Amit has von Neumann-Morgenstern utility u(x)=√ over levels of wealth r. He has 4 lollars initially, but faces a 50 percent chance of losing all 4 dollars. Barbara offers to sell nsurance to Amit. If Amit purchases q units of insurance at a price of p dollars per unit, hen Barbara will pay him q dollars if the loss occurs. Barbara is risk neutral and cares only ■bout maximizing her profit from the sale of insurance. (a) Find Amit's optimal demand for insurance for each price p > 0. Assume that Amit cannot sell insurance, so that he must choose q ≥ 0. Solution: The optimal demand is 4(1-P) if p≤ 1 and 0 otherwise. (b) Suppose that trade is determined according to the following sequential game. Barbara first chooses the price p>0 at which she offers insurance to Amit. Amit then chooses the quantity q> 0 of insurance to buy at the offered price. Find a subgame perfect equilibrium of this game. Solution: Using backward induction, the optimal strategy for Amit is to use the demand from part (a). Given this strategy, Barbara's expected profit is marimized by choosing p=1/√2. Hence the SPE is (p=1/√2.q(p) = max{4(1-p)/p,0}). (e) Now suppose instead that first Amit offers a price p>0 and quantity q 20 of insurance to buy. After hearing his offer, Barbara chooses whether to accept it or reject it. If she