فرمول Riesz-Kantorovich و نظریه تعادل عمومی

Let L be an ordered topological vector space with topological dual L′ and order dual L~. Also, let f and g be two order-bounded linear functionals on L for which the supremum f∨g exists in L. We say that f∨g satisfies the Riesz–Kantorovich formula if for any 0≤ω∈L we have This is always the case when L is a vector lattice and more generally when L has the Riesz Decomposition Property and its cone is generating. The formula has appeared as the crucial step in many recent proofs of the existence of equilibrium in economies with infinite dimensional commodity spaces. It has also been interpreted by the authors in terms of the revenue function of a discriminatory price auction for commodity bundles and has been used to extend the existence of equilibrium results in models beyond the vector lattice settings. This paper addresses the following open mathematical question: ⋅ Is there an example of a pair of order-bounded linear functionals f and g for which the supremum f∨g exists but does not satisfy the Riesz–Kantorovich formula? We show that if f and g are continuous, then f∨g must satisfy the Riesz–Kantorovich formula when L has an order unit and has weakly compact order intervals. If in addition L is locally convex, f∨g exists in L~ for any pair of continuous linear functionals f and g if and only if L has the Riesz Decomposition Property. In particular, if L~ separates points in L and order intervals are σ(L,L~)-compact, then the order dual L~ is a vector lattice if and only if L has the Riesz Decomposition Property — that is, if and only if commodity bundles are perfectly divisible.

It has for sometime been well-understood that one cannot hope to prove the existence of general equilibrium — or establish the validity of the welfare theorems — under the standard finite dimensional assumptions when the commodity space is infinite dimensional and consumption sets lack interior points. In this literature, the commodity space is most often a Riesz space (vector lattice) and primitive data of the economy are supposed to satisfy various assumptions known as “properness conditions” (see Aliprantis et al., 1990Aliprantis et al., 2000). A distinctive feature of this literature is the non-trivial use of the lattice structure of the commodity space. Indeed, Aliprantis and Burkinshaw (1991) show that when the commodity space is a vector lattice, the lattice structure of the dual space is basically equivalent to the validity of the welfare theorems.1 Furthermore, the various proofs in this literature can be delineated by means of the Riesz Decomposition Property of the commodity space. For example, Mas-Colell (1986) and Aliprantis et al. (1987) use the Decomposition Property to facilitate a separating hyperplane argument, while Yannelis and Zame (1986) use the property to show the continuity and extendibility of the equilibrium prices of truncated economies. This is in sharp contrast to the case where consumption sets are assumed to have interior points and where the existence of a continuous quasi-equilibrium price can be proven with little reference to the lattice structure of the commodity space (see for example, Bewley, 1972; Florenzano, 1983). In the more recent approach of Mas-Colell and Richard (1991) and Richard (1989) (see also Deghdak and Florenzano, 1999; Podczeck, 1996; Tourky, 1998 and Tourky, 1999) the Decomposition Property is used in an indirect manner. Here, the authors consider economies in the more general setting of a Riesz commodity space that need not be locally solid. In this setting a supporting hyperplane argument in the space of allocations furnishes a list of prices and the crucial part of the proof is showing that the supremum of these prices is indeed the required supporting (equilibrium) price. In this second group of papers, the Decomposition Property is used through two of its consequences. First, the fact that the order dual of the commodity space is a vector lattice and second the Riesz–Kantorovich formula for calculating the supremum of any two order-bounded linear functionals. It is also quite clear that the decentralization arguments in these more recent papers go through with little fuss if both of these properties are present, and without regard to whether the commodity space is a Riesz space or has the Riesz Decomposition Property. This observation was recently made by Aliprantis et al. (1998), who extended the literature on the existence of equilibrium and on the welfare theorems in infinite dimensional spaces to commodity spaces that are not lattice ordered. They were able to drop the requirement that both the commodity space and the price space be lattice ordered by introducing a new class of non-linear prices based on the Riesz–Kantorovich formula. They also provide concrete economic interpretations to the Riesz Decomposition Property and the Riesz–Kantorovich formula. The Riesz–Kantorovich formula coincides with the revenue function of a discriminatory price auction — a generalization of the revenue function of the single commodity US Treasury Bill Auction. They interpreted the Riesz Decomposition Property (and its extension termed “Consumption Decomposability”) as the perfect divisibility of commodity bundles and showed that in the presence of such perfectly divisible bundles the revenue function of the auctioneer is always linear. Motivated by these observations, we address in this paper the following long standing open mathematical question. ⋅ Is there an example of a pair of order-bounded linear functionals for which the supremum exists but does not satisfy the Riesz–Kantorovich formula? We consider an arbitrary ordered vector space L with order unit and weakly compact order intervals. We show if f and g are continuous linear functionals on L and f∨g exists in the order dual L~, then f∨g must satisfy the Riesz–Kantorovich formula. Therefore, we provide a negative answer to our question in the important setting of an ordered vector space with order unit and σ(L,L~)-compact order intervals. We also show that if, in addition, L is locally convex with weakly compact intervals, then f∨g exists in L~ for any pair of continuous linear functionals f and g if and only if L has the Riesz Decomposition Property (see also Andô, 1962). In particular, if L̃ separates points in L and order intervals are σ(L,L~)-compact then L~ is a vector lattice if and only if L has the Riesz Decomposition Property — hence, if and only if commodity bundles are perfectly divisible. Commodity spaces with order units often arise in the study of economies with infinitely many commodities even when the underlying commodity space does not have an order unit. Consider an exchange economy with an ordered vector space L as a commodity space, with total endowment ω≥0, and consumption sets that coincide with the positive cone of L. Since all economic activity takes place inside the Edgeworth box [0,ω], it is often useful to restrict the commodity space to the ideal Lω generated by ω. This ideal consists of all bundles that are dominated by multiples of ω, i.e., The space Lω is the canonical example of a commodity space that satisfies our assumptions. For an extensive analysis of economies truncated to Lω, see Aliprantis et al. (1987). The paper is organized as follows. The mathematical preliminaries are outlined in Section 2. Our main results are in Section 3. In Section 4we show the usefulness of our main results in the theory of value with non-linear prices as developed by Aliprantis et al. (1998). We show that the perfect divisibility of commodity bundles, the linearity of the non-linear prices, and the lattice ordering of the order dual are equivalent.