## Prices and Demands (Part I)

You might remember, that we are looking for general laws describing the behavior of markets. To that purpose, at first glance, it is not self-evident to describe such markets, i.e. prices ${p}$ of and demands ${d}$ for goods, by observables acting on some Hilbert space ${X}$. However, as we have seen in Commutation Relations in Markets and in Scientific Laws there might be good reasons for doing this and if we do it, the observables ${p}$ and ${d}$ have to satisfy the following axioms:

• (MA1) The price ${p_i}$ of good ${i}$ is a positive observable on ${X}$ for all goods ${1\leq i\leq n}$.
• (MA2) The demand ${d_i}$ of good ${i}$ is an observable on ${X}$ for all goods ${1\leq i\leq n}$.
• (MA3) The endowment ${\omega_i}$ of good ${i}$ is a real number ${\omega_i \in\mathbb{R}}$ for all goods ${1\leq i\leq n}$.
• (MA4) Prices ${p_i}$ and demands ${d_j}$ interact according to

$\displaystyle \left[p_i, d_j\right]=i \mu_i p_i \delta_{i,j}$

for a fixed real ${\mu_i\in\mathbb{R}}$.
Up to now, we have also learned that the demand ${d}$ essentially is the generator of the price-scaling group and that mathematicians often represent groups as linear operators acting on vector spaces to get an idea of what is going on. Let us just do that and find some ‘matrices’ to represent price-scaling. Maybe we then get a better understanding of such market descriptions.

Unfortunately, there is a famous result of H. Wielandt stating that if linear operators ${A,B}$ satisfy a commutation relation ${[A,B]=\textnormal{id}_X}$, then not both can be bounded simultaneously. Since there are no unbounded linear operators on finite dimensional vector spaces, the Hilbert space then must be infinite dimensional.

Just as a side note, you might notice that (scalar multiples of) momentum and position in quantum dynamics satisfy the above commutation relation and this is the reason why we cannot represent these observables as matrices. On our hunt for market laws that might be a little setback. Certainly not for mathematicians or physicists, but probably for economists. As far as i know, infinite dimensional Hilbert spaces are up to now not part of their syllabus.

Today we are not doing in quantum mechanics and thus there is still hope. If we divide market axiom 4 by ${i \mu}$ we observe that our commutation relation is ${[A,B]=A}$ which is certainly different from the quantum situation.
Let us do some mathematics.

Assume ${A,B}$ to be bounded linear operators (acting on some Banach space) with ${A^n\neq 0}$ for all ${n\in \mathbb{N}}$. Assume furthermore ${[A,B]= A}$ and as induction hypothesis ${[A^n,B]= nA^n}$. Then ${[A^{n+1},B]= A[A^n,B]+[A,B]A^n=nA^{n+1}+A^{n+1}=(n+1)A^{n+1}}$. The norm estimate ${n \|A^n\| = \|[A^n,B]\|\leq 2 \|A^n\| \|B\|}$ yields a contradiction since ${A^n\neq 0}$ for all ${n\in\mathbb{N}}$. Therefore, either at least one of the operators ${A}$ and ${B}$ is unbounded and/or the commutation relation ${[A,B]=A}$ does not hold.

In our situation the commutation relation holds as stated in market axiom 4. Since, by market axiom 1, the observable ${p_i}$ is positive we obtain ${p_i^n\neq 0}$ for ${n\in\mathbb{N}}$ and hence, at least one of the operators ${p_i}$ and ${d_i}$ is unbounded.

Oh, oh … not nice. But, it could have been worse. Next week I choose a Hilbert space and give you a representation of price and demand as unbounded operators on this Hilbert space. Things will look much more down-to-earth then. Promised …