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Facts in Energy and Environmental Economics

Exhaustible Resources and Classical Theory

Ressources épuisables et théorie classique
Christian Bidard et Guido Erreygers
p. 419-446


Smith, Ricardo, Marx et Sraffa ne faisaient pas de distinction théorique entre ressources épuisables e tterres.Néanmoins, la notion d’épuisement peut être opposée à celle des “pouvoirs indestructibles de la terre” (Ricardo)etappelleuneanalysespécifiquedistinctedecelledelarente.La diversité des tentatives contemporaines de traiter cette question dans un cadre classique témoigne d’une grande variété dans l’interprétation des caractéristiques méthodologiques de la théorie classique. Trois points cruciaux apparaissent : d’abord, le traitement des prix, qui sont invariants dans la théorie classique mais qui, selon la règle d’Hotelling, varient dans le temps pour les ressources épuisables; ensuite, la notion et la mesure du taux de profit; enfin, la relation entre l’analyse économique et une approche plus historique et sociologique soulignant le rapportdeforcesentrelesclasses.Lepointdedépartdenotreapproche est un modèle simple, dit modèle blé-guano, où le guano est l’unique ressource épuisable. Nous étudions la dynamique de cette économie, tant pour les prix que pour les quantités. Les leçons de cette étude servent de base à une extension aux modèles multisectoriels. Enfin, nous proposons un examen critique de quelques approches alternatives développées par des auteurs sraffiens.

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1When classical economists discussed natural resources, they mostly thought of land. Although they did occasionally refer to what we now call exhaustible resources, e.g., when dealing with mines, they did not perceive these as sufficiently different from land to deserve specific theoretical attention. A typical example is David Ricardo, who defined rent as “that portion of the produce of the earth, which is paid to the landlord for the original and indestructible powers of the soil” (Ricardo, [1817] 1951, 67). He criticised Adam Smith’s inconsistent views on rent (ibid., chapter XXIV) but argued that the principle of differential rent which he applied to the case of lands was “precisely the same” (ibid., 85) for the case of mines. In the same vein, Marx ([1867] 1962) and later Sraffa (1960) paid little attention to exhaustible resources in their economic theories. Sraffa, for instance, referred to “[n]atural resources which are used in production, such as lands and mineral deposits, and which being in short supply enable their owners to obtain a rent” (1960, 74), seemingly ignoring the theoretical specificity of exhaustible resources which are not produced but, unlike land, are destructible in the long run.

2For a long time, therefore, those working in the tradition of classical political economy acknowledged the existence of exhaustible natural resources, but considered them to be insufficiently distinct from land to merit special attention. This changed in the 1980s. The first to squarely address the question of the integration of exhaustible resources into classical theory was Sergio Parrinello (1983). His aim was to examine the compatibility of Sraffa’s formalisation of classical theory and the Hotelling rule with regard to exhaustible resources, which is a result associated with neoclassical theory. The integration of exhaustible resources is an important methodological question: since the Hotelling rule obliges us to go beyond the Sraffian framework, one must identify the main characteristics of classical theory in order to preserve them in an enlarged approach. In that respect, we retain three main features of classical theory:

  1. The primacy given to production. Classical theory ignores the notion ofutility, and it is often assumed that final demand is given independently of prices.

  2. The theory of distribution is linked with a vision of a capitalist societydivided into classes: capitalists own the means of production and receive profits, workers supply labour and receive wages, and landlords make their lands available for production and receive rents. If landlords are ignored for simplicity, the distribution of the net product between these two classes results from a balance of power. While Ricardo and Marx took the level of the wage as a historical datum, Sraffa chose the rate of profits as the exogeneous distribution variable.

  3. A long-term view of the economy. The Classicals did not ignore theinfluence of demand on prices, but considered that the relevant economic question is that of the determination of long-run prices, once production is adapted to the “normal” level of demand. The price of a commodity is then explained by its difficulty of production. The most well-known form of that conception is the labour theory of value, modified by Sraffa into a theory of prices of production, in which relative prices and wages are constant from period to period and which takes into account the influence of distribution on prices. Departing from Sraffa himself, most Sraffians have interpreted the notion of long run in a restrictive sense and identified it with that of a steady state or of a regular growth path.

3Taking into account exhaustible resources leads us beyond the traditional Sraffian framework for two reasons: from a physical standpoint, the intertemporal path cannot be the same before and after exhaustion of a resource and, as far as prices are concerned, the Hotelling rule entails an intertemporal change in the price of the resource, and therefore a change in the prices of all commodities directly or indirectly produced by it. As a result, the dynamics of quantities and prices are more complex than those considered by Sraffa and his followers. The innovative (or is it explosive?) nature of Parrinello’s question becomes apparent now: how can Classical theory be adapted to the study of exhaustible resources?

4Following Parrinello’s pioneering article, several authors inspired by Sraffa’s ideas have attempted to come to grips with the issue of exhaustible resources. A first wave of contributions was made by Neri Salvadori (1987), Bertram Schefold (1989) and Heinz Kurz and Neri Salvadori (1995; 1997;

  • 1 Kurz and Salvadori (1995, 366-368) however studied an energy-oil model close to the cornguano model (...)

52000). In 2001 a peak was reached when Metroeconomica published a “Symposium on exhaustible natural resources and Sraffian analysis”, edited and introduced by Ian Steedman. This included papers by Christian Bidard and Guido Erreygers (2001a; 2001b), Eiji Hosoda (2001), Heinz Kurz and Neri Salvadori (2001), Christian Lager (2001), Sergio Parrinello (2001) and Bertram Schefold (2001). Since then, Marco Piccioni and Fabio Ravagnani (2002), Heinz Kurz and Neri Salvadori (2002; 2009; 2011; 2015), Sergio Parrinello (2004), Fabio Ravagnani (2008) and Biao Huang (2018) have further refined the analysis. We will return to some of these contributions later in the paper. The 2001 symposium found its origins in the debate sparked by the corn-guano model, a pedagogical device which we developed in order to deal with the issue of exhaustible resources in a simple and transparent way. We remain convinced that a simple economic model can be useful as a first step in the process of analysing a complex problem and, in this respect, our approach differs from the one followed by all other authors.1 Of course, further steps are required to verify which properties remain valid in a more general framework (see also Bidard and Erreygers, 2020).

6We begin by explaining the background and properties of the cornguano model (section 1). Next, we discuss what in our view constitute major obstacles for the generalisation of the model (section 2). We then develop a few ideas for the multisector version of the model (section 3). We also provide a critical account of alternative answers given by other economists of classical inspiration, as the point at stake behind a seemingly rather technical question is the very understanding of classical theory (section 4). We end with a few concluding remarks.

1 The Corn-Guano Model

7Corn models have been used frequently by authors of classical inspiration (Skourtos, 1991). Sraffa famously attributed a “corn-ratio” theory of profits to Ricardo (Sraffa, 1951, xxxi-xxxiii), an interpretation which has been challenged. Whether or not Ricardo had such a simplified economic model in mind, we find it appropriate to start with a brief reminder of the corn model (see chapter 1 of Bidard, 2004, for a more extensive presentation). We then introduce exhaustible resources and the corn-guano model.

1.1 The Corn Model

8Let us assume an extremely simple economy in which there exists only one commodity, corn, which is produced by means of itself and labour. Let us furthermore assume that the same production process is used year in, year out, and that it can be described schematically as follows:

Image 100002010000028B0000002066B1C6806E284AA0.png

Two economic properties can be enunciated:
(C1) Maximum rate of profits: Even if labourers “could live of the air” (Marx, [1894] 1964, chapter 15, section II, 257), corn must be invested in order to produce corn, and therefore the rate of profits Image 100002010000000B00000019F0FF5EAE3B5B67D7.png is finite. Its maximum level Image 100002010000001300000019224324FE439D3FA8.png can be interpreted in physical terms:

Image 100002010000028B0000001CD6893BE9DD20ED62.png

(C2) Ricardian trade-off: In the interval [0,Image 100002010000001300000019224324FE439D3FA8.png ] the variations of the rate of profits are inversely related to those of the real wage.

9Formally, these properties follow from the price equation associated to the operation of process (1), which is written as:

Image 100002010000028B0000001D50980679D5293CCA.png

  • 2 Equation (3) is in line with the classical tradition according to which wages are advanced (we adop (...)

where Image 1000020100000010000000190C4BDEDEE6C32574.png stands for the price of corn and Image 10000201000000120000001985CCB4829F562D45.png for the wage.2 One way of studying this equation is to take corn as numeraire Image 100002010000004F000000193735361FCD011D79.png , in which case equation (3) determines the relation between the rate of profits and the real wage expressed in units of corn (i.e., Image 100002010000002B000000192A87070A5891013D.png ). Alternatively, we can choose the wage as numeraire Image 100002010000005E00000019976EC8D620F73BBC.png . We then obtain a relation between the rate of profits and the price of corn expressed in labour (i.e., Image 100002010000002C0000001931D4948695C31BB7.png , the inverse of the wage expressed in corn). Whatever the numeraire—a mixed corn/labour numeraire would yield exactly the same result—we always find the same relation between the “real rate of profits” and the “real wage” and the above-mentioned two properties hold.

10If several processes of type (1) are available, the operated process is the least expensive of all, given the rate of profits. We have the following choice-of-technique property:

(C3) Wage-maximisation: For a given rate of profits, the operated process coincides with the process that maximises the real wage.

1.2 The Corn-Guano Model (I)

11We now introduce exhaustible resources by transforming the corn model into the corn-guano model. This model is conceived as a methodological tool: its analytical simplicity allows us to shed light on the distinct economic features linked to the introduction of exhaustible resources. Its structure is rich enough to initiate the reader to the study of the dynamics of models with exhaustible resources. The dynamics of the more general models which we will explore in section 2 are certainly more complex, but some of the intricacies are already present here.

The study of exhaustible resources is inexorably linked to the “Hotelling rule”, a seminal result first derived by Harold Hotelling (1931). Basically, an exhaustible resource owner faces a choice between selling one unit of his resource at date t, or letting it lie idle and selling it at date Image 100002010000003200000019EEA0E525E092006F.png . In the first case the sale gives the owner an immediate revenue Image 100002010000002600000019DD530414CE463881.png , i.e. the price (or royalty) of one unit of the resource at time Image 100002010000000900000019D39B26010CAA4B4B.png , which can then be invested at the rate Image 100002010000000C00000019DEB7B9D6C180E05E.png . In the second case the owner simply waits and obtains revenue Image 10000201000000510000001930D0A3EF8C9C8FF3.png at time Image 100002010000003200000019EEA0E525E092006F.png . If one of the two options were more profitable than the other, all resource owners would have an incentive to follow the most profitable course of action, which would mean that either the whole supply of the resource would be exploited at time Image 100002010000000900000019D39B26010CAA4B4B.png , or that none of it would. But since in any period of time before exhaustion, part of the supply is exploited and part of it is conserved, the two options must be equally profitable in equilibrium. As the first option allows the owner to obtain a rate of return r, then so must the second. This implies that the royalty Image 100002010000002600000019DD530414CE463881.png must increase at the rate Image 100002010000000C00000019DEB7B9D6C180E05E.png , a result known as the Hotelling rule.

12In the corn-guano model, there is only one produced commodity, called corn, and one exhaustible resource, called guano. Corn can be produced in a one-period time either by means of the “guano method”:

Image 100002010000028B0000001F9EE1A6FADB9DB743.png

or by means of the “backstop method” (which will be necessarily used after the exhaustion of the stock of guano):

Image 100002010000028B0000001C1C4026AD1A99E23A.png

A unit of guano not used up to date Image 100002010000000900000019D39B26010CAA4B4B.png remains available at date Image 100002010000003200000019EEA0E525E092006F.png , which we represent by means of the “guano conservation method”:

Image 100002010000028B0000001ACC425AC0788E5B86.png

13These three processes admit constant returns.

As is usual in Sraffian models, we treat the rate of profits Image 100002010000000C00000019DEB7B9D6C180E05E.png as exogenously given. If, during period Image 100002010000000900000019D39B26010CAA4B4B.png , which starts at date t and ends at date Image 100002010000003200000019EEA0E525E092006F.png , process Image 100002010000000A000000197114B0F785BE4DE6.png is operated (i.e., its activity level is strictly positive), that process breaks even at the given rate of profits. Non-operated methods, by contrast, pay extra-costs.

  • 3 Ex post, one must check that the last assumption is consistent with the analysis of prices, i.e. on (...)

We assume that if guano were free, the guano method would be less costly than the backstop method. Otherwise, none would ever want to use guano for the production of corn. If in a given period the two corn methods are operated simultaneously, the price paid for the use of guano, or royalty, can be interpreted as a kind of differential rent: it is equal to the difference in the production costs of the two methods when guano is ignored. For simplicity, we assume that the period Image 1000020100000012000000191033DD12A7ED8587.png when the stock of guano becomes exhausted is known. The underlying hypotheses may be that the initial stock and the demand for corn are exogenously given and that the guano method is continuously used until exhaustion.3

In this section we analyse the first version of the model (Bidard and Erreygers, 2001a), characterised by the choice of corn as numeraire: Image 100002010000005600000019C0C1B4075A9A7F2A.png for any Image 100002010000000900000019D39B26010CAA4B4B.png . Since the royalty and the wage change with time, time indications are added. The following properties can be established easily:

14(CG1) Maximum rate of profits: The rate of profits has a finite upper bound. (CG2) Ricardian trade-off: In every period the rate of profits and the real wage are inversely related.

(CG3) Maximum royalty: In the period of exhaustion Image 1000020100000012000000191033DD12A7ED8587.png , the royalty Image 100002010000002F000000193FD6141D65A5B02F.png is equal to the differential rent between the two methods of corn production.

(CG4) Hotelling rule: The royalty at date Image 100002010000000900000019D39B26010CAA4B4B.png is equal to Image 1000020100000091000000191946251F8EFDDD22.png .

(CG5) Continuous use of guano: For any Image 100002010000004C0000001900F68BE4C5066817.png , the guano method is cheaper than the backstop method. Therefore the guano method is used first, until exhaustion of the stock of guano.

15(CG6) Translation principle: Let us consider two economies which only differ by their initial stock of guano. When the exhaustion dates are chosen as origins of time, the dynamics of prices and quantities are identical, except that the past is truncated in the economy with the smaller stock.

(CG7) Fall of the real wage: For a given rate of profits, the real wage decreases from date 0 to Image 1000020100000012000000191033DD12A7ED8587.png and reaches its long term level at date Image 1000020100000012000000191033DD12A7ED8587.png .

16(CG8) In a corn-guano model with several alternative guano methods, the cost-minimising method at each date before exhaustion coincides with the wage-maximising method, given the rate of profits and the level of the royalty at that date.

17A few comments are in order:

(i) The upper bound of Image 100002010000003900000019062962A825EDEC7A.png is equal to min Image 1000020100000076000000194054AB93B0110DCB.png . Property (CG1) is in accordance with property (C1) of the corn model: the rate of profits cannot exceed the maximum level which can be sustained by either of the two methods of corn production.

18(ii) Property (CG2) is an immediate generalisation of property (C2).

19(iii) A simple economic argument for (CG3) is:

At the moment of exhaustion ... we expect the backstop method to be used alongside the guano-method. Only by fluke would the then remaining supply of guano be sufficient to satisfy the whole demand for corn by means of the guano process: normally the remaining quantity will be too low, and the backstop process must be operated to fill the gap. (Bidard and Erreygers, 2001a, 249)

The coexistence of the two processes in the period of exhaustion requires that they are equally costly at that time. That condition determines the royalty at the date Image 1000020100000012000000191033DD12A7ED8587.png of exhaustion, when it is equal to the differential rent between the two corn methods. In this basic version of the corn-guano model, the simultaneous operation of the two methods occurs in the period of exhaustion, but not in other periods.

20(iv) As long as guano is not exhausted, the preservation process (6) is operated, which implies that the price of guano rises at a rate equal to the rate of profits. Property (CG4) is nothing but the Hotelling rule.

21(v) Property (CG5) justifies ex post a simplifying assumption made at the beginning.

22(vi) According to property (CG6), once the period of exhaustion is known, everything that precedes it can be determined using the principle of backward induction.

(vii) The distribution of the net product between profits, wages and royalties changes over time as a result of the non-constant price of guano. At a given rate of profits, it is therefore not surprising that the increase of the price of guano entails a decrease of the real wage. From a theoretical point of view, the important thing to note is that the real wage Image 100002010000002D0000001919653B373712672C.png changes from period to period for no other reason than the future exhaustion of guano:

Image 100002010000028B000000386431717316191B36.png

23This is an unexpected property from a classical perspective, and property (CG7) constitutes the most significant difference between the corn model and the corn-guano model.

24(viii) Property (CG8) in an extension of property (C3) of the corn model, the difference being that, for a given rate of profits, the wage-maximising method is uniquely defined in the corn model, whereas it changes with the level of the royalty in the corn-guano model.

2 Multisector Models and the Measurement of Profits

25Our aim is to examine whether the properties of the corn-guano model carry over to multisector models with one exhaustible resource. Up to now, we have identified the corn model with a one-good model, and the cornguano model has been obtained by introducing one exhaustible resource into it. In a more faithful historical interpretation, a corn model is a model in which corn is the only basic commodity. Sraffa (1951) pointed out that the move from the corn model to general multisector models introduces the question of values, i.e. prices. In a multisector setting with an exhaustible resource, the Hotelling rule and the fact that the exhaustible resource enters directly or indirectly in the production of other goods implies that the relative prices of goods will also change over time, and that evolution is the source of conceptual problems which need to be tackled first. That issue is not exclusively linked to the Hotelling rule: if relative prices change for another reason, the same problems occur. Here, we highlight three intimately connected aspects: the measurement of the rate of profits, price effects, and real effects.

2.1 The Numeraire and the Rate of Profits

  • 4 All vectors represent column vectors; transposition is indicated by a prime.

Consider a multisector model and suppose that at date Image 100002010000000900000019D39B26010CAA4B4B.png we invest a basket of inputs equal to Image 100002010000000E00000019437FD6379B21643C.png and obtain at date Image 100002010000003200000019EEA0E525E092006F.png a basket Image 100002010000000C00000019D55871E98A036374.png of outputs. If Image 100002010000000E00000019437FD6379B21643C.png and Image 100002010000000C00000019D55871E98A036374.png are proportional Image 100002010000006200000019DA1F00642AC1039C.png , the rate of profits can be interpreted in physical terms and amounts to Image 100002010000003D0000001941C237D275D79555.png . Similarly, if the relative prices Image 1000020100000010000000190C4BDEDEE6C32574.png at dates Image 100002010000000900000019D39B26010CAA4B4B.png and Image 100002010000003200000019EEA0E525E092006F.png are the same (a hypothesis retained in the classical theory of value), the rate of profits is calculated by means of these prices and amounts to Image 100002010000009D00000019280454B978D583F4.png .4 But outside a steady state or a regular growth path, be it for the study of out-of-equilibrium paths (“gravitation problem”, with different sectoral rates of profits at date Image 100002010000000900000019D39B26010CAA4B4B.png ) or that of equilibrium paths (uniform rates but with varying relative prices, as in the presence of exhaustible resources), the definition and measurement of the rate of profits sets a problem. From now on, we add time labels to distinguish relative price vectors at different moments of time, i.e. Image 10000201000000290000001973E1C65D19EB2443.png at date Image 100002010000006300000019672D2BF657B3459F.png at date Image 100002010000003200000019EEA0E525E092006F.png , etc. Since inputs and outputs may also vary from period to period, we write Image 1000020100000081000000192246EA22D97A8A90.png , etc.

For a multinational firm which trades in dollars and euros, with a flexible exchange rate between the two currencies, the yearly rate of profits differs according to the currency chosen as numeraire (it is higher when calculated in terms of the one which depreciates). A similar phenomenon occurs in multisector models for intertemporal production with changing relative prices: the rate of profits of the firm is not the same according to the commodity or basket of commodities chosen as numeraire, and therefore the references to a “rate of profits” and to its uniformity across industries depend on the prior definition of a numeraire. The natural numeraire is money. In non-monetary models, a commodity or a basket Image 100002010000001000000019E4D11A76FE4ECA15.png of commodities is used as numeraire, which means that the prices Image 100002010000002900000019ED23087B4AD060D8.png and Image 100002010000005100000019D38E99E6B6281025.png at dates Image 100002010000000900000019D39B26010CAA4B4B.png and Image 100002010000003200000019EEA0E525E092006F.png are normalized by setting Image 100002010000006E000000195DB8932CDC00C925.png and Image 10000201000000980000001911F88AD100042C6E.png . The apparent rate of return Image 1000020100000018000000198C2E5DD771969585.png , i.e. the one that appears when using the given numeraire Image 100002010000001000000019E4D11A76FE4ECA15.png for purposes of valuation, compares the nominal value of the investment Image 1000020100000027000000197C264861ED87DCA5.png to the nominal value of the outcome Image 100002010000004F000000196DD90EC644DF0427.png :

Image 100002010000028B00000038A09288AC3378A29E.png

That rate is independent of the numeraire if relative prices are constant, a common hypothesis in long-run models. If they change, there are as many apparent rates of return as numeraires, and one may wonder if some numeraire would be more significant than the others and could be used to define an absolute rate of return. This question of measurement and the search for the definition of “absolute values” pervades the history of economic thought: Ricardo, for instance, expected that gold could be such a standard by means of an appropriate management of money, and his very last writings were about the search of a commodity whose difficulty of production would be constant and, therefore, could be used as an invariant standard (Ricardo, [1823a] 1951; [1823b] 1951). In neoclassical theory, production is conceived as being oriented towards the satisfaction of needs, and the representative consumption basket is a quite natural standard. In any case, one would like to take “what really counts for us” as a standard of measure. If we trade coal but consume only corn, then it would certainly interest us to compare the units of corn we sacrifice at time Image 100002010000000900000019D39B26010CAA4B4B.png by buying coal to the units of corn we earn at time Image 100002010000003200000019EEA0E525E092006F.png by selling coal. For this both the investment and the outcome must be expressed in corn, which in this case represents what matters.

Let us assume there exists a specific basket Image 100002010000000B00000019FAFEC99C00C41308.png , called the standard of value, which captures what really counts for us. At time Image 100002010000000900000019D39B26010CAA4B4B.png , the real (or absolute) rate of return Image 1000020100000015000000195D17CA69B52400BA.png , based upon a comparison of the real investment and the real outcome, is defined by formula (8) assuming that the values Image 100002010000002A00000019B74D41C74E716409.png and Image 100002010000004E0000001957E4B1D870E58C3E.png are those corresponding to the specific numeraire Image 100002010000003B000000193BD2549408CD7A64.png . Alternatively, the real rate of return can also be derived from the apparent rate of return by means of the factor of appreciation of the standard of value.

Let Image 1000020100000044000000190D77F7C75EAFD615.png be the factor of appreciation of the standard of value s in terms of the prices defined by the numeraire Image 100002010000001000000019E4D11A76FE4ECA15.png , i.e.

Image 100002010000028B00000036B0447556E00783C3.png

26The relationship between the real and the apparent rates of return is then such that:

Image 100002010000028B00000032B1678E46C9076C3A.png

The fact that the value of the rate of return depends on the numeraire was pointed out long ago by Irving Fisher: “the rate of interest is always relative to the standard in which it is expressed” (Fisher, 1930, 41). We assume here that all investors adopt the same standard of value, although we acknowledge this should not to be taken for granted, as Keynes (1936) observed. In that case, the analysis is simplified by assuming that the numeraire is equal to this shared standard of value, implying that the apparent rate of return is also the real rate of return. The intertemporal price equations of a system of production Image 1000020100000053000000191406D87F8C52B801.png can then be written as Image 10000201000000B7000000199AF9087AE7F4EDB5.png Image 10000201000000D5000000190135FDD5A82E3208.png . It is possible to choose a different numeraire, but then the intertemporal price equations must be written as Image 10000201000000AA000000193DA8B5DCCEA20A72.png Image 10000201000000A000000019A2F42B6DE372D518.png Image 100002010000008400000019506F3EFF31EABA87.png . In what follows, we assume that the numeraire coincides with the standard of value.

We end this discussion on the influence of the numeraire by drawing attention to an unexpected phenomenon which occurs when labour is chosen as numeraire, the rate of profits Image 100002010000000C00000019DEB7B9D6C180E05E.png being given. In a corn model with changing prices, the dynamics of prices are then defined by Image 10000201000000CE00000019AD662774CD11CE81.png Image 100002010000004A000000192F190105091871E3.png , with Image 10000201000000A100000019CAF27A8A6849DC32.png . Therefore, the price of corn at date Image 100002010000000900000019D39B26010CAA4B4B.png amounts to

Image 100002010000028B0000001FBFDDA93CB7276112.png

where Image 100002010000001800000019653679A298D78E2C.png is the long-run price defined by equality Image 10000201000000EA000000191FB8380899AB7DC0.png . When time passes, the sequence of prices tends towards its long-term level.

But let us look at the past Image 1000020100000056000000195F8D634937AE7095.png if Image 10000201000000690000001911CFA228CA2503D2.png , the prices must have been negative at some point; if Image 1000020100000066000000193074271B1540A085.png , they were positive but arbitrarily high, so that the real wage was negligible. But then the effective rate of profits is Image 100002010000007900000019E6AB96625ECA89F0.png , when it is upposed to be equal to Image 100002010000000C00000019DEB7B9D6C180E05E.png . This shows that the choice of labour as numeraire, which is quite natural when studying long-term positions, is problematic in a dynamical framework.

2.2 Price Effects

For given but different price vectors Image 100002010000002900000019F7EECC660C4BA98B.png and Image 100002010000004A000000192F190105091871E3.png , the rate of profits in an industry depends on the numeraire used to measure it. A dual aspect of the same phenomenon is that, for a given rate of profits across industries, the relative prices depend on the numeraire. The following calculations illustrate that point.

Let there be a multisector model with a unique technique Image 100002010000004B000000190FDC628E35B9DABC.png , where Image 100002010000001300000019E8E2F40EFBAAB6F7.png is the square matrix of material input coefficients (constant returns are assumed) at date Image 100002010000000900000019D39B26010CAA4B4B.png , Image 100002010000000E00000019664E0243DC65B214.png the labour vector and Image 100002010000000E00000019664E0243DC65B214.png the output matrix at date Image 100002010000003200000019EEA0E525E092006F.png .

Let us choose a basket Image 100002010000005F000000197FC50104B79AB369.png as numeraire. For a given rate of profits Image 100002010000000C00000019DEB7B9D6C180E05E.png , the price vector and the wage evolve according to the rule:

Image 100002010000028B0000001E4029216B8FBF5584.png

After pre-multiplication by Image 100002010000001200000019314F77E54C69F7DF.png , we get:

Image 100002010000028B0000001EBC9A56CE05402E71.png

and since Image 10000201000000EB00000019A5AF16CEF1F00A41.png , we obtain:

Image 100002010000028B00000022AC403058CA05B379.png

Formula (14) shows that, for a given rate of profits, the price vector Image 100002010000002900000019127B064B7165CE15.png at date Image 100002010000000900000019D39B26010CAA4B4B.png determines the wage Image 100002010000002C000000199D90CE66445D1891.png , hence the next price vector Image 100002010000005300000019F8CFBB198ABBBF81.png by formula (12):

Image 100002010000028B0000003037217BBEC73CA9E4.png

Formula (15) makes the influence of the numeraire on the evolution of relative prices explicit. The sequence of relative prices is stable in the special case Image 100002010000006100000019D992065AE840D9CD.png where Image 100002010000001800000019653679A298D78E2C.png is defined by equalities Image 100002010000007C00000019F0539659372512A8.png Image 100002010000007800000019AF00E02BDF44F98E.png and Image 100002010000005600000019DA18475924E69991.png . But the conditions of convergence of the sequence Image 100002010000002900000019127B064B7165CE15.png towards Image 100002010000001800000019653679A298D78E2C.png are modified. In the present framework, that condition is written:

(H) The dominant eigenvalue of matrix Image 100002010000007A00000019D05D8F2B9686AFFE.png is smaller than Image 10000201000000590000001960CAA9B42A02CF53.png .

Condition (H) differs from the usual hypothesis which refers to the input matrix Image 100002010000001400000019348A8779BF057A15.png only and, for instance, the eigenvalue with maximum modulus of Image 100002010000001100000019F24B9BDB4F71EB71.png may be complex, as matrix Image 100002010000001100000019F24B9BDB4F71EB71.png is not semipositive.

2.3 Real Effects

27In the presence of several methods of production in the same industry, the very fact that the numeraire affects prices implies that the relative costs of two alternative methods may change with the numeraire. As a consequence, the cost-minimising method may also change, and therefore the competitive intertemporal path: it is easy to build a one-period example with such a switch of method according to the numeraire.

Moreover, an indeterminacy problem arises. Given the numeraire and knowing the prices Image 1000020100000029000000192A46E89117B0FE2F.png , two phenomena occur: on the one hand, the operated technique determines the wage of the period by (14); on the other hand, the prices and the wage determine the costs of production which, in a competitive framework, are minimum. Because of that cross determination, it is unclear whether the real and the value dynamics are well determined (existence and uniqueness). The following result provides a positive answer:

Wage-maximisationproperty. Given the rate of profits, the numeraire basket and the price vector Image 1000020100000029000000192A46E89117B0FE2F.png at date Image 100002010000000900000019D39B26010CAA4B4B.png , the operated technique is the wagemaximising technique.

28That property (Bidard and Erreygers, 2020) is a non obvious extension to the dynamic framework of a standard result when relative prices are constant through time. It is also an extension of property (CG8) of the cornguano model. A significant gap with the standard result is that the wage is maximum in terms of the numeraire, which may differ from the workers’ effective consumption basket.

3 Dynamics and the Natural Path

3.1 The Corn-Guano Model (II)

29The question of the measure of profits arises in multisector models as soon as the relative price of two commodities changes through time. We know that, in the corn-guano model, the wage evolves with time. Therefore, a similar problem of measure is met in the corn-guano model when a given combination corn and labour is chosen as numeraire. Hence the second version of the model (Bidard and Erreygers, 2001b), which is a pedagogical device allowing us to maintain a very simple analytical framework (only one produced commodity) and to explore the difficulties met when one tries to extend properties (CG1) to (CG8) to multisector models. The emphasis is on the dynamics.

It is now assumed that the numeraire is a given combination of d units of corn and Image 100002010000000E00000019BC8C13A3E02640CD.png unit of labour, with both Image 100002010000000E00000019957F688E2A91D3DD.png and Image 100002010000000E00000019BC8C13A3E02640CD.png positive (by contrast, in our first version we assumed Image 100002010000003A00000019DB0B9A63C2A84DC8.png and Image 100002010000003C00000019A6873AF8DCA61577.png ). Hence, we have Image 10000201000000D600000019CBBE15BBACAFEA63.png . There is no harm in assuming Image 100002010000004500000019B2E55536BB78BAA9.png . The dynamics of prices and wages until exhaustion are defined by the equations:

Image 100002010000028B0000007956B040365CAFBED4.png

That is, there are Image 100002010000004400000019A3DC5BDCAEF94B13.png equalities to determine Image 100002010000004400000019E20AC4028D00956E.png unknowns (Image 100002010000003600000019F0CC524E2261322D.png prices, Image 100002010000003600000019F0CC524E2261322D.png wages and Image 100002010000003600000019F0CC524E2261322D.png royalties). It thus appears that there now exists one degree of freedom in the determination of the price dynamics: it might be the inital price of corn, the initial royalty, the final royalty, etc.

When corn was the numeraire (in other words Image 100002010000006B000000198DB296306C38B890.png ), equality Image 1000020100000082000000198C53DA28175C74F4.png Image 10000201000000940000001984A887DB3FB20822.png held at date Image 100002010000001200000019C302F4A4D1E633C4.png and showed that the wage and the price at the exhaustion date Image 100002010000001200000019C302F4A4D1E633C4.png immediately reached their long-run values Image 1000020100000064000000193A9569DC432B8F4B.png associated with the backstop method. It may be tempting to introduce a similar hypothesis in the new framework, and that additonal condition would serve as the missing equation for a full determination of the dynamics. The following argument shows that this cannot be the case.

Let the final price Image 10000201000000310000001923D9951C84D9386E.png be arbitrarily given. The forward and the backward dynamics are then fully determined. Let us look at the backward dynamics: the wage Image 10000201000000360000001978BC392B98D231BE.png is determined by the numeraire equation (19), then the royalty Image 100002010000003000000019DFEBBFA120378239.png by (17). According to the Hotelling rule (17), the royalties Image 1000020100000026000000197F10499358807F3B.png at any date Image 100002010000004E00000019C50A53CD90A0A48B.png are known. And, once the price and the wage at date Image 100002010000000900000019D39B26010CAA4B4B.png are known, those at date Image 10000201000000330000001988E6B3D326B679C0.png are obtained by solving the price equation (16) and the numeraire equation (19). The price of corn Image 100002010000000E0000001946C55A0D06936961.png periods before exhaustion is given explicitly by formula:

Image 100002010000028B0000001F0136B69DF088391E.png

where Image 1000020100000019000000193EC7477A8B29D14E.png would be the long-run price of corn associated with the guano method if guano were free, Image 100002010000009500000019927D9C49C6EE82EA.png and Image 10000201000000130000001938C0FACE042A8E9B.png is a constant calculated in order that, for Image 100002010000004300000019FA70152F5F0BB3BA.png , formula (20) fits with the already known values of Image 10000201000000320000001968BED29FBBBB289F.png and Image 100002010000002F00000019781F19C8AC5C8E97.png . It is assumed that Image 100002010000009400000019C7462D91B8BC311E.png (the condition is necessary to have Image 1000020100000052000000190AD2658F68D716CF.png ). Suppose that the initial stock of exhaustible resource is great enough, so that Image 100002010000000D000000198532D123AF699697.png can take large values. If Image 10000201000000130000001938C0FACE042A8E9B.png is negative, the same holds for the value of Image 100002010000002C0000001947AB380D764C9BAB.png ; if Image 10000201000000130000001938C0FACE042A8E9B.png is positive, the value of Image 100002010000002C0000001947AB380D764C9BAB.png exceeds Image 10000201000000210000001997031F8C882E25F7.png and therefore that of Image 100002010000003100000019767EA05CB0F2402F.png is negative. The conclusion is that the path defined by the value Image 100002010000004200000019D3B0BDA700BEA6DD.png is the only one compatible with any initial stock of guano. Let us call it the “natural path”.

On the natural path at date Image 100002010000001200000019C302F4A4D1E633C4.png , the price of corn, the wage and the royalty are uniquely defined by the three equalities (18), (19) and (20) with Image 100002010000003B00000019FB40913F5DEC209E.png and Image 100002010000004200000019D3B0BDA700BEA6DD.png . There is no reason why Image 10000201000000320000001968BED29FBBBB289F.png should be equal to the longrun price determined by the backstop method. As a consequence, it will continue to evolve according to the forward dynamics defined by the backstop method: in simple words, the price of corn evolves initially because guano will become exhausted, and eventually because guano has been exhausted! Property (CG7) does not hold any longer for a mixed corn-labour numeraire.

  • 5 See Bidard and Erreygers (2001b) for more details.

30For a given stock of guano, the natural path is only one of the infinitely many feasible paths obeing the dynamics. However, when the stock of the exhaustible resource is large, all feasible paths tend to converge to the natural path, as shown by formula (20).5

3.2 The Dynamics of Multisector Models

We now consider a true multisector model with Image 100002010000004200000019A5C8C32AD7FF0B09.png commodities, with one exhaustible resource (guano) used in one sector (agriculture). The wage-maximisation property (CG8) has a general validity, independently of the question of exhaustible resources. The translation principle also holds because, if one knows the prices, the wage and the royalty at the exhaustion date, the forward and the backward dynamics of values are uniquely defined up from that date, and this also holds in the presence of several guano methods: suppose that, for some stock, a guano method 1 is used for 35 years, then another guano method 2 for 17 years, guano being then exhausted; then, if the economy starts with a lower stock of guano sustaining production for 30 years only, method 1 will be used for 13 years, followed by method 2 for 17 years, with the same dynamics on a lower time interval.

  • 6 Note that property (CG8) then holds, but not for the corn-guano model with a mixed numeraire.

31The question we examine in the present section is whether the longrun prices are reached at the exhaustion date (as in the first version of the corn-guano model) or not (as in the second version) when the numeraire is either one commodity or a basket of commodities, labour being excluded.6

Let Image 100002010000000E0000001916EEFC2DBF237E30.png be the numeraire basket, which may consist of a single commodity. The Image 100002010000000F000000199B2F53EE25AA34AC.png prices Image 10000201000000310000001975AD61C473802BAD.png at date Image 100002010000001200000019C302F4A4D1E633C4.png , the wage Image 100002010000003500000019B33EBBB675FB0421.png and the final royalty Image 100002010000002F00000019877DEFA363BC74AB.png are linked by two equalities: the numeraire equation Image 100002010000007400000019B12140C961E8A180.png and the property that the royalty at date Image 100002010000001200000019C302F4A4D1E633C4.png can be considered as a one-shot differential rent, the level of which equalises the costs of production of corn by the guano and backstop methods. There remain Image 100002010000000F000000199B2F53EE25AA34AC.png degrees of freedom. Let us assume, for simplicity, that the guano method is continuously operated before exhaustion. With Image 1000020100000060000000191E8046258CC4E76A.png representing the operated technique, the value dynamics are defined by the equations:

Image 100002010000028B0000004A956D09B53FDB2108.png

In equality (21), Image 100002010000002F00000019877DEFA363BC74AB.png is now a vector whose first component—which corresponds to the agricultural sector—is the royalty at date Image 100002010000001200000019C302F4A4D1E633C4.png and the other components are zero. We can now combine the calculations made above in sections 2.2 and 3.1: for a value of the royalty at date Image 100002010000001200000019C302F4A4D1E633C4.png considered as a parameter, equality

Image 100002010000028B000000212CBAB5FA6EC26B56.png

allows us to determine the level of Image 100002010000002D00000019E63A6E8D46A0F4CF.png . Inserting that value in (21) allows us to define Image 100002010000005400000019EC93C4B7CF4C070C.png as a function of Image 100002010000002900000019A9E27A44D2A5AA3F.png . If corn is not a part of the numeraire (e.g. because another commodity is chosen as numeraire), we have Image 1000020100000070000000195D88DE7135C4E769.png and formula (14) holds exactly. If corn is the numeraire or a part of the numeraire basket, the affine relationship (14) is modified by the presence of a term including the royalty at date Image 100002010000003B000000195AB6F7DE54354AF6.png , but that term is negligible as long as guano is far from being exhausted. Consider now the backward dynamics: the value of Image 100002010000002900000019A9E27A44D2A5AA3F.png is obtained from that of Image 100002010000005400000019EC93C4B7CF4C070C.png and, by induction, one obtains a more complex formula analogous to (20)

Image 100002010000028B000000211D1B36D83CBCC660.png

In that formula, matrix Image 100002010000001900000019E27C7DA12BDBADA0.png is equal to Image 1000020100000065000000198BEDD2AFE9C1E2CE.png is a vector and Image 100002010000005B00000019C136A97497EDFBFC.png another vector which tends to zero when Image 100002010000003400000019ED6AA9A2D75F4F69.png increases to infinity. Assumption (H) of section 2.2 ensures the convergence of the long-run dynamics of values of the guano technique, when guano is free. Condition Image 100002010000004500000019A4F7B8DBE9B11C0C.png is the way to ensure the positivity of prices at any date before exhaustion of guano, when the initial stock of guano is large. It defines the natural path, and all feasible paths converge to it. Since there is no reason for the prices at date Image 100002010000001200000019C302F4A4D1E633C4.png to be equal to the long-run prices associated with the backstop technique, the prices will continue to fluctuate after exhaustion. To sum up, the origin of the difficulties mentioned in section 3.1 does not lie in our adopting a rather bizarre numeraire in that section: it is a general phenomenon in multisector models. From that point of view, the corn-guano model with corn as numeraire is the exception.

4 Alternative Approaches

32In the preceding sections we have explored how exhaustible resources could be integrated into Classical theory starting from a simple economic model. However, not everyone working in the Classical tradition has embraced our approach. In this section we reflect on some of the alternatives that have been put forward and assess how they have tackled the issues. The presence of exhaustible resources poses difficult challenges to which various responses have been given. Instead of writing an exhaustive survey of the literature, we examine the alternative models with three questions in mind: What are the critiques addressed to the corn-guano and its extensions? What is the conception of classical economics underlying the construction? Is that construction consistent from a mere logical point of view?

4.1 Parrinello: Abandoning Perfect Foresight

33Exhaustible resources would not be a problem for economic theory if their exhaustion were of no concern to economic agents, for instance if they foresaw that the supply of these resources would be forever sufficient to cover demand (e.g., because the resource becomes obsolete after a certain date). So the interesting case arises when economic agents do worry about exhaustion. Two points of view may be adopted here. Either one assumes that agents acknowledge that exhaustion will be on the agenda some time in the future, but do not have a clue about the date at which exhaustion will occur. Or, alternatively, one assumes that agents know the date of exhaustion exactly. Let us designate these hypotheses as those of “imperfect” and “perfect foresight”, respectively. There is no doubt that the hypothesis of imperfect foresight is more realistic. From a theoretical point of view, however, it has the disadvantage of making the determination of prices subject to fragile hypotheses on expectations. By contrast, the hypothesis of perfect foresight is certainly heroic, but it allows us to determine prices.

34Following Hotelling (1931) and a large part of the literature on exhaustible resources, we have assumed perfect foresight in our corn-guano model. In his 2001 contribution, Parrinello explicitly rejected this hypothesis and assumed that the date of exhaustion is unknown. In order to close the model he needed to come up with an alternative assumption. The trouble is that Parrinello’s assumption—a rank condition—is of a purely mathematical character, and may be in conflict with other assumptions of the model.

35Parrinello’s oil-corn model is many respects similar to the corn-guano model. As in our model, Parrinello assumed that the rate of profits is given, that corn serves as the numeraire, and that before exhaustion, two processes are available for the production of corn (in Parrinello’s model the processes can change over time, but this is a non-essential variant). It is easy to show that, in order to arrive at a determinate solution, in exactly one period two processes must be operated simultaneously while in all others only one process (the cheapest of the two available) is used. If two processes were operated in more than one period, the system of prices would be over-determined; and if in no period two processes were used, it would be under-determined. In our corn-guano model a simple economic argument is invoked to state that the two-process period must be the period of exhaustion. Parrinello did not address this economic argument and, in his oil-corn model, the period of exhaustion is unknown and the period of coincidence may be any period before exhaustion. Towards the end of the article, Parrinello seemed to opt for the solution that the two-process period must be the initial period, on the grounds that “[t]he future cannot affect the past” (Parrinello, 2001, 311). But that leads to a problem of time inconsistency: since the period he refers to is “today”, which moves as time passes by, Parrinello’s rule (coincidence in the present period) would be wrong tomorrow if it were true today. That contradiction does not occur when the period of coincidence is defined as that of exhaustion.

36In 2004 Parrinello proposed a revision of his theory of exhaustible resources. He abandoned the rank condition, and instead introduced the notion of the “effectual supply” of an exhaustible resource, i.e. the limited quantity of the resource available for production in every period. He claimed that if both the path of effectual demand for goods and the path of effectual supply of exhaustible resources were known, it would be possible to extend the classical theory of prices, and in particular Sraffa’s equations for the determination of rent, to the case of exhaustible resources.

4.2 Schefold on Long-Period Analysis

37The corn-guano model is a theoretical tool developed in order to shed light on the problems that arise when one tries to integrate exhaustible resources in a classical approach. It proceeds by building a bridge between the corn model, which belongs to the Ricardian tradition, and Hotelling’s seminal model on exhaustible resources. Like its basic bricks, it is an economic abstraction and its ambition is methodological. Its main feature is to proceed by mixing the simplest characteristics of two models: three equations are sufficient and their treatment is transparent. The substantial differences between the solution of the corn-guano model and that of the standard corn model can be attributed unambiguously to the presence of an exhaustible resource. For instance, in the corn-guano model the relationship between the wage and the rate of profits is not time invariant, despite the fact that the same production process remains in use as long as the stock of guano is not exhausted. This result is at odds with the “objective” point of view defended by the Classical economists and Sraffa, according to whom the knowledge of the operated methods and the real wage suffices to infer the level of the rate of profits.

38Once it is acknowledged that the introduction of exhaustible resources leads to qualitatively different results, a second step consists of examining the degree of generality and the robustness of the laws derived from the simple model (for instance: is the exhaustible resource always used continuously until exhaustion?), and of questioning key concepts (how is the notion of rate of profits defined in the presence of changing prices?). This justifies the analysis of more complicated multisector models. In our mind, models of exhaustible resources are simple cases of models characterised by time-varying prices, with the cause of changing relative prices lying in production (as opposed to psychological motives, such as the consumer’s impatience). Therefore, the study of the corn-guano model is the first step in the elaboration of a research program of Classical inspiration. It is not at all meant as an attempt to describe a “Peruvian” economy. When Schefold (2001) criticised our model for its unrealistic features, he was obviously right, since our aim is theoretical consistency rather than empirical accuracy.

  • 7 It should be noted that for an unknown reason Schefold shifted terminology and considered the extra (...)

39A related, but more specific, criticism raised by Schefold concerns the lack of distinction between guano in situ and guano extracted. For reasons of simplicity, our model assumes that guano in situ can be used without further processing or effort in the production of corn. According to Schefold this is nonsense: guano can be used as a fertiliser in the production of corn only after it has been extracted and transformed. Hence he stressed the need to make a distinction between exhaustible resources in the ground and exhaustible resources above the ground.7 The issue at stake is whether the distinction makes a significant difference. It does not: a simple extension of the corn-guano model with an additional process describing the extraction of guano is basically all that is needed. The price of in situ guano still follows the Hotelling rule, whereas the price of extracted guano follows a slightly modified Hotelling rule.

  • 8 Kurz and Salvadori attribute to Ricardo the following implicit hypothesis: “For each exhausted depo (...)

40Schefold also stresses the differences of quality across mines. Again, there is no denying that this is a well-established empirical fact, but what matters from a conceptual standpoint is to underline the specificity of exhaustible resources in comparison to land and to identify the notion of royalty as distinct from that of rent. In that respect, a model which ignores the heterogeneity of mines is suitable and justified.8

41A puzzling feature of Schefold’s alternative formalisation concerns the way prices change: production by means of exhaustible resources is presented as akin to production by means of lands of different fertility, in the sense that the normal prices of produced commodities “will rise and fall in steps, as in Sraffa’s rendering of Ricardo’s theory of rent” (Schefold, 2001, 320). More specifically, Schefold divided time in successive “long periods”—“decades” in his terminology—during which prices of produced commodities remain at their normal levels. Normal prices change spasmodically at the instant of time which separates one decade from the next. Schefold does not explain, however, why such changes are necesary and why they occur only between two decades. In the theory of rent, a change of normal prices follows an increase in demand which meets a scarcity constraint and requires the introduction of a new marginal method of production. As no criterion of that type holds in Schefold’s model, it is unclear why prices are frozen for long periods of time, and then change suddenly. In his formalisation, the precise length of a decade is an essential characteristic, the theoretical determination of which is left open.

42Another issue which lends itself to contention is that according to Schefold, different rules apply to the prices of commodities (including extracted guano) on the one hand, and to the price of the in situ resource on the other. Commodity prices remain at their normal levels during each period, but “an essential change in the price of the resource takes place within each period” (ibid.). No economic reason is given for this asymmetrical treatment. In the absence of a strong argument to the contrary, we believe that all prices should be allowed to change within a period, not only those of the resources in the ground.

43Schefold’s alternative model can be criticised on several points. It is worth mentioning that if the prices of produced commodities (including the produced guano) are stable for a decade while that of in situ guano changes, numerous opportunities for arbitrage are open, be they between guano in situ and the other commodities during a decade, or between commodities before and after the periodic break. Schefold’s implicit thesis is that a competitive economy cannot adapt itself smoothly to the presence of exhaustible resources and suffers a dramatic crisis at the end of every decade. The definition of a decade, which is essential for the determination of the resulting intermittent dynamics, remains unclear.

4.3 Kurz and Salvadori on the Concept of Profit

44For many years Kurz and Salvadori have worked on a theory of exhaustible resources of a Sraffian inspiration or, they claim, “with classical features”. Their early contributions (Salvadori, 1987; Kurz and Salvadori, 1995; 1997; 2000) have been instrumental in our motivation to develop the cornguano model, and the number of pages they have devoted to the topic justifies a detailed analysis of their framework, with which we shall explain the reasons of our disagreement. Their work is also the alternative attempt whose formalisation is the most developed, which can therefore be submitted to a more precise criticism than the others.

45Let us begin by Kurz and Salvadori’s (2001) treatment of the corn-guano model. Instead of assuming that the (real) rate of profits is given, as we did, they started from the assumption that the real wage is given. We have argued that the two cases can be examined just as easily (Bidard and Erreygers, 2001a, 251-252). Our position is that in the case of a given real wage we also need to specify a standard of value, otherwise the rates of profits that will be determined by the model have no “real” meaning. Let corn be the numeraire, i.e. let us take one quarter of corn as the unit of prices:

Image 100002010000028B0000001AE8658774CFBB3705.png

The model then determines a unique sequence of royalties Image 100002010000003F00000019CFA30A1825CDC70F.png and of real profit rates Image 100002010000003D00000019D51175ABF9219C8A.png , just as the original corn-guano model determined a unique sequence of royalties and real wage rates (see section 1.2). By contrast, rather than specifying the standard of value, Kurz and Salvadori followed a different route: “The sequence of nominal rates of profit Image 10000201000000290000001922D4E8806CDC907D.png is assumed to be given” (Kurz and Salvadori, 2001, 284). Implicitly, this procedure—choosing the nominal profit rates without specifying the numeraire—boils down to using a sequence of changing numeraires, so defined that they yield the desired rates of profits. This is how it goes. Let Image 10000201000000C90000001920E01DFC96BE22F2.png be the sequence of real profit rates obtained by taking a quarter of corn as the numeraire, and let Image 100002010000008B0000001992141824CE729D5E.png be another arbitrary sequence. Suppose now that at time Image 100002010000000900000019D39B26010CAA4B4B.png the numeraire consists of Image 100002010000002C000000198B5116B95321C2AF.png quarters of corn, i.e. Image 100002010000008000000019CFCCE93B895DD18D.png . Then, if the sequence of numeraires Image 100002010000002C000000198B5116B95321C2AF.png evolves through time according to the rule Image 100002010000005B000000196E60D230719933C8.png and

Image 100002010000028B0000003605707C2E4A4273FE.png

we obtain the sequence Image 100002010000008B0000001992141824CE729D5E.png as nominal profit rates of the model. As we have argued before (Bidard and Erreygers, 2001a, 246), we do not believe that assuming given rates of profits without specifying the numeraire is the right choice.

We now examine Kurz and Salvadori’s own approach and assess their claims concerning the classical features of their construction. Their formalisation has evolved, but the basic equation attached to the working of the technique Image 100002010000006C00000019184B231690E2FD00.png is always written

Image 100002010000028B0000001DAA3EBFB5F6B05C13.png

46(the equality is replaced by an inequality for a non-operated method). When applied to an exhaustible resource, the Hotelling rule

Image 100002010000028B0000001C3DA3B9AF88D132A1.png

is obtained as a particular case. The significant gap with our approach is that Kurz and Salvadori do not set the numeraire equation Image 100002010000006700000019816E6DC17D651B38.png . Kurz and Salvadori (1995) followed the standard interpretation of Sraffian models and considered the rate of profits as a real magnitude but, in 1997, they realised that, by changing Image 100002010000002900000019213EC909928F92B3.png into Image 1000020100000016000000196FAD957D87C85842.png and Image 100002010000002D000000191E4553708BB50123.png into Image 100002010000001A000000192DFD0E4118F9576A.png defined by the transformation formulae

Image 100002010000028B00000038C71539C7E757FB6B.png

where the sequence Image 100002010000002B0000001918B536B0F36A6575.png is defined by Image 100002010000005B00000019418DCE0CA3D11CD4.png and

Image 100002010000028B000000324CEC0C9A7A05288D.png

47equality (27) becomes

Image 100002010000028B0000001C4F61D634C2C4FC7E.png

Thanks to the conversions (29) and (30), a sequence of price-and-wage vectors sustaining the rate of profit Image 100002010000000B0000001949D4AEBA95B262B6.png is thus transformed into another sequence sustaining the rates Image 100002010000001300000019F3F921411B0EAE9D.png . The change in the rates of profit has no effect on the relative prices at each date and there is no upper bound to the rates of profits. These two phenomena are unexpected in a Sraffian framework and led them to reinterpret the magnitude Image 100002010000000B0000001949D4AEBA95B262B6.png of equality (27) as a “nominal rate of profit”, an idea they have since defended. The transformation allows the study of prices to be decomposed into two steps. First, by setting Image 100002010000003D000000193445AAF2C83799F8.png in (27), the simplified equality

Image 100002010000028B0000002076BA77BA93D0C0FD.png

is obtained. Second, once it has been shown that some problem admits a solution Image 1000020100000075000000192B68477B69B7F751.png for a “zero nominal rate of profit”, the formulae (29)(30) provide a solution to the general problem corresponding to any positive nominal rate of profits Image 100002010000000B0000001949D4AEBA95B262B6.png , and even to any arbitrary sequence Image 10000201000000290000001922D4E8806CDC907D.png of nominal rates.

Kurz and Salvadori (2000) make use of the simplified equation (33) (since they assume that the real wage basket is given and incorporated in the input matrix, the equation they consider is even reduced to Image 100002010000007300000019D03F8C1E76EBD3B8.png Image 100002010000004500000019436B03B975681FB5.png to establish their main result, which is an existence property. That result can be stated as follows. Consider a multisector model with one or several exhaustible resources and, possibly, extraction costs and capacity constraints. Let there be a given sequence Image 10000201000000290000001922D4E8806CDC907D.png of nominal rates of profits and a basket Image 100002010000000E000000198CCA34B6C8455C14.png . Then, there exist a sequence of prices and a sequence of activity levels with the following three properties: (i) the operated methods at date Image 100002010000000900000019D39B26010CAA4B4B.png do yield the nominal rates, whereas the others pay extra-costs;

(ii) they produce a final demand basket Image 100002010000007500000019E54892FC9662D5DC.png at each date; and, (iii) the scarcity constraints on the initial endowments are met. The scalar Image 100002010000000E00000019A10D2016AD29CE63.png is endogeneously determined and is defined by a maximality property.

Rather than examining the proof, let us return to the remarkable equality (33), because it is immediately recognised by theoreticians: it coincides with Walras’ “ni bénéfice ni perte” condition ([1874] 1988, 284), as written in the neo-Walrasian approach of intertemporal production (Malinvaud, 1953; Arrow and Debreu, 1954; Debreu 1959). More explicitly, let Image 1000020100000031000000199BD0C7E6ED557D63.png be the present price of the dated commodity Image 1000020100000009000000199EF48EA7E1071302.png , i.e. the price paid today for the delivery of one unit of commodity Image 1000020100000009000000199EF48EA7E1071302.png at date Image 100002010000000900000019D39B26010CAA4B4B.png , and Image 100002010000002D000000191E4553708BB50123.png the present wage for one unit of labour available at date Image 100002010000000900000019D39B26010CAA4B4B.png . Equality (33) is a non-arbitrage condition for entrepreneurs, under the constant returns hypothesis. In that framework, the Hotelling rule is that the present value of the royalty is constant.

48Because there is no harm in assuming that the “nominal rates of profit” are zero and that Kurz and Salvadori’s equations cannot then be distinguished from neo-Walrasian equations, the formal properties established in a neo-Walrasian framework hold for the other. In the Appendix, we show that the results established by Kurz and Salvadori follow immediately from neoclassical theory, thus reducing the number of required equations from sixty in their 2000 paper (and up to a hundred-and-thirty in Huang, 2018) to zero.

49The point, however, concerns the economic interpretation of equality (33). Walras and his followers distinguished the entrepreneur, who combines factors of production he does not own, the owners of capital goods and the workers. The no-profit condition (33) means that the entrepreneur is left with no income once he has paid workers (who receive wages) and the owners of the capital goods (the capitalists, in the classical sense, who receive profits). The vanishing of “pure profits”, which are similar to a rent, has no relationship with that of profits, and equality (33) can in no sense be interpreted as a zero rate of profits equality, be that rate either “real” or “nominal”. The same for the more general equality (27) which, when isolated from the reference to a numeraire, is based on a conceptual confusion between the Classical notion of profits and the neoclassical notion of pure profits.

50The reader is invited to look at chapter 16 of Kurz and Salvadori (2015) for quite different analytical and historical views on the same story and on the nature of Classical economics.

4.4 Ravagnani: The Role of Socio-Historical Factors

51Ravagnani (2008) stresses the unrealistic character of the perfect foresight hypothesis used in theoretical models, for instance concerning the date of exhaustion of the resource. The historical evidence drawn from the US oil industry is that the bargaining between landowners and oil companies led to long-term contractual arrangements in which the lessee pays the landowner a fixed percentage of the crude oil produced on his tract of land, that percentage being subject to changes over time due to the evolution of the respective bargaining positions. The evidence of the role of socio-historical factors in the determination of distribution is in line with the Classical approach, as exemplified by the determination of the “natural” wage in that construction. Ravagnani thus invites the reader to consider the royalty as another independent distribution variable, and claims that there is no analytical difficulty in adapting Sraffa’s price equations “by taking the share of the resource price attributed to landowners as a ‘given’ coefficient reflecting the (persistent) share paid on average in actual economies” (Ravagnani, 2008, 91). The model that Ravagnani has in mind for the determination of royalties is Marx’s theory of absolute rent (Piccioni and Ravagnani, 2002), introduced in connection with the thorny “problem of transformation of values into prices”, a construction that most theoreticians consider as frail, if not contradictory with the notion of pure competition.


52In this paper we have examined different attempts to integrate exhaustible resources into Classical theory. The main problem is the reconciliation of two seemingly opposing logics: on the one hand, changing relative prices induced by the Hotelling rule; on the other, constant relative prices characteristic of the long-term approach. As soon as the possibility of changing relative prices is admitted, the choice of the standard of value and the measurement of the rate of profits are no longer trivial matters. We have tried to show that the corn-guano model constitutes a good starting point to analyse these points, even if the perfect foresight hypothesis is a strong limitation of the approach. The model is a simplifying device which brings conceptual problems rather than technical points to the fore and provides a yardstick for further generalisations.

53The great diversity of the points of view expressed by Sraffian scholars is quite striking. The driving force of the debate on how exhaustible resources should be integrated is not in the first place a difference of opinion about what the relevant facts are. As we have pointed out, there is criticism of the lack of realism of some of the models which have been proposed; nevertheless, correspondence to empirical reality has not been the main preoccupation. The core of the debate is theoretical: it concerns the very understanding of basic economic concepts and the characterisation of the main features of Classical theory.

We are grateful to the reviewers and editors of the journal for their helpful comments, which have allowed us to ameliorate the paper.

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Given a stock of scarce resources today (“endowments”: capital goods, labour, lands, exhaustible resources...) and the technology, the set Image 1000020100000013000000191824184AE7665D8C.png of overall feasible products is compact and convex in Image 100002010000001E00000019FC2933A37364DA35.png . Productive efficiency is reached when it is impossible to increase the production of some good without decreasing that of another, i.e. when the product belongs to the outer frontier Image 100002010000001400000019345946514AAA96F9.png of feasible productions. A necessary condition for efficiency is that the allocation of inputs between industries is such that the relative marginal productivities of any two inputs be the same in all industries, otherwise an adequate cross transfer of these inputs between industries would sustain an increase of the overall product for an unchanged total amount of inputs. A well-known economic argument in favour of a competitive organisation of production is that such an efficient allocation of resources is reached in a decentralised way when firms maximise individual profits, because each firm then equalises its own marginal rate of substitution between any two inputs with their relative price. In the debates on the economic organisation of socialism, that argument was considered to be strong enough to suggest that socialist firms should mimic a competitive behaviour (Lange, 1936; 1937).

The precise relationship between a given price vector Image 1000020100000010000000193A684F349B90B0D7.png and the corresponding overall product Image 100002010000000E000000199286BF0912961202.png is

Image 100002010000028B0000002B9CFAB90369994A26.png

It means that Image 100002010000000E000000199286BF0912961202.png is efficient and that the price vector Image 1000020100000010000000193A684F349B90B0D7.png is normal to Image 100002010000001400000019345946514AAA96F9.png (or to Image 1000020100000013000000191824184AE7665D8C.png ) at point Image 100002010000000E000000199286BF0912961202.png . Note that, when the notion of preference is introduced, the maximality property in physical terms is required for the Pareto optimality of a competitive equilibrium.

For many years, Kurz and Salvadori have used the same theoretical framework to deal with more and more complex technical points related to exhaustible resources (costs of extraction, capacity constraints on extraction, etc.; see also Huang (2018) for the introduction of research-seeking activities) in more and more complex models. The question they set is that of the existence of a competitive price vector for given “nominal rates of profit”. Their formalisation may look like a convoluted version of Sraffian models, but two hypotheses should have attracted the readers’ attention: (i) the initial amounts of exhaustible resources are given, and also those of commodities at date 0; Kurz and Salvadori note that “[the] second initial condition ... is perhaps less obvious” (2000, 362) and later justify it since “the analysis is not a long-period one” (Kurz and Salvadori, 2015, 294); (ii) in standard Sraffian models, the consumption basket d is given; in the version they retain, the final demand in every period is proportional to Image 100002010000000E000000199286BF0912961202.png and amounts to Image 100002010000001B00000019C723BA8877684170.png , where Image 100002010000000E00000019C524C89D7D7B439B.png is an endogenously determined scalar, defined as the maximum feasible level of constant consumption with direction Image 100002010000000E000000199286BF0912961202.png .

An arbitrary demand basket Image 100002010000003F000000191D5B486A88D7054B.png need not be either efficient or even feasible, but the line of direction Image 100002010000000E000000199286BF0912961202.png cuts the efficiency frontier at some point Image 100002010000001B00000019C723BA8877684170.png Image 100002010000005800000019CFE92D36FE27195B.png , and the price vector normal to the efficiency frontier at that point sustains the production Image 100002010000001B00000019C723BA8877684170.png . The almost equivalence between productive efficiency and competitiveness also holds, after minor adaptations, in an intertemporal framework. A gap is that, in an intertemporal economy, a part of the product is (re)invested, therefore the product at some date is no longer identified with the basket available for final demand. The maximality property now concerns intertemporal final demand. Then efficiency is obtained when the rates of substitution between inputs and those between outputs at the same date are identical across all industries: this is indeed the case at equilibrium with perfect foresight under competitive conditions, the price of a “dated good” being its present price. The case of infinite horizon sets a problem because the number of dated goods becomes infinite with the horizon. In a remarkable contribution to capital theory, Malinvaud (1953; 1962) studied that point and derived an extension of the equivalence under some additional conditions. As a consequence, for a given basket Image 100002010000003F000000191D5B486A88D7054B.png , consider the maximal scalar Image 100002010000004200000019DA66E75C06EFF637.png such that the intertemporal demand Image 10000201000000FD000000197B94A58268122A50.png is feasible: by definition, the corresponding production is efficient, therefore it is sustained by an intertemporal price vector. Hence, the existence result.9

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1 Kurz and Salvadori (1995, 366-368) however studied an energy-oil model close to the cornguano model but with different properties (e.g., the method using the exhaustible resource and the backstop method are used simultaneously during several periods, a phenomenon excluded in the corn-guano model). We shall not examine that model, as the authors have changed their formalisation since and work on multisector models directly.

2 Equation (3) is in line with the classical tradition according to which wages are advanced (we adopt the same hypothesis later for royalties). By contrast, Sraffa’s convention is that the wage is paid post factum. All formulae are easily adapted to one or the other hypothesis, and the qualitative properties of the models are not affected by these choices.

3 Ex post, one must check that the last assumption is consistent with the analysis of prices, i.e. one must check that, up to date T, the guano method is cheaper that than the backstop method.

4 All vectors represent column vectors; transposition is indicated by a prime.

5 See Bidard and Erreygers (2001b) for more details.

6 Note that property (CG8) then holds, but not for the corn-guano model with a mixed numeraire.

7 It should be noted that for an unknown reason Schefold shifted terminology and considered the extracted resource (“above the ground”) rather than the in situ one (“in the ground”) to be the exhaustible resource of his own model

8 Kurz and Salvadori attribute to Ricardo the following implicit hypothesis: “For each exhausted deposit of the resource another one with exactly the same characteristics is discovered” (2009, 5). That interpretation, which is not sustained by a precise textual reference, is another way to assume away the specificity of exhaustible resources

9 Incidentally, Kurz and Salvadori’s proofs are wrong. They are based on the assumption that, for a given basket d, the number γ = γ(y) of produced units of d is a linear function of the activity levels y. But let d0 = (1,1), and the net product for activity levels y1 (respectively y2) be (1,2) (respectively (2,1)). Then γ(y1) = γ(y2) = 1 but γ(y1 + y2) = 3: the fundamental theorem of marginalist theory on efficient production is grounded in convex analysis, not in linear programming

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Christian Bidard

EconomiX, Université Paris Nanterre.

Guido Erreygers

†Department of Economics, University of Antwerp.

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