Hotelling’s rule

  • By Alain Grandjean
  • Updated on 31 July 2020

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Harold Hotelling (1895 – 1973) is one of the best-known environmental economists of the 20th century, along with Arthur Pigou (the ecotax), Garrett Hardin (the tragedy of the commons) and Ronald Coase (one of the inspirers of the “rights to pollute” market). 1 “). His 1931 article 2 on commodity pricing was the basis for the eponymous rule, which is still in use today. 3 intellectual benchmark on this issue. After briefly explaining the meaning of this rule, we’ll look at its limitations and argue in favor of abandoning it altogether.

This article has benefited from the observations of Antoine Missemer, CNRS researcher at Cired, whom we would like to thank; however, he is in no way responsible for this version.

The “Hotelling rule” in a nutshell

Using a mathematical model, Hotelling determined the value of a stock of an exhaustible resource, the evolution of this value and the rate of extraction of the resource according to the economic regime in force (competition, duopoly, monopoly). From this, a rule has been deduced, according to which the unit price 4 of a non-renewable natural resource must grow at a rate equal to the return on other economic assets (to simplify, their interest rate). In Hotelling’s initial model, this return is assumed to be known and unique, which is obviously a simplification: we know that returns depend on assets and their level of risk.

This rule in fact implies that non-renewable natural resources will only be exhausted at the end of time, their demand decreasing to zero as prices rise. It also states that, from the operator’s point of view, there is an optimal rate of change in the price of a natural resource, and an optimal “path” to its depletion. This rule was later used to justify that it can be economically optimal to eventually deplete a natural resource, forgetting in the process that such “optimal” depletion for each individual user would lead to the collapse of the economy (not to mention human beings) at the macroeconomic level (i.e., by generalizing to all natural resources), and thus ultimately to their ruin.

Finally, it should be noted that empirical observations concerning oil show that the exploitation of a deposit follows a “bell curve”: production increases up to a ceiling (the “production peak”) and then decreases. The basic version of Hotelling’s model does not account for this phenomenon.

A reference model with questionable representations and assumptions

Hotelling’s model is “globally” neoclassical; it aims to optimize the intertemporalutility of an economic actor, assumed to be rational: the latter would optimize the discounted sum of 5 (This bold hypothesis alone invalidates the model’s ability to represent reality, but we won’t go into its generality here. Let’s just mention the case of the island of Nauru, which once had a very large phosphate deposit: its operators (the island’s inhabitants) have exhausted it and, after having been among the richest in the world, are now living in misery.

The model also includes specific assumptions, some of which are understandable in an initial model, others less so, but all of which necessarily lead us to see that it does not at all and by far represent the complexity of the world.

  • It represents only a mine, a market, a resource and producers who may or may not be in competition.

In general, there are several deposits for a given raw material or energy source. Most of these are publicly owned (though not everywhere: in the USA, for example, private owners of the soil also own the subsoil), and are exploited by an operator (possibly a consortium) to whom exploitation is granted in return for a rent-sharing agreement. What’s more, there are usually several markets (even in a case as exceptional as oil) and several substitutable resources.

  • It does not take into account the financialization of commodity markets (and the logic of financial players is not that of a mining operator), which creates price volatility that is observable and absent from the model’s conclusions.
  • Reserves 6 are assumed to be perfectly known, when in fact they are not, or are very poorly known.

In 2023, economists 7 introduced uncertainty into a variant of Hotelling’s model: “instead of postulating that the resource in question is spread out on the ground and all you have to do is bend down to pick it up, [they postulate] that it’s hidden in deposits that need to be discovered before it can be exploited.” Their conclusion is clear: “Mathematical analysis shows that there is no price signal. […] It’s only when the resource is fully extracted, and therefore stored, that we find ourselves in Hotelling’s model and the price increases exponentially.”

  • It greatly simplifies the issue of transport costs and their variability, which is sometimes a determining factor in the economics of the resource in question (as in the case of gas or coal).
  • It considers only one alternative for the operator: spend to produce in the mine or invest the money, whereas in practice an entrepreneur’s behavior is much more complex and he can consider many other alternatives.
  • The return on the alternative investment is assumed to be known and constant (at the average interest rate for assets), whereas the returns available are fluctuating and, what’s more, unknown ex ante. Furthermore, not all players have access to the same returns (due to their ability to access information and their financial power).
  • It describes a situation of equilibrium: a mining entrepreneur will make the same profits, in this configuration, whether he keeps his resources in the ground by speculating on rising prices, or sells them to generate income that can then be invested in the financial markets. This situation does not arise in reality.
  • Production costs are assumed constant 8 whereas the reality is more complex: for a given technology, it becomes more and more expensive to extract a resource as it becomes scarcer (yields are decreasing), but technical progress can change this, at least for a while. Note that this last assumption can be corrected to make it more consistent with empirical reality. 9 .

A complete lack of empirical validity

Let’s recall the rule derived from Hotelling’s work: the unit price of a non-renewable natural resource must grow at a rate equal to the return on other economic assets.

It may seem odd that a rule that is simply expressed mathematically and links measurable data is not abandoned when observation invalidates it.

Here are a few examples:

Price of a barrel of oil (in US dollars) 1861 – 2019

MISSING DATAVIZ: prix-du-baril-de-petrole-en-dollars-us-1861-2019

Source BP Statistical Review of World Energy 2020

From the beginning of the twentieth century until the oil crisis, the price per barrel remained at a constant $20. Since then, it has fluctuated between $20 and $120; it even went into negative territory during the COVID19 crisis. These variations are not linked to changes in the yield on financial assets.

A final example, from Frog Syndrome by Ivar Ekeland (Odile Jacob, 2015).

Colin Clark, for example, has shown that there was no guarantee that those who exploited the whale (a non-renewable resource beyond a certain hunting pressure), the whaling industrialists (Norwegian, Japanese), and who opposed all regulation, would preserve it (in their own interests). As with La Fontaine, they could have perfectly well killed the goose that laid the golden eggs, contrary to popular intuition, as it was more profitable to kill all the whales and invest the profits with interest, rather than manage the whale stock sustainably. At that time, money simply “worked” more efficiently than whales reproduced. Clearly, given the demand for profitability from private capital, conservation of the “whale resource” required continuous public monitoring and control of physical yield and stock status. Regulations followed. When it comes to the financial and non-whaling industries, competition is based directly on financial performance.

Ivar Ekeland

In this example, without public intervention, the non-renewable resource would have been exhausted well before the model’s prediction.

It should be noted that, over the last few years, attempts to refine the model have produced results that are closer to observations, but in specific situations, which makes the basic model less general.

A fundamental ambiguity common to neoclassical economists

Harold Hotelling’s “demonstration” is a mixture of attempted real-world description and normative recommendation. True, his model is logically coherent. But if it were relevant and its hypotheses proven, its results would be in line with observed data, and it would enable us to make predictions. 10 . We could then infer that the price of a natural resource follows the “rule”. But neither the model nor its assumptions are proven. Should we then fit the real world into the model’s ideal?

This may seem surprising, but it’s the conclusion neoclassical economists generally come to. It was Léon Walras, one of the founders of this school of thought, who first formulated this ideological bias: “Mr. P(areto) believes that the aim of science is to get closer and closer to reality by successive approximations. And I believe that the ultimate goal of science is to bring reality closer to a certain ideal; that’s why I formulate this ideal.” 11

This ambiguity is reflected in Kenneth Arrow’s and Gérard Debreu’s “demonstration” of the two theorems of welfare economics. 12 . This “demonstration” is based on an abstract mathematical formalization of reality, which leads neoclassical economists to make free competition an ideal to be attained, since in theory it is an efficient mechanism… The approach is the same for theCO2 quota market: through a modeling exercise, it is possible to show that it is an optimal system, so it must be implemented… 13 . This is also what is suggested to reduce unemployment, which is said to be due to imperfections and rigidities in the real world, to be eliminated in order to bring it into line with theory. The reasoning is always the same: if there are imperfections and rigidities in the real world, they must be eliminated.

Let’s return to Hotelling’s rule: it doesn’t tell us anything about reality as far as natural resource management is concerned. In the uses to which it has been put since the 1970s, however, it has established itself as unquestioned and normative – which was undoubtedly not Hotelling’s intention.

Questionable neutrality

As we have seen, Hotelling’s rule is based on the use of a discounting calculation to optimize an intertemporal trade-off (whether to exploit now or later). This calculation is sometimes presented as “self-evident”, i.e. as a neutral, even economically optimal, or at any rate “rational” practice. This is the practice of financial markets and “big” companies. On the contrary, this “practice” implicitly refers to political and ethical issues surrounding the intertemporal allocation of resources. Hotelling was well aware of this, as shown in an article by Marco P.V. Franco et al. 14

We can therefore deduce that, regardless of the empirical and technical issues involved, Hotelling’s rule cannot be presented as “natural” or legitimate because it is neutral; in fact, it evacuates essential questions that we find in the question of choosing the discount rate.

A rule to forgo

Harold Hotelling’s work has been the subject of numerous refinements to bring it more closely into line with the real world. This work can only nuance Hotelling’s generalization and render it meaningless. Nevertheless, as we have seen, this rule has endured, since it still serves as a benchmark for setting the value of carbon to be used in economic calculations.

Analysis shows, however, that this kind of modeling has no connection with the real world, and also presupposes ethical and political choices that are not made explicit and do not impose themselves “naturally”. We therefore recommend abandoning it to the museum of intellectual curiosities.

The social, economic and ecological stakes involved in the current transition are too high to be satisfied with ethereal intellectual exercises and to grant them, out of habit or intellectual laziness, an authoritative status.