## Commutation Relations in Markets

To derive commutation relations in microeconomics we first have to reach sure ground. What is a minimal set of assumptions we need to derive something interesting, but still comprehensive enough to describe something meaningful?

In a market, it is definitely safe to assume that we have ${n}$ goods for some number ${1\leq n \in \mathbb{N}}$. This goods are being traded and therefore we need to talk about prices and demand. Call ${p_i}$ the price of good ${i}$ and ${d_i}$ the demand for good ${i}$ and let ${1\leq i \leq n}$. What else do we need?

Sure, we need a lot more, but not now! As we have seen in Scientifc Laws all we need now is a symmetry between price and demand. The key to this symmetry is found in any basic text book like e.g. Microeconomic Theory by A. Mas-Colell, M.D. Whinston and is called invariance of demand under price-scaling. What is meant by that? Let me just give you an example. When continental europe introduced the Euro currency, many nations swapped their national currency for the new Euro. In Germany, 1 Euro was worth 1.95583 Deutsche Mark. All prices, wages, debts aso. where scaled by ${\frac{1}{1.95583}}$. The day after, no increase of demand for fridges, cars, credits aso. was observed. That was no surprise for economists. Where should a change of demand come from? A redefinition of the currency is not enough to generate demand. That is generally believed and a pillar in the following argumentation.

Just for the sake of completeness let me emphasize that price scaling, as introduced above form a group. Whenever we scale by a factor ${\alpha\in\mathbb{R}_{>0}}$ and then scale by a factor ${\beta\in\mathbb{R}_{>0}}$ we obtain a scaling by the factor ${\alpha\beta}$. Scaling with 1 is the neutral element and for each scale factor ${\alpha\in\mathbb{R}_{>0}}$ we can go back by scaling with ${\frac{1}{\alpha}}$.

As mathematicians, we often represent abstract groups (like the above price-scaling) as linear operators acting on some vector space. To that purpose, we choose the state of the market to be given by a non-zero vector ${\xi}$ in a Hilbert space ${X}$ with inner product denoted by ${\left\langle \cdot | \cdot \right\rangle}$. Of course, in the moment you can think of ${X}$ as a finite dimensional Hilbert space ${\mathbb{R}^n}$ or ${\mathbb{C}^n}$. On the other hand, it is always good to be suspicious and fixing the dimension to be finite might be premature. Observables are self-adjoint operators on this Hilbert space and 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 ${1\leq i\leq n}$.
A positive observable ${a}$ on ${X}$ is an observable with ${\langle a\xi | \xi \rangle>0}$ for all ${0\neq\xi}$ in the domain ${D(a)}$ of ${a}$.

By a famous result of E. Noether, symmetries and invariants are closely tied together. What are the market invariants of the asymmetric market under the price-scaling symmetry? To see this, let ${\left(U_i(\alpha)\right)_{0 < \alpha\in \mathbb{R}}}$ be a strongly continuous family of unitary operators on ${X}$ such that

$\displaystyle U_i^{-1}(\alpha)p_i U_i(\alpha)=\alpha p_i.$

The family ${U_i(\cdot)}$ satisfies the following properties for all ${\alpha>0}$ and ${\beta>0}$:

• ${U_i(1)= \textnormal{id}_X}$
• ${U_i(\alpha)U_i(\beta)=U_i(\alpha\beta)=U_i(\beta)U_i(\alpha)}$
• ${U_i^{-1}(\alpha) = U_i\left(\frac{1}{\alpha}\right)}$

Define ${T_i(t):=U_i(e^t)}$ and observe

• ${T_i(0)= \textnormal{id}_X}$
• ${T_i(t)T_i(s)=T_i(t+s)=T_i(s)T_i(t)}$
• ${T_i^{-1}(t) = T_i(-t)}$

This yields ${T_i}$ to be a strongly continuous group of unitary operators acting on ${X}$. Thus, the theorem of Stone ensures the existence of a skew-adjoint generator ${A_i}$. Set ${\alpha = e^t}$ and with ${U(\alpha)=T(\ln \alpha)}$ it follows that

$\displaystyle \begin{array}{rcl} p_i & = & \frac{d}{d\alpha}\left(U_i^{-1}(\alpha)p_i U_i(\alpha)\right) \\ & = & \frac{d}{d\alpha}\left(T_i(-\ln \alpha)p_i T_i(\ln \alpha)\right) \\ & = & - \frac{1}{\alpha} T_i(-\ln \alpha) A_i p_i T_i(\ln \alpha) + \frac{1}{\alpha} T_i(-\ln \alpha)p_i A_i T_i(\ln \alpha). \end{array}$

Evaluation at ${\alpha=1}$ yields

$\displaystyle \left[p_i, A_i\right] = p_i. \ \ \ \ \ (1)$

Since a generator commutes with the strongly continuous group it generates it is easily seen that ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ also commutes with ${U_i(\alpha)}$ for any ${\beta_i,\gamma_i\in\mathbb{C}}$. Hence ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ represents a market invariant under price-scaling.

Now we derive an economic interpretation of ${A_i}$. We know already that ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ represents a market invariant under price-scaling for any ${\beta_i,\gamma_i\in\mathbb{C}}$. Since ${A_i}$ is skew-adjoint and ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ needs to be an observable, we get that ${\beta_i = i \mu_i }$ and ${\gamma_i = \omega_i}$ for some ${\mu_i, \omega_i \in\mathbb{R}}$. Furthermore, since scaling of one price does not influence scaling of the others (i.e., ${\left[p_i, U_j(\alpha)\right]=0}$ for ${i\neq j}$) we can use (1) and obtain

$\displaystyle \left[p_i, i \mu_i A_j - \omega_i \textnormal{id}_X\right] = i \mu_i p_i \delta_{i,j}.$

The operator ${i \mu_i A_i + \omega_i \textnormal{id}_X}$ is an observable and is invariant under price-scaling. Economic intuition therefore leads us to identify this operator with the demand respectively excess demand for good ${i}$ if ${\mu_i\neq 0}$. The real parameter ${\omega_i}$ is identified as endowment. The other real parameter ${\mu_i}$ represents a new feature. Intuitively it measures the difference of first selling and then buying a good versus first buying and then selling that good.
The observations in the last paragraph yield the final axioms.

• (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 ${\mu_i\in\mathbb{R}}$.
There are still a lot of things to say, e.g. on how measurements are done, on the dimension of the Hilbert space ${X}$, on representations of demand ${d_i}$ and price ${p_i}$ as operators and on a comparison to the commutation relations of quantum mechanics. Stay tuned …