Assume that you own an exhaustible resource that is sold competitively

Assume that you own an exhaustible resource that is sold competitively. The price of the resource is:

Pt + 1 – C = 1.08(Pt – C),
where t = 0 at the beginning of 2005, P = price in dollars per ton, and C = marginal cost of extraction (fixed over time). It is also known that the demand for the resource is:
Q = 1,000,000 – 25,000 P,
where Q represents output in tons per year. If the beginning of 2005 price is $30 per ton and the marginal cost of extraction is $10 per ton, what will the price be at the end of 2009? What is the user cost of production in 2009? Is it different from the user cost for 2005? Explain. How much of the resource will be extracted in 2009? What is the market rate of interest on money? Explain.

ANSWER

The price at the end of 2009 will be determined from equation (1).

time(t) Net Price
beginning 0 30 – 10 = 20
end of 2005 1 P1 – 10 = 21.600
end of 2006 2 P2 – 10 = 23.328
end of 2007 3 P3 – 10 = 25.194
end of 2008 4 P4 – 10 = 27.210
end of 2009 5 P5- 10 = 29.390

Thus, the end of 2009 price is P. = 23.39 + 10 = $39.39/ton. The user cost is the difference between the selling price of 39.39 and the marginal cost of extraction of 10.000 or 29.39/ton. This user price is higher in 2009 than in 2005 reflecting the fact that more of the resource has been extracted by 2009 than by 2005, and the value of each remaining unit has risen.

At the price of $39.39 per ton, the quantity extracted in 2009 is:
Q = 1,000,000 – 25,000(39.39 ) = 15,250 tons/year
The market rate of interest on money is the same rate as the rate at which Pt – C increases each year. In this problem, 1 + R = 1.08; therefore, R = 0.08 or 8 percent per year.