GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.91

On the Theory of Exhaustible Resources: Ricardo vs. Hotelling

Heinz D. Kurz and Neri Salvadori

October 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanOn the Theory of Exhaustible Resources:
Ricardo vs. Hotelling*

By
Heinz D. Kurz and Neri Salvadori1

1. Introduction
Modern contributions to the economics of exhaustible natural resources, such as oil or coal,
generally start from one form or another of the famous ‘Hotelling Rule’, first put forward by
Harold Hotelling (1931). The Hotelling rule is an application of the concept of a competitive(uniform) rate profits to all processes in the economy, whether these are conservation or
production processes. In the classical economists this rule is not yet to be found. Does thismean that their analyses are of necessity defective, incomplete or inferior? Or does it only
mean that their argument relates to a world characterised by conditions that are different fromthose contemplated by the Hotelling Rule? Or is the rule implicit in their analyses and what is
missing is only an explicit reference to royalties as something different from profits?
The paper answers these questions. As regards the classical economists we will focus
attention on David Ricardo, the most ‘classical’ of all classical authors, and deal with
Adam Smith only in passing. Takashi Negishi in the introduction to a collection of*Paper given at a seminar on 11 September 2009 at ISER (Institute of Social andEconomic Research) of Osaka University by Heinz D. Kurz. The paper is a slightly
revised version of Kurz and Salvadori (2009); all additions and changes are my
responsibility. I should like to thank the participants at the seminar and especially
Yoshiyaso Ono for most valuable discussions and useful comments.

1

Neri Salvadori thanks Francesco Chioni for the discussions they had while Chioni
worked on his Laurea Thesis under Salvadori’s supervision.
1

essays devoted to the history of economic thought stressed that ‘it is necessary to study
theories that are regarded as past ones from the point of view of other research
programmes.’ (Negishi, 1994, p. xi) Alternatively, one might study theories that are
regarded as incorporating the most recent vintages of economic knowledge from thepoint of view of earlier approaches to the problem at hand. Comparing old and new canbe expected to shed new light on both and improve our understanding on what is truly
novel, what is only an old result in a new garb and on what has been lost sight of in thecourse of time. As Negishi put it succinctly in another contribution: ‘The history of ourscience should be used as a mirror in which the current theory reflects the knowledge of
how it failed to succeed in the past. To learn from past theories does not impede theprogress of our science. Progress often means, however, sacrificing something old. To
make sure that we are going in the right direction, it is always necessary to see whether
we have sacrificed something in error.’ (Negishi, 1992, p. 228)
The Hotelling Rule, as it is typically presented, concerns the fact that the prices ofresouces in situ need to increase over time at a rate that is equal to the competitive rateof profits. This fact seems in turn to imply another fact, namely, that all prices need tochange over time. As mentioned in the above, the first fact follows from the requirementthat the conservation of a resource is an economic activity which ought to yield to the
proprietors of deposits of the resource the same rate of profits as it is obtained from any
productive activity. The second fact is not immediately obvious.
The contribution of this paper may thus be summarized in the following way. While it iscommonly thought that Ricardo’s treatment of exhaustible resources is to be found firstand foremost in the barely three pages of Chapter III of the Principles, ‘On the Rent ofMines’, this is actually not so. This chapter is in fact confined to a discussion of the rentsof differently fertile mines in complete analogy to the rents of differently fertile lands.
These rents arise because the exploitation of mines is typically subject to capacity
constraints which imply that mines with different costs of extraction have to be
operated at the same time. Ricardo develops an analysis of exhaustible resources rather
in the context of a discussion of the difference between rent and profits. He begins this
discussion in Chapter II, ‘On Rent’, in which he also criticizes Smith’s cavalier andconfusing use of the two concepts. Then, in Chapter XXIV, ‘Doctrine of Adam Smith
concerning the Rent of Land’, he elaborates on his criticism of Smith’s doctrine. He
illustrates the fecundity of his own, Ricardo’s, rigorous conceptualizations of these two
2

important analytical categories in political economy and shows that Smith’s analysis isbound to end in a muddle. As usual, Ricardo is ‘desirous only to elucidate theprinciple’ at work (Works, vol. I, p. 121), as he stresses in another context, and therefore
bases his argument on strong assumptions. These assumptions, which we will explicate
below, imply that the exhaustion of each and every deposit of an exhaustible resource
will nevertheless leave the prices of all produced commodities unaffected over time. In
this way Ricardo manages to isolate a particular phenomenon at hand and put it into
sharp relief. In the context we are interested in, this refers to the distinction between
differential rent and profits, where what Ricardo calls profits comprises what we
nowadays call royalties. Hence royalties are there in Ricardo’s analysis, but they are noteasily identifiable as such.
The composition of the paper is the following. In Section 2 we specify two fundamental
assumptions required in order to be able to establish the fact that all prices need tochange over time because of the Hotelling Rule. We then confront these assumptions
with alternative ones which, it will be argued, are characteristic of the analyses of Smith
and Ricardo. Section 3 provides some evidence in support of this proposition in terms of
passages taken from Ricardo’s Principles. The main differences between the world inwhich all prices need to change over time because of the Hotelling Rule and the worldabout which Ricardo wrote are the following: (i) While Smith and Ricardo were awareof the exhaustibility of each and every deposit of a resource, they did not yet
contemplate the case of the exhaustibility of the resource as a whole. (ii) Ricardo
assumed that in order to meet the effectual demand for a resource, several deposits
typically have to be worked simultaneously, because with regard to each deposit there isa capacity constraint that limits the time rate of raising the resource. Section 4 provides amathematical formulation of the Ricardian point of view which allows one to compare
the latter with the one underlying the Hotelling Rule. It concludes that Ricardo may
have well come up with the modern interpretation of the Hotelling Rule had heconsidered the case of the exhaustion of a resource in its entirety as a realistic
possibility, which apparently he did not. Hence, a modern formulation of Ricardo’s view
may complement the view expressed by the Hotelling Rule and thus render the overall3argument about exhaustible resources more complete. Section 5 contains someconcluding remarks.2

2. Different assumptions – different worlds
The modern interpretation of the Hotelling Rule presupposes that the following two
assumptions hold:(H1)

The resource is available in homogeneous quality and in an overall quantity thatis limited and that at any moment of time is known with certainty.

(H2)

The amount of the resource that can be extracted in a given period of time, ayear, for example, is only constrained by the amount of it left over from thepreceding period.

(H stands, of course, for Hotelling.) In case one of these assumptions, or both, are notmet, the Hotelling Rule has to be modified according to circumstances. It portrays a boldcase of a resource whose exhaustion is actually foreseeable with certainty. The Rule

2

Aiko Ikeo has drawn our attention to the interesting paper by Kemp and Long (1984),
who in the context of a discussion of the conventional 2x2 Heckscher-Ohlin trade model
replaced the usual assumption of two non-depletable original factors of production(‘Ricardo’s indestructible powers of the soil’) and allowed instead for one or two
exhaustible resources (‘Hotelling's destructible power of the soil’). They thus also
combined ideas of Ricardo and Hotelling. However, their overall set-up differs
markedly from the present one. In particular, they adopted a partial equilibrium
framework by taking relative world market prices of final goods as given to the small
open economy. In the case of exhaustible resources it is assumed that their exhaustion
affects only production conditions in the economy under consideration, but not world
market prices. They were also concerned only with homogeneous factors of production
and thus not with extensive differential rent. The problem of whether Hotelling’s
fundamental idea is somewhere hidden in Ricardo’s argument is not touched upon bythem.
4does not rigidly fit (m)any cases in the real world.3 Yet it expresses an importantprinciple at work that contributes to our understanding of what is going on in the latter.
It can be objected that despite the fact that today we have a much clearer idea of what isstill there of certain resources at a given moment of time and are possessed of much
improved techniques to discover hitherto unknown deposits of resources, assumption(H1) is typically not met with regard to any single exhaustible resource. It is also not
clear whether knowing precisely what is still there would mean much, because technicalprogress typically affects the economic importance of a resource. The discovery of new
ways to use known substances as well as the discovery of the useful properties ofhitherto unused substances may lead to substitution processes and in the extreme replacesome given resource entirely by new ones. Also assumption (H2) is never strictly met.
Typically, there are capacity constraints that limit the time rate of exploiting a deposit.
These constraints are very often binding with regard to any single deposit of theresource, so that many deposits have to be exploited simultaneously in order to meeteffectual demand.
We might go to the opposite extreme and postulate instead of assumptions (H1) and(H2) the following:(R1)

For each exhausted deposit of the resource another one with exactly the samecharacteristics is discovered and the cost of the search, in terms of labour andcommodities, is always the same.

(R2)

The working of each deposit is subject to a capacity constraint that limits theamount of the resource that can be extracted in a given period of time.

3

This is confirmed by Krautkraemer’s survey article (Krautkraemer, 1998). He
maintains, among other things: ‘For the most part, the implications of this basic
Hotelling model have not been consistent with empirical studies of nonrenewable
resource prices and in situ values’ (p. 2066). ‘Other factors have overshadowed finite
availability of the resource as determinants of the observed dynamic behavior ofnonrenewable resource prices and in situ values’ (p. 2087). And: ‘It does seem to be arecurring tendency to overestimate the imminence of nonrenewable resource
exhaustion’ (p. 2103; emphasis added). Vis-à-vis the evidence provided by
Krautkraemer the classical approach to exhaustible resources could be said to farebetter.
5

(R stands, of course, for Ricardo.) In case assumptions (R1) and (R2) replace assumptions(H1) and (H2) we are in a world that is much closer to that of the classical economists. Its
properties are obviously different from those invoked in modern interpretations of theHotelling Rule and therefore it should come as no surprise that Ricardo came up with a viewwhich at first sight sits uncomfortably with modern interpretations of Hotelling’s analysis.
However, the reason is not that one of the analyses is right and the other wrong, but that they
cover vastly different cases.
It goes without saying that there are intermediate cases beteen the two extreme ones: (H1)
may be combined with (R2) or (H2) with (R1). Many additional cases could be studied whichtake into account, for example, that the discovery costs of new deposits are not constant orthat the capacity constraint may depend on the amount of the resource that is still in situ.
We shall refrain from elaborating a richer typology of cases followed by a comparative
investigation of them all. We focus attention rather on the case that was most probably at theback of Ricardo’s mind. For this purpose we discuss, in the following section, what Ricardo
wrote about exhaustible resources, the distinction between profits and rent and his criticism ofAdam Smith’s views on the matter.

3. Ricardo on exhaustible resources
In the Principles Ricardo defines rent rigorously in the following way:
Rent is that portion of the produce of the earth, which is paid to the landlord for theuse of the original and indestructible powers of the soil. (Works, vol. I, p. 67;
emphasis added )He continues:
It is often, however, confounded with the interest and profit of capital, and, in popular
language, the term is applied to whatever is annually paid by a farmer to his landlord.
If, of two adjoining farms of the same extent, and of the same natural fertility, one had
all the conveniences of farming buildings, and, besides, were properly drained andmanured, and advantageously divided by hedges, fences and walls, while the other hadnone of these advantages, more remuneration would naturally be paid for the use ofone, than for the use of the other; yet in both cases this remuneration would be called
rent. But it is evident, that a portion only of the money annually to be paid for the6

improved farm, would be given for the original and indestructible powers of the soil;
the other portion would be paid for the use of the capital which had been employed inameliorating the quality of the land, and in erecting such buildings as were necessary
to secure and preserve the produce. (Ibid.)
Adam Smith, Ricardo goes on to argue, did not stick to a rigorously defined concept whenusing the word rent. In Part II of Chapter XI of Book I of The Wealth of Nations, ‘Of the
Produce of Land which sometimes does, and sometimes does not, afford Rent’, Smith gives
an example of the timber business, timber clearly being a reproducible resource, in which heconfounds the concepts of profits and rent (see WN I.xi.c.5):
He [Smith] tells us, that the demand for timber, and its consequent high price, in the
more southern countries of Europe, caused a rent to be paid for forests in Norway,
which could before afford no rent. Is it not, however, evident, that the person who paid
what he thus calls rent, paid it in consideration of the valuable commodity which wasthen standing on the land, and that he actually repaid himself with a profit, by the saleof the timber? If, indeed, after the timber was removed, any compensation were paid
to the landlord for the use of the land, for the purpose of growing timber or any otherproduce, with a view to future demand, such compensation might justly be called rent,
because it would be paid for the productive powers of the land; but in the case statedby Adam Smith, the compensation was paid for the liberty of removing and selling thetimber, and not for the liberty of growing it (p. 68; emphasis added).
Ricardo’s criticism extends to Smith’s dicussion of coal mines and stone quarries:
He [Smith] speaks also of the rent of coal mines, and of stone quarries, to which thesame observation applies—that the compensation given for the mine or quarry, is paid
for the value of the coal or stone which can be removed from them, and has noconnection with the original and indestructible powers of the land. (Ibid.)
In Ricardo’s view the distinction between profits and rent is crucial, because as capital
accumulates and the population grows the two component parts of the social surplus are
typically affected differently:

7

This is a distinction of great importance, in an enquiry concerning rent and profits; for
it is found, that the laws which regulate the progress of rent, are widely different fromthose which regulate the progress of profits, and seldom operate in the same direction.
In all improved countries, that which is annually paid to the landlord, partaking ofboth characters, rent and profit, is sometimes kept stationary by the effects of opposing
causes; at other times advances or recedes, as one or the other of these causes
preponderates. In the future pages of this work, then, whenever I speak of the rent ofland, I wish to be understood as speaking of that compensation, which is paid to theowner of land for the use of its original and indestructible powers. (Ibid., pp. 68-9;
emphasis added)
Hence what Smith called ‘rent’ of coal mines or stone quarries is to Ricardo profits and not
rent. But does Ricardo not contradict himself by giving Chapter 3 of the Principles the title‘On the Rent of Mines’? Scrutiny shows that this is not so. Chapter 3 is actually devoted to
the rent of mines precisely in the sense Ricardo intended. The problem is the following: Why
are mines possessed of different ‘fertilities’ operated simultaneously? Why is not the most‘fertile’ mine exploited in full first, followed by the second fertile mine, and so on? The
answer is straightforward: Several mines have to be worked at the same time because each
one is typically subject to a capacity constraint that limits the amount of the coal or ore thatcan be extracted per unit of time. This constraint itself is seen to depend typically also on theamount already extracted. Effectual demand cannot be satisfied in the given circumstances by
operating exclusively the most ‘fertile’ mine, because the required rate of output in order tomeet effectual demand cannot be generated in this way. The amount of the resource ‘which
can be removed’ (ibid., p. 68) will generally fall short of the resource in situ at the beginningof the extraction period. The same argument applies in the case in which there are several
equally fertile minds. Yet, ‘If there were abundance of equally fertile mines, which any one
might appropriate, they could yield no rent; the value of their produce would depend on thequantity of labour necessary to extract the metal from the mine and bring it to market’ (ibid.,
p. 85). This is generally not the case and differently fertile mines will have to be wrought
simultaneously. The situation may change due to innovations, as Ricardo emphasizes withregard to coal: ‘by new processes the quantity should be increased, the price would fall, andsome mines would be abandoned’ (ibid., p. 331).

8

The absence of an abundance of equally fertile mines and the presence of a capacity
constraint limiting the yearly output of any single mine in general necessitate the utilization ofmines of different fertility in order to meet the effectual demand for the resource. In suchcircumstances, Ricardo stresses, it is the ‘relative fertility of mines [which] determines the
portion of their produce, which shall be paid for the rent of mines’ (ibi., p. 330). Ricardo
concludes that ‘the whole principle of rent is here … as applicable to land as it is to mines’(ibid., p. 330). When mines of different fertilities need to be wrought simultaneously, thenthis makes room for the emergence of (extensive) rents, exactly as in the case of the
agricultural cultivation of land. This is rent in the true sense of the word and has nothingwhatsoever to do with what nowadays we call ‘royalties’. What we call ‘royalties’, Ricardo
actually calls ‘profits’.
Ricardo’s use of the concept of profits for ‘the compensation ... paid for the liberty ofremoving and selling the timber’ is not surprising: timber can be sown and grown again, it isclearly not an exhaustible resource, but a reproducible good, and to the extent to which it isused as a produced means of production it is capital. But the use of the word profits for thecompensation paid for the liberty of removing and selling coal or stones may be surprising:coal cannot be reproduced by men, neither can stones. However, new coal pits can always be
expected to be discovered and the cost of the search is equal to the value of the mine, a valuethat decreases with the amount of the resource that has been removed. In other words, Ricardo
did not need the word royalties since the minerals and ores etc. as such were not considered to
be fully exhaustible in the foreseeable future. Both in Ricardo and in Smith we encounter timeand again references to the finding of new deposits with no serious consideration given to thefact that such deposits, taken as a whole, are limited. This is the reason why Ricardo did notneed a new concept in order to be able to deal with the case under consideration. The conceptof profits was all that was required.
The fact that Ricardo did not elaborate what now is called the Hotelling Rule cannot therefore
be considered an expression of a failure and a lack of analytical profundity on his part. It
simply expresses a concern with a world in which the total exhaustion of certain resources
was not yet considered a possibility worth studying.
What we now call royalties are a sub-category of profits. Profits are proportional to the valueof capital invested or possessed, and in conditions of free competition the rate of profits9

obtained in oder to conserve the mineral in the ground has to be equal to the rate of profits
obtained from any other production or conservation process.
If assumption (R1) held true, while each deposit would be exhaustible, the resource as such
would not; and each deposit could in fact be treated as if it were a (reproducible) machine: theprice of the new machine equals the cost of the search and the price of an old machine of age tequals the value of the deposit after t periods of utilization (see Kurz and Salvadori, 1995, pp.
359-60). The price of the resource in situ would change as predicted by the Hotelling Rule,but the price of the extracted mineral would be constant over time.4 In the next section weassume that (H1) and (R2) apply. We will show that also in this case the changes of the pricesof the resources in situ may not need the introduction of intertemporal equilibria. But themodel elaborated is more general and is of some interest in itself. It consists essentially of amodified version of a model we put forward in Kurz and Salvadori (2000). The novelty in the
new formalization compared with the original one is to be seen first and foremost in the
introduction of capacity constraints with respect to the exploitation of each single deposit of a
resource.

3. A formalization
The formalization suggested here is based on the following simplifying assumptions. A finite
number n of different commodities, which are fully divisible, are produced in the economy
and a finite number m (> n) of constant returns to scale processes are known to produce them.
Let pt be the vector of prices of commodities available at time t ! ! 0 and let xt be the vectorof the intensities of operation of processes at time t ! ! . A process or method of production
is defined by a quadruplet (a, b, c, l), where a ! ! n is the commodity input vector, b ! ! n is
the output vector, c ! ! s is the exhaustible resources input vector, and l is the labour input, ascalar; of course a ≥ 0, b ≥ 0, c ≥,_ 0, l ≥ 0. The production period is uniform across all

4

Adam Smith wrote about the discovery of new mines: ‘In this search [for new mines]
there seem to be no certain limits either to the possible success, or to the possible
disappointment of human industry. In the course of a century or two, it is possible thatnew mines may be discovered more fertile than any that have ever yet been known; andit is just equally possible that the most fertile mine then known may be more barren than
any that was wrought before the discovery of the mines of America’ (WN I.xi.m.21).
10

processes. It is important to remark that the inputs referred to in vector c are inputs of theresources as they are provided by nature; for example, extracted oil is not contained in c, but
in b, if (a, b, c, l) is an extraction process, or in a, if (a, b, c, l) is a process that uses it, unlessthe extraction costs are nil. The m existing processes are defined by quadruplets(aj, bj, cj, lj). j = 1, 2, ... , m
Then define matrices A, B, C and (now) vector l as follows:5" a1T %
" b1T %
" c1T %
"l1T %
$ T'
$ T'
$ T'
$ T'
$a2 '
$b 2 '
$c2 '
$l2 '
$ ! '
$ ! '
$!'
$!'
A = $ ', B = $ ', C = $ ', l = $ ' .
$ ! '
$ ! '
$!'
$!'
$ ! '
$ ! '
$!'
$!'
$ T'
$ T'
$ T'
$ T'
#a m &#b m &#c m &#lm &Assume that the annual consumption of commodities by profit (and royalty) recipients is
proportional to a vector d, which, for simplicity, is assumed to be given and constant overtime, that is, independent of prices and quantities, including the quantities of the exhaustible
resources left over at the end of each production period. In addition, the real wage rate,
defined by a commodity vector w, is taken to be given and constant over time. y is the vectorof royalties earned with respect to the various natural resources; q is the vector of rents
obtained in exploiting different deposits of them; z is the vector of the amounts of resources
available. Technical innovations of any kind are set aside. All exhaustible resources are
private property. In conditions of free competition there will be a (tendency towards a)
uniform nominal rate of profits rt across all production activities in the economy. This implies
that, for each time t ! ! 0 , the following inequalities and equations are to be satisfied:
Bp t +1 ! (1+ rt ) ( Ap t + Cy t + Cqt ) + lwT p t +1

(1)

xTt +1Bp t +1 = xTt +1 !"(1+ rt ) ( Ap t + Cy t + Cqt ) + lwT p t +1 #$(2)

y t +1 ! (1+ rt ) y t(3)z Tt +1y t +1 = (1 + rt ) z Tt +1y t

(4)

(

)

xTt +1 B ! lwT " xTt +2 A + # dT

5

(5)

Transposition of a vector or a matrix is denoted by superscript T.
11

(

)

(

)

xTt +1 B ! lwT p t +1 = xTt + 2 A + " dT p t +1

(6)

z Tt ! xTt +1C + z Tt +1

(7)

(

)

z Tt y t = xTt +1C + z Tt +1 y t(8)z Tt ! z Tt +1 + t T

(9)

(

)

z Tt qt = z Tt +1 + t T qt

(10)

! > 0, p t " 0, y t " 0, q " 0, z t " 0, x t +1 " 0 .

(11)Inequality (1) means that nobody can get extra profits by producing commodities available attime t + 1. Equation (2) implies, because of inequalities (1) and (11), that commodities
available at time t + 1 will only be produced if the ruling nominal rate of interest is obtained.
Inequality (3) means that nobody can get extra profits by storing exhaustible resources fromtime t to time t + 1. Equation (4) implies, because of inequalities (3) and (11), that exhaustible
resources will be stored from time t to time t + 1 only if the ruling nominal rate of interest will
be obtained by this storage activity. Inequality (5) implies that the amounts of commodities
produced are not smaller than the amounts of commodities required, and equation (6) implies
that if an amount is larger, then the price of that commodity is zero. Inequality (7) implies thatthe amounts of exhaustible resources available at time t are not smaller than the amounts of
exhaustible resources available at time t + 1 plus the amounts of exhaustible resources utilized
to produce commodities available at time t + 1, and equation (8) implies that if an amount islarger, then the price of that exhaustible resource is zero. Inequality (9) implies that at eachtime t extraction of resource j cannot be larger than t T e j , and equation (10) implies that if it issmaller, then the rent obtained by the owner of the deposit of resource j is zero. The meaningof inequalities (11) is obvious.
The difference with a world in which there are no capacity constraints in the extraction of
resources is close at hand: the elements of vector t are so high that inequality (9) is always
satisfied as a strict inequality, then equation (10) implies that qt = 0 : in this case the model
collapses to that analyzed in Kurz and Salvadori (2000).
The following observations are perhaps apposite. First, as the system gradually uses up its
given stocks of exhaustible resources, moving from deposits that are less costly to operate tomore costly ones, the overall (i.e. direct and indirect) amount needed of any such resource to12

produce one unit of the various commodities and indeed also to extract one additional unit of
the resource itself may, and generally will, go up. Therefore, with a given net output vector
the total amounts of the exhaustible resources extracted per period will increase over time: As
the remaining stocks of the resources get smaller, the quantities used up get larger, at least forsome time. Second, due to the decrease in the economic system’s overall productivity,
reflecting diminishing returns in the extraction industries, the rate of profits can be expected
to fall over time. This mimicks Ricardo’s result in his theory of ground rent. Third, there is no
reason to presume that all resources will be fully exhausted. Since costs of extraction can be
expected to rise, a point may come where it is no longer advisable to exploit deposits. (Thatthis may very well be the case can be seen with reference to the extreme case in which moreof a resource would be needed, directly and indirectly, than is being produced.) Fourth,without any technical progress or some deus ex machina, our economy would be doomed toextinction, at least in the long (or very long) run. This brings us to a discussion of the role of a‘backstop technology’ in the economic system under consideration, whose role is preciselythat of a deus ex machina or saviour of the world.
Despite the changes introduced in the above model, the procedure to prove the existence of asolution to the model of 2000 in the presence of a backstop technology can be applied also
here. More precisely, let the processes ( A, B, 0, l ) be obtained from (A, B, C, l) by deleting all()

the processes using directly some natural resource (i.e., process eTi A, eTi B, eTi C, eTi l is in theset of processes ( A, B, 0, l ) if and only if eTi C = 0 ) and let us assume that there is a scalar r*
and there are vectors x* and p* which solve the system()
x ( B ! A ! lw ) p = d p
xT B ! A ! lwT " dT
T

TT

Bp ! (1+ r *) Ap " lwT p

xT Bp ! xT #$(1+ r *) Ap " lwT p%&
x ! 0, p ! 0, dT p = 1 .
Let us assume, further, that the processes corresponding to positive elements of vector x*,(Â, B̂, 0, l̂ ) , are exactly n and that in the absence of exhaustible resources the economy would
converge to these processes. Then the procedure used by Kurz and Salvadori (2000) is able to13

construct a solution also for system (1)-(11) despite the introduction of capacity contraints on
the extraction of resources. Such a proof follows exactly the same lines and we will notprovide it here.
What we want to stress here, on the contrary, is that the introduction of capacity constraints
on the extraction of resources may introduce a further reason in support of the view that
whereas the prices of resources in situ are bound to change at a rate equal to the rate ofprofits, all other prices will remain constant or will at any rate not follow the path of theformer. We do not wish to assert that this fact will in fact happen, but that it may happen. Thisis so because the owners of deposits of resources get not only royalties, but also rents. The
sum of royalties and rents for a given deposit may be constant even if royalties are changing,since rents are changing in equal amounts but in the opposite direction. To see this, consider
an economy where capacity constraints are so high that production of consumption d requires()

the operation of the backstop technology and therefore processes Â, B̂, 0, l̂ are activated. As
a consequence, vector p is determined and it may also be constant over time (it is so in thelong run). This does not mean that production goes on in the same way year after year. Not at
all: production changes potentially every year since the use of the resources reduces theiravailability. But the reduction of the availability of resources reduces also the rents and since
the price of the extracted mineral is constant over time, the rent is reduced in exactly the sameamount in which royalties are increased because of Hotelling’s Rule.

5. Concluding remarks
The world to which applies the Hotelling Rule in its modern interpretation and the world towhich applies the classical, especially Ricardian analysis are rather different. While theHotelling Rule presupposes that a scarce natural resource is available in a known quantity andits extraction is not subject to any capacity constraints, Ricardo’s treatment of exhaustible
deposits does not contemplate the case of the exhaustion of the resource as a whole andallows for capacity constraints that limit extraction per unit of time with respect to eachdeposit actually known at a given moment of time. Both types of analyses are valuable andimprove our understanding of the properties of economic systems that make use of wasting
assets. It is then argued in terms of a model with exhaustible resources that incorporates what14

we think are the premisses from which Ricardo begins his reasoning, that the Hotelling Rulecan be considered to be implied by it: the Rule concerns the resources in situ and requests
their prices to change at a rate that is equal to the competitive rate of profits. However, thesechanges need not affect the prices of the other commodities, including the prices of theresources that are actually extracted. Hence, in Ricardo’s argument the Hotelling Rule may besaid to be effective, but its effects may be limited to changing prices of the conserved amountsof exhaustible resources only. The owners of deposits will obtain both royalties and rents, inthe course of time rents will fall and royalties rise, and the sum of both may remain constant.
If this condition is met, then these price changes will not affect any other prices in theeconomic system. Ricardo’s argument may be said to implicitly correspond to this case.

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Negishi, T. (1994). The Collected Essays of Takashi Negishi, Volume II: The History of
Economics, Aldershot: Edward Elgar.
Ricardo D. (1951): On the Principles of Political Economy and Taxation, first edn 1817, in
The Works and Correspondence of David Ricardo, edited by Piero Sraffa with thecollaboration of Maurice H. Dobb, Vol. I, Cambridge: Cambridge University Press.
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number, paragraph number.

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