The cone condition,properness, and extremely desirable commodities

Summary This note links two conditions which have generated some interest in the literature and have an important role in proving the existence of an equilibrium, the second welfare theorem and the core equivalence theorem in infinite dimensional commodity spaces: These are thecone condition introduced in Chichilnisky and Kaiman (1980), and theproperness condition in Mas-Colell (1986a), which were studied also in Yannelis and Zame (1986), Aliprantis et al. (1987a, b) and (1989), Aliprantis and Burkinshaw (1988), Mas-Colell (1986b), Chichilnisky and Heal (1984, 1992), and Rustichini and Yannelis (1991) among others. I establish that these two conditions are the same. Indeed, the cone condition coincides also with the assumption of anextremely desirable commodity used in Yannelis and Zame (1986) and Rustichini and Yannelis (1991). The motivation for studying these conditions comes from the same economic application, showing the need to bring within the scope of equilibrium theory commodity spaces whose positive orthants have empty interior, a typical situation in infinite dimensional linear spaces.I thank the participants of several seminars and workshops, two anonymous referees, CD. Aliprantis, D. Brown, D. Cass, D. Foley, G. Heal, H-M. Wu, A. Khan, R. Radner, A. Rustichini and N. Yannelis, for their comments and suggestions. Hospitality and financial support from the Stanford Institute for Theoretical Economics during the summer of 1991 is gratefully acknowledged.