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 [3] 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.

2.1 – 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).

2.3 – Money and temporary disequilibrium

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.

3.1 – 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 a_ij 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 p^e_i and p^e_j 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 q^e_i:

34

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 p^e_iq^–_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

37

38where d_i is the total demand of good i. The product p^e_id_i is the monetary expense on market i, and s_i is the quantity of good i brought to the market. [8]

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 p_i ¤ p^e_i ⇔ d_i ¤ s_i: 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

41

42holds by construction. Or, in formal terms,

43

44hence q^–₁(p^e₁ − p₁) + q^–₂(p^e₂ − p₂) = 0. The sum of monetary balances is zero and p^e₁ £ p₁ ⇔ p^e₂ ¤ 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

47

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:

50

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? [9] 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 e_i which is not accumulated in any sector and is excluded (hence notation ‘e’) from accumulation:

55

56Consider now producer i. The value v_i he will not invest is equal to:

57

58By combining these relationships with the similar formulas relative to good j and sector j, one obtains the following relation (which can be generalized to any number of commodities): [10]

59

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.:

61

62The factor of profit in sector i (R_i = 1 + r_i) as calculated at the replacement costs amounts to

63

64The three equivalent forms of the noteworthy property we point at for our model (and others [11]) are:

65

66In our model, the producer i who gets a positive monetary balance accumulates all of his effective profits (v_i = 0) and the commodity he produces in the current period is entirely accumulated (e_i = 0). [12] For the other agent j, equality (7) shows that v_j = p_je_j, e_j and v_j 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.

67

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.

69

70A decreasing homographic relationship between S_i and S_j is obtained by eliminating the ratio q_i / q_j between equality (8) and the similar equality for S_j. The same relationship between R_i and R_j is obtained by eliminating p_i / p_j between equality (9) and the similar equality for R_j. It follows that S_j = R_j if S_i = R_i. 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 (R_i = R_j) is an alternative expression of the uniformity of the rates of surplus (S_i = S_j).

Real dynamics

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.

Torrens

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:

78

79They determine the sectoral accumulation rates g_i, 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.