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A classic characterization of competitive equilibria views them as feasible allocations maximizing a weighted sum of utilities. It has been applied to establish fundamental properties of the equilibrium notion, such as existence, determinacy, and computability. However, it fails for economies with missing financial markets. We give such a characterization for economies with a single commodity and missing financial markets, by an amended social welfare function. Its parameters capture both the relative importance of households welfare, through the classic welfare weights, as well as the disagreements among them as to the value of the missing markets. As a by-product, we identify the dimension of the set of interior equilibrium allocations.

If no financial markets are missing, following Lange (1942) and Allais (1943), interior allocations of given resources rr are competitive equilibria if and only if they solve the program maxΣxh=rWδ(x)maxΣxh=rWδ(x) for some strictly positive δδ, with WδWδ being the social welfare function1 equation(1) Wδ(x)≔Σδhuh(xh).Wδ(x)≔Σδhuh(xh). Turn MathJax on The parameters δδ in Lange’s social welfare function capture the relative importance of households’ welfare. This characterization has been applied to establish fundamental properties of the equilibrium notion: existence, Negishi (1960) and Bewley (1969), determinacy with infinitely long living households, Kehoe and Levine (1985), and computability, Mantel (1971). If some financial markets are missing as in Radner (1972), however, this equivalence fails: some interior competitive equilibria need not solve the program maxΣxh=rWδ(x)maxΣxh=rWδ(x) for any strictly positive δδ. Moreover, no natural social welfare function WW has been found that would rescue this implication. We extend the characterization to economies with some missing financial markets, by amending the social welfare function. Thus interior allocations of given resources are competitive equilibria if and only if they solve the program maxΣxh=rWδ,μ(x)maxΣxh=rWδ,μ(x) for some parameters δ∈D,μ∈Mδ∈D,μ∈M living in certain spaces, with Wδ,μWδ,μ being the social welfare function equation(2) View the MathML sourceW(x)≔Σδhuh(xh)−Σμh⋅x1h. Turn MathJax on Here, the social evaluation of allocations is described by the usual weights δδ on households’ welfare, and by new charges μμ on their future consumption. The parameter δδ is interpreted classically, whereas μμ is interpreted as the “disagreement” among households as to the “value” of the “missing financial markets”, as justified below. Why does it fail, the equivalence of competitive equilibria and maxima of (1), if some financial markets are missing? On the one hand, any allocation xx that maximizes this is Pareto efficient. Indeed, if yy were Pareto superior to xx, i.e. (uh(yh))>(uh(xh))(uh(yh))>(uh(xh)), then Wδ(y)>Wδ(x)Wδ(y)>Wδ(x) for any δ≫0δ≫0, so xx could not be a maximum for any δ≫0δ≫0. On the other hand, some allocations xx that are competitive equilibria of incomplete financial markets are Pareto inefficient. Indeed, for almost every initial allocation, every competitive equilibrium allocation is Pareto inefficient; for an exposition of this well-known fact, see Magill and Quinzii (1996).2 So some competitive equilibria fail to maximize (1) for any δ≫0δ≫0. We explain in what sense the parameter μμ is the “disagreement” among households as to the “value” of the “missing financial markets”, by clarifying each of these terms. By “missing financial markets” we mean the orthogonal complement a⊥a⊥ of the span of the existing financial instruments aa. By “value” of the missing financial markets we mean a linear functional v:a⊥→Rv:a⊥→R. The separating hyperplane theorem implies that any linear functional on a finite-dimensional vector space can be represented uniquely as the inner product against a unique element of the vector space—call this element View the MathML sourcevˆ∈a⊥, so that View the MathML sourcev(m)=m⋅vˆ. If each household hh thinks such a valuevhvh, the disagreement is then the differences from the mean, View the MathML sourceμh≔vˆh−mean(vˆ1,…,vˆH). When so defined, the disagreement μ=(μh)μ=(μh) satisfies two properties: (i) μh∈a⊥μh∈a⊥, because it is a linear combination of points View the MathML sourcevˆh∈a⊥ in a vector space, and (ii) Σμh=0Σμh=0, because these are differences from the mean. In sum, imagining that each household has its own vhvh, an opinion as to the value of the missing financial markets, then this is the sense of the new parameter in our social welfare function (1)—a matrix μ=(μh)μ=(μh) satisfying conditions (i), (ii). Our main contribution is a fine characterization of the set XX of interior competitive equilibrium allocations of all the economies parameterized by initial endowment distributions (eh)h(eh)h of given state-contingent, aggregate resources rr. In doing so, besides rr, we take as given smooth preferences uu as in Debreu, 1972 and Debreu, 1976, and the asset structure formed by finitely many financial instruments aa. The characterization is accomplished in steps, establishing three results. The first result (Theorem 1) is that an allocation x≫0x≫0 is an equilibrium allocation if and only if it solves the program maxΣxh=rWδ,μ(x)maxΣxh=rWδ,μ(x) for some (δ,μ)∈D×M(δ,μ)∈D×M, where View the MathML sourceD≔{δ∈RH∣δ≫0,Σ1δh=1}M≔{μ∈(a⊥)H∣Σμh=0}. Turn MathJax on We see that the “welfare” parameter δδ is normalized in a standard way, and the “disagreement” parameter μμ reflects properties (i) and (ii) above. The second result (Proposition 1, part A) identifies the (δ,μ)(δ,μ) from the equilibrium allocation as being equation(3) View the MathML sourceδh(x)=1Dx0uh(xh) Turn MathJax on View the MathML sourceμh(x)=vˆh−mean(vˆ1,…,vˆH)with vˆh≔Dx1uh(xh)Dx0uh(xh). Turn MathJax on Thus δhδh is the inverse of the marginal utility of present consumption, as usual, and μμ is, as interpreted above, the disagreement among households as to the value of the missing financial markets, where each household’s “value” View the MathML sourcevˆh is concretized as the marginal rates at which it substitutes consumption in future states for consumption in the present state. Here, the abstract notion of “value” as a linear functional v:a⊥→Rv:a⊥→R is made concrete by the idea of marginal willingness to pay as View the MathML sourceΔ↦Δ⋅MRS, the inner product of the infinitesimal change View the MathML sourceΔ in future consumption against the marginal rates of substitution View the MathML sourceMRS. The third result refines the first two. Theorem 2 establishes that the relation x↔(δ,μ)x↔(δ,μ) between XX and D×MD×M is a bijection, smooth in both directions. This implies immediately that the dimension of XX equals the dimension of D×MD×M, which is easily shown to be (H−1)(1+m)(H−1)(1+m), where mm is the number of missing financial markets. This nests a well-known fact about complete markets, where m=0m=0: the interior Pareto optima (which are XX by the two welfare theorems) have dimension H−1H−1; see proof 5.2.4 in Balasko (1988). We focus attention on an exchange economy that has a single good per state and in which asset payoffs are denominated in the numéraire. Although restrictive, this context is interesting both theoretically and for financial applications. Theoretically, because it allows one to concentrate on financial markets, leading aside issues concerning spot markets. As for applications, a single good suffices to embody, in a general equilibrium model, classical models of asset pricing such as the CAPM. Extensions of our characterization to economies with multiple goods and spot markets are substantially available in two other works, Siconolfi and Villanacci (1991) and Tirelli (2008), whose understanding of the geometry of the equilibrium set is instrumental, respectively, in the study of the indeterminacy and of the welfare properties of equilibria also in the sense of Geanakoplos and Polemarchakis (1986). With respect to these papers, our single-good characterization is simpler, proposing a welfare function that differs from the complete market analogue, Wδ(x)Wδ(x), only by a term that is linear in the allocation xx. As often happens, a greater level of simplicity implies a loss of generality. Yet, the methodology carefully spelled out in this paper to derive our equilibrium characterization extends to other possible formalizations of the welfare function, such as those proposed in the existing literature. The rest of the paper is organized as follows. Section 2 spells out the model and assumptions. Section 3 develops the characterization. Section 4 refines the characterization, computing the dimension of XX. Section 5 contains the more formalistic and less insightful proofs.