## 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 Schrödingers 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 …

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## Scientific Laws

September 2, 2009

As I have told you earlier, my guest is very sceptical about our scientific achievements. What follows are the notes I took, when he gave me a short summary of what he considers ‘our strategy’.

In modern understanding of science, the fundamental laws seem to be consequences of various symmetries of quantities like time, space or similar objects. To make this idea more precise scientists often use mathematical arguments, thereby choosing some set ${X}$ as state space encoding all necessary information on the considered system. The system then is thought to evolve in time on a differentiable ${n}$-dimensional path ${x_i(t)\in X}$ for all ${t\in\mathbb{R}}$ and ${1\leq i \leq n\in\mathbb{N}}$. Quite frequently there is a so-called Lagrange function ${L}$ on the domain ${ X^n \times X^n \times \mathbb{R} }$ and a constraint function ${W}$ on the same domain. The path ${x(\cdot)}$ is required to minimizes or maximizes the integral

$\displaystyle \int_0^T L\left(x(s),\dot{x}(s),s\right)ds$

under the constraint

$\displaystyle W\left(x(s),\dot{x}(s),s\right)=0.$

(Under some technical assumptions) a path does exactly that, if it satisfies the Euler-Lagrange equations

$\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{x}_i}-\frac{\partial L}{\partial x_i}=\lambda \frac{\partial W}{\partial \dot{x}_i}$

for some function ${\lambda}$ depending on ${X^n \times X^n \times \mathbb{R}}$.

Define ${y_i:=\frac{\partial L}{\partial \dot{x_i}}}$ and observe that (under suitable assumptions) this transformation is invertible, i.e. the ${\dot{x}_i}$ can be expressed as functions of ${x_i, y_i}$ and ${t}$. Next, define the Hamilton operator

$\displaystyle H(x,y,t) = \sum_{i=1}^n \dot{x}_i(x,y,t) y_i - L(x,\dot{x}(x,y,t),t)$

as the Legendre transform of ${L}$. The Legendre transformation is (under some mild technical assumptions) invertible.

Now, (under less mild assumptions, namely holonomic constraints) two things happen. The canonical equations

$\displaystyle \frac{d x_i}{d t} = - \frac{\partial H}{\partial y_i} \left(=[x_i, H]\right), \frac{d y_i}{d t} = \frac{\partial H}{\partial x_i}\left(=[y_i, H]\right),\frac{d H}{dt} = -\frac{\partial L}{\partial t}$

are equivalent to the Euler Lagrange equations. Here ${[\cdot,\cdot]}$ denotes the commutator bracket ${[a,b]:= ab-ba}$. Furthermore, if ${L}$ does not explicitly depend on time, then ${H}$ is a constant. That is the aforementioned symmetry. ${H}$, the energy, is invariant under time translations.

Given all that, the solution of the minimisation or maximisation problem can then be given (either in the Heisenberg picture) as

$\displaystyle x(t) = e^{t H} x(0) e^{-t H}, y(t) = e^{t H} y(0) e^{-t H}$

or (in the in this case equivalent Schrödinger picture,) as an equation on the state space

$\displaystyle u(t)= e^{t H}u(0).$

This description is equivalent (under mild technical assumptions) to the following initial value problem:

$\displaystyle \dot{u}(t)=H u(t), u(0) = u_0\in X.$

where the operator ${H}$ is the ‘law’. More technically, the law is the generator of a strongly continuous (semi-)group of (in this case linear and unitary) operators acting on (the Hilbert space) ${X}$. As an example of this process he mentioned the Schrödinger equation governing quantum mechanical processes.

His conclusion was that the frequently appearing ‘technical assumptions’ in the above derivation make it highly unlikely for laws to exist even for systems with, what he calls, no emergent properties. ‘If that was true’, I thought ‘then … bye bye theory of everything!’ He explained further, that under no reasonable circumstances it is possible to extrapolate these laws to the emergent situation. I am not sure, whether I understand completely what he means by that, but his summary on how we find scientific laws is in my opinion way too simple. It can’t be true and I told him.

With just a couple of ink strokes he derived the commutation relations for exchange markets from microeconomic theory. That left me speechless, since I always thought, that there cannot be ‘market laws’. Markets are on principle unpredictable! They are, or?