New concepts and techniques for equilibrium analysis
In the study of the existence of an equilibrium for a private ownership economy, one meets with the basic mathematical difficulty that the demand correspondence of a consumer may not be upper semicontinuous when his wealth equals the minimum compatible with his consumption set. One can prevent this minimum-wealth situation from ever arising by suitable assumptions on the economy; for example, in K. J. Arrow and G. Debreu [1], Theorem I, it is postulated that free disposal prevails and that every consumer can dispose of a positive quantity of every commodity from his resources and still have a possible consumption. However, assumptions of this type have not been readily accepted on account of their strength, and this in spite of the simplicity that they give to the analysis. Thus A. Wald [11, (Section II)]; K. J. Arrow and G. Debreu [ 1, (Theorem II or II')]; L. W. McKenzie [7], [8], [9]; D. Gale [4]; H. Nikaido [10]; and W. Isard and D. J. Ostroff [5] permit the minimum-wealth situation to arise but introduce features of the economy that nevertheless insure the existence of an equilibrium.