Supplementary Exercises > Exhaustible and Recyclable Resources

Allocating multiple grades over 5 periods

Suppose an exhaustible resource is to be allocated across five periods. Demand for the resource in each period i is given by Pi = 150 – Qi. The data in the table below has been collected about the resource’s distribution. You may assume that all areas where the resource might be found have been explored and that there are no undiscovered deposits. You may also assume that the interest rate is zero.

Grade of oreAB C
D
E
MEC
10
20
30
40
50
Tons 300
200
200
100
100
Please answer the following questions:
  1. Find the efficient allocation of the resource across the five periods. What will the price, quantity, marginal extraction cost and royalty be in each period? Be sure to show all your work. What would the US Geological Survey (USGS) report as "proven reserves" at the beginning of the first period? Will the resource be completely exhausted by the end of the fifth period? Explain why or why not.
  2. Now suppose that just before extraction begins in period 1, news arrives that a new backstop technology has been invented. The marginal cost of the backstop is $20. Determine how this changes the allocation of the resource across the periods. Does this change what the USGS would report as proven reserves? Explain and discuss.
  3. Over the years much use has been made of "reserve ratios" or "years of supply" calculations to indicate whether we’re about to run out of various exhaustible resources. Is this a good measure? Explain why or why not.
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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 05/04/2006