Question

Suppose you have access to 50 units of oil. Your goal is to maximize the net present value associated with selling the 50 units of oil in two time periods (t=0, t=1). The inverse demand for oil in each period is given by p_t = 60- q_t, where p_t is the price in dollars and q_t is the quantity sold in period t. Since you can earn a 10% rate of return by investing the proceeds of oil in any time period, the discount rate is r= 10%.

a. How much oil should you sell in each period to maximize the net present value? Show that the hotelling rule applies to the prices across the two time periods.

Answer 1

sale_year0 = 60q - q^2

sale_year1 = 60(50-q) - (50-q)^2

NPV = 60q - q^2 + (60(50-q) - (50-q)^2)/1.1

= 60q - q^2 + (60(50-q) - (50^2+q^2-100q)/1.1

= (66q -1.1q^2 + 3000 -60q -50^2 - q^2 + 100q)/1.1

maximizing NPV, dNPV/dQ = 0

q = 26.67

quantity in period 0 = 26.67

quantity in period 1 = 23.33

Price in period 0 = 60-26.67 = 33.33

Price in period 1 = 60-23.33 = 46.67

Prices increase according to Hotelling rule i.e. an increase of 10%