UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIQN BOOKS I ACKS \l^ ^ I CENTRAL CIRCULATION BOOKSTACKS The person charging this material is re- sponsible for its renewal or its return to the library from which it was borrowed on or before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each lost book. THeit, mutilation, and underlining of books ore reasons for disciplinary action and may result In dismissal from the University. TO RENEW CALL TELEPHONE CENTER, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN OCT 1 9 1395 When renewing by phone, write new due date below previous due date. L162 STX COPY 2 FACULTY WORKING PAPER NO. 1204 On Approximate Decentralization of Pareto Optimal Allocations in Locally Convex Spaces M. AH Khan Rajiv Vohra College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 1204 College of Commerce and Business Administration University of Illinois at Urbana-Champaign November, 1985 On Approximate Decentralization of Pareto Optimal Allocations in Locally Convex Spaces M. Ali Khan, Professor Department of Economics Rajiv Vohra Brown University This research was supported, in part, by an NSF Grant to Brown University, Abstract . We present approximate versions of the second fundamental theorem of welfare economics in the setting of dual pairs of locally convex spaces. We do not require the commodity space to have a posi- tive cone with a nonempty interior. We also present a sufficient con- dition under which our approximate results hold exactly. Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/onapproximatedec1204khan I. Introduction In a pioneering paper, Debreu [4] extended the second fundamental theorem of welfare economics to economies with a topological vector space of commodities. Specifically, he showed that corresponding to a given Pareto optimal allocation of such an economy, there exists a price system at which the given allocation can be sustained as a valuation equilibrium. Debreu formalized the notion of a price system as a non-zero element of the topological dual of the commodity space and assumed, in particular, that the aggregate production set of the economy has a nonempty interior. Under this assumption and given con- vexity, Debreu could prove his result as a consequence of the sup- porting hyperplane theorem. Recent work in mathematical economics (see Mas-Colell [11] and the references therein) has questioned Debreu's justification of the interiority assumption. This work has emphasized the importance of several spaces, none of which has a positive cone with a nonempty interior. Thus, the economic assumptions of "free disposal" or "monotonic preferences" do not lead, as in the case of [4] or in the subsequent work of Bewley [2], to a set which needs to be supported and which has a nonempty interior. Indeed, it now appears that from the point of view of economic theory, R and L may be the only interesting spaces for which this is true. Furthermore, we now have simple examples of economies for which Debreu's theorem does not hold; see Jones [8] and also [11, 12]. Thus, there are two natural directions in which one can proceed. The first is to look for additional conditions on the underlying -2- parameters of the economy under which Debreu's theorem can be proved. This is the approach of Mas-Colell [11, 12] and of subsequent work that utilizes his concept of "proper preferences." Alternatively, one can look for satisfactory approximate versions of Debreu's result as is done in [9] and in another context by Aliprantis, Brown and Burkinshaw [1]. This is the direction that we pursue in this paper, although we also present a sufficient condition on an individual production set under which we obtain an exact result. Our work uses in an essential way recent results in functional analysis. For our approximate results, the basic idea is to find an "approximate" support to the closure of a set with an empty interior. This then leads to the question of finding sufficient conditions that are attractive in terms of economic theory and under which the sum of closed convex sets is closed. We present two such conditions. The first is due to the collective efforts of Choquet [3], Dieudonne [6] and Ky Fan [7] and it allows us to set our results in real, Hausdorff locally convex spaces. The second condition leads us to restrict the commodity space to those spaces which are strictly hypercomplete [15, Sections 12-2 and 12-3]. Our exact result, perhaps not surprisingly for functional analysts, draws on the Bishop-Phelps theorem, specifi- cally a result [13] whose importance for economic theory was first seen by Majumdar [10]. More generally, however, our results bring out the relevance of functional analytic methods for a rather basic problem in mathematical economics. The paper is organized as follows. Section 2 presents the model and results and Section 3 the proofs. -3- 2. The Model and Results For the basic terminology and notation we follow Wilansky [15]. However, for ease of exposition we recall some basic concepts for the reader. Let [E, F, v] be a dual pair , i.e., E, F are vector spaces over the reals and v is a bilinear functional on ExF and written v(x,y) = [x, y] and such that (a) [x, y] = for all y in F implies x = and (b) [x, y] = for all x in E implies y = 0. In what follows, E will be given the interpretation of a commodity space, F that of a price space and v a valuation functional. Thus [x, y] denotes the value of a commodity bundle y in E at the price x in F. As is customary, we shall abbreviate a dual pair to be [E, F] and assume that some bilinear functional is specified. In the special case that F CT E* where E* is the algebraic dual of E, the bilinear functional is given by = y(x) for any x in E and for any y in F. For any dual pair [E, F], a Hausdorff locally convex topology T on E (respectively F) is said to be compatible with the dual pair if the topological dual of E, E' (respectively F'), is F (respectively E). We shall respectively denote the weak, strong and Mackey topolo- gies, say, on E as a(E, F) , 6(E, F) and x(E, F). We shall refer to the topology a(E', E) as the weak * topology and to 8(E', E) as the strong * topology. Thus, for example, a weak * closed subset of E' should be taken to mean a a(E', E)-closed subset of E. -4- When we refer to a set satisfying a particular topological pro- perty without specifying the topology, it is to be understood that the said property holds in any compatible topology. In our view, it is one of the advantages of our approach that we can state the assump- tions underlying our results in any compatible topology. In the sequel we shall assume that E is a partially ordered vector space and F is endowed with an ordering induced by that on E. Given the recent work of Aliprantis et. al. [1], it may be worth mentioning that neither E nor F is assumed to be a lattice. We now turn to our basic economic concepts. Definition 1 . An economy £ is given by {[E, F] , _>, (X(t), ^, e(t)) t£T , (Y(j)) jep ) where (i) [E, F] is a dual pair with (E, _>) a partially ordered vector space, (ii) T is a finite set of consumers such that for each consumer t, X(t) CL E, ~ is a binary relation on X(t)xX(t) and e(t) £ X(t), (iii) P is a finite set of producers such that for each producer j, Y(j) u > u . v > a b means a ~ b and not b ~ a. t t t Without any confusion, we denote the induced ordering on F also by >. For any p, q in F, let p > q denote (p-q) _> and (p-q) * 0. Let N(E), respectively N(F), denote the set of neighborhoods of zero in E, respectively F. Definition 4 . For any real number 5 > and any U e N(E), a (6 ,U)- approximate valuation equilibrium of & consists of a pair (p, (x,y)) such that (i) p e F, p > and there exists z e U such that [z, p] = -5 . (ii) (x, y) is an allocation of £ such that (a) z ~ x(t) implies [z, p] _> [x(t), p] - 6 for all t in T, (b) z e Y(j) implies [z, p] _< [y(j), p] + 5 for all j in P. Thus, an approximate valuation equilibrium consists of a price system and an allocation such that (i) each consumer is approximately maximizing his preferences subject to a budget set and (ii) each pro- ducer is approximately maximizing his profits. In each case, the degree of approximation is measured by 5 and U, the latter controlling for the fact the price system is not normalized to render this approxi- mation trivial. -6- Our final concept relates to an economy modelled on the dual pair (I/, L) where L is a semi-normed space. Definition 5 . A valuation equilibrium of £ is a pair (p, (x,y)) such that (i) p e L" P > 0, II pi! = 1. (ii) (x,y) is an allocation of & such that (a) z ~ x(t) implies _> for all t in T. (b) z £ Y(j) implies _< for all j in P. For our results we shall need the following assumptions on the economy £ = {[E, F] , >, (X(t), ~, e (t)) teT » (Y(j)). p }. For any closed, t •* convex subset K of E, A(K) denotes the asymptotic cone of K and, follow- ing [3], is given by ( \ A(K-x) where x is any particular point in K. X>0 Let E = {x e E: x _> 0} and E_ = -E . Al. (a) For all t in T, X(t) is a closed convex subset of E. (b) There exists convex K C E such that for all t in T, X(t) CK and K has a lower bound for >. A2. For all t in T, for any y in X(t), the set R (y) = {x e X(t): x ~ y} 1 t is closed, convex and contains y. A3. (a) For all j in P, Y(j) is a closed, convex subset of E. (b) For all j in P, e Y(j) and (E_) and any U e N(E), for any Pareto optimal allocation (x*,y*) of £, there exists p* e F, such that (p*,(x*,y*)) is a (6-U) -approximate valuation equilibrium of a. Corollary * Theorem 1 is valid if either the Choquet or the Dieudonne Condition is replaced by the requirement that the x(F,E)-interiors of K and Y are nonempty . The Corollary makes clear the fact that the hypotheses relating to K are no restriction for an economy modelled on the dual pair [L',L], L a Banach space such that L has a nonempty norm interior. The hypotheses relating to Y are however a restriction even in these circumstances and our next two results attempt to cope with this by restricting the class of locally convex spaces. They also require the following conditions on X(t), Y(j) and on the mutual position of the sum of these sets. -8- Dieudonne Condition II . (i) There exists a in T such that Z X(t) is locally compact. (ii) There exists f e P such that Z Y(j) is locally compact. Condition A . For any 8(E,F)-bounded set B, ( Z X(t))H ( Z Y(j)+B) is teT jeP 8(E,F)-bounded. We can now state Theorem 2 . Let S satisfy Al to A4, Condition A and the Dieudonne Con - dition II. If B (E,F)-sequentially closed convex sets are 8(E,F)-closed and [F,E] is a quasibarrelled dual pair, then the conclusion of Theorem 1 holds . Our final result of this genre substitutes the following condition for the Dieudonne Condition II and, in so doing, allows us to drop Al(b) and A3(b). Ky Fan Condition . (i) For any Pareto optimal allocation (x,y), n t-1 o R (x(t)) u p| (T(F,E)-interior ( Z R (x(i))r) * 0, t = 2,...,|t|. i=l n i_1 n (ii) (Y(i)) U Pl (T(F,E)-interior ( Z Y(j)) U ) t 0, i = 2,...,|p|. J-l We can now state Theorem 3 . Theorem 2 is valid without Al(b) and A3(b) if the Ky Fan Condition is substituted for the Dieudonne Condition II . For our next result we model our economy £ on the dual pair [L',L] where L is a real, locally convex Hausdorff space and L is a normal -9- cone for 8(L',L), (see [14, p. 215] for a definition). Moreover, we shall need to substitute A3' for A3 where A3'. Y is a closed, convex subset of L'. We shall also need Condition B . For any U e N(L), ( E X(t)) p\ (( E e(t) + E Y(j))+U°) teT teT jeP is strong * bounded. We can now state Theorem 4 . Let £ satisfy Al, A2, A3' and A4. If, in addition, & satisfies Condition B and L is strictly hypercomplete and quasi - barrelled, then the conclusion of Theorem 1 holds . It is well known that a strictly hypercomplete quasibarrelled space is barrelled; see, for example, [15; Theorem 12-4-3 and Remark 10-4-13]. We do not know if Theorem 4 is true for barrelled spaces. Our final result presents a sufficient condition under which Pareto optimal allocations can be exactly decentralized. This sufficient condition formalizes the requirement of uniformly bounded marginal rates of substitution of any one production set. Condition M . There exists f in P and there exists p e L, p > such that any supporting hyperplane with unit norm, p e L, l|p| = 1, to the set Y(f) satisfies p > p. Condition M is an assumption as the parameters of an individual agent and does not assert that a supporting hyperplane necessarily exists at a particular boundary point of Y(f). -10- Theorem 5 . Let 8 be modelled on (L',L) with L a Banach space and let & satisfy Al, A2 , A3', A4 and Conditions B and M. Then corresponding to any Pareto optimal allocation (x*,y*) of &, there exists p* e L '' such that (p*,(x*,y*)) is a valuation equilibrium of £ . Remark 2 . Note that Theorem 5 has nothing to say about an exchange economy, i.e., one where Y(j) = {0} for all j. Remark 3 . We have presented our exact result in the context of the setting of Theorem A. It should be clear from an inspection of the proofs that we could equally well have presented it under the setting of our other approximate results. 3. Proofs . We begin with the following elementary result. Lemma . Let Z.. and Z„ be two subsets of E. Then (a) for any real number X ^ 0, X(Z i n Z 2 ) = XZ -O XZ- (b) for any z in E, (Z.n 1^ - (z) = (Z 1 - z) C\ (Z„ - z); (c) if Z 1 and Z~ are closed convex and such that z n z 2 * 0, A(Z n z 2 ) = a(z )rs a(z 2 ). Proof of Lemma We begin with (a). For any z in X(Z 1 P\Z„), there exists k in Z 1 r\ Z„ such that z = Xk which implies z is in (XZ, C\ XZ„). On the other hand, if z is in \(Z.A Z n ), there exist z. e Z. such that z = 12' 11 Xz, = Xz„. Since X ^ 0, z, = z„ and hence z e XZ.. O XZ„. As regards (b), k e (Z O Z„) - (z) if and only if (k+z) e Z ± (1-1,2), if and only if k e Z. - z (i=l,2), that is k e (Z - z) O (Z 2 - z). -11- Let ksZ.H Z . Then by (a) and (b), z e [f^XiZ - k)] ^ L X>0 [P\X(Z 9 - k)] if and only if z e [C\\((Z C\ Z ? ) - k) ] . X>0 X>0 Proof of Theorem 1 . Let (x*,y*) be a Pareto optimal allocation of £ and let W = E R (x*(t)) - E Y(j) - E x*(t) + E y*(j). teT jeP teT jeP where R (x*(t)) is as defined in A2. The essential part of the proof is to show that W is a closed set in E. We begin with the case when the Choquet Condition holds. Since K is bounded from below for _>, there exists b e E such that K - b is a subset of E . By the Choquet Condition, K - b is a a(E,F)-complete subset of E. Since E is an ordered vector space, E contains no straight line. By A2 , R (x*(t)) is closed, convex for all t in T. We can now appeal to Choquet 's theorem [3] to assert that £ R (x*(t)) is teT closed. Next we show that Y contains no straight line. Suppose it did, i.e., there exist x, y in E, x * y such that Xx + (l-X)y is in Y for all real numbers X. This implies (l-X)(y-x) e A(Y) for all X. On choosing X = and X = 2, we can conclude that (y-x) is in A(Y) and in -A(Y) = A(-Y). On using (c) of the lemma, we contradict the fact that Y^(-Y) = {0}. Hence by a second appeal to Choquet's theorem, we can assert that E Y(j) is closed. For the final step, observe that A3(b) implies that E + CL -Y. Since (K-b) C. E + , K C b - Y. Since E R (x*(t)) teT is contained in K, we can assert that the set (b-Y) contains -12- E R (x*(t)). Since Y is o(E,F)-complete by the Choquet condition, teT so is b - Y. We can now make a third appeal to Choquet' s theorem to assert that E R (x*(t)) - Y + b is closed. This implies that W is teT closed and completes the proof of our claim. Next, we turn to the case when the Dieudonne Condition holds. Since R (x*(t)) are subsets of a locally compact set, they are locally compact. Since R (x*(t)) are subsets of K and K is bounded from below for _>, there exists b e E such that R (x*(t)) -bCE for all t in T. Since R (x*(t)) - b is closed and convex, we can take its asymptotic cone. Since E A E_ = {0}, A(R (x*(t) ) C\ A(R (x*(s)) = {0} for any t,s in T. We can now appeal to Dieudonne' s theorem to assert that R (x*(t)) + R (x*(s)) is closed for any t,s in T. The same argument can be repeated for a given R (x*(u)) and R (x*(t)) + R (x*(s)) Proceeding in this way, we can show that E R (x*(t)) is closed. teT Since Y(j) c Y and Y is locally compact, Y(j) is locally compact for j e P. Furthermore Y(i)A (-Y(j)) = {0} for all i * j in P as a consequence of Y A. (-Y) = {0}. Hence, by a second appeal to Dieudonne 's theorem, we can assert that Y(i) + Y(j) is closed. By repeating this argument for another Y(k), we can show that E Y(j) jeP is closed. Next, we assert that YAE = {0}. If not, there exists y in Y, y e E and such that y * 0. By A3(b), -y e Y. This implies y e (-Y) and we contradict the fact that "Y Aj(-T) = {0}. Since K - be E , so is its closure K - b. This implies that Y A\(K-b) = {0} which implies that A(Y) A, A(K) = {0}. Since A( E R (x*(t))C A(K), and Y teT -13- is locally compact, a final appeal to Dieudonne's theorem completes the proof of our claim that W is closed. Next, we claim that W C\ E_ = {0}. This is a simple consequence of A4 and the fact that (x*,y*) is a Pareto optimal allocation of &. By A2 and A3, W is a nonempty, convex set. For any U e N(E), choose z e (U r}~E_). We can now apply the second separation theorem [13, II*9*2] to assert the existence of p e F, p * such that (1) [w,p] > [z,p] for all w e W. Since e W, [z,p] < 0. Now let p* = -5p/[z,p]. Then we can rewrite (1) as (2) [w,p*] > -6 for all w e W. Now for any t in T, and any x in R (x*(t)), let v = (x-x*(t)). Cer- tainly v e W, and hence (3) [x,p*] > [x*(t),p*] - 6. Similarly, for any j e P, and any y e Y(j), u = (-y+y*(j)) is an ele- ment of W. This yields (4) [y,p*] < [y*(j),p*] + 5. Finally, we show that p* > 0. Suppose there exists v in E such that < 0. By A5 , (x*(t)+kv) e R (x*(t)) for any positive integer k. This implies k[v,p] > -5 for any k, an obvious contradiction. -14- Proof of Corollary . Since Ky Fan's Theorem 1 [7] shows that nonempty T(F,E)-lnteriority of K implies that K is a(E,F)-complete, assuming the former property for K and Y requires no changes in the proof. Proof of Theorem 2 Let W be defined exactly as in the proof of Theorem 1. Our first claim is that W is 8 (E,F)-closed. Since W is convex, we need only show that W is 8 (E,F)-sequentially closed. Towards this end, choose a sequence {w } from W such that the 8(E,F)-limit of w is w. We have to show w e W. Since w e W, there exist x (t) e R (x*(t)), y (j) e Y(j) such that (5) E (x n (t)-x*(t)) - E (y n (j)-y*(j)) = w. teT jeP Since {w } is a 8 (E,F)-bounded set, there exists a 8 (E,F)-bounded set B such that (6) (w n + E x*(t) - E y*(j)) e B for all n. teT jeP This implies that (7) ( E x n (t)) e ( E X(t))r\(-Y + B) teT teT We can now appeal to condition A to assert that {Ex (t)} is a teT 8(E,F)-bounded set. Since F is quasibarrelled in any compatible topology, {Ex (t)} is equicontinuous and hence, by Alaoglu-Bourbaki , teT a(E,F)-relatively compact. This implies that there exists a subnet {Ex (t)} of { E x (t)} which converges in a(E,F) to a limit, say x. teT teT It is clear from inspection of the relevant arguments in the proof of -15- Theorera 1, that our appeal to Dieudonne's theorem requires local com- pactness of all but one of R (x*(t)). Hence, Z R (x*(t)) is closed teT by Dieudonne Condition II(i). Hence x e Z R (x*(t)). teT We can now rewrite (5) as (8) Z y V (j) = Z (x V (t)-x*(t)) + Z y*(j) - w V jeP teT jeP On taking a(E,F)-limits of both sides and on observing that a(E,F)-limit of {w } is w, we can show that Z y (j) also has a a(E,F)-limit , say y. jeP However, as in the proof of Theorem 1, a second application of Dieudonne's theorem along with Dieudonne Condition II (ii) and A3 yields the fact that Z Y(j) is closed. Hence there exist y(j) e Y(j) such jeP that Z y(j) = y. This implies (9) w-x- Iy(j)- Z x*(t) + E y*(j). jeP teT jeP This completes the proof of the assertion that w e W. We can now follow the remaining steps in the proof of Theorem 1 to complete the proof of the theorem. Proof of Theorem 3 Given Ky Fan Condition (i), we can make successive appeals to Theorem 1 in [7] to assert that Z R (x*(t)) is closed. Also given teT Ky Fan Condition (ii), successive appeals to Theorem 1 in [7] allows us to conclude that Y is closed. We can now repeat the argument in the proof of Theorem 2 and conclude that x e Z R (x*(t)) and teT -16- — - -» ye E Y(j) without any appeal to Dieudonne Condition II. The re- jeP maining steps in the proof of Theorem 2 which call for the repetition of the corresponding steps in the proof of Theorem 1, remain unchanged. Proof of Theorem 4 Let W be defined exactly as in the proof of Theorem 1. Our first claim is that W is weak * closed. Since L is strictly hypercomplete, all we need to show is that W Pi U is weak * closed for any U e N(L), [15, 12-3-7 and 12-2-3]. Towards this end, pick any net {w } from WH U and such that the weak * limit of {w } is w. We have to show that we W Pi U . Since w V e W, there exist x V (t) e R (x*(t)), y (j) e Y(j) such that (10) E x V (t) = w V + E y V (j) + £ x*(t) - E y*(j) teT jeP teT jeP Since w e U , and since (x*,y*) is an allocation, we can appeal to Condition B to assert that {Ex (t)} is strong * bounded. We can teT apply [15, Theorem 10-1-11] to assert that {Ex (t)} is equicon- teT tinuous. By Alaoglu-Bourbaki , this implies that there exists a weak * convergent subnet { E x (t)}. Let the weak * limit of this subnet teT be x. Since {Ex (t)} is a strong * bounded set, so is { E x (t)}. teT teT Furthermore, since x (t) e X(t), there exists b e E such that (x (t) - b) J> for all t in T and all p. Since L is a normal cone, we can appeal to [14, Theorem 3.1, p. 215] to assert that {x (t)} is strong * bounded. Since L is quasibarrelled, by a second appeal to -17- [15, Theorem 10-1-11], we can assert that {x (t)} is equicontinuous. By Alaoglu-Bourbakl, this implies that there exists a weak * conver- gent subnet, also denoted by {x (t)}, which tends to a limit x'(t). Since {x (t)} is a net from R (x*(t)) and the latter is a closed set, x'(t) e R (x*(t)). By taking as many subnets as necessary, we can conclude that E x'(t) e E R (x*(t)) and that x = S x'(t). teT teT teT Now we can rewrite (10) as (11) E y P (j) = -w P + E (x P (t)-x*(t)) + E y*(j). jeP teT jeP Since the right hand side tends to a weak * limit, the left hand side tends to a weak * limit, say y. Since E Y(j) is weak * closed by A3', jeP it contains y. This implies that (12) w = x - y - E x*(t) + E y*(j) teT jeP which in turn leads us to conclude that we WflU . We can now follow the remaining steps in the proof of Theorem 1 to complete the proof. Proof of Theorem 5 Since L is strictly hypercomplete [15, 12-3-3], we can proceed just as in the proofs of Theorem 4 to assert that W is weak * closed and that is in the norm boundary of W. If the convex set W can be supported at 0, the proof can be easily completed. If not, we can appeal to Phelps' theorem [13, Theorem 1] to assert, for each e > 0, the existence of w e W, II w II < e and p e L/{0} such that -18- (13) 2 < w . P > for a11 w e W. Since w e W, there exist x (t) e R (x*(t)) and y (j) e Y(j) such that (14) 2 < E x e Ct) - s y e (j), P £ > teT jeP for all w e ( E R (x*(t)) - E Y(j)) teT jeP Now consider the producer f described in Condition M and deduce from (14) by appropriate choice of w that (15) < for all z e Y(f) This implies that p supports Y(f) at y (f). Without loss of generality we can assume |p | =1 and thus by Condition M, p > p > 0. 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