## General Equilibrium in a Nutshell

February 17, 2010

My goal today is to tell the story of textbook general equilibrium theory and relate it to some of the things I have done so far in this blog.

Let me first quote from Stephen Smale, Mathematical Problems for the next Century, 1998:

The following problem is not one of pure mathematics, but lies on the interface of economics and mathematics. It has been solved only in quite limited situations.

Extend the mathematical model of general equilibrium theory to include price adjustments.

What is the mathematical model of general equilibrium theory? Usually there are ${n}$ goods and prices ${p_i}$ attached to each good for ${1\leq i\leq n}$. Furthermore there is an excess demand function ${z:\mathbb{R}^n_+\rightarrow\mathbb{R}^n}$ (considered as the difference of demand and supply) from the set of prices to the set of goods. Excess demand satisfies three axioms:

1. Homogeneity of degree zero: ${z(\lambda p) = z(p)}$ for all ${0<\lambda\in\mathbb{R}}$ and all ${p\in \mathbb{R}^n_+}$.
2. Walras’ law: ${\sum_{i=1}^n p_i z_i(p)=0}$.
3. Positive demand for a free good: ${z_i(p)>0}$ if ${p_i=0}$.

For this vector field Hopf’s theorem ensures the existence of an equilibrium price vector ${p^*}$ such that ${z(p^*)=0}$. This is often refered to as “supply equals demand”. The problem now is to find a dynamical model explaining the time evolution of prices maybe even as actions of agents acting in the market. This model should preferable be compatible with the existing equilibrium theory.

How does this relate to what I have said so far in this blog? The essential question:

Is (excess) demand indeed a function of price?

To my knowledge there is hardly any economic evidence/experiment to settle this question. “What else could it be?” is certainly not a valid viewpoint. While there are reasons to believe this in the stationary setting, as soon as time hits the scene a functional relation needs justification. Agents may learn or exhibit other hard to explain behavior.

If we do not assume excess demand to be a function of price, how far do we get? Quite far, actually. The reason is axiom 1. Its content in everyday speech: demand does not depend on price-scaling. Since price-scalings form a group we are in a position to find a representation, for example, as linear operators on a vector space. Doing this we can derive a kind of uncertainty principle for demand and price. They cannot be measured simultaneously with arbitrary precision. This is certainly necessary if we wanted to define a functional relation between them. Moreover, as an invariant of the group action we get a two parameter family of operators. One of these parameters is readily identified as “endowment”, the other represents a willingness-to-pay/willingness-to-accept discrepancy. Goods are considered more valuable if we own them compared to if we want them. We get what economists call the endowment effect for free.

All this could be done without the other axioms. As soon as this problem is tackled I comment on these too. Not today however …