Question

Consider a competitive oil market where a producer faces costs to get the oil out of the ground. Suppose that it costs $10 dollars per barrel to extract oil from the ground. Let ?? denote the price of oil in period ? and let r be the interest rate. a. If a firm extracts a barrel of oil in period ?, how much profit does it earn that period? b. If a firm extracts a barrel of oil in period ? + 1, how much profit does it earn in period ? + 1? c. What is the present value of the profits from extracting a barrel of oil in period ? + 1? What about period ?? d. If the firm is willing to supply oil in each of the two periods, what must be true about the relation between the present value of profits from sale of a barrel of oil in the two periods? Express this in an equation. Explain how this relates to the no-arbitrage condition.

Answer 1

Part (a)

Profit = (sale price - cost price) x 1 barrel = (P_t - 10)

Part (b)

Profit = (P_{t + 1} - 10)

Part (c)

PV of profit in the year t + 1 = (P_{t + 1} - 10) / (1 + r)^{t + 1}

PV of profit in year t = (P_t - 10) / (1 + r)^t

Part (d)

The two PVs should be equal.

(P_{t + 1} - 10) / (1 + r)^{t + 1} = (P_t - 10) / (1 + r)^t

Hence, (P_{t + 1} - 10) = (P_t - 10) x (1 + r)

If the two present values are not equal, it will lead to an arbitrage. If the above condition is not true, then

• If the present value of oil produced in year t > PV of oil in year t + 1; then all the firms will produce all the oil in year t
• If the present value of oil produced in year t < PV of oil in year t + 1; then all the firms will defer the production of all the oil to year t + 1

This in a way relates to the no arbitrage condition that if production has to be uniform across the two periods then, prices in the year t + 1 must increase or grow by a factor same as the interest rate. Thus expected future price = spot price x (1 + r)