Leon Walras and Fisher

7/28/2019 Leon Walras and Fisher

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American Academy of Political and Social Science

Geometrical Theory of the Determination of PricesAuthor(s): Irving Fisher and Lon WalrasSource: Annals of the American Academy of Political and Social Science, Vol. 3 (Jul., 1892),pp. 45-64Published by: Sage Publications, Inc. in association with the American Academy of Political and SocialScience

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GEOMETRICAL THEORYOF THEDETERMINATION OF PRICES.*

Translator's Note.The following article of Professor Walras is the first by that famouswriter to appear in English. It is not too much to say that ProfessorWalras, together with Jevons and Menger, marked an epoch in poli-tical economy. These three writers independently and almost simul-

taneously developed and applied the idea of rarete or marginal utility,the corner-stone of the so-called " Austrian School " and of mathe-matical economics. Those who are not already familiar with theservices which the Economie Politzque pure has rendered will beinterested to read the praise of Jevons (Theory of Pol. Econ. 3d Ed.Preface to 2d Ed. p. xxxviii.).The problem which Professor Walras has here set himself is a diffi-cult one, viz.. to present a geometric picture of the causation of theprices of all commodities, recognizing the fact that these prices aremutually dependent. I have never seen it attempted elsewhere.In his Economie Politique pure he treated this problem analyticallyand was, I feel confident, the first to do so. Auspitz and Lieben alsoin their much praised " Unlersuchungen ilbecr die Theorie desPreises " use their ingenious diagrams only for a single commod-ity and content themselves with an analytical treatment as soon asthey pass to several commodities. The present writer is about topublish in the transactions of the Connecticut Academy of Arts andSciences a Mathematical Theory of Value and Prices, in which is asolution of this same problem, by the aid of an instrument quite differ-ent from either the analytic or graphical method. An actual mechan-ism is constructed so as to exhibit the complex market adjustment asan automatic equilibrium of a system of liquids.

* Of the three parts of which this paper is composed, the first consists of amemoir read before the Society of Civil Engineers of Paris, October 17,189o,printedin the Bulletin of that Society, January, 1891. In it the author has, however, madecertain modifications, of which one is rather important, because it simplifies thefundamental demonstration of the theorem of maximum satisfaction. The lasttwo consist, with certain modifications necessitated by what preceded, of a paperprepared for the Recuezl inaugural of the University of Lausanne. The editorsof the Recuezl kindly authorized the transmission of the MS. of this article,previous to its publication in Lausanne, to the American Academy of Politicaland Social Science, with a view to insertion in its ANNALS.(45)

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46 ANNALS OF THE AMERICAN ACADEMY.A few words of comment on the present article may be appropriate :(T) Although Professor Walras clearly recognizes that the demdndand supply of each commodity is a function of the prices of all com-

modities, he omits to state that the rareth to a given individual of agiven amount of one commodity is a function of the quantities notonly of that commodity, but of all others. Hence the curves he em-ploys are not independent, but the shape of the A curve in Fig. Iwill change according to what point is selected on the B curve. TheA curve could not be said to be given until the demand for B, C, D,etc., was each given. The utility of bread, it is true, decreases withthe amount of bread, but the law of that decrease depends on theamount of butter; in general the utility of the same quantity of breadincreases as the amount of butter increases.

(2) With a slight change of phraseology the methods and curvesused by Professor Walras would apply to the rate of consumption intime. Thus the quantities Oq,, Oqb, Oq,, etc. (Fig. I), could be taken tomean the amounts of each commodity annually produced by thegiven individual and disposable for consumption and exchange. Thevertical length, used in Fig. 2, qa+ qb pb + qc pc+ etc., would thenindicate his annual income measured in terms of the commodity A.I am aware that many economists object to such an introduction ofthe time element, but their objections disappear as soon as the notionof a statical market is admitted. To limit price analysis to a singleinstant and the supply to the amounts of stock then on the market,unnecessarily restricts its range of application, and " dealing infuitures " makes its meaning vague. What can be meant by theamount of petroleum on the New York market at an instant ? Manygallons are fiowing through pipes and in twenty-four hours morethan a million gallons will be added. The rate of production andnot the stock is the well defined quantity, and so with all staplearticles.

(3) Professor Walras goes unnecessarily out of his way in makinghis special supposition that in order to have a common cost price equalto the selling price suppliers must produce equal quantities (by whichpresumably is meant equal quantities per unit of time). The rate ofproduction regulates the cost of production, that is, the marginalcost or sacrifice, and it is quite possible for this marginal cost (meas-ured in money) to be the same for a small cobbler as for a large shoemanufacturer, though the fixed and running expenses may dividethemselves very differently. Moreover, it is rather misleading tosay that the equality of [marginal] cost and selling price impliesneither gain nor loss. There exists a normal gain or "producer'srent," as Marshall calls it, which is quite distinct from and inde-



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THEORY OF THE DETERMINATION OF PRICES. 47pendent of any speculative gains or losses, Q2 (,,,--pb) etc., due toa disturbanceof the equilibrium between cost and selling price.IRVING FISHER.I.

THE EXCHANGE OF SEVERAL COMMODITIES AMONGTHEMSELVES.

In my ElMments 'iconomiepolitiquepure, * passing fromthetheory of the exchange of two commodities to the theory ofthe exchange of several commodities among themselves, andseeing that in that case the demand or the supply of each ofthe commodities by each of the traders is a function, notonly of the price of that commodity, but also of the price ofall the others, I believed it was necessary to adopt exclu-sively the analytic method of expression and do without thehelp of diagrams. But since then I have found a means,which I will indicate briefly, of elaborating the theory inquestion by the method of geometrical representation.Suppose a party to the exchange with the quantities q,, q,q,, q,. . . of the commodities (A), (B), (C), (D), representedby the lines Oq., Oqb, OOq,Oqp. . . (Fig. i) and having forhim the utility expressed by the curves a.ar,qir /r,~qy/r) rI proceed to describe these curves which are the essen-tial and fundamental basis of all the mathematical theoryof social wealth.

We may say in ordinary language: " The desire that wehave of things or the utility that things have for us, dim-inishes in proportion to the consumption. The more a maneats, the less hungry he is; the more he drinks, the lessthirsty; at least in general and saving certain regrettableexceptions. The more hats and shoes a man has, the lessneed has he of a new hat or a new pair of shoes; the morehorses he has in his stables, the less effortwill he make to pro-cure one horse more, always neglecting the action of impulseswhich the theory has the right to neglect, excepting whenaccounting for certain special cases." But in mathematicalterms we say: "The intensity of the last desire satisfied, is* Lausanne. F. Rouge. 1889.



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48 ANNALS OF THE AMERICAN ACADEMY.a decreasing function of the quantity of commodity con-sumed," and we represent these functions by curves, thequantities consumnedby the ordinates and the intensity of thelast desire satisfiedby the abscicsas. For example, take thecommodity (A), the intensity of the desire of our consumerwhich would be Oa, at the beginning of the consumption,would be nil after the consumption of a quantity Oaq,theconsumer having then arrived at satiety. That intensity ofthe last desire satisfied, for the sake of brevity, I call raretd.The English call it the final degree of utility, the GermansGrcnznutzen. It is not an appreciable quantity, but it isonly necessary to conceive it in order to found upon the factof its diminution the demonstration of the great laws ofpure political economy.For the present let Pb,P, P . . . be the prices of (B), (C),(D), in terms of (A) determined at random on the market.The first problem that we have to solve consists in deter-mining the quantities of (A), (B), (C), (D) . . ., x,y, z,w... the first positive and representing the quantitiesdemanded, the second negative and representing the quanti-ties offered, which our trader will add to the quantitiesq?,q&,qb, d, of which he is already possessed or which he willsubtract from them, so as to consume the quantities q, + x,qb + Y, + z, qd + w..

. represented by the lines Oa, Ob,Oc, Od ... Just as we employed the general hypothesisabove, of a party to the exchange for whom the raretddecreased with the quantity consumed, so here we employthe general hypothesis of a party to the exchange who seeksin the exchange the greatest possible satisfaction of hisdesires. Now the sum of the desires satisfied by a quantityOa of commodity (A), for example, is the surface Oap. a,.The effective utility is the integral described by the raretf inrelation -to the quantity consumed. Consequently theproblem, whose solution we are seeking, consists preciselyin determining Oa, Ob, Oc, Od. ... under the condition thatthe sum of the shaded areas Oap, a,, Obb, Pr, Ocp, ,, Od4, ,r...be a maximum.



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FiFig. 1. d d9 BEd Fig.2. Fig. 5aa a

O r. O ra a, r B

0oN ' Bp PSBoe

b BI i Fig4 7

P - r . r.7bO rb- r,

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C' AI', 0- 4r 0 _7j>Pi, B ?p 01 II I

I x

o0 ra , o r o r o ti,pb B pO Pj K, pL.Waras.

AVILt RINTINGCO., PHILA.

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THEORY OF THE DETERMINATIONOF PRICES. 49In order to furnish that solution very simply in the

geometric form, I subject the curves of utility or desire PqP,7qY, , r -.. to the following transformation. I lay off fromthe origin 0, upon the horizontal axes, the new abscissasequal to 1 of the old abscissas. Also, upon the parallelsto the vertical axes drawn through the extremities of thenew abscissas, I lay off from the horizontal axes the newordinates equal to f times the old ordinates. In the figure,Pb =2, p 3,~ = ... As is easily seen, the new2curves 'q ', , Y',,Y', 'q6'. represent the utility of (A),as spent for (B), for (C), for (D) ..., or, in other words, thedesire the exchanging party has of (A), in order to procuresome of (B), of (C), of (D)... In short, if we consider theareas OPr,, Ordy,, 06q6 ... as the limits of sums of rectanglesinfinitely small, we may consider the surfaces, O3'q, ',, Or', y',O6',q'6, as the limits of equal sums of rectangles infinitelysmall, each base being / times less, and each height p timesgreater. Now, each of the rectangles of the former sumrepresents the effective utility of an increment of commodity;each of the rectangles of the latter sum represents, in thesame way, the equal effective utility of the p increments of(A), with which that increment of commodity may bebought.The curves aqar,,)',',, '/qyr, d',', being placed beside eachother, I take a vertical length OQa, representing the equiva-lent in (A) of the quantities q,, qb, q,, qd, of (A), (B), (C),(D) . .. at the prices I, Pb,, d . viz : q, + qbPb+ qcP, +qd + .. . and I advance it from right to left, in order tosatisfy the varying desires in the order of their intensity,until it is sub-divided among the curves into the ordinatesra,, = Oa, rB = Ob', rC = Oc', rD = Od', . . . correspond-ing to a like abscissa. Now, that abscissa Or, will represent,in terms of (A), the raret/ of (A), of (B), of (C), of (D) . ..say, ra, corresponding to the maximum of effective utility.The ordinates Oa, Ob',Oc',Od'..., will represent, in terms



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50 ANNALS OF IHE AMERICAN ACADEMY.of (A), the quantities to be consumed, of (A), of (B), of (C),and of (D), the only commodities to be consumed being thosefor which the intensity of the first desire to be satisfied isgreater than r,.If we carry back the abscissas Or. = r, Orb=pbra,, Or, =/cra, Ord --~drr ,.. to the curves aq a,, / /Pr, 6'q, r, q .r . . weobtain the ordinates Oa, Ob, Oc, Od ... representing thequantities of (A), of (B), of (C), of (D),... to be con-sumed. And so, in a state of maximum satisfaction, the rareAtsare proportional to the frices, accordingoto the equations :

r. rb , r,,I Pb P0 PdThus it is, that, given the quantities possessed and theutilities of the commodities, we determine for a party in theexchange the demand orsupply of each of the commodities atprices taken at random, which will affordthe maximum satis-faction of his wants.Having given the demand and supply of commodities byall the parties in the exchange at prices taken at random,it remains to determine the current prices at equilibrium,under the condition of the equality of the total effectivedemand and supply. The solution of the second problemmay also be furnished geometrically.Let us., for an instant, neglect P~,A . . and seek at firstto determine, provisionally, pb. And, for that purpose,let us inquire how (pb,pd ... being supposed constant) thevariations of Pb influence the demand and supply of (B).If y is positive, that is to say, if the trader is in need of(B), an augmentation of fib can only diminish y. In short,if he takes at a higher price an equal quantity, he stillowes a difference which he cannot pay, without diminish-ing the quantities of (A), (C), (D) ... But, then, hewill augment the rare/ts of these commodities; and, inconsequence, the condition of maximum satisfaction will beless perfectly fulfilled. Hence, the quantity y is too greatfor a price higher than Pb.If y is negative, that is to say, if the party is a supplier of



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THEORY OF TaE DETERMINATIONOF PRICES. 51(B), there are three possible results. The party, being sup-posed to supply an equal quantity at a higher price, asurplus is due him, and, by means of that surplus, hecan augment his quantities, and consequently diminishhis raret~s of (A), (C), (D)... Then, one of threethings occurs: Either the surplus is insufficient to re-establish the condition of maximum satisfaction, or it is justsufficient, or it is more than sufficient; and, in consequence,at a price higher than fb, the party must supply a quan-tity of (B), either greater than, equal to, or less than y.It is certain, that he will find himself in one of these threecases, according to the amount of the enhancement offb. For, if, on the one hand, that enhancement of Pbconstantly diminishes the ratio b by increasing the denomi-nator, that same increase of Pb admits, on the other hand, ofa continual lowering in r,r. .. by a diminution of theI'P' Pdnumerators r,, r rd, . . . which may finally cause a decreaseof the numerator rb itself.The variation of Pb, from zero to infinity, therefore, causesthe party to the exchange to pass from the side of demandto that of supply; then, from an increasing supply to a de-creasing supply. At the price zero, the demand is equalto the excess of the quantity necessary perfectly to satisfy thewants over the quantity possessed; at the price infinity,the amount offered is nil. In the case of the exchange ofseveral commodities, as in the case of the exchange of twocommodities with each other, the tendencies may be repre-sented geometrically, for a party to the exchange, by a curve*bd b b . (Fig. i).

* [The ordinates of this curve have the same meaning as the ordi-nates of the utility curve for (B) on which it is superposed, but its ab-scissas represent the price of (B) in terms of (A.) Thus in order that theindividual may consume Ob, that is buy qb b in addition to his origi-nal stock Oq, the price must be by (in this case 2). If the pricerises above q bphe ceases to buy and begins to sell. But the curve



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52 ANNATS OF THE AMERICAN ACADEMY.All the parties to the exchange being not identical, but

similar in their tendencies, as far as concerns the commodity(B), it is clear that all the partial curves of demand must beunitedt in a total curve that continually decreases, Bd Bp (Fig.2), and all the partial curves of supply in a total curve NP, suc-cessively increasing and decreasing, from zero to zero, if wetake it positively, by making NP' turn around the horizontalaxis, so as to bring it to the position NP. The abscissa OP'b,of the point of intersection B of the two curves B B, andNP, will be provisionally the current price at equilibrium forwhich the total effective supply and demand of (B) will beequal. Furthermore, the intersection of the two curves, Bd Bpand NP, may take place either when the second curverises, or when it falls.It follows from the nature of the curves, that we shall ob-tain the provisional current price of (B) by raising it in caseof a surplus of effective demand over effective supply, andlowering it, on the contrary, in case of a surplus of effectivesupply over effective demand. Passing then to the determi-nation of the current price of (C), then to the current priceof (D) ..., we obtain them by the same means. It is quitetrue that, in determining the price of (C), we may destroythe equilibrium in respect to (B); that, in determining theprice of (D), we may destroy the equilibrium in respect to(B), and in respect to (C), and so on. But, as the determi-reaches a minimum point and then approaches qb~ as an asymptote.That is as the price rises beyond a certain point, he ceases to stinthimself in the enjoyment of (B) and parts with less and less of it.-TRANSLATOR.]t [The total demand curve for (B) (B, B, Fig. 2) may evidently befound from the partial curves such as bd bpb, by selecting on theseindividual curves the points which correspondto a given abscissa(price) and constructing a corresponding point in the total curvewhich shall have the same abscissa,but whose ordinate shall be thesum of the ordinates of the individual curves measured aboveqb P asyp b. In like manner the ordinates of NP' are the sum of the indi-vidual ordinates measured belowqbp and corresponding to like ab-scissas.--TRANSLATOR.]



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THEORY OF THE DETERMINATION OF PRICES. 53nations of the prices of (C), (D) ... in respect to the demandand supply of (B), will result in a contrary way, we shallalways be nearer the equilibrium at the second trial than atthe first. We enter here on the theory of trial and error,such as I have developed in my work, and by virtue ofwhich we arrive at the equilibrium of a market by raisingthe price of commodities, he demandfor which is greater thanthe supply, and byl lowering the price of those, the supply ofwhich is greater than the demand.It is due to the concurrent employment of analytic expres-sion and geometric representation that we have here, in thecase of the exchange of several commodities among them-selves, not only the idea but the picture of the phenomenonof the determination of prices upon the market. And withthis, it seems to me, we possess at last the theory. Somecritics, however, laugh at the number of pages I use indemonstrating that we may arrive at a current price by rais-ing in case of an excess of the demand over the supply, andlowering in case of an excess of the supply over the demand."And you," I said once to one of them, "how do youdemonstrate it ?" " Well," he answered me, a little sur-prised and embarrassed, "is there any need of demon-strating it ? It seems to me self-evident. " "There is nothingevident except axioms, and this is not an axiom. But onenaturally follows the mode of reasoning which Jevons hasformulated so clearly in his little treatise on PoliticalEconomy, that a rise, making necessarily a diminutionof the demand and an augmentation of the supply, causesequality in case of a surplus of the one over the other."" Precisely." " But there is an errorthere. A rise neces-sarily diminishes the demand; but it does not necessarilyaugment the supply. If you are a supplier of wine, it maywell be that you supply less at a million, than at a thousandfrancs, less at a billion than at a million, simply be-cause you prefer to drink your wine yourself, rather thanuse the surplus which you could procure by selling itbeyond a certain limit. The same is true of labor. We easily



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54 ANNALSOF THE AMERICANACADEMY.conceive that a man, who supplies ten hours a day of histime at the price of one franc an hour, would not supply morethan four at the price of io francs, or than one at ioo francs.We see, every day, in the large towns, that the laborers,when they earn 20 or 25 francsa day, do not work more thanthree or fourdays a week." "But if that is so, how is raisingit a means of reaching the current price ?" " It is this thatthe theory explains. Two individuals, who have separated,may meet again, either by moving each, in an opposite direc-tion to the other, orby one going faster than the other. Sup-ply and demand equalize themselves, sometimes in one way,sometimes in another." Is it not worth while to demon-strate rigorously the fundamental laws of a science?We count to-day I do not know how many schools of po-litical economy. The deductive school and the historicalschool; the school of laisser-faire and the school of state-intervention,or socialisme de la chaire, the socialistic schoolproperly so-called, the catholicschool, the protestant school.For me, I recognize but two: the school of those who do notdemonstrate, and the school, which I hope to see founded, ofthose who do demonstrate their conclusions. It is in demon-strating rigorously the elementary theorems of geometry andalgebra, then the theorems of the calculus and mechanicswhich result from them, in order to apply them to experi-mental ideas, that we realize the marvels of modern industry.Let us proceed in the same way in political economy, and weshall, without doubt, succeed in dealing with the nature ofthings in the economic and social order, as they are dealtwith in the physical and industrial order.

II.THE EXCHANGEOF PRODUCTSAND SERVICESWITH

EACH OTHER.It is my present purpose to apply to the theory of produc-tion and the theory of capitalization the exclusively geo-metric method of demonstration according to which I havesketched the theory of exchange in the preceding paragraph.



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THEORY OF THE DETERMINATION OF PRICES. 55Now, in formulating the theory of exchange, we suppose

the quantities of the commodities to be a given, not an un-known element of the problem. To begin with, in order toarrive at a theory of production, it is necessary to considercommodities as the products resulting from combined pro-ductive services, and, in consequence, it is necessary tointroduce the quantities of manufactured products into theproblem, as so many unknown quantities, adding as is proper,an equal number of determining mathematical conditions.That is what I wish to do here, referring to my Elements offure political economy or definitions and notations.*Suppose, then, the services of land, labor and capital [in thenarrow sense] (T),(P), (K)... susceptible of being utilized,either directly as consumable services, or indirectly as pro-ductive services, that is to say, in the form of the productsof the sorts (A), (B), (C), (D)... The first problem thatwe have to solve consists in determining, for each consumer,the supply of services and the demand for services, either inthe form of consumable services, or as products. Now thesolution of the problem is furnished us by the theory ofexchange.Given, then, a consumer possessed of the quantities q,, q,,q,, of the services (T), (P), (K) ... andhaving a desirefor theseservices and a desirefor the products, (A), (B), (C), (D) ... ex-pressed by the curves of utility or desire. Given, also,

~t,pA, Pk . . , ,-b) , ?-. the prices (taken at random) of(T), (P), (K) ... andof (B), (C), (D) ... in termsof (A). Wewill transform the curves of utility or desire of services andproducts into curves of utility (measured in (A)) of (T), (P),(K) . . . (B), (C), (D). . . or, in other words, into curves ofthe desire for (A) to be used in procuring some of (T), (P),

* [These definitions of conceptions peculiar to the author and con-stituting essential elements of his system of Economics are to be foundin the preface to his work (pp. XII-XVI). I have translated bycapitals the word capitaux (which includes all things material orimmaterial that are used more than once) and by services the wordservices (the successive uses of the capitals).-TRANSLATOR.]



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56 ANNALS OF THE AMERICAN ACADEIMY.(K), . . . (B), (C), (D) . . . This is done by dividing the ab-scissas and multiplying the ordinates by market prices.The curve of the utility or desire of (A) and the trans-formed curves of utility, or desire of (T), (P), (K)... (B),(C), (D)... being placed one under the other, we mayadvance a vertical line of the length 02 = q, +q,~P, + qkPk+ ... from right to left, until it distributes itselfamong all the curves into ordinates corresponding to alike abscissa ra. By carrying back the abscissas f, ra,ip ra, ra . . . r., b a r, a d ar . . . to the primitive curves,we obtain the ordinates representing the quantities oflabor (T), (P), (K) ... and of products (A), (B), (C),(D). .. to be consumed. It is evident that in the state ofmaximum satisfaction, the rare/ts will be frofortional to theprices according to the equations :rt= r -_ rr r rb- e rd_

Pt p pk I b 7o WOur pricesp,,f, ... , ) d .. for servicesandpro-ducts are supposed to be taken at random. We will nowsuppose, that there have been manufactured, say, thequantities sa,ba , ,d . . . of (A), (B), (C), (D) .. ., atrandom, and, leaving pf, p,,Ak. .. as they are, let us deter-mine the prices of (B), (C), (D) . . . by the condition thatthe demand for the products shall be equal to their supply,that is to the quantity manufactured. The solution of thesecond problem is likewise furnished us by the theory ofexchange. Suppose, then, Ab,represented by the ordinate

Ab (Fig. 3), to be the total demand for (B) at the pricesjust supposed for services and products. We know, by thetheory of exchange, that if, disregarding at first the pricesof (C), (D) ... and seeking to determine provisionally theprice of (B), we cause the price to vary from zero to infinity,the demand for (B) will diminish always according to thecurve BdB,. Hence, there exists a price, r'b, correspondingto the equality of the demand for (B), with the supply 'b,which is > b, if, at the price -,, the demand for (B) isgreater than the supply, and which is < Wb, if, at the price



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THEORY OF THE DETERMINATION OF PRICES. 57br,',he supply of (B) is greater than the demand. We shalllikewise find a price ,', corresponding to the equality of thedemand for (C), with the supply ?o, a price d', correspond-ing to the equality of the demand for (D), with the supplyQd, and so on. After the first experiment we proceed to asecond, to a third still, and so on, until we have obtained aseries of prices, rb", [, -rd" . . at which the demands for (B),(C), (D) . . . will be equal to the supplies 'b, , d * * Weconclude then that in the matter of production, as in the mat-ter of exchange, we reach the equilibrium of market of pro-ducts in raising the price of those, the demandfor which isgreater than the supply and lowering the price of those, thesupply of which is greater than the demand.

r.b, = o,bI)d. . are thus the selling prices of the quantities,

2b,)s2, 2d. . . of (B), (C), (D) ... But from the prices, p,1,,p,...of the services, (T), (P), (K) ... result certaincost prices,Pb, A,PA... of the products, (B), (C), (D).. .* And the differ-ence, positive or negative, between the selling price and thecostprice, in the production of (B), (C), (D) . .. results in thegain or loss, "b(b --Pb), ( --Pc) d (d -Pd) . . . It isnow necessary to determine the manufactured quantities of(B), (C), (D) ... by the condition that the selling priceand costprice be equal, so that there may be neither gain nor* It is true that, in order to suppose a cost price, common to all the undertakers,it is necessary to suppose that the fixed expenses distribute themselves among anequal quantity of products, in order to allow us to make them correspondto the proportional expenses; that is, it is necessary to suppose all the partiesmanufacturing equal quantities of products. The hypothesis is no more real thanthat of the absence of gain or loss, but it is as rational. If, in short, at a givenpoint, a certain quantity of manufactured products corresponds to the absence ofgain and loss, the parties in the transaction, who manufactured less, take thelosses, restrain their production and finish by liquidating, those who manufac-tured more take the gains, develop their production and attract to themselves thebusiness of the others; thus, owing to the distinct nature of proportional expenseand fixed expense, production in free competition, after being engaged in a greatnumber of small enterprises, tends to distribute itself among a number less greatof medium enterprises, then among a small number of great enterprises, to endfinally, first in a monopoly at cost price, then in a monopoly at the price of maxi-mum gain. This statement is corroborated by the facts. But during the wholeperiod of competition and even during the period of monopoly at cost price, it isalways permissible, in order to simplify the theory, to suppose the undertakersmanufacturing equal quantities of products and to make the fixed expense cor-respond to the proportional expense.



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58 ANNALS OF THE AMERICAN ACADEMY.loss to the undertakers. This third problem is the especialproblem of the theory of production, and may also be solvedgeometrically as follows:Let OPb Fig. 4) be an abscissa representing the costpnce,Pb. Let Or"bbe an abscissa representing the selling brice,b", and 7r"bB'an ordinate representingthe quantity ?bof(B), manufactured at random, and demanded at the price7b". If we supposep, , fP,,'f "P,, 7C ) d". . determined andconstant, and that we may vary the price of (B), from zeroto infinity, it is certain that the demand for (B) will diminish,always following a curve Bd'Bp'. Consequently, there existsa demand ,b',corresponding to a selling price, equal to thecostprice b, which is


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THEORY OF THE DETERMINATION OF PRICES. 59

Our prices of services , ,, A. . . have always beendetermined at random. There remains to us a fourthand last problem to solve, which is to determine theway in which the quantities demanded and the quanti-ties supplied are equal. Now, at the point where weare, there are quantities supplied of (T), (P), (K) . . .U, U,, U . . . which are determined by the conditionof maximum satisfaction, conformably to the solutionof our first problem. And, in view of the quantities sup-plied, there are quantities demanded which are com-posed of two elements: first, the quantities demandedby the consumersin the way of consumable services u,,up, Uk...which are also determined by the condition of maximum sat-isfaction; then, the quantities demanded by the under-takers in the way of productive services, Dt, Dp, D, . . .which are determined by the quantities manufactured of theproducts (A), (B), (C), (D)... the demand for which is equalto the supply, and the selling price equal to the cost price,conformably with the solution of our second and third pro-blems. We may demonstrate exactly as in the theory of ex-change, that if, everything else remaining equal, we causep tovary from zero to infinity, (i), the demand for (T), D, + It4willdiminish, always following a curve Td T, (Fig. 5); {2), thesupply of (T) will, starting from zero, increase, then dimin-ish and return to zero, following a curve QR ; and that, con-sequently, there exists a price p',, at which the supply anddemand of (T) are equal, which is > p, if at the price p,,the demand for (T) is greater than the supply, and


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THEORY OF THE DETERMINATION OF PRICES. 6Ibeen said here, I take the liberty of referring the reader tosection III of my Eliments.

III.THE EXCHANGE OF SAVINGS FOR NEW CAPITALS.

In order to simplify, let us suppose for the present, theequilibrium established as regards the quantities of commo-dities manufactured as well as the prices of commodities andof services, and let us neglect the changes which may becaused in this equilibrium by our investigation of the specialeqiulibrium of capitalization. Let us, in the same way, neg-lect the cost of the redemption and insurance of the capitals.The elements in the equilibrium of capitalization are thequantities produced of new capitals and the rate of inter-est whence results the prices of the capitals following thegeneral formula r = ~

Suppose, there are produced at random, the quantitiesD,, Dkl, Dkl... of capitals of the sorts (K), (K'), (K")...and that there is a rate of interest at random, i. At thatrate each man engaged in exchange determines the excessof his income over his consumption, and the total of theseindividual excesses forms a total excess E, which is thequantity of cash at hand to buy new capitals or the demandof the new capitals in cash at the rate of interest, i. Onthe other hand, at the current prices for their use, A,1~,An,supposed to be determined and constant, the quanti-ties Dk,,Dk1, Dk . . of the capitals (K), (K'), (K") . .. givea total income DkPk + Dkl pkl + DkllPkll -- . . , and possess atotal value

DkPk + DklPkl + Dk,11kll+. .which is the quantity of cash demanded in exchange for thenew capitals or the supply of new capitals at the rate of inter-est i. If, by chance, the two quantities of cash are'equal,the rate i will be the rate of the equilibrium of the interest,



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62 ANNALS OF THE AMERICAN ACADEMY.but generally they will be unequal and it remains to renderthem equal.Now, we may assume that the excess of the income overthe consumption is at first nil, at a nil rate, then it mul-tiplies and augments at a positive and increasing rate,then diminishes and returns to zero, if the rate tends to be-come infinitely great; that is to say, if, with a minimumsaving, one may gain a very great increase in his income.In other words, the rate of interest, being an abscissaon the axis OI (Fig. 6), the excess of income over con-sumption will be the ordinate of a curve, successively in-creasing and decreasing, ST. As to the value of thenew capitals it evidently increases or decreases, accordingas the rate of interest decreases or increases. In otherwords, the rate of interest being an abscissa on the axisO I, the value of the new capital may be an ordinate of acurve continually decreasing, U V. Hence, we see im-mediately that it is necessary to raise the price of the newcapitals by lowering the rate of interest if the demand for newcapitals in cash is greater than the supply, and to lower theprice of the new capitals by raising the rate of interest, if thesupply of the new capitals in cash is greater than the demand.At this time, there are the cost prices Pk,, Pkl Pkl. of the new capital (K), (K'), (K") . . . besides the sellingprices ~k, l7k. ... The question is to reduce the sellingand cost prices to the equality which generally does notexist between them. Now we may regard as establishedby the previous demonstrations that in augmenting ordiminishing the quantity of a capital (K), we diminish oraugment the rarete and the price of its use, and conse-quently the selling price of this capital, and that is to say,that the curve of the quantity in relation to the selling priceis the constantly decreasing curve K, K, (Fig. 7). And weare equally justified in concluding that in augmenting ordiminishing the quantity of the same capital (K), we aug-ment or diminish the rare/l and the prices of the productiveservices which enter into the making of the capital and con-



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THEORYOF THE DETERMINATIONF PRICES. 63sequently its cost price; that is to say, that the curve ofquantity in relation to the cost price is the constantlyincreasing curve, X Y. Hence, we see immediately andwithout the necessity of reproducing here the exposition ofthe successive approximations in regard to the quantities ofcapital (K), (K'), (K")... that it is necessary to augmentthe quantity of a new capital, whoseselling price exceeds its costprice and diminish thequantity of that whose costprice exceedsits selling price.The equilibrium of capitalization once established, ashas just been explained, we have:

Pk=Ir Pkp)k7 Pk1 =kor Ak- tk1t k _Pk Pk1 k11that is to say, the rate of interest is the same for all capitalsaved.We may demonstrate geometrically in a very simple man-ner, at least as far as concerns capitals in consumable services,that this identity of rate of interest is the condition of themaximum utility of new capitals.There are two problems of maximum utility relating to theservices or use of new capitals; that connected with the distri-

bution by an individual of his income among his differentkinds of desires, and that connected with the distribution bya society of the excess of its income over its consumptionamong many varieties of capital. The first is solvedby means of the construction which was made in the theoryof exchange, and referred to at the beginning of the theory ofproduction, involving the proportionality of the raretg of aspecies of capital to the price paid for its use, according tothe equations :rk rkl_ rk11k Al llIt will be understood, without difficulty, that the secondproblem would be solved by a construction exactly similar



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64 ANNALS OF THE AMERICAN ACADEMY.to the former. Instead of transforming the curves of thedesires for the various services of capitals by dividing the ab-scissas, and multiplying the ordinates by the prices for theiruse, pk, k,, Pk . . . we should divide the one and multiplythe other by the cost prices P,, Pk,, Pk11 . ., involving theproportionality of the raret/s to these prices, viz:

rk rkr k11 -Pk Pk1 P,11or, dividing the latter system by the former:

Pk PkI kiwhich expresses the identity of the rates of interest of allcapital. LPON WALRAS.University of Lausanne..

(Translated under the supervision of IRVING FISHER,of YaleUniversity.)

Leon Walras and Fisher

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