The Encaisse Désirée of Léon Walras

Reymerswaele's Banker


The Quantity Theory is among the oldest economic propositions, dating long before Adam Smith. It is in this respect that Keynes (1936) rightly referred to all pre-Keynesian theorists as the "classicals" - for in breaking the Quantity Theory he was attacking a theory cherished by theorists long before the Neoclassical revolution. Thus, pre-1871 Classical theorists and the Marginalists who followed were both objects of Keynes's critique.

However, one must admit a certain peculiarity about this situation. Why did the Neoclassical Marginalists hold on to the Quantity Theory when breaking with the rest of the pre-1871 Classical theory? How did they fit its old propositions in the new theoretical schema?

The Quantity Theory, whatever its incarnation, proposes that as the supply of money rises, whether endogenously or exogenously, the price level - and, in the longer run, only the price level - will also rise proportionally. Although arguments for it were largely inductive, quantity theorists recognized from the very beginning that the main focus of future research should be to state precisely how this happens. Fisher provided one version by stressing the mechanism of bidding away excess balances. Wicksell another through the more indirect mechanism of credit operating on supply-constrained factor markets. However intuitive, neither explanation bases itself explicitly on the principles of Neoclassical economics. There is no direct or obvious relationship between these explanations and the choice-theoretic foundations of Neoclassical theory.

However, Léon Walras (1874) did attempt to construct a consistent theoretical rationale for this - in an approach better known as "encaisse désirée" or "desired cash balance".

Recall that most old Quantity Theorists regard money as a commodity with a difference: it is not desired for itself but really only for its purchasing power. If so, then it should not be modelled in a choice-theoretic framework which concentrates exclusively on the exchange of commodities to satisfy wants. But the whole idea of desired money balances is that they are, in effect, demanded. It is therefore a "choice" problem and thus some microfoundations must be given to that choice. Fisher (1911) circumvented the problem by arguing that it was not really a choice as much as it was an imposition. If there is a cash constraint ・la Clower (1967), in the sense that nothing but money is accepted for exchange, then the exchange of commodities (the real, underlying economic action) requires some amount of money to carry it through. It is not desired in any way but rather is required solely by convention. In contrast to the Cantabrigian approach, this is a "mechanical" rather than a "behavioral" theory of money demand. (Of course, the cash-constraint is, however, quite debatable given the wide use of "inside money" which, while still defined as "money" obviously does not impose a constraint in this manner; recognizing the existence of "endogenous" money blows this approach apart even further).

To some extent, Léon Walras conceived of money differently. Primarily, he treated money in a manner analogous to capital. Capital, he recognized, is not a regular commodity since it is not desired for itself but rather for its service as a facilitator of intertemporal production and thus allocation. It is in providing such a service that it is demanded.

"[Capital goods] are demanded because of the land-services, labour and capital-services they render, or better, because of the rent, wages and interest which these services yield."

(L.Walras, Elements of Pure Economics, 1874: p.267)

Similarly, money is also not desired for itself. However, again like capital, money provides a stream of services, in the form of overcoming transactions costs or permitting the temporal break-up of purchases and sales - as the Cantabrigians argued (and Fisher did not). Thus, it is only in yielding these "services" to consumers that we can say that money yields utility.

It is the store of value function that Walras identifies as the "service" of money. Therefore, money is identified with capital as an intertemporal reallocation mechanism with the only difference being that the returns (i.e. services) yielded by capital are now paid "not in kind but in money." (Walras, 1874: p.320).

We should note, perhaps prematurely, that Wicksell objected to this conception:

"It is not true that `money is only one form of capital', that the lending of money constitutes a `lending of real capital goods in the form of money', etc. Liquid real capital (i.e. goods) are never lent (not even in a system of simple merchandise credit); it is money which is lent, and the commodity capital is then sold in exchange for this money."

(K. Wicksell, Interest and Prices, 1898: p.xxvi).

Nonetheless, let us adhere to Walras's conception. By yielding "services of availability", money does improve utility and therefore must be placed in the consumer's utility function. Thus, let m = M/P, where M is nominal money balances and P is the price level. Identifying x as a vector of commodities, then the utility of the consumer can be written:

U = U(x, m)

where m (= M/P) is the amount of real money balances desired by the consumer, i.e. the "encaisse désir&eactue;e". Maximizing this function with respect to a budget constraint, we obtain two "marginal utilities of money". The first is merely the Langrangian multiplier. This is really the marginal utility of spending an extra "dollar" of income on goods and is assumed constant (cf. Hicks, 1939: p.26). However, our main interest is the other marginal utility of money, dU/dm, i.e. the marginal utility of money services. This can be shown easily (from Samuelson, 1947: p.117-22). Defining the budget constraint as pxe = px + rm, where xe is an endowment vector and r is the rate of interest on alternative assets (i.e. the opportunity cost of holding real money balances), then the problem becomes, in Lagrangian form:

L = U(x, m) + v(pxe - px - rm)

where v is the Lagrangian multiplier (or the "marginal utility of income"). We obtain the first order condition for m:

dL/dm = dU/dm - vr = 0

so that:

(dU/dm)/r = オ

Or the (perpetually) discounted stream of money services (i.e. (dU/dm)/r) will be equal to the marginal utility of income (v). Or alternatively, we can write this as:

dU/dm = vr

The convenience gained by holding an extra dollar in the form of money (dU/dm) is some proportion of the opportunity cost of doing so (r).

The essence of Walras's encaisse désirée is that money yields utility by providing the service of being a temporary abode of income between sales and purchases - if purchases are for goods to be delivered at time periods in the future. The amount of money held will be to the point where capitalized marginal stock value of money ((dU/dm)/r) is equal to the marginal flow value (v). Alternatively, what we are essentially referring to is that the marginal benefit of holding money is equated to the marginal cost (in this case, some proportion of it).

There are several things we must note. The first peculiarity of the system, stressed by Patinkin (1956: p.549) is that it does not make clear why precisely money is placed in the utility function if the only usefulness is that it permits future transactions to take place. In a sense, it is almost a Fisherian transactions approach to money: individuals who have some trade in the future must hold money. The choice-theoretic framework is severely weakened by such a proposition.

If there are other commodities which can also perform this function, then we might indeed consider it a choice problem. But this brings us to our second comment: recognizing that money, like capital, is defined as a time-unifying commodity - that is the service it yields, by Walras's own argument - it is not made clear why money is necessary. Hicks (1933) argued that if money is demanded solely "as a means of making future payments", then:

"If the date and amount of future payments are absolutely certain, money does not need to be held in order to meet it; it is more profitable to lend it out, until the date when the payment has to be made." (J. Hicks, 1933)

Thus, if the Walrasian equilibrium is defined as an intertemporal one with perfect foresight (which it is), then holding money does not make sense since one is foregoing interest on alternative assets - which are also time-unifying. Why anyone should hold money remains a mystery.

The third major problem with Walras's analysis is that there seems little sense in placing money in the utility function and then going on to assume that money is, in some sense, a veil. Walras's propositions here are tricky. Firstly, to understand his system, we need to set up the budget constraint. Recall, as we had written before, that pxe = px + rm. As we saw, m was M/P implying that the price of money is 1/P. Thus, we can rewrite the constraint as:

pz + rzm = 0 where z is a vector of excess demands and zm the excess demand for money. Or, more explicitly: pz + r(Md - Ms)/p = 0 where Md and Ms are money demand and money supply respectively. Now if, by the process of tatonnement (holding the money market constant), z = 0, so that the good markets clear, then it must be that rzm = 0. Now, if r is established by the initial tatonnement in the goods market (by the equilibrium in the capital market), then zm = 0. However, zm is merely the excess demand for money multiplied by the price of money stock, i.e. 1/P. Now, unless P is infinite, the only way that zm = 0, is if the demand for money is equal to the supply of money.

Let us now displace this curious story. Let the supply of money increase (in some manna-from-heaven style), then actual money holdings would exceed the desired real money holdings, i.e. Md < Ms. The implicit assumption in Walras is that r and the real price vector, p, would remain fixed in such a circumstance so that zm would have to move to back to zero by itself. The logic Walras proposed was that P, the price level (not real price vector) would change, depreciating the value of real money supply, until Md/P = Ms/P.

However, there is really no explanation for this in Walras. If the money market is in disequilibrium, why does Walras assume that neither r nor p will help with the adjusting? In other words, a disequilibrium in the money market disequilibrium is cured by movements entirely within the money market, with no argument shown as to what makes that happen.

The curious part of the story is that we cannot intelligently speak of nominal money demand and supply not being equal. We get a story which immediately requires that zm = 0, that money demand and supply are, from the outset, the same. If there is a discrepancy, there will be an internal adjustment in the money market (without reference to exactly how) to bring it into equilibrium. However, the curious question is whether there can ever be a disequilibrium. By Walras's Law:

pz + rzm = 0

so if pz = 0 (by Walras's own assumption), then zm = 0. There can never be zm > 0 or zm < 0 , since this would imply that some "real" market in the z vector would be in converse disequilibrium implying some adjustment in prices. But since Walras assumes that neither the real price vector p nor r ever changes, then z cannot be in disequilibrium. By the way the story is told, the money market is in equilibrium at all times. Thus, an increase in the money supply immediately (somehow) raises the price level and eliminates itself. (Morishima notes that the problem might simply be that Walras "did not know Walras's Law" Morishima (1977: 126)). Furthermore, we can note that this implies that there is no determinate price level. Since money demand is equal to money supply at all times, then whatever P happens to be is irrelevant. It may rise or fall at will, it will not lead to a disequilibrium. Thus, Walras's story does not extend to the aggregate demand-aggregate supply world of Neoclassical macroeconomics.

As we can see, Walras's story of encaisse désirée is full of holes. Morishima (1977) makes note that it really should only be considered in the context of a growing economy, but does admit that the implied imposition of Say's Law is quite intolerable. However, Walras did provide the first tentative steps to a choice-theoretic approach to a monetary economy. Unfortunately, this approach was buried by the much simpler, and more intuitive, Quantity Theory within the aggregative equation of exchange.

Ludwig von Mises (1912) did attempt to construct a marginal utility theory of money, but noted immense difficulties leading to his conclusion that because money has no utility, then it must arise from an "objective exchange value". How is this to explain fiat money? Mises merely argues that the value of money today is determined in part by the value of money yesterday, thus, imputing back several decades (nay, centuries), to the time when money was actually a commodity, then the value of money today is related, to a great extent, to the value it had when it was gold.

This apparently improbable story, what has become known as the "Mises Regression Theorem", however, was the only attempt at creating a marginal utility theory of money until John Hicks (1935) called for a "marginal revolution" in the theory of money. Hicks's argument was simple. Given the Cantabrigians' attempt to construct a theory of money where, being a store of value as well as overcoming transactions costs, money yields utility through its services, then the individual's demand for money should be considered in the same way as any other utility-yielding good. In other words, through marginal analysis of individual demand. This suggestion was jumped on by Hicks (1935) who articulated a careful suggestion for a "marginal utility theory of money", a choice-theoretic theory of money. This was followed again by Hicks in his Value and Capital (1939) but subsequently sidelined in the wake of Keynes's General Theory (1936) only to be resurrected later by Tobin and others as the Neo-Keynesian revolution began to run out of steam.

Hicks's (1935) "Suggestion for Simplifying the Theory of Money" was that the choice of money holdings was merely a part of a more generalized theory of choice. Now, choice theory is perfectly capable of determining the intertemporal allocation decision which will establish the amount of income that is consumed today and saved for tomorrow. However, of savings, which are merely additions to the stock of wealth of the consumer, how much is precisely to be allocated to money, bonds, capital and other stores of wealth is an issue which can also be easily solved by assuming the same marginal benefit-marginal cost framework of marginalist theory. (Why hold money? Hicks (1935) overcomes his previously mentioned 1933 position by recognizing the existence of frictions and uncertainty).

We can follow Hicks (1935), Metzler (1951) and Tobin (1961) and rectify our position by recognizing that we should expand the utility function to include other assets, such as bonds (B) and equity (K):

U = U(x, M/P, B/P, K)

Of this, we obtain a general intertemporal allocation, including a decision of how much wealth to hold. Thus, let us define wealth, W, as being a portfolio to be allocated among the three stores of value:

W = M/P + B/P + K

The decision then boils down to finding, at given rates of return, transactions costs and given liquidity premia, the portion of the portfolio which is held in the form of real money balances, bonds and capital. Thus, since wealth demand must be equal to the wealth stock, then we can rewrite this as:

(Md/P - Ms/P) + (Bd/P - Bs/P) + (Kd - Ks) = 0

which show the excess demand for money, bonds and capital in the form a Walras's Law stock constraint. Disequilibria between any set of asset markets is cured by interest rate movements. Thus, this approach implies that the optimal level of holdings of each type of assets will be found where:

MB(M) = MB(B) = MB(K)

the marginal benefits MB(.) of each asset are equated (and, for the liability side, the marginal costs are equated). In this case, there will be no incentive to move from any one type of asset to another, thus the excess demand in each of the three asset markets will be zero. Details were filled in for more complicated scenarios on numerous occasions by James Tobin (e.g. Tobin (1969) and Brainard and Tobin (1968)). To a large extent, this approach resembles Keynes's theory of liquidity preference, where there is a portfolio choice between money and alternative assets (but the rate of return on money "rules the roost").

The only difficulty with the Hicks-Tobin approach is that there is still an implicit division between the consumption-saving decision and the portfolio allocation decision. Milton Friedman (1956) was among the first to make a direct objection and attempt to include commodity durables also as a "store of wealth". However, Friedman's "Monetarist transmission mechanism" also involved somewhat a confusion between stock and flow. The correction was (belatedly but remarkably) recognized by James Tobin (1982) who reworked his general, multi-asset approach to allow for the savings-consumption decision to play a significant role. This truly makes it a more generalized choice-theoretic approach since it allows for a single set of simultaneous decisions: no longer is there a two-step decision, but asset market disequilibrium can affect the consumption-saving decisions and consumer market choices can affect the asset market equilibrium. However, while it does make vast strides towards Walras's idea of an integrated theory of money and value, the quantity theory results are no longer obtained.

The Quantity Theory was (partly) reinstated by the resurrection of the Walrasian theory of money by Patinkin (1956). Most of the inconsistencies between the two approaches were attempted a reconciliation and the internal contradictions of the Walrasian system was smoothed out. However, the theory is not really very faithful to Walras's original system of encaisse désirée, thus it may be useful to regard it as an entirely independent attempt to integrate the quantity theory of money and the Walrasian equations of general equilibrium. Thus, we give it a section of its own.



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