Excess demand function

As excess demand function is called in economics a mathematical function for describing a market. It indicates for a given budget, how much his demand for each good its original equipment with the relevant item exceeds (individual excess demand function).

Formal definition

Be the vector of prices of all n goods of the economy and is the initial endowment of consumer i is the Good k The amount of consumers. It then summarizes the individual equipment to an individual vector features together. Let further

The Marshallian demand of household i ( with respect to all goods).

Definition: The (individual ) excess demand function of household i, is given by

And the aggregate excess demand function of the economy, by

Given.

If the k- th element of negative, prevails in the economy an excess supply of good k You can (somewhat less common ) directly define a goods -specific excess demand function. These are for a given good k at how high the related excess demand in the economy; formal:

Properties

Properties of the aggregate excess demand function: Let the utility function of each consumer steadily, monotonically increasing and strictly quasikonkav; should also. Then:

The properties 1 and 2 follow directly from the properties of Marshallian demand function. The homogeneity of degree zero is obvious, because the unit prices for the excess demand should be irrelevant: While Will approximately determined that there is no euros but only euro cents, as prices rise in the observation unit by a factor of 100, however one would not expect that this will affect the quantities of goods that are offered or asked for. Property 3 holds because

Where the last equality follows because for every consumer, the budget constraint must be satisfied with equality ( strict monotonicity of the utility function ). Intuitive: A benefit -maximizing household will always exert a demand to the extent of its total assets, which is why the value of its individual excess demand must be zero; but must be zero as a consequence, the value of the aggregate excess demand of all households.