1
$\begingroup$

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:

enter image description here

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

enter image description here

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

enter image description here

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

What does the concept of index means?

$\endgroup$
  • 2
    $\begingroup$ 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. $\endgroup$ – Michael Jul 16 at 22:59
  • $\begingroup$ @Michael I have never heard of the index theorem in topology before. Is it the Atiyah-Singer index theorem? $\endgroup$ – Bruno Schiavo Jul 16 at 23:04
  • 1
    $\begingroup$ 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. $\endgroup$ – Michael Jul 16 at 23:11
  • 1
    $\begingroup$ I would try Differential Topology by Milnor ("degree modulo 2 of a map"), and as I said, Math SE. $\endgroup$ – Michael Jul 16 at 23:37
  • 1
    $\begingroup$ You may want to also add the mathematical economics tag. $\endgroup$ – Michael Jul 17 at 2:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.