Finally Top Ten!

October 31, 2009

Last week I was teaching 8 hours mathematics a day, but that could not prevent me from checking my e-mail. What I found was a link to the following top ten list:

Top Ten List

It contains a joint work with Christian Schwarz on 10th place which is not overly remarkable. The paper is about a fairly general proof of the non-existence of market equilibria.

The twist is that the list also contains a ‘handful’ of papers proving the existence of said market equilibria.

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 Schrödinger 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 …