Prices and Demands (Part II)

October 19, 2009

In part one we have seen that in microeconomics if we treat demand and price as observables on a Hilbert space, then not both of them can be bounded linear operators. Especially, since all linear operators on a finite dimensional Hilbert space are bounded, the state space for our market cannot be finite dimensional. All this is a consequence of a single assertion, namely demand invariance under price-scaling and somehow resembles the situation in quantum theory.

So far our considerations concerning prices and demands were quite abstract. To do some real computations we need a representation of these concepts. As promised last week, I now provide such a representation for the observables price ${p_i}$, demand ${d_i}$ and excess demand ${z_i}$ on an appropriate Hilbert space.

The results so far, lead to the following approach: The Hilbert space is given as ${X = L^2(\mathbb{R}^n)}$. This, in a way, is the simplest non-finite dimensional Hilbert space and therefore an unsurprising first choice. Thus, the state of the market with ${n\in\mathbb{N}}$ goods is described by a function ${\xi\in X}$. Let the coordinates of ${\xi}$ be denoted as ${(x_1,\ldots,x_n)\in \mathbb{R}^n}$, then the demand ${d_i: D(d_i)\rightarrow X}$ is given as a differential operator

$\displaystyle d_i \xi = -i \mu_i \frac{d}{d x_i} \xi$

with domain

$\displaystyle D(d_i) = \{ \xi\in X: \xi \textnormal{ absolutely continuous and } \xi' \in X \}.$

The excess demand operator ${z_i=d_i-\omega_i \textnormal{id}_X}$ has the same domain as ${d_i}$.
Define the function ${e_i:\mathbb{R}^n\rightarrow \mathbb{R}}$ as ${e_i(x)=e^{x_i}}$. Then, the price operator ${p_i:D(p_i)\rightarrow X}$ is given as a multiplication operator

$\displaystyle p_i \xi = e_i \cdot \xi$

with domain

$\displaystyle D(p_i) = \{ \xi\in X: e_i \cdot \xi \in X \}.$

All operators ${p_i, d_i, z_i}$ are self-adjoint, ${p_i}$ is positive, the commutator satisfies

$\displaystyle \left[p_i,z_i\right]\xi = \left[p_i,d_i\right]\xi = -i \mu_i \left(e_i \cdot \frac{d}{d x_i}\xi - e_i \cdot \xi - e_i \cdot \frac{d}{d x_i} \xi\right)= i \mu_i p_i \xi$

and thus the market axioms are fulfilled.

That still might look a little abstract if one is not used to Functional Analysis. The corresponding representation in quantum mechanics however lead to major new insights into the field.

Next time I shall give a formal comparison of microeconomics and quantum mechanics. As you can imagine by now, they are similar on some abstract level. However, there are also some striking discrepancies like the different commutation relations and thus the step towards the desired SchrÃ¶dinger type equation for markets is not straight forward.

Stay tuned …

Prices and Demands (Part I)

October 5, 2009

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 …

• Commutation Relations in Markets

September 8, 2009

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 …