Hotelling's rule

Not to be confused with Hotelling's law.

Hotelling's rule states that the most socially and economically profitable extraction path of a non-renewable resource is one along which the price of the resource, determined by the marginal net revenue from the sale of the resource, increases at the rate of interest. It describes the time path of natural resource extraction which maximizes the value of the resource stock.

The rule is derived from a 1931 paper by U.S. economist Harold Hotelling, which laid the foundations for further research in the field of non-renewable resource economics.

History

During the late nineteenth and early twentieth century, conservation in the United States was significantly visible. This conservation movement was concerned about the possible overexploitation of non-renewable natural resources and called for regulation. Harold Hotelling responded to this call with his paper. In fact, he began his paper with an introduction to this problem by stating in 1931 that:

“Contemplation of the world’s disappearing supplies of minerals, forests, and other exhaustible assets had led to demands for regulation of their exploitation. The feeling that these products are now too cheap for the good of future generations, that they are being selfishly exploited at too rapid a rate, and that in consequence of their excessive cheapness they are being produced and consumed wastefully has given rise to the conservation movement.”

Before the 1970s, serious attention was not given to Hotelling’s views regarding economics of exhaustible resources. This was because when Hotelling came up with this hypothesis, the media’s emphasis was towards issues like the Great Depression and World War II.

After World War II, there were concerns about the sufficiency and adequacy of natural resources, especially in the United States. This led to the establishment of the President’s Materials Policy Commission, called the Paley commission, the report of which was published in 1952. It was an outcome of the Resources of the Future, a part of the Washington think tank. However this did not help in any way in recognizing Hotelling’s work. In 1963, Harold Barnett and Chandler Morse published their paper "Scarcity and Growth", which was one of the first of its kind in analysing the long-run measures of scarcity of a number of natural resources in a systematic fashion. A passing reference to Hotelling’s paper was made in that paper. This leads to the conclusion that a lack of attention and interest to Hotelling’s work cannot be attributed to the lack of public concern for the issues relating natural resources before the 1970s as there is no sufficient evidence to prove this point.

Hotelling’s paper was difficult and given the level of sophistication in economic profession concerning mathematics at that time, it might have been a reason why his contribution went unnoticed. In fact Hotelling himself stated that the Economic Journal had rejected his paper due to high difficulty level of mathematics involved in it.

The publication by Dennis Meadows et al. The Limits to Growth in 1972 set off another period of intense public concern for natural resources (Robert Solow, 1974 AER). According to the publication, unless economic growth was cut back, there would be disastrous consequences on the level of natural resources. This made Hotelling rule, which consisted of optimal control theory, seek attention from the academic economists. In 1973 “The Allocation of Energy Resources" was published by William Nordhaus. It sought to solve the problem of usage of different energy resources by applying the Hotelling model. Nordhaus also published in the next year a critical analysis of The Limits to Growth in American Economic Review. The subject of natural resource economics got further contribution from Robert Solow, 1974 AER, who committed his 1974 Ely Lecture to the American Economic Association. In the same year, Partha Dasgupta, Geoffrey Heal, Tjalling Koopmans, Joseph Stiglitz, and Robert Solow contributed to the topic of exhaustible resources which was published by Review of Economic Studies. All these events provided greater weight and prominence to natural resource economics as a field of research. Hotelling’s rule moved to centre stage, and still continues to maintain this position.

The Rule

In Jeffrey A. Krautkraemer’s words, “Hotelling’s formal analysis of nonrenewable resource depletion generates some basic implications for how the finite availability of a nonrenewable resource affects the resource price and extraction paths.”

Hotelling’s rule primarily addresses one basic question of the owner or agent involved in the exploitation of the non-renewable resource: How much of the asset should I consume now and how much should I store for the future? In other words, the agent has to choose between the current value of the asset if extracted and sold and the future increased value of the asset if left unexploited. This simple rule can be expressed by the equilibrium situation representing the optimal solution

$\frac {P'(t)}{P(t)} = \delta,$

where $P(t)$ is the unit profit at time t and $\delta$ is the discount rate (the inverse of rate of return).

The stock of a non-renewable resource, being an asset, holds a market value which yields returns to its owner at a certain rate. This rate of return can be determined by three components:

1. Flow of product generated by the marginal unit of the resource, Marginal Productivity or Dividend rate.
2. Change in the physical characteristics of the asset over time.
3. The rate at which market value of the asset will change over time.

The equality of this rate of return to the rate of return of alternative investments (i.e., if the yield obtained from asset’s sale is invested elsewhere) determines the asset market’s equilibrium. Considering a nonrenewable resource, say a stock of oil in the ground, which is subject to two characteristics; one, it has a fixed size which cannot be increased over time and two, the in-situ asset is unproductive. This makes the first component, marginal productivity, nil. Assuming that holding the asset in-situ will not lead to its depreciation, even the second component is rendered zero. Remainder is the rate of appreciation of the asset’s value which is hence, the only determinant of rate of return of the stock of oil.

Framework

The value of a marginal unit of the natural resource held in-situ, less the cost of taking it out of the ground, is the value the resource would have in the flow market. If $p(t)$ is the price which the extracted resource can fetch on the market and $c(t)$ is the marginal cost of extracting it at time $t$, then its marginal value must be:

$\Pi(t) = p(t) -c(t),$

where $\Pi(t)$ represents the asset price of the resource at time t.

Denoting the rate of interest by r, then asset markets equilibrium is established when

$\frac{ \Pi'(t) }{ \Pi(t) } = r$.

This is the famous Hotelling’s rule which states the asset market equilibrium condition. It states that the net price of the natural resource must grow at the rate of interest.

Assuming that the marginal cost of extracting the resource does not depend on the rate of extraction and does not vary over time, then the market price of the asset over time would be

$\frac{ P'(t) }{ P(t) } = r \left( 1 - \frac{ c }{ p(t) } \right),$

If this were the case then the rate of growth of the observed price of non-renewable resource would be inclined towards the rate of interest as the proportion of costs in price decreases and the rent brought about by the scarcity of the natural resource increases.

But this is not in agreement with observations over more than a century. Statistics concerning U.S. price data for the period 1870–2004 for copper, lead, zinc, coal, and petroleum, 1880–2004 for tin, 1900–2004 for aluminium and nickel and 1920-2004 for natural gas, reveal that the rate of change of prices of these resources is influenced by a high degree of volatility. But the more important phenomenon is that the volatility seemed to be centered at 0. In fact the mean rate of change of price in none of the ten cases is considerably different from 0. Thus the actual price of the resources does not seem to be following any particular trend, and definitely not the path of the positive trend as suggested by Hotelling’s rule.

Drawback

While the rule predicts exponentially increasing resource prices, the results of empirical studies do not confirm the rule. The results so far have showed either declining or constant resource prices over time. In response to this failure, many economists and researchers have tried to bridge this gap by including more variables to see their influence on resource price. Alternatively, some authors have tried to derive Hotelling's rule within a dynamic general equilibrium model.