Hotelling's rule
Encyclopedia
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.
This rule is derived from the work of a U.S. Economist Harold Hotelling
in his seminal paper “The Economics of Exhaustible Resources” (Hotelling 1931), published in the Journal of Political Economy. This paper has laid the foundation for further research in the field of non-renewable resource economics.
has been significantly visible. This Conservation Movement
was concerned about the possible overexploitation of the non-renewable natural resources and called for a regulation. Harold Hotelling
responded to this call with his seminal paper. In fact, he begins his paper with an introduction to this problem by stating that:
Before 1970s serious attention was not given to Hotelling’s views regarding economics of exhaustible resources. This was due to the fact that when Hotelling came up with this hypothesis, the media’s emphasis was towards issues like the Great Depression
and the World War of the 1940s.
After World War II
, there were concerns about the sufficiency and adequacy of natural resources especially in the US. 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 kinds 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 takes us 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 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 got 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 ,” Limits to Growth”, in the year 1972 set off another period of intense public concern for natural resources. 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 was a vital piece of study which 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 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. The now famous Hotelling’s rule acquired the centre stage and still continues to maintain its position.
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.
:
when P(t) is the unit profit
at time t and δ 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:
i. Flow of product generated by the marginal unit of the resource, Marginal Productivity or Dividend rate.
ii. Change in the physical characteristics of the asset over time.
iii. 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.
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 date t, then its marginal value must be:
Π(t) = p(t) − c(t),
which represents the asset price of the resource.
.
Denoting the rate of interest by r, then asset markets equilibrium is established when
Π'(t)/ Π(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 doest not vary over time, then the market price of the asset over time would be
p'(t)/ p(t)= r (1 - c/ p(t))
If this was 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 gets smaller and the rent brought about by the scarcity of the natural resource gets higher.
But this is not in line with the statistics observed 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 the rate of change of prices of these resources being influenced by a high degree of volatility. But the more important phenomena 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 a particular trend and definitely not the path of the positive trend as recommended by Hotelling’s Rule.
Non-renewable resource
A non-renewable resource is a natural resource which cannot be produced, grown, generated, or used on a scale which can sustain its consumption rate, once depleted there is no more available for future needs. Also considered non-renewable are resources that are consumed much faster than nature...
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.
This rule is derived from the work of a U.S. Economist Harold Hotelling
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
in his seminal paper “The Economics of Exhaustible Resources” (Hotelling 1931), published in the Journal of Political Economy. This paper has laid the foundation for further research in the field of non-renewable resource economics.
History
During the late nineteenth and early twentieth century, the Conservation in the United StatesConservation in the United States
Conservation in the United States can be traced back to the 19th century with the formation of the first National Park.-Philosophy of early American conservation movement:During the 19th century, Americans developed a deep and abiding passion for nature...
has been significantly visible. This Conservation Movement
Conservation movement
The conservation movement, also known as nature conservation, is a political, environmental and a social movement that seeks to protect natural resources including animal, fungus and plant species as well as their habitat for the future....
was concerned about the possible overexploitation of the non-renewable natural resources and called for a regulation. Harold Hotelling
Harold Hotelling
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
responded to this call with his seminal paper. In fact, he begins his paper with an introduction to this problem by stating 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.” - Harold Hotelling (1931)
Before 1970s serious attention was not given to Hotelling’s views regarding economics of exhaustible resources. This was due to the fact that when Hotelling came up with this hypothesis, the media’s emphasis was towards issues like the Great Depression
Great Depression
The Great Depression was a severe worldwide economic depression in the decade preceding World War II. The timing of the Great Depression varied across nations, but in most countries it started in about 1929 and lasted until the late 1930s or early 1940s...
and the World War of the 1940s.
After World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, there were concerns about the sufficiency and adequacy of natural resources especially in the US. 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 kinds 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 takes us 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 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 got unnoticed. In fact Hotelling himself stated that the Economic Journal had rejected his paper due to high difficulty level of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
involved in it.
The publication by Dennis Meadows ,” Limits to Growth”, in the year 1972 set off another period of intense public concern for natural resources. 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 was a vital piece of study which 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 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. The now famous Hotelling’s rule acquired the centre stage and still continues to maintain its position.
The Rule
As mentioned earlier, and 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.
:
when P(t) is the unit profit
Profit (economics)
In economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...
at time t and δ is the discount rate(the inverse of rate of return).
The stock
Stock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
of a non-renewable resource
Non-renewable resource
A non-renewable resource is a natural resource which cannot be produced, grown, generated, or used on a scale which can sustain its consumption rate, once depleted there is no more available for future needs. Also considered non-renewable are resources that are consumed much faster than nature...
, 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:
i. Flow of product generated by the marginal unit of the resource, Marginal Productivity or Dividend rate.
ii. Change in the physical characteristics of the asset over time.
iii. 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 the marginal unit of the natural resource held in-situ less the cost of taking it out of the ground is what its value of the resource would be 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 date t, then its marginal value must be:
Π(t) = p(t) − c(t),
which represents the asset price of the resource.
.
Denoting the rate of interest by r, then asset markets equilibrium is established when
Π'(t)/ Π(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 doest not vary over time, then the market price of the asset over time would be
p'(t)/ p(t)= r (1 - c/ p(t))
If this was 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 gets smaller and the rent brought about by the scarcity of the natural resource gets higher.
But this is not in line with the statistics observed 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 the rate of change of prices of these resources being influenced by a high degree of volatility. But the more important phenomena 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 a particular trend and definitely not the path of the positive trend as recommended by Hotelling’s Rule.