Hotelling Revisited: Oil Prices and Endogenous Technological Progress

Abstract

This article examines the Hotelling model of optimal nonrenewable resource extraction in light of empirical evidence that petroleum and minerals prices have been trendless despite resource scarcity. In particular, we examine how endogenous technology-induced shifts in the cost function would have evolved over time if they were to maintain a constant market price for nonrenewable resources. We calibrate our model using empirical data on world oil, and find that, depending on the estimate of the initial stock of reserve, oil reserves will likely be depleted some time between the years 2040 and 2075.

Appendix: Proofs

Proof of Proposition 1

When there are no stock effects (i.e., $\frac{\partial C}{\partial S}\left( \cdot ,\cdot ,\cdot \right)=0$), then condition [#2] yields the Hotelling rule that the shadow price rises at the rate of interest:

$$ \frac{\frac{dp(t)}{dt}}{p(t)}=\rho. $$

(21)

When combined with condition [#1], this means that the market price minus marginal costs must increase at the rate of interest:

$$ \frac{\frac{d}{dt}\left( P(t)-\frac{\partial C}{\partial E}\left( S(t),E(t),t\right)\right)}{P(t)-\frac{\partial C}{\partial E}\left( S(t),E(t),t\right)}=\rho, $$

(22)

which yields, after rearranging terms, the following equation for the growth rate of market price in the absence of stock effects:

$$ \frac{\frac{dP(t)}{dt}}{P(t)}=(1-\theta(t))\rho +\theta (t)\frac{\frac{d}{dt} \left(\frac{\partial C}{\partial E}\left( S(t),E(t),t\right)\right)}{\frac{\partial C}{\partial E}\left( S(t),E(t),t\right)} $$

(23)

where the weight θ(t) is defined as:

$$ \theta (t)\equiv \frac{\frac{\partial C}{\partial E}\left( S(t),E(t),t\right) }{P(t)}\text{.} $$

(24)

When marginal extraction costs are nonzero, θ(t) ≥ 0. Moreover, from [#1] and the non-negativity of the shadow price, θ(t) ≤ 1. Under A1, when there are no stock effects and costs are linear in extraction, Equation (23) reduces to:

$$ \frac{\frac{dP(t)}{dt}}{P(t)}=(1-\theta (t)) \rho +\theta (t)\frac{\frac{dh(t)}{dt}}{h(t)}. $$

(25)

In order for market price to be constant (i.e., $\frac{dP(t)}{dt}=0$), we need h(t) to rise at rate $\left(1-\frac{1}{\theta (t)}\right) \rho ,$ which is nonpositive since θ(t) ≤ 1.

Proof of Lemma 2

(i) $g(t)\equiv \frac{\frac{d}{dt}F(S(t))}{F(S(t))} =\frac{-F^{\prime}(S(t))E(t)}{F(S(t))}=\frac{\left\vert \frac{\partial C}{ \partial S}\left(\cdot \right) \right\vert}{\frac{\partial C}{\partial E} \left(\cdot \right)}.$ (ii) $g(t)\equiv \frac{\frac{d}{dt}F(S(t))}{ F(S(t))}=\sigma E(t).$

Proof of Proposition 3

Under A1, B1, and B2, [#1] can be written as $p(t)=\overline{P}-F(S(t))h(t),$ which implies $h(t)=\frac{\overline{P}-p(t)}{F(S(t))}.$ Taking the derivative with respect to time yields:

$$ \begin{aligned} \frac{dh(t)}{dt}&=\frac{-\frac{dp(t)}{dt}}{F(S(t))}-\frac{\overline{P}-p(t)} {F(S(t))}\frac{\frac{d}{dt}F(S(t))}{F(S(t))} \\ &=\frac{-\frac{dp(t)}{dt}}{F(S(t))}-g(t)\frac{\overline{P}-p(t)}{F(S(t))}. \end{aligned} $$

Thus,

$$ \begin{aligned} \frac{\frac{dh(t)}{dt}}{h(t)}&=\frac{\frac{dp(t)}{dt}}{p(t)-\overline{P}} -g(t)\\ &=\frac{g(t)\left( p(t)-\overline{P}\right) +\rho p(t)}{p(t)-\overline{P}} -g(t), \end{aligned} $$

where the second line comes from [#2]. Further simplification yields the desired result.

Proof of Corollary 4

The closed-form equation (19) for h(t) is the solution to the linear first-order differential equation (18). Condition [#1] and the non-negativity of p(t) implies $h(t)\leq \frac{\overline{P}}{\Uppsi}e^{\sigma S_{o}-gt}\ \forall t ,$ so $h(0)\leq \frac{\overline{P}}{\Uppsi}e^{\sigma S_{o}} .$ Non-negativity of costs implies $h(t)\geq 0 \ \forall t ,$ which then implies that $h(0)\geq \frac{\rho}{ \rho +g}\frac{\overline{P}}{\Uppsi}e^{\sigma S_{o}}\left(1-e^{-(\rho +g) \frac{S_{0}}{\overline{E}}}\right).$

Proof of Proposition 5

Similar to proof of Proposition 3.