Coalitional extreme desirability in finitely additive exchange economies

Abstract

We define a new notion of extreme desirability for economies in coalitional form. Through this, we obtain a finitely additive core-Walras equivalence theorem for an exchange economy with a measure space of agents and an infinite dimensional commodity space, whose positive cone has possibly empty interior.

Appendix

In this Appendix we will give an explanation of the fact that, in Theorem 3.1 it is possible to choose $G_1\in \Sigma ^+$, $G_1\subseteq F_1$ such that:

$$\begin{aligned} P(G_1)\Vert x_1\Vert \le \displaystyle \frac{\rho }{3}; \end{aligned}$$

$$\begin{aligned} \frac{x^*\beta (G_1)}{P(G_1)}\ge \frac{x^*\beta (F_1)}{P(F_1)}. \end{aligned}$$

Since we are assuming that P is strongly nonatomic, P satisfies the Darboux Property, that is for every $\tau >0$ and every $E\in \Sigma$ one can decompose E into finitely many disjoint $\Sigma$-measurable sets, each with probability P less than $\tau$.

All allocations are assumed to be absolutely continuous with respect to P in the $\varepsilon -\delta$-sense; hence $x^*\beta$ will also fulfill the Darboux Property. Since $x^*$ is a positive functional, we are reasoning on a ${\mathbb R}^2_+$-valued finitely additive measure on an algebra $\Sigma$.

By means of a Stone argument, $\Sigma$ is transformed into a pure algebra (i.e., containing no countable unions), the Stone algebra, where therefore P and $x^*\beta$ transfer to countably additive measures which we shall denote by $\widetilde{P}$ and $\widetilde{x^*\beta }$ . By a standard argument in measure theory, one can extend each of these two set functions to a nonnegative measure on the generated $\sigma$-algebra, and this measure will automatically inherit the Darboux Property; let us denote by $\widetilde{\widetilde{P}}$ and $\widetilde{\widetilde{x^*\beta }}$ these two further extensions (see Martellotti 2001, Sect. 2, for a complete treatment of this construction).

Now we are in a countably additive setting, where all forms of nonatomicity are equivalent.

The Liapounoff’s Theorem then implies that the range of the pair $\left( \widetilde{\widetilde{P}},\widetilde{\widetilde{x^*\beta }}\right)$ is a compact convex subset of the positive orthant which contains the origin and is symmetric w.r.t the middle point of the line joining the origin with $\left( \widetilde{\widetilde{P}}(\Omega ),(\widetilde{\widetilde{x^*\beta }})(\Omega )\right)$. Classically, the range of a finite dimensional, nonatomic, countably additive measure is called a zonoid (see Bolker 1969).

Furthermore, we know that the image under $(\widetilde{P},\widetilde{x^*\beta })$ of the Stone algebra is dense in this zonoid, as well as we know that the image of the Stone algebra under $(\widetilde{P},\widetilde{x^*\beta })$ precisely coincides with the range of $(P,x^*\beta )$.

In conclusion, the image of $\Sigma$ under the pair $(P,x^*\beta )$ is dense in a zonoid of ${\mathbb R}^2$.

Due to its symmetry properties, a two-dimensional zonoid (which, in the case of nonnegative measures, is a subset of the positive orthant) will look like the “leaf” in the picture below.

The above argument can be analogously applied to the coalition $F_1$ instead of the whole grand coalition $\Omega$.

Hence the image of the trace algebra $\Sigma _{F_1}$ under the pair $(P,x^*\beta )$ is dense in a form of the type in the following picture, where the endpoint P has coordinates $(P(F_1),x^*\beta (F_1))$.

Now it is enough to note that the ratios involved in the inequality:

$$\begin{aligned} \frac{x^*\beta (G_1)}{P(G_1)} \ge \frac{x^*\beta (F_1)}{P(F_1)} \end{aligned}$$

represent the slope of the segments joining O with Q, if $Q = (P(G_1), x^*\beta (G_1))$, and O with P.

Hence, the two requirements simply reduce to finding a set on the upper part of the leaf, so that the slope of the joining segment is greater than that of the diagonal, and with first coordinate $P(G_1)$ small enough.