The inhabitants of Fantasia live for two periods, 0 and 1. They consume a nonrenewable resource called Fantasium in each period. Fantasium has to be extracted from the ground and the (constant) marginal cost of extraction is $4/unit. The total available supply of Fantasium in the ground is 22 units and you should assume that the entire amount will be consumed over the two periods. The market demand curve for Fantasium in period 0 is given by P0 = 10 - Q0 + M and in period 1 this demand curve is given by P1 = 10 - Q1 + M, where Pi is price in period i, Qi is quantity demanded in period i (i = 0, 1), and M is Fantasia's national income in each period. Assume that there are many identical firms extracting Fantasium in every period, each firm has perfect foresight about prices, each firm discounts profits at the rate of 50%, and M = 20.
(a) Use Hotelling's Rule to get an equation which relates the equilibrium price in period 0 (i.e., P0) to the equilibrium price in period 1 (i.e., P1).
(b) Use the market demand curve in each period and your answer to 3(a) to get an equation which relates the equilibrium quantities Q0 and Q1. Use this equation and the constraint on total available supply of Fantasium to solve for the equilibrium quantities of Q0 and Q1.
(c) Use your answers to 3(b) in the demand curve for each period to get equilibrium values of P0 and P1.
(d) What would be the equilibrium values of Q0 and Q1 if the rate of discount was 100% instead of 50% ?
(e) What would be the equilibrium values of Q0 and Q1 if the rate of discount was 50% but the marginal cost of extraction of Fantasium was $14/unit instead of $4/unit ?
(f) What would be the equilibrium values of Q0 and Q1 if the rate of discount was 50%, the marginal cost of extraction of Fantasium was $4/unit, but M was 40 instead of 20 ?