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588 MATHEMATICS: G. DEBREU PROC. N. A. S. 1 A. Borel, "Sur la cohomologe des espaces fbres prncpaux et des espaces homogbnes des groupes de Le compacts," Ann. Math., 57, No.

1 588 MATHEMATICS: G. DEBREU PROC. N. A. S. 1 A. Borel, "Sur la cohomologe des espaces fbres prncpaux et des espaces homogbnes des groupes de Le compacts," Ann. Math., 57, No. 1, 115, 1953; "Homology and Cohomology of Compact Connected Le Groups," these PROCEEDINGS, 39, No. 11, , M. Morse, "The Calculus of Varatons n the Large," Math. Colloq. Pub., 18, E. Cartan, "La G6ometre des groupes smples," (Euvres, 2, Part I, 793; also papers referred to theren. See also E. Stefel, "tyber ene Bezehung..." Comm. Math. Helv., 14, , C. Chevalley, Theory of Le Groups I (Prnceton, N.J.: Prnceton Unversty Press, 1946). 6 R. Bott, "Non-degenerate Crtcal Manfolds" (to appear n Ann. Math.). VALUATION EQUILIBRIUM AND PARETO OPTIMUM* By GERARD DEBREU COWLES COMMISSION FOR RESEARCH IN ECONOMICS Communcated by John von Neumann, May 6, 1954 For an economc system wth gven technologcal and resource lmtatons, ndvdual needs and tastes, a valuaton equlbrum wth respect to a set of prces s a state where no consumer can make hmself better off wthout spendng more, and no producer can make a larger proft; a Pareto optmum s a state where no consumer can be made better off wthout makng another consumer worse off. Theorem 1 gves condtons under whch a valuaton equlbrum s a Pareto optmum. Theorem 2, n conuncton wth the Remark, gves condtons under whch a Pareto optmum s a valuaton equlbrum. The contents of both theorems (n partcular that of the frst one) are old belefs n economcs. Arrow' and Debreu2 have recently treated ths queston wth technques permttng proofs. A synthess of ther papers s made here. Ther assumptons are weakened n several respects; n partcular, ther results are extended from fnte dmensonal to general lnear spaces. Ths extenson yelds as a possble mmedate applcaton a soluton of the problem of nfnte tme horzon (see sec. 6). Its man nterest, however, may be that by forcng one to a greater generalty t brngs out wth greater clarty and smplcty the basc concepts of the analyss and ts logcal structure. Not a sngle smplfcaton of the proofs would ndeed be brought about by restrcton to the fnte dmensonal case. As far as possble the mathematcal structure of the theory has been dssocated from the economc nterpretaton, to be found n brackets. 1. The Economc System.-Let L be a lnear space (on the reals R).3 The economc system can be descrbed as follows: The th consumer ( = 1,..., m) chooses a pont x [hs consumpton] n a gven subset X [hs consumpton-set] of L. [x completely descrbes the quanttes of commodtes he actually consumes, to be thought of as postve, and the quanttes of the varous types of labor he produces, to be thought of as negatve. X s determned by constrants of the followng types: quanttes of commodtes consumed (labor produced) must be nonnegatve (nonpostve), and, moreover, they must enable the ndvdual to survve. ] There s on X a complete orderng, denoted by < [correspondng to the preferences of that consumer].4 x 0 s a saturaton pont of X, f, for all x E X, one has x < x,0.

2 VOL. 40, 1954 MATHEMATICS: G. DEBREU 589 The th producer ( = 1,., n) chooses a pont y [hs producton] n a gven subset Y [hs producton-set] of L. [y s a complete descrpton of all hs outputs, to be thought of as postve, and hs nputs, to be thought of as negatve. YJ s determned by technologcal lmtatons. ] Denote x = x, y = y; they are constraned to satsfy the equalty x - y =, where v s a gven pont of L. [t corresponds to the exogenous resources avalable (ncludng all captal exstng at the ntal date). x - y s the net consumpton of all consumers and all producers together. It must clearly equal I. ]5 A (m + n)-tuple [(x), (y)], one x for each, one y for each, s called a state of the economy. [It s a complete descrpton of the actvty of every consumer and every producer.] A state [(x), (y)] s called attanable f x E X for all, y e Y for all, x-y = P 2. Valuaton Equlbrum.-v(z) wll denote a (real-valued) lnear form on L.6 [It gves the value of the commodty-pont z. When L s sutably specalzed, ths value can be represented by the nner product p z, where p s the prce system.] A state [(x ), (y 0)J s a valuaton equlbrum wth respect to v(z) f: (2. 1) [(x20), (y0) ] s attanable. (2.2) For every " x e X, v(x). v(x ) ' mples ' x < x0o. [Best satsfacton of preferences subect to a budget constrant.] (2.3) For every ' y e Y, mples < v(y) < V(yv P ). [Maxmzaton of proft subect to technologcal constrants.] 3. Pareto Optmum.-The set X1 X... X X. of m-tuples (x), one x for each, s (partally) ordered as follows: (x') > (x) f and only f x' > x for all. A state [(x ), (y0) ] s a Pareto optmum f: (3.1) [(x), (y0) ] s attanable. (3.2) There s no attanable state [(x), (y)] for whch (x) > (x0). [It s mpossble to make one consumer better off wthout makng another one worse off.] 4. A Valuaton Equlbrum Is a Pareto Optmum.-The followng assumptons wll be made: I. For every, X s convex. II. For every, x' e X, x' e X, x' < x * mples * x' < (1 - t) x' + S tx' for all t, 0 <t< 1 a. These two axoms on the convexty of the consumpton-sets and the convexty of preferences have been used by Arrow and Debreu7 n a dfferent context. THEOREM 1. Under assumptons I and II, every valuaton equlbrum [(x0), (y0) ], where no x s a saturaton pont, s a Pareto optmum. Proof: (4.1) 4 x e X and x > x0 mples t v(x) > v(x0). Ths s a trval consequence of defnton (2.2). (4.2). x e X and x x0 ' mples av(x) > v(x0).> Snce x0 s not a saturaton pont, there s x' e X, such that x' > x, hence x' > x. Consder x(t) = (1 - t) x + tx'. By assumpton II, for all t, 0 < t < 1, t x(t) > x, hence x1(t) > x0, so (by [4.1]) v(x0) < v(x1(t)) = (1-t) v(x) + tv(x'). Let t tend to zero; n the lmt v(x ). v(x).

3 Soo MATHEMATICS:. DEBRUP. PROC. M. A. S. To complete the proof we consder a state [(x), (y) ], where xf e X for all, ye Y - for all, and show that f (x) > (x0), the state s not attanable,.e.,x-yy. (x) > (xa) means that for all, x > x, and for some ', xz, > x4,; so by (4.1) and (4.2) Ev(x) > Ev(xO),.e., v(x) > v(xo). On the other hand, (2.3) mples v(y). v(yo), so v(x) -v(y) > v(xo) - v(yo). Snce x0 - y -, v(x - y) > v(r), whch rules out x-y y = 5. A Pareto Optmum Is a Valuaton Equlbrum.-In ths secton L s a topologcal lnear space.8 Let x', x' be ponts of X; we defne I(x', x') = { ti t)x1' + txg] [(1 e x}. When X s convex, I(x', x') s a real nterval wth possbly one or two end-ponts excluded. In addton to assumptons I and II, three further assumptons are needed here. III. For every, x, x', x" n X the sets {t e I(x', x) I (1 - t)x' + tx' > x} and { t e I(x', x") (1 -t)x' + txst < x are closed n I(x', x'). Ths weak axom of contnuty for preferences has been ntroduced by Hersten and Mlnor9 n another context. We defne Y = Y (the set of all y = Ey, where y, e Y1 for all ). IV. Y s convex. [The assumpton that the aggregate producton-set s convex s strctly weaker than the assumpton that the ndvdual producton-sets Y are all convex. ] V. L s fnte dmensonal and/or Y has an nteror pont. [The assumpton that Y has an nteror pont wll be shown n secton 6 to be mpled by free dsposal of commodtes. ] THEOREM 2. Under assumptons I-V, wth every Pareto op-mum [(x0), (y0)], where some x0 s not a saturaton pont, s assocated a (nontrval) contnuous lnear form v(z) on L such that: (5.1) For every xf X, x > x ' mples " v(x) > v(x0).> (5.2) For every ' y E Y ' mples z v(y). v(yo). Proof: From assumptons I, II, and III follows: a) 4 x', x' n X, x' < x' mples ' for all t, 0. t < 1, x' < [(1 - t)x' + txoi] e x.> By assumpton III, the set {t e I (xe', x')i (1 - t) x' + tx' < x'} s open n I(x', x"). Its ntersecton wth the nterval ]0, 1 [ (end-ponts excluded) s open. We wsh to show that ths ntersecton s empty. If t were not, t would contan two numbers t1 < t2. Take the correspondng ponts xl, x2. Then x' < x' < x/. By assumpton II, x' < x" gves x' <x I x X1 X2 x" Smlarly, X2 < x' gves X12 < x', a contradcton. As an mmedate consequence of (a), for all, the sets X() = X e X X > 4} and X() {X E X1IX > xo} are convex. x} and -$(X)7{OFeXF XtX{ } C~ ar

4 VOL. 40, 1954 MATHEMATICS: G. DEBREU 591 Let ' be a value of for whch x', s not a saturaton pont, and consder the set Z = X'(X!,) + Ex(xZ ) - Y f Z, ths s the defnton of a Pareto optmum [(x0), (y/') ]. Z s convex as t s the sum of convex sets. If Y = E Y has an nteror pont, Z also has one. The Hahn-Banach theorem10 can therefore be appled to Z and I. There s a (nontrval) contnuous lnear form v(z) on L such that v(z). v(r) for all z e Z,.e., snce x -Evo v (x-xo) (Y-Y10) > for all x, E X (x?), x e X(xo?) (for $ '), y e Y (for all ). In ths statement kx(xo?) can be replaced by X,(xo,), for every x, e X rx,, x,, can be exhbted, as n the proof of (4.2), as a lmt of ponts belongng to X,'(x,). Therefore, b) v(x - x) + Z v(yo - y) > 0 for all x e X(x0), y e Yp.. By makng all but one of the x, y equal to the correspondng x, yf,, one proves that for the remanng term n (b) v(x - x) > 0 for all x e X(x 0) (or v(yo - y). > o for all y e Y) whch s precsely the statement of Theorem 2. (5.2) s dentcal to (2.3), but (5.1) does not necessarly mply (2.2), and Theorem 2 does not qute correspond to the ttle of ths secton. The followng Remark, due to Arrow1' n ts essence, tres to fll ths gap: REMARK. Under assumptons I and III, f there s, for every, an x' e X such thatv(x') < v(x?), then (5.1) mples (2.2). Consder an x e X, v(x) < v(x0). Let x(t) = (1 - t) x + txt'. For all t, O <t < 1, v(x(t)) <v(x,0) and thus, by (5.1), x(t) <x. The set {t ei (xf, x')i (1 -t) x + tx/' < x } contans the nterval ]0, 1 [; snce t s closed n I(x, x') (by assumpton III), t contans 0,.e., x x0. [The condton that there s x' e X such that v(x') < v(x) means that the consumer does not have such a low v(x 0) that wth any lower value he could not survve. ] 6. The Free Dsposal Assumpton.-An example wll show the economc ustfcaton of assumpton V when L s not fnte dmensonal. Suppose that there s an nfnte sequence of commodtes [because, for example, economc actvty takes place at an nfnte sequence of dates, a case studed by Malnvaud'2 wth dfferent technques]. The space L wll be the set of nfnte sequences of real numbers (Zt) such that Sup I Zh! < + Xo. L s normed byzik = Sup I ZhI. The assumpton of free dsposal for the technology means that f y e Y and yh' < y, for all h, then y' e Y [f an nput-output combnaton s possble, so s one where some outputs are smaller or some nputs larger; t s mpled that a surplus can be freely dsposed of ]. Wth ths assumpton, f Y s not empty, t clearly has an nteror pont: select a number p > 0 and a pont y e Y; consder y' defned by yh' = yā p for all h. The sphere of center y', radus p, s contaned n Y. Other examples of lnear spaces n economcs are provded by the case where there

5 592 MATHEMATICS: G. DEBREU PROC. N. A. S. s a fnte number 1 of commodtes, and tme and/or locaton s a contnuous varable. The actvty of an economc agent s then descrbed by the 1 rates of flow of the commodtes as functons of tme and/or locaton. The space L s the set of 1- tuples of functons of the contnuous varable. In any case, f L s properly chosen, the exstence of an nteror pont for Y wll follow from the free dsposal assumpton. Then applcaton of Theorem 2 wll gve a contnuous lnear form v(z). * Based on Cowles Commsson Dscusson Paper, Economcs, No (January, 1953). Ths artcle has been prepared under contract Nonr-358(01), NR between the Offce of Naval Research and the Cowles Commsson for Research n Economcs. I am grateful to E. Malnvaud, staff members and guests of the Cowles Commsson, n partcular I. N. Hersten, L. Hurwcz, T. C. Koopmans, and R. Radner for ther comments. I K. J. Arrow, "An Extenson of the Basc Theorems of Classcal Welfare Economcs," Proceedngs of the Second Berkeley Symposum (Berkeley: Unversty of Calforna Press, 1951), pp G. Debreu, "The Coeffcent of Resource Utlzaton," Econometrca, 19, , A real lnear space L s a set where the addton of two elements (x + y) and the multplcaton of a real number by an element (tx) are defned and satsfy the eght axoms: 1. Forallx,y,znL,(x +y) +z = x +(y +z). 2. There s an element 0 e L such that for every x e L, x + 0 = x. 3. For every x el, there s an x' e L such that x + x' = Forallx,ynL,x+y=y+x. Forallx,ynL,t,t'nR, 5. t (x + y) = tx + ty, 6. (t + t')x = tx +t'x, 7. t(t'x) = (tt')x, 8. lx = x. 4An order s a reflexve and transtve bnary relaton (generally denoted by _). x x' means x _ x' and x' < x, whle x < x' means x _ x' and not x' < x. The order s complete (as opposed to partal) f for any x, x' one has x < x' and/or x' < x. One may obect to completeness of the preference orderng as well as to ts transtvty. The reader must therefore note that, wth slght modfcatons of the defntons and the assumptons, Theorems 1 and 2 can easly be proved for arbtrary bnary relatons on the X. 6 Usually the net consumpton s only constraned to be at most equal to the avalable resources. But ths mples that any surplus can be freely dsposed of. Such an assumpton on the technology should be made explct (see sec. 6) whle requrng at the same tme x - y = P. 6 For all x, y, v(x + y) = v(x) + v(y). For all t, x, v(tx) = tv(x). v(z) s sad to be trval f t vanshes everywhere. 7K. J. Arrow, and G. Debreu, "Exstence of an Equlbrum for a Compettve Economy," Econometrca, A topologcal lnear space s a lnear space wth a topology such that the functons (x, y) x + y from L X L to L and (t, x) - tx from R X L to L are contnuous. For defnton of a topology, of the topology on a product, of a contnuous functon see N. Bourbak, Elements de mathematque (Pars: Hermann, et Ce, 1940), Part I, Book 3, chap.. For the representaton of contnuous lnear forms on L see S. Banach, Theore des operatons lndares (Warsaw, 1932), n partcular, chap. v, sec I. N. Hersten and J. Mlnor, "An Axomatc Approach to Measurable Utlty," Econometrca, 21, , In a real topologcal lnear space, f Z s a convex set wth nteror ponts, r a pont whch does not belong to Z, there s a closed hyperplane through, boundng for Z. (See for example, N. Bourbak, Elements de mathematque [Pars: Hermann et Ce, 1953', Part I, Book 5, chap., n partcular, sec. 3.) '1 Arrow, op. ct., Lemma E. Malnvaud, "Captal Accumulaton and Effcent Allocaton of Resources," Econometrca 21, , 1953.

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1) Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A