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Analysis of Harold Hotelling’s Theory

| April 7, 2014


Economists have long been concerned with the extraction of natural resources. This paper presents an evaluation or analysis of Harold Hotelling’s theory that asserts that the most socially and economically profitable extraction track 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

The paper presents a model of the Hotelling rule and examines its applicability to real life phenomena. The paper finds that while the Hotelling theory had contributed to the economics of nonrenewable resources and the rise of the conservationism movement, the assumptions laid out by the theory are not applicable to the real world. In conclusion, the paper suggests the need to relax the assumptions in order to explain the real-world phenomena.


The efficient use of scarce natural resources, both renewable and non-renewable sources, has long been a concern of natural resource economics (Shogren 2000). For example, Adam Smith explored on the natural progress of opulence and suggested that for a country to achieve an optimum economic progress, it had to allocate capital to land, fisheries and mines (Barnett & Morse 1963).

In a similar vein, Ricardo explored on the significance of land quality on economic rent. Robert Malthus raised concern about the dangers of population growth, asserting that the increasing population was likely to preclude the endless progress towards a utopian society (Barnett & Morse 1963). Similarly, Jevons raised concern about the consequences of coal depletion on population growth (Shogren 2000).  A feature shared by all these economists is their treatment of natural resources as a free factor of production. That is, they all treat natural resources as provided freely by nature.

But towards the beginning of the 21st century, a shift in mindset occurred as economists began treating natural resources as something more distinct than just a free factor of production (Shogren 2000). Theorists such as Hotelling and Gray particularly pointed out to the additional intertemporal cost of extracting natural resources (Shogren 2000). In this regard, this paper presents an evaluation or analysis of Harold Hotelling’s theory that asserts that the most socially and economically profitable extraction track 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 (Harold 1931).

Harold Hotelling’s theory

In 1931, Hotelling used differential calculus to derive the optimal extraction of a fixed resource over time (Bradley 2007). He began by recognizing the inadequacy of the standard economic analysis in the industry in which production was bound to decline (Bradley 2007). Hotelling then asserted that if the total resource base and capital investments were fixed and efficient extraction methods used, then the marginal net revenue of extraction of non-renewable resource would rise at the rate of interest over time (Hotelling 1931). This premium price was referred to as the ‘resource rent’ or rather the ‘hotelling rent’.

In other words, under a perfectively competitive market, the market price of a non-renewable resource minus the marginal costs must grow in tandem with the interest rate. As such, the price of the natural resource should increase with time, provided that the marginal costs are kept constant (Chakravorty et al. 2009)

Harold Hotelling’s theory postulated that the most socially and economically profitable extraction track of a non-renewable resource was one in which the price of the resource, determined by the marginal net revenue from the sale of the resource, increased at the rate of interest (Harold 1931). The theory thus proposed the time track of natural resource extraction that most increases the value of the resource reserve.

But at the time, his analysis was way ahead of time as mathematics had not yet been incorporated into economics. As a result, his seminal analysis was originally rejected for this reason.  However, Hotelling’s rule later came to be known as the pillar of the theory of nonrenewable resource economics through its rich insight on the long-run behaviour of resource price and scarcity of extraction and its contribution to the rise of conservatism movement (Gaitan et al. 2007).

Modeling of the Hotelling rule

To illustrate the Hotelling rule, let us consider as basic model where in So denotes an economy’s total stock of resource and Rt denotes the total extraction at time t (Gaitan et al. 2007). The utility of consumption would be denoted by U(Rt). The objective is to maximize the marginal net revenue of extraction of the non-renewable resource. Assuming that the extraction is carried out with constant unit costs,

Then the optimal path of extraction of the natural resource would be found by the following equation

………………………………………………………………….. (i)

But the maximization must meet the condition

……………………………………………………………………………………. (ii)

Assuming that private and social discount rates are the same and that there are no externalities (Gaitan et al. 2007), then the solution to equation (i) and (ii) can be found using optimal control theory as shown below:

……………………………………………………. (iii)

But conditions require the optimal extraction path to fulfill the following relation (Gaitan et al. 2007),

……………………………………………………….. (iv)

The opportunity cost or rather the shadow price at time t, Yt, is in the present case constant. We thus have

Equation (iv) can thus be rewritten as follows:

…………………………………………………………………………….. (v)

And given that in a market economy, then it can be seen that equation (iv) reflects the Hotelling rule that the marginal price of the natural resource increases with increase in the rate of discount.

Hotteling’s predictions of a decline in non-renewable resources with time such as oil, minerals and forests led to demands for regulation of exploitation of such exhaustible assets (Rothband 2000). The selfish exploitation of natural resources at a rapid rate gave rise to the conservation movement (Rothband 2000). This theory has formed the basis of the conservationist movement and has been influential to the point that prohibitions against oil and mineral mining and deforestation in certain government lands have been justified on this ground (Hotelling 2007).

Criticism of Hotelling’s theory

An important point to emphasize in the Hotelling model is that the market price of non-renewable resources must increase with time, provided that costs remain time-invariant (Chakravorty et al. 2009). However, in reality such an increase in the price of non-renewable sources may not persist as many short-run factors such as regulation and speculation in commodity markets may come into play resulting in alternative phases of upward and downward price movements (Chakravorty et al. 2009).

For example, if the Kyoto Treaty was to impose a target of 450 PPM of carbon, energy prices would be expected to rise but fall soon after the constraint becomes binding (Chakravorty et al. 2009). But when there is a fall below the 450 ppm level, the prices are bound to rise again (Chakravorty et al. 2009). Such a cyclical behaviour in the prices of non-renewable resources is not covered under the Hotelling model.

As a result of such kind of price variations, Harold’s Hotelling theory has come under criticism. Economists have argued that Hotelling’s theoretical prediction of a rise in scarcity and relative prices of nonrenewable resources over time is not borne out of facts (Watkins 1992). For example, Krautkraemer (1998) argues that for the most part, Hotelling’s theoretical predictions have been inconsistent with empirical studies of non renewable resource prices and in situ values.

Over the past 100 years or so, there hasn’t been a persistent increase in the prices of nonrenewable resources (Krautkraemer 1998). Also, economic indicators do not provide evidence of an increase in scarcity of these resources (Krautkraemer 1998). Although the popuplar view among the general public is that exhaustion of non-renewable resources is progressing at a faster rate, this view is highly debatable. In fact, economic indicators have shown that there has been growth in nonrenewable resource supply as new deposits continue to be discovered and the extraction technology continues to progress (Krautkraemer 1998). This has been argued to mitigate the scarcity effect (Krautkraemer 1998).

But the above criticism may not necessarily hold as just like other economic analysis on this subject, Krautkramer’s analysis has been conducted at the global level (Wright & Czelusta 2002).   Although it may seem appropriate to test Hotelling predictions at the global level, such analysis leaves open the possibility that the depletion may have been staved off at the global level, through the discovery new and underexplored territories (Wright & Czelusta 2002).

Nonetheless, the popular view that the world is likely run out of natural resources in the near future may not necessarily be true. If resources are considered to be scarce, then there is a higher likelihood of its real price rising (Braddley 2007). In the event of an increase in real price, producers are likely to be induced by the high prices to explore for more reserves resulting in an increase in resource stock (Braddley 2007). The lifetime measures of most resources can thus be assumed to remain constant over time

Another error that Hotelling made was linking his highly conditional analysis to the real world (Braddley 2007). Hotelling faulted laissez-faire for deviating from his derived optimality in extraction of non-renewable resources, stating that the extractive industries had discrepancies which resulted in wasteful forms of exploitation (Braddley 2007). He also argues that such wasteful forms of exploitation would have been regulated in the interest of the general public (Braddley 2007).

Whilst Hotelling was quick to recognize market failure, he failed to account for what is currently known as government failure (Braddley 2007). As can be seen with the long history of petroleum regulation in the US, government intervention has generally been lacking in information and has been highly problematic in practice (Adelman 1993). Only an omniscient planner would know the specifics of demand, supply, price, cost, interest rates, and entreprenurial alertness needed to arrive at an optimal extraction solution (Braddley 2007).

Role of logistics

Harold’s hotelling theory can be applied to the logistic industry. According to the hotelling theory, the most profitable extraction is one in which the price of the resource, determined by the marginal net revenue from sale of the resource, increases at the rate of interest. In this respect, to reduce on the marginal cost of extraction, it would require that an industry be located close to the extraction point. This would not only decrease the transport costs, but will also increase efficiency in the supply chain and logistics.

Logistics thus has an important role to play in determining the optimal extraction solution. The lower the transport costs, the more likely are the returns to scale. Whilst the transport costs are account for a small percentage of the total costs, the optimal extraction solution must also take into account the total logistics costs. Logistics have an impact on a firm’s optimal location behaviour and thus must be taken into account of when determining the optimal extraction solution of a natural resource.


In Summary, the Hotelling theory has contributed to the economics of nonrenewable resources. It has formed the conceptual and theoretical framework used by economists to model the supply and the prices of nonrenewable resources. More so, it has contributed to the conservationist movement. However, the Hotelling theory, though elegant, seem somewhat misplaced. The model points out to a rise in trajectory of net prices of non-renewable resources along with the rate of interest yet there is a lack of empirical evidence to back this pricing behaviour.

Moreover, the assumption of an increase in scarcity of non renewable resources is highly debatable. These assumptions seem not applicable to the real world.  Perhaps, to explain the real-world phenomena, it would be helpful to relax these assumptions. For example, in order to explain the price of oil, it would be necessary to discard all assumptions of inevitable increase in price and the assumption of a fixed stock.


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