the only terms which should appear in the smallest discount factor are $c(s^*)$ and $v(s^*)$.)
