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Article Excerpt
The motivation for this paper is the Walrasian general equilibrium model of economy, as formulated by Arrow and Debreu [Arrow, K., G. Debreu. 1954. Existence of an equilibrium for a competitive economy. Econometrica 22 264-290]. The problem considered takes the form of a system of variational inequalities on a reflexive Banach space as the infinite dimensional commodity space. The conditions sufficient for the existence of solutions are provided by means of the theory of pseudomonotone multivalued mapping due to Browder and Hess [Browder, F. E., P. Hess. 1972. Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11 251-294], and the Fenchel duality theory combined with the Galerkin method. The analysis is carried out without any lattice considerations and the commodity space is not required to have interior points. The substantial difference of the presented approach in comparison with currently applied methods is that the preferences are not bound by any variant of the [omega]-properness assumption and the consumption sets are not required to have a cone structure. This paper affords new existence results for both the finite and infinite dimensional setting.
Key words: general equilibrium theory; variational inequalities; pseudo-monotonicity; duality; Galerkin method
1. Introduction. Let us consider a competitive market economy consisting of m consumers (indexed by j [member pf] J = {1, ..., m}), n firms (indexed by i [member of] I := {1, ..., n}), and s goods (indexed by l [member of] L := {1, ..., s}). In such an economy, society's initial endowments and technological possibilities (i.e., the firms) are owned by consumers. The initial endowment of j's consumer is given by [[omega].sub.j] [member of] [R.sup.s.sub.+]. In addition, we suppose that consumer j owns a share [[kappa].sub.ji] of firm i, where [[summation].sub.j [member of] J] [[kappa].sub.ji] = 1. Denote by [Y.sub.i] [subset] [R.sup.s] the production set associated with i's firm.
Recall that allocation ([x.sup.*.sub.1], ... [x.sup.*.sub.m], [y.sup.*.sub.1], ... [y.sup.*.sub.n]), [x.sup.*.sub.j] [member of] [R.sub.s.sub.+], j [member of] J, [y.sup.*.sub.i] [member of] [R.sup.s], i [member of] I, and price vector [p.sup.*] [member of] [R.sup.s.sub.+] constitute a competitive (or Walrasian) equilibrium if the following conditions are satisfied (Mas-Colell et al. [23]):
Profit maximization. For each firm i [member of] I, [y.sup.*.sub.i] solves
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]; (1)
Disutility minimization. For each consumer j [member of] J, [x.sup.*.sub.j] solves
min{[V.sub.j]([x.sub.j]): [less than or equal to] + [summation over I] [[kappa].sub.ij] , [x.sub.j] [member of] [R.sup.s.sub.+]}; (2)
Market clearing.
[summation over j [member of] J] [x.sup.*.sub.j] = [summation over j [member of] J] [[omega].sub.j] + [summation over i [member of] I] [y.sup.*.sub.i]. (3)
Now observe that the market-clearing condition in the case of positive prices [p.sup.*] > ([p.sup.*.sub.l] > for each l [member of] L)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4)
If we set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],
then (4) can be expressed in the form of the variational inequality
+ [PHI](p) - [PHI]([p.sup.*]) [greater than or equal to] 0, [for all] p [member of] [R.sup.s.sub.+], (5)
where [PHI] and [[phi].sub.j] are convex, nonnegative valued, positive homogeneous of degree 1 functions.
This observation has inspired the study of a more general problem in which the market-clearing condition (3) is no longer required and is replaced by the variational inequality (5). It states that the market clears for a commodity if its equilibrium price is positive. Otherwise, there may be an excess supply of the commodity in equilibrium, and then its price will be zero.
Thus, a more general problem can be stated: Find [{[x.sup.*.sub.j]}.sub.j [member of J] [subset] [R.sup.sm.sub.+] and [p.sup.*] [member of] [R.sup.s.sub.+] such that Disutility minimization. For each consumer j [member of] J, [x.sup.*.sub.j] solves
min{[V.sub.j]([x.sub.j]): [less than or equal] [[phi].sub.j]([p.sup.*]), [x.sub.j] [member of] [R.sup.s.sub.+]}; (6)
Balance condition.
+ [PHI](p) - [PHI]([p.sup.*]) [greater than or equal to] 0, [for all] p [member of] [R.sup.s.sub.+]. (7)
The purpose of this paper is to establish the existence of solutions for Problems (6)-(7) formulated in the setting of a reflexive Banach space.
There is a large literature on the general equilibrium model as given by (1)-(3) with both finite and infinite dimensional commodity spaces. For this issue, we refer the reader to Neumann [35], Nash [33], Arrow and Debreu [6], Arrow and Intrilligator [7], Aliprantis et al. [2], Chichilnisky and Heal [13], Chichilnisky [12], Nagurney and Siokos [28], Mas-Colell et al. [23], Mas-Colell [21], Mas-Colell and Richard [22], McKenzie [24], Gale and Mas-Colell [18], Negishi [34], Scarf [40], Eaves [16], Hirsh and Smale [19], Smale [41], Aliprantis et al. [5, 4, 3], and the references therein.
Let us recall that the main difficulty when studying the existence of competitive equilibria in an infinite dimensional setting is that for important Banach spaces the corresponding positive cone (commodity space) has an empty interior. In such a case, direct application of separation arguments for establishing equilibrium is not possible. To overcome this inconvenience a condition on preferences has been introduced, called [omega]-properness (Mas-Colell et al. [23], Chichilnisky and Kalman [14]) that compensates for the absence of interior points in the positive cone. Following this idea, many important results have been obtained in the setting of the Riesz commodity-price duality due to Aliprantis and Brown [1] by applying the lattice-theoretic techniques. In Aliprantis et al. [3] the vector lattice ordering of a commodity space has been weakened by assuming that the ordering is not of the lattice type.
Concerning extensions of the classical model, we refer to Aliprantis et al. [5] where the notion of generalized prices (nonlinear prices) has been introduced and new existence results involving this notion have been shown. In Aliprantis et al. [4] some examples of economy have been constructed in which there is no linear price equilibrium while the generalized price equilibrium does exist.
Another important direction concerning the development of the General Equilibrium Theory is related to nonconvex economics with infinite-dimensional commodity space satisfying the Asplund property (see Mordukhovich and Shao [27], Mordukhovich [25, 26], Borwein and Jofre [10], and the references quoted therein).
In all the aforementioned results on existence of equilibrium in topological ordered spaces the crucial point is the Mas-Colell's [omega]-properness condition or its variants. Moreover, the consumption sets are required to be of conic type or to span the commodity space (cf. Mas-Colell et al. [23]).
The presented approach differs considerably in the methods employed. The replacement of the market-clearing condition (3) by the variational inequality (5) clears the way for using variational and nonvariational techniques of nonlinear and convex analysis, such as the theory of pseudomonotone multivalued mappings, Fenchel duality, and Galerkin methods. These techniques have not yet been markedly applied in the discussed context. It is worth it to point out that by application of these methods we are able to deal successfully with some advanced problems, which could not be solved with the "order" techniques. We mean here, for instance, equilibrium problems in which preferences are determined by disutility functions that may attain their minima in consumption sets, or the situations in which the prices in equilibrium are so unfavorable for a consumer that his budget is equal to zero (Nockowska [36]). In such a case, a condition can be formulated on the preferences and the budget function for that consumer to prevent such an unfavorable situation.
With this new approach we are able to establish existence results in reflexive Banach space without assumptions that were fundamental for the currently used methods. The substantial difference of the presented approach is
(i) preferences are not required to satisfy any kind of [omega]-properness assumption and they may not be monotone. Indeed, disutility functions may attain their minima in the consumption sets;
(ii) The consumption sets are not required to have a cone structure. They are represented by the effective domains of the corresponding disutility functions that are assumed to be convex and lower semicontinuous. Thus the effective domains may be very small. The only requirement here is that zero is their cluster point.
Let X be a reflexive Banach space with its dual [X.sup.*] and the pairing over [X.sup.*] x X denoted by . Assume K [subset] X to be a closed, convex, pointed cone (i.e., K [intersection] (-K) = 0) with the associated positive dual [K.sup.+] = {[tau] [member of] [X.sup.*]: ([tau], x) [greater than or equal to] [for all] x [member of] K}. It must be stressed that K is not required to have nonempty interior. The problem to be studied reads as follows: Find or [pi] [member of] [K.sup.+] and [x.sub.j] [member of] K, j = 1, ..., m, such as to satisfy the conditions
(PM) [V.sub.j]([x.sub.j])=min{[V.sub.j](x): [less than or equal to] [[phi].sub.j]([pi]), x [member of] K}, j=1, ..., m,
(PE) + [PHI]([tau]) - [PHI]([pi]) [greater than or equal to] 0, [for all] [tau] [member of] [K.sup.+],
where [V.sub.j]: K [right arrow] R [union] {+ [infinity]}, [PHI]: [K.sup.+] [right arrow] R [union] {+ [infinity]} are convex lower semicontinuous functions, [[phi].sub.j]: [K.sup.+] [right arrow] [R.sub.+] are convex, continuous, positively homogeneous of degree 1 functions with nonnegative values on [K.sup.+].
It is worth pointing out that our study will be carried out without any lattice considerations. We do not assume disutility functions [V.sub.j] to be strictly convex and to have the whole K as their effective domains. Moreover, they are allowed to attain their minima on K.
A system (PM)-(PE) has been first studied in the finite dimensional setting in Naniewicz [29] under the assumption that [[phi].sub.j]([tau]) [greater than or equal to] [[differential].sub.j], [[differential].sub.j] > and in Naniewicz and Nockowska [31], where [[phi].sub.j, j = 1, ..., m, have been assumed to be positive homogeneous of an arbitrary positive degree [[theta].sub.j] > 0. See also Naniewicz [30], where existence results for such a system have been established in reflexive Banach spaces for weakly continuous [[phi].sub.j], and under stronger hypothesis concerning the regularity of [V.sub.j].
In this paper, special attention is paid to the case in which [[phi].sub.j], j = 1, ..., m, are convex positive homogeneous of degree 1 functions and [PHI] = [[summation].sup.m.sub.j=1] [[phi].sub.j], which is important for the study of competitive equilibrium problems.
The main feature of the aforementioned problem is that the feasible set for the unknowns [pi], [x.sub.j], j = 1, ..., m, is nonconvex and, in consequence, the standard theory of variational inequalities (cf. Kinderlehrer and Stampacchia [20], Ekeland and Temam [17]) does not work. The approach presented here is based, roughly speaking, on analyzing constrained minima ([x.sub.j]) as a function of [pi]. The resulting multivalued mapping R turns out to be pseudomonotone on the shifted price cone [K.sup.+] + [[pi].sub.0] (in the sense of Browder and Hess [11]). The theory of nonlinear monotone type mappings (cf. Browder and Hess [11]) allows us to establish the existence of solutions for the suitably constructed approximation problem (63)-(67) without a very restrictive hypothesis on the existence of nonempty interior of K. The limit procedure leads to Theorem 6.2 involving rather strong assumptions on [V.sub.j] and [[phi].sub.j]. In the following part of the paper these hypotheses are reduced by making use of the Yosida approximation technique and Galerkin method. Such procedure allows us to establish Theorem 8.3 as the main existence result. Some ideas concerning the treatment of nonmonotone variational inequality problems, developed in Naniewicz and Panagiotopoulos [32], appear to be very useful in this analysis.
The main result of the paper is Theorem 8.3. The following conclusions follow directly from this theorem:
COROLLARY 1.1. Assume X to be a reflexive Banach space and [PHI] = [[summation].sup.m.sub.j=1][[phi].sub.j]. Suppose the existence of a finite dimensional subspace [F.sub.0] of X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]; and assume ([A.sub.1]) [member of] cl(Dom([bar.[V.sub.j]])), j = 1, ..., m; ([A.sub.8]) [[summation].sup.m.sub.j=1][[phi].sub.j]([tau]) [greater than or equal to] [parallel][tau][parallel], [for all] [tau] [member of] [K.sup.+], [gamma] > 0; ([A.sub.9]) for at least one j [member of] {1, ..., m}, [not member of] [differential][bar.[V.sub.j]](0); ([A.sub.12]) ([[summation].sup.m.sub.j=1] [differential][bar.[V.sup.*.sub.j]]*(0)) [intersection] [DELTA] = [empty set], ([PHI](*) = [sup.sub.y [member of] [DELTA]] );
then there exist a nonzero price vector or [pi] [member of] [K.sup.+]\{0}, commodity bundles ([x.sub.j]) [member of] [K.sup.m], and a positive number r [member of] (0, 1] such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
and
+[PHI]([tau]) - [PHI]([pi]) [greater than or equal to] 0, [for all] [tau] [member of] [K.sup.+], (9)
where [??] is given by
j [member of] [??] [??] ([differential][bar.[V.sup.*.sub.j]](0) = [empty set], or > [[phi].sub.j]([tau]) for any [tau] [member of] [K.sup.+]\{0} and y [member of] [differential] [bar.[V.sup.*.sub.j]](0), if [differential] [bar.[V.sup.*.sub.j]](0) [not equal to] [empty set].
Moreover, if [??] = {1, ..., m}, then r = 1.
Now the question arises when the foregoing result implies the existence of a competitive equilibrium understood in the classical sense, i.e., when r = 1 and (9) reduces to the market-clearing condition. To answer this question, let us introduce [K.sup.+.sub.0] as a subset of price cone [K.sup.+] such that
[pi] [member of] [K.sup.+.sub.0] [??] ((x [member of] K and <[pi], x)=0) [??] x=0).
It is not difficult to check that if ([pi], ([x.sub.j])) is a solution of (PM)-(PE) and
(i) [XI] = {1, ..., m};
(ii)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for some [y.sub.j] [member of] [y.sub.j];
(iii) [pi] [member of] [K.sup.+.sub.0];
then ([pi], ([x.sub.j]), ([y.sub.i])) is a competitive equilibrium in the classical sense, i.e., (9) becomes the market-clearing condition [[summation].sub.j[member of]J][x.sub.j] = [[summation].sub.j[member of]J][[omega].sub.j] + [[summation].sub.i[member of]I][y.sub.i].
The requirements for the obtained result can be easily verified and interpreted from an economics point of view. They are certainly realistic for market equilibrium problems.
Since [differential][[bar.V].sup.*.sub.j](0) consists of all minimizers of [V.sub.j] on K, the condition j [member of] [XI] means that either [V.sub.j] does not attain its infimum on K, or for each nonzero price vector p the income [[phi].sub.j](P) of j's consumer is strictly lower than the value of his globally optimal commodity bundle at price p (the minimizer of his disutility function on K). Condition (iii) is related to what can be defined as strictly positive prices (in the infinite dimensional setting). It implies that if equilibrium price [pi] is strictly positive, then (PE) reduces to the market-clearing condition.
The rest of the paper is organized as follows: Section 2 contains all necessary definitions and preliminary material to be used in the study. In particular, problem (PM)-(PE) is reduced to the variational inequality with [pi] as the basic unknown. This inequality involves the multivalued operator R: [K.sup.+] [right arrow] [2.sup.X] playing the crucial role in the study. Section 3 is devoted to pseudomonotonicity of R on a shifted cone [K.sup.+] + [[pi].sub.0]. In [section]4, the existence result for the basic variational inequality on [K.sup.+] + [[pi].sub.0] has been established by applying the theory of multivalued pseudomonotone mappings due to Browder and Hess [11]. Section 5 deals with the approximation problem related to small parameter [epsilon] > 0, obtained by suitable extension of [V.sub.j] to the space X x R. In [section]6, the limit procedure has been performed to establish the existence of solutions for (PM)-(PE) under rather strong regularity hypotheses imposed on [V.sub.j] and [[phi].sub.j]. In [subsection] 7 and 8, these hypotheses are reduced by making use of the Yosida approximation technique and Galerkin method. Theorem 8.3 is established as the main existence result. The concluding [section]9.1 presents, among others, how to deduce from this theorem the existence of competitive equilibria in the classical sense both in finite and infinite dimensional settings.
2. Statement of the problem and preliminary results. Let X be a reflexive Banach space with its dual [X.sup.*]. The pairing over [X.sup.*] x X will be denoted by (*,*). Let K [subset] X be a closed convex cone of X. Denote by [K.sup.*] = {[tau] [member of] [X.sup.*]: [less than or equal to] [for all] x [member of] K} its negative polar cone and by [K.sup.+] = {[tau] [member of] [X.sup.*]: < [greater than or equal to] [for all] x [member of] K} its positive polar cone. It will be supposed that K [intersection] (-K) = {0}.
Throughout the paper we let
[V.sub.j]: X [right arrow] R [union] {+[infinity]}, j=1, ..., m, (10)
be convex, proper, and lower semicontinuous functions; and
[[phi].sub.j]: [K.sup.+] [right arrow] R with [[phi].sub.j] ([tau]) [greater than or...
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