The Economics of Exhaustible Resources

Abstract

We consider in this chapter Hotelling’s classic model of the economics of exhaustible resources and its extensions. The social planner’s problem in this model is to maximize the discounted social benefit from consumption of a homogeneous resource under the constraint that cumulative consumption is no greater than the initial resource stock. The marginal resource rent for the optimal extraction path grows at the real rate of interest, and the equilibrium resource price meets this condition, known as “Hotelling’s rule”. This rule modifies under the model extensions to heterogeneous resources: the marginal rent falls if extraction switches from low-cost to high-cost resource stocks. We consider the models of resource pricing in the presence of a backstop technology, the Herfindahl principle that a lower-cost resource stock depletes before extraction switches to a higher-cost stock, and the principle of comparative advantage for a more general case of heterogeneous resources and consumers.

Appendices

Appendices

1.1 A.1 Equation (2.44)

Rewrite condition (2.42) as:

$$ {v}_{10}={v}_{20}+\left({c}_2-{c}_1\right){e}^{-r{T}_1}. $$

(2.A1)

We have it that ${v}_2\left({T}_1\right)=\left(\alpha -{c}_2\right){e}^{-r\left({T}_2-{T}_1\right)}$ due to (2.12) and because p₂(T₂) = α. Consequently, ${v}_{20}=\left(\alpha -{c}_2\right){e}^{-r{T}_2}$. The market-clearing price for 0 ≤ t ≤ T₁ is: p₁(t) = α − βq₁(t), hence, from (2.40a), (2.A1), (2.43):

$$ \beta {q}_1(t)=\alpha -{p}_1(t)=\alpha -{c}_1-{v}_{10}{e}^{rt}=\alpha -{c}_1-\left(\left(\alpha -{c}_2\right){e}^{-r{T}_2}+\left({c}_2-{c}_1\right){e}^{-r{T}_1}\right){e}^{rt}=\alpha -{c}_1-\left(\left(\alpha -{c}_2\right){e}^{-r\left({T}_2-{T}_1\right)}+{c}_2-{c}_1\right){e}^{r\left(t-{T}_1\right)}= $$

$$ \left(\alpha -{c}_1\right)\left(1-{e}^{r\left(t-{T}_1\right)}\right)-\left({c}_2-\alpha +\left(\alpha -{c}_2\right){e}^{-r\left({T}_2-{T}_1\right)}\right){e}^{r\left(t-{T}_1\right)}= $$

$$ \beta {q}_1^{\prime }(t)+\left(\alpha -{c}_2\right)\left(1-{e}^{-r\left({T}_2-{T}_1\right)}\right){e}^{r\left(t-{T}_1\right)}=\beta {q}_1^{\prime }(t)+\beta {q}_2\left({T}_1\right){e}^{r\left(t-{T}_1\right)}=\beta {q}_1^{\prime }(t)+\beta {q}_1^{\prime \prime }(t). $$

1.2 A.2 Inequalities (2.45)

Due to (2.15) and (2.44), the resource constraint for the low-cost site is written as:

$$ \frac{\alpha -{c}_1}{\beta r}\left(r{T}_1+1-{e}^{-r{T}_1}\right)+\underset{0}{\overset{T_1}{\int }}{q}_1^{\prime \prime }(t) dt={S}_{10}. $$

Insert here ${q}_1^{\prime \prime }(t)={q}_2\left({T}_1\right){e}^{r\left(t-{T}_1\right)}$and rearrange, using (2.43), the left-hand side of this equation:

$$ \frac{\alpha -{c}_1}{\beta r}\left(r{T}_1+1-{e}^{-r{T}_1}\right)+\frac{\alpha -{c}_2}{\beta}\left(1-{e}^{-r\left({T}_2-{T}_1\right)}\right)\underset{0}{\overset{T_1}{\int }}{e}^{r\left(t-{T}_1\right)} dt= $$

$$ \frac{\alpha -{c}_1}{\beta r}\left(r{T}_1+1-{e}^{-r{T}_1}\right)+\frac{\alpha -{c}_2}{\beta r}\left(1-{e}^{- rT\left({S}_{20}\right)}\right)\left(1-{e}^{-r{T}_1}\right)={S}_{10}. $$

From this equation, we have it that ∂T₁/∂S₁₀ > 0 and by the implicit function theorem:

$$ \frac{\partial {T}_1}{\partial {S}_{20}}=-\frac{\left(\alpha -{c}_2\right){e}^{- rT\left({S}_{20}\right)}{T}^{\prime}\left({S}_{20}\right)\left(1-{e}^{-r{T}_1}\right)}{\left(\alpha -{c}_1\right)\left(1+{e}^{-r{T}_1}\right)+\left(\alpha -{c}_2\right)\left(1-{e}^{- rT\left({S}_{20}\right)}\right){e}^{-r{T}_1}}<0, $$

since T^′(S₂₀) > 0.

1.3 A.3 Equation (2.51)

Equation (2.49) is fulfilled for the times of switching T_g1 and T_g2:

$$ {c}_{g1}+{v}_{g0}{e}^{r{T}_{g1}}={c}_{c1}+{v}_{c0}{e}^{r{T}_{g1}} $$

$$ {c}_{g2}+{v}_{g0}{e}^{r{T}_{g2}}={c}_{c2}+{v}_{c0}{e}^{r{T}_{g2}}. $$

This is rewritten as

$$ {c}_{c1}-{c}_{g1}=\left({v}_{g0}-{v}_{c0}\right){e}^{r{T}_{g1}} $$

$$ {c}_{c2}-{c}_{g2}=\left({v}_{g0}-{v}_{c0}\right){e}^{r{T}_{g2}} $$

and yields

$$ \frac{c_{c2}-{c}_{g2}}{c_{c1}-{c}_{g1}}={e}^{r\left({T}_{g2}-{T}_{g1}\right)}. $$