### G-84-08

# Exogeneous Versus Endogeneous Variables: A General Approach to Mixed Systems

## Edouard Wagneur

The problem of exogenity in economics is a highly relative one. For example in microeconomics (production theory or consumer theory) quantities are exogeneous with respect to prices (and income in the case of consumer theory) and conversely. In other words, as long as there is an invertible relation between quantities and prices (and income), whether we consider prices (and income) or quantities as the signal to which the agents react is rather a matter of taste or opportunity. The question of exogenity really arises when we want to know if, say, the prices of the first k goods together with the quantities of the last n-k ones may be considered exogeneous with respect to the other variables (e.g. the quantities of the first k goods and the prices of the n-k last ones, together with income). The mathematical setting of the problem can be given the following form: Given a functional relation φ between x = (x^{1}, ..., x^{n}) ∈ ℝ^{n} and y = (y^{1}, ..., y^{n}): y = φ(x), what are the properties of φ which allow for the existence of a map X such that

i) (y^{1}, y^{2}, ..., y^{k}, x^{k+1}, ..., x^{n}) = X(x^{1}, ..., x^{k}, y^{k+1}, ..., y^{n})

ii) X and φ have same graph in ℝ^{2n}.

In consumer theory, the problem has been dealt with recently to give rise to what is known as "mixed demand" (C. Bronsard and L. Salvas Bronsard (1979), J.P. Chavas (1983).

Published **May 1984**
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14 pages