Consider the case of a car dealer. Demand for cars depends on the state of the economy. In particular, it is higher in the case of growth than in the case of a recession. The car dealer\'s consultant has given the car dealer the following (accurate) estimates of demand for each state of the economy as:

QG = 400 - 20PG and QR = 200 - 10PR where QG = quantity in growth, QR = quantity in a recession, and P, the price charged, is measured in thousands of dollars. The probability of growth and of a recession are: Pr (G) = .6 and Pr(R) = .4. Assume that the car dealer must incur $150,000 in fixed costs per year and obtains cars from the manufacturer at $10,200 each. Hence, the car dealers cost (in thousands of dollars) is C(Q) = 150 + (10.2)Q .

a. Naturally, the car dealer must order their cars to sell for the year before they know the direction of the economy. Find the car dealers expected inverse demand function and the profit maximizing quantity. What is the car dealer\'s profit in a growing economy and in a recession? What is the car dealers expected profit?

b. Suppose that the car manufacturer allows the car dealer to return all unsold cars at the end of a recessionary year. What is the car dealer's profit in a growth year and in a recession? What is their expected profit?

The flexible arrangement described above comes at a cost for the car dealer. In particular, the car manufacturer requires the car dealer to pay a franchise fee to the manufacturer in order to have this flexibility. Suppose that both the manufacturer and the car dealer are risk neutral. How much would the car dealer be willing to pay for this flexibility?