The Natural Projection and Envelope Theory

Yves Balasko

Chapter 8 in Foundations of the Theory of General Equilibrium, 2016, pp 143-164 from World Scientific Publishing Co. Pte. Ltd.

Abstract: A large part of the results of the previous two chapters can be summarized by saying that equilibrium prices do not show significant qualitative changes as long as the initial endowments vary within the same connected component of the set of regular economies. Discontinuities in terms of prices as well as changes in the number of equilibria can be observed only when initial endowments are varied in order to come across singular economies. The main goal of qualitative comparative statics is therefore one of improving our understanding of the nature of these discontinuities of the price system. This requires a thorough investigation of the singularities of the natural projection, which goes much further than the questions investigated in Chapters 6 and 7 where only the most elementary tools of singularity theory had to be used. More precisely, the properties of equilibria that have been stated so far are consequences of the economic interpretation of such general mathematical concepts as those of critical points, singular values or degree, applied to the natural projection. Almost no use has been made of more specific features among which the fiber bundle structure of the equilibrium manifold should undoubtedly play a prominent role.

Keywords: Economics; Economic theory; Equilibrium; Singularities; Equilibrium Manifold; General Equilibrium; Natural Projection; Arrow-Debreu Model; Walras Equilibrium; Consumer Theory; Multiple Equilibria; Unique Equilibrium (search for similar items in EconPapers)
JEL-codes: D50 (search for similar items in EconPapers)
Date: 2016
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