10.4 Exhaustible resources and durable assets: Hotelling and Gibson

Motivation

Two old ideas explain two different things a gold or oil trader is told. Hotelling (1931) says the price of an exhaustible resource should rise at the rate of interest. Gibson's paradox, explained by Barsky and Summers (1988), says the real price of gold should fall when real interest rates rise. The first mostly fails in data. The second is the theoretical root of the "real yields drive gold" argument of 10.6.

10.4.1The Hotelling rule#

Setup

A competitive owner of a deposit of size $R_0$ can extract at constant marginal cost $c$ . The price $p_t$ is taken as given; the interest rate is $r$ . Extraction is reversible in timing but not in quantity: a barrel sold today is not available tomorrow.

Proposition 10.7Hotelling rule

In a competitive equilibrium with extraction in every period, the scarcity rent $p_t - c$ grows at the rate of interest:

p_t - c = (p_0 - c)\, e^{rt}. \tag{10.13}

With zero extraction cost, $p_t = p_0 e^{rt}$ .

Proof 1arbitrage

Compare extracting a marginal unit at $t$ and investing the rent until $t + dt$ , worth $(p_t - c)e^{r\,dt}$ , with leaving it in the ground and extracting at $t + dt$ , worth $p_{t+dt} - c$ . If the first were larger, every owner would extract now, depressing $p_t$ and raising $p_{t+dt}$ ; if smaller, no one would extract now. With extraction in both periods the owner must be indifferent: $p_{t+dt} - c = (p_t - c)e^{r\,dt}$ . Letting $dt \to 0$ , $\frac{d}{dt}(p_t - c) = r(p_t - c)$ , whose solution is (10.13).

Proof 2optimal control

A planner (or the market, by the first welfare theorem) chooses the extraction path $q_t \ge 0$ to maximise $\int_0^\infty e^{-rt}[B(q_t) - c q_t]\,dt$ subject to $\dot R_t = -q_t$ , $R_t \ge 0$ , where $B$ is gross consumer benefit with $B'(q) = p(q)$ , the inverse demand. Form the present-value Hamiltonian $H = e^{-rt}[B(q) - cq] - \lambda_t q$ . Pontryagin: $\partial H/\partial q = 0 \Rightarrow e^{-rt}[p(q_t) - c] = \lambda_t$ ; costate equation $\dot\lambda = -\partial H/\partial R = 0$ , so $\lambda$ is constant. Hence $p_t - c = \lambda e^{rt}$ : (10.13) with $\lambda = p_0 - c$ . The terminal condition (the resource is exhausted exactly when $p$ reaches the choke price $\bar p$ at which demand is zero) pins down $\lambda$ through $\int_0^{T^*} q_t\,dt = R_0$ .∎

Worked example

Inverse demand $p = 100 - q$ , $c = 0$ , $r = 5\%$ , $R_0 = 1000$ . Then $p_t = p_0 e^{0.05t}$ until it hits 100 at $T^*$ , so $p_0 = 100e^{-0.05T^*}$ , and cumulative extraction $\int_0^{T^*}(100 - 100e^{0.05(t-T^*)})dt = 100T^* - 2000(1 - e^{-0.05T^*}) = 1000$ . Solving numerically, $T^* = 23.97$ years, $p_0 = 30.17$ , initial extraction $69.8$ a year. (Problem 7 asks you to derive this.)

Where it fails Real prices of oil and metals have not risen at anything like the real interest rate over the last century (Krautkraemer 1998 surveys the evidence). Reasons: reserves are not fixed but grow with exploration and price; extraction costs fall with technology (US shale since about 2010 is the obvious case); market power (OPEC+) sets prices above the competitive path; and backstop technologies (renewables, EVs) cap the choke price. For a trader the useful residue of Hotelling is the arbitrage logic: a producer with spare capacity and reserves compares selling now with selling later, so expected price growth, interest rates and inventory decisions are linked. OPEC+ spare capacity plays that role in the crude market.

Gold is not a Hotelling resource Gold is not consumed. Almost all gold ever mined still exists, so the relevant stock is above ground and the annual flow is small relative to it. Its price behaves like the price of a durable asset, which is the next model.

10.4.2Gibson's paradox and gold as a durable asset#

Gibson's paradox Under the classical gold standard (roughly 1821–1913 in Britain), the price level and long-term interest rates were positively correlated. Keynes named this after A. H. Gibson and called it one of the most completely established empirical facts in economics. Standard monetary theory had trouble with it: a higher price level should mean a higher inflation rate, not a higher interest level.

Barsky–Summers explanation (1988) Under a gold standard the nominal price of gold is fixed by law. So the general price level $P$ is the reciprocal of the real price of gold, $p_g = P_{\text{gold}}/P$ : if $p_g$ falls, $P$ rises. Gold is a durable asset. Its real price is the present value of the services it provides (monetary, ornamental, industrial). A higher real interest rate lowers that present value, so lowers $p_g$ , so raises $P$ . Hence rates and the price level co-move.

Proposition 10.8real price of a durable asset

Suppose a unit of gold yields a constant real service flow $\delta$ per year forever, and the real discount rate is $\rho > 0$ . Then $p_g = \delta/\rho$ , and the semi-elasticity of the real gold price to the real rate is

\frac{\partial \ln p_g}{\partial \rho} = -\frac{1}{\rho}. \tag{10.14}

If the service flow grows at $g < \rho$ , $p_g = \delta/(\rho - g)$ and the semi-elasticity is $-1/(\rho - g)$ .

Proof

$p_g = \int_0^\infty \delta e^{(g-\rho)s}\,ds = \delta/(\rho - g)$ ; differentiate $\ln p_g = \ln\delta - \ln(\rho - g)$ .∎

This is a duration: gold behaves like a perpetual real bond with duration $1/(\rho - g)$ . The model is crude (the "service flow" is not observable, and at low $\rho$ the duration is enormous), but it gives the sign and the opportunity-cost logic of 10.6: when the real yield on a 10-year TIPS rises, the cost of holding a non-yielding asset rises. Problem 8 shows how sensitive the level is to $\rho - g$ , which is one reason empirical gold–real-yield betas are unstable.