1. Let S0 = 500 be the total quantity of barrels of oil contained in the field. (There is no uncertainty in this problem.) A barrel of oil sells at the given unit price of $50 on the market, i.e., p0 = p1 = 50. The total cost of extraction per period is C(Rt) = Rt2/20, or ' Rt represents the quantity extracted from barrels in t. The net profit per period is, therefore, π t = ptRt − C(Rt).

Assuming a constant discount rate r between periods, write Ms. McBain's present value maximization problem V0and determine the optimal extraction levels R0∗ and R1∗ when r = 10%.

Marginal rent is defined by Pt = pt − C′(Rt). Calculate the value of the marginal pension in each period. At what rate does this pension increase between the two periods? How much would V0 increase if the initial size of the field increased by an additional barrel, from 500 to 501? (NB No recalculation is required to meet the latter.)