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A restatement of Walras' theories of capitalisation and money.

Abstract: The primary aim of this paper is not to comment on Leon Walras' theories of capitalisation and money, but to restate them in a way that avoids ambiguities and inconsistencies. Two other groups of agents are introduced together with consumers and producers: the owners of fixed capital goods, who finance their purchases by issuing securities, and the owners of circulating capital goods, who finance their purchases by issuing money. The principal monetary implications are: there are two quantitative relations (which concern transactions during the period and transactions at the end of the period respectively); the theory of money is integrated with the theories of exchange, production and capitalisation; securities and money are a veil; bank deposits can be introduced in place of paper money; and the length of the period matters.

1 Introduction

Walras introduces general equilibrium theory in successive stages, with the theory specified in each successive stage containing the theory developed in the preceding stage. These stages are: the theory of exchange of two commodities for each other (Part II of the Elements); the theory of exchange of several commodities for one another (Part III); the theory of production (Pan IV); the theory of capitalisation and credit (Part V); and the theory of circulation and money (Part VI).

The analysis of the exchange and production of fixed capital goods was presented by Walras in the second part of the first edition of his Elements (1877). This theory, though misunderstood by some readers, is perfectly consistent. (3) A version of it, slightly different from the original Walrasian version, is given in Section 3: a kind of security, which is issued and exchanged, is introduced in place of the Walrasian notional good, perpetual net income.

Most of the discussion, however, will be devoted to the theory of money, which is the most controversial part of the Walrasian system. Walras introduced three different versions of the theory of money in the successive editions of his Elements. (4) In the first edition the Walrasian theory of money consists of a transaction equation, which is a completely analogous formulation to that developed later by Irving Fisher. In the second and third editions the notion of circulation a desservir is replaced by the notion of encaisse desiree, of desired cash balance, which is a formulation analogous to the one developed later by Alfred Marshall. (5) In the fourth and definitive edition Walras presents a very elaborate theory of money which is connected, unlike the preceding ones, to the remaining part of his theoretical structure, because of the link between money and circulating capital goods (consumer goods and raw materials). The service of money consists of obtaining goods which may be made available for use in production and consumption. In other words, the quantity of money demanded depends on the marginal utilities of services that derive from the availability of goods and, for producers, the coefficients of production.

Walras' theory of money has been less successful than his theories of exchange and of production. The theory of money has been neglected, even though some economists have appreciated it, and criticisms, developments and discussions of it have not been lacking. (6) I believe that Walras' mature theory of money contains a very interesting core that provides a general equilibrium approach to the analysis of money as the medium of exchange, but which is only possible in an idealised world devoid of friction, uncertainty and illusion. In other words, Walras' theory concerns the pure transactions role of money. This approach can be taken as a starting point for more realistic analyses, which could be obtained by introducing friction (in particular, transaction costs, thus considering the transaction demand for money) and uncertainty (thus considering the precautionary and speculative demand for money). (7)

Walras' theory of money, however, is tarnished by some ambiguities and inconsistencies. Moreover, the literary account of the theory does not always correspond to its mathematical formulation (on this aspect, for instance, see Hall 1983). The primary aim of this paper is not to comment on Walras' theory of money, but to restate it in a form that avoids the original ambiguities and inconsistencies. Obviously, any new formulation (of which many are possible) (8) implies some alteration of the original theory. The restatement presented in this paper retains, I believe, the most significant aspects of Walras' theory of money, showing, moreover, its importance in view of further analyses.

2 Some Observations on Walras' Theories of Capitalisation and Money

Walrasian theories of exchange and production do not take into account the exchange and production of fixed capital goods and the inventories of products and money. These are, respectively, the objects of the theory of capitalisation and credit and of the theory of circulation and money, which can be very briefly summarised, with particular regard to Walras' explicit and implicit hypothetical properties and their respective inconsistencies, as follows.

Fixed capital goods are owned only because income is expected from the sale of their services. Assuming that expectations are for stationary prices and that capital goods produced in the period under examination can be used only in the following period, a uniform rate of net return results for all fixed capital goods, which are, as a consequence, perfect substitutes with infinitely elastic demands (that is, the demand function for a specific fixed capital good is infinitely elastic with respect to its price, which is determined by relationship (2) in Section 3), while the total investment in fixed capital goods is determined, according to consumer preferences, as a function of all prices that comprise the rate of net return.

The theory of circulation and money concerns the 'service of availability' (service d'approvisionnement) given by inventories of commodities and by the quantity of money available during the equilibrium period under examination. More specifically, the 'service of availability' provides the capacity to consume commodities before the end of the period of production. Formally, inventories and money are demanded by consumers according to the utility of their services of availability and by entrepreneurs according to their coefficients of production. Holding inventories and money implies a cost, which consists of giving up the income attainable by holding fixed capital goods. In equilibrium, the value of the service of availability equals the cost of holding inventories.

Given that the original formulation of Walras' theory is affected by inconsistencies, it is appropriate to discuss his theory in some detail before proposing a coherent new formulation.

Walras' theory of circulation and money is based on a number of hypothetical properties. The first is that all uncertainty over prices and the dates of the exchanges undertaken during the period under examination are excluded. (9) This will be referred to as the property of certainty in the period. The second is that goods produced during the period under consideration cannot provide the 'service of availability' during the same period: this service can only be supplied in the successive period. (10) This will as the property of invariability of circulating capital in the period. The third, which is implicitly assumed by Walras in his theory of capitalisation, will be referred to as the expectations of stationary prices property. In fact, Walras' theory concerns the temporary equilibrium, that is, . time is divided into periods connected to each other also through the presence of durable goods. There is an equilibrium for each period (Donzelli 1986, pp. 264-8, Montesano 1970-71, pp. 710-12, and Witteloostuijn and Maks 1988). Temporary equilibrium implies that investment depends on expectations and Walras' equations on investment (equations (8). Walras 1954, p. 281) require expectations of stationary prices across periods.

Certainty in the period and expectations of stationary prices implies that inventories and money are not demanded for a speculative or precautionary motive, that is, with regard to possible changes in prices or possible delays in the availability of commodities. Inventories are held (and are useful and productive) because there are asynchronies within the period under examination, as indicated by Marget (1935, pp. 160-1) and Kuenne (1963, p. 288f); that is, not only does production require time (for outputs to be available after the correspondent inputs are introduced), (11) but products are also used by consumers and producers after they have been made available through production. Consequently, inventories are not constant during the period but vary according to asynchronies. Moreover, for all agents, there is at least one instant within the period under examination when the inventory for each specific commodity and money is zero. No-one will hold a permanent inventory because the property of certainty in the period makes it unnecessary. (12)

The property of invariability of the circulating capital in the period means that the endowment of circulating capital goods is a datum. This implies that inventories are used once and only once by consumers and producers, that is, the owners of inventories consume directly or sell to other agents the commodities they have, which are wholly consumed during the period under examination. Products produced within this period are stored and their service of availability is only supplied in the following period. Thus, the remuneration for services of availability supplied by inventories must be referred to the whole length of the period, independently from the particular instants in which commodities are given and returned.

One problem, which was not considered by either Walras or by scholars who have commented on his theory, is the fact that the goods consumed during the period derive from both inventories and current production. To these two groups of commodities, however, different values correspond, since the value of goods from inventories also includes the cost of the service of availability (in addition to the cost of production). This situation is inconsistent with the condition requiring only one price for any one commodity (that is, the law of one price). Thus, we must require not only that products produced in a given period do not supply their service of availability in the same period of their production, but also that they cannot provide consumer or productive services within this period. (13) Consequently, only commodities existing within initial inventories can be used during the period, while the commodities produced in the period can be used in the following period. This hypothetical property (not assumed by Walras, but determined by the logic of consistency), which may be more formally stated as the proposition that produced goods are not made available before the end of the period, provides a sound basis for the hypothesis of invariability of the circulating capital in the period.

In relation to the role of money, the principal point of Walras' theory seems to be the proposition that consumers and producers can maintain purchasing power during the period only through money, (14) that is, non-monetary exchanges are not possible during the period. Moreover, Walras assumes that money can be lent at the current interest rate. Consequently, agents must use money for their purchases; the quantity of money is related to current payments; and the cost (or the proceeds) due to the payment of interests is proportional to the excess (or the lack) of payments over the endowment of money for any agent. It is appropriate in this scheme to assume that agents can lend money only at the beginning of the period, that is, loans are not possible after the beginning of the period. Otherwise, every agent would hold money only in the instant when he purchases something, so that no-one would detain money for more than an infinitesimal length of time. Thus, a very small quantity of money would be sufficient for a large number of exchanges and the value of money, the quantity of which is not nil by assumption, would be negligible. (15) Given the impossibility of loans after the beginning of the period, agents who have a sufficient quantity of money in their endowments continue to hold a quantity equal to the payments envisaged for the period under examination and they lend the outstanding quantity. On the contrary, agents who have no money in their endowments (or have an insufficient quantity) borrow the quantity of money they need for their payments. In this way money supplies a 'service of availability' which is identical to that supplied by inventories of goods, also taking into account that the hypothetical property of certainty in the period determines the partition of the quantity of money for every agent in his purchases.

In Walras' theory, however, the demand for the 'service of availability' in money is related to commodities consumed within the period, since the commodities that constitute agents' endowments supply the 'service of availability' in kind. If we assume, as proposed above, that products are not available before the end of the period, then money has no role, unless we modify Walras' original description, as will be done in the following section.

Neither does Walras consider the service of availability in terms of money and the service of availability in terms of commodities as perfect substitutes for all agents (that is, in a given period, a consumer is indifferent to owning the quantity of a commodity that he plans to consume or to owning the amount of money that gives him the power to purchase that quantity of a commodity) and, thus, that different demand functions for the 'service of availability', in commodities or in money, do not exist. (This is Dei Vecchio's principal criticism, 1909, pp. 269-72, of Walras' theory. See also Hall, 1983, p. 250).

Moreover, Walras introduces the theory of circulation and money as if inventories and money could directly supply their service of availability through their presence: that is, he does not treat explicitly the sale of stored goods to consumers and producers or the utilisation of money in exchanges, which is the circulation of goods and money. (16)

Finally, Walras does not introduce the process of determination of the quantity of money between successive periods of time, (17) that is, not only is the quantity of money a datum for the period under examination, but it is also constant in lime if there are no exogenous changes. This constancy is not consistent with the expectation of stationary prices in the case of a progressive stale in which the quantities of goods and services grow (considered by Walras to be the most appropriate for his theory). However, expectations of stationary prices are not rational in the case of a progressive state if there are non-producible goods, like land. (The von Neumann model (1937) is a model in which expectations of stationary prices in a progressive state are rational.)

3 A New Formulation of Walras' Theories of Capitalisation and Money

After having indicated the foundations and the inconsistencies of Walras" theory of circulation and money, it is time to propose a new formulation which avoids those inconsistencies and includes all Walrasian theories. The following five hypothetical properties are assumed:

a) certainty in the period:

b) expectations of stationary prices;

c) produced goods are not made available before the end of the period,

d) non-monetary exchanges are not possible during the period (these exchanges concern circulating capital goods; for all other transactions--concerning fixed capital goods and their services, the goods produced in the period, securities and their yields, interest and principal on loans--are settled at the end of the period under examination with monetary payments made only for the balance due, or balance to be paid, by any agent):

e) loans are not possible after the beginning of the period.

Walras' original model assumes hypothetical properties a) and d) explicitly, and hypothetical properties b) and e) implicitly (that is, required in order to justify its equations). While the model does not assume properly c), one implication of this property is explicitly stated, namely, the invariability of the circulating capital in the period.

In this new formulation two other groups of agents are introduced together with consumers and producers. They are the owners of fixed capital goods (including, in this formulation, also land and the old fixed capital goods, which were produced in past periods and are not produced in the period under consideration) and the owners of circulating capital goods. The owners of fixed capital goods buy and sell the services of land and other fixed capital goods relative to the period under examination, which are delivered and paid for at the end of the period; they finance their purchases by issuing securities. These decisions are taken in order to maximise expected profit (equal to the difference between the net income of capital goods and the yields of issued securities). The owners of circulating capital goods (that is, goods which can be used by consumers and producers only once) sell these goods during the period under examination for money; at the end of the period they buy the circulating capital goods produced in the period; and they finance their purchases by issuing money. These decisions are taken in order to maximise expected profit (equal to the difference between the proceeds of sales and the cost of purchases). Consequently, consumers, whose endowments include all the money (which has been issued at the end of the period before the one under examination), retain the amount of money necessary for purchasing consumer goods during the period and they lend the rest to producers, who use it for their purchases of productive goods.

Let us now formally specify these economic relationships with respect to the aforementioned four groups of agents.

A Owners of Fixed Capital Goods

The endowments of the owners of fixed capital goods are composed of assets represented by a vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of quantities of goods bought in the past and of liabilities represented by securities [bar.B]. Securities are perpetual and their quantity is measured by their yields: thus, assuming expectations of stationary prices, the unitary security is expected to yield one unit of money at the end of every future period, so that the quantity of securities is determined by the expected perpetual yield of fixed capital goods that correspond to securities. If the prices of fixed capital goods are indicated with vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have the relationship

(1) [p.sub.kk][bar.K] = [p.sub.B]z[bar.B]

where [bar.B] is the quantity of securities issued in the past (this quantity is determined with respect to the yield of capital goods as expected in the past), [p.sub.B] is the market value of a unitary security, and z[bar.B] is the quantity of securities issued in the past but determined with respect to the yields expected in the period under examination, so that z is a factor by which the quantity [bar.B] is multiplied in order to obtain the equality between the value of assets [p.sub.kk][bar.K] and the value of liabilities [p.sub.B]z[bar.B]. In other words, since in each issue the value of securities equals that of assets, when some unexpected changes in prices occur between successive periods, then a windfall gain or a loss emerges for the owners of fixed capital goods. Consequently, the value of the securities changes, since securities represent the ownership of fixed capital goods and the no-arbitrage condition requires that the market value of securities equals the market value of capital goods. Since we have chosen to measure the quantity of securities by means of their yields and changes in the prices of real assets modify their yields, then the quantity of securities is modified by changes in the prices of capital goods, so that the variable z [greater than or equal to] 0 is introduced and the quantity [bar.B] of securities issued in the past becomes z[bar.B] in the period under examination.

Owners sell the services of fixed capital goods at the prices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which are paid at the end of the period, when the yields are also paid. The profit relative to the period under examination of the owners of fixed capital goods is

[[PI].sub.k] = ([p.sub.k] - [??] [p.sub.kk]) [bar.K] - z[bar.B]

where [??] is a diagonal matrix representing the coefficients of depreciation and risk (called [mu] + v by Walras). Since the owners of fixed capital goods do not use the services, their supply is totally inelastic.

The owners of fixed capital goods invest (that is, they demand fixed capital goods) by maximising the present value of expected profits, defined by the difference between the proceeds of the sales of services and the costs represented by the yields of the securities issued for financing the purchases of the corresponding capital goods. Representing [d.sub.kk] as the quantities of demanded capital goods and [S.sub.BB] as the securities offered, with superscript e indicating the expected values, and with subscript t the future periods, the present value of expected profits is [[pi].sup.e.sub.k] = [[summation].sup.[infinity].sub.(t=1)] [v.sub.t] ((p.sup.e.sub.k,t] - [??] [p.sup.e.sub.kk,t]) [d.sub.kk] - [S.sub.BB]), where [v.sub.t] is the factor of discount for time t, under the finance constraint [p.sub.B][s.sub.BB] = [p.sub.kk][d.sub.kk]. Consequently,

[[pi].sup.e.sub.k] = [[summation].sup.[infinity].sub.t+1] [v.sub.t] ([p.sup.e.sub.k,t] - 1/[p.sub.B] [p.sub.kk] - [??] [p.sup.e.sub.kk,t]) [d.sub.kk]

Since the expectation of stationary prices requires [p.sup.e.sub.k,t] = [p.sub.k] and [p.sup.e.sub.kk] = [p.sub.kk] and the condition of perfect competition requires [[pi].sup.e.sub.k] = 0, we obtain

(2) [p.sup.e.sub.k] = ([??] + 1/[p.sub.B] I) [p.sub.kk]

where I is the identity matrix, i.e. rates of net return of fixed capital goods (which are for [p.sup.i.sub.k]/[p.sup.i.sub.kk] - [v.sup.i] for i = 1, ..., [n.sub.k]) and of securities (which is 1/[p.sub.B]) are all equal. (18) Consequently, at prices satisfying relationship (2), capital goods are perfect substitutes and their demand is infinitely elastic. (19) Equations (1) and (2) imply that the current profit [[PI].sub.k] of the owners of fixed capital goods is zero.

The last equation concerning the owners of fixed capital goods is their total finance constraint

(3) [p.sub.kk] ([X.sub.kk] - [??] [bar.K]) = [p.sub.b][X.sub.BB]

where vector [X.sub.kk] [greater than or equal to] 0 represents their demands for fixed capital goods (that is, [p.sub.kk] [X.sub.kk] is their total gross investment and [p.sub.kk] ([X.sub.kk] - [??] [bar.K]) is their total net investment), while [X.sub.BB] is the quantity of securities issued for financing investments. At the end of the period, the owners of fixed capital goods have vector (I - [??]) [bar.K] + [X.sub.kk] of fixed capital goods while the total quantity of securities is z[bar.B] + [X.sub.BB].

The number of relationships (1)-(3) is [n.sub.k] + 2, where [n.sub.k] is the number of fixed capital goods.

B) Owners of Circulating Capital Goods

The endowments of the owners of circulating capital goods are composed of assets represented by a vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of quantities of goods produced in the preceding period and of liabilities represented by the quantity of money [bar.M]. Circulating capital goods are 'revenues' in Walras' sense, that is, as goods which do not outlast their first use, owing to their intrinsic physical characteristics. Some of them (for example, wheat) are storable for more than one period, in order to be used in one of the future periods. However, in this model storage of circulating capital goods for future periods is excluded by economic reasons, which depend on the assumption of expectations of stationary prices. In particular, during the period under examination, the owners of circulating capital goods offer the whole quantity of their goods, while consumers and producers demand these goods using the whole quantity of money available to them. In fact, on the one hand, consumers and producers prefer not to save money since it would be better for them, respectively, to hold securities and not to borrow money. On the other hand, there is no advantage for the owners of circulating capital goods in holding a speculative store of goods, since they expect stationary prices.

Consequently, the owners of circulating capital goods offer, during the period, the whole quantity of goods and receive correspondingly the whole quantity of money, so that

(4) [bar.M] = [p.sub.c][bar.C]

where vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents the prices of circulating capital goods.

The owners of circulating capital goods invest (that is, they demand circulating capital goods) by maximising expected profit. Representing the quantities of demanded goods with [d.sub.c] and the money offered with [S.sub.M], the expected profit is [[pi].sup.e.sub.c] = [p.sup.sub.c][d.sub.c] - [S.sub.M] under the finance constraint [S.sub.M] = [p.sub.c][d.sub.c]. Consequently,

[[pi].sup.e.sub.c] = ([p.sup.e.sub.c] - [p.sub.c]) [d.sub.c]

The hypothetical property concerning expectations of stationary prices implies that [[pi].sup.e.sub.c] = 0, that circulating capital goods are perfect substitutes, and that their demand is infinitely elastic at the current prices [p.sub.c]. (20) (Equation (4) and the condition that consumers and producers spend the whole quantity of money [bar.M] also implies that the current total profit of the owners of circulating capital goods is zero, but it does not require that the profit of every owner is zero. We assume for simplicity that these possible profits, the sum of which is zero, do not affect consumers' incomes).

The last relevant equation concerning the owners of circulating capital goods is given by their total finance constraint

(5) [p.sub.c][X.sub.cc] = [bar.M] + [X.sub.MM]

where vector [X.sub.cc] [greater than or equal to] 0 represents their demands (that is, [p.sub.c][X.sub.cc] is their total gross investment and [p.sub.c]([X.sub.cc] - [bar.C]) is the total net investment), and [bar.M] + [X.sub.MM] is the total amount of money issued at the end of the period ([X.sub.MM] is the increment, positive or negative, in the quantity of money with respect to the preceding period).

The number of relationships (4) and (5) is 2.

C) Households

Households (indicated also as consumers) have endowments composed of personal capitals of which they sell the services (labour), securities and money. They lend some of their money to producers with the residual used to purchase consumer goods from the owners of circulating capital goods during the period under examination. Thus, consumers:

* demand consumer goods during the period, with immediate payment in money;

* lend money to producers, with payment of interests and reimbursement of the principal at the end of the period;

* supply the services of labour to producers, with wages paid at the end of the period:

* demand services of fixed capital goods from their owners, with payment at the end of the period;

* demand securities and money, in order to have an income in the future and the liquidity sufficient for buying consumer goods in the following period.

Let us introduce an intertemporal utility function for any consumer, of the type

U = [PHI]([d.sub.k], [d.sub.l], [d.sub.c], [d.sub.k, 1], [d.sub.c,l], ..., [d.sub.k,t], [d.sub.l,t], [d.sub.c,t], ...)

where vectors [d.sub.k,t], [d.sub.l,t], [d.sub.c,t], with t = 1, ..., indicate respectively the services of fixed capital goods, the labour services and the circulating capital goods which the consumer plans to use in the t-th period after the one under examination. Taking into account that there are expectations of stationary prices, the liquidity constraints are: for the current period [p.sub.c] [d.sub.c] = [bar.m] - [s.sup.h.sub.M] and for the following periods [p.sub.c][d.sub.c,t] = [m.sub.t] - [s.sup.h.sub.M,t], with t = 1, ..., where [bar.m] is the endowment of money at the beginning of the period under examination, [s.sup.h.sub.M] is the amount of money lent to producers in the same period, [m.sub.t] is the quantity of money at the beginning of period t and [s.sup.h.sub.M,t] is the amount lent to producers in period t. The budget constraints (which concern payment at the end of each period) are, for the current time period:

[p.sub.k][d.sub.k] + [p.sub.l][d.sub.l] + [p.sub.B][d.sub.B] + [bar.m]+[d.sub.M] = [p.sub.l][bar.l] + z[bar.b] + (1 + i) [s.sup.h.sub.M]

and for the foilowing periods

[p.sub.k][d.sub.k,t] + [p.sub.l][d.sub.l,t] + [p.sub.B][d.sub.B,t] + [m.sub.t]+[d.sub.M,t] = [p.sub.l][[bar.l].sub.t] + [b.sub.t] + (1 + i) [s.sup.h.sub.M,t] t = 1, ...

where [d.sub.B,t] is the excess demand for securities at time t, [d.sub.M,t] is the excess demand for money, [[bar.l].sub.t] is the vector of available labour services, (21) and [b.sub.t] are the securities at the beginning of period t. There are also some constraints linking variables of successive periods: [m.sub.t] = [m.sub.t-1] + [d.sub.M,t-1] and [b.sub.t] = [b.sub.t-1] + [d.sub.B,t-1], with t = 1, ..., [m.sub.0] = [bar.m], [b.sub.0] = z[bar.b], [d.sub.M,0] = [d.sub.M], and [d.sub.B,0] = [d.sub.B].

The maximisation of the utility function with respect to [d.sub.k], [d.sub.l], [d.sub.c], [d.sub.B], [s.sup.h.sub.M], [d.sub.M], [d.sub.k,l], [d.sub.l,l], [d.sub.c,l], [b.sub.l], [d.sub.B,l], [m.sub.l], [s.sup.h.sub.M,l], [d.sub.M,l], ..., [d.sub.k,t], [d.sub.l,t], [d.sub.c,t], [b.sub.t], [d.sub.B,1], [m.sub.t], [s.up.h.sub,M,t], [d.sub.M,t], ... subject to the aforementioned constraints leads to a choice (22) which can be expressed, for the current period (the only one relevant in the analysis of a temporary equilibrium), by the relationships

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These relationships can be aggregated for all consumers. We obtain the demand functions for circulating capital goods

(6) [X.sup.h.sub.c] = [F.sup.h.sub.c] ([p.sub.k], [p.sub.l], [p.sub.c], i, z)

the corresponding offer of money during the period under examination

(7) [X.sup.h.sub.M] = [p.sub.c][X.sup.h.c]

the offer of loans to producers

(8) [X.sup.f.sub.M] = [bar.M] - [p.sub.c][X.sup.h.sub.c]

the demand functions for services of fixed capital goods

(9) [X.sup.h.sub.k] = [F.sup.h.sub.k] ([p.sub.k], [p.sub.l], [p.sub.c], i, z)

the supply functions of labour services

(10) [bar.L] - [X.sup.h.sub.l] = [[bar.L] - [F.sup.h.sub.l] ([p.sub.k], [p.sub.l], [p.sub.c], i, z)

the condition of equal returns for loans and securities

(11) [p.sub.B] = 1/i

and the demand for securities and money (available at the end of the period), which are perfect substitutes

(12) [p.sub.B][X.sub.BB] + [bar.M] + [X.sub.MM] = z[bar.B] + (1 + i)[X.sup.f.sub.M] - [p.sub.k][X.sup.h.sub.k] + [p.sub.l] ([bar.L] - [X.sup.h.sub.l])

The number of relationships (6)-(12) is [n.sub.k] + [n.sub.t], + [n.sub.c] + 4.

D) Producers

Producers buy circulating capital goods, labour services and services of fixed capital goods required as inputs and sell the circulating and fixed capital goods obtained as outputs. Since the circulating capital goods used as inputs are paid during the period (while outputs and other inputs are paid at the end of the period) producers must borrow money at the beginning of the period, which will be given back with the corresponding interests at the end of the period.

Producers maximise profit

[[PI].sub.f] = [p.sub.c] [X.sup.f.sub.cc] + [p.sub.kk] [X.sup.f.sub.ff] - [p.sub.l][X.sup.f.sub.l] - (1 + i) [X.sup.f.sub.M]

where vectors [X.sup.f.sub.cc], [X.sup.f.sub.kk] indicate the quantities of products: [X.sup.f.sub.k], [X.sup.f.sub.l] the demands for services of fixed capital goods and of labour services, the amount of which equals in equilibrium the differences [bar.K] - [X.sup.h.sub.k] and [bar.L] - [X.sup.h.sub.l]; and [X.sup.f.sub.M] the quantity of money borrowed, subject to the liquidity constraint

[p.sub.c][X.sup.f.sub.c] = [X.sup.f.sub.M]

where [X.sup.f.sub.c] indicates the demands for circulating capital goods, the amount of which equals in equilibrium [bar.C] - [X.sup.h.sub.c] so that

(13) [p.sub.c] ([bar.C] - [X.sup.h.sub.c]) = [X.sup.f.sub.M]

and to the constraints which require all vectors to be non-negative and compatible with the technical possibilities of production. Assuming for the sake of simplicity fixed coefficients of production, these constraints require

(14) [A.sub.kc][X.sub.cc] + [A.sub.kk] [X.sub.kk] = [bar.K] - [X.sup.h.sub.k] [A.sub.l][X.sub.cc] + [A.sub.l] [X.sub.kk] = [bar.L - [X.sup.h.sub.l] [A.sub.cc][X.sub.cc] + [A.sub.ck] [X.sub.kk] = [bar.C] - [X.sup.h.sub.c]

where [A.sub.kc], [A.sub.kk], ... are matrices of coefficients. Taking into account these constraints and the competitive condition by which maximum profit is zero, we obtain the following relationships

(15) [p.sub.k] [A.sub.kc] + [p.sub.l][A.sub.lc] + (1 + i)[p.sub.c][A.sub.cc] = [p.sub.c] [p.sub.k][A.sub.kk] + [p.sub.l][A.sub.lk] + (1 + i)[p.sub.c][A.sub.ck] = [P.sub.kk]

The number of relationships (13)-(15) is [2n.sub.k] + [n.sub.l] + [2n.sub.c] + 1.

System (1)-(15) is composed of [4n.sub.k] + [2n.sub.l] + [3n.sub.c] + 9 equations, (23) two more equations than the variables, which are

[X.sup.h.sub.k], [X.sup.kk], [p.sub.k], [p.sub.kk], [X.sup.h.sub.l], [p.sub.l], [X.sup.h.sub.c], [X.sub.h.sub.M], [X.sup.f.sub.M], [X.sub.MM, [X.sub.BB], i, [p.sub.B] and z.

Correspondingly, there are two dependent relationships since Walras' law holds both for the transactions during the period and for the transactions with payments at the end of the period. In fact, beside the usual dependence (among relationships (1), (2), (3), (5), (12), (13), (14) and (15)), we find that the sum of relationships (4), (8) and (13) is zero.

4 Implications of the Restated Walrasian Theory of Money

Although based on Walras' principal hypotheses, the theory presented in Section 3 differs substantially from the original one developed by Walras. In particular, the theory of circulation and money differs notably from Walras, bur not the theory of capitalisation and credit. In fact, Walras' original theory of capitalisation and credit is readily obtained if we assume that households are also owners of fixed capital goods. (24)

The principal implications of the restated Walrasian theory of money are as follows.

a) There are two quantitative relations. Relationships (4) and (5) are two quantitative monetary equations: the former concerns transactions during the period under examination (the circulating capital goods produced in the preceding period and stored are sold by their owners to consumers and producers); the latter concerns transactions at the end of the period (the circulating capital goods produced in the period are sold by producers to the owners of circulating capital goods). The causal nexus of these relations can be synthesised as follows. Relationship (4) indicates that the value of money (that is, the general level of prices) depends on the quantity of money [bar.M], which is a predetermined variable. Relationship (5) indicates that the quantity of money issued at the end of the period [bar.M] + [X.sub.MM] depends on the value of money (that is, on the expected level of prices, which is predetermined and unchanged because of the assumption of stationary expected prices). Naturally, complete causal nexuses are not so simple.

b) The theory of money is fully integrated with the theories of exchange, production and capitalisation. The theory of money is integrated with the theory of exchange since the quantity of money, its value and the rate of interest influence consumers' choices, even if money is not directly useful. On the one hand, loans yield an income and, on the other hand, money is indirectly useful because of the liquidity constraints. The theory of money is integrated with the theory of production since producers are subject to the liquidity constraint and there is, among costs, the interest on monetary loans. The theory of money is integrated with the theory of capitalisation since, at equilibrium prices, securities and money loans are perfect substitutes in consumers' portfolios and give the same yield. Moreover, the analysis of stability of monetary equilibrium can be undertaken as usual by considering excess demands and the difference between selling prices and cost prices of products, and the presence of money matters. (25)

c) Securities and money are a veil. Securities and money exist only because real goods are owned by households through other agents. If these agents (that is, the owners of fixed capital goods and the owners of circulating capital goods) are not introduced and the property is given directly to households, securities and money do not exist. In this case for all the real goods we find equilibrium conditions which are identical to those obtained when there are also the owners of capital goods. (26) In this sense, securities and money are a veil. Anyhow, without friction, uncertainty and illusion, securities and money must perforce be a veil: they can neither improve nor worsen the allocation of resources, which is a Pareto optimum (the Pareto optimum of temporary allocations).

d) Paper money or bank deposits? In the formulation proposed in Section 3, money, which is paper money yielding nothing, is issued by the owners of circulating capital goods in order to finance the purchase of circulating capital goods. This assumption may seem unrealistic since it excludes the presence of banks. An alternative formulation considers the group of owners of circulating capital goods as representative of two types of agent: the owners of stores and the banks. They can be formally introduced and money defined as bank deposits, yielding an interest. This situation can be described in the following way. At the beginning of the period under consideration, banks' assets are represented by a credit towards the owners of stores, to whom they have lent the necessary sum for buying their initial inventories, and banks' liabilities are represented by a debit of the same amount towards households. This debt is represented by bank deposits. Assets and liabilities give a yield (in equilibrium at the current rate of interest), which is credited at the end of the period. During the period banks give credit to producers for their purchases of inputs. Whenever producers buy circulating capital goods from the owners of stores, the producers' debt increases and that of the owners of stores decreases to the same extent. When consumers buy consumer goods the debt of owners of stores decreases and consumers' deposits decrease to the same extent. At the end of the period, producers sell products and reimburse their debt, yield included. In the meantime, the owners of stores have also reimbursed their debt, but they, at the end of the period, buy products and in this way a new debt is created. At the end of the period, households receive the balance between proceeds and payments and the yield of securities and bank deposits, which composes their new deposits. The total amount of bank deposits will again be equal to the debt of the owners of stores. Equilibrium is not substantially modified by the distinction between owners of stores and banks. The only modification is represented by the course of the prices of circulating capital goods during the period: prices are not constant bur increasing, according to the relationship

p([tau]) = p [1+i]/[l+(1-[tau])i]

where i is the rate of interest and 0 [less than or equal to] [tau] [less than or equal to] 1 is the instant of the payment within the period under examination. This relationship is determined by an arbitrage condition taking into account that bank money gives a yield. With these prices nobody can profit by anticipating or postponing the sale of goods in the period. (For instance, if a sale is anticipated the seller cashes a lower price but he reduces in advance his debt, thus paying a lower interest.) (27)

e) The length of the time period. The length of the period is relevant in Walras' theory of money. It is a synthetic representation of the asynchronies which justify the existence of money. In fact, in this theory, if the length of the period is varied according to a factor [theta], then, ceteris paribus, since agents need money for purchasing goods during the period, the quantity of money must vary according to the same factor; that is, we need a quantity of money multiplied by [theta] in order to maintain unchanged prices. (28) The assumption that loans have a predetermined maturity and that their reimbursement coincides with the payments at the end of the period, even if unnecessary for giving value to money, influences its value, since otherwise agents would need a different quantity of money for operating current purchases. A more realistic description would require the specification of asynchronies, that is, to specify the length and kind of productive process and the time intervals among all acts of production, consumption and investment.

If money is defined as bank deposits, instead of paper money, the predetermined maturity of loans is an unnecessary assumption and the theory can also be referred to continuous time. In this case, bank deposits are always equal to the value of circulating capital goods existing in storage or bought by producers (the quantities of these capital goods depend on the length of production processes and on asynchronies among production, consumption and investment).

5 The Model with Only One Good (Circulating Capital) and Money

Should there be only one good (circulating capital) and money, we find the following system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the solution of which requires

i = 1/[a.sub.cc] - 1

[p.sub.c] = [bar.M]/[bar.C]

With regard to the two quantitative relations implied by the restated Walras' model, the former, which determines the price with respect to the transactions during the period under examination, is represented by the equation

[p.sub.c] = [bar.M]/[bar.C]

the latter, which determines the quantity of money issued at the end of the period under the assumption of a stationary expected price, by the equation

[bar.M] + [X.sub.MM] = [p.sub.c][X.sub.cc]

where [X.sub.cc] = 1/[a.sub.cc] ([bar.C] - [F.sup.h.sub.c]([p.sub.c], 1/[a.sub.cc] - 1)).

The stability analysis (which is the modern version of the Walrasian tatonnement) takes into consideration the excess demand for the stored good

[E.sub.c] = [bar.M]/[p.sub.c] - [bar.C]

the excess demand for loans

[E.sub.M] = [p.sub.c][a.sub.cc][X.sub.cc] - ([bar.M] - [p.sub.c][F.sup.h.sub.c]([p.sub.c], i))

and the difference between the selling price and the cost price of the product

G = [p.sub.c] - (1 + i) [p.sub.c] - (1 + i) [p.sub.c][a.sub.cc]

Assuming money to be the numeraire, we can refer to these variables respectively the adjustment of [p.sub.c], i and [X.sub.cc], in order to find conditions of stability. For instance, with

[dp.sub.c]/dt = [k.sub.1][E.sub.c]

[d.sub.i]/dt = [k.sub.2][E.sub.M]

[dX.sub.cc]/dt = [k.sub.3][G.sub.c]

where [k.sub.1], [k.sub.2], [k.sub.3] are three positive parameters, the conditions for local stability (which require that the eigenvalues of the Jacobian matrix of functions [E.sub.c], [E.sub.M] and [G.sub.c] with respect to [p.sub.c], i and [X.sub.cc] have a negative real part) are satisfied if [partial derivative][F.sup.h.sub.c]/[partial derivative]i < 0, that is, if the demand of households for the circulating capital good is a decreasing function of the interest rate.

If the owners of the circulating capital good are identified with households, so that money is excluded and loans are made in kind. then the model with only one good, assuming this good to be the numeraire, shows the functions

[E.sub.c] = [F.sup.h.sub.c](i) + [a.sub.cc][X.sub.cc] - [bar.C]

[G.sub.c] = 1 -(1 + i)[a.sub.cc]

therefore generally implying different conditions of stability. However, with

di/dt = [k.sub.1][E.sub.c]

[dX.sub.cc]/dt = [k.sub.2][G.sub.c]

the conditions of local stability again require [dF.sup.h.sub.c]/di < 0 since this model is so simple that only the demand function for the circulating capital good matters.

When introducing bank deposits in place of paper money, the results of the model under examination are modified in an essentially irrelevant way. In fact, the first two equations are substituted by the following ones: for the owners of stores

[p.sub.c][bar.C] = [bar.D]

[p.sub.c][X.sub.cc] = [bar.D] + [X.sub.D]

where [bar.D] is their debt towards banks at the beginning of the period and [bar.D] + [X.sub.D] is the debt at the end; for the banks

[bar.D] = [bar.M]

[bar.M] + [X.sub.MM] =[barr.D]+ [X.sub.D]

where [bar.M] is again the quantity of money (now, bank deposits) at the beginning of the period and [bar.M] + [X.sub.MM] that at the end, In the remaining equations the quantity [X.sup.f.sub.M] (loans of households to producers) indicates the debt of producers towards banks before the sale of products (this debt corresponds to deposits of households). Paying attention to the sales of the stored good, by indicating with [X.sub.c],j] and [[tau].sub.j] respectively the quantity sold at instant [[tau].sub.j] and this instant (with O [less than or equal to] [[tau].sub.j] [less than or equal to] 1 and [[summation].sub.j][X.sub.c,j] = [bar.C]), the proceeds of sellers are [p.sub.c]([[tau].sub.j]) [X.sub.c,j] where [p.sub.c]([[tau].sub.j]), is the price at instant [[tau].sub.j]. With the course

[p.sub.c]([tau]) = [p.sub.c] [1+i]/[1+(1-[[tau].sub.j])i

the owners of stores will have a null balance at the end of the period whatever instants [[tau].sub.j] and quantities [X.sub.c,j] may be. In fact, taking into account their initial debt, proceeds and interests, we find

[bar.D](1 +i) - [p.sub.c][[summation].sub.j][X.sub.c,j][p.sub.c]([[tau].sub.j])(1 + (1 - [[tau].sub.j])i) = ([bar.D] - [p.sub.c][bar.C])(1 + i) = 0

The introduction of bank deposits in place of paper money allows for the analysis in continuous time (so avoiding the predetermination of the maturity of loans). In continuous time, assuming that the production lag is equal to a unitary period of time, the owners of stores have inventories

C(t) C(0)+ [[integral].sup.1.sub.0] ([X.sub.cc]([tau]) - [X.sup.f.sub.c] ([tau]))d[tau]

and debt towards banks

[D.sup.c](t) = [D.sup.c] (0) + [[integral].sup.1.sub.0] [p.sub.c]([tau])([X.sub.cc]([tau]) - [X.sup.h.sub.c]([tau]))[e.sup.i([tau])(t - [tau])]d[tau]

where [X.sub.cc]([tau]), [X.sup.h.sub.c]([tau]) and [X.sup.f.sub.c]([tau]) are respectively the current intensity of production, of consumer demand and of producer demand and i([tau]) is the current rate of interest for a unitary period of time; households have bank deposits (since

M (t) = [D.sup.c] (t) + [D.sup.f] (t) for every t [greater than or equal to] 0)

M (t) = M(0) + [D.sup.f](t) - [D.sup.f](0) + [[integral].sup.1.sub.0] [p.sub.c] ([tau])([X.sub.cc] ([tau]) - [X.sup.h.sub.c]([tau]) - [X.sup.h.sub.c] ([tau]))[e.sup.i([tau])(t - [tau])]d[tau]

demand for consumption

[X.sup.h.sub.c](t) = [F.sup.h.sub.c] ([p.sub.c](t), i(t))

and the budget constraint

dM(t)/dt + [p.sub.c](t)([X.sup.h.sub.c](t) = i(t)M(t)

while for producers there are the relationships

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In case of a stationary equilibrium, where the only asynchrony consists of the production lag and money is necessary only in order to finance production, we have, for every t, C = 0, [D.sup.c] = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This system can be easily solved with respect to [p.sub.c], i, [X.sup.h.sub.c], [X.sup.f.sub.c], and [X.sub.cc] (while M is a datum).

6 Concluding Remarks

In the restated Walrasian theory of money, the quantity of money, its value and its movement depend on the assumed institutional framework, that is, on the assumptions that the owners of circulating capital goods buy products at the end of each period and sell them, exclusively in exchange for money, during the following period; and that loans are reimbursed and all other payments are effective at the end of each period, and so on. We have already emphasised the importance of these hypotheses which govern, even if they are not all equally relevant, the function of money in transactions. This theory explains how money can exist, have value and be relevant in agents' plans of utility and profit even in a world without friction, uncertainty and illusion. In fact, money is the exclusive purchasing power on circulating capital goods (the life of which lasts only one period); it is issued at the end of each period, when the owners of circulating capital goods buy it from their producers; and consumers and producers spend during each period the whole quantity of money existing at its start. Nevertheless, money is not socially useful, since it is a veil.

However, the function of money in transactions is insufficient for describing actual monetary economies in any realistic sense, not only because there is a permanent store of money, but also because monetary exchanges are not actually determined by an institutional constraint (stated in a theory by an assumption) but by mutual convenience of agents, which is not analysed at all. (29) In other words, friction, uncertainty and illusion, although indispensable for explaining the emergence of a monetary economy, (30) are not indispensable for describing a monetary economy. Walras' theory of money really describes a monetary economy without friction, uncertainty and illusion: an economy where money has only the transactions role. Walras' original version of the theory of money is inconsistent not because uncertainty is disregarded but because of the imperfections described in Section 2. The restatement proposed in Section 3 sets out to be a consistent version of Walras' theory. Even if this theory supplies only the mechanics of a monetary economy, without explaining the emergence of such an economy, it is, however, in my opinion, a worthy skeleton, on which meaningful monetary general equilibrium theories can be built, by introducing, with friction and uncertainty, the reasons why money emerges and becomes socially useful.

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Notes

(1) I wish to thank the participants in the Troisieme colloque, Association internationale Walras (Lyon, 20 September 2002) and two anonymous referees for their helpful comments. This paper is an in-depth revision and extension of previous studies (Montesano 1986 and 1991).

(2) Obviously, data introduced in a subsequent stage contribute to determine variables introduced in the preceding stages. For instance, the coefficients of production (introduced in Part IV) contribute to determine the prices of consumer goods (introduced in Parts II and III). On the contrary, Negishi (1977, pp. 602-3, 611) believes that money does not affect variables except the general level of prices, because money is introduced in Part IV, after the determination of the relative prices in the preceding Parts. This argument is similar to Nogaro's (1906, pp. 687-8).

(3) A debate on the Walrasian theory of capitalisation took place among Italian economists in the sixties (Montesano 1970 and 1971). The Walrasian theory of capitalisation was discussed, among others, by Jaffe (1953), Floss (1957), Morishima (1960), Collard (1973), Walker (1984), Eatwell (1987) and van Daal (1998).

(4) These three versions are proposed respectively by Walras (1874-77), Walras (1889, 1896) and Walras (1900, 1926). An outline is given by Jaffe (Walras 1954, pp. 600-602) and Porta (1980, pp. 18-34).

(5) It has been discussed whether this theory anticipates the Cash-Balance Approach, as Marget (1931) declares, or it provides only a Cash-Balance Equation, as Patinkin (1965, pp. 542-6) asserts (in this case a demand for money determined by its marginal utility would not exist).

(6) Pareto and his followers disregarded Walras' theory of money. Schumpeter (1954, pp. 1020-6, 1082-3) approved of it. Criticisms were given by Nogaro (1906), Del Vecchio (1909), Hicks (1933) and Patinkin (1965). Aupetit's (1901) theory is a development of Walras' theory. Walras' theory of money was discussed, among others, by Marget (1931 and 1935), Rosenstein-Rodan (1936), James and Lecoq (1961), Kuenne (1961, 1963), Collard (1966), Morishima (1977), Negishi (1977), Jaffe (1980), Hall (1983), Walker (1991), Bridei (1997) anal Rebeyrol (1998).

(7) The inclusion of money in the general equilibrium theory is not a simple task. There is a vast literature on this topic: see, for instance, Ostroy and Starr (1990), Hahn (1990), Duffie (1990) and Walsh (1998).

(8) An example is given by the so-called neowalrasian theory of money, which is declared neowalrasian only because money is considered in the general equilibrium scheme (albeit a macroeconomic one) and it is included in the utility function. The origin of this theory can be attributed to Hicks (1935); its more significant elaboration is provided by Patinkin (1965). Reformulations nearer to Walras' theory are those proposed by Aupetit (1901), Kuenne (1963) and Morishima (1977).

(9) Walras (1954, p. 317): 'If, however, we suppose these data [that is, the data of the problem] constant for a given period of lime and if we suppose the prices of goods and services and also the dates of their purchases and sale to be known for the whole period, there will be no occasion for uncertainty'.

(10) Walras (1954, pp. 318-9):

'We propose to solve the problem of equilibrium of the mechanism of circulation in the same general way as we solved the problem of equilibrium of the other mechanisms previously considered. Thus, we shall imagine an economy establishing this equilibrium ab ovo over a given period of lime during which no change take place in the data of the problem. We shall, accordingly, endow our land-owners, workers and capitalists, viewed as consumers, with random quantities of circulating capital and money, just as we endowed them before with random quantities of fixed assets in the form of landed capital, personal capital and capital proper. Furthermore, we shall suppose our entrepreneurs to borrow the circulating capital goods and the money they need for production, just as we previously supposed them to borrow the fixed capital goods they required. [...] The new capital goods, both fixed and circulating, which are made available during the second phase [that is, the static phase] [...] are not put to use until the third phase [that is, the dynamic phase]. This should be clearly understood from the above definitions and constitutes the first change in the data of our problem.'

At this point Walras quotes his [section]251 where, with respect to the theory of capital formation, Walras (1954, p. 283) writes: 'Although the economy is becoming progressive, it remains [for the time being] static because of the fact that the new capital goods play no part in the economy until later in a period subsequent to the one under consideration'.

(11) Walras (1954, p. 242) presenting his theory of production, where he assumes away the time element, writes that the production lag gives a rationale for the theory of circulation and money, as summarised in [section]273 (Walras 1954, pp. 316-18).

(12) The hypothesis of expectations of stationary prices excludes not only permanent inventories of circulating capital goods produced in the period under examination, but also of circulating capital goods produced in the preceding period, but no more produced in the period under examination since their production cost exceeds price. In fact, such goods would be stored without being wholly consumed within the period under examination only if the expected price for the following period is at least (1+i) limes the current price, where i is the interest rate.

(13) In Walras' (1954. p. 320) system of equations there is only one budget constraint for every consumer. It implies that all payments are supposed to be simultaneous, at least from the economic point of view. The budget constraint includes both the expenditure for the commodities produced in the period (that is, [d.sub.a] + [d.sub.b][p.sub.b] + ...) and the net earning for the service of availability (that is, [o.sub.a'] [p.sub.a'] + [o.sub.b'] [p.sub.b'] + ... + [q.sub.m][p.sub.m'] + ... + [o.sub.u'] [p.sub.u]'). Consequently, the goods consumed during the period and produced in the same period are bought at the prices [p.sub.a] = 1, [p.sub.b], ..., while the prices of the same goods produced in the preceding period and inventoried include also the price for their service of availability. If, disregarding the equations, we follow Walras' literary presentation, a similar inconsistency comes out from the assumption that exchanges of commodities produced in the period occur during the period, as well as exchanges of inventoried commodities. For instance, let us suppose that a firm buys the service of availability just after the starting of the current period. The owner of these circulating capital goods gets some money, which he can use, before the end of the period, for purchasing commodities produced in the period. Thus, this consumer obtains commodities before the end of the period without paying any cost for their service of availability (this cost would be paid, as an opportunity cost, if the money used for purchasing these commodities comes out from the savings of the preceding period). Moreover, the assumption that products can be consumed in the same period of production, while they cannot be stored and supply the service of availability, seems excessively strong.

(14) In his theory of circulation and money Walras assumes that the payments regarding the productive services and the products of the period of time considered are made in money. Walras (1954, p. 316): 'The payments for these services, evaluated in numeraire, will be made in money at fixed dates. The delivery of the products will also begin immediately and will continue in a given manner during the same period. And the payments for those products, evaluated in numeraire, will also be made in money at fixed dates'. This implies a cash-in-advance condition, as Walras promptly deduces (1954, pp. 316-7): 'It is readily seen that the introduction of these conditions makes it necessary, first, so far as consumers are concerned, that they have on hand a fund of circulating or working capital consisting of : (1) certain quantity of final products [...1; and (2) a certain quantity of cash on hand [...]; and, secondly, so far as producers are concerned, that they have on hand a fund of circulating and working capital, consisting, in this case, of: (1) certain quantities of raw materials held in stock for future use and certain quantities of finished products placed on display for sale [...]; and (2) a certain quantity of cash on hand [...]'.

(15) This reasoning is the basis of Hicks' criticism (1933, pp. 446-8) of Walras' theory of money. Also Rosenstein-Rodan (1936, pp. 271-2) follows this reasoning, while hinted at earlier by Knight (1921, pp. 193-4f). The above assumption has been used by Kuenne (1963, pp. 298-9).

Here money is paper money, as Walras clearly states (1954, pp. 320 and 325): 'Let (U) be money which we shall first suppose to be an object without any utility of its own, but given in quantity, distinct from (A) [that is, the numeraire], having a price of its own [p.sub.u], and a price for its service of availability [p.sub.u'] = [p.sub.u] i. We reserve the right, however, later to identify (U) with (A)' and 'To start with, let us suppose, as we have already done [that is, at p. 320], that (U) is money, but is neither a commodity nor anything that can serve as the numeraire. It is easy to imagine such a situation. It would be true, for example, of a country where money consisted of inconvertible paper francs, but where prices were quoted in metallic francs or gold or silver. In Austria and Italy at the present time, for instance, money consists of inconvertible paper florins and liras: but under certain circumstances prices could be quoted in these countries in terms of gold or silver florins and liras'. The use of credit money (bank deposits) changes the description of the problem: this possibility will be briefly analysed in Section 4.

(16) The need to consider money as the medium of exchange is emphasised by some economists, particularly by Clower (for instance, Clower 1967), against the neowalrasian theory of money. However. neither Clower (1967) nor his follower Howitt (1973) take into account that Walras (1954, pp. 316-7) assumes monetary exchanges during the period.

(17) Aupetit (1901, pp. 149-51) is aware of the need to introduce an endogenous change in the quantity of money. However, Aupetit's theory refers to a metallic money produced under the same conditions as other capital goods. But in Walras' theory paper money is considered, the cost of production of which is zero.

(18) Diewert (1978, pp. 78-9) believes that this equality is incorrect, indicating that the original Walras relationship [p.sup.i.sub.kk] = [p.sup.i.sub.k]/1/[p.sub.B] + [v.sup.i] must be substituted by [p.sup.i.sub.kk] = [p.sup.i.sub.k] (1 + 1/[p.sub.B])/1/[p.sub.B] + [v.sup.i]. Diewert also extends this relationship to circulating capital goods (which would require [v.sup.i] = 1 and [p.sup.i.sub.kk] = [p.sup.i.sub.k]), thereby neglecting Walras' exclusion of circulating capital goods from the theory of capitalisation and credit since they are the object of the theory of circulation and money. Walras' relationship is correct with reference to fixed capital goods the moment we consider that they are assumed to be used only from the period subsequent to that of their production, so that their present value is [p.sup.i.sub.kk] = [[pi].sup.i.sub.k]/1 + 1/[p.sub.B] + [[pi].sup.i.sub.k]/[(1 + 1/[p.sub.B]).sup.2] + ... = [[pi].sup.i.sub.k]/1/[p.sub.B]

where net income [[pi].sup.i.sub.k] is [[pi].sup.i.sub.k] = [p.sup.i.sub.k] - [v.sup.i][p.sup.i.sub.kk]. We can extend the relationship under examination also to circulating capital goods by including the service of availability: consequently we have [p.sup.i.sub.kk] = [p.sup.i.sub.k] + [p.sup.ia.sub.k]/1/[p.sub.B] + [v.sup.i] where [p.sup.ia.sub.k] is the value of the service of availability, [v.sup.i] = 1, and [p.sup.i.sub.kk] = [p.sup.i.sub.k], so that [p.sup.i.sub.k] = [p.sup.i.sub.k] + [p.sup.ia.sub.k]/1/[p.sub.B] + 1 which requires [p.sup.ia.sub.k] = [p.sup.i.sub.k]/[p.sub.B] exactly the relationship proposed by Walras in the theory of circulation and money.

(19) Investment is not infinitely elastic if expected prices are not stationary. If the demand for fixed capital goods is not infinitely elastic and capital goods are not perfect substitutes, then equations (2) are substituted by demand functions of the type [D.sub.kk] = [F.sub.kk] ([p.sub.k], ...). Morishima (1977, pp. 100-22), on the contrary, maintains conditions (2) even when a non-infinitely elastic investment is introduced, consequently obtaining an inconsistent system.

(20) Investment is not infinitely elastic if expected prices are not stationary. In this case the prices of circulating capital goods produced in the period under examination and exchanged at the end of the period do not necessarily equal the prices of circulating capital goods produced in the preceding period, available in stores and exchanged during the period.

(21) Vectors [bar.l] and [[bar.l].sub.t] are assumed to be data. However, we can assume a set of possible labour services represented by a function g(l, ..., [l.sub.t]) = 0. Vector l will be determined by maximising the utility function with respect to l (and all other variables) subject to g(l, ..., [l.sub.t], ...) = 0 (and all other constraints).

(22) The first order conditions are represented by the constraints and by the equations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [p.sub.B] = 1/i, where [lambda] is a Lagrange multiplier. Since [p.sub.B] = 1/i, money and securities are perfect substitutes.

(23) All relationships are indicated as equalities. Taking into account land, old fixed capital goods and that some goods could be free goods, some relationships would be represented as inequalities. For simplicity, this possibility is not introduced: however, it is useful only when we must demonstrate the existence of a solution, a problem which goes beyond the scope of this work. With reference to land and old fixed capital goods no longer produced, the number of which is here indicated with [n.sub.k]', the [n.sub.k]' corresponding variables in vector [X.sub.kk] are equal to zero and the [n.sub.k]' corresponding equations in subsystem (15) are inequalities, that is, [p.sub.k][A.sub.kk] + [p.sub.l][A.sub.lk] + (1 + i) [p.sub.c] [A.sub.ck] [greater than or equal to] [p.sub.kk]. If we like to take into account also the fixed capital goods of a new type (which are produced in the period under consideration for the first time), then their quantities in vectors [bar.K] and [X.sup.h.sub.k] are zero, as well as the corresponding coefficients [A.sub.kc], and [A.sub.kk], the corresponding equations in subsystem (2) do not determine the current prices of their services, but they indicate the expected prices for the future periods, and the corresponding functions [F.sup.h.sub.k](.) in subsystem (9) are identically equal to zero: consequently, for the new fixed capital goods, there are no corresponding equations in subsystems (2) and (9) and no corresponding variables [p.sub.k] and [X.sup.h.sub.k].

(24) If owners of fixed capital goods are identified with households, then relationships (1) and (3) must be merged with relationship (12). We obtain, also taking into account relationships (2) and (8)

[p.sub.kk] + [X.sub.kk] + [X.sub.MM] = [p.sub.k]([bar.K] - [X.sup.h.sub.k]) + [p.sub.l] ([bar.L] - [X.sup.h.sub.l]) + i[bar.M] - (1 + i) [p.sub.c][X.sup.h.sub.c]

which is the last of Walras' equations (2) (Walras, 1954, p. 279) once the variables considered by the theory of circulation and money are disregarded. The same goes for the other equations.

Walras' original theory identifies owners of fixed capital goods with households (so that saving is identical to investment). Other theories (like Keynes' theory) identify them with producers. Both these positions are compatible with the formulation presented in this paper and determine the same equilibrium conditions. The infinite elasticity of investment (which is not present in Keynes' theory) is not determined by the lack of distinction between the motivations to save and motivations to invest, but by the hypotheses of expectations of stationary prices and of perfect competition (with free entry and exit).

(25) Patinkin (1965, p. 571) asserts that Walras' theory of money is not fully integrated with the theory of production and capitalisation and refers this missed integration to the lack of interdependence between the tatonnement on the money market and that on the other markets. Patinkin's criticism does not apply to the restated model owing to the change of assumptions with respect to Walras' original model.

(26) The equilibrium system without securities and money is composed of relationships (14), (15) and

[p.sub.k] = ([??] + iI)[p.sub.kk]

[X.sup.h.sub.c] = [F.sup.h.sub.c]([p.sub.k], [p.sub.l], [p.sub.c], i)

[X.sup.h.sub.k] = [F.sup.h.sub.k] = ([p.sub.k], [p.sub.l], [p.sub.c], i)

[p.sub.kk] [X.sub.kk] + [p.sub.c] [X.sub.cc] = [p.sub.k] ([bar.K] - [X.sup.h.sub.k]) + [p.sub.l] ([bar.L] - [X.sup.h.sub.l]) + (l + i) [p.sub.c] ([bar.C] - [X.sup.h.sub.c])

This system is homogeneous of degree zero with respect to [p.sub.kk], [p.sub.c], [p.sub.k], [p.sub.l].

(27) Owners of stores must sell all the inventories, otherwise, with expectations of stationary prices (that is, with the same course of prices in the following period), they will incur losses.

(28) The above proportionality (which depends on the condition ceteris paribus) excludes seasonal production. Of course, if the length of the period increases by factor [theta], then also the quantities of inputs, products and services change in the same proportion.

(29) This criticism applies to all models which assume a monetary constraint on exchanges (presented, together with others, by Arcelli 1975). Schumpeter (1939, pp. 547-8) criticises, with precise reference to the theory of money, this kind of assumption, which I would like to justify by considering that the elements, disregarded by the theory, which determine some features of the real word, can be assumed as institutional data. This method is very common in economics: for instance the same distinction between consumers and producers, currently assumed by theory, derives from ignoring the elements which actually determine it.

(30) Let us disregard the case where money is the only durable good, thus store of value, demanded for carrying purchasing power over the period.

Aldo Montesano, Bocconi University, Milan, Italy. Email: aldo.montesano@unibocconi.it.
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Date:Jan 1, 2008
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