Abstract
Exhaustible resources are non-producible and in that respect different from renewable resources, considered in the next chapter. Optimal depletion over many periods is about the tradeoff between the net benefits of extraction today and the net benefits of extraction in the future. The key insight here is that extracting a unit of the resource today carries an opportunity cost beyond the cost of the inputs used in extracting, the value that might have been obtained by extracting at some future date. The question is: What is the time pattern of extraction that maximizes the net present value of the resource in the ground? This is a constrained optimization problem, solved using elementary calculus techniques to obtain the result that the opportunity cost, the difference between price and marginal extraction cost, also known as the royalty, must grow at a rate equal to the rate of interest (here understood as the theoretically correct rate rather than an empirical rate which might be influenced by political manipulations or economic imperfections). Equivalently, the present discounted value of the royalty must be the same across all periods. If this were not so, some gain could be had by shifting a unit of extraction from a lower value period to a higher, so the initial configuration would have been neither efficient nor an equilibrium. Further results are developed in a variety of models ranging from a simple two-period model to extraction over many periods, including how to find the optimal exhaustion date for a mine, the effect of shocks and market imperfections, and finally to the case of continuous extraction. Simple algebraic and geometric analyses and many worked examples are used, along with graphical presentations. A major focus throughout is the relation between the socially efficient rate of extraction and the market-determined rate. For the continuous case, the mathematics of dynamic optimization, in particular optimal control, is introduced and applied to the resource problem, yielding additional insight into the solution. This will prove useful also in the next chapter on renewable resources such as forests, characterized by continuous growth.
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Notes
- 1.
Our reference year does not need to be the present day—it could also be next year or 10 years from now; what is important is that all costs and benefits be valued relative to the same reference year.
- 2.
Remember the formula for a geometric series for any \(y < 1\) summed infinitely:
$$\begin{aligned} 1 + y + y^2 + y^3 + \cdots = \frac{1}{1-y} \end{aligned}$$$$\begin{aligned} PV = x \left( 1 + \left( \frac{1}{1+r}\right) + \left( \frac{1}{1+r}\right) ^2 + \left( \frac{1}{1+r}\right) ^3 + \cdots \right) = x \left( \frac{1}{1-\left( \frac{1}{1+r}\right) } \right) = x \left( \frac{1}{\frac{1+r-1}{1+r}} \right) = x \left( \frac{1+r}{r}\right) \end{aligned}$$.
- 3.
That is, if the investment stream is the above minus x:
$$\begin{aligned} \frac{x}{1+r} + \frac{x}{(1+r)^2} + \frac{x}{(1+r)^3} + \cdots \end{aligned}$$.
- 4.
A note on the algebra:
$$\begin{aligned} \frac{\partial }{\partial q_0}\int _0^{nq_0}{p(q)dq} =\frac{\partial }{\partial q_0} \left[ P(q)\left| _0^{nq_0} \right. \right] \text { where } P(q) = \int {p(q)dq} \end{aligned}$$$$\begin{aligned} =\frac{\partial }{\partial q_0} [ P(nq_0) - P(0)] =\frac{\partial }{\partial q_0}P(nq_0) \text { because }P(0) \text { is constant } \end{aligned}$$$$\begin{aligned} =\frac{dP}{d(nq_0)} \cdot \frac{d(nq_0)}{dq_0} = p(nq_0) n \end{aligned}$$.
- 5.
I am replacing parentheses by subscripts, thus \(\lambda (t)\) becomes \(\lambda _t\), and so on.
- 6.$$\begin{aligned} \frac{\mathrm{d} (\lambda _tx_t)}{\mathrm{d} t}= & {} \lambda _t \dot{x} + x_t\dot{\lambda } \\ \int ^T_0\frac{\mathrm{d}(\lambda _tx_t)}{\mathrm{d} t}\mathrm{d} t= & {} \int ^T_0\lambda _t\dot{x}\mathrm{d}\, t + \int ^T_0\dot{\lambda }x_t\mathrm{d}\, t \\ -\int ^T_0\lambda _t\dot{x}\mathrm{d}\, t= & {} \int ^T_0\dot{\lambda }x_t\mathrm{d}\, t - \int ^T_0\frac{\mathrm{d} (\lambda _tx_t)}{\mathrm{d} t}\mathrm{d}\, t = \left. \int ^T_0\dot{\lambda }x_t\mathrm{d}\, t - \lambda _tx_t\right| ^T_0 \\= & {} \int ^T_0\dot{\lambda }x_t\mathrm{d}\, t - [\lambda _Tx_T - \lambda _ox_o]. \end{aligned}$$
- 7.
See Hanley et al. (1997), ch. 9 for discussion of imperfect competition of various kinds: monopoly, duopoly, cartel.
- 8.
Note: this is not unrealistic. World oil reserves have gone from 70 billion bbl in 1950 to 1000 billion bbl in 2000. Japan uses approximately 1/2 the energy per capita used in the U.S.
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Fisher, A.C. (2020). Optimal Depletion of Exhaustible Resources. In: Lecture Notes on Resource and Environmental Economics. The Economics of Non-Market Goods and Resources, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-48958-8_2
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DOI: https://doi.org/10.1007/978-3-030-48958-8_2
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