The fundamental theorems of welfare economics without proper preferences (Q908833)

\textit{G. Debreu} in a classical paper [Proc. Nat. Acad. Sci. USA 40, 588- 592 (1954; Zbl 0055.380)] showed that, in infinite-dimensional commodity spaces, Pareto-optimal allocations can be supported by prices as a valuation equilibrium. (This is known as the second welfare economics theorem.) This result was proved by using the Hahn-Banach (H-B) theorem, and the assumption that the production set has an interior point played an important role (i.e., was needed to make the H-B theorem applicable). Recently, \textit{A. Mas-Colell} [in: Contributions to mathematical economics, 317-331 (1986; Zbl 0664.90016)] generalized Debreu's result. In particular, he replaced the interiority assumption of Debreu by the uniform properness condition. \textit{M. A. Khan} and \textit{R. Vohra} [J. Approximation Theory 52, No.2, 149-161 (1988; Zbl 0645.90008)] showed that even without interiority or uniform properness conditions a Pareto-optimal allocation can be approximately price supported as a valuation equilibrium. The authors of the present paper build on the work of Khan-Vohra in order to characterize Pareto-optimal allocations in the absence of interiority or uniformity conditions. In particular, they show that an allocation is weakly Pareto-optimal if and only if it can be approximately price supported as a valuation equilibrium. The paper uses extensively results from the authors' books on Riesz spaces [see in particular ``Positive operators'' (1985; Zbl 0608.47039)] and it will be of interest to economists and mathematicians.