1. Suppose that a seller wishes to sell one unit of a good to a single buyer, whose outside option has utility zero. The cost of producing the unit is zero and the seller is risk-neutral. The buyer can be of two types, having a monetary valuation for the good of either L or H ; H > L > 0. The seller does not know the buyer's type. The agent's utility from consuming the good and paying the amount T to the principal depends upon type in the following way:

L T if = L, U(1,) =

log(1+H T) if=H. (Note that the agent's level of risk aversion is increasing in valuation.)

Show that in the optimal contract, the seller always sells the good, leaves no information rent to either type of buyer, and imposes no costs of risk in equilibrium, and does all this by using a random scheme. Explain the intuition behind the use of the random scheme.