scholarly journals Regular Infinite Economies

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Enrique Covarrubias
Keyword(s):  
Demand Function ◽  
Excess Demand ◽  
Natural Projection ◽  
Banach Manifold ◽  
Equilibrium Manifold ◽  
Equilibrium Set ◽  
Regular Economies

The main contribution of this paper is to place smooth infinite economies in the setting of the equilibrium manifold and the natural projection map à la Balasko. We show that smooth infinite economies have an equilibrium set that has the structure of a Banach manifold and that the natural projection map is smooth. We define regular and critical economies, and regular and critical prices, and we show that the set of regular economies coincides with the set of economies whose excess demand function has only regular prices. Generic determinacy of equilibria follows as a by-product.

The Broad Picture

2011 ◽  
Author(s):  
Yves Balasko
Keyword(s):  
Demand Function ◽  
Exchange Model ◽  
Natural Projection ◽  
Demand Functions ◽  
Regular Economies

This chapter shows that the m-tuple (fi) of demand functions defining the exchange model belongs to ε‎r, i.e., the demand function fi is bounded from below (B) for every consumer and satisfies desirability (A) for at least one consumer. These additional properties will give to the natural projection the very important property of properness. The combination of smoothness and properness will suffice to yield what is now known as the theory of regular economies.

The Exchange Model

2011 ◽  
Author(s):  
Yves Balasko
Keyword(s):  
General Equilibrium ◽  
Smooth Manifold ◽  
Comparative Statics ◽  
Exchange Model ◽  
Natural Projection ◽  
Equilibrium Models ◽  
Manifold Structure ◽  
Equilibrium Manifold ◽  
Complex Models ◽  
Smooth Map

The exchange model is the simplest of all general equilibrium models. Studying it will show us the directions to follow when studying more complex models like those that include production or take explicitly into account time and uncertainty. This chapter introduces the exchange model defined by the equilibrium manifold and the natural projection. It presents proof that the equilibrium manifold is indeed a smooth manifold. The smooth manifold structure implies that the natural projection is a smooth map. In terms of comparative statics, this tells us that equilibrium prices can be considered as depending linearly on the fundamentals defining an economy in sufficiently small neighborhoods of regular equilibria.

The Fine Picture

2011 ◽  
Author(s):  
Yves Balasko
Keyword(s):  
Demand Function ◽  
Terms Of Trade ◽  
Exchange Model ◽  
Full Measure ◽  
Revealed Preferences ◽  
Trade Intensity ◽  
Demand Functions ◽  
Regular Economies ◽  
Uniqueness Of Equilibrium ◽  
Weak Axiom

This chapter restricts the m-tuple (fi) of demand functions defining the exchange model to belong to ε‎c. In addition to the assumptions made in the previous chapters (recall that ε‎c is a subset of ε‎r), the demand function fi satisfies the weak axiom of revealed preferences for every consumer, and the slightly stronger negative definiteness of the Slutsky matrix for the consumer whose demand function satisfies desirability (A). These stronger assumptions are aimed at giving more economic flesh to the exchange model. As a consequence, the natural projection inherits much stronger properties that give a specificity of its own to the exchange model. The most important properties of the exchange model with (fi) ɛ ε‎c are the regularity of the no-trade equilibria, the openness and full measure (a.k.a., the genericity) of the set of regular equilibria as a subset of the equilibrium manifold, the inclusion of the set of equilibrium allocations in one and only one connected component of the set of regular economies, the uniqueness of equilibrium for all economies belonging to that component, and the interpretation of that property in terms of trade intensity.

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