Postulates of economic theories

In the abstract language of mathematics

This is the second entry in a series on mathematical economics. Calibrate your expectations.

Last week’s post looked at a very abstract description of economic state space. Canonical coordinates (q,p) may be hard to wrap your head around, but anything we can say in their terms applies to any model of economics, in any coordinate description. Read my previous post about this here:

Arbitrage and dimensional analysis

With those concepts in mind, let me postulate three facts common to market economic models.

1. There is a concept of value: and more specifically a measure of value-added;

2. (Dynamic) equilibrium is characterized by absence of arbitrage: cycles of trades produce no added value in equilibrium;

3. Endogenous economic processes preserve equilibrium. If no arbitrages exist, trading won’t create them.

Remember, anything that follows from these statements will apply to any economic system where quantities can be transformed into other products or traded between agents.

Mathematical economics like you’ve never seen

This formulation is intrinsically mathematical. That is of course why I phrase it in this way. This post will let me do some advanced math … Hold on to your hats!

1. Value as symplectic structure

Our formulation permits a symplectic structure, We denote the value-added by θ, and last week identified it with the tautological one-form (Wiki), so write ω≝-dθ for the symplectic (two-)form.

2. Arbitrage as geometric measure

The equilbrium condition identifies isotropic sub-manifolds. Using Stokes’ theorem (Wiki):

\(\oint_{\partial \sigma} \theta = \iint_\sigma d\theta = -\iint_\sigma \omega\)

and so if the loop-integral on the left is 0 (i.e. no arbitrage), then ω must vanish on the enclosed area σ. The requirement {ω=0} is the geometric definition of an isotropic submanifold.

Intermezzo

Let me give an economic interpretation of what this means before continuing. That dθ=0 suggests¹ there is a function U(q): Q → R depending only on q so that dU=θ. This is slightly more understandable in coordinates:

\(dU ≝ \sum_i \frac{\partial U}{\partial q^i}dq^i = \theta ≝ \sum_i p_i dq^i \quad \text{so for all }i \quad p_i = \frac{\partial U}{\partial q^i}\)

since coefficients must correspond. This means economically U(q) is a utility function and the pᵢ’s become its associated shadow prices. Conversely, if there is a utility function then equilibrium/no-arbitrage requires that prices equal shadow prices.

3. Dynamics

Endogenous processes preserve submanifolds where {ω=0}. Furthermore, it can also be shown² that for a process to preserve this it is sufficient that there exists a generating function G(q,p,s) such that flows—per-time, noted with a dot—are described by

\(\dot{q} = \frac{\partial G}{\partial p}, \quad \dot{p} = -\frac{\partial G}{\partial q} -p \frac{\partial G}{\partial s}, \quad \dot{s} = p\frac{\partial G}{\partial p}-G \qquad (*)\)

(for any coordinate pair, dropping the indices except in the last one where we sum). That is, the function G entirely determines the dynamics of the model! We see it depends on s, units [€], which measures utility or value. The units of G itself (we see from the third relation) are [€]/time: a per-time income or production function.

(Deutero-)Canonical transformations

In physical Hamiltonian systems, we are familiar withh a Hamiltonian function H(q,p) which determines the dynamics by:

\(\dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q}\)

A question of practical merit is what requirements change of coordinates (q,p)→(Q,P) needs to satisfy in order to still have this dynamic description for new Hamiltonian K(Q,P). This leads to the concept of a canonical transformation (Wiki)³ which can be used to derive the Hamilton Jacobi equation (Wiki)⁴. The notation introduced last time allows us to describe this more concisely.

Strictly speaking, the structure we are interested in preserving is not the symplectic structure of Hamiltonian mechanics, but the contact structure. The structure-preserving dynamics give contact transformations rather than symplectic transformations. Superficial knowledge of the books in the Bible suggests calling (*) a deutero-canonical transformation!⁵

A Hamiltonian usually satisfies dH/dt=0 meaning it is a constant of motion. This is not our situation, however using (*) gives us:

\(\frac{dG}{dt} = \frac{\partial G}{\partial q}\dot{q} + \frac{\partial G}{\partial p}\dot{p} + \frac{\partial G}{\partial s}\dot{s} = \ldots = -\frac{\partial G}{\partial s} G\)

which is in general not 0. When ∂G/∂s is positive, this means G approaches the manifold where {G=0} exponentially through time.

This has shown that (forgive some hand-waving) for an economic system satisfying the initial postulates, there is a generating function that describes how it will develop through time, according to the rules (*). That’s enough for today.

1. I am being circumspect because the subject of my PhD as with most of geometry is studying spaces by where this observation is not true. In the context here it is called the Poincaré lemma which you should only read if you’re interested in a PhD in geometry.

2. I’m sorry to say it, I hate it when others do so, but showing this goes beyond what I can explain to a lay audience.

3. Known to mathematicians as a symplectomorphism (Wiki)—I kid you not.

4. Classical mechanics, the first area of physics to be tackled with calculus, has various classical approaches: Newtonian, Lagrangian, Hamiltonian and Hamilton-Jacobi. This article gives a geometric discussion of the different perspectives.

5. Because it is known to mathematicians as a contactomorphism—srsly.