Yoann Verger Sraffa

Yoann Verger
email: yoann.verger@uvsq.fr

Sraa and ecological economics

est destinée au dépôt

under the condition that a standard numeraire is used, there is a linear relationship between the rate of prot and the wages. Thus the neoclassical theory of production and of distribution is attacked: the revenues of the dierent factors of production do not depend on their marginal productivity. Moreover he shows that there is no need for a market equilibrium where supply meets demand to construct the prices.

2. Second, on a methodological level, Sraa refuses to introduce expectations and subjective data in his model. There is no assumption on the dynamics of the system, as there is no assumption on constant returns to scale, and no assumption on the behavior of the agents of production. He is thus arguing against the neoclassical methodology based on subjective preferences, rational optimization and perfect knowledge of the future from the agents. He is also warning that static comparison based on prices is meaningless as soon as technical change occurs (there is no constant standard of value, Sraa (1960), 93).

3. Third, on the level of history of thought, Sraa's analysis calls for a return to classical political economy, where the center of the analysis is to understand how to reproduce the system and how to allocate the social surplus. This is a completely dierent paradigm than the neoclassical one: there is scarcity only in the case of non-produced resources, such as natural resources. All the rest can be produced in order to answer the necessary needs to achieve human well-being, which can be dened in absolute terms (whereas in neoclassical theory, everything is relatively scarce, because humans are never satised, and thus growth is the only possibility to increase human well-being -see [START_REF] Martins | The place of capability approach within sustainability economics[END_REF] on this point). Moreover there is no connection between use-value and exchange value, as the latter depends on the conditions of production and on the distribution of the surplus (whereas in the neoclassical theory, in a free market, prices express the marginal use value of the commodity). And nally there is the possibility to reduce prices of every commodities into a sum of prot weighted quantities of labor [START_REF] Pasinetti | The notion of vertical integration in economic analysis[END_REF]), even in case of joint production, and thus to link Sraa with the political economy of Smith, Ricardo and Marx, where, in a capitalist system of production, human labor is the substance of exchange value.

Sraa's methodology starts by observing the quantities of commodities produced and exchanged on the market by the industries at a given period of production (usually a year), as well as the quantity of labor used by each industries, and it deduces from this observation the prices. Sraa denes price as exchange-values in this way: each commodity, which initially was distributed between the industries according to their needs, is found at the end of the year to be entirely concentrated in the hands of its producer. There is a unique set of exchange-values which if adopted by the market re-stores the original distribution of the products and makes it possible for the process to be repeated [START_REF] Sraa | Production of Commodities by Means of Commodities[END_REF], 1).

There is actually a unique set of prices because he assumes that all processes are interdependent, i.e. there is at least one basic commodity needed directly or indirectly for the production of all commodities 1 . This interdependence also explains why only one rate of prot is possible: this does not mean that the system is at equilibrium, it is just a mathematical property of the system of equations (see [START_REF] Sinha | Listen to sraa's silences: an new interpretation of sraa's production of commodities[END_REF] for a demonstration of this point) 2 .

Then Sraa shows that, if there is a surplus, in order to solve the system of price, an hypothesis on the distribution of this surplus between the agents of production has to be xed. The surplus is composed by the commodities that are not needed directly for system to reproduce -indirectly, part of the surplus needs to be consumed by the workers in order to reproduce their labor force (Sraa thus assumes that there is always a surplus, as soon as he introduces human labor as an input for the system of production). When one of the distribution variable, the rate of prot or the wage, is xed in an exogenous way, the price system can be solved. Sraa's prices thus express the necessary exchanges needed for the reproduction of the system and for answering the given claims on the distribution of the surplus. Prices vary as soon as the hypothesis about the distribution variables changes.

It is important for the following of the paper to understand that Sraa is describing a capitalist system of production: owners of processes ask for prots, workers sell their labor force and receive wage. Commodities are exchanged on a market at the end of the period of production and receive each a unique price.

The price assessed by the owners should allow them to buy new inputs for the next period of production (including labor) and to receive a prot, which is a part of the surplus produced by the economic system. Following this, in Sraa's methodology, there is a price for a good or a service only when one agent of production sells it on the market. He is thus claiming that he wants a return on the good or the service he is producing. If nobody claims for a return, the good does not appear in the equations of production, and thus no price can be assessed on this good. This is why a common critique in ecological economics 1 Sraa only dierentiate between two types of commodities: basics, which are needed directly or indirectly for the production of all goods, and non-basics. The same distinction can be found for the processes.

2 And this interdependence also explains why Sraa's theory is not a cost of production theory: it is desirable at this stage to explain why the ratios which satisfy the conditions of production have been called 'values' or 'prices' rather than, as might be thought more appropriate, 'costs of production'. The latter description would be adequate so far as nonbasic products were concerned, since ... their exchange ratio is merely a reection of what must be paid for means of production, labour and prots in order to produce them -there is no mutual dependence. But for a basic product there is another aspect to be considered. Its exchange ratio depends as much on the use that is made of it in the production of other basic commodities as on the extent to which those commodities enter its own production ... A less one-sided description than cost of production seems therefore required. Such classical terms as 'necessary price', 'natural price' or 'price of production' would meet the case, but value and price have been preferred as being shorter and in the present context (which contains no reference to market prices) no more ambiguous (Sraa (1960), 7). about Sraa's system, arguing that he is violating the physical laws of mass and energy conservation [START_REF] Patterson | Commensuration and theories of value in ecological economics[END_REF]; [START_REF] Christensen | Recovering and extending classical and marshallian foundations for post-keynesian environmental economics[END_REF]), is not relevant. Sraa's system is designed to represent only exchanges of marketed commodities and services, not to represent physical exchanges between industries.

3

Critical review of literature in ecological economics3

This literature can be divided in three parts: rst a big part of the literature is engaged in the valuation of nature debate. Second, some researchers try to develop a neo-Ricardian approach of ecological conicts. And then several works try to use the neo-Ricardian knowledge in the analysis of physical interdependence between processes, in particular for the assessment of CO 2 emissions.

3.1

The valuation of nature debate Sraa was rst used, in the ecological economics literature, as a way to construct prices when the neoclassical theory could not do it. I refer here to the large part of literature on ecological prices, and specically on the energy theory of value.

Authors arguing for the pricing of nature could be separated in two camps: those who think that Sraa's method could be included in the neoclassical theory, and those who see it as a complementary approach to neoclassical theory -both camps actually do not acknowledge that Sraa's rst intention was to attack the neoclassical theory.

On the rst camp, authors (a non exhaustive list of their works could be [START_REF] Costanza | Embodied energy and economic valuation[END_REF]; [START_REF] Costanza | Embodied energy and economic valuation[END_REF]Neill (1981, 1984); [START_REF] Costanza | Dealing with the "Mixed Units" Problem in Ecosystem Network Analysis[END_REF]; [START_REF] Hannon | How might nature value man?[END_REF]; [START_REF] Farber | Economic and ecological concepts for valuing ecosystem services[END_REF]; [START_REF] Sciubba | From engineering economics to extended exergy accounting: A possible path from monetary to resource-based costing[END_REF]; [START_REF] Winkler | Valuation of an ecosystem goods and services, part 1: an integrated dynamic approach[END_REF]) start from the position that a price is not yet dened on the environment because nature is not privately owned, but that pricing the environment would reveal at the same time its relative scarcity and its marginal use value. As a consequence, they assume that the market can correctly manage environmental resources as soon as they receive a price. On the second camp, authors (a non exhaustive list of their works could be [START_REF] Perrings | Economy and Environment[END_REF]; [START_REF] Judson | The convergence of neo-ricardian and embodied energy theories of value and price[END_REF]; [START_REF] Patterson | Commensuration and theories of value in ecological economics[END_REF][START_REF] Patterson | Ecological production based pricing of biosphere process[END_REF]; [START_REF] Patterson | Ecological pricing and transformity: A solution method for systems rarely at general equilibrium[END_REF]; Kemp-Benedict (2014)) do not think that ecological price could reveal the marginal utility of natural resources and are more skeptical on the benets of the market solution. Nevertheless they assume that it would help the protection of natural resources if a price is calculated on them through Sraa's method. They argue that it would help to account for specic natural resource's contribution in the reproduction of ecosystems and in the production of economic wealth. For instance, it is unlikely that the value of protozoa in the ecosystem would be measured in a [Willingness-To-Pay] survey, whereas in ecological pricing, the value of protozoa in the ecosystem would be taken account of by the ... linkages they have with other components of the system [START_REF] Patterson | Ecological production based pricing of biosphere process[END_REF]).

Sraa's theory gives both camps the possibility to build a theory of ecological prices even without a market, because we can represent physical exchanges within environmental processes, and between economic and environmental processes. In order to set prices on natural resources, they rst insert natural resources and processes into the equations of production. Then they include a primary input which is not produced by the system: usually this primary input is solar energy, as in [START_REF] Costanza | Embodied energy and economic valuation[END_REF]; [START_REF] Costanza | The energy embodied in products of the biosphere[END_REF]; [START_REF] Costanza | Dealing with the "Mixed Units" Problem in Ecosystem Network Analysis[END_REF]; [START_REF] Hannon | How might nature value man?[END_REF] but it is possible to have several primary inputs, as in [START_REF] Patterson | Commensuration and theories of value in ecological economics[END_REF][START_REF] Patterson | Ecological production based pricing of biosphere process[END_REF]. And nally they calculate the exchange ratios that ensure the reproduction of the system. If the non-produced input is energy, the prices are said to reect the energy embodied in each commodity.

The problem of this methodology is twofold: rst nature does not function as a capitalist system of production, thus the analytical framework is not easily justied. Second Sraan prices do not reveal the importance of one input, so this framework leads to a result which is not in accordance with the goal of this authors.

On the rst point, if one wants to express the prices in a Sraan system with ecological processes and ecological resources, he has to represent the environment as a set of capitalist processes of production, in the sense that natural processes are privately owned by agents who sells natural resources on the market as commodities, with a unique price set on each natural resource. This capitalist vision of nature has been criticized by several authors [START_REF] Daly | Postscript: Unresolved Problems and Issues for Further Research[END_REF];

O'Connor (1993); [START_REF] Burkett | The value problem in ecological economics[END_REF][START_REF] Burkett | Marxism and ecological economics: toward a red and green political economy[END_REF]): this is a simplistic view of nature, a view that can allow for the capitalist appropriation of nature, and furthermore, a capitalized nature does not ensure that the economy will be more sustainable (see especially [START_REF] O'connor | Value system contests and the appropriation of ecological capital[END_REF] on this point). Furthermore, the value of the non-produced inputs must be equal to the value of the surplus (as there is no prot in ecological pricing models). This is understandable in the context of a capitalist production system where distribution is needed in order to sell the surplus to the agents of production. But it is much more dicult to justify this in the context of ecological production: it looks like an arbitrary assumption, as the surplus is not sold to the owners of the non-produced inputs (see 7).

The second part of the problem with this representation is that it tries to express the value of nature from the economic point of view, or the importance of one input (for instance solar energy) for the production of economic goods.

But Sraan prices represent the exchanges needed to reproduce the system of production: they do not express the importance of one commodity for the reproduction of the system. Actually, in a set of basic commodities (and nature is usually represented as a set of basic process, with the economic processes using the outputs of natural processes in order to produce the economic surplus), if one commodity disappears, the system will collapse. Of course, it is truly important to understand the linkages between natural resources and between nature and the economy -physical input-output analysis could indeed help to do that. But drawing an economic theory based on the importance of this link is extremely perilous 5 , and it is not possible if one wants to stay in the Sraan framework.

The dierent methods trying to assess the importance of nature for the process of exchange value creation using a neo-Ricardian framework [START_REF] Costanza | Embodied energy and economic valuation[END_REF]; [START_REF] Patterson | Ecological production based pricing of biosphere process[END_REF]; Kemp-Benedict ( 2014)) are actually representing the result of a specic political decision about the valuation of nature and about the distribution of the surplus 6 .

Other Sraan approaches in ecological economics

The other Sraan approaches in ecological economics can be separated in two parts. A rst part of the literature tried to develop an ecological political economy, with a focus on ecological conicts (a non exhaustive list of their works could be [START_REF] England | Production, distribution, and environmental quality: Mr. sraa reinterpreted as an ecologist[END_REF] 2013)). The goal is to represent the society as a place where conicting systems of value compete and where power relationships play an important role in order to assess the dynamics of the economic production. This representation leads to questions about who are the agents of production claiming for a return on the environment, or stated dierently, who are the relevant social actors defending the environment [START_REF] England | Production, distribution, and environmental quality: Mr. sraa reinterpreted as an ecologist[END_REF]; [START_REF] Martinez-Alier | Distributional issues in ecological economics[END_REF]). The value of nature can not be reduce to exchange value, but a price on natural processes is sometime needed, for instance when reparation for a damage is asked.

The level of this price is the result of political decision, and depends on the power relationships between the social actors included in the debate. There is an acknowledgment that a deliberation able to express all dierent systems of value could help to solve ecological conicts. They connect their works with the classical economy of Ricardo and Marx, but also with the capability approach developed by Sen [START_REF] Sen | Development as Freedom[END_REF], see the work of [START_REF] Martins | The place of capability approach within sustainability economics[END_REF] on this point).

The second part of the literature follows a completely dierent track, centered on subsystem analysis. A subsystem is dened by Sraa in Appendix A of his book [START_REF] Sraa | Production of Commodities by Means of Commodities[END_REF]):

a system can be subdivided into as many parts as there are commodities in its net product, in such a way that each part forms a smaller self-replacing system the net product of which consists of only one kind of commodity. These parts we shall call 'sub-systems '.

5 Already one century ago, Ostwald argued that a strong relationship existed between the use of energy and economic growth, but warned that we would err if we measured value only in proportion to the amount of free energy [START_REF] Ostwald | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF], p. 164, cited in Georgescu-Roegen (1979)).

6 We can relate this attempts to Ricardo's argument against Lauderdale's idea that if water becomes the exclusive possession of an individual, then national wealth would increase (Ricardo (1817), reported in [START_REF] Douai | Value theory in ecological economics: The contribution of a political economy of wealth[END_REF]). Ricardo answered that, as human production has not changed, the national wealth will just be distributed in a dierent way. Thus the price of water will only express this dierent distribution, and not the part of national wealth that can be related to the use of water.

4 Critical review of literature in neo-Ricardian economics 7

A subsystem represents then all the quantities needed to produce, directly and indirectly, one specic commodity or a group of several specic commodities.

The construction and the analysis of these subsystems have been investigated

by [START_REF] Pasinetti | The notion of vertical integration in economic analysis[END_REF][START_REF] Pasinetti | Growing subsystems, vertically hyperintegrated sectors and the labour theory of value[END_REF]) in order to understand the dynamics of a system of production. His decomposition of a system into vertically integrated subsystems is used by Alcantara and Padilla [START_REF] Alcántara | Input-output subsystems and pollution: An application to the service sector and co2 emissions in spain[END_REF]) in order to investigate the induced emissions of CO 2 on all sectors of an economy due to the consumption of one specic commodity -they take as an example the service sector in Spain. This analysis of emissions per subsystems is deepened by [START_REF] Butnar | Structural decomposition analysis and inputoutput subsystems: Changes in co2 emissions of spanish service sectors (2000-2005)[END_REF]), who study the determinants of changes of CO 2 emissions on the subsystem level, through structural decomposi- tion. Finally Kemp-Benedict (Kemp-Benedict ( 2014)) describes also a vertically integrated structure of an economy, but this time the goal of this representation is to highlight aggregated markups, on labor on one hand and on rent from the exploitation of natural resources on the other hand, and to deduce some observations about taxes on rents, natural resource productivity and resource return on investment.

Critical review of literature in neo-Ricardian economics

All authors trying to connect Sraa with ecological economics come from the ecological economics eld (or the Marxist one). As a consequence, it is not possible to say that a connection exist between the neo-Ricardian authors and the authors of ecological economics. Nevertheless, among the neo-Ricardian school, some authors try to develop a Sraan environmental approach.

Sraa touches upon the subject of natural resources in his book. Natural resources are not produced therefore they do not receive a price, but if they are privately owned and in short supply, their owners can receive a rent, i.e.

a share of the total value of the surplus. His model allows for the assessment of dierential rents, when natural resources are of dierent qualities. He left entirely open the questions about the ways the changing availability of natural resources and services could inuence the price system and the distribution of the surplus. On the contributions that tackle this problem, some try to assess the waste treatment problem and how to take into account recycling in a Sraan framework [START_REF] Hosoda | Recycling and landlling in a dynamic sraan model: application of the corn-guano model to a waste treatment problem[END_REF]; [START_REF] Kurz | Goods and bads: Sundry observations on joint production, waste disposal, and renewable and exhaustible resources[END_REF]); others to take into account the pollution of a natural resource and to implement a polluter-pay principle in a Sraan model (Quadrio Curzio and Pellizzari ( 2003)); and others to describe the management of renewable resources, when their extraction could lead to their extinction (salmon model in [START_REF] Kurz | Theory of Production[END_REF], sh farming vs.

sh in the wild in Erreygers ( 2014)). Furthermore a debate has arisen about the good way to take into account depleting resources, and especially how a royalty on the extraction of non-renewable natural resources could appear and how to properly assess the level of that royalty [START_REF] Kurz | Economic dynamics in a simple model with exhaustible resources and a given real wage rate[END_REF]; [START_REF] Bidard | The corn-guano model[END_REF]; [START_REF] Schefold | Critique of the corn-guano model[END_REF]; Parrinello (2004); [START_REF] Ravagnani | Classical theory and exhaustible natural resources: Notes on the current debate[END_REF]).

Part of the debate concerns the possible integration of the Hotelling's rule [START_REF] Hotelling | The economics of exhaustible resources[END_REF]) into the Sraan framework. The assumptions on which Hotelling's rule is based are: the exhaustible resource is privately owned; the owner wants to maximize the present value of his future prots; the market rate of interest γ expresses the average degree of impatience in the economy and it is equal to the society's rate of time preference; there is perfect competition 9 ;

the quality of the resource is uniform and constant at any time; the stock of the underground natural resource is known; there is perfect foresight with respect to demand for the resource; it is possible to calculate the present social value of the resource, i.e. the total value enjoyed in present and future periods by the consumers of the resource; and nally, quantity produced and price p (t) depend on demand and supply functions of the resource. Given these assumptions, the resource price that maximizes the present social value, when there is no extraction cost, is given by: p (t) = p (0) e γt . This is the price of the resource in the ground, also called 'scarcity rent' or 'royalty'. This royalty should be added to the extraction cost in order to nd the price of the extracted resource.

These assumptions runs clearly against the criticism raised by Sraa and the methodology he adopted. It involves assumptions on the way people behave paradigm from the neoclassical one, and a genuine critique of the market framework from the economic point of view. It actually redenes the boundaries between economics, politics and ethics, as it shows that the valuation process is intrinsically a political process, a result of social choice and power relationships.

As a result, it could help to support certain views on ecological economics, for instance the view developed by Spash [START_REF] Spash | New foundations for ecological economics[END_REF]) that ecological economics should become a more politically and ethically engaged movement.

Furthermore, the Sraan framework is not contradictory to the core beliefs of ecological economics expressed by Røpke in this way: environmental problems are critical; we face basic ignorance in front of them so we need transdisciplinarity, pluralism and system thinking in order to grasp them; nature as value in itself; the economy is embedded in a broader social and cultural system [START_REF] Røpke | Trends in the development of ecological economics from the late 1980s to the early 2000s[END_REF]). If Sraa says nothing about the rst two core beliefs, he acknowledges that the economic variables are dependent on social (the political choice for the distribution of the surplus) and physical relationships (the necessary exchanges in order to produce the commodities). And he proves that prices express the allocation of private property rights on resources and processes, the exchanges needed to produce the commodities and the adopted distribution. He thus shows that prices do not express social wealth or intrinsic values of the commodities.

Nature could be inserted into a Sraan framework, but the price on natural resources will not reveal their use-values: it will just reveal the way nature is appropriated. Nature as a value in itself that can not be expressed by using exchange values.

For the second question, ndings come from the debate around the value in ecological economics and around Hotelling in the neo-Ricardian eld on one hand, and from Sraa's initial goal on the other hand. We have to take into account all levels of criticism Sraa tries to express, mathematically, methodologically and historically, in order to build a real eco-Sraan framework. First, Sraa's theory is non-reconcilable with the neoclassical theory. If one adopt the Sraan point of view, he has to abandon marginal thinking, supply and demand reasoning and equilibrium assumptions. The neoclassical theory on the value of capital should also be abandoned, and thus Hotelling's rule should not be used.

We have to reject as well the idea that prices reect sustainability or utility: they only reect the conditions of production and of distribution during one period of production.

Second we have to adopt Sraa's methodology, starting from empirical observations of the system of production of marketed commodities. It thus appears that nature should not be included in a Sraan framework, unless it has eectively been capitalized. In the general case, the Sraan framework is unable to assess a price on natural resource, as it is not produced: the level of the rent that an owner of a natural resource can claim should be assessed in an exogenous way (through empirical observations, for instance). This does not involve that consequences of depletion or pollution of natural resources on the economy could not be assessed through a Sraan framework. The physical consequences could be integrated in the physical exchanges already needed for the economic production, and thus consequences on the maximum rate of prot and on prices 10 could be observed. But from that kind of observations, no intrinsic value of the environment could be deduced.

Sraa shows that no comparison of prices could be meaningfully drawn as soon as technical change is involved (no point of comparison will exist, so the value of one commodity is only relevant during one period of production). Thus comparisons between dierent periods of production should be done on the physical level, not on the value level. Is the physical rate of surplus (equal to the maximum rate of prot) changed? Is the productivity of one hour of labor dierent? These are physical variables that can be compared among dierent periods of production. Thus studies on the physical properties of system of production are really important, and especially about consequences of interdependence between processes. Analyses in terms of vertically integrated sectors looks promising in this respect (see [START_REF] Garbellini | Blocks and circularity in labour requirements: An interplay between clusters and subsystems in the eu[END_REF] for a recent neo-Ricardian development on that subject).

And nally the analysis must be included in the classical framework of Ricardo, Marx and Sen. In classical political economy, the economy is seen as a circular process of sustainable reproduction, where the surplus should be used eciently, in a non-wasteful way [START_REF] Martins | The place of capability approach within sustainability economics[END_REF]). Thus the classical approach, interested in the reproduction of the economic and natural systems in order to answer objective basic needs and bypass absolute scarcity, is much more compatible with sustainability analysis than the neoclassical. This approach also allows for political and ethical considerations to take part in the denition of those basic needs and the protection of nature could enters in the discussion as a valuable ethical point of view.

Conclusion

Sraa's rst intention was to criticize the neoclassical marginal thinking. After thirty years of work, he only succeeded to write a prelude, but it was sucient to generate a lot of enthusiasm about his project. Unfortunately, he did not engage himself enough in the debate following the publishing of his book. Thus some points of his thinking are still unclear nowadays, and this may explain why the neo-Ricardian school did not grow enough to really contest the neoclassical paradigm. It could nevertheless be a valuable foundation for heterodox ecological economics. It recalls that valuation is a political act, that economic valuation does not express the social wealth, and that nature can not be seen as a produced capital.

Up to now, the major discussion about Sraa in ecological economics was about nature's valuation. The initial assumption of people calculating ecological prices is that it would help the protection of the environment. This is in line with the tragedy of the commons notion that asserts that the non-pricing of natural resources and the non-assignment of private or state property rights to these resources explains why they are over-exploited (Burkett (2006), p. 45).

But it actually promotes the idea that natural resources can be substituted with xed capital. Sraa's analysis helps to understand why natural resources pric-ing does not account for their part played in value creation: they can be seen as basic commodities, supporting our economic system of production. Thus it is not possible to adequately calculate their contribution, as in a set of interdependent processes, if one basic commodity disappears, all processes collapse. The contribution of nature is innite, and its price is a matter of political decision.

This conclusion may have emerge sooner if a connection between researchers of the two elds, ecological and neo-Ricardian economics, were existing. This connection may leads to fruitful results, if in both schools a clear commitment is made to develop a completely dierent paradigm from the neo-classical one.

The reborn of classical economics is a source of interesting developments for sustainability economics and this is why ecological economics authors should denitely pursue their Sraan interest.

7 Annexe: Distribution in a Sraan framework Sraa's equations of production can be written in this way:

(1 + π) Ap + wl = Bp

(1)

With A and B the matrices of input and output coecients, p the column vector of prices, l the column vector of labor coecients, π the uniform rate of prot and w the uniform wage.

In ecological pricing, the usual price system is the following 11 :

Ap + p e e = Bp (2)
With e a column vector, which represents the coecients of a non-produced input and p e the price of the non-produced input.

If there is no surplus, prices express the exchanges in the market that must happened to allow the identical reproduction of the system. Thus, in Sraa, the prot rate and the wage are equal to zero, and in ecological pricing, either the value of the non-produced input is equal to zero, or it is not possible to solve the system. Thus the energy-value theory collapses, as it states that the value of all commodities can be deduced from its embodied quantity of energy: if energy has no value, all commodities should have no value.

If there is a surplus, in Sraa's analysis, it is also sold in the market. As the companies does not need this surplus, it is bought by the agents of production (capitalists or workers). But now, they must be granted with a revenue (prots or wages), otherwise they could not by this surplus. Thus the value of the 11 An exception is [START_REF] Kemp-Benedict | The inverted pyramid: A neo-ricardian view on the economy-environment relationship[END_REF], who asserts absolute rents on natural resources extraction, and explicitly says that the level of the rent is given in an exogenous way. His goal is to express the part of GDP which is based on the use of environmental resource, as he is trying to reveal Daly's inverted pyramid (natural resources extraction account for only 5% of the global GDP but actually all production is related to this initial extraction, [START_REF] Daly | Consumption and welfare: Two views of value added[END_REF].

But what he is actually revealing is by how much the value of the GDP has increased when a rent on natural resources is added. An other exception is [START_REF] Sciubba | From engineering economics to extended exergy accounting: A possible path from monetary to resource-based costing[END_REF], but again with a strong assumption on distribution: in his case, energy, labor and capital are remunerated in the same way.

surplus is distributed to the agents of production, so they can buy the surplus (the question of the equality of this distribution between agents is a matter of political struggle). So equation 1 is justied from an economic point of view, in order to close the system.

In ecological pricing models, there is no prot on natural processes. Thus, if there is a surplus (as for instance in [START_REF] Costanza | The energy embodied in products of the biosphere[END_REF][START_REF] Patterson | Ecological production based pricing of biosphere process[END_REF]), the value of the surplus is distributed to the non-produced input: this is actually justied by the ecological pricing authors on the footing that the real value added comes from the sun, or from other primary inputs. But what they do not acknowledge is that this distribution means that the owners of the non-produced input are able to buy the whole surplus: the total value of non-produced inputs becomes equal to the value of the surplus. Conrmation of this statement can be found in [START_REF] Patterson | Ecological production based pricing of biosphere process[END_REF], where Patterson nds that the net inputs (solar and geothermal energy, fossil fuels, uranium) have a total value of 0.98 times the value of the net output (global GDP): this is actually an imposed result of the analytical framework. And if this analytical framework is justied in a market analysis where companies needs to sell their production, it is hardly justied in ecological pricing, when there is no agents of production and no market.

  and how they take decisions, where Sraa only looked at the consequences of these decisions. It involves expectations about future demand, where Sraa was only interested in the present period of production. It involves the belief that social value could be assessed through an economic reasoning, where, in classical political economy, use value and exchange value are separated. And nally these assumptions allows for the capitalization of the exhaustible natural resources, as the resource is privately owned and a claim for a price (or royalty) on that resource can be stated by the owner. The debate within the neo-Ricardian school is not closed however -some argue that Ricardo was actually close to Hotelling's thinking on the subject of exhaustible resources[START_REF] Kurz | Ricardo on exhaustible resources, and the Hotelling Rule[END_REF]) 10. Thus there is a real interest into environmental questions in the neo-Ricardian eld of research, and it could be possible to connect their works to Sraan approaches in ecological economics. In the next section, I will draw the main lines around which this connection should be constructed. 5 Key ndings Is Sraa's analysis interesting for the ecological economics eld? And how to use Sraa in ecological economics? For the rst question, we can go back to the characteristics of the ecological economics eld, as a multidisciplinary eld, open to heterodox theories. Sraan economics are economics, with very few openness from the neo-Ricardian researchers to other scientic elds. Nevertheless it is an heterodox stream of thought within the economic science, a dierent 9 This assumption is later relaxed in Hotelling's model, but this is never done in the related neo-Ricardian models. 10 But this argument leaves open the question about the dierences between the Sraa's and Ricardo's thoughts on that subject.

I am referring to authors writing in the Journal of Ecological Economics, or specically writing about the ecological economics eld, and at the same time referring to Sraa.


Yoann Verger
email: yoann.verger@uvsq.fr

A Critique of Attempts to Introduce Hotelling's rule in Sraa's Theory

Keywords: Sraa, Rent, Royalty, Hotelling's rule, Exhaustible resource

.

Sraa and exhaustible resources

As is well known, Sraa's theory [START_REF] Sraa | Production of Commodities by Means of Commodities[END_REF] of prices deals only with produced commodities. In the case of non-exhaustible natural resources such as land, Sraa has however shown that rent of land can also be consistently explained by his theorythe existence of rent implies the existence of more than one technique in use to produce a commodity at a moment (see Sraa, 1960, Ch. 11). In the case of exhaustible natural resources such as oil in the ground, [START_REF] Sraa | Production of Commodities by Means of Commodities[END_REF], however, remains silent. The silence of Sraa on the question of how to explain the positive value of exhaustible natural resources has led many neo-Ricardian scholars to introduce Hotelling's rule into Sraa's theory to ll this gap in his theory. In this section I argue that all such attempts contradict Sraa's fundamental theoretical position. This should be expected since Hotelling's rule is built on neo-Classical foundations, which does not t with Sraa's theoretical foundations. I further go on to provide an alternative explanation for positive value of non-produced exhaustible resources, which I contend is consistent with Sraa's theoretical foundations. In section 2, I briey describe the literature that has attempted to introduce Hotelling's rule into Sraa's theory; in section 3, I show that introduction of Hotelling's rule changes the system of prices in Sraa's theory; in section 4, I show why these attempts are in contradiction to Sraa's theoretical foundations; in section 5, I present my alternative explanation, which I think ts with Sraa's theoretical foundations; in section 6, I present the conclusion of this paper.

2 The neo-Ricardian attempts to introduce Hotelling's rule in Sraa's theory

Before taking up the leading neo-Ricardian attempts to incorporate Hotelling's rule in Sraa's theory, it may be helpful to take note of the assumptions on which Hotelling's rule is based. These assumptions are: the exhaustible resource is privately owned; the owner wants to maximize the present value of his future prots; the market rate of interest γ expresses the average degree of impatience in the economy and it is equal to the society's rate of time preference; there is perfect competition;

2 the quality of the resource is uniform and constant at any time; the stock of the underground natural resource is known; there is perfect foresight with respect to demand for the resource; it is possible to calculate the present social value of the resource, i.e. the total value enjoyed in present and future periods by the consumers of the resource; and nally, quantity produced and price p (t) depend on demand and supply functions of the resource. Given these assumptions, the resource price that maximizes the present social value, when there is no extraction cost, is given by: p (t) = p (0) e γt . This is the price of the resource in the ground, also called 'scarcity rent' (e.g. in [START_REF] Solow | The economics of resources or the resources of economics[END_REF] or 'royalty' (e.g in [START_REF] Devarajan | Hotelling's "economics of exhaustible resources": Fifty years later[END_REF]. This royalty should be added to the extraction cost in order to nd the price of the extractible resource.

In this paper we do not take up the problems with Hotelling's rule itself.

3 Here we focus only on the reasons why it is not correct to integrate Hotelling's rule inside Sraa's framework. Among the major contributors in this area are [START_REF] Kurz | Exhaustible resources in a dynamic inputouput model with 'classical' features[END_REF][START_REF] Kurz | Economic dynamics in a simple model with exhaustible resources and a given real wage rate[END_REF], 2001, 2009), Schefold [START_REF] Schefold | Mr Sraa on joint production and other essays[END_REF][START_REF] Schefold | Critique of the corn-guano model[END_REF], Bidard and Erreygers (Bidard and Erreygers, 2001a,b) and Parrinello [START_REF] Parrinello | Exhaustible natural resources and the classical method of long-period equilibrium[END_REF][START_REF] Parrinello | The price of exhaustible resources[END_REF][START_REF] Parrinello | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF] 4 . Kurz and Salvadori ask the question: why the owners of such [exhaustible] resources do not sell the whole amount that they own at current prices in order to invest the proceeds in an industry where the normal rate of prot can be obtained [START_REF] Kurz | Exhaustible resources in a dynamic inputouput model with 'classical' features[END_REF]). Their answer is: the storage of natural resources, which may be considered as a `conservation industry', cannot be operated if it does not yield a `royalty' to the owners of the resources [START_REF] Kurz | Exhaustible resources in a dynamic inputouput model with 'classical' features[END_REF]. They justify this precisely because they assume a long-term equilibrium framework:

Hotelling's rule [see [START_REF] Hotelling | The economics of exhaustible resources[END_REF]] is nothing but the application of the concept of a uniform rate of prot to all processes in the economy, whether these are conservation or production processes [START_REF] Kurz | Classical economics and the problem of exhaustible resources[END_REF]. Thus Kurz and Salvadori develop a dynamic Input-Output model under the assumption of free competition (uniform rate of prot), constant returns to scale and a given real wage rate where the Hotelling's rule applies.

For Bidard and Erreygers: a consistent theory of exhaustible resources is needed just as much as a consistent theory of prices (Bidard and Erreygers, 2001a). They accept Hotelling's rule and apply it in the Sraan framework without providing any justication:

In order of analytical complexity, the question of exhaustible resources comes next to that of land with, however, a qualitative gap: the price of an exhaustible resource changes with time, in order to compensate his owner for waiting (Hotelling's rule) (Bidard and Erreygers, 2001a).

3 There could be rst an empirical criticism about the reality of this rule. Some empirical facts are not taken into account (market imperfections, dierence in ore qualities, capacity and investment constraints, exploration costs, e.g. see [START_REF] Krautkraemer | Nonrenewable resource scarcity[END_REF] and empirical analyses of price do not always reect the Hotelling's rule (e.g. see [START_REF] Hart | The elephant in hotelling's room[END_REF].

4 We could also add to this list [START_REF] Ravagnani | Classical theory and exhaustible natural resources: Notes on the current debate[END_REF] who criticizes both backstop technology models supposing perfect foresight (models of the type of [START_REF] Kurz | Economic dynamics in a simple model with exhaustible resources and a given real wage rate[END_REF]Bidard and Erreygers, 2001a;[START_REF] Schefold | Critique of the corn-guano model[END_REF], and intensive rent models making strong assumptions on the existence of a second process at each period of time allowing to calculate the rent (models of the type of [START_REF] Parrinello | The price of exhaustible resources[END_REF]. He shows that in the case of oil industry in the US, contracts between managers and owners do not show that royalties are increasing following the rate of prot: they are actually higlhy dependent on norms, previoulsy created through bargaining and power relationships. These norms can be broken when power relationships change and, in case of State ownerships, royalties depend on public policy and international relationships. So he argues that royalties should be considered as an exogenous variable, on the same ground than the prot rate (or the wage rate).

They acknowledge that the Hotelling's rule may be based on weak empirical basis, but they only care about theoretical consistency. They work with a small system with one commodity and one exhaustible natural resource, with no extraction cost and constant returns to scale. In my view their model has been successfully generalised by Schefold [START_REF] Schefold | Critique of the corn-guano model[END_REF].

Schefold [START_REF] Schefold | Mr Sraa on joint production and other essays[END_REF][START_REF] Schefold | Critique of the corn-guano model[END_REF] is ambiguous on the use of Hotelling's rule in a Sraan framework. On the one hand, he thinks that Ricardo's theory of rent can be applied to the extraction of most exhaustible resources [START_REF] Schefold | Critique of the corn-guano model[END_REF] and rightly nds that the assumptions behind the Hotelling's rule (perfect foresight on future demand, technology and stocks) are not compatible with a Classical framework if the natural resource is a basic good:

if the resource really gets exhausted over an intermediate time span, its price must rise; if it is a basic commodity, relative prices of other commodities must change, complicated patterns of substitution in the processes of production and of consumption may ensue in all other periods and perfect foresight becomes implausible [START_REF] Schefold | Critique of the corn-guano model[END_REF].

However, on the other hand, while playing the game of logic and [trying] to reconstruct a theory of exhaustible resources within the context of classical theory, Schefold assumes that the rise of the price of the exhaustible resource, in the absence of technical progress, is the expression of a future diminution of the surplus available for consumption and investment [START_REF] Schefold | Critique of the corn-guano model[END_REF]. He also justies this assumption on the ground that a classical model is a long period model [START_REF] Schefold | Critique of the corn-guano model[END_REF]. He then constructs a model with constant returns to scale, an exhaustible resource (which could be a basic commodity)

and the Hotelling's rule on the price of the resource in the ground (i.e. before extraction).

Parrinello [START_REF] Parrinello | Exhaustible natural resources and the classical method of long-period equilibrium[END_REF][START_REF] Parrinello | The price of exhaustible resources[END_REF][START_REF] Parrinello | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF] follows an interesting approach: the royalty in his case is equivalent to an intensive rent, whereas for other neo-Ricardians the royalty is seen as the prot of the conservation process. Furthermore, instead of assuming rst that the royalty increases and then looking at the consequences on quantities (as in the rst three models), he assumes that quantities are given and then looks at its consequence on the royalty. But then how is it possible to face the contradiction that the royalty should follow the Hotelling's rule? In fact, for Parrinello, Sraa's system of production is assumed to be the result of prot maximizing choices under conditions of competition and long period equilibrium [START_REF] Parrinello | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF].

He follows here the interpretation of [START_REF] Garegnani | On a change in the notion of equilibrium in recent work on value and distribution: a comment on Samuelson[END_REF] that a process of gravitation is able to generate a uniform rate of prot and a choice of technique such that the system of production is square. But this process of gravitation is seriously questioned by [START_REF] Sinha | A sraan critique of the classical notion of center of gravitation[END_REF]. Furthermore, there is no reason to accept a priori that there is competition and prot seeking behavior [START_REF] Parrinello | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF], because the uniformity of the rate of prot is deduced from the equations of the system of production [START_REF] Sinha | Listen to sraa's silences: an new interpretation of sraa's production of commodities[END_REF]. So from my point of view, Parrinello's model is good (no assumption regarding constant returns to scale, no price on the resource not yet extracted) but there is no need to justify the Hotelling's rule with it. I will now concentrate my criticism on the rst three models in sections 3 and 4, before presenting a modied version of Parrinello's model in section 5.

3 Introducing Hotelling's rule inside Sraa changes the price system

Let us start with Kurz and Salvadori's model, which does not include rent [START_REF] Kurz | Economic dynamics in a simple model with exhaustible resources and a given real wage rate[END_REF].

There are n commodities, and m > n processes operating with constant returns to scale. The price column vector during the period of production t is written p t and the line activity vector (describing the scale of operation for each process) is written x t . The quantity of natural resources in the ground at t is represented by a line vector z t . The period of production is uniform for all processes. A process j is dened by quadruplet (a j , b j , c j , l j ), with a j the line vector of inputs, b j the line vector of outputs, c j the line vector of natural non-renewable resources (before extraction) used and l j the labor coecient. The wage rate w is given in real terms (a line vector of given commodities). Assuming that there is perfect competition, the rate of prot r is uniform for all processes. Then Kurz and Salvadori write down their system of equations, with y t the column vector of royalties on natural resources and γd the share of the surplus going to prot and royalties (given in proportion to a constant column vector of commodities d):

for each time t ∈ N 0 , the following [. . . ] equations are to be satised:

x t+1 Bp t+1 = x t+1 [(1 + r t ) (Ap t + Cy t ) + lwp t+1 ] (2) Bp t+1 (1 + r t ) (Ap t + Cy t ) + lwp t+1 (3) z t+1 y t+1 = (1 + r t ) z t+1 y t (4) y t+1 (1 + r t ) y t (5) x t (B -lw) p t = (x t+1 A + γd) p t (6) x t (B -lw) x t+1 A + γd (7) z t y t = (x t+1 C + z t+1 ) y t (8) z t x t+1 C + z t+1 (9)
γ > 0, p t 0, y t 0, z t 0, x t+1 0 (10) [START_REF] Kurz | Economic dynamics in a simple model with exhaustible resources and a given real wage rate[END_REF].

Eq. 2 means that there is production only if the equilibrium rate of prot is obtained. Eq. 3 means that it is not possible to obtain more than the equilibrium rate of prot by producing a commodity. Extracting a natural resource is here seen as producing a commodity, so extraction costs are possible, and there is extraction only if the equilibrium rate of prot is earned from this activity.

Eq. 4 means that there is storage of natural resources only if the equilibrium rate of prot is earned from it. Eq. 5 means that it is not possible to earn more than the equilibrium rate of prot by storing natural resources. Eq. 6 and Eq. 7 mean that if a commodity is produced without being used in production or consumed in the next period, its price falls to zero. Eq. 8 and 9 mean that if a natural resource is stored without being used in production or stored again in the next period, its royalty falls to zero. Furthermore Kurz and Salvadori assume given amounts of commodities and exhaustible resources available at time 0. So they are building a neo-Classical system of linear programming, with an optimization program in order to choose the most ecient technological pathway (we can nd a similar model in [START_REF] Pasinetti | Lectures on the Theory of Production[END_REF], Appendix to chapter 6). Then here we are not talking about Sraan production price, but about allocated price, because initial resources are given and thus prices reect the ecient allocation of scarce goods.

Given the initial set of natural resources and commodities, their problem is to choose the technology so that they obtain a uniform rate of prot and zero price for excess supply. But in the context of Sraa, if an input becomes less available, the problem that arises is not a question of technological change but a question of change in the required production levels. Characteristics of both systems are really dierent. In Kurz and Salvadori model, prices and royalty depends on the allocation of a given set of resources and can be seen as a kind of scarcity index (price is equal to zero if there is overproduction and royalty is equal to zero if there is overconservation). Whereas in Sraa, prices depend only on the production level and on the distribution of the surplus (between wage and prot) at the given period and there can be a strictly positive price even if there is overproduction [START_REF] Pasinetti | Lectures on the Theory of Production[END_REF], Appendix to chapter 6). Now from equation Eq. 4 and equation Eq. 8 we get:

z t+1 y t+1 = (1 + r t ) (z t -x t+1 C) y t (11)
Then we can add equation Eq. 4 and equation Eq. 11, and we get, through simplications:

x t+1 Bp t+1 + z t+1 y t+1 = (1 + r t ) (x t+1 Ap t + z t y t ) + x t+1 lwp t+1 (12)
Now from equations Eq. 2 and Eq. 6 we get:

x t+1 Ap t = x t (B -lw) p t -γdp t (13)
Then we can add equations Eq. 12 and Eq. 13, and we get:

x t+1 Bp t+1 + z t+1 y t+1 = (1 + r t ) [x t (B -lw) p t -γdp t + z t y t ] + x t+1 lwp t+1 (14)
This equation, i.e., Eq. 14, shows the relation between the value of inputs and the value of outputs of the system. This relation is close to the one suggested by [START_REF] Schefold | Critique of the corn-guano model[END_REF], itself a generalization of Bidard and Erreygers proposition (Bidard and Erreygers, 2001a). With only one natural resource entering into the production of only one commodity i, Eq. 14 is written in Schefold's system as:

p i,t + z t+1 y t+1 = (1 + r) [a i p t + z t y t ] + w t l i (15)
But while in Kurz and Salvadori model the value of the rent depends on initial endowments, in Schefold model the value of the rent depends on a backstop technology (the technology which can replace the technology using the exhaustible resource) and the expected life of the mine. The rent is calculated at the period of exhaustion, then backward until the initial period through the assumption of Hotelling's rule. However, in some sense, both the models are back-stop technology models: in the rst one the existence of a backstop technology is just a hypothesis in order to save the economy from destruction whereas in the second model, a backstop technology is mandatory in order to solve the system.

But the fact that a future technology is needed to solve the value problem in the present period is clearly in contradiction with the way Sraa constructed his model, prices being entirely dependent on present technology and present distribution. [START_REF] Parrinello | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF] says that in Kurz and Salvadori and Schefold models, the determination of equilibrium prices is not self-contained in each period because of special intertemporal link. In Kurz and Salvadori, value is related to the initial endowments (that is why we can talk about allocated prices, and not about prices of production) and in Schefold, value is related to the future back-stop technology (a change in this technology causes a change in the value of the royalty in the present).

So we can see that:

1. Kurz and Salvadori, as well as Schefold and Bidard and Erreygers, start from the fact that royalties exist, but they do not explain why it is so, and their models do not show it either.

2. Then they say that it is possible to earn a rate of prot from the natural resource storage activity, because otherwise owners of such capital should sell everything at the rst period.

3. Then equilibrium assumption ensures that the rate of prot that is earned by storing natural resources is the equilibrium one. Thus the Hotelling's rule is deduced from this assumption (albeit from two dierent methods). 4. Finally this has an impact on relative prices, especially if one natural resource is a basic (see subsection 7). I will explain in the next subsection why the rst three points (existence of royalties, prot on conservation industry and equilibrium assumption) cannot cope with the Sraan framework.

4 Is introducing Hotelling's rule a change in the Sraan direction?

Theory of value

The reasons why the Hotelling's rule cannot be applied in a Sraan framework lies in the notion of `royalty' in Hotelling and the way Sraa denes value. In Hotelling, royalty comes from the fact that natural resource (not yet extracted) is seen as capital.

5 But why it is seen as capital? It is so because in the future it may be possible to earn something by extracting the natural resource. Value comes from expectations -a subjective consideration.

Furthermore royalty is sometimes renamed as the cost of use of the exhaustible resource, or the user cost. 6 It means that we are giving up future prots by using the resource now; the royalty is rewarding owners of the natural resource for future consumer's satisfaction, otherwise everybody would extract everything now. But Sraa's prices reect only past and present exchanges, not future ones.

Actually the work of Sraa goes as this (I follow here the new interpretation of Sinha, 2012):

1. It is a description of the real market at one moment of time;

2. The only magnitudes taken into account are objective magnitudes: physical quantities of commodity inputs and employed human labor used and physical quantities of outputs;

3. Then the system of value can move with one degree of freedom: this is closed when the wage (or the rate of prot) is xed in an exogenous way.

Thus there is no room for expectation from the agents: no subjective data are taken into account. So the essential theory of value behind the Hotelling's rule does not t with Sraa's theory.

5 How much of the proceeds of a mine should be reckoned as income, and how much as return of capital? (Hotelling, 1931, p. 139).

6 The exhaustible resource is commodity n [. . . ] Its value in use is the same as that of its future substitute in that both render the same services to all users of the commodity [START_REF] Schefold | Critique of the corn-guano model[END_REF]. The cost of production of the commodity is equal to its cost of extraction plus a user cost (due to the prospective exhaustion of the ore in the ground) [START_REF] Schefold | Critique of the corn-guano model[END_REF].

The conservation industry

Furthermore royalties in Hotelling can be seen as the prot of the conservation industry [START_REF] Kurz | Exhaustible resources in a dynamic inputouput model with 'classical' features[END_REF]. This is a strange industry: there is no worker, and no need for inputs from other industries. Apparently there is no creation of value by keeping a resource in the ground, but in the Hotelling theory, owner of a stock of exhaustible resources should earn royalties on conservation. This is not in accordance with Sraa's value theory. Pasinetti shows that it is possible to reduce prices of every commodities into a sum of prot weighted quantities of labour [START_REF] Pasinetti | The notion of vertical integration in economic analysis[END_REF], even in case of joint production. But this reduction is not possible anymore if an industry is not using labor. The implication of this is that political economy, from the point of view of Marx and Sraa, should be abandoned. In their approaches, the economic value comes from socially approved labor: by giving value to a resource kept in the ground without labor, we are changing the theory of value. Now every good can be a source of economic value as long as it can be a source of user value: and then every owner of such goods could ask for a reward on his conservation industry.

If we allow the Hotelling's rule, then we allow the neoclassical theory of value starting from Menger, Jevons and Walras, and the distribution theory behind it. It is oered as a preliminary and there is no a priori reason why, on the basis of it, an equilibrium system should be built: there is some room left for it, as this is confessedly indeterminate; but the question is whether there is room enough for the marginal system.

The result of equilibrium due to perfect competition

(D3/12/46:20, dated 2 April 1957; quoted in [START_REF] Sinha | The new interpretation of sraa's prices: a response to heinz kurz[END_REF] So in a Sraan framework prices should not be inferred from subjective appreciation, but only from observable data (quantities exchanged). Sraa was so determined in this direction that he was even reluctant to use the amortisation 7 The main argument for the theory of equilibrium in Sraa's work is that he was reasoning with a uniform rate of prot. But actually this uniform rate of prot in Sraa's equations is not the result of a gravitation process but the inevitable result of the system of equations. It is not an equilibrium rate of prot, but rather an ex-post accounting rate of prot of the real economy under study during one period of production [START_REF] Sinha | Listen to sraa's silences: an new interpretation of sraa's production of commodities[END_REF]. 10 technique on xed capital [START_REF] Kurz | Removing an 'insuperable obstacle' in the way of an objectivist analysis: Sraa's attempts at xed capital[END_REF]. But the Hotelling's rule y t+1 = (1 + r)y t is not an observable data. This rule may be inferred from observable data: this is precisely what [START_REF] Parrinello | The notion of eectual supply and the theory of normal prices with exhaustible natural resources[END_REF] tries to do, incorporating royalty as an intensive rent, and then showing that a given technological path could allow the rent to evolve in a way consistent with the Hotelling's rule. But the Hotelling's rule cannot be given in the premises without breaking Sraa's position on the theory of value.

5 What could be a Sraan proposition on exhaustible natural resources I will argue here that Sraa's system, without introducing any subjective notion and assumption on equilibrium, is able to explain the exhaustible resource problem.

The question of constant return to scale

One criticism against this system is about the use of constant returns to scale (e.g. see [START_REF] Solow | Comments on Scott Carter[END_REF]. In this regard, an easy criticism was put forward by [START_REF] Menger | Principles of Economics[END_REF]: if constant returns to scale is assumed then given an exhaustible natural resource of uniform quality and uniform location, the resource price after extraction will stay the same even if it becomes exhausted. But Sraa's model should not be confused with Leontief one, or with the theory of Ricardo: Sraa warned explicitly that no assumption regarding constant returns to scale is made in his work (see Sraa, 1960, p. v).

Should we assume constant returns to scale in the case of extraction of exhaustible resources?

8 Firstly, constant returns are extremely rare, if not non- existent, in the case of natural resource extraction. The general law is that the more a resource becomes scarce, the more dicult is its extraction (this concerns all extractive industries, but also forestry, shery, etc.). 9

Secondly, here is what happens when the natural resource becomes scarce:

either demand stays at the same level or the demand decreases (the commodity is substituted with another good). If the demand stays up, either a new technique of production will be introduced in order to satisfy the demand, giving rise to a rent on the more protable technique, or the system will break down. So

Menger's criticism is not valid because, for any given exhaustible resource, it is really unlikely that we observe constant returns to scale for its extraction and, at the same time, that no second technique for the extraction would appear to answer the demand.

Below I present a model of an exhaustible resource, say coal. A commodity is produced, so a natural resource becomes a commodity only after extraction:

8

The assumption of zero cost of extraction is just one example of the assumption of constant returns to scale in this case.

9 One exception could be the case where all lands are cultivated and some disappear because of urban expansion.

Tab.

Two processes, constant returns to scale

Let us rst have one mine of Coal, privately owned. Let us say that Extracted

Coal is a basic commodity, entering in the production of its own process of production as well as in the process of production of wheat; and let us rst assume that there are constant returns to scale operating in both industries.

Table 1 shows the system of production.

The productivity of the extraction of Coal depends on the initial amount of

Coal in the mine, C (0). It is assumed that we have the extraction function Eq.

16:

b 22 = αC (0) (16)
With α a constant and 0 < α < 1.

Then prices of production are:

b 11 p 1 = (1 + r) (a 11 p 1 + a 12 p 2 ) + l 1 w (17) αC (0) p 2 = (1 + r) (a 21 p 1 + a 22 p 2 ) + l 2 w (18)
With r the uniform rate of prot (as it must be uniform in Sraa's equations, and not because of an equilibrium assumption, [START_REF] Sinha | Listen to sraa's silences: an new interpretation of sraa's production of commodities[END_REF], and w the wage.

Then we have an expression of the price of extracted coal which is independent both of time and, more importantly, of the quantity of resource left:

p 1 = (1 + r) a 12 p 2 + l 1 w b 11 -(1 + r) a 11 (19)
p 2 = (1 + r) a 21 l 1 w b 11 -(1 + r) a 11 + l 2 w / αC (0) -(1 + r) a 22 - (1 + r) 2 a 12 a 21 b 11 -(1 + r) a 11 (20)
Here we cannot have any information about the fact that the natural resource is scarce or not: but as I said earlier, this model is highly implausible. In real systems, cost of extraction should be correlated with the quantity of the 10 This idea comes from the economics of renewable resources [START_REF] Gordon | The economic theory of a common-property resource: The shery[END_REF][START_REF] Schaeer | A study of the dynamics of the shery for yellown tuna in the eastern tropical pacic ocean[END_REF], following a presentation from [START_REF] Erreygers | Renewable resources in a long-term perspective: the corntuna model[END_REF] on this subject.

the owner of the mine or by the workers. We will assume that the new technique is less protable and that rent ρ goes in the pocket of the mine owner:

b 11 p 1 = (1 + r) (a 11 p 1 + a 12 p 2 ) + l 1 w (23) αC (0) p 2 = (1 + r) (a 21 p 1 + a 22 p 2 ) + l 2 w + ρ (24) b 32 p 2 = (1 + r) (a 31 p 1 + a 32 p 2 ) + l 3 w (25)
The price of the Extracted Coal/Synthetic Coal and the rent are then:

p 2 = (1 + r) a 31 l 1 w b 11 -(1 + r) a 11 + l 3 w / b 32 -(1 + r) a 32 - (1 + r) 2 a 12 a 31 b 11 -(1 + r) a 11 (26) ρ = αC (0) -(1 + r) a 22 - (1 + r) 2 a 12 a 21 b 11 -(1 + r) a 11 p 2 - (1 + r) a 21 l 1 b 11 -(1 + r) a 11 + l 2 w (27)
This rent again is not time-dependent.

Three processes, non constant returns to scale

But now if we want to add both modications, when the extraction function is represented by Eq. 21, the system of price becomes:

p 2 = (1 + r) a 31 l 1 w b 11 -(1 + r) a 11 + l 3 w / b 32 -(1 + r) a 32 - (1 + r) 2 a 12 a 31 b 11 -(1 + r) a 11 (31) ρ (t) = αC (t) -(1 + r) a 22 - (1 + r) 2 a 12 a 21 b 11 -(1 + r) a 11 p 2 - (1 + r) a 21 l 1 b 11 -(1 + r) a 11 + l 2 w (32)
Now an interesting feature of it is that the price of Extracted Coal and Synthetic Coal is not time-dependent, as its price depends on the third technique (the least protable one -Synthetic Coal production). Thus only the rent is time-dependent, and the rent decreases as the extraction cost increases. If the extraction cost gets too high, the Synthetic Coal process could become more protable than extracting coal from the ground: at that point, the rent will become negative, meaning that it will be earned by the owner of the Synthetic Coal process.

Concluding comments

The price of the Coal mine could depend on the rent that the buyer expects to get (using the rate of prot as a depreciation rate for future prot). The current rent and prot rate can be calculated by Sraa's method. In case of an arbitrage between two competitive investments, a calculation of the value of the Coal mine could be:

p mine = ∞ 0 [r (t) αC (t) p 2 (t) + ρ (t)] x 2 (t) e -r(t)t dt (33)
With x 2 (t) is the intensity of the extraction process. And from that one could argue that the value of Coal in the ground is v Coal = p mine /C(t). But this calculation cannot be done without some expectations on future rate of prot, future demand and future prices. On that subject Sraa cannot help, so the price of the mine, as well as price of Coal as a natural resource not extracted yet (or price of a land not cultivated yet) should stay indeterminate within the bare Sraan framework.

A convention about what future rates should be applied may be conventionally given. These rates don't have to be the same as the rates that actualize in the future; for example, 20 years of current prevailing rent used to be the convention to determine the price of land at the present. Then, starting from this convention, price of natural resource into the ground can be found. But then we have a conventional price and it should be pointed out that it does not reect the true value of the resource, neither its cost of use.

We found here again the dierence between the value theory of the Austrian school and of Hotelling and the value theory of Sraa. For the rst ones, if one land is used in one process of production, thus, not only this land, but all land have a value, dened by the consumer's satisfaction that we can create with this land (now and in the future that is why even a not yet used land can receive a royalty). Hotelling was in the same line of thought when he is assuming a royalty on conservation process. But for Sraa, value derives only from production, and land not used in production cannot receive a rent.

The analysis in subsection 5 can easily be widened to other natural resources: land of course, but also renewable resources. There is still lot of work to be done in order to understand the relationships between rents and biophysical limitations of natural resource extractions.

Appendix

Here I prove that there is a change in relative prices when the Hotelling's rule is introduced in a Sraan framework, starting from Kurz and Salvadori's model, described in subsection 3.

Let us suppose that wages are set to zero (thus the rate of prot r t becomes the maximum rate of prot R t ). As coecients of production are constant, we can clearly see that the price system is changing when we add the Hotelling's rule. If there was no Hotelling's rule then Eq. 12 would be written as:

x t+1 Bp t+1 = (1 + R t ) x t+1 Ap t (34)
The scarcity of the natural resource is not anymore taken into account: if the resource gets scarcer the activity level will be lower for those processes using (directly or indirectly) the natural resource. But as there are constant returns to scale, the price relation would not change. As the technology is not changing, the maximum rate of prot has no reason to change from time to time, so we can change the system into:

x t+1 Bp t+1 = (1 + R) x t+1 Ap t (35)
This maximum rate of prot R is the standard rate of prot. Assuming the Hotelling's rule, royalties increase from period to period and then prices may change from period to period. However, is it possible that introducing royalties does not change the standard system, and thus does not change relative prices and R? From equation Eq. 12 we should get:

x t+1 Bp t+1 = (1 + R) (x t+1 Ap t + z t y t ) -z t+1 y t+1 (36) If introducing royalties does not change relative prices, we should have an equality between relations Eq. 35 and Eq. 36:

(1 + R) x t+1 Ap t = (1 + R) (x t+1 Ap t + z t y t ) -z t+1 y t+1 (37) Simplifying we get:

(1 + R) z t y t = z t+1 y t+1 (38) And from equation Eq. 4 we have:

(1 + R) z t y t = z t+1 y t (39) So in order that prices do not change with the introduction of royalties, it has to be that z t = z t+1 ; i.e., that there should be no decrease in the stock or in other words, exhaustible natural resources should actually not be used. So we can conclude that royalties do not inuence the system of price only if natural resources are not basic goods.

Finally, for

 for Hotelling and his followers, perfect competition gives us a uniform market rate of interest and this allows us to adequately discount future prots and to nd the Hotelling's rule. But for Sraa there is no equilibrium a priori. 7 This is best illustrated by this draft 'Preface' of Sraa's book, Production of Commodities by Means of Commodities: This is not proposed as a complete system of equilibrium. The data assumed are not sucient to determine either distribution or values. Only the eects of hypothetical, arbitrarily assumed extra data (such as wages, or the rate of prots) are discussed. . . .

  1: system of production of wheat and Extracted Coal, constant returns to

	scale.
		Input			Outputs
	Wheat	Extracted Coal	Labor	Wheat	Extracted Coal
	a 11	a 12	l 1	b 11	0
	a 21	a 22	l 2	0	b 22
	in our case I make the distinction between the natural resource conserved in the
	ground Coal, which bears no price, and the commodity Extracted Coal.

This assumption is later relaxed in Hotelling's model, but this is never done in the following neo-Ricardian models.