Mrs. McBain is the proud owner of an underground oil deposit. Mrs. McBain has only two periods left to live, periods t = 0 and t = 1. (Think of each period as 30 years in length. Each person will live for 90 years and will pass through youth (0 to 30), middle age (30 to 60), and old age (60 to 90). period 0.) According to the customary law of her village, which preceded the adoption of the civil code, Mrs. McBain could not sell her oil deposit because it was attached to the surface land and the latter could not be resold. Mrs. Mcbain having no heir, she will have to return her deposit to the village after her death. (a) Let S0 = 500 be the total quantity of barrels of oil contained in the deposit. (There is no uncertainty in this problem.) A barrel of oil sells for the given unit price of $50 on the market, or p0 = p1 = 50. The total cost of extraction per p period is C(Rt) = Rt2/20, where Rt represents the extracted quantity of barrels in t. The net profit per period is therefore t = ptRt C(Rt). Assuming a constant discount rate r between periods, write Mrs. McBain's present value maximization problem V0 and determine the optimal extraction levels R0 and R1 when r = 10%. (b) We define marginal rent by Pt = pt C(Rt). Calculate the value of the marginal rent at each period. At what rate does this rent increase between the two periods? By how much would V0 increase if the initial size of the deposit increased by an additional barrel, i.e. from 500 to 501? (NB No new calculation is necessary to answer this last one.) (c) Solve the same problem as in (a) but now assuming S0 = 1200. Interpret your results using a graph similar to the one used in class where the length of the abscissa corresponds to S0. (d) Suppose that after the death of Mrs. McBain at the end of period 1, life on earth continues, say for only one more period t = 2. For interg In the period, the government requires Mrs. McBain to leave 200 barrels of oil in the ground for the period t = 2, where p2 = 50. We always assume S0 = 1200. How will this additional constraint affect the choices R0 and R1 of Mrs. McBain? Justify. (e) The government is considering replacing customary law with the civil code. This will allow Mrs. McBain to resell her deposit as she sees fit. We seek to predict whether this will affect his extraction choices R0 and R1 and if so, how. We assume S0 = 1200 and p2 = 50. Explain how this simple institutional change, even if it has no effect on technology and prices, can greatly increase the opportunity costs of extracting in t = 0 and t = 1. (Hint: Suppose Mrs. Mcbain wants to sell her asset at the beginning of period 1 at the highest possible price.)