(Adverse Selection) There is one seller who has a single object to seller. Her value of the object is normalized to 0. There is one buyer whose value of the object is denoted by v, which is his private information. We assume v can be high (v_h) with probability pi_h (0; 1), or low (v_l) with probabilitypi_l (0;1),where v_h > v_l > 0 and pi_h + pi_l = 1.

The contract variables are (p;t), where p denotes the probability the seller sells the object to the buyer, and t stands for the payment of the buyer. The payoff of the seller is t; the buyer's payoff is pv - t.

Suppose the seller designs the optimal contract to maximize her expected payoff. The outside opportunity utility level U₀ is normalized to 0.

(1) (1 mark) Suppose the seller knows v. Characterize the first-best contract, i.e., the solution to the seller's problem which maximizes her expected payoff under the participation constraint of the buyer.