Sonnenschein–Mantel–Debreu theorem

As sunshine Coat Debreu theorem is known in microeconomics and there, especially in the theory of general equilibrium one goes back to Hugo Sonnenschein, Rolf coat and Gerard Debreu theorem. It says to put it simply, that the aggregate excess demand functions that belong to one constructed using standard assumptions of general equilibrium model, only about some certain general properties have, but otherwise no concrete statements about their shape are possible.

Classification

After mainly from the mid-1950s, starting with Arrow and Debreu (1954 ), a variety of existence theorems for Walrasian ( ie: competitive ) equilibria had been formulated and was furthermore shown variously that the resulting equilibrium allocations regularly sweeping assumptions in any case are finite in number, the question whether from the parameters of economics underlying further conclusions about the nature of the resulting equilibrium can be derived set. Specifically, this problem takes on an equilibrium analysis by excess demand functions of the following form: Which properties can be obedient from the common assumptions pure exchange economy

For the characterization of aggregate excess demand

( with the Marshallian demand of i with respect to all goods) to meet?

Several general properties of aggregate excess demand can be derived from common assumptions, such as the following lemma shows.

The Sonnenschein -Mantel- Debreu theorem states simplified, but that it is not possible to derive further properties of aggregate excess demand, without putting more restrictive assumptions.

Theorem

Theorem: Let steadily, homogeneous of degree zero in and satisfies the Walras law. Then, for every k consumers with a continuous, strictly quasikonkaven and not falling utility function and an equipment vector whose excess demand function is, and this for all prices, valid for that for all.

This is both a generalization and a fit in the Arrow- Debreu framework, based on an earlier proof of mantle (1966 ), which dates back to the turn assisted by Sonnenschein ( 1973). Shell version of the theorem reads as follows ( with H the set of all for which it holds that, and with the multitude of those who produce non-negative cost for any price vectors, ie ):