History of Economic Analysis part 101 potx

THE WALRASIAN THEORY OF GENERAL EQUILIBRIUM * In this section we shall analyze the logical structure of Walras’ system of the conditions or relations equations that are to determine the

this universal interdependence was demonstrated by numerous investigations into the theory of joint and composite demand and supply and of the values of related commodities in general which produced some of the most illuminating passages of Book

V of the Principles and which were further developed by Edgeworth In fact, it stands to

reason that the comprehensive but gaunt and colorless idea of the universal interdependence that subsists between all elements of the economic system—and so easily provokes sneers about everything’s depending upon everything else—can be brought home and made alive to the many by means of concrete cases about the relations between the values of beef and mutton or again of tea and sugar—the relations between the values of ‘competing’ or ‘completing’ commodities (Fisher) And this can be done without violating the restrictions inherent in the methods of partial analysis We do indeed in such cases, sometimes with a slight disregard of strict logic, go beyond direct effects and take into account also indirect ones; but still we do so only within small sectors that do not produce significant effects upon the whole economy, at least not effects significant enough to affect the quantities, such as national income, that determine the setting of the small sector In such cases the relations, in the small sector which can

be managed by partial analysis, illustrate or exemplify to a limited extent the relations in

the whole of the economic cosmos.12

But they do so only up to a point Beyond this point the methods and results of partial analysis remain inadequate and may become even misleading Marshall realized this It is very instructive to observe how carefully he watched his step wherever his argument rose into the realm of the ‘general’ theory of distribution.13 However, it is obvious from his appendix (note XXI) that, had he wished to go further, he would have sought the necessary comple-

work of his group in its true light, namely in the light of a special proposition within the theory of marginal utility: however emphasized, it remained, as later on with Cassel, a supplementary

principle that was added to, instead of being derived from, the fundamental theory of values and costs

12

Of Edgeworth’s many contributions in this line—contributions that treat partial analysis cases but illustrate general relations by doing so—I shall mention only one: his famous taxation paradox

which the reader had better study in Professor H.Hotelling’s version, ‘Edgeworth’s Taxation

Paradox and the Nature of Demand and Supply Functions,’ published in the Journal of Political Economy, October 1932 The extent to which partial and general analysis may co-operate in the

sense explained in the text, shows up well in Marco Fanno, ‘Contributo alla teoria dell’ offerta a

costi congiunti,’ Giornale degli Economisti, October 1914, and in a later work of the same author which belongs here, ‘Contributo alla teoria economica dei beni succedanei,’ Annali di Economia,

1926

13

See in particular Principles, pp 587 et seq and the manner in which he arrived at the ‘general

theorems’ on pp 609 and 611 As regards the former passage it is noteworthy that he did not postulate the existence of social production functions (i.e production functions that are to apply to the economy as a whole) but, after having carried out his analysis within the individual industry or even firm, confined himself to stating that ‘the substance of the problem is the same in every industry’ (p 588) We shall return to this in sec 8

ments of partial analysis in the methods of general microanalysis of the Walrasian type rather than in a separate body of aggregate analysis (macroanalysis)

We shall see (Part V, ch 5) that it is the latter solution which appeals to many economists of our own day, especially to members of the Keynesian group These divide

up economic theory into a theory of the individual firm and a macroeconomic theory that

is to take care of the relations between aggregate consumption, investment, employment, and so on It is therefore worth our while to point out, first, the historical connection that exists in this respect between Marshall and his apparently so rebellious followers of the 1930’s and, second, the extent to which this combination of the theory of the individual firm and macroanalysis has been anticipated during the period under survey

As regards the first point, the fact that Marshall made the small industry his cheval de bataille in Book V of his Principles should not be allowed to obliterate the not less

important fact that much of his analysis of industries was really carried out in terms of the economy of the individual firm,14 and that even beyond what this implies Marshall assembled practically all the bricks and all the mortar required for the theory of the individual firm, including even a fairly complete assortment of all those circumstances that prevent the sweeping generalizations of pure theory from working out in real life and have been repeatedly adduced as objections against his own generalizations (see in

particular Principles, Book VI, ch 8 and the notion of normal profit there developed,

especially pp 696 and 700) Therefore, so soon as the concept of the industry gave way under modern criticism, the one of those divisions of economic theory lay ready at hand while the desirability of the other imposed itself much more obviously on his pupils than

it would have imposed itself upon pupils of Walras

As regards the second point, it cannot be too often repeated that Marshall himself gave

no lead toward macroanalysis But macroanalysis itself and its combination with

microanalytic explanations of individual behavior were old Quesnay’s tableau is a

macroanalytic description of a stationary circuit flow of economic life, and Quesnay supplemented it, as we have seen, by a microanalytic theory of exchange In the following period, Ricardo did much the same thing: his distributive shares are aggregates but the reason why they behave as they are supposed to do is derived from a fragmentary micro-

14

The bridge between the theory of the small industry and the theory of the individual firm was his Representative Firm (later on reformulated, by Pigou, into the Equilibrium Firm) This curious construct embodies a most interesting attempt to resolve or to circumvent the difficulties that arise when we try to describe industrial processes by means of concepts developed from the life of individual firms It is neither an average nor a marginal nor a leading firm but one in the position and structure of which the conditions of the industry are at any given time reflected in such a manner that certain propositions hold with respect to it that do not hold with respect to any actually existing firm or with respect to any industry as a whole Marshall’s authority as a teacher secured mechanical acceptance of the concept But it received neither the criticism nor the development it deserves

analysis During the period under survey Böhm-Bawerk also did much the same thing: he started with a theory of individual behavior and with a theory of exchange that is based upon it; but, on the highest floor of his building there is almost nothing left but aggregates such as (value of) the sum total of wage goods, (value of) total output, and an aggregative ‘period of production’ to boot Similarly, Wicksell reasoned on a social production function without displaying any symptoms of critical discomfort And there is hardly any need for adding that this Quesnay-Ricardo-Böhm-Wicksell method is also that

of Lord Keynes

7 THE WALRASIAN THEORY OF GENERAL EQUILIBRIUM *

In this section we shall analyze the logical structure of Walras’ system of the conditions

or relations (equations) that are to determine the equilibrium values of all the economic variables, to wit: the prices of all products and

*

[This section on the Walrasian Theory of General Equilibrium was written in the last year

(possibly in the last few months) of the author’s life The material in subsections (a), (b), and (c) was found in typescript (unread by J.A.S.), whereas subsections (d) and (e) were in manuscript J.A.S probably intended only 4 subsections; his subsection (b) included what is now (c) The pages were not numbered and there were no titles for the subsections but the intended order seems

perfectly clear and agrees with the order in Walras’ table of contents (Éléments, pp 489–91) There

was no opportunity for the inevitable minor revisions and corrections usual in any work of this kind, but the final writing shows every sign that the author knew what he wanted to say He had started and abandoned many other attempts before the final one

He had not made up his mind, however, concerning the title and the brief introductory paragraphs There were at least three unfinished introductions, one of which appears below in this note and a second as the first two paragraphs of the text There were also three different titles suggested: the one actually used above for this section, the one given below, and a third, ‘General Analysis: The Walrasian System.’

The following unfinished introduction may have been the last one:

‘7 Walrasian Microanalysis In this section I shall sketch the main features of the Walrasian

system, reformulating certain points for convenience of exposition and leaving a number of others for closer consideration in section 8 and in the appendix to this chapter This system that Walras

embodied in a system of equations will be discussed verbally Barring a brief remark, we shall

assume pure competition all round

‘We consider a closed domain that does not act upon, or experience any influence from, the outside world In this domain there are households which sell productive services (we neglect for the sake

of brevity services that are consumed directly, such as personal services, except those that are consumed by their owners in the form of leisure or pleasure grounds) and purchase products; and firms that purchase productive services and sell products But whereas the households sell their services only to firms, the firms sell products not only to households but some firms also produce certain products (raw materials and equipment) for sale to other firms In order to make the

essential problems stand out clearly, we shall at first disregard these intermediate products and reason as if firms did nothing but combine labor and services of natural agents into products for sale to households and then introduce…’]

factors and the quantities of these products and factors that would be bought, in perfect equilibrium and pure competition, by all the households and firms Let us notice at once that, since the determination of these quantities implies the determination of individual as well as group and social incomes, this theory also includes all that is covered by the concept of Income Analysis and that the conditions or relations to be considered, though they are fundamentally microanalytic in nature (they refer fundamentally to the quantities bought and sold by individual households and firms), also include macroanalytic aspects, for example, as regards total employment in the society It cannot be too strongly impressed upon the reader that it is not correct to contrast income or macroanalysis of, say, the Keynesian type with the Walrasian microanalysis as if the latter were a theory that neglects, and stands in need of being supplemented by, income and macroanalysis Attention should also be drawn at once to three other points First, I have spoken above of prices of products and factors But Walras’ theory of pricing, primarily and on

the ground floor, refers to prices of services of products and factors This amounts to the

same thing only as regards products and factors that do not serve more than once For all the others, the problem of pricing the products and factors themselves is a distinct problem that is solved on a second floor, as we shall see It would be unnecessarily pedantic, however, to insist on this where no misunderstanding is to be feared Second, I have spoken of prices ‘that would be paid in perfect equilibrium and pure competition.’ This manner of speaking is not Walrasian: Walras, much like J.B.Clark, conceived these equilibrium prices to be, normally, the actual level around which prices oscillate in real life,1 which involves a claim which I do not wish to make Third, Walras grouped his productive services into services of land, labor, and ‘capital proper,’ 2 but this does not spell acceptance of the old triad of factors: actually Walras admitted an indefinite number

of means of production and services [This piece of manuscript breaks off here.]

(a) Walras’ Conceptualization

The description of the economic pattern

1

Like Clark, he used the analogy with the ‘level’ of a lake in order to convey his idea—the old idea

of A.Smith

2

As we know, Walras defined capitaux, in a wider sense, as all ‘goods’ that serve more often than once and, in a narrower sense, as durable goods that are themselves produced (capitaux proprement dits) Their services he called revenus, no matter whether they are consumed by the owner (e.g as leisure in the case of ‘personal capital’: this leisure is still travail) or used productively This

conceptual arrangement which Walras derived from his father, Antoine Auguste Walras (1801–66;

Théorie de la richesse sociale, 1849) and which was (substantially) adopted by Irving Fisher, has

its logical advantages but is for us important only in so far as it must be borne in mind if Walras’

reasoning is to be correctly understood (leçon 17) For the same reason, I repeat that the capitals, in

addition to rendering services that are directly consumed or are transformed into products, may also

render a service d’approvisionnement that may in turn be consumable or productive Leçons 18 and

19 describe minutely Walras’ set-up of the process of production and the accounting system of his firms—matters that have not attracted the attention they deserve

that Walras’ equations were to express is contained in Éléments,3 leçons 17–19 The functioning of this pattern is further illustrated by the tableau économique presented in leçon 35, where he also indicated his opinions concerning the oscillations that occur

around the equilibrium state.4 We are introduced to his entrepreneur and, by means of a most useful analysis of a simplified accounting system, to the structure of a typical firm’s operations This analysis dovetails with a list of assets 5 that determines much, if not everything, in Walras’ theoretical organon For our present purpose it will be useful to note, or to note again, some of the salient features of this list of assets As we know, the Walrasian entrepreneur is the agent (a physical person or a corporation) 6 that buys raw materials from other

3

[Éléments d’économie politique pure ou théorie de la richesse sociale (1st ed 1874–7; 4th ed 1900; the édition definitive, 1926, is the one quoted throughout unless a different one is mentioned

specifically).]

4

These views do not differ essentially from those of A.Smith Walras’ analogy of market

equilibrium with a lac agité par le vent, which is so characteristic for his belief in the reality—

normality even—of the equilibrium level of values, has been repeated by J.B.Clark It should be emphasized once more that this uncritical belief, undoubtedly held widely at that time, is untenable; but that this fact does not render analysis of the properties of those equilibrium levels either superfluous or practically useless (see above, sec 3 on the ‘dreamland of equilibrium’) It should

also be emphasized that Walras (see, e.g., Éléments, p 370), though he did underestimate the

distance between his theory and the facts of capitalist reality, was by no means unaware of its existence And no indictment at all can be leveled against Pareto on this score

5

Every object that enters the range of economic consideration, even ‘labor power’ (Marx) or the

capitaux personnels (Walras), may be treated as an asset, if we halt the economic process for a

moment and list every element of it Presently Walras sets the process into motion when the difference between funds and flows asserts itself, and we are told how the assets are kept

reproducing themselves There are thirteen kinds of assets: the ‘capitals’ (all things that serve more often than once, including land, labor power, and produced capitals) that produce ‘services’ for direct consumption (including leisure); the ‘capitals’ (land, labor power, and produced capitals consisting of plant and equipment) that produce productive services; and in addition to these six

items, the produced capitals (plant and equipment) that are ready for sale in the hands of their producers and yield no services as yet (the capitaux neufs); the stocks of consumers’ goods that

serve once only in the hands of the consumers; the raw materials and semi-finished goods in the hands of the producers, who are going to use them; the stocks of (transient) consumers’ goods and raw materials in the hands of the producers who hold them for sale; and finally three types of money stocks, namely money held by consumers to finance consumers’ transactions, money held

by producers to finance producers’ transactions, and monnaie d’épargne The Keynesian

controversy makes the translation of the last item a delicate task I think that ‘money earmarked for the purpose of investment’ comes nearest to rendering Walras’ meaning

6

Although Walras blamed English economists for confusing the entrepreneurial function with that

of the capitalist and French economists for confusing it with that of labor (entrepreneurship being a kind of labor), his theory of entrepreneurship does not go much further or deeper than J.S.Mill’s or J.B.Say’s All he did was to

entrepreneurs, hires land from landowners, personal aptitudes (facultés personnelles)

from workmen, capital goods from capitalists, and sells the products that result from the co-operation or combination of their services for his account.7 Into this and the meaning

of the concept of entrepreneurs who, as such, neither make nor lose, we need not go again Important is it, however, to notice three other things

First, Walras was careful—much more so than other writers—to construct theoretically, and to identify practically, the various ‘markets’ through which his economic mechanism works and the interaction of which constitutes his analytic organon Simplifying and combining as much as we can, we have the two fundamental markets, those of the products and of the productive services, and in addition the market that determines the prices of the capitals, hence also the rate of new revenue and the market of means of payment The reader may be somewhat surprised at my emphasizing this apparently trivial matter But the strict association of every part of the argument with an identifiable market, even on the highest level of abstraction, is an essential feature of Walras’ procedure that starts in each of these four cases with a theoretical solu-

isolate the ‘combining function’ more clearly As is shown by the fact that he admitted corporations into the circle of entrepreneurs, his conception was one belonging to the range of ordinary business routine and is roughly equivalent to Marshall’s fourth productive agent, organization

7 As a result of Walras’ strict distinction between capitals (capitaux), i.e goods (including labor power) that serve more often than once, and services (or revenus)—a distinction which lapses in the

case of goods that serve but once—the Walrasian theory of pricing runs on two levels: immediately (on the first level) we have to do with the pricing of services only (which includes the pricing of transient goods) On another level we then meet the problem of the pricing of these capitals

themselves (from which the pricing of the labor power, unless enslaved, is of course in practice excluded) All incomes uniformly result from the sale of services, a conceptual arrangement that creates no difficulty in the case of ‘land’ (permanent factors) and labor but, in the manner to be explained, begs the question of the existence of a net income in the case of produced and durable goods that wear out in time For the moment we note again that Walras really admitted an indefinite number of productive services, although he gave in to tradition by grouping them into services of

various kinds of land, labor power, and produced capitals and thus seems to accept the old triad of

factors We must also note that, foreshadowing later developments, he stated at once (p 197) that only land and labor power (plus plant and very few items of equipment) are hired in kind Most of the durable instruments of production are hired by entrepreneurs not in kind but in money, which is what capitalists save and lend, although at first, before having introduced money into the

productive process, Walras allows capital goods to be let in kind This seems to involve a stricter parallelism between lending money and lending capital goods than, as we shall see Walras was prepared to admit: actually his capitalists have money and not goods to lend to entrepreneurs; and it

is only for perfect equilibrium in pure competition that the process is supposed to go on as if

capitalists were owners of produced durable goods This subtle point must be kept in mind— mathematically, it makes all the difference between an identity and an equilibrium condition— particularly if we are to see the affinity between the Walrasian and the Keynesian system

tion of an equilibrium problem and then investigates the manner in which this theoretical solution works out ‘practically’ in the corresponding market.8

Second, we observe when going over Walras’ list of assets that very considerable emphasis is placed upon stocks or inventories: there are inventories of new capital goods, consumers’ goods’ inventories held by households and by firms, raw-material inventories held by both their producers and their users, and also, as we have seen, stocks of money (cash holdings) of various types Since the existence of these inventories presupposes a certain past behavior of the people concerned and since their current reproduction presupposes certain expectations, the system—even if perfectly stationary—still depicts a process in time and might therefore be called ‘implicitly dynamic.’ If Walras did not feel like this and if we agree with him in calling it static, this is only because of a device that was perhaps justified by the purpose of exhibiting the logical skeleton of economic life

but is highly artificial all the same: he tried to build up an equilibrium state ab ovo in the

manner in which it would be built, if smooth and instantaneous adaptation of all existing goods and processes, to the conditions obtaining at the moment, were feasible His households do not purchase consumers’ goods or sell productive services outright Nor

do his firms (entrepreneurs) purchase productive services and offer products outright

They all merely declare what they would respectively buy and sell (produce) at prices criés au hasard, that is, announced experimentally by some agent in the market, and are

free to change their minds if these prices do not turn out to be the equilibrium prices: other prices are thereupon announced, other declarations of willingness to buy or sell

(and to produce) are written down on bons—pieces of paper that do not carry any

obligation—until equilibrium values emerge, namely prices such that no demand willing

to pay them and no supply willing to accept them remain unsatisfied And the only mechanism of reaction to these variations of experimental prices that Walras recognizes

is to raise the prices of commodities or services, the demand for which at these prices is greater than the supply, and to reduce the prices of commodities or services, the supply of which at these prices is greater than the demand.9 I shall not stay in order to proffer the obvious arguments that may be adduced in mitigation of such heroic theorizing

8

Each of four problems—pricing of products, pricing of productive services, pricing of capital goods, and ‘pricing of money’—is thus solved twice: in each case we have first a proof of the existence of an equilibrium solution and second the proof that this solution is the one which the market mechanism under pure competition tends to establish or, slightly more technically, we have

in each of the four cases two distinct proofs (or attempts at proofs), the one of the existence of an

equilibrium solution, the other of the tendency toward it Since the latter proof involves the

statement that, if the equilibrium solution be once hit upon, it would not be departed from without the intervention of an additional force, we equate the proof of an equilibrium tendency to a proof of

the stability of the equilibrium solution

9

Edgeworth’s method of arriving at equilibrium prices and quantities by means of ‘recontracting’ comes of course to the same thing

[Apparently there is no discussion of the last of the ‘three other things’ that it is important

to notice.]

[(b) The Theory of Exchange.]

Since the equilibria in the two basic markets, the consumers’ goods and the service markets, and the way in which they interlock—simultaneously determining one another—are of decisive importance for the strength of the Walrasian structure we shall now consider these two basic markets separately For this purpose we neglect both saving

and the production of capitaux neufs,10 a procedure which involves the assumption that the produced capitals are just as permanent and indestructible as is ‘land.’ Further, in

order to emphasize the steps in the procedure, we shall indeed introduce a numéraire—

the standard commodity in terms of which all exchange relations are to be expressed— but no money that actually circulates or is being held.11 Several questions that cannot be answered without impeding the progress of our argument will be reserved for section 8

We know already that Walras based his structure on an elaborate theory of exchange which fills two distinct roles: first it was to describe the fundamental features of economic logic which, with Walras, amounts to the same thing as the fundamental mechanism of competitive markets in general; second, it was to yield the behavior equations (maximizing equations) of the households As regards the first role, Walras’ theory of economic logic issues into a marginal utility explanation of economic value that will be discussed, in its historical setting, in the appendix to this chapter Here we are not interested in such questions as whether there is any sense in speaking of marginal utility

as the ‘cause’ of value, but immediately proceed to a discussion of the second aspect of the Walrasian theory of exchange We can do this because, as has been pointed out by Pareto,12 the concepts of marginal and total utility are redundant so long as we merely wish to formulate equilibrium conditions On other features of this theory of exchange a few comments are nevertheless desirable

Making ample use of the concepts that we have just voted superfluous, Walras first developed brilliantly the theory of (competitive) exchange of two

10

For brevity, we also postulate that firms do not purchase raw materials from one another: they all

of them simply combine ‘services’ into products for sale to households Unfortunately, we cannot similarly throw out the services that are directly consumed by their owners

11

This simplifying measure must not, however, be interpreted, either with reference to Walras himself or with reference to our own presentation, as the theory that money does not enter into the fundamental process of determining values and is merely a technical device or ‘veil.’ All that we mean is that we shall posit this question separately, meanwhile reserving the right to scrap or modify the results at which we are aiming just now, if it should turn out that the intervention of money requires us to do so

12

Implicitly this was already seen by Antonelli, Boninsegni, and others For Pareto’s statement see

e.g., Manuel p 542 Equation 9 on that page does not only without marginal utility but also without any ‘index function’: the first 76 paragraphs of the appendix to the Manuel render the Paretian version of Walras’ leçons 5–16

commodities The point to notice is that he fully recognized the possibilities that there may be no solution to the problem or else multiple equilibria which in his set-up reduce

to three, two stable and one unstable, whereas in general no such situation will occur and unique equilibrium prices will practically always emerge if there are many commodities

in the market

[This piece of manuscript ends here but the next one seems to follow with no serious break in the argument.]

[(c) Determinateness and Stability of Simple Exchange.]

Since the theory of exchange, besides providing the theoretical description of the behavior of consumers (households), also serves to display the fundamental properties of economic action in general (the logic of choice), there is point in raising right here the questions of determinateness and of stability of simple exchange in a perfectly competitive market, indirect exchange (arbitrage) being duly taken into account and a

standard commodity (numéraire) but no money being used.13 We raise these questions in the same sense as did Walras, except for one point that will appear presently

People—say n of them—endowed with definite tastes and possessing, to begin with, arbitrary quantities of an arbitrary number of well-defined commodities, say m in all,

appear on the market, in order to take advantage of the possibilities this market may offer

to them of improving upon the satisfaction of their wants as guaranteed by their original possessions.14 We thus accept Walras’ manner of speaking of a tendency on the part of all participants to maximize their satisfaction.15 We also accept the usual assumptions about 13

The arbitrage operations are supposed to be carried out in terms of the numéraire It should be

repeated, however, that they are supposed to be organized in such a manner as not to deflect any

quantity of the standard commodity from its uses as a commodity If people hold any part of the numéraire commodity this would turn it into money

14

Those original possessions of every participant in the market are data that are subject to certain conditions such as that the quantities originally possessed should all be non-negative, that at least one of them should be greater than zero, and that the original distribution should not violate the

hypothesis of pure competition For the rest, in leçon 14, Walras establishes that, in full

equilibrium, prices would not change if the commodities were redistributed between participants so

long as the sum of the possessions of each participant remains equivalent in terms of numéraire (théorème des répartitions équivalentes) I mention this theorem, which space does not permit us to

discuss, to give an example of Walras’ awareness of the necessity of establishing every point in his schema by formal proof It is this awareness (whatever the success or shortcomings of his proofs) that made him the teacher of all theorists of the future

15

As already stated, this is not necessary But it was the almost universal practice of Walras’ generation, not only of the mathematical economists such as Edgeworth and Marshall but also of the Austrians, most explicitly so of Böhm-Bawerk The questions that are now before us are not affected by our lapse into primitive utility theory We do not imply measurability, and maximizing

an index of satisfaction would do just as well

continuity and differentiability, at least of the resulting market ‘curves.’ Finally we assume for the moment, as did Walras, that the marginal utility functions of every participant, for every commodity, not only exist but are functions of the quantity of this commodity alone, that is, independent of whatever other commodities he might possess

They are all monotonically decreasing We then have: n(m−1) Behavior Equations expressing for all n participants the quantities (including zero quantities) they will give

away or acquire at any given system of exchange relations (or prices in terms of the

numéraire) by virtue of the condition that they will go on exchanging until no further

exchange can increase their individual satisfactions; 16 n equations such that all the

quantities the participants acquire and give away, each quantity multiplied by its price in the standard commodity, must add up to zero, if we give minus signs to quantities given

away and plus signs to quantities acquired (Individual Balance Equations); finally m

equations such that, for every commodity, the total amount of quantity given away must equal the total amount of quantity acquired for the market as a whole (Market Balance Equations).17 These are m(n+1) conditions or equations But, as is easily seen, one of

them, for example the last one of the set of market balance equations, may be shown to follow from the rest of these and from the household balance equations and must

therefore be thrown out as not independent Thus we are left with m(n+1)−1 independent ones by which to determine the variables or ‘unknowns,’ namely, the m equilibrium prices and the mn quantities exchanged by the households Now, we may say either that, since the price of the numéraire commodity in itself is of necessity always equal to unity, there are only m−1 prices to determine; or that, since the two first sets of equations (the

behavior and the household balance equations), considered by themselves, are homogeneous of zero degree in the prices, it is only the exchange ratios and not the absolute prices which we can determine, though we can then translate these ratios into

absolute prices by means of the numéraire-price identity.18 The reader should make sure that he understands the perfect equivalence of these two ways of putting the matter and also the

16

This means, as we know, that they will continue to exchange until the marginal utilities to them

of the quantities of all commodities that can be had for a unit of numéraire (if the numéraire is

cigarettes and the unit a package, then the marginal utilities of a package’s worth of every

commodity) are equal

17

The prices for which the last group of equations is verified are the market equilibrium prices In

the terminology foreshadowed by Walras but definitively established by Professor Hicks (Value and Capital, p 63) we can express this last group of equations also by saying that, for every

commodity, excess demand must be zero

18

A function x1=f(x2, x3…x r) is called homogeneous of zero degree if, λ being any positive arbitrary constant, the dependent variable remains the same when the independent ones are all multiplied by

λ, so that x1=f(λx2, λx3…λx r ) Putting now λ equal to, say 1/x2, we get x1=f(1, x3/x2…x r /x2), that is to

say, a relation in which the former independent variables of which there are r, are replaced by ratios

of which there are only r−1