Nonrenewable resource economics
Assume the exploitation of a nonrenewable stock resource to be separated into two periods, period 1 and period 2. Let the initial resource size be S. Assume further the quantity q(t) the be exploited each period, t = 1, 2. Then 


Assume a constant unit price p and the extraction cost, c, to be a function of extracted quantity, q. Net revenue of period t then is


and the marginal net revenue is


We assume a discount rate r. Then the present value of the net revenue of the two periods is

The present value is maximized when the discounted marginal net revenues equal each other:


which may be rewritten to


Maximization of present value is obtained when the relative change (percentage change) between the two periods equals the discount rate, r, or:

The immediate gain of extracting one unit more should equal the long term discounted loss of not being able to extract this unit in the future.

Since r > 0 the optimal solution is obtained  with an increasing nominal value of net revenue by time, as illustrated in the figure below.

The multi-period case

Consider the multi-period case. In principle there are no differences between the multi-period case and the two-period case. The results above is still valid when changing 1 and 2 by t and t + 1. We then have


and


The terminal condition is


when the average net revenue equals the marginal net revenue.

Non-renewable resource extraction

Hotelling's rule

Now let us assume the resource owner to have some degree of market power and that the demand is decreasing by increasing quantity extracted. The broad picture from the reasoning above still remains and we have


In 1931 Harold Hotelling showed how the optimal price was described by a curve defined by this equation. The simplest version of Hotelling's rule is expressed by


The price is a function of time.




Faustman's equation (simple version):

Delta i the interest rate an t the optimal length of the rotation period.


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