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By Donald G. Saari (auth.), Prof. C. D. Aliprantis, Prof. O. Burkinshaw, Prof. N. J. Rothman (eds.)

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Read Online or Download Advances in Equilibrium Theory: Proceedings of the Conference on General Equilibrium Theory Held at Indiana University-Purdue University at Indianapolis, USA, February 10–12, 1984 PDF

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Additional resources for Advances in Equilibrium Theory: Proceedings of the Conference on General Equilibrium Theory Held at Indiana University-Purdue University at Indianapolis, USA, February 10–12, 1984

Example text

As for [f], Example 6 (which has monotone preferences) will do: EXAMPLE 6. See Figure 7. Both preference relations are monotone. , 0 and so p·z > 0 = P·xi). Summing up: [Xl Under [dl. [el anti [f] if :x: is a quasiequiZibril4Tl üJith respeat to p;l O. then :x: is an equiZibril4Tl üJith respeat to p (hence. :x: is an optimum). 6. EQUILIBRIA PROPERTIES OF OPTIMA To what an extent can we now reverse the direction of our conc1usions and assert equi1ibrium properties for arbitrary optima? g. Debreu, 1959) and that, therefore, the proper setting for the question is one where the convexity hypotheses are made throughout.

F(a*) and suppose, to the contrary, that a* E (K'~). *)ä = o. *} implies that there exists ~* E Q(>"*) such that ~~ < O. *. , o E Q(X*) •• 5. , the collection of commodities that each brings to trade), and their preferences.

Once it is known that U is convex, the second claim follows immediately from the Supporting Hyperplane Theorem •• The previous Proposition is illustrated in Figure ll(c). Note that, as u' in the figure, an optimum may not be supportable by a A» 0, even in the strictly monotone case. Observe also that the set of supporting weights for a given u is always a convex cone. Taking stock of developments so far, we see that in the case where preferences admit concave utility representations, any weak optimum can be supported by prices (results [XII] and [XIII]) and by a vector of utility weights (result [XIV]).

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