The Hotelling Rule in Non-Renewable Resource Economics: A Reassessment

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The Hotelling Rule in Non-Renewable Resource

Economics: A Reassessment

Roberto P. Ferreira da Cunha, Antoine Missemer

To cite this version:

Roberto P. Ferreira da Cunha, Antoine Missemer. The Hotelling Rule in Non-Renewable Re-

source Economics: A Reassessment. Canadian Journal of Economics, 2020, 53 (2), pp.800-820.

�10.1111/caje.12444�. �hal-03216483�

- Authors’s post-print -

- published in the Canadian Journal of Economics (2020), 53(2), 800-820 -

THE HOTELLING RULE IN NON-RENEWABLE

RESOURCE ECONOMICS: A REASSESSMENT

Roberto P. FERREIRA DA CUNHA

Antoine MISSEMER

†

Full reference:

FERREIRA DA CUNHA, Roberto P. and MISSEMER, Antoine. 2020. “The Hotelling Rule in Non-

Renewable Resource Economics: A Reassessment”. Canadian Journal of Economics, 53(2), 800-

820.

[https://doi.org/10.1111/caje.12444]

The pagination of the published version is indicated in the margin.

Abstract

Harold Hotelling’s 1931 contribution is known for providing a basic principle—the Hotelling rule—to the

economics of non-renewable resources. Nearly 90 years later, empirical tests conclude the rule lacks

empirical validity, requiring strong amendments to describe the long-term, aggregate behaviour of its

target object. On the basis of Hotelling’s unpublished archival material, this paper revisits the place given

to the Hotelling rule in non-renewable resource economics. Our reconstruction shows that Hotelling’s

1931 paper has been misinterpreted: from the outset, the Hotelling rule was not valid for mineral

resources. In contrast, the consideration of two inherent geological constraints, alongside exhaustibility,

offered the opportunity for an alternative basic framework, capable to generate bell-shaped and U-shaped

equilibrium trajectories for supplies and prices, respectively. Inspired by this unknown aspect of

Hotelling’s work brought to light by our archival investigation, we sketch this alternative basic model,

enabling non-renewable resource economics to circumvent the empirical shortfalls of the Hotelling rule.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[In 2020] Berkeley Research Group, LLC. Rua da Quitanda 86, 2° andar – Centro, Rio de Janeiro, RJ, Brazil.

E-mail: rcunha@thinkbrg.com

†

[In 2020] CNRS, CIRED Paris – Centre international de recherche sur l’environnement et le développement

(UMR 8568 CNRS/ENPC/EHESS/AgroParisTech-University-of-Paris-Saclay/CIRAD), 45 bis av. de la Belle

Gabrielle, 94736 Nogent-sur-Marne Cedex, France. E-mail: missemer@centre-cired.fr

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1. Introduction

“The Economics of Exhaustible Resources,”

published in April 1931, is considered a seminal

contribution for non-renewable resource

economics. Harold Hotelling proposed an

extensive analysis of the intertemporal

optimization of extraction paths under various

competitive conditions. He laid the foundations

for many research questions, including the role

of taxation and public regulation in extractive

sectors. Even if the Hotelling model is no longer

taught to students in the original form, the

lessons drawn from the 1931 article are still at

the ground of contemporary research and

teaching, in which the equation p

= p

• e

r•t

plays

the role of basic principle.

In 1974, Robert M. Solow paid tribute to

Hotelling, labelling this equation as “the

Hotelling rule, the fundamental principle of

natural-resource economics” (Solow 1974,

p. 12). As far as we know, that was the first

occurrence of the expression “Hotelling rule,”

and Solow’s impulse, within the favourable

context of the oil price shock, reignited research

on the subject of non-renewable resources

(e.g., Levhari and Liviatan 1977, Dasgupta and

Heal 1979, Devarajan and Fisher 1981, Farzin

1984, 1992). Since then, the literature retains

finiteness, in the form of Hotelling’s hard

constraint

∫

∞

q • dt ≤ a, as the central defining

factor of the class of non-renewable natural

resources.

Tests on the empirical validity of the

Hotelling rule, by itself, have concluded it is

unable to stick to reality, both in terms of

assumptions and capacity to predict real market

prices. Gaudet (2007), analyzing the price

evolution of 10 mineral resources throughout the

20th century, concluded the rate of change in

prices is approximately centred at zero, and not

at the interest rate. Livernois (2009) reviewed

34 empirical studies on the Hotelling rule,

concluding data does not provide overwhelming

support for it. Slade and Thille (2009), reviewing

roughly the same studies, concluded for the

rejection of the basic model by data. Further

empirical tests, of specific aspects of Hotelling’s

model, or of elementary extensions of the basic

model, reached the same conclusion (e.g., Hart

and Spiro 2011, Gaugler 2015, Atewamba and

Nkuiya 2017, Karp 2017).

The rejection of the basic model by data has

been remediated by the consideration of

additional real-world factors. The main ones

capable of bringing the basic model closer to

reality are technological progress (Slade 1982),

degradation costs (Pindyck 1978), durability

(Levhari and Pindyck 1981), market structure

(Salant 1976) and uncertainty (Kemp 1976).

Each extension inaugurated a modern branch in

non-renewable resource economics, expanding

and detailing the level of sophistication of each

factor within the original basic framework.

Nonetheless, the implementation of different

additional factors leads to questions about the

robustness and readability of the basic theory: by

varying the choice of factors to implement on the

model, one can attain diverging results. By

requiring strong amendments to represent its

target object, its value as a theoretical starting-

point comes into question.

This paper inquires whether a deeper

understanding of Hotelling’s 1931 contribution

can help us circumvent the empirical invalidity

of the Hotelling rule. We hypothesize the

prevailing understanding stems from a

problematic interpretation of the original

contribution, inaugurated through Solow’s 1974

labelling of Hotelling’s basic result as a rule. To

test this hypothesis, we investigated Hotelling’s

project on non-renewable resources through the

history of economic thought methodology,

allowing us to leverage unpublished materials

towards demonstrating potential theoretical gaps

left behind along the evolution of thought,

possibly changing the way we understand and

use theory going forward.

We based our investigation on an

unprecedented exploration of Hotelling’s

archives, stored at Columbia University

(hereafter HHP).

These materials seem to have

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Many other extensions of Hotelling’s model have been

proposed in the literature over the past decades. For the

most recent years, we can notably cite Holland (2008),

Venables (2014), Okullo et al. (2015) and Anderson et al.

(2018). This list is not exhaustive.

Previous works in the field of history of economic thought

referring to non-renewable resources include Robinson

(1980, 1989), Kula (1998) and Missemer (2017, 2018).

HHP: Harold Hotelling Papers, Rare Book and Manuscript

Library, Columbia University, USA. Catalogue available at

findingaids.cul.columbia.edu/ead/nnc-rb/ldpd_4078401.

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been consulted only by a few other scholars for

previous research (e.g., Crabbé 1986; Darnell

1988, 1990; Hands and Mirowski 1998), but

none of these works offer a comprehensive view

on Hotelling’s project regarding the economics

of non-renewable resources. The documents

stored at Columbia University consist of several

thousand sheets, filed in 58 boxes. Items related

to exhaustible resources are scattered in different

boxes. Thanks to a systematic review, we found

about 20 to 30 significant unpublished drafts,

preparatory documents and letters from 1924 to

1925 onwards worthy of interest for our inquiry.

Our paper is organized as follows. In

section 2, we investigate what sort of research

Hotelling was conducting in the 1920s, showing

his hesitation between a theoretical abstract

ambition and empirical concerns for what he

named at the time exhaustible resources.

This

section allows a clearer picture on the area of

validity, from the outset, for his basic result—

which is different than what has been retained

from his article. In section 3, we closely review

Hotelling’s work on geological constraints for

his 1931 article. We reveal a neglected

fundamental role for them in his analysis.

Hotelling considered cumulative production x(t)

and the change in production q’(t) as

fundamental variables, and not additional

factors, constituting a potential alternative basic

model capable of attaining U-shaped trajectory

for prices—the now empirically accepted

stylized fact for non-renewable resources. In

section 4 we provide our concluding remarks,

reassessing the Hotelling rule as the starting

point of mineral resource analysis.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

This paper is part of the research project “Bifurcations in

Natural Resource Economics” (BNRE), sponsored by the

European Society for the History of Economic Thought

(ESHET). The objective has been to explore the genesis of

modern natural resource economics, notably through the

creation of a digital database of Hotelling’s economic

archives (7000+ images taken of 3000+ documents). The

project team mobilized researchers in natural resource

economics and history of economic thought over 2.5 years,

producing to date several coordinated articles and working

papers on complementary aspects related to Hotelling’s

research (e.g., Franco et al. 2019, Gaspard and Missemer

2019, Missemer et al. 2020, Missemer and Nadaud 2020).

What Hotelling reffered to as “exhaustible resources” are

now called “non-renewable resources.” In the following, we

use the modern term, except when it comes to evoking and

analyzing Hotelling’s project in its historical context.

2. Hotelling’s research on exhaustible

resources in the 1920s

2.1. From (natural) resources to assets, and

vice versa

Until the submission of his paper to the

Journal of Political Economy, most likely in the

fall of 1930,

Hotelling’s archives reveal,

regarding exhaustible resources, that he actually

worked along two approximately parallel lines of

research: a generic project on the valuation of

exhaustible assets, in the line of his theory of

depreciation, and a specific project on

exhaustible natural resources (minerals, fossil

fuels), more empirically grounded. In his final

article, Hotelling uses several terms to describe

his research: “exhaustible (natural) resources,”

“exhaustible assets,” “irreplaceable (natural)

resources,” “irreplaceable assets,” “mineral

content,” “resources of the earth,” etc. By

definition, the set of “natural resources” here

contains both fossil energy sources (coal, oil,

gas) and minerals. The set of “assets” is broader,

containing “natural resources” as a subset. A

mineral stock is an asset. A financial bond is also

an asset, but obviously not a natural resource.

Focusing on “natural resources” suggests that the

precise subject of research is related to concrete

forms of natural items. On the opposite side,

focusing on “assets” opens the reflection to other

case studies, beyond natural items.

Hotelling’s drafts, correspondence and notes

throughout the 1920s confirm the co-existence of

two levels of analysis: a generic level, illustrated

by the term “exhaustible assets,” and a more

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Two letters from Hotelling to R. A. Fisher and

Chamberlin (dated July 23, 1930, and Aug. 8, 1930,

respectively) indicate that Hotelling was working on the

final version of the paper in the summer of 1930 (RAFP,

digitized version online; Guicherd 2017, p. 179). A letter

from Mills (dated Jan. 17, 1931) shows that the final

version of the paper was already circulating at the very end

of 1930 (HHP, Box 1, Mills, Frederick C.). Arrow (1974,

p. 1103; 1980; 1987, p. 670) reports that Hotelling (who

was his PhD supervisor) submitted his paper to the

Economic Journal before submission to the Journal of

Political Economy. As shown in Gaspard and Missemer

(2019), there is, however, no clear evidence that he did so.

This statement is not retrospective. In the early 20th

century, Irving Fisher (1906) had already discussed the

generic nature of the words “assets” and (not necessarily

natural) “resources.”

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specific level, with the term “exhaustible natural

resources.” We found Hotelling hesitated

between which of the two to use in his final

paper. Until only a few weeks before

submission, the title of the paper was “The

Economics of Exhaustible Assets,” as indicated

in a letter to the statistician Ronald A. Fisher:

The book [on statistics] is gradually

progressing, though I am leaving it alone for a

few weeks at present to finish a paper on “The

Economics of Exhaustible assets” which I have

been at for five years and for which various

people have been calling.

Only in January 1931, four months before

publication, had the final title been chosen, as

another letter from Fredrick C. Mills (NBER)

testifies to:

Schultz was kind enough to let me read a copy

of your paper “The Economics of Exhaustible

resources.” I returned this to him. It seemed to

me to be an extremely suggestive attack upon a

problem which economists have largely

neglected.

Hotelling therefore finally chose an intermediary

title, aiming at covering both his generic,

theoretical ambition and his specific, more

concrete concerns. The archives further reveal

that from 1924/1925 to the final version,

Hotelling constantly changed the title of his

research. A first draft from 1924 was entitled

“Exhaustible Assets under Competition.”

Another draft from 1925 was labelled “The

Exploitation of Irreplaceable Resources.”

December 1929, a full abstract was named “The

Economics of Exhaustible Assets.”

The contents of these drafts are not exactly

the same. When assets are considered, the

analysis makes a few mentions of mineral

resources or concrete questions, instead focusing

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Letter from Hotelling to Fisher, July 23, 1930. Online

digitized version. RAFP: Ronald A. Fisher Papers, Special

Collections Library, University of Adelaide, Australia.

Catalogue available at digital.library.adelaide.edu.au/

dspace/handle/2440/3860.

Letter from Mills to Hotelling, Jan. 17, 1931. HHP, Box 1,

Mills, Frederick C.

HHP, Box 10, AMS Reports and Correspondence (3).

HHP, Box 42, Exploitation of Irreplaceable Assets.

on the use of advanced mathematics to deal with

abstract principles.

When natural resources, or

minerals, are scrutinized, references to real-

world situations are more present, as shown by

the 1925 draft entitled “The Exploitation of

Irreplaceable Resources”:

The complete monopoly and the perfectly fluid

competition of traditional economics are

extremes between which lie all the cases which

actually occur in the world.

The generic subject (exhaustible assets)

offered opportunities to draw theoretical

principles, such as the equation p

= p

• e

r•t

(i.e., the Hotelling rule). The specific subject

(exhaustible natural resources), a particular case

of exhaustible assets, required more attention to

the real world.

The chronology of the archives helps us to

reconstruct Hotelling’s early treatment of

exhaustible resources. The first draft on the

subject dates back to 1924; it was written for a

meeting of the American Mathematical Society

in Chicago. These notes outline the close

connection between Hotelling’s research on

depreciation and on exhaustible assets, as

already mentioned.

The work on asset

valuation was published in 1925 under the title

“A General Mathematical Theory of

Depreciation.” As noted by Crabbé (1986),

assuming Hotelling’s asset valuation function

from Hotelling (1925) is a negative exponential

with a constant parameter—the rate of interest—

yields the Hotelling rule published in 1931.

We found that it was in the winter of 1925

that Hotelling considered for the first time

engaging with dedicated research on exhaustible

assets. It is also when he decided to explore in

more detail the calculus of variations to

“simplify” the issue:

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

For instance, in the draft “Exhaustible Assets under

Competition” from 1924, Hotelling states his problem as

“applying Euler’s equation.” HHP, Box 10, AMS Reports

and Correspondence (3).

HHP, Box 42, Exploitation of Irreplaceable Assets.

HHP, Box 10, AMS Reports and Correspondence. For a

full inquiry into the intertwining of Hotelling’s projects on

depreciation and exhaustible assets, see Missemer et al.

(2020).

Monthly Report from Hotelling to the Directors of the

FRI. Mar. 2, 1925. HHP, Box 41, Agriculture III.

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The recent work on the economic of

irreplaceable assets had during the past month

been undergoing revision and extension. The

revision is made necessary by a simplification

of the Calculus of variations method which is

found to be applicable to the problem, and also

by the publication of a paper by G. C. Evans

(Proc. Nat. Acad. Sci., Jan. 1925, pp. 90-95)

touching upon the subject.

It appears that Hotelling clearly regarded his

generic project—i.e., the theory of exhaustible

assets—as an extension of his theory of

depreciation. Interestingly, another draft from

the same time, March 1925, indicates that he had

meanwhile started to investigate the more

concrete and real-world side of the problem of

exhaustible resources, with special attention to

minerals, oil and natural gas in the case of an

oligopoly. This confirms not only the

entanglement of the two research lines but also

their synchrony:

The problem is inderteminate because after one

mine is exhausted the other has a monopoly;

and since we have assumed the demand

inelastic the profits of the monopoly are

unlimited.

In summary, Hotelling’s economics of

exhaustible resources in the 1920s was

composed of two levels of analysis. Because the

1931 article mixed the two levels, it became

difficult for readers to identify the purposes and

limits of the different arguments, including the

area of validity of the Hotelling rule. The

archival material helps provide a clearer

understanding: the apparent discrepancy between

Hotelling’s concrete subject (non-renewable

resource depletion) and abstract proposals

(e.g., the Hotelling rule) is due to the existence

of two initially distinct lines of work developed

in parallel, and later merged in the same

contribution. For the treatment of real-world

resources, Hotelling needed more information on

the actual dynamics and issues pertaining to

extractive sectors. Archival materials helped us

identify where he found this information.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

HHP, Box 42, Exploitation of Irreplaceable Assets.

2.2. Conservation (or not) and energy policy

Gaudet (2007) states Hotelling wrote his

1931 paper in reaction “to the demand for

regulation of the exploitation of exhaustible

resources by the conservation movement, which

had been particularly strong in the U.S. during

the period 1890 to 1920” (p. 1034). Livernois

(2009) asserts, “Hotelling’s seminal article was a

response to the conservation movement’s claims

that unregulated private markets would lead to

the overexploitation of non-renewable resource

stocks” (p. 23). Their conclusion is likely based

on the opening paragraph of “The Economics of

Exhaustible Resources,” where Hotelling

textually frames his research scope in relation to

the “conservation movement” (p. 137). How-

ever, he mentions the movement only one more

time (p. 143), in section 3.

The reference to conservation is intriguing

because, in the 1920s, among economists, the

movement was not as popular as in the 1900s

and 1910s, when Lewis C. Gray (1913, 1914)

and Richard T. Ely (1918) wrote on the subject.

Of course it was still an important trend in the

reflections about natural resources (e.g., Ise

1925). However, the importance given by

Hotelling to the conservation movement as a

counterpoint to his analysis and as a source of

information about on-going issues in extractive

sectors is not clear at all and deserves to be

examined in detail.

The archives reveal that early drafts make no

mention of the conservation movement, even

along the specific, more empirical research line.

In a preliminary introduction of the paper, in

December 1929, and which was very close to the

final version, the catchphrase is already written

(apart from the last addition on taxation), but the

part on conservation is missing. Instead,

Hotelling insists on competitive settings and on

exhaustible assets as a specific case of a wider

issue, in line with his previous drafts starting in

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On the economic thought of the first conservation

movement, see for instance Smith (1982), Ramos Gorostiza

(2003), Kula (1998) and Missemer (2017). Gray worked on

conservation when he was at the University of

Saskatchewan (Canada), initiating a long tradition of

Canadian research in resource and environmental

economics (see Copeland and Taylor 2017). On Gray’s

contributions, see Crabbé (1983).

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1924/1925:

Contemplation of the world’s disappearing

supplies of minerals, forests, and other

exhaustible assets has led to demands for

regulation of their exploitation, by taxation or

otherwise. On the other hand the existence of

monopolies and quasi-monopolies has raised

the same problems of excessively curtailed

production here as in other industries, but in a

more complex form. The economics of the

exploitation, that is, the study of the rates of

exploitation which on the one hand tend to take

place under commercial conditions, and of

those which on the other hand ought in the

public interest to take place, has however

received no serious attention [our emphasis].

This provisional introduction reveals that

conservation was not so directly decisive in

Hotelling’s undertaking.

If not in the conservation literature

(especially the economic literature), where did

Hotelling find information about the concrete

dynamics of extractive sectors (minerals, oil and

gas)? Several allusions disseminated in the 1931

article find justification in issues of the 1920s

not directly related to conservation:

The prohibitions against oil and mineral

development and cutting timber on certain

government lands have this justification, as

have also closed seasons for fish and game and

statutes forbidding certain highly efficient

means of catching fish (p. 137).

On the other hand, certain technical conditions

most pronounced in the oil industry lead to

great wastes of material and to expensive

competitive drilling, losses which may be

reduced by systems of control which involve

delay in production (p. 138).

The government of the United States under the

present administration has withdrawn oil lands

from entry in order to conserve this asset, and

has also taken steps toward prosecuting a group

of California oil companies for conspiring to

maintain unduly high prices, thus restricting

production (p. 138).

Daniel Yergin’s 1991 work on oil history

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

HHP, Box 42, Exploitation of Irreplaceable Assets.

illuminates the events Hotelling refers to. The

1920s in the United States were marked by fast

growing oil demand brought by gasoline-fuelled

cars (Yergin 1991, p. 208). Meanwhile, very few

new oil discoveries had been registered on the

1917–1920 period, leading oil prices to triple in

the period. Within this context took place in the

early 1920s the “Teapot Dome Scandal.” This

episode of corruption was about the exploitation

of several oil fields in Wyoming initially

preserved by the US government as naval oil

reserves under the Taft and Wilson

administrations (Yergin 1991, p. 211). In 1924,

when Hotelling started to work on exhaustible

resources, the presidential election was marked

by the scandal. The mention of government

ownership over lands and resources in the 1931

paper (above quotes) matches this episode.

Hotelling’s allusion to technical conditions

leading to waste can also be related to legal

debates in the 1920s. The so-called “Rule of

Capture” attributed ownership over oil resources

to the owners of the land immediately above it.

When several properties spanned a single

reservoir, operators would rush to produce as

rapidly as possible in order to avoid letting the

neighbours drain the volumes under their

properties through their wells. This ultimately

led to an excessive number of wells and lowered

the ultimate recovery of oil from the reservoir.

Debates took place mostly under the Coolidge

administration (1923–1929), leading to the

establishment of the Federal Oil Conservation

Board, in 1924 (Yergin 1991, pp. 220–23).

Therefore, the remarks on minerals, oil and

gas disseminated by Hotelling in his final article,

not fully related to the academic debates on

conservation, actually correspond to various

affairs linked to the US federal policy. His

allusions to practical cases were not thought

experiments, or pretexts to explore fictive

situations; they were based on real political and

economic experiences that marked American

public life in the 1920s.

In this context, the Hotelling rule, although

essential for the generic project on exhaustible

assets, and which supposed perfect competition

and no State intervention for achieving the social

optimum, was clearly established as inadequate

for the analysis of concrete, specific natural

resources (minerals, oil and gas), as

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demonstrated by the closing of section 2:

...[T]here are in extractive industries

discrepancies from our assumed conditions

leading to particularly wasteful forms of

exploitation which might well be regulated in

the public interest (pp. 143–44; our emphasis).

The word “discrepancies” is significant as a

strong indicator of the inoperability of the basic

model for describing concrete situations. As

soon as these situations are involved, the “real

economic world” (Hotelling 1931, p. 171)

matters, up to assigning a role to regulators in

the extractive sector, in contrast to what is

usually retained from Hotelling’s analysis in

terms of State intervention.

Theoretically

speaking, empirical realities require further

developments beyond the basic result.

An insight emerges from the realization that

Hotelling did not see what Solow (1974) labelled

“the Hotelling rule” as a rule for non-renewable

resource economics because it could not describe

what took place in reality in extractive sectors, as

Hotelling found in concrete experiences from the

1920s. The area of validity of the rule was

limited to finitely available assets, in the absence

of some realities he proceeded to analyze.

Among these realities, apart from alternate

market structures and taxation issues, we

discovered Hotelling paid close attention to

geological constraints, which we will show he

concluded to be necessary conditions to depict

natural resources.

3. Hotelling and geological contraints

3.1. Geological constraints in the 1931

article

“The Economics of Exhaustible Resources” is

known for its basic model, which is described in

sections 2 and 3 of the 1931 article. The final

paper, nonetheless, contains 15 sections. Archive

research revealed the basic result dates back to

the first drafts, circa 1924/1925. Subsequent

drafts deal with more substantial models, which

one can find in the less-known sections of the

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Regarding Hotelling’s conception of State intervention,

some insight can be found in Franco et al. (2019).

1931 article. Slade and Thille (2009)

acknowledge that “Hotelling derived several

variants of his model” (p. 241; our emphasis).

These “variants” actually represented the core of

Hotelling’s work, at least as reflected in the

archives.

From sections 8 to the end of his analysis,

Hotelling deals with his specific research line on

concrete mineral resources, directing his

attention to the realities of mining extraction.

There he leaves aside the basic equation and

turns to the issue of “cumulative production

affecting price” (p. 152). The “net price”

function becomes p(x(t),q(t),t): cumulative

production x(t) =

∫

q(t) dt is now a variable,

affecting the formation of net prices. Hotelling

justifies the need for this variable on the cost

side. It turns out resources are located under the

ground, in varying depths. A first geological

constraint (GC1) other than finiteness thus

appears: “the cost of extraction increases as the

mine goes deeper” (p. 152).

In section 12, “The Need for Steadiness in

Production” (p. 162), Hotelling further expands

his net price function to portray the rate of

change in production q’(t) as an additional

variable. The cost of increasing production,

Hotelling argues, depends on the magnitude of

the increase. In developing a mine, producers

have to make capital investments, which makes

sudden increases in production more expensive

than gradual ones. This is due to the uneven

distribution of concentrations of resources under

the ground—a second geological constraint

(GC2) other than finiteness, which imposes a

cost for drawing the labour force to or from the

site of the mine, making large variations in

output significantly costlier than small

variations. Hotelling closes the section by

noting:

The cases considered in the earlier part of this

paper all led to solutions in which the rate of

production of a mine always decreases. By

considering the influence of fixed investments

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

The word “mine(s)” is also employed a few times in the

sections dedicated to the basic model, but to designate

stylized mines, for which finiteness is the only

characteristic considered. Hotelling makes clear from

section 8 onwards that concrete mines do not fall under the

same analysis.

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and the cost of accelerating production at the

beginning, we may be led to production curves

which rise continuously from zero to a

maximum, and then fall more slowly as

exhaustion approaches. Certain production

curves of this type have been found statistically

to exist for whole industries of the extractive

type, such as petroleum production (p. 164; our

emphasis).

In section 15, the closing section of the paper,

Hotelling comes back to the discussion between

perfect competition and monopoly, presenting

the case of duopoly, in which a few competing

sellers can share the market’s demand. In this

model, Hotelling avoids the complex

formulation of section 12, excluding for

simplicity the influence of the rate of change in

production as a determinant of the price.

However, at this point, Hotelling makes

reference to a third geological constraint (GC3),

other than finiteness, namely the uncertainty on

the actual size of reserves in the beginning and

the need to acquire geological information

through drillings:

The problems in exhaustible resources involve

the time in another way besides bringing on

exhaustion and higher prices, namely, as

bringing increased information, both as to the

physical extent and condition of the resource

and as to the economic phenomena attending

its extraction and sale (p. 174).

Therefore, aside from market structure and

taxation, Hotelling’s 1931 displays another,

often overlooked, clear axis of progression from

his basic result, towards the incorporation of

GC1, GC2 and GC3, notably through x(t) and

q’(t), as fundamental variables. While the

number of firms that operate on a given market

or its tax regime can in principle be affected by

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

In all mineral activities, the underground location of

mines is initially unknown, and each mine or well drilled

brings increased geological knowledge that facilitates the

discovery of the next underground occurrence. Costs are not

known with certainty from the start, which carries

substantial investment risks. Only upon drilling wells and

evaluating their output or how they respond after another

well is drilled into the same oil reservoir, it it possible to

infer engineering properties such as reservoir pressure and

porosity and, as a result, determine the total number of

wells needed to optimally exploit the resource.

policy, geological constraints are something no

one can do anything about. Therefore, they

should not be considered as “variants” or

“possibilities,” as mentioned by Slade and Thille

(2009), nor as additional factors, but as

fundamental inalterable characteristics with the

same status as finiteness, affecting all mineral

resources. In addition to the already mentioned

closing of section 2, in the opening of section 8,

Hotelling states the influence of x(t) on the

exploitation of resources (which results from

GC1) is a certainty, not a mere possibility: “[t]he

net price...depends...on past production,”, i.e., it

depends on it, not may depend on it (p. 152; our

emphasis).

Therefore, sections 8 to 15 of Hotelling’s

article discuss concrete real-world factors that

define the category of mineral resources within

the class of non-renewable resources, itself a

subset of non-renewable (exhaustible) assets.

Once more, in contrast to modern interpretations

that keep the basic rule as a central principle to

be sophisticated, it clearly emerges that this

theoretical principle does not apply here. The

rule governs only generic finite assets in the

absence of geological constraints other than

finiteness (GC1, GC2 and GC3). It therefore

does not govern mineral resources (minerals, oil

and gas), as presently overstated.

3.2. Hotelling’s preparatory work in geology

and engineering

By contrasting the archival materials with the

final paper, we discovered the relative degree of

effort Hotelling devoted to the topic of

geological constraints. Two drafts from

November 1928 are dedicated entirely to the

issue of “strong variations,” in which Hotelling

includes for the first time cumulative production

x(t) as a fundamental variable, and devote

significant effort to understanding the

implications to the overall equilibrium.

Furthermore, in the draft abstract written in

1929, Hotelling does not present the basic model

at all, starting with the model with x(t), using

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

In other parts of the article (e.g., section 12), Hotelling

uses less clear-cut vocabulary for the occurrence of

geological factors, but he always argues they have to be of

the utmost importance.

HHP, Box 42, Exploitation of Irreplaceable Assets.

810

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empirical examples to justify his formulation:

If the price obtainable varies with the amount

which has been extracted, as with gold and

diamonds, or if the cost of production increases

as the shaft sinks deeper, the problem is to

maximize

∫

e^(-

t) p(x,q,t) qdt.

By 1930, Hotelling continued to focus on

how geological constraints other than finiteness

could affect the net price function. A noticeable

preparatory work is his correspondence with

Stanley C. Herold, professor of petroleum

engineering at Stanford University—Hotelling

was still at Stanford in 1930.

Herold wrote to

Hotelling on July 7, 1930, in a friendly tone,

providing empirical data on how, in the case of

petroleum, costs increase as shafts sink deeper

(GC1); see table 1.

Table 1 shows a non-linear (quadratic)

relationship between cost and depth; costs

increase more than proportionately with well

depth (figure 1).

GC2 and GC3 are also addressed in the letter,

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

HHP, Box 42, Exploitation of Irreplaceable Assets.

Hotelling becase professor of economics at Columbia

University in 1931 after being approached by Wesley

Mitchell (letter from Mitchell to Hotelling, Jan. 21, 1931.

HHP, Box 50, Columbia Employment).

and some remarks can be traced to Hotelling’s

statements in his final article. For production

speed at the beginning of the extraction, Herold

writes:

The cost mounts more rapidly than the footage

on account of the fact that deeper holes require

heavier rigs and equipment, also more time is

consumed in running in and out the holes.

Indeed, because different oil and gas

reservoirs vary in depth and many other

geological conditions, their unit production costs

vary. Producers seek to developthe lowest cost

ones first (Herfindahl 1967), although sometimes

they are unable at first to discover the best

reservoirs when only limited geological

information is available (Uhler 1976). This leads

to Hotelling’s remark on GC2, when he

mentions “the cost of accelerating production in

the beginning” (1931, p. 164).

Still in the same letter, Herold describes the

spatial element affecting costs, thus insisting on

the heterogeneity of wells:

...[H]oles at Kettleman Hills are costing a little

more than shown here on account of

transportation of equipment and general

operating expenses, longer delays on

replacement of equipment, etc.

This explains Hotelling’s remarks in section 12

about costs related to the relocation of

production factors, in particular the labour force:

Under the term “capital” might possibly be

included the costs, both to employers and to

laborers, in drawing laborers to the mine from

other places and occupations (Hotelling 1931,

p. 163).

The example of Kettleman Hills is significant

because it also highlights the variability of costs

resulting from new discoveries: when promising

wells are discovered—the Kettleman Hills oil

fields, in particular around the North Dome,

were a major discovery in California in the late

1920s—they attract capital and labour, with

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Letter from Herold to Hotelling, July 7, 1930. HHP, Box

42, Exploitation of Irreplaceable Assets.

Letter from Herold to Hotelling, July 7, 1930. HHP, Box

42, Exploitation of Irreplaceable Assets.

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potentially abrupt variations in both production

and costs.

Therefore, Hotelling’s work between 1928

and 1930, as reflected in the archives, dealt

almost exclusively with the impact of geological

constraints on his basic model from 1925. The

importance given to the incorporation of x(t) as a

variable is unrivalled, comparable only to the

distinction between monopoly and perfect

competition. In addition to Herold’s 1930 letter

to Hotelling, the 1928/1929 drafts, and how

these documents were reflected in the final

paper, demonstrate that geological constraints

are a necessary factor to describe real-world

minerals and fossil fuels. All such resources are

subject to geological constraints.

3.3. Towards an alternative basic Hotelling

model?

The constraints GC1 and GC2 are at the

foundation of the technical understanding of

what we see today as bell-shaped supply

trajectories insofar as they explain the specific

dynamics of extraction in relation to reserve

degradation and to uncertainty about reserve

size. We know that Hotelling was aware of

C. E. Van Orstrand’s 1925 article “On the

Empirical Representation of Certain Production

Curves” because Hotelling refers to it in his own

1931 article (p. 164). In his contribution, Van

Orstrand pays particular attention to extractive

sectors and depicts their dynamics by drawing

three bell-shaped graphs (1925, pp. 26 and 32),

including two with a long tale, depending on

estimations. Hence, Hotelling knew of the

existence of bell-shaped representations of the

exploitation of exhaustible resources. Hotelling’s

description of extractive industries in his 1931

article corresponds to this kind of production

trajectory, as shown in the earlier quotation:

By considering the influence of fixed

investments and the cost of accelerating

production at the beginning, we may be led to

production curves which rise continuously

from zero to a maximum, and then fall more

slowly as exhaustion approaches. Certain

production curves of this type have been found

statistically to exist for whole industries of the

extractive type, such as petroleum production

(p. 164).

More intriguing is precisely what Hotelling

retained from his reading of Van Orstrand. In the

archival material, we found a short memo,

entitled “Mine Economics,” that contained a full

reference to Van Orstrand’s paper as well as a

handwritten graph, quickly drawn, representing a

bell-shaped curve with a long tale (figure 2).

Unlike Van Orstrand, who specified on the axes

of his graphs the dates and volumes of

production from real data on West Virginia,

Ohio and Pennsylvania oil fields and so on,

Hotelling sketches only the general slope,

without the original figures, merely noting

“production of petroleum” next to the vertical

axis and “t” (for time) next to the horizontal axis.

This is not anecdotal and suggests that his

vision of the bell-shaped curve was generic and

not contingent on this or that field. This type of

approach—to transform specific quantified cases

into generic stylized facts—reminds us of

Cournot, who was the first in the history of

economics (1838) to draw stylized curves from

specific cases, in particular for demand

functions. We know that Cournot played an

important role in Hotelling’s mathematical

economics training (Hotelling 1929, p. 51;

Darnell 1990, pp. 12 and 14), and we found

references to Cournot in drafts and notes from

the 1920s.

Hotelling’s ambition with the bell-

shaped curve therefore can be related to the

description of a stylized fact for mineral

resources, in Cournot’s tradition.

The fact that “fixed investments and the cost

of accelerating production at the beginning...

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

“Talk on ‘Mathematical Economics’ before Math. Club,

May 1928.” HHP, Box 26, Mathematical Economics. We

thank Marion Gaspard for drawing our attention to the

connection between Cournot and Hotelling.

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may...le[a]d [us]” to trajectories like the one

sketched in Hotelling’s hand-drawn graph

suggests he knew that the incorporation of q’ on

the net price function would change the model

towards a bell-shaped equilibrium. We were able

to reproduce it ourselves through a simulation

exercise, adjusting the linear net price function

in section 8 to p = a-bq-cq’-dq’’, where q is the

cumulative production at time t, assuming

Hotelling’s original parameters and d = 10.000

(see box 1 and figure 3).

Results indicate a

bell-shaped equilibrium for production q’ and an

U-shaped equilibrium for market prices

p = a-cq’.

In 1931, Hotelling did not have at his

disposal mathematical tools such as optimal

control theory (developed in the 1950s),

allowing him to propose an alternative model to

fit with this kind of observations. The hand-

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

We make g = 0 in order to avoid masking the U-shaped

price path in equilibrium by an increasing demand as a

function of time.

drawn graph, however, confirms his awareness

of bell-shaped supply trajectories for the case of

extractive industries, such as petroleum. His

stylized-fact drawing, supplemented by his

quotation, confirms that his basic model leading

to the Hotelling rule was not relevant for

concrete mineral resources. To the best of our

knowledge, the suggested model in section 12, as

described in box 1, has not been depicted in the

literature to date. While its complete

characterization might be insightful, it is beyond

the scope of the present paper.

4. Conclusion

Based on unprecedented archival inquiry, this

paper has reconstructed Hotelling’s project on

non-renewable resources in the 1920s leading to

the publication of “The Economics of

Exhaustible Resources” in 1931. We showed that

Hotelling actually worked along two lines of

research: a generic one on asset valuation, in the

line of his theory of depreciation, and a specific

one on concrete mineral natural resources,

requiring more empirical insights from the real

world. In contrast to what is often argued, his

source of information for implementing concrete

situations in his framework was not the

conservation movement but rather specific

experiences in US policy throughout the 1920s.

We found Hotelling carried out substantial

preparatory work to circumscribe the geological

conditions of mineral resource extraction. This is

hardly noticeable to the naked eye in the 1931

article. However, after consulting the archives,

we presented a new reading of the article where

the importance of geological constraints for the

treatment of minerals and fossil fuels is

significantly increased. After considering the

implication of their presence, Hotelling became

aware of, notably, bell-shaped trajectories linked

to U-shaped price trajectories.

In addition to presenting these novel findings

in the history of economic thought, our

investigation allowed us to derive theoretical

implications. After nearly 90 years of

developments since Hotelling’s 1931 article, we

still attribute the title “Hotelling rule” to his

basic result p

= p

• e

r•t

and consider it to be the

starting-point principle of non-renewable

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resource economics. This seems problematic

given the rule lacks empirical validity and

requires strong amendments in order to describe

the long-term, aggregate behaviour of its target

object. Our reconstruction of Hotelling’s

research project sheds light on this theoretical

matter.

In particular, we established that Hotelling

never saw his basic result p

= p

• e

r•t

as a rule

for non-renewable resource economics. Instead,

he stated his basic result referred to a

hypothetical stylized asset marked exclusively

by finiteness, which is not the case of mineral

resources. For this category, elaborating on his

preparatory work in geology and engineering,

Hotelling concluded it was necessary to consider

as fundamental factors at least the influences of

cumulative production x(t), because costs

increase as mines go deeper (GC1), and the rate

of change in production q’(t), because strong

variations in output are costlier than small ones

(GC2).

By sketching the implied model, we were

able to confirm it is possible to obtain bell-

shaped trajectories for supplies and U-shaped

ones for prices, as described by Hotelling, at

least for given values of the parameters. Future

research should investigate the formal

mathematical demonstration of this result. If

confirmed, two inherent geological constraints

(GC1 and GC2), in addition to exhaustibility,

would be sufficient to reflect the long-term

equilibrium of mineral resources. Hotelling’s

1931 contribution would thus remain the starting

point of non-renewable resource economics,

albeit in a form different from that advocated to

date.

Acknowledgements

This paper benefited from the support of the

European Society for the History of Economic

Thought (ESHET), through the project

Bifurcations in Natural Resource Economics

(1920s-1930s). We thank Marion Gaspard,

Marco P. Vianna Franco, Thomas M. Mueller

and Franck Nadaud for stimulating discussions

in the working-group. Thanks also to the

participants in the Facts in Environmental and

Energy Economics, Models and Practices, Past

and Present workshop (Paris, October 2018) for

feedback. We are grateful to the staff of the Rare

Book and Manuscript Library of Columbia

University for their help during our visit to

Hotelling papers archival collection, to the

Hotelling familiy for encouraging our work and

to three anonymous referees and the editor of the

journal, who substantially helped us to improve

the paper. The views, thoughts and opinions

expressed in the text belong solely to the authors

and not necessarily to their employers,

organizations, committees or other groups to

which they are affiliated. Any errors or

omissions are ours alone.

Supporting information

Supporting information is available in the online

version of this article (publisher’s website).

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