# Existence of Equilibria

## Abstract

Demonstrating the existence of competitive equilibria is the first and most fundamental check whether a model based on the notion of a competitive equilibrium is well defined. One would like to be able to do this for the most general assumptions. There are various ways how to proceed in proving the existence of competitive equilibria. In this book we follow the so called *excess demand approach*. See Debreu [1982] for a survey of approaches to prove existence of competitive equilibria. The excess demand approach is interesting in itself since it is based on deriving properties of excess demand that are useful also for more elaborate questions like uniqueness of equilibria. To get the idea of the existence theorem consider the graph of an excess demand as depicted in Figure 2.1. Suppose there are just two commodities so that a zero excess demand of the commodity considered in this graph is by Walras Law already a competitive euqilibrium. Now suppose that the graph is continuous and that it satisfies the boundary behavior. Then from the intermediate value theorem there exists an equilibrium. In this chapter we show how this idea can be made more general and more precise also. The discussion begins with a series of examples which show that, if markets are incomplete, the standard Walrasian assumptions on the agents’ characteristics are no longer sufficient to guarantee Walras Law, Continuity or Boundary behavior of excess demand.

## Keywords

Asset Price Excess Demand Competitive Equilibrium Boundary Behavior Incomplete Market## Preview

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