1This paper aims at contributing to an extension to disequilibrium of the theories of value of a Classical inspiration.  The exclusion of disequilibrium from most contemporary research is all the more questionable since disequilibrium is intrinsically linked to the social division of labour which is at the very basis of market economies: usually, the agents’ expectations and plans are not met, hence the need for an analysis of quantitites and prices outside the equilibrium path.
2The development of the theory of value beyond equilibrium must not be confused with a theory of stability. A reflection on the “tendency to equilibrium” was at the core of the first theoretical contributions in Political Economy, as early as in the first part of the eighteenth century, and its aim was and still is to justify the theoretical primacy given to equilibrium analysis. In that approach, the study of disequilibrum states is a secondary problem, whereas it becomes a central issue in a more general theory of value.
3Marx (1867) stressed that the social division of labour goes with a divergence between the private and the social evaluations of produced commodities. This is the starting point of an analysis of reproduction in disequilibrium. By following that path, we are closer to Torrens’s (1821) and Marx’s (1885) traditions than to Ricardo’s and Sraffa’s (1960). Torrens and Marx took into account the capitalists’ accumulation targets and the budget constraints. We go farther in that direction by considering a monetary economy, and explicitly taking into account two economic phenomena: first, we introduce a market mechanism, suggested by Cantillon (1755), for the determination of monetary prices and the allocation of inputs; second, we consider an institutional device for the settlement of monetary imbalances. These two phenomena lead to the effective allocations and productions.
4The model takes into account some features which are often neglected in modern Classical theory, namely: transactions in disequilibrium as well as in equilibrium; the formation of prices; individual real and monetary disequilibria; the role of price expectations; money and institutional rules for monetary settlements; and the real effects of the integration of money in the theory of value. The disequilibrium results from the contradiction arising between the capitalists’ accumulation desires and the physical constraints as represented by the wage basket, the methods of production and the existing quantities. The study of those dynamics shows that the economy may converge towards the natural quantities and the natural prices as defined by the Classicals or towards a limit cycle.
5The paper extends some of our previous analyses (Benetti et al., 2012 and 2014), which studied reproduction in a bisector model, respectively without and with money. Here, we return to the monetary model with minimal formalization and develop its economic interpretation. The main innovation is the connection of the rule of formation of prices, which plays a central role in our treatment of disequilibrium, with Marx’s analysis of the division of labour. We show how this rule may solve some of the problems that Marx left open. Other significant innovations are: (i) the stress on the divergence between an approach based on our notion of temporary disequilibrium and the Ricardian approach which considers disequilibrium as an accidental departure from the natural position; (ii) a detailed study of the “noteworthy property” mentioned in Section 3.3, which can be expressed in three equivalent forms. One of them states the equality between the rate of profit in an industry and the rate of surplus of the commodity it produces: this generalizes to disequilibrium a property of the Sraffian standard system; (iii) the introduction of a monetary constraint in the relationship between the bank and the agents; (iv) a critical assessment of physical reproduction in Torrens’s model.
6The paper is organized as follows. Section 1 reminds us of the diversity of the Classical studies in their approaches to disequilibrium and gives a central role to Marx’s study of the division of labour. Marx elaborated a theory of value in disequilibrium, however limited it is. Section 2 proposes a study of an economy in disequilibrium in which the reproduction of capital deals with a process of exchange inspired by Marx’s analysis and which may help to solve the problem he left open. Section 3 studies a Classical model of a monetary bisector economy in temporary disequilibrium illustrating our approach. The dynamics of that economy are sketched in Section 4.
1 – Disequilibrium and Classical tradition: a broken promise
1.1 – Equilibrium and disequilibrium in the Classical tradition
7Today, the dominant theory is an equilibrium theory on all markets. Only equilibrium is considered as amenable to a rigorous analysis. That requirement (the “discipline of equilibrium,” in Lucas’s words) discards a field of research and implicitly expresses a faith in the automatic equilibrium of markets.
8Surprisingly enough, a methodological parallel can be drawn with the best-known modern version of the Classical theory. The Sraffian version of Ricardo’s theory is centred on the analysis of equilibrium seen as a “normal” or “long-term” position defined by the technique and the state of distribution. Sraffa himself assumed the uniformity of the rates of profit without any justification (1960, § 4). That condition unduly restricts the theory of value to the definition of permanent prices, be they labour contents or production prices. Ricardo (1821, pp. 91-92) argued that the uniformity results from persistent causes, whereas “accidental” causes move the market magnitudes away from their natural positions: in that approach, market prices are non-theoretical magnitudes. Modern attempts to formalize and justify the Smithian intuition echoed by Ricardo follow the theory of gravitation. These models make a link between the changes in prices through time and the gaps between the rates of profit. That justification of a long-run convergence towards the uniformity of the rates of profits meets some difficulties: first, in the absence of an explicit capital market, the formalization of transfers of capital is a problem (for instance, where does the information on the rates of profit come from?); second, the reaction coefficients are arbitrary: there is no theory of the sensitivity of prices to excess demands; and, third, it is not ensured that the scheduled produced quantities fit with the available resources and their distribution. This is all the more open to criticism since the aim is to describe the effective dynamics, not virtual ones. The point is that the economic mechanisms at work in the gravitation models are a kind of black box.
9However, an alternative approach, also rooted in the Classical tradition, is possible. For the understanding of economic crises, some Classical authors have indeed attempted to improve the analysis by integrating the dynamics of prices and quantities in a more convincing way. In that line of thought, attention can be drawn to Torrens’s (1821) and Marx’s reproduction schemes which stressed that reproduction “is not only a replacement of value, but also a replacement in material” (Capital, Book II, 1893, p. 241). Torrens was the very first to point at the physical constraints imposed on the reproduction of capital when the sectoral profits are totally  reinvested in the same branch. Unlike Ricardo, Torrens considered the non-uniformity of the profit rates as the expression of the same systematic causes as those which govern equilibrium. He elaborated the first theory of value which is not restricted to the study of equilibrium prices. His intuitions can be formalized by means of a system of equations which determines the rates of accumulation, the rates of profit in each sector and the prices, in equilibrium as well as in disequilibrium (see Section 4 for a critical assessement of Torrens’s analysis). Our aim is to determine the prices, the rates of profit and the produced quantities within an explicit disequilibrium framework.
1.2 – Marx and the social division of labour
10The social division of labour is at the basis of market economies. Smith distinguished between the social division of labour and its technical division inside the factory. Only the first (which we now just call division of labour, to simplify) makes the exchange relationship become systematic (“Every man thus lives by exchanging, or becomes in some measure a merchant, and the society itself grows to be what is properly a commercial society,” Smith, 1776, I.4.1), by creating interdependency relationships between producers: “In civilized society he stands at all times in need of the cooperation and assistance of great multitudes” (Smith, 1776, I.2.2). An individual’s wealth then depends on his production decision and on prices he does not control.
11A few exceptions apart, the Classical theory of value is centred on the determination of equilibrium prices as the solution of a system of equations in which the division of labour is reduced to the differences between products and the interdependencies between sectors. However, in the main two works on political economy published in his lifetime (A Contribution …, 1859, Capital, Book I, 1867), Marx’s analysis of capitalism partly moved aside from that tradition. He started by constructing a theory of value in disequilibrium, which is conceived as a necessary consequence of the division of labour and not as a departure from an equilibrium position due to “accidental causes” à la Ricardo. His remarkable analysis of the division of labour is unfortunately made obscure by his project to elaborate a labour theory of value that he intended to oppose to Ricardo’s. When one gets rid of that ambition, Marx’s analysis becomes the main historical reference for the analysis of market (or exchanges) (see Benetti and Cartelier, 1999).
12Marx enhanced the Smithian concept of the division of labour: “As a general rule, articles of utility become commodities, only because they are products of the labour of private individuals or groups of individuals who carry on their work independently of each other” (Capital, Book I, Chapter 1, p. 47, see also p. 30). In such a society every agent decides his production independently of the others, and without knowing what they are doing, whereas the result of everybody’s action depends on the others. Individual freedom and interdependencies are two sides of the same coin: “The seeming mutual independence of the individuals is supplemented by a system of general and mutual dependence through or by means of the products” (ibid., pp. 72-73). The consequence is what, in his colourful language, Marx designated as “the salto mortale of the commodity” when it is sold (ibid., p. 71, and A Contribution, p. 47), resulting from the ignorance of exchange values, itself a consequence of the division of labour. Disequilibrium occurs because actual prices differ from those on which the agents based their calculations, and not because of price rigidities. The amounts of goods obtained after trade, as well as the prices, are not those that had been foreseen. An analysis restricted to market equilibria states does not address that issue: any contradiction between private and social evaluations (viz. expected and market prices) disappears and the economy behaves as if it was centralized or, in Marx’s words, as “the patriarchal industries of a peasant family” or “a community of free individuals, carrying on their work with the means of production in common” (Capital, p. 50).
13Marx’s analysis of an economy with division of labour relies on the labour theory of value. His own interpretation of the division of labour is based on the “twofold character of the labour embodied in commodities” (Capital, p. 29). Labour is simultaneously private and social. What is “private” is the concrete and heterogeneous labour provided by individuals engaged in the production of use-values, the quantities of which result from individual decisions taken independently from each other. Social labour, which is homogeneous and abstract, is the substance of value, and its quantity determines the exchange values. Marx explained the relationship between the two notions of labour in the following terms: “The point of departure is not the labour of individuals considered as social labour, but on the contrary the particular kinds of labour of private individuals, i.e., labour which proves that it is universal social labour only by the supersession of its original character in the exchange process. Universal social labour is consequently not a ready-made prerequisite but an emerging result” (A Contribution …, p. 16). The question is then to determine values inside the unity of production and circulation, viz. the amounts of social labour which are formed through exchanges by starting from given quantities of private labours. The basic weakness of an approach based on the labour theory of value was noticed by Marx himself: “Thus a new difficulty arises: on the one hand, commodities must enter the exchange process as materialized universal labour-time, on the other hand, the labour-time of individuals becomes materialized universal labour-time only as the result of the exchange process” (ibid.). Hence a contradiction: the “universal social labour” is simultaneously the condition and the result of exchange.
14To the best of our knowledge, neither Marx nor the Marxists have ever solved that problem. That is why Marx’s theory of value eventually admits the primacy of production. The canonical model of labour values determines values as the solution v = (I − Α)-1l of a system of linear equations. But that loophole is not acceptable, as it contradicts the whole of Marx’s analysis of the division of labour: vector l cannot represent heterogeneous private labours when vector v is a vector of values and therefore of social homogeneous labour, and l cannot be a vector of social labour either since, as Marx clearly showed, that vector is not a datum analogous to the matrix A of technical coefficients.
15As a consequence, Marx’s labour theory of value does not keep its promise to determine exchange values in an economy with division of labour in which, except by fluke, the agents are in disequilibrium. That failure is harmful to Marx’s theory inasmuch it deprives his analysis of the commodity (or of the division of labour) of its indispensable theoretical ground that neither philosophy nor history can provide. In the subsequent developments, we follow Marx’s idea of an opposition between the “private” and the “social” evaluations of commodities, which is at the very basis of market economies, but detach it from Marx’s own interpretation in terms of the labour theory of value.
2 – Towards a theory of market disequilibrium
16Starting from Marx’s analysis of the division of labour, we propose a solution to the indetermination of social values, independently of any reference to the labour theory of value and to the notion of exploitation, which is not relevant for the problem at stake. That answer introduces two devices inseparable from the division of labour itself: one is money, the other is a market mechanism which determines prices and allocations. Each producer is assumed to take, before the opening of the market, irrevocable decisions which lead him to ask a bank for some amount of money and to bring to the market some amount of product. The market mechanism then determines monetary prices and allocations which, as in Marx’s analysis, usually differ from those expected by the agents. In a competitive economy, that mechanism is of a social nature and prices are not chosen by the firms, as it might be the case in an imperfect competition framework.  The market prices are unique, but the temporary disequilibrium we consider differs from a Hicksian temporary equilibrium. In Hicks’s theory, expected prices make a link between present and future and their aim is to overcome the limits of what Hicks calls the “static theory”: “supplies (and ultimately demand too) are governed by expected prices as much as by current prices” (Hicks, 1938, p. 116). Then, the supplies and demands at date t depend on prices at t and on the expectations of prices at date t + 1. In our approach, by contrast, demands and supplies do not depend on the price expectations for the next period, but on those that the agents had in mind before the opening of the “Monday” market. They then take irrevocable decisions on the basis of which the market mechanism will determine the effective market prices.
2.1 – Monetary economy
17Money is the institutional expression of the division of labour. It allows each agent to act independently of the others (to buy without depending on the sale of his product. The existence of a universal means of exchange on all markets is the condition for the consistency of market prices. We give priority to that function (Wicksell (1906, p. 7): “Of the three main functions, only the last [medium of exchange] is in a true sense characteristic of money”). The quantity of money is endogeneous, and it is issued by a bank at the producers’ requests. The amount of money demanded by each agent is equal to the expected value of his supply of goods, which itself is equal to the expected value of his purchases. Each agent commits himself to reimburse the money he received from the bank (this is why money does not appear in the ex ante budget constraint), and money is destroyed after having accomplished its circular flow. As we assume that the creation of money, its circulation and its destruction take place within a very short time interval, the rate of interest is ignored. There is neither financing of present purchases by future resources nor financing of future purchases by today’s money. In short, money is not a specific good in the initial endowments; it is neither a credit nor a store of value.
2.2 – Prices and division of labour
18Our analysis of the exchange process relies on the notion of the “twofold character of price,” which is inspired by the “twofold character of the labour embodied in commodities” we have seen in Marx. We distinguish two notions:
- the “expected prices” are used by the producer to define his production plan: given his expected budget constraint, he determines his money requirement and his demand for inputs.
- the “market prices” result from the market mechanism and are those at which transactions do occur on the market.
19That distinction is an echo of that mentioned by Marx: in the exchange process, “money only circulates commodities which have already been ideally transformed into money, not only in the head of the individual but in the conception held by society (directly, the conception held by the participants in the process of buying and selling). This ideal transformation into money is by no means determined by the same laws as the real transformation” (1857-61, p. 187). A few pages later, Marx wrote that “the sum of money exchanged for a commodity is its realized price” (1857-61, p. 198).
20Each producer has a given technique of production and owns the product obtained at the end of the previous period. His expected prices result from a personal interpretation of the state of the market, i.e. a personal opinion on the evaluation by the other agents (the society) of his own product, and a personal estimate of the value of the other goods. At these expected prices, a producer determines the quantities he intends to produce and the inputs required for that production. We retain the hypothesis that each capitalist aims at maximizing accumulation in his own sector.  Individual decisions must satisfy the expected budget constraint: the expected value of the purchases must not exceed the expected value of the sales (credit is discarded by hypothesis).
21Individual productions receive a social evaluation on the market. Which are the prices resulting from the whole set of individual actions governed by expected prices?
22“Market” is conceived as the set of places of exchanges, or trading posts, one for each commodity. We follow Cantillon (1755) by assuming that the market price of good i is equal to the ratio between the aggregated monetary expense and the quantity of the good brought to that market.  The exchange value of a good thus results from the whole set of private estimates. That mechanism determines simultaneously the market price and the allocations, which, except by fluke, differ from those expected by the agents. Consequently, their plans are not fulfilled and real disequilibria appear.
2.3 – Money and temporary disequilibrium
23Since the availability of a means of exchange allows each agent to buy goods independently of his sales, monetary imbalances also appear and must be settled. Let us have a closer look at monetary disequilibrium. The amount of money that an agent spends at a given date is the total expected value of his product, whereas the amount he actually receives is the value of that product at market prices. These two amounts result from distinct evaluations (the first is private, the second social) of the same product and their difference gives a balance, which may be either positive or negative. The cause of an agent’s negative balance is that his expected price was higher than the market price: the total value of purchases exceeds sales. By contrast with the so-called “monetary economy” in disequilibrium as described by Arrow and Hahn (1971, pp. 337 et sq.), monetary imbalances occur because the means of exchange is not an initial endowment. The budget constraint, which was met ex ante when calculations were made at expected prices, is violated ex post at market prices. An agent who has not financed the whole of his purchases is not the ultimate owner of the goods he has got on the market (a liquidator can seize them). Different institutional rules can be considered for the settlements of the monetary imbalances. We assume that ex post transfers are used to balance the effective budget (details in Section 3).
24To sum up, in our scheme, money takes the place that labour had in Marx’s theory. Individual evaluations of production are made in monetary terms (at expected prices), not in terms of concrete labour. Similarly for the social evaluation of products, which is made in monetary terms, not in those of social labour. The mysterious passage from private labour to social labour, which in Marx’s theory is deemed to occur through the exchange process, disappears. The question of the incommensurability between heterogeneous private labours and homogeneous social labour, which is a source of difficulties in Marx’s approach, does not appear here since the monetary magnitudes associated with private evaluations are of the same nature as the monetary magnitudes socially determined on markets. Our market model is therefore a potential candidate for analysing an economy characterized by division of labour.
25After the balance settlements—still to be described—all magnitudes defining the temporary disequilibrium are determined. The available goods which were initially in their producer’s hands are redistributed across sectors and used as inputs (and, for some good, partly excluded from accumulation). The effective production of each sector is then determined. Since the market prices are known, so are the rates of profit, which are not uniform.
26In an economy with division of labour, disequilibrium is the “normal” state and not an accidental departure from the natural position. In such an economy, the divergence between private and social estimates sets the question of the determination of prices and quantities in disequilibrium. It is after that step that the dynamical analysis can start. To that end, a hypothesis on the formation of price expectations will allow us to build a sequence of temporary disequilibria. We will check that the dynamics are not explosive. The similarities in the mathematical tools used in that analysis and in that of the stability of equilibrium should not hide the differences between our approach to disequilibrium and that of the Ricardian tradition.
2.4 – Equilibria
27A full equilibrium (or long-period position) of an economy corresponds to a specific (and, in fact, exceptional) state defined by the implementation of two conditions: (i) market temporary equilibrium, that is the equality between expected and market prices; and (ii) reproduction equilibrium, that is the equality between the two rates of accumulation.
28Market temporary equilibrium implies that all production plans are met and that monetary balances are nought. It is not the result of an underlying stable adjustment process (as in the Hicksian temporary equilibrium (Hicks, 1938)). That is why the dynamics of our model are those of a sequence of temporary disequilibria.
29Reproduction equilibrium assumes that the structure of production is constant. Full equilibrium is market equilibrium combined with the full success of reproduction. At full equilibrium, the economy is on a von Neumann maximum growth path. In that case, the rate of profit is uniform and equal to the maximum growth rate of the economy, and the market prices coincide with the Classical prices of production. The structure of relative prices and the proportions in production are given by the right and left dominant eigenvectors of the input-output matrix.
3 – A Classical model
30The model  we study is based on the following hypotheses: (i) Labour does not appear explicitly, every worker being replaced by the corresponding wage basket, which is incorporated in the means of production; (ii) there are two sectors and one method of production per sector, constant returns prevail, production takes one period, all capital is circulating and all goods are perishable; (iii) goods can be disposed of freely; (iv) each capitalist aims at maximizing accumulation in his own sector.
31We moreover assume for simplicity that all producers have the same monetary price expectations. That hypothesis is quite natural as soon as one admits static expectations (as we shall do).
3.1 – Prices and market disequilibrium
32Each agent asks for and gets from the bank a quantity of money equal to the expected value of his product. As each producer maximizes his production plan under the budget constraint he foresees, the expected value of his product is equal to the expected value of the inputs he intends to get. He shares the money he gets among the two markets in accordance with his production plan, which takes into account the technical coefficients and the expected values of the inputs. In our model, the whole production goes to the market: each agent i (i = 1, 2) sells on the market the whole of his product, then buys again the amount of input i he needs (in that “wash sales” version, producer i intervenes on both sides of market i, as both a seller and a buyer). The application of the Cantillon rule then determines the monetary market price of good i as the ratio between the aggregate amount of money on market i (stemming from both producers) and the quantity of good i supplied on that market by agent i. These physical quantities and the money exchange for each other on both markets.
33The corresponding formalization is simple: let aij be the quantity of input j per unit of product i, q–i the quantity produced at the end of the previous period and brought to the market at the beginning of the current period and, eventually, let pei and pej be the monetary prices expected by the producer of commodity i. Given these data, agent i’s plan reflects his desire to maximize accumulation subject to his expected budget constraint. This determines his expected production plan, denoted qei:
35Formula (1) shows that plans depend on the relative price, not on monetary prices. As a consequence of the budget constraints, there is a Walras law at expected prices.
36The producers get from the bank the quantity peiq–i of money and spend it on both markets. The Cantillon rule relative to the price formation then determines the market prices and the market allocations of commodities
39Flukes apart, prices expectations are not met. The production plans are not fulfilled and the capitalists are in real disequilibrium. That disequilibrium is, in Marx’s terms, “private” since the agents’ plans reflect their personal point of view. Formula (2) shows that pi ¤ pei ⇔ di ¤ si: the undervalued commodity is in excess demand and is called “scarce,” while the overvalued commodity is in excess supply and is called “superabundant.” The allocations differ from demands: the quantity of some good the producers receive on a market is either smaller or greater than that he had scheduled.
40These individual disequilibria have a social counterpart, which takes the form of monetary disequilibria: since each agent spends a monetary sum equal to the expected value of his sales, the money he receives differs from his expectations whenever his price expectation is wrong. The agents who underestimated their good receive more money than they have spent and have therefore a positive monetary balance; and conversely for the other agents. As money is endogenous and is a pure means of exchange, the equality
42holds by construction. Or, in formal terms,
44hence q–1(pe1 − p1) + q–2(pe2 − p2) = 0. The sum of monetary balances is zero and pe1 £ p1 ⇔ pe2 ¤ p2: if agent 1 underestimates his product, agent 2 overevaluates his. It is worth noting that, even in the simple case of uniform price expectations, exchanges in disequilibrium result in monetary imbalances. Those common expectations are not self-fulfilling. The individual monetary disequilibrium is the expression of the “salto mortale” of the commodity when it is sold, i.e. transmuted into money.
3.2 – Balances settlement and effective production
45The monetary disequilibrium puts the agents in an asymmetric position and, in particular, the capitalist with a negative balance meets a serious problem: he is unable to reimburse the bank. The institutional rule we admit is that this producer pays back the bank by transferring some of his real assets to the agent with a positive money balance. The physical composition of the transferred basket is determined by the capitalist with a positive balance, who is in a position to impose his choices, and the transferred goods are evaluated at market prices. Then the effective allocations are fully determined. The capitalist with a positive monetary balance (i.e. the one who underestimated the value of his production) gets the inputs allowing him to accumulate the whole of his profits, beyond his initial plan based on lower expected profits. Except in the pathological case referred to in note 11 below, the whole of his product is fully accumulated (see Benetti et al., 2014). As for the agent in the red, he will ultimately get a basket of inputs that he will not be able to use in its totality for production because the proportions differ from his (given) technical coefficients. For these two reasons (reimbursement of the debt and inadequate proportions), his profit is not totally accumulated and the corresponding product will be lower than expected.
46Let capitalist 1 be the one with a positive monetary balance after the market. He accumulates all the inputs he got either on the market or for the settlement of monetary balances. Given his budget and the value of the inputs per unit produced, his effective production will amount to
48and is greater than the notional production given by (1).
49Producer 2, by contrast, must deliver some inputs to capitalist 1 for the settlement of his negative monetary balance. The basket he delivers being in the proportion desired by producer 1, the proportions of the input basket remaining to producer 2 do not fit the technical coefficients of industry 2, therefore an input is not totally used and the effective production of good 2 is expressed by means of the min function:
51The production of good 2 will be lower than its producer expected and, in the normal case (see note 11 below), the initial quantity q–2 is not fully accumulated.
52Does the mechanism we consider violate the principle of voluntary exchange, which expresses that no agent is obliged to sell or buy more than he had planned at market prices?  The question is sensible in a model in which plans and exchanges are made at the same fixed price (see for instance Bénassy, 1986). But it is irrelevant in the present model because the agents’ plans are based on expected prices while exchanges are ruled by the initially unknown market prices. A comparison between expected and effective positions is meaningless when these prices differ. Furthermore, in our model, the exchange remains incomplete as long as monetary imbalances have not been settled by means of an institutional procedure. After the market and before that settlement, no agent can know if he bought too much or too little of a good. The very notion of voluntary exchange does not fit with that framework.
3.3 – A noteworthy property
53A noteworthy property of the model we study may be expressed under three equivalent forms. These forms are most conveniently derived from the following relationships, which hold in any bisector classical model.
54Let us consider good i. The balance of its uses and resources determines the quantity ei which is not accumulated in any sector and is excluded (hence notation ‘e’) from accumulation:
56Consider now producer i. The value vi he will not invest is equal to:
60By definition, the surplus factor of good i is equal to the ratio between its produced quantity and that used in the economic system, i.e.:
62The factor of profit in sector i (Ri = 1 + ri) as calculated at the replacement costs amounts to
66In our model, the producer i who gets a positive monetary balance accumulates all of his effective profits (vi = 0) and the commodity he produces in the current period is entirely accumulated (ei = 0).  For the other agent j, equality (7) shows that vj = pjej, ej and vj being positive magnitudes. Therefore, for any producer, the receipts he does not devote to accumulation are equal to the value of the amount of his good excluded from accumulation.
68The equivalence with equality (i) results from (5) and (6). That relation expresses the equality between the values of external productive resources: the value of the quantity of input i used in industry j is the same as the value of inputs j used in industry i. In spite of the asymmetry between agents, the indices i and j play symmetric roles in equality (ii) and can be substituted for each other. That property is basic in the study of the dynamics.
70A decreasing homographic relationship between Si and Sj is obtained by eliminating the ratio qi / qj between equality (8) and the similar equality for Sj. The same relationship between Ri and Rj is obtained by eliminating pi / pj between equality (9) and the similar equality for Rj. It follows that Sj = Rj if Si = Ri. On the other hand, rewriting (5) and (6) by using (8) and (9) respectively, we get and , that is to say the equivalence with (i). A consequence is that, in our model, the classical condition concerning the uniformity of the rates of profit (Ri = Rj) is an alternative expression of the uniformity of the rates of surplus (Si = Sj).
4 – Dynamics
71That section summarizes the results obtained by Benetti et al. (2014). The effective productions being determined within a given period, an expectation hypothesis allows us to link the periods to each other and therefore to study the dynamics of the model. One may then wonder whether or not the dynamics converge towards a full equilibrium state. For simplicity, we assume static expectations. Even under such a hypothesis, an analytical study of the dynamics seems out of reach because the dynamics depend on the nature and the amounts of goods excluded from accumulation as described by the functions min, and which may vary from a period to the next. In our model, however, that study remains possible because of the existence of a simple linear relation between the relative price and the relative quantity, which comes from equality (ii) above:
73That relation allows us to express the real dynamics by an induction equation relative to either the proportion of productions or the relative price.
74When the physical proportion is that of the Perron-Frobenius row-eigenvector of the technical matrix and the price expectations are the corresponding column-eigenvector, all goods are accumulated, the price and quantity expectations are met and the economy follows a regular growth path at maximum rate (von Neumann growth rate). Otherwise, the physical proportion and the relative price vary from a period to the next, there are balance settlements and some goods are excluded from accumulation. It can be shown that these dynamics are never explosive, but several evolutions remain possible.
75The dynamics of the relative quantities depend on the sign of the determinant of the technical matrix: if the determinant is positive, the system converges towards a von Neumann growth path; if it is negative, either that convergence is local (and may be global), or there exists a limit cycle of order two, depending on the ratio between the second and the first (or dominant) eigenvalue of the matrix.
76Thanks to relation (10), the dynamics of the relative price are basically the same as those of the relative proportion. In particular, in case of convergence, the relative price tends towards the right eigenvector of the technical matrix A, i.e. towards the price of production as defined in the Classical theory.
77We may compare these results with Torrens’s theory (1821), a classical reference for the physical reproduction of capital and the accumulation process. In Section VI of Chapter VI of his book, Torrens made use of numerical examples to study the reproduction of an economy where capitalists accumulate all profits in their own sector. The available production is totally accumulated and the corresponding equations can be written as:
79They determine the sectoral accumulation rates gi, which by definition are equal to the rates of profit. Then disequilibrium prices are also determined. The equilibrium state requires “good proportions” (in Torrens’s terms) between sectors, which are those corresponding to the left dominant eigenvector of the input matrix A and coincide also with the von Neumann proportions. Torrens shows that this equilibrium is unstable and the dynamics result in what Torrens called a “general glut.” This explains Torrens’s original position in the debate on Say’s law: since supply creates its own demand only if the system is in the “good proportions,” Say’s law is neither always wrong, as claimed by Malthus, nor always right, as claimed by Ricardo.
80Interpreted in the terms of our model, the total accumulation of profits in Torrens’s model presumes that price expectations are met at each date, which means that all capitalists are in equilibrium but the economy itself is in disequilibrium: that configuration, which is exceptional in our model, is the rule in Torrens’s. Unlike Torrens’s model, ours is stable because the market mechanism we consider partly discards some good from accumulation.
Nominal dynamics and monetary constraints
81Rather unexpectedly, there exists an asymmetry between the dynamics of quantities and those of nominal prices. The rule adopted for quantities makes reference in equation (4) to function min and is not amenable to a simple analytical study.
82A doubling of nominal prices at date 0 leads to the same proportional change at any date. In the case of convergence of relative prices and quantities, the nominal prices at equilibrium are also doubled. Consider now the effects of an exogenous non-proportional shock at some date on expected monetary prices when the economy is assumed to be at equilibrium at that date, all goods being entirely accumulated. As the relative expected price is changed, the agents’ demands are modified: some good becomes superabundant and is not totally accumulated, monetary imbalances requiring monetary settlements appear and effective productions are modified. The economy is thus submitted to real and monetary disequilibria. These phenomena last all along the transitional dynamics until a new equilibrium is reached in the long run. The economy then recovers its initial von Neumann rate of growth (no long-run rate effect), but the real effects of the short-term shock in terms of levels of production are permanent. There is a long-run level effect, with no catch up.
83The main novelty of our model is to introduce money and a market mechanism in a Classical framework taking disequilibrium into account. The grounds of our approach are Marx’s analysis, which contrasts private and social valuations of commodities. However, we substitute a reference to monetary magnitudes, with social valuations being determined by a market mechanism, for Marx’s distinction between private and social labour, and reconsider the problem of exchange in that framework.
84Money is considered as a pure means of exchange and is issued by a bank at the agents’ requests (endogenous money). Why does money matter? The behaviour of a monetary economy differs from that of a barter economy having otherwise the same characteristics (Benetti et al., 2012). Even if money is neutral at equilibrium, its presence modifies the relative price and the physical magnitudes in disequilibrium. Money matters because monetary imbalances must be settled at each period. The presence of a means of exchange requires the definition of institutional rules related to its issue and to balance settlements, which affect the allocation of inputs between agents and the dynamics. The model calls for variants concerning legal rules and for extensions, beyond the peculiar rules we have adopted: for instance financing monetary balances by transferring securities instead of real capital or, alternatively, introducing credit and interest rates.
85Our approach differs from the post-Sraffian long-period analysis, of which it is a complement. The post-Sraffian analysis only goes half of the way when it considers regular growth paths and assumes the uniformity of the rates of profit. The full equilibrium may be the result of the dynamics of market prices and proportions. In any case it is fruitful to complement the construction with a theory of prices and allocations in temporary disequilibrium, which is the normal state of an economy characterized by the division of labour. It is our endeavour to go in that direction by taking into account expectations and money. The model may serve as a basis for a macroeconomic study inspired by a Classical approach.
C. Benetti: University Paris Ouest Nanterre La Défense, EconomiX; email@example.com
C. Bidard: University Paris Ouest Nanterre La Défense, EconomiX; firstname.lastname@example.org
E. Klimovsky: Universidad Autónoma Metropolitana – Azcapotzalco (Mexico),
A. Rebeyrol: University Paris Ouest Nanterre La Défense, EconomiX; email@example.com
With acknowledgements to the participants of the colloquium “What have we learnt on Classical economics since Sraffa?” and to two anonymous referees for their comments on a previous version of the paper.
A recent development of Torrens’s ideas allows for a partial accumulation of the excess product and proposes two models which differ by the hypothesis retained for the sectoral distribution of the value of the non-accumulated part of production: one of them generalizes the Torrens system, the other the Sraffa system (see Benetti et al., 2013).
See for instance Salvadori and Signorino (2013).
The idea of maximizing accumulation is quite usual in the Classical tradition, connected with the intuition that accumulation is the best way to get future profits. In our model, the hypothesis that capitalists invest in their own sector is justified by the absence of a capital market. That hypothesis is also found in Torrens (1821) and in Marx’s (1885) enlarged reproduction schemes.
The notion of “trading post” was introduced in the modern theory of strategic market games by Shapley and Shubik (1977). These authors make use of a similar Cantillon rule to determine the market price.
For a complete study of the model, see Benetti et al. (2014).
Equation (2) shows that, if the bank does not supply the whole of the monetary demands addressed to it, and if these restrictions are equiproportional for both agents, they are neutral: the agents downsize their plans but the market prices fall proportionally so that their real balances remain unchanged, as do the quantities they will produce. Real effects appear when the monetary constraints are not proportional.
In general strategic market games, it may also be the case that an agent receives more than he had asked for. Shapley and Shubik (1977, p. 947) pointed at that phenomenon but only noticed that “it is a matter of letting one’s stomach rather than one’s purse absorb the fluctuations.”
Equality (7) is not a Walras law: equality piei = vi does not mean equilibrium.
The same property also holds in: (i) the “Torrens model” in which the whole production of both sectors is by hypothesis totally accumulated; (ii) a model which generalizes it, in which the rates of accumulation are positive and exogeneous (by hypothesis vi = eipi > 0 ∀i) (see Benetti et al., 2013, model 1); (iii) a bisector temporary disequilibrium model without money, in which the effective quantities which intervene in relation (6) are evaluated at expected prices (see Benetti et al., 2012).
At least in the “normal” case. One can identify a “pathological” case in which relation (i) does not hold because the totally accumulated commodity is not the one produced in the sector in monetary excess.