3. Suppose a medical company (i.e., the agent) supplies vaccine to = a benevolent planner (i.e., the principal). The company makes a profit t0q when q 0 is the quantity produced and t 0 is the transfer made. Suppose with probability v = = the company is efficient with cost parameter 0 1 and with probability it is inefficient with cost parameter = 2. Suppose that the planner offers a menu of contracts after the company observes its type. Suppose the company has to report its type 8 {1,2} to be eligible for the contract intended for the reported type. Suppose also that the planner can use an audit technology at a cost c(p) = p to detect with probability p = [0, 1] the true cost parameter 0 = {1,2} whenever the company misreports it, 0 0. The planner's surplus is given by 2ln(q) - t - p when the probability of detection is p [0, 1]. Suppose that, as part of its audit mechanism, the planner can charge the company up to 1 amount of payoff as a punishment P 0 whenever the company is caught misreporting its parameter. (a) Write down the planner's optimization problem by clearly stating the objective function and the constraints. (b) Simplify this problem by noting binding and non-binding con- straints, and making necessary substitutions. (c) Find the efficient type company's intended contract (t, q, p, P) as part of the optimal audit mechanism. (d) Find the inefficient type company's intended contract (t, q, p, P) as part of the optimal audit mechanism. (e) Find the planner's expected payoff and each type of company's information rent.