A Formal Proof of Walras Law

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A Formal Proof of Walras Law

Set out below is a formal demonstration of Walras Law using summationnotation.

(If you are a little rusty on summation notation and/or summation calculus youmay like to look at one or both of these sites .)

We begin with a proposition known as Walras Identity.

Walras Identity

Imagine an economy in which there are n commodities (the number ofcommodities in most macro models is equal to five: goods and services, labour,

bonds, money and foreign exchange).

Now, whether or not prevailing market prices are such as to equate demand withsupply for each commodity, the money value of all commodities which anindividual transactor (i.e. a household, a firm, or the government) plans to buy inany period must be equal to the money value of all commodities offered for sale

by that transactor at the same time. For example, if an individual plans to purchase 200 dollars worth of commodities (let us say, 50 commodities at 2dollars per commodity and 2 bonds at 50 dollars per bond), then, simultaneously,he or she must also plan to sell commodities to the value of 200 dollars (forexample, 50 hours of labour at 4 dollars per hour). (Tacitly, our model assumesthat transactors are not thieves, extortioners, embezzlers - or philanthropists!)

Moving from the particular to the general, the total money value of what the j'thindividual transactor plans to purchase can be written symbolically as:

where P 1, P 2. . . P n are the prices of the n commodities, and D1j , D 2j . . . D nj arethe quantities of those commodities that the j'th individual plans to purchase.

Similarly, the total money value of what the j'th individual plans to sell can bewritten symbolically as
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where S1j , S2j , . . . S nj are the quantities of the n commodities that the j'thindividual plans to sell.

Since the money value of all the commodities that the j'th individual plans to buymust always be equal to the money value of all the commodities that individual

plans to sell, we may write for a single intransactor:

(1)

This condition is written as an identity (an identity is a proposition which isalways true, usually because of the way we have defined things) since we haveassumed that no individual transactor in our model will be so misguided as tosuppose that he or she can acquire something for nothing. This being so, (1) is, ineffect, a statement of the budgetary constraint under which individuals formulate

their purchase and sales plans.

Granted that each individual's planned market transactions satisfy condition (1),it follows as a matter of simple arithmetic that the aggregate money value of thequantities demanded by all individuals must be equal to the aggregate moneyvalue of the quantities offered for sale by all individuals. We can see this bysumming condition (1) over all (m) individual transactors to obtain:

Factoring out the price variables from each side of this expression yields:

However, the expression in parentheses on the left-hand side is simply the totalmarket demand for the i'th commodity, since it is the sum of the individualtransactors demands for that commodity. We will write this total market demandfor the i'th commodity as D i. Similarly, the expression in parentheses on theright-hand side is simply the total market supply of the i'th commodity, since it is

the sum of the individual transactors' supplies of that commodity. We will writethe total market supply of the i'th commodity as S i.

Thus, we arrive at the conclusion that:

(2)


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D n = S n which implies immediately that the n'th market is also in equilibrium.

To recapitulate verbally, we have shown that if all but one of the markets in aneconomy are in equilibrium, then that other market also must be in equilibrium.In the next section we shall see what Walras Identity implies when at least onemarket is in disequilibrium.

Second Implication: Walras Law

We now look at the implications of Walras Identity for dis-equilibrium.

Assume that one market (the n'th market) is in dis-equilibrium. This may take the

form of either (positive) excess demand (where P iD i > P iSi) or excess supply,also known as negative excess demand (where P iD i < P iS i).

It is an implication of Walras Identity that for all markets taken as a whole therecan be neither excess supply nor excess demand when we sum over all markets.We can see this by rearranging (2) to give:

(3)

In order for this condition to be satisfied in the presence of disequilibrium in then'th market, it must be the case that there is on 'off-setting' dis-equilibrium in atleast one other market. This result is known as Walras Law . Walras Law statesthat the sum of excess demands over all the markets in the economy must equalzero and this applies whether or not all markets are in (general) equilibrium. So ifthere is excess supply in one market (that being negative excess demand) thenthere must corresponding to this be positive excess demand in at least one othermarket. (But it is important to notice that the excess demands and supplies aremeasured as differences between planned (or notional) demands and supplies andnot necessarily actual demands and supplies).


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An aside: What can we say about prices when markets are indisequilibrium?

We have been assuming that all transactors in any market face the same price. (Ifyou look at the equations set out above, none of the 'price' variables have a jsubscript.) In his (justly) famous 1937 article titled Economics and Knowledge first

published in the journal Economica in 1932, Friedrich Hayek pointed out that thenotion that all transactors in a market face a single (ie the same) price is adefining characteristic of equilibrium, but is not, in the nature of things, a featureof disequilibrium. Fortunately, since we are interested in aggregate expendituresand not in modelling individual prices, neither Walras Identity nor Walras Lawwould be affected if we dropped this assumption. (To see this you could workthrough the proofs given above for the case where P also has a 'j' subscript exceptwhere we are explicitly modelling equilibrium.)

References

R Crouch (1972), Macroeconomics, New York, Harcourt Brace Jovanovitch,Section 6.2. (The proof set out above is based on Robert Crouch's book.)

D Patinkin, 'Walras Law' in The New Palgrave, A Dictionary of Economics,Macmillan, London, 1987, Volume 4, pp 864 - 8.

T Sargent, Macroeconomic Theory: Academic Press, Boston, 1987, passim.

D Gale, Money in Disequilibrium: Cambridge University Press, Cambridge,1983.

F Hayek, "Economics and Knowledge," Economica, Volume 4, 1937, pp 33 - 54.Reprinted in F Hayek, Individualism and Economic Order, Chicago: Universityof Chicago Press. 1948.

Department of Economics, University of Melbourne

Created:
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