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5 - Valuation equilibrium and Pareto optimum

Published online by Cambridge University Press:  05 January 2013

Gerard Debreu
Affiliation:
University of California, Berkeley
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Summary

For an economic system with given technological and resource limitations, individual needs and tastes, a valuation equilibrium with respect to a set of prices is a state where no consumer can make himself better off without spending more, and no producer can make a larger profit; a Pareto optimum is a state where no consumer can be made better off without making another consumer worse off. Theorem 1 gives conditions under which a valuation equilibrium is a Pareto optimum. Theorem 2, in conjunction with the Remark, gives conditions under which a Pareto optimum is a valuation equilibrium. The contents of both theorems (in particular that of the first one) are old beliefs in economics. Arrow and Debreu have recently treated this question with techniques permitting proofs. A synthesis of their papers is made here. Their assumptions are weakened in several respects; in particular, their results are extended from finite dimensional to general linear spaces. This extension yields as a possible immediate application a solution of the problem of infinite time horizon (see sec. 6). Its main interest, however, may be that by forcing one to a greater generality it brings out with greater clarity and simplicity the basic concepts of the analysis and its logical structure.

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Chapter
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Mathematical Economics
Twenty Papers of Gerard Debreu
, pp. 98 - 104
Publisher: Cambridge University Press
Print publication year: 1983

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