# Index of an Excess Demand Vector

Mas-Colell, Whinston and Green, in Microeconomic Theory (third edition), postulate the concept of an index for an excess demand vector, which is later used in the Index Theorem:

A regular equilibrium of the economy is defined as the following:

$$D$$ denotes derivative (here, in respect to price), while $$z$$ denotes the excess demand matrix:

Where $$p$$ denotes price and $$\omega$$ denotes endowment.

What does the concept of index means?

• Index theorem is a result in differential topology. What MWG has is a very informal discussion for empirical economists. Loosely speaking, an index is how many times a map on a manifold "winds around". The map in this particular case is the excess demand function. You should consult an introduction to differential topology and consider asking the question on Math SE, where it's much more likely to find people qualified to address this. – Michael Jul 16 '19 at 22:59
• @Michael I have never heard of the index theorem in topology before. Is it the Atiyah-Singer index theorem? – Bruno Schiavo Jul 16 '19 at 23:04
• No, it's more elementary, I believe. Atiyah-Singer says that the topological index of a manifold---an intrinsic quantity of the manifold, computed via Chern classes, or what have you---is the Fredholm index of certain canonical Fredholm operator acting on vector bundles on that manifold. A (very) special, and basic, case is the Toepliz operator on the circle. OTOH, the index theorem being applied in GE is a statement about the index of a map between manifolds. They are different statements. – Michael Jul 16 '19 at 23:11
• I would try Differential Topology by Milnor ("degree modulo 2 of a map"), and as I said, Math SE. – Michael Jul 16 '19 at 23:37
• You may want to also add the mathematical economics tag. – Michael Jul 17 '19 at 2:06