Demand Theory of the Weak Axiom of Revealed Preference

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The Demand Theory of the Weak Axiom of Revealed Preference

Richard Kihlstrom; Andreu Mas-Colell; Hugo Sonnenschein

Econometrica, Vol. 44, No. 5. (Sep., 1976), pp. 971-978.

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Econornetrica, Vol. 44, No. 5 (September. 1976)T H E D E M A N D T H E O RY O F T H E W E AK A X IO M O FR E V E A L E D P R E F E R E N C E

BY RICHARDKIHLSTROM, NDREUMAS-COLELL,ND H U G O SONNENSCHEIN'In this paper we provide a statement of the relationship between the weak axiom ofrevealed preference (WA) and the negative semidefiniteness of the matrix of substitutionterms (NS D). As a corollary we determine the relation between WA and the strong axiomof revealed preference (SA). Th e latter is equivalent t o N SD and the sym metry of thematrix of substitution terms. The former, WA, implies NSD but is not implied by NSD.Also, WA is implied by the condition that the matrix of substitution terms is negativedefinite (ND), but it does not imply ND. Application of these results yield an infinity ofdeman d functions which satisfy WA but n ot SA.

INTRODUCTIONTHEPURPOSE OF THIS PAPER is to provide a final step in the solution to a problemwhich has been central to the foundation of consumer demand theory sinceSamuelson's introd uc tio n of the weak axiom of revealed preference in 1938 [ l l ] .Acharacterization, via classical restrictions, of the set of demand functions whichsatisfy the weak axiom is obtained .Demand functions have as their arguments an 1-dimensianal price vector pan d a wealth v ariable w.' They assign to ea ch p an d w a n 1-dimensional vector ofcom mo dity amounts, called a dem and. We assume that dem ands are nonnegativeand satisfy the b alance condition that for each (p ,w), the inner product of p withthe de m and a t (p, w) is w. In add ition , we will assume th at dem and functions ar ehom ogen eous of degree zero in (p, w). It is a classical result (Joh nso n [9] an dSlutsky [15]), th at dem and fu,nctions w hich are gen erated by maxim izing a quasi-concave (smooth) utility function satisfy the conditions of negative semidefinite-ness (NS D ) an d symm etry (S) of a m atrix of com pen sated price derivatives (sub-stitution term^).^ Th e converse theorem is also classical in bac kgrou nd (Antonelli[I]) and b oth a re central to the dem and theory presented in Samuelson's Foun-dations of Economic Analysis [12].4In 1938 Samuelson [ l l ] proposed a new foundation for the theory of consumerbehavior. He defined the direct revealed preference relation R by xRy if x is thedem and in a situation where y # x can be afforded, an d to ok as his centra l axiomthe a sym metry of R (called the weak axiom of revealed preference). Sam uelson leftunanswered the q uestio n of whether his weak axiom was equivalent t o the utilityhypothesis ; i.e., the assum ption th at dem and is generated by maximizing a quasi-concave utility function. His efforts yielded the conclusion that the weak axiom

Most of the results in this paper were obtained in the sum mer of 1973 at a w orksho p at Amherst,Massachusetts. The authors wish to thank the Mathematics Social Science Board and the NationalScience Foundation for their support.Static Demand Theory by D. Katzner [lo] provides an excellent introduction to demand theory.


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972 R . KIHLSTROM, A. MAS-COLELL, A N D H . SONNENSCHEINimplies N S D ; however, he seemed aware that S was proba bly not im plied by hisaxiom , except for the special case of two com mod ities.Th e question of a possible equivalence between th e weak ax iom a nd the utilityhypothesis remained unansw ered for over twenty years before an exam ple du e t oGale [4] demonstrated that there exists a demand function satisfying the weakaxiom which can no t be generated by utility-maximizing behavior. In the m eantimeHouthakker [6]show ed th at the acyclicity of R (called the strong axiom of revealedpreference) is equivalent to the utility hypo thesis. Still, the question of the preciserelationship between the weak axiom and the strong axiom (or equivalently theutility hypothesis) has remained open. Because of Gale's example they are notequivalent. Uzawa [17, p. 1331 introduced a condition under which they areequivalent; however, it is marred by the fact that it looks very much like thestrong axiom itself.In this paper we provide a definitive statement of the relationship between theweak axiom and NS D. As a corollary we determ ine the relationship between theweak and strong axioms. Th e strong axiom is equivalent to N S D and S. Th e weakaxiom implies NSD but is not implied by NSD; however, the weak axiom isimplied by the condition that the matrix of substitution terms is negative definite(N D ). Application of these results yields a n infinity of G ale-type examples andproves tha t a n an alog of N D for excess dem and functions, since it implies theweak axiom , is sufficient for the stability of com petitive equilibrium .

THE THEORYLet V = P x (0 , co) be an open convex co ne of positive vectors in R 1+ '. Th efirst 1 coordinates of points in V are interpreted as price variables and the re-maining coordin ates as an income variable. Fo r this reason a generic element of

V is den ote d by (p, w) where p E P and w > 0. A demand function h : V -+ R' is acontinuously differentiable function which satisfies the following: (i) for all@, W ) E V, h(p, w) 3 0 ; (ii) for all (p, w) E V, for all A > 0, h(p, w) = h(Ap, Aw)(homogeneity); (iil) for all (p, w)E V,p .h(p, w) = w (balance).Given a V, A(h)(p, w) will de no te the 1emand function h and (p, w ) ~ x 1matrix with generic entry

It follows from (ii) an d (iii) tha t p lA(h)(p,w)p = 0 for every (p, w) E V. Define alsothe relation R by xRy if x = h(px, wx), x # y, and px. y Q wx. Som e conditionswhich may be satisfied by a d em and function a re now stated. The first three involvecon dition s on derivatives, while the rema ining three a re finitistic.


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973EAK AXIOM OF REVEALED PREFERENCECONDITION (N D ) : Fo r every (p ,w )E V, A (h ) (p ,w) is negative definite on T,;i.e., for a ll v E R1,v # 0, vlA(h)(p , )v < 0.5CONDITION(S ) : Fo r every (p ,w )E V,A(h)(p ,w ) is a sym metric ma trix.CONDITION ( W W A ): For every @, 8), @, G )E V, if h(p,8 )# h(p, G ) andp . h(p, G ) < 8 , then p . h@ ,G) > w . ~CONDITION( W A ): R is asym metric o r, equivalently, for every @, 8 ) ,@, G )E V,if h(p,8 )# h@, G ) and p . (p,G ) 6 8, then p . h(p,8 )> G (weak axiom).CONDITION( S A ):R is acyclic (strong a xiom).Obviously, N D implies N SD and W A implies W W A .Wewill show th at W W Aimplies NSD (compare with Samuelson : W A mplies NSD [12, p. 11 ]), N Dimplies W A , and N SD implies W W A . In add ition, we will show th at W W A doesnot imply W A .LEMMA1 : Assume that h is a demand function, (p a,wo)E V, v E R 1, pa + v E P ,and xo = h(po,wo).De jne p( t) = pa + tv , and ,for ail vectors p E P de jn e E(xo,p) =h(p,p .xo) . Then

and(2 ) lim l ( l l t 2 )W - A t ) ). 5(x0 , O ) - 5(x",p(t)))- 1 ai j (h)(p0,w0)vi~jl= 0 .t + O j iIf, in addit ion, h satis jes W W A , hen

PROOFStatement (1) follows from a chain rule differentiation an d th e definitionof a i j ( h ) W , o).A variant of ( 2 ) is implicit in [lo, p. 1101, and is attributed toL. Hurwicz. It is established as follows.Since each 5' is continu ously differentiable an d Ilp(t)- pa 1 1 = t llv 1 1 ,

In view of the fact that p8A(h)(p,w ) = 0 and A(h)(p,w )p = 0 for all ( p , w )E V , N S D is equivalentto " A ( h ) ( p ,w ) is negative semidefinite on R'" and ND is equivalent to "u'A(h)(p,w)v < 0 for all v # 0,v/llul/ # p/llplln. The latter is the definition of ND given in an earlier version of the paper; however,T. Rader pointed out to us that it leads to some difficulties in exposition which are avoided by thepresent formulation.


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974 R. KIHLSTROM, A. MAS-COLELL, AND H. SONNENSCHEINSince tv , = pi(t) - pp, multiplying by Ilvll(pi(t)- pp)/t yields

i = l

and summing over i together with ( 1 ) complete the proof of ( 2 ) .Statement ( 3 )combines p ( t ) . ( < ( x O ,O)- l ( x O , ( t) ))= 0 from the definition of 5and pO. (4 (x0 , O)-


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975EAK AXIOM OF REVEALED PREFERENCETheorems 1 an d 2 com bine to yield the result that a de m and function h satisfiesN D if an d only if there exists a neighb orhoo d of h such that all dem and functionsin that n eighbo rhood satisfy WA.' Thus, N D is an o pen infinitesimal a nalo g of

WA . This fact mak es it very easy t o exhibit de m and functions which satisfy WAbu t can not be ge nerated by utility maximization. To achieve this take any d ema ndfunction (for at least three comm odities) which satisfy N D and S. All dem andfunctions sufficiently close to h (and equ al t o h outside a com pact set of prices)will satisfy N D bu t m ost will not satisfy S. Hence, they mu st satisfy WA bu t can -not come f rom utili ty rn ax im i~ at io n .~pplying Theorem 2 in an approximationargum ent yields the following theorem which, together w ith Theo rem 1,establishesthe equivalence of N S D and WW A.THEOREM: If the demand function h satisfies NSD, then it satisfies W W A.PROOF:Assum e tha t h satisfies N S D but th at WW A is violated. Then for som e

@?6) , (p, 3 ) E V we h ave h(p, 6 ) # h(p, %), a . h(p, W) < G, and p . h e , 6 ) 6 $3. Re-placing $3 by w' > %, close enough to %, we have h@,6) # h(p, %), p . h(p, w') < 6 ,and p . h@, 6 ) < w'. Therefore, w itho ut loss of generality, we can a ssume h@,%) #h@, W), p h(p, W) < %, and p .h@, 6 ) < %,Let V' c V be a n open convex cone such th at (p, 6) , (p,%) E V' and any limitpoint of V' different from 0 is contain ed in V .Let f :Rf\{O) -+ R be a continuouslydifferentiable function which is homogeneous of degree one and such that forevery p E R1\{O), D' f (p), the Hessian matrix off at p, satisfies the conditionV'D'f (p)v < Ofor every v G R', v # 0, vlllvll # pIIIpII.(Forexample, letf (p)= - llpll.)F o r sufficientlysmall i j > 0, if g < i j the function h, :V' + R' given by h,(p, w) =h(p, w - gf (p))+ gDf (p) is well defined. It is immediately seen to be a demandfunction ; i.e., to satisfy (i), (ii), an d (iii). M oreover,and so h, satisfies N D an d, by Th eorem 2, also W A. H owever, for g sufficientlysmall, p . h,(p, W) < 6 and p .h,@, 6 ) < 3,a contradiction. Q.E.D.

If h satisfies S, then it is know n t ha t h ca n be locally integrated.'' Using thisfact Hurwicz and U zawa [8] show tha t if h satisfies S in addition to N S D then hmust satisfy WA. Since we have already demonstrated that WWA is equivalentto NSD, it follows that given S the conditions WWA, NSD, and WA are allequivalent. Since we are not concerned with integrability theory we will notpursue the above line; however, we will demonstrate by example that, in theabsence of S, WA an d W WA are not equivalent. In particular, the example showsBecause of the openness of the domain, neighborhoods should be defined by the Whitney topology.The idea of perturbing a demand function which satisfies ND and S to obtain a demand functionwhich satisfies WA but does not come from utility maximization was employed by Gale [4] (and isattributed by him to Samuelson). Since the question of the equivalence of the weak axiom and the


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976 R. KIHLSTROM, A. MAS-COLELL,ND H. SONNENSCHEINthat WWA cannot be replaced by WA in Theorem 3, and thus the theorem cannotbe strengthened.

: Let h :(0 ,XAMPLE + R3 be defined by

The function h is not positive valued, but this is inessential for our purpose. Itsatisfies NSD (hence, WWA) since for every ( p , w ) E V,A ( h ) ( p ,w ) is skew-symmetric,and this implies vlA(lz)(p,w)v = 0 for every v E R'. ~owkver ,h does not satisfyWA: consider ( p ,@) = ( 1 , 1 , 1 , 1 ) and ( p ,E) = (2 , 1 , 1 , 2 ) .We conclude with a discussion of the corresponding theory for excess demandfunctions. An excess demand function f : P + R' is a continuously differentiablefunction which is positive homogeneous of degree zero ( f ( A p )= f ( p ) for allp E P and A > 0 ) and satisfies the balance condition that p .f C p ) = 0 for all p E P.Note that if 1 2 : V -+R' is a demand function and w is a positive vector in R', thenh(p ,p .o)- o defines an excess demand function. The conditions analogous toND and WA (for brevity we shall not be concerned here with NSD and WWA)are the following : (ND') for every p E P, D f ( p ) is a negative definite matrix onT P n { v ~ R i : v . f ( p ) = O ) ,.e., if v # O , v . f ( p ) = O , v T p , then v ' D f ( p ) u < 0 ; 1 2(WAe) or every p, p E P , iff @) # f @) and p .f ( p )6 0 , then p . f @) > 0 ,

The following result is an analog of Theorem 2 and is proved in a similar way.THEOREM: I f f is an excess demand funct ion which sa ti sf ies ND e, then f sa t i s jesW A e .

As an application, let f be an excess demand function and consider the dif-ferential equation p = f ( p ) . This defines a tgtonnement price dynamics. Arrow-and Hurwicz [3] have shown that iff satisfies WAe, then f is globally stable. Com-bining this fact with Theorem 4 yields the result that iff satisfies NDe andf(j)0 for some p, then p is a globally stable equilibrium.13 This strengthens aresult proved by Arrow and Hurwicz [2 ,Theorem 41 since our hypothesis requiresnegative definiteness of Df (p) only on a proper subset of p.14' The function h can be altered to have position values on an arbitrary compact subset of its domain.We do not know if there exists a function with the properties of h defined on (0, wI4.l 2 Since p'Df (p)= -f(p) and Df (p)p = 0 for all p, NDe is equivalent to "if cf (p)= 0, ti # 0,

till1oll # plllpll, then o'Df(p)v < 0".l 3 Let g:R" 4 Rn be a continuously differentiable function with g(0) = 0. Hartman and Olech[5 , p. 5481 have established that the following condition is sufficient for the asymptotic stability of

,t = g(x) at x = 0: for all 0 # v E R", x E R", if v . g(x) = 0, then v'Dg(x)v < 0. Notice the strikingsimilarity between this condition and NDe. This suggests that the weak axiom WAe is the finitisticequivalent of the Hartman-Olech condition.l 4 Our analysis may shed some light on the question of whether every economy with sufficientlysimilar individuals will satisfy WA' and thus generate a globally stable excess demand function. Sincearbitrarily small perturbation of a given excess demand function which satisfies WA' cannot beguaranteed to satisfy WAe, there is some problem in obtaining an affirmative answer to the question.


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977EA K AXIOM OF REVEALED PREFERENCEA RELATED CONJECTURE(with W ayne Shafer)

This concerns the theory of the "nontransitive consume r". Let W be a relatio,no n R (the closed positive orthant of R'), T be defined by xWy and not yWx,W(x) = {x' E R :x' Wx), W - ' (x) = {x' E R :x W x'). T he following three axiomsprovide the founda tion for the theory of the nontransitive consum er:

AXIOM1 : W is strongly connected.AXIOM2 : F or all x, y, z ER, X,y E W (Z), and x # y imply tx + (1 - t)yTz,O < t < l .AXIOM3 : For all x, W(x) and W -'(x ) are closed.It is known (Sonnenschein [16] and Shafer [14]) that under these axiomscompact competitive budgets contain unique W-maximal elements and thatdemand functions exist and are continuous. It is also trivial to prove that non-transitive co nsum er de m an d functions satisfy WA. W . Shafer [14] has also show nthat the nontransitive consum er dem and functions need no t satisfy S : considerk(x, y) = y;*x$ + In x, - x;*y$ - In y, an d define xW y by k(x, y) > 0. Thedem and function for the first two com modities g enerated by W isAs we have stated,

(4) N SD a nd S o trong Axiom o Utility HypothesisSince we have just shown ND, NSD oW eak Axiom," an d since it is easilyestablished, Weak Axiom (.Nontransitive Consumer, the following conjectureis natura l.

CO NJE CT UR E:iven a de m an d function h satisfying WA, there exists a non -transitive co nsum er who gen erates h : i.e.,( 5 ) N D , N S D o Weak Axiom o Nontransit ive Cons um er,a cou nterpa rt of (4).

University of Illinois at Urbana, Un iversity of California a t Berkeley, and Northwestern University

Manuscript received June, 1974.


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978 R . KIHLSTROM, A. MAS-COLELL, A N D H. SONNENSCHEINR E F E R E N C E S

[I] AN TON ELLI, . B.: "Sulla T eoria M atematica della Economia Politica," 1886. Reprin ted inGiornale degli Economisti e Annuli di Economia, Nuo va Serie, 10 (1951), 233-263. Eng lishtranslation in Preferences, Utility and Demand, ed. by J. S. Chipman, L. Hurwicz, M. K.Richter, and H . F. Sonnenschein. New Y ork : H arco urt Brace Jovanovich, 1971.[2] ARROW,K. J . , AN D L. HURWICZ:The Stability of Competitive Equilibrium, I," Econometrica,26 (1958), 522-552.

[31 - . "Some Remarks on the Equilibrium of Economic Systems," Econometrica; 28 (1960),6 4 M 4 6 .[4] GAL E ,D .: "A No te on Revealed Preference," Economica, 27 (1960), 348-354.[S]HAR T M AN,.: Ordin ary Differential Equations. New Y ork: W iley, 1964.[6]HOUT HAKKE R ,. S.: "Revealed Preference and the Utility Function," Economica, 17 (1950),159-174.[T HURWICZ,.: "On the Problem of Integrability of Demand Functions," in Preferences, Utilityand Demand, ed. by J. S. Chipman, L. Hurwicz, M . K. Richter, and H . F. Sonnenschein. NewYork: Harcourt Brace Jovanovich, 1971.[8] H u ~ w i c z ,L., AN D H . U Z A W A :On the Integrability of Demand Functions," in Preferences,Utility and Demand, ed. by J. S. Chipman, L. Hurwicz, M . K . Richter, and H . F. Sonnenschein.New York : Harc ourt Brace Jovanovich, 197 1.[9] JOHNSON,W. E.: "The Pure Theory of Utility Curves," Economic Journal. 23 (19 13). 483-5 13.Reprinted in Precursors in Mathematical Economics: An Anthology, ed. by William J . Baumoland Stephen M. Goldfeld. London: London School of Economics and Political Science, 1968.[lo] KATZNER, . W. : Static Demand Theory. New York: MacMillan, 1970.[ l l ] SAMUELSON,. A,: "A Note on the Pure Theory of Consumer Behavior," Economica, 5 (1938),61-71.[I21 -. . Foundations of Economic ~ n a i ~ s i s .ambridge, Mass.: Harvard University Press, 1947.[13] - . "The Problem of Integrability in Utility Theory," Economica, 17 (1950) , 355--385.

[14] SHAFER,W. : "The Nontransitive Consumer," Econometrica, 42 (1974). 91 3-920.[IS] SLUTSKY,.: "Sulla Teoria del Bilancio del Consumatore," Giornale degli Economisti e ~iuistadi Statistica, 51 (19r5). 1-26. English trans lation . "O n the Theo ry of the B udget of theConsumer," in Readings in Price Theory, ed. by G. J. Stigler and K . E. Boulding. Hom ewood,Illinois: Richard D . Irwin, 1952.[16] SONNENSCHEIN,. F .: "Demand Theory Withou t Transitive Preferences, With A pplications tothe Theory of Competitive Equilibrium," in Preferences, Utility and Demand Theory, ed. byJ . S. Chipman, L. Hunvicz, M. K. Richter, and H. F. Sonnenschein. New York: HarcourtBrace Jovanovich, 1971.[I71 UZAWA, .: "Preference and Rational Choice in the Theory of Consumption," in MathematicalMethods in the Social Sciences, 1959, ed. by K . J. Arrow, S. Karlin, an d P. Suppes. Stanford,California: Stanford University Press, 1960.


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You have printed the following article:

The Demand Theory of the Weak Axiom of Revealed Preference

Richard Kihlstrom; Andreu Mas-Colell; Hugo Sonnenschein


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[Footnotes]

9 A Note on Revealed Preference

David Gale

Economica, New Series, Vol. 27, No. 108. (Nov., 1960), pp. 348-354.

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References

2 On the Stability of the Competitive Equilibrium, I

Kenneth J. Arrow; Leonid Hurwicz

Econometrica, Vol. 26, No. 4. (Oct., 1958), pp. 522-552.

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3 Some Remarks on the Equilibria of Economic Systems

Kenneth J. Arrow; Leonid Hurwicz

Econometrica, Vol. 28, No. 3. (Jul., 1960), pp. 640-646.

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4 A Note on Revealed Preference

David Gale


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6Revealed Preference and the Utility Function

H. S. HouthakkerEconomica, New Series, Vol. 17, No. 66. (May, 1950), pp. 159-174.

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9 The Pure Theory of Utility Curves

W. E. Johnson

The Economic Journal, Vol. 23, No. 92. (Dec., 1913), pp. 483-513.

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11 A Note on the Pure Theory of Consumer's Behaviour

P. A. Samuelson

Economica, New Series, Vol. 5, No. 17. (Feb., 1938), pp. 61-71.

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13 The Problem of Integrability in Utility Theory

Paul A. Samuelson


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14 The Nontransitive Consumer

Wayne J. Shafer


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