Mechanics and Markets

November 25, 2009

When we talk about markets we often use terms like equilibrium or even market force. We choose this terminology for a reason. The analogy to the well established theories of mechanics and quantum mechanics is intended and the pictures we have in mind are a pendulum or even a simple spring. Their restoring forces seem to model the market forces and therefore we frequently observe argumentations very similar to:

if prices increase, then demand decreases and vice versa finally, because of some process still to be described, the market settles down in an equilibrium (called Walrasian price equilibrium).

As a start, that sounds convincing. There just remains one big question. Is that a good picture? Or, even more to the point:

Are there any justifications for the existence of market forces?

Rather than answering this question (regular readers know my standpoint anyway) I would like to justify why this question is actually reasonable and should be asked and answered. In physics this question is answered to the positive, in economics the situation is a little blurry to say the least. I continue by comparing mechanics with economics in catchwords. Thereby pointing out similarities, but also discrepancies and, in a way, recalling ‘the story so far’.

Basic notions

Let me start with two of the fundamental notions in mechanics, namely position and momentum. In earlier posts we have identified their counterparts in economics as price and demand.

Symmetries

In mechanics the intuition is that momentum is invariant under translation of position. In economics we need demand invariance under price-scaling.

Commutation relations

These symmetries lead to commutation relations of the form ${[A,B]=\text{id}}$ in quantum mechanics and ${[A,B]=A}$ in economics (cf. here). This difference is essential and has a huge impact, albeit not immediately.

Bounded representations

Both commutation relations imply that the symmetry groups do not have representations on a finite-dimensional vector space (cf. here).

Unbounded representations

While there are no bounded representations, we get unbounded representations on the Hilbert space ${L^2(\mathbb{R}^n)}$ of square integrable functions. Momentum and demand operators are differential operators, whereas position and price are (different) multiplication operators (cf. here).

Uncertainty principle

The uncertainty principle of quantum mechanics is well-known. So far I didn’t write about that here in the blog, but in economics the commutation relations imply inequalities which can also be interpreted as some sort of uncertainty principle. I shall come back to this later.

Time evolution

As described in scientific laws to get the time evolution in quantum mechanics one chooses an action, one uses Legendre transform to obtain the energy, one derives the canonical equations and essentially plugs in the above representation to obtain SchrĂ¶dingers equation governing the time evolution of a quantum system. That surely sounds more complicated than it actually is.

Why can’t we just do that for markets and obtain market equations governing their time evolution? Now, there are a couple of technical difficulties. The most prominent probably is that the Legendre transform of a market action is not invariant under time translation. Hence, in markets there is no conservation of energy. This fact alone makes the usage of a term like market force a little obscure. What is meant by force if there is no energy or at least no energy conservation?

That essentially is the programme for the rest of the year. I shall spell out the maths behind the uncertainty principle for markets and then delve into the technical details of obtaining a time evolution for markets.

Stay tuned …

The Origin of Life

November 11, 2009

Tim Gowers proposes a polymath project to answer the question about the origin of life scientifically (mathematically).

… and please start to breathe again.

That is incredibly ambitious and fits perfectly into the scope of this blog. Go there and contribute!

I am way to pessimistic to do it myself. Is the axiomatic ideology we follow able to lead to insights concerning this question? I really doubt that. So far mathematicians were remarkably unsuccessful to even define notions like ‘life’ and ‘intellligence’. We were so unsuccessful, that it might be allowed to ask whether a definition is actually possible.

Mathematicians introduce notions by description (like set and element) or by definition (like function and number). ‘Life’, like many other notions, does not fit into this approach. It is so hard to grasp the meaning of ‘life’ by definitions and descriptions that one is lead to the idea that there are probably other ways to introduce notions in a precise sense.

Indeed you can think of processes to introduce notions which are not definitions and descriptions. Unfortunately, once you start to do that, you have to give up a lot of what makes us feel comfortable with contemporary mathematics. That sounds vague, but I have a feeling that I catch upÂ  writing this stuff here in the blog maybe even years before Tim finishes the above project.

… just kidding …

The plan now is to elaborate a bit further on scientific laws and then I explain how to ’emerge’ new notions from old ones.