Pure Choice

7/27/2019 Pure Choice

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The Pure Theory of Consumer's BehaviourAuthor(s): N. Georgescu-RoegenSource: The Quarterly Journal of Economics, Vol. 50, No. 4 (Aug., 1936), pp. 545-593Published by: Oxford University PressStable URL: http://www.jstor.org/stable/1891094.

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THEQUARTERLY JOURNALOF

E ONOM I SA UGUST, 1936

THE PURE THEORY OF CONSUMER'S BEHAVIORSUMMARY

I. Introduction, 545.- II. The main postulates of the theory ofchoice, 5-18.- Analytical treatment of classical scheme, 551.- Dis-cussion of stability conditions, 553. -III. Introduction of a saturationregion, 557.- Analytical treatment of the case involving two com-modities, 558.- IV. The theory of choice applied to fixed incomes, 562.- Correspondence between two and three dimensional cases, 564.-V.Indifference varieties vs. integral varieties, 565.- Consistent preferen-tial fields, 565.-- Transitivity condition, 567.- VI. Analytical treat-ment of an alternate scheme leading to indeterminateness of exchangeequilibrium, 568.- Individual and market demand laws, 583.-VII.Conclusions, 584.- Appendix. Mathematical note on the conditions ofstability in an exchange with constant prices, 588.

IAll essential differences between static and dynamiceconomics center upon the fundamentally distinct ways inwhich their mathematical treatments are elaborated. Most,if not all, of the functional relations used in the set-up ofstatic problems are likely, because of the rationale of theproblem itself, to be resolved into some simple type of generalfunction in the sense given to this concept by Dirichlet.1 Theuse of such functions in dynamic problems has not yet beenjustified by any analysis. The functions used in many

attempts at a mathematical treatment of dynamic economics,besides involving the time element introduced in a way thatseems to imply a certain causal relation between facts and1. In this sense, u is said to be a function of x, y, z, .. . if to any setof values of the latter there corresponds unequivocally one value of u.Another type of function may be defined so as to make the value ofu depend not only upon the values x, y, z, ... but also upon the pathby which any particular set of these values is reached.

545


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546 QUARTERLY JOURNAL OF ECONOMICStime, have been limited also in their structural properties forconvenience in carrying out the necessary transformations,For the sake of this rationale of the general set-up, theintroduction of the concept of marginal utility was regardedas an advance over Cournot's approach to the static problemwhich uses only the tools of demand and supply curves. Thedemand and supply laws appear today to be derived con-cepts, and their justification is sought in terms of the ulti-mate considerations that find their place within the frame ofeconomic science; i.e., the reasons that induce individuals toproduce and exchange goods.

For the same reason, the introduction of indifference vari-eties or of the marginal rate of substitution were regardedas new refinements of the set-up. And thus the theory ofstatic equilibrium, in so far as it is concerned with the posi-tion of consumers, which is beyond doubt the most delicateside of the problem, has reached a degree of rigorousness andexactness that was considered, not very long ago, as theappanage of the natural sciences only. One need but readthe admirable papers of R. G. D. Allen and J. R. Hicks2 inorder to be convinced of the vast possibilities that are opento the mathematical treatment of static economics.The method of economics remains - and it seems that itwill remain despite many attempts in the opposite directionthat of the mental experiment aided by introspection.There are well known attacks directed against this procedurefor supporting scientific laws. Nevertheless, we may defendour position by arguing that, so far as we deal with the con-sumer's position, introspection is justified by the problemitself.At the same time we may seek a safer line of approach.This might be reached, for instance, by formulating ourmental experiment in such a way as to suggest, and directstep by step, the pattern of an actual experiment which maybe carried out in the future, subject to technical possibilitiesin the matter.

2. A Reconsideration of the Theory of Value, Economica, 1934,Nos. 1 and 2.


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PURE THEORY OF CONSUMER'S BEHAVIOR 547Such a scheme will implicitly require, if we want to pro-ceed in a scientific way, that all the results of the mentalexperiment that are not evident a priori be stated as addi-tional assumptions. We can arrive in this way at the formu-lation of a necessary and sufficient set of assumptions forhandling the problem, and thus obtain a kind of measure ofthe extent to which our mental experiment may diverge froma similar actual investigation. For maintaining further thisparallelism between the mental and actual experiment, theformulation of our postulates in such a way as to outline ina straightforward fashion the corresponding physical inves-tigation is undoubtedly the most advisable procedure.3In so far as our analysis is of the type just mentioned, caremust be taken not to postulate the very thing we want toexplain. A phenomenon may be regarded as justified merelyby its existence, if this can be experimentally established. Ifthis is not possible, we have to seek for another kind ofexplanation. It is true that the meaning of explanation is

in this case formal only. Nevertheless, the advantage of anexplanation, even formal, of the exchange equilibrium interms of concepts outside economics cannot be denied.Pareto seems to have realized the advantage of such aposition, for we find in his latest writings an attempt todescribe how the existence of indifference directions mightpossibly be tested by an actual experiment. But he failedto state explicitly the essential assumption thus introduced,and because of this the whole problem appeared to have aunique issue.'The failure to state explicitly one's assumptions in con-nection with the individual's behavior accounts also for the

3. It is far from my belief that all the points connected with thenature of indifferenceelements can be elucidated by an actual experi-ment. For the subtlety of some of these - perhaps of the most im-portant ones - is beyondthe usualdegreeof accuracy of our measure-ments. But, as is clearly seen from what precedes, my advocacy ofa parallelism between the two kinds of experiments has entirely adifferentaim.4. Manuel, p. 542. Economie Math6matique, Encyclopddie desSciencesMath6matiques,tome I, Vol.4, fasc. 4, pp. 596-598.5. See below, Sections II and VI.


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548 QUARTERLY JOURNAL OF ECONOMICSapparent paradox to which the non-integrability case led.It is the purpose of the present paper to undertake thetask of such an analysis as that outlined above. I shall takethis opportunity for correcting some errors that have slippedinto the most recent papers on the subject and, finally, fordeveloping an alternate theory of the nature of indifferencecurves.

III shall not go back as far as Jevons' concept of total andmarginal utility, for this way of presenting a theory of

exchange has been proved to constitute a minor view incomparison with the latest developments, which, however,it must not be forgotten, had Jevons' construction as astarting point. Consequently, I shall confine myself to theconsideration of the most recent presentations, as they arefound in the writings of Edgeworth, Irving Fisher and Pareto,namely:1. The theory of indifference varieties.2. The theory of choice.Since the second way of approaching the exchange problemhas lately been regarded as an improvement over the first,I shall begin by considering the theory of choice. The theoryof indifference varieties will not be taken into account untilthe point of view just mentioned has been analyzed.I shall assume perfect continuity of the entire field underconsideration and make also the further assumption that themathematical functions which are introduced have deriva-tives up to the order desired. The first assumption is onlyformal, for the discontinuous fields may, by a suitable inter-polation, be converted into continuous ones. The secondassumption is more delicate; it may, however, be lookedupon as the mathematical interpretation of the regularityof human behavior.Let S be an ordinal and continuous set of combina-tions, i.e., a set such that any combination (C7) belongingto the set may be completely characterized by its rank r,and vice versa. Let us assume further that the combination


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PURE THEORY OF CONSUMER'S BEHAVIOR 549(Ca) is always preferred to (Ca) if a> f3. When this last con-dition is fulfilled, the set (Ca) will be called preferential. Ifthe individual is already in possession of a third combination(T) which is preferred to (Ca) if


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550 QUARTERLY JOURNAL OF ECONOMICSthe trouble to move from M to M'(xl + Ax1, X2+ AX2), if hecan do so without further conditions, no matter how smallthe positive increments Ax, and Ax2are. On the contrary,M will be preferred to any combination within the angleCMB.Let us consider all combinations lying on a positivelyinclined straight line cow'. These combinations form a pref-erential set. As there are positions such as V', that are

Ao, B

FIGURE 1

preferred to M and others, and such as V, that are non-preferred to M, there will be a unique point p, on the linecow' which will represent an indifference combination inrespect to M. As the line cow'moves parallel to itself towardsmm', the direction Mit will tend towards a definite positionMw. This position will be the tangent at M to the locus of p,.By repeating the same argument for the line 01X1, weshall obtain another limiting direction Mw1.


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PURE THEORY OF CONSUMER'S BEHAVIOR 551Any direction within the angle w1Mw, such as Mv, willbe a preference direction; in other words the individual willmove away from M on any such direction if he has the oppor-tunity of doing so. The directions within the angle wMwi,

such as Mv', will be non-preference directions.We shall further assume that:C. The limiting directionsMw, Mw1are vertically opposite.In this case the element defined at the point M by theslope of ww1 s the indifference element. The last assumptionexpresses the fact that the individual will exchange eitherX1 for X2 or X2 for Xi, at any given rate of exchange, withthe exception of that rate which equals the slope of the cor-responding indifference element. This may be looked uponas an interpretation of the fact that the converse of what weprefer is always non-preferred.Nothing has yet been said about how the indifferencedirection will vary with the change of the slope (co) of the

line coco'A. Clearly, two alternatives are possible. However,in order to follow the classical point of view, I shall intro-duce the assumption:D. The indifference direction at any point is uniquelydetermined.Subject to assumptions A, B, C, D, the individual's tastescan be described by the differential equation

(i(XI, X2)dxl+ (xi, X2)dX2=0 (1)where (pi, 0 are so far subject only to the conditions

?O? p>0 (2)the = signs not being taken simultaneously.Proceeding to the case of three commodities, and applyingthe same kind of reasoning to the case where the choice ofthe individual is limited to the combinations represented by

6. I speak here only of the slope of a straight line, altho the locusof the combinations forming a preferential set is not necessarily astraight line. But on such a locus, only the infinitesimal element aroundthe indifferent combination u matters; and this can be always assimi-lated to a linear element.


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552 QUARTERLY JOURNAL OF ECONOMICSa plane which passes thru a preference direction positivelyinclined with respect to all coordinate axes, we reach theresult that the indifference element is represented by thetotal differential equation

(pldxl 02dx2+ 03dx3= 0. (3)It is clear that we could not possibly have any other formfor this element that would be consistent with the choice ofthe individual in a two-dimensional space as constructedabove. The linearity of the differential equation definingthe indifference element, once established for two dimensions,

must be extended to the case of n dimensions. Therefore,the indifference element will be in general defined by2i(pidxi O. (4)

The meaning of this is that a direction defined by the increments Ax,, Ax2, . * Ax. will be one of preference, indifference or non-preference, according to whether:lifijAXi>, =,


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PURE THEORY OF CONSUMER'S BEHAVIOR 553nately an error has slipped into his mathematical treatmentof it.1Let M' be a point infinitesimally near to M, such thatMM' be an indifference direction for M. The stability ofequilibrium of exchange with constant prices will be securedif the direction MM' is a non-preference direction for M'.It is clear that this condition is necessary and sufficient.Let us write B2, B3, * * BA,instead of S02/S01, (P3/S&1 . aePn/S02; the indifference element corresponding to the pointM'(xi+Axi) may be written, abstracting from infinitesimalsof higher order,

dx,+ 2i(Bj+ABj)dxj=? (6)2if Axi are sufficiently small. And, because MM' is an indif-ference direction for M, we have (5),

L = Ax,+ iBiAxi= 0. (7)2According to (5), the condition of stability will therefore be

Ax,+ 2i(Bi+ ABi)Axi< ? (8)2Owing to (7), this last relation becomes

2i BikAXiAXk


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554 QUARTERLY JOURNAL OF ECONOMICS0 1 B2 B3 ... Bn1 0 B2,1 B3,1 Bn)1B2 B2,1 2B2,2 B2y3+B3,2. B2)n+Bn)2D= B3 B3,j B3y2+B2)3 2B3,3 B3)n+Bn)3 (10)Bn Bnyl Bn)2+B2,n Bn)3+B3)n . 2Bnyn

are, starting with the third order, alternatively positive andnegative; 2 i.e.,

0 1 B2 B30 1 B2 2I B11 0 B2,1 >0; 1 0 B2,1 B,


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PURE THEORY OF CONSUMER'S BEHAVIOR 555BR9=Bo B3=Bo Bn=Box1+ 23iBxi= xA+ ZAA (14)2 2

we conclude that in a given planexl - A+ ZAB(xi - A) =0 (15)2(i.e., for a given budget equation) there is only one pointwhose indifference element is contained in this plane.It is, however, true that from (12) it follows that thestability condition is fulfilled for some indifference directions.For instance, in the case of three commodities, from

|1 B2 1 B2 B30B2y, B2,2 B3,1 B3,2 B83,3

it follows that the marginal rate of substitution between Xiand X2 decreases along the corresponding indifference direc-tion. A simple numerical verification is sufficient to provethat from the above conditions it does not necessarily followthat either of the inequalities

1 B3 B2 B3B3,1B3,3 < Oa B3/ Oa B3 0(P (Si1-X) + (S2' - X2)> 0it follows that


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PURE THEORY OF CONSUMER'S BEHAVIOR 559confine myself to the case where there is only one saturationpoint, this case being susceptible to a simpler and more com-plete analytical solution. It is however possible to justify,to a certain extent, the elimination of the other cases by thefact that the uniqueness of the indifference combinations inan ordinal set (assumption B) will harmonize better withthe uniqueness of the saturation point. Besides, if we wantto maintain a certain parallelism between the two casesconsidered so far, it is necessary to assume the uniquenessof the saturation point in order to reach the fundamentalresult obtained above, namely that on any straight linethere is only one point of equilibrium. But these are onlyformal considerations, and consequently cannot be regardedas sufficient logical reasons for disallowing the possibleexistence of a saturation segment or a saturation area.For simplicity in the calculations to follow, let us choosethe saturation point as the origin of coordinates. The onlyconditions imposed then upon wp,sp2are the condition ofstability

a (f:2 O(


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560 QUARTERLY JOURNAL OF ECONOMICSfor any positive value of a. Hence, either pi(O,0) =0 or(PI(0,0) is indeterminate. In any case, on any path avoidingthe origin and joining the two points (+a, 0), (-a, 0) theremust be at least one zero of pi. Consequently the curve,=0 approaches infinitesimally close to the origin, but wecannot decide if it passes through it or not. Along thiscurve, except perhaps at the origin, we deduce from (20)that

f


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PURE THEORY OF CONSUMER'S BEHAVIOR 561

2 _+[?TY2?+ s7)2]dy, =O (25)where T(yn)= (po?+(s? + s1)Y2?+ 0?2Y2 (26)Owing to (23), the trinomial T has imaginary roots, andtherefore there are only two cases to be considered.

* I % I~~~~~~~~~~~~~~X

(a) ' (b)FIGURBE21. (Po2 sOi The integral of (24) will be

log[ x2TQ )]?2(p2 -2) tan'[ 1 T'(X2)] =const. (27)where 62 = 4(1p02l - (sn2 + s001)2* The integral curves willenvelop the saturation point in the same way as logarithmicspirals turn round their asymptotic point (Fig. 2a).

2. ? = R. The term involving tan-' in (27) will dis-appear and thus the equation of integral curves near theorigin will beS(iXP ? ( S12 +P)x1x2? +22 = const. (28)They will be convex closed curves enveloping the saturation

point as a center (Fig. 2b)11. The preceding argument is only an intuitive sketch of proof. Thereplacement of (1) by (24) in a small domain around the origin for find-ing out the shape of the integral curves calls for further restrictions tobe imposed upon the functions S%1,2. These restrictions could be con-sidered as one of the aspects of the regularity of human behavior.


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562 QUARTERLY JOURNAL OF ECONOMICSThe second case has been consideredprevious to the presentpaper, but its treatment has always been approached thruthe utility concept. The first one is most curious. As weshall presently see, it has great theoretical importance.Suffice it to point out now that in the case where the inte-gral curves are ellipse-like, an index of ophelimity can bebuilt analytically. This is no longer possible in the casewhere the integral curves are spiral-like. For if it were, thenby keeping the amount of one commodity, say x2, constant,we would obtain an index of ophelimity (I), the variation ofwhich with respect to x1is indicated by Fig. 3.

I

0 ok ~~~~~~~~~FIGURE 3

IVLet us consider the planexI+?Lx2+J3Ox3 = x?+BOxA+BL 0 (29)For a rigorous and complete discussion of the problem under con-sideration the reader is invited to refer to H. Poincare, Oeuvres, tome I,p. 17 and to M. H. Dulac, Points singuliers des equations diffdrentielles,M6morial des Sciences Math6matiques, fasc. LXI.


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PURE THEORY OF CONSUMER'S BEHAVIOR 563which contains the indifference element corresponding to thepoint Mo(xo2 2, x0). The intersection of this plane with theindifference elements.

dxi+B2dx2+B3dx3= 0 (30)determines in each point of (29) an indifference direction.The theory of choice can be applied to the movements ofthe individual in this plane exactly as in the case of twocommodities, for the number of dimensions is the same inboth cases. We must therefore expect the indifference direc-tions thus obtained to fulfill the conditions established in theprevious chapter. An analysis in this direction may beregarded as one way of checking the conclusions alreadyobtained.If we project the indifference directions on the planeX20X3, their equation will be

(iB2-BR)dx2+(B3-B)dx3=? (31)where for brevity1i = Bi[xo+ Bo 20-X2) + Bo xo- x3), 2, x31 (32)Any point M(x1,x2,x3) in the plane (29) will satisfy theinequality

x1- x?+ B2 X - X20) + W3x3 X?3 0 (33)for the direction (x1 x1, x2 x1, x3 x) being an indiffer-ence direction for Mo, will be a non-preference direction forM(18). Owing to (29), the last inequality yields

(B2h20))(x2-x2) + (W3-B?) (X03-x3) O (34)which shows that mo(xo2,A) is a saturation point for (31).This is the only saturation point. Indeed, the correspondingconditions

B2-B =? B3-B33 = 0are satisfied only when x2=x?, 3=x0 (13). Since, on theother hand, Mo(x, xA,A0) is the equilibrium point corre-sponding to the budget equation (29), we see that the equi-librium position is one of relative saturation. This is trueof any equilibrium position.


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564 QUARTERLY JOURNAL OF ECONOMICSFrom the condition of stability

(B2, 1dx1+ B2,2dx2+ E2,3dX3)dX2+ (B3,idxl+ 13,2dx2+B3,3dx3)dx3< 0when

dxi+112dx2+s3dx3 = 0joined todxi+B~dx2+bIdx3= 0

we obtain the other necessary condition (20)B2-h2 B3

aX2 (73-Jv?) a70


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PURE THEORY OF CONSUMER'S BEHAVIOR 565of the non-integrable case is made possible. At the sametime, it is seen that the difference between the case involvingtwo commodities only and that of more than two was onlyapparent and due to the fact that the problem concerninga saturation point had not been worked out to a sufficientextent. The impossibility of an analytical construction ofan ophelimity index is a feature that belongs to both thesecases mentioned.

VThus far, we have mainly been concerned with a geo-metric development of the four postulates set out at thebeginning of the paper (Section II), and to this extent thewhole analysis has refrained completely from any inter-pretation of the results obtained. We may now proceedfurther and analyze the implications of the theory of choiceand its standing as a satisfactory explanation of the unique-

ness and stability of exchange equilibrium.According to the postulate A, we can, given a direction(co) and a point Mo(x?), construct a surfaceF. (xi; 4) = 0 (39)

the locus of all combinations u(xi) that are indifferent incomparison to Mo. The subscript w is used in order to em-phasize the dependence of the indifference surface thus con-structed upon the slope of the line cocw'Section II, Fig. 1).The last relation can be written in the formIf (x ; x?)=f -(Al; xi (40)

which shows that Al belongs to (39).So far we have succeeded in constructing a preferentialfield. In this field the result of the comparison of N(yi)with M(xi) is always known. Symbolically we may write

(N= M). if f. (yi; xi) ==f.(xi; xi)(N > M). if f )(yi; xi) >f.(xi; xi) (41)(N


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566 QUARTERLY JOURNAL OF ECONOMICSIn the case of two commodities the curves (p) will all betangent to the integral curves (I) of

dxi+1B2dx2 = 0the point of contact being precisely the corresponding pointMo (Fig. 4). A similar picture can be drawn for the case ofthree commodities, with the difference that the integral sur-faces (I) may not exist.

x,XI))

' '

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FIGURE 4The preceding considerations show that even if the indif-ference elements are integrable, no meaning can be attachedto the integral varieties without introducing further assump-

tions. Two points belonging to the same integral surface donot necessarily represent equivalent combinations. Theintegrability of indifference elements alone does not implythat the individual will be able to tell, independently of anyrestriction, which combinations give him the same satisfac-


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PURE THEORY OF CONSUMER'S BEHAVIOR 567tion.A For the field (co) is in general inconsistent. Indeed,we may find a path going always in a preference directionfrom M to P, and still have M


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568 QUARTERLY JOURNAL OF ECONOMICSThis shows that the tangential element corresponding to anypoint of the variety (,u) is independent of xi. In other words,(,u) and (I) coincide; the existence of an integral-indifferencevariety is thus proved. The equation (40) becomes

A~xi) Ax?) (49)This equation will, according to the postulate D, be inde-pendent of w. It follows also that the transitivity is validwith respect to the preference and non-preference changesand that the comparisons are reversible. In other words, ifM=N, then N =M.

The preceding analysis shows clearly that without thetransitivity postulate the integral varieties and the indiffer-ence varieties are two distinct things. If a point of saturationexists, the indifference varieties will be always concaveclosed surfaces (postulate A), while the integral ones neednot necessarily be so. The much-discussed paradox of thenon-integrable case is due to the confusion of these two con-cepts.The existence of indifference elements satisfying certainconditions expressed by inequalities is sufficient for an expla-nation of the uniqueness and stability of equilibrium in anexchange with constant prices. As R. G. D. Allen haspointed out, the integrability condition is too severe.4 Ishould add and without any meaning outside the transi-tivity condition.VI

The main argument levelled against the existence ofintegral-indifference varieties consists in denying the possi-bility of a mental comparison at a finite distance. This argu-ment was logically questioned because of its failure in thecase of two commodities. But despite this fact, it was gen-erally accepted that a mere introspection is enough to proveour hesitation if faced with a problem of choice. As wasshown in the previous section, this failure was only appar-ent, and the paradox was due to a confusion between the4. The Foundations . . ., p. 223.


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PURE THEORY OF CONSUMER'S BEHAVIOR 569integral varieties and integral-indifference varieties. Thisdifficulty being eliminated, the standing of the argumentmentioned is very much improved.Hesitation, however, does not necessarily mean impossi-bility of choice. We might, for instance, be in doubt aboutwhich one out of two masses is heavier, if we have to judgetheir weight by lifting them only. From this it does notfollow that we are unable to distinguish with certainty dif-ferences in weight.Let us suppose that we have recorded the number of timesan individual decided that A>B in a series of mental com-parisons so conducted as to avoid as much as possible anyhereditary influence from test to test. It is a well knownfact that the relative frequence p of the decision A>B isperfectly stable and that, ceteris paribus, it depends only onA and B. What appears to be haphazard in a single case isno longer so in a series of similar attempts. If we denoteby q the complementary frequency; i.e., that of the casesin which the decision is A


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570 QUARTERLY JOURNAL OF ECONOMICS

yo y Y1 X(a)

YO y x txldx Y1 x(b)

FIGURE 5

jf(y, x)dx=J f(y, x)dx (52)and

P(Y'Y)=q(y, 1) (53)2The last relation constitutes the definition of physicalequality. It is the basis of all means of detecting the equalitybetween two quantities.The interval (yo, Yi) is known as the psychological thresh-old. Outside of it, there is no doubt as to which stimulusis greater. The length of the interval (yo, yi) will dependupon many causes, but the most important factor in thisrespect is the interval of time the individual is allowed to


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PURE THEORY OF CONSUMER'S BEHAVIOR 571perceive the two stimuli before formulating his judgment.The greater this interval of time, the smaller will be thepsychological threshold. At the limit when the time ofexperimenting is infinite, the threshold is zero. The latter isessentially the basis of postulate A. Its use in a static anal-ysis is justified by the very characteristic of such a system,which is that of being independent of time. A dynamicsystem could not be analyzed on the same basis. The indi-vidual's doubt as to which of two combinations he willfinally choose would have to be taken into account andpostulate A modified accordingly.

Let us adopt now the following postulate:A1. If the individual has already experimented during acertain interval of time with two combinations of commoditiesM1(xi), N(yi) there is a probability expressed by the functionco(xi;yi) that he will consider the combination M(xi) preferableto N(yi).For obvious reasons X&iwill satisfy the following relationCl)Xi;piJ+ c)(yi; xi) = 1 (54)The only conditions to be imposed for the time being uponc) is that-i > 0 if l)


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572 QUARTERLY JOURNAL OF ECONOMICSXi i

n

FIGURE 6

within this angle, the individual might be a seller as well asa buyer of either of the two commodities involved (Fig. 6).As the price line rotates from the position II' to JJ', xriwill decrease monotonically from 1 to 0, while W2will increasein the same manner from 0 to 1.6 It follows that all thecurves

o (xi; ) =7r (O< r< 1) (56)6. The question of measuring the physical quantities of goods is left,in the present scheme, to some outside scales to which the individualrefers. He knows that x < x' by using the scales and not as a consequenceof his feelings of the equal or different degrees of utility he can derivefrom the consumption of x or x'. The individual's preference for x+dxin comparison to x is justified, that, altho the individual will not feelthe addition of 1/100 apple daily, he will certainly appreciate an incre-

ment of three apples yearly. The individual's behavior appears there-fore as a resultant of two different types of measurement: a physicalone, which is supposed to tell him the exact amounts of commodities,and a psychological one, which is his possibility of comparing satis-factions. The fact that these two kinds of measurements are bothinvolved in the present scheme constitutes an important point in theproblem.


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PURE THEORY OF CONSUMER'S BEHAVIOR 573xi varying, will pass trough N(yi), and consequently &i(xi;yi)is indeterminate, for xi=yi. The value of 6i(yi; yi) willdepend upon the path the point M(xi) follows toward N(yi).If we keep x2 constant and let only xl vary, the variationof 6(xi; yi) will be represented by a graph similar to that ofFig. 5a; i.e., 1i will be a monotonically-increasing functionfrom 0 to 1. The probability that, trying to find on thestraight line x2= const. an equivalent combination toN(yi), the individual will choose a point between xi, andxl+dxl, is (51)

Ol(x1,X2; Yi, Y2) dxj=f(xi, X2; y', y2)dxi (57)ax,If we now perform the same operation, but keeping x1constant and letting x2 vary, we shall have instead (mutatismutandis) the probability

4i(Xly, X2;YlyY2) dx2=g(x1, x2; y', y2)dx2 (58)dx2The same considerations apply to any path along which

Z is monotonically increasing. If (-y) is such a path, thenthe probability analogous to (57) or (58) will be expressedin terms of a directional derivative,7 i.e.,ds= (f cos a+g sin a)ds (59)

The probability that the position equivalent to N will beon the path (-y) below8the point M(x1, x2) of this curve willbe, according to (50),L) ds=C5(xi; yi) (60)

If the path is a straight line nNn' passing thru N (Fig. 6)Xi-yl+p(X2-Y2) = 0 (61)

the formula (59) will be valid for the separate directions7. See W. F. Osgood, Advanced Calculus, New York, 1933, p. 143.8. By a point below M is meant here any point for which thecorresponding value of Wis smaller than Pi(xi; yi).


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574 QUARTERLY JOURNAL OF ECONOMICSnN and n'N. In this case, the probability that from N theindividual will move to any point on Nn or Nn' will berespectively

ll= lim U(y1-ds cos a, y -ds sin a; yi, Y2)ds=O (62)IH'=lim W(y1+ds cos a, Y2+ds sin a; Yi, Y2)ds-So

The directions Nn, Nn' will consequently be tangentrespectively to the curves ii(xi; yi) =11, -i xi; yi)=I1'. Theprobabilities H and I' will then be identical with r1 and 7T2defined above in connection with the price line nn'. Thecurves (56) will present a cusp point in N, except in the case__1where 7T = 732= 2

By joining two branches of the curvesi3(xi; ji) = 7T Z5(xi;yi) = 1-7T (63)

we form two curves that have no longer any irregularity inN (Fig. 6). We shall refer to such curves as (ir) curves,where (ir) is the greater of iT and 1- 7T. We shall also intro-duce the term limiting curves for designating those forwhich (iT) = 1.The angle of indifference (II', JJ') can be represented bya quadratic differential equationD= flldx2i+ 2 p12dx&dx2+ p22dx2 0 (64)(Pll (12, (P22 being given functions of x1, x2. This equationmay be regarded at the same time as the differential equa-tion of the limiting curves.By a suitable choice of signs in (64), the conditions im-posed upon the functions up, an be written

A = f11P12 2-t'12< O (65)'P11, 'P12, p22> (

In order to determine the nature of a given direction it isconvenient to decompose the form D into linear factors


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PURE THEORY OF CONSUMER'S BEHAVIOR 575= Pi(dxl+aldx2)(dxl+a2dx2) (66)where a,, a2>0.9

The direction Ax1, Ax2will be an indifference, preferenceor non-preference direction according to whetherAx1+ajAx2>O Ax1+a2Ax20 Ax1+a2Ax2> 0 preferenceAx1+ a1jx2


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576 QUARTERLY JOURNAL OF ECONOMICSdirections (Fig. 7). The analytical formulation of theserestrictions is

d 0 when dxi+aidx2=O if dxl+a2dx2< 0dcb


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U1=_+ 2=(1 log)22 -2Q(P12) (70)a2 a2 IOX2 (011 (9x, (011

U2= 21+E2= _( ogfll -2a 8(12)al a2 &IX \ 022 aX2 em22and further

| UO Ul | I tEE (71)U1 U2 EE1, a2

ButUo= 2 u.U-22 U2 (72)

'P11 'P11and introducing this last relation in (71) we get

1 2sPii - 2 012 U1U2 + 22022=- |* ,E2 (73)

a, a2It is easily seen now that the conditions (67) may bereplaced by Ul


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578 QUARTERLY JOURNAL OF ECONOMICSThe curves A, and A2 are, according to the assumptionsmade, negatively inclined (65) and convex toward the co6rdi-nate axes (67).Two curves belonging to different families cannot inter-

sect more than once. Indeed, if they intersected in m and n,we should have (j2) > $X2) in one point anddxj a,1 dxj a2(&2) < dX2 in the other. But this would mean thatUdxi al _dxi a2there are points for which

(dx2)(dx)Udxi~al \dxl af2which is in contradiction with our main condition (65).As on the other hand, through each point N(y,) passesonly one curve of each family, such curves divide the planeinto four regions, the meaning of which is evident (Fig. 6).Owing to these properties of the curves A, a one-to-one

C d

0LFIGURE 8


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PURE THEORY OF CONSUMER'S BEHAVIOR 579correspondence can be established between the points of theplane X1 OX2 and those of the plane aiOa2 (Fig. 8). To eachpoint M(xi) in the former, there will correspond an imagem(a,, a2) in the latter. In this last plane, the straight linesal-const. and a2= const. will correspond to the curvesAl, A2. Taking a,, a2 as new variables (66) becomes

D= (a,,a2)dada2 (p1>0) (78)The indifference directions being characterized by I'0, and consequentlythe latter can lead only to some point within the quadrantamd. Similar reasoning can be applied to a path going alwaysin a non-preferred direction (Fig. 8). The inequality ofsatisfaction is transitive.The equality between different combinations is, in general,not transitive. Thus from

m=n n=pit follows that m=p (Fig. 8); but, although n=p,' p' ispreferred to m. In other words, the equality between differ-ent combinations will be transitive if they can be connectedby a path going always in an indifference direction and suchas along it the direction of the movement, d2-, be continu-dxious in all points. If this is not true, we cannot tell whetherthe transitivity is valid or not.

Consequently fromM(xi) =N(yi) N (yi) = P(zi)if xi>Ys zi, we can always conclude that M=P.

The lack of transitivity with respect to the indifferencesituation is easily explained. What we really mean by MI N


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580 QUARTERLY JOURNAL OF ECONOMICSis that there is a probability co for M to be considered prefer-able to N. The comparisons between M and N, N and P,and M and P are three independent phenomena. There isno way of establishing a relation between the probabilitiesconnected with each of them.

* * *An equation similar to (64) will define any curves (Xr)

01ndX2+2ji2dxidx2+ J22dx2 = 0 (79)where 4in, P12, 422 are functions of x1, x2,AT,atisfying the con-ditions011022-%12A have the same probability equal to 2

According to the principle of decreasing marginal rate ofsubstitution the curves (79) will have to be convex towardthe coordinate axes and consequently the functions 4 willhave to satisfy relations analogous to (76). Each curve (X)is divided by any curve of complete indifference into twoparts, a superior and an inferior one. All the combinationssituated on the (ir) curve passing through N will be preferredto N with the frequency (Xr) if they belong to the superiorpart, with the frequency 1-(Xr) in the other case.

* * *I shall proceed now to examine shortly the consequenceswhich the scheme described in the present section has for theequilibrium of exchange with constant prices. Let us con-


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BN

QIS

Mo A e

FIGURE 9

sider an individual whose income is x4and who can obtainone unit of X2 in exchange for p units of X1. The propertiesof the limiting curves as outlined above lead to the followingpattern:As the individual starts to move on the budget lineX1 =X1+pX2

from A towards B (Fig. 9), he will - if AB is not an indiffer-ence direction for A - move for a certain interval in a prefer-ence direction. There will be on AB a point S(e1, e2) forwhich SE is a limiting direction between the preference andindifference regions, and further - if BA is not an indiffer-ence direction for B - another point T(T1, T2) for which TBwill be the limiting direction between the indifference andnon-preference regions. Between S and T there will be onlypoints for which AB is in both senses an indifference direc-


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tion; AB will be a non-preference direction for any pointsituated between B and T. This shows that any pointbetween S and T might be an equilibrium point and thatthe system defining it isxi=X +pX2

P25Pll-2p P12+ P22


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PURE THEORY OF CONSUMER'S BEHAVIOR 583established in Q'. The function P will be defined by theintegral equation'

P(x2)= JP(Y2)k(Y2, X2)dy2 (82)But this law of probability could not be used for a deter-mination of the final equilibrium position in a single bargainany more than we can use a mortality table to determinehow long a given individual will live.In fact, P(X2)dX2 s the probability that under the givencircumstances the quantity of X2 demanded will lie between

X2 and X2+dX2. The demand law can no longer be picturedas a curve relating prices and quantities, but as a bivariatedistribution between the same variables.Let us denote by Pi(x/p)dx the probability that the quan-tity demanded at the price p by the individual (i) lie betweenx and x+dx and let si(p) be the variance of the distributionthus defined. The probability connected with the collectivedemand of n individuals is easily worked out in terms of Pi.Let P(X/p)dX be the probability that the market demandat the price p be comprised between X and X+dX. We haveP Xp)dX=JP1(X1/P)P2(X2/P) .. Pn(x/p)dxidx2 . .. dxthe integration being subject to the condition

Xl+X2+X3+ * * * +Xn=XIt is obvious that the range of indeterminateness of themarket demand is equal to the sum of the correspondingranges of the individual demands. The variance of P is,according to a well-known theorem

S=sI+s2+ * * * +Snand consequently the larger the market the greater will bethe variance of its demand.Only relative values, like the relative deviation (X -X being the mean value of X, or the quantity demanded

1. G. Darmois, Analyse et comparison des series statistiques quise d6veloppent dans le temps, Metron, vol. viii, No. 1-2, 1929,pp. 238-239.


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584 QUARTERLY JOURNAL OF ECONOMICSper capita X/n will present a smaller variance as the sizeof the market increases.2The transformation of quantitative data into some kindof relative values as the preliminary step of a statisticaltreatment of such series is thus shown to be not only justi-fied by also necessary.

VIIThe aim of this last section will be both to summarize themain results reached thus far - presenting them at the sametime in a more concise form - and to advance further con-siderations upon the bearing of some of these points on theexchange problem.1. Four main points have been shown to affect any analyt-ical construction of a theory of choice. Each of these pointspresenting two alternatives, there will be as many versionsof such a theory as there are consistent combinations of thepostulates to which these alternatives give rise. Among the

consistent aspects of the theory we find the theory of utilityvarieties of Edgeworth as well as the theory of choice ofPareto.Pure analysis can do no more in this direction. Whichform of the postulates A, B, C and D we have to accept is aquestion that can be decided by actual experiment only.Some of these postulates, however, namely C and D, are veryunlikely to lend themselves successfully to such a treatmentThis fact is rather discomforting, for the uniqueness of theindifferenceelements - and therefore that of the equilibriumof exchange - depends precisely on these points. Yet notall the roads are completely barred. The question as towhether the indifference combinations are transitive or notplays an important role regarding the conclusions of such ananalysis. It seems that this point could be easily submittedto an experimental verification. We should really lose allhope in this direction only if the answer to such an investiga-

2. The variance of (X-X)/X which is o-=S/K2 will, under verygeneral assumptions, decrease as n increases. The variance of X/n isS/n2 and will also decrease as n increases if si has an upper limit.


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PURE THEORY OF CONSUMER'S BEHAVIOR 585tion should be negative. For, as has been shown (Section V),while the transitivity confirms the postulates C and D, fromthese it does not necessarily follow that the indifferent com-binations have this special property.3

Nevertheless, many significant formal considerations couldbe advanced as to the standing of these postulates.Once the invariance of the tangential element in M to thevariety (p,) in respect to the variation of w is accepted, Icannot think of any objection compatible with the spirit ofthe static pattern which could be leveled against the extensionof this invariance to the entire variety. The postulates Cand D are sufficient for an analytical explanation of theuniqueness of exchange equilibrium, but they fail to allowthe construction of a consistent picture of human behavior.Can the illusion described above (Section V) be inter-preted? We have been willing to consider - and later toembody in the theory - the impossibility of accurate mentalcomparison at a finite distance, on the mere grounds thatanybody can verify this assumption by a simple introspec-tion. But this type of argument may be reversed. Thus,why should we not reject a scheme which everybody findsparadoxical? One has to realize that what we aim at is atheory molded on a type of individual that really exists andnot on a necessary and sufficient one.

3. So far as I know, there has been only one attempt at an experi-mental investigation of the nature of the indifference curves. Cf.L. L. Thurstone, The Indifference Function, The Journal of SocialPsychology, vol. ii, May, 1931, p. 139.Professor Thurstone's experiment is, however, very unlikely to helpus in deciding anything about the forms of the postulates here analyzed.The investigation having been carried out by way of questions andanswers, we cannot be sure whether the prices ruling on the market atthe time of the experiment had or had not influenced the subjects intheir answers. Some of the diagrams in Professor Thurstone's paper,namely 13 and 17, suggest on the contrary that they had.Besides, the result of mere visualization cannot be relevant to atheory concerned with an actual choice, unless the combinations usedin the experiment are those with which the subject is familiar becauseof his latest experience. This last condition restricts the range of theexperiment to a degree which simply makes the investigation useless.It seems that we cannot avoid the necessity of letting the subjectexperience the satisfaction before making his choice.


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586 QUARTERLY JOURNAL OF ECONOMICSThese considerations might, in extremis, be simply ignoredin economics, were they confined to the explanation of theexchange process only. But they are encountered again andagain under the same form in connection with items no lessimportant, such as the marginal utility of money, priceindices, individual and social welfare, etc. A consistent theoryof choice forms a basic part of economic science.2. The necessary and sufficient analytical conditions forthe stability of exchange equilibrium have been establishedin their correct form, and the integrability of indifferenceelements has been shown to be necessary for their resolution

into the principle of decreasing marginal rate of substitution.The geometric interpretation of these conditions is thatthe curvature of the integral varieties - whenever theyexist - is at any point greater than that of a linear variety,i.e., greater than zero. It is worth emphasizing here that theconditions thus obtained are valid only for an exchange withconstant prices. Their form will be entirely different if thebargain is not perfected on constant price basis, i.e. if theindividual supply curve is not perfectly elastic. In the lastcase the budget equation will no longer represent a linearvariety. The stability conditions will require that the curva-ture of the integral varieties be greater than that of thevariety represented by the budget equation. The last con-dition is the general one.3. The analysis also showed the necessity of distinguishingbetween indifference and integral varieties. The confusionof these two concepts accounted for the paradox of the non-integrability case. From the mere existence of the integralvarieties we are not entitled to deduce that these representthe loci of constant ophelimity. The integrability of theindifference elements has less to do with the existence ofconstant ophelimity varieties than the condition of transi-tivity has. This argument is intuitively illustrated by thespiral-like integral curves (Section III).4. Finally, a scheme derived from a different version of thepostulates A and C was proposed and analyzed. The maincharacteristic of this scheme is the existence of a threshold


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PURE THEORY OF CONSUMER'S BEHAVIOR 587in the comparison of satisfactions. As a direct consequence,a certain range of indeterminateness was shown to exist inconnection with the equilibrium of exchange. The demandlaw appears, in this case, as a multi-variate distributionbetween prices and quantities, or, to use a term introducedby Professor Taussig in connection with a similar problem,as a penumbra.4From this scheme, by assuming the interval of timeallowed for the mental comparison to be infinite, we obtainas a limiting case the classical static scheme formulated interms of utility varieties (Edgeworth). In a dynamic analysisthis last supposition can no longer be accepted and such ananalysis will have to drop the assumption of a demand lawbeing a rigid connection between prices and quantities.The penumbra of demand makes way for an explanation ofmany facts that are observed in actual markets. In this waywe may account for certain aspects of the imperfection ofmarkets.

The general means used by the entrepreneurs for inducingpeople to buy more of one commodity - which necessarilyimplies less of some others - might be separated into twogroups: those that aim at influencing the position of thebuyer within the existing penumbra and those that seek tomodify his tastes. The immediate task of the salesman,special sales, loss leaders, etc. belong to the first group. Thesecond group consists mainly of advertising on a large scaleand for a long time, in other words, consists of a real propa-ganda to convince buyers of the advantages of consumingmore of the advertised commodity.Staple commodities, like meat, bread, milk, etc., that aremore regularly consumed and consequently experimentedwith longer, will naturally present a smaller threshold andthus a smaller penumbra of demand. This seems to be incomplete agreement with the fact that the first group ofactions on the part of the seller is practically absent in themarketing of these commodities. Such actions will be espe-

4. F. W. Taussig, Is Market Price Determinate? , Quarterly Jour-nal of Economics, vol. xxxv, May, 1921, p. 394.


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588 QUARTERLY JOURNAL OF ECONOMICScially important and conspicuous in the case of those com-modities whose use is less frequent.There is thus a sense in which two different commodities,not necessarily competitive, can be considered as competingagainst each other in the short run as well as in the long.

MATHEMATICAL NOTEON THE CONDITIONS OF STABILITY OF EQUILIBRIUM

IN AN EXCHANGE WITH CONSTANT PRICES1. The condition of stability is (9)

Xi zkBi,k'AXiAXk< ? (i)2 2subject to (7)

Ax,+ IiBiAxi = ? ii2Eliminating Ax1from these two relations we getF = 1i2k(Bi,k-Pi,n 1Bk) AXiAxk< 0 (iii)2 2

F is a homogeneous quadratic form in n-1 variables Ax2,IAX3,.. Ax, which may be written under the form

F=2 1 i 1kfik-AXiAXk- (iv)22where

fik=fki=JBik+PkI-BiilEk-Bk,1Bi (v)Except for a constant factor of proportionality, the dis-criminant of the form F is

f22 f23 . . . An

| =f32 f33 . . . f3n(ifn2 fn3 . . . fnn

The necessary and sufficient condition for F to be definite


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PURE THEORY OF CONSUMER'S BEHAVIOR 589and negative (iii) is that the principal minors of IF I shouldbe alternatively negative and positive:'

s 6 : ~f22 ,f23(Vif22 I< 0;

f223>0;...- (vii)f32 f3 3

By bordering the determinant (vi), F may be written0 1 B2 * Bn1 0 O* *0

F l B2 0 (viii)fik

Bn ?

By adding to the elements of third, fourth, . * (n+l)throw those of the first row multiplied respectively by B2,1,B3,1, . .. BnRl and by performing a similar operation inrespect to the columns, we obtain, according to (v),

0 1 B2 B3 ... Bn1 0 B2, B3,1 . . . Bnxl

F - B2 F2,1 2F2,2 B2,3+B3,2 * B2jn+RBn2 (ix)

Bn nil Dn.2+Th2jn Bnj3+B3n . . 2RnnSince, thru the same procedure, we obtain

0 1 B2f22 |= 1 0 B2,1

R2 132,1 2B2,25. See, for instance, T. J. I'A. Bromwich, Quadratic Forms and TheirClassification by Means of Invariant Factors, Cambridge Tracts, No. 3,p. 19.


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0 1 B2 B,3f22 f23 1 0 1B2,1 B3,1fn f33 B2 B2,1 2B2,2 B2,3 IB3,2B3 B3,1B3,2+B2,3 2B,33

the conditions (vii) are shown to be equivalent with (11)i.e.,0 1 B2 0 1 B2 B31 0 B2,1 >0; 1 0 By,1 B3,1 fik (

2n-1 (Xx1)?.fikiBnY1 ny2. . . _'nyn

Returning now to the stability conditions (vii) or (xi), wesee that if these are satisfied, from (xxxi) it follows that weshall also have1 B2 1 B2 B3< 0; B2,1 B2,2 B2,3 >0; . . . (xxxii)B2,1 B222 B3,1 B3,2 B3,3

but that the reciprocal proposition is not true.The two sets of conditions are equivalent when and onlywhen all 'M O. But this cannot happen unless gik =0.Indeed, the quadratic formG= gi kgikAxi AXk (Xxxiii)2 2is transformed by the linear substitution (xii) intoG= 2r 2sv3rsYrN (xxxiv)2 2

and if the last one is identical null, so must be the formerThe conditionsgik = Bk-Bki+Bk,~iBi- 13,Bk =0 (xxxv)

are easily recognized to be those of the integrability of thetotal differential equationdxl+ 2dx2+B3dx3+** +Bndxn = 0 (xxxvi)

and thus we reach the conclusion that the sets of conditions(xi) and (xxxii) are equivalent only when the differentialequation of the indifference elements is integrable; otherwisethe latter is a consequence of the former but not vice versa.This differencewill therefore not exist in the case where n = 2.

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