<p>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:</p>

<p><a href="https://i.sstatic.net/lvMRB.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/lvMRB.jpg" alt="enter image description here"></a></p>

<p>A regular equilibrium of the economy is defined as the following:</p>

<p><a href="https://i.sstatic.net/yTkxY.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yTkxY.jpg" alt="enter image description here"></a></p>

<p><span class="math-container">$D$</span> denotes derivative (here, in respect to price), while <span class="math-container">$z$</span> denotes the excess demand matrix:</p>

<p><a href="https://i.sstatic.net/s9OWo.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/s9OWo.jpg" alt="enter image description here"></a></p>

<p>Where <span class="math-container">$p$</span> denotes price and <span class="math-container">$\omega$</span> denotes endowment.</p>

<p>What does the concept of index means?</p>

CC BY-SA 4.0
2019-07-16T22:59:37.753Z
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
CC BY-SA 4.0
2019-07-16T23:04:32.507Z
@Michael I have never heard of the index theorem in topology before. Is it the Atiyah-Singer index theorem?
Incognito
CC BY-SA 4.0
2019-07-16T23:11:06.700Z
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
CC BY-SA 4.0
2019-07-16T23:25:20.027Z
@Michael Where can I find this theorem about the index of a map between manifolds? I could not find anything in Wolfram MathWorld.
Incognito
CC BY-SA 4.0
2019-07-16T23:37:11.933Z
I would try Differential Topology by Milnor ("degree modulo 2 of a map"), and as I said, Math SE.
Michael
CC BY-SA 4.0
2019-07-17T02:06:50.183Z
You may want to also add the mathematical economics tag.
Michael
<microeconomics><mathematical-economics><general-equilibrium>
Index of an Excess Demand Vector
Incognito
# Index of an Excess Demand Vector

(Asked by: username_0 on 2019-07-14)

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](https://i.sstatic.net/lvMRB.jpg)](https://i.sstatic.net/lvMRB.jpg)

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

[![enter image description here](https://i.sstatic.net/yTkxY.jpg)](https://i.sstatic.net/yTkxY.jpg)

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

[![enter image description here](https://i.sstatic.net/s9OWo.jpg)](https://i.sstatic.net/s9OWo.jpg)

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

What does the concept of index means?

**Tags:** microeconomics, mathematical-economics, general-equilibrium

> **Comments:**
> **username_1**: 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.  
> **username_0**: @username_1 I have never heard of the index theorem in topology before. Is it the Atiyah-Singer index theorem?  
> **username_1**: 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.  
> **username_0**: @username_1 Where can I find this theorem about the index of a map between manifolds? I could not find anything in Wolfram MathWorld.  
> **username_1**: I would try Differential Topology by Milnor ("degree modulo 2 of a map"), and as I said, Math SE.  
> **username_1**: You may want to also add the mathematical economics tag.  

---

## 0 answers

