I am thinking that the temporal dynamics of an economic system could be modeled by:

and

when the system is in equilibrium

so that the economic system has an equilibrium point, phase transition and critical exponent, if there are interaction between individual agents. These equilibrium point are solution of the dynamics, when the system does not oscillate due to dispersion, or when the initial condition is (a constant economic potential).

When the extensive quantity are near a critical point, then the equilibrium points change a lot with small perturbations, and so the trajectories change a lot with small perturbations: the economic system is unstable in the temporal evolution, and the economic system is near a critical point.

I am learning the Landau theory of phase transition, that has an approximation of the thermodynamics potential in a Taylor series near a critical point (instead of an approximation of a differential equation, that could give a dynamics).

A possible differential equation for a critical point is but it is not a differential equation obtained from a Lagrangian, so that an analogy with physical system with potential and kinetic energy is not possible.

Perhaps it might be possible to model thermodynamic systems that vary over time: thermodynamic oscillations or irreversible transformations.

I didn’t quote Fisher’s thesis. I’ve tried to make that clearer.

]]>Thanks! By the way, about this:

For example, any monotonically increasing transformation of the utility function preserves the rank-ordering. If you wanted to make an analogy to physics, you might consider such monotonic transformations a “gauge freedom.” The reason you can’t add utilities of different individuals is because what you get is physically meaningless, not merely unimportant (it reminds me of Feynman complaining about a textbook which asked students to work out the “total temperature” of two stars). What would it mean to add two rank orderings?

I was trying to allude to this “gauge freedom” when I wrote:

Furthermore, if we believe we can reparametrize the utility of each agent without changing anything, it makes no sense to add utilities.

Of course I was keeping a more agnostic attitude when I wrote “if we believe”. I just didn’t want to commit to using “utility” in the way many economists do, where it’s purely a way of referring to a preference relation. I wasn’t wanting to argue *against* using the word this way, either.

Please let me know if you come out with a paper on thermodynamics and economics.

]]>That said, there is one economically meaningful way to add up people’s desires: you add up people’s willingness to pay. The maximum you are willing to pay for X is the value of X to you, and this is a property of your preference relation. This is basically what economists talk about when they talk about consumer surplus, producer surplus etc. The disadvantage is it treats everyone’s dollars the same. You could construct a measure that gave greater weight, say, to a poor man’s dollars to a rich man’s, but economic theory necessarily tells you nothing about what the relative weight should be since from an economic standpoint all dollars are interchangeable — you have to step outside economics and make an arbitrary value judgement.

Finally, I will cryptically remark that I have my own ideas of how to apply thermodynamic ideas in economics very different from the conventional analogies but I will not say what they are because I want to write a paper about it!

]]>Maybe, but there’s a funny thing going on here. In classical mechanics we use symplectic (and contact) geometry to describe dynamics, but in classical thermodynamics we use them to describe statics: that is, equilibrium situations. The analogy to economics discussed in this paper by Smith and Foley mainly concerns statics!

In statics, Hamilton’s equations play a different role than we’re used to. In classical mechanics they’re called the ‘Maxwell relations’. I tried to explain this here:

• Classical mechanics versus thermodynamics (part 1).

• Classical mechanics versus thermodynamics (part 2).

I guess it can still make sense to study complete integrability in classical thermodynamics. But I don’t have much of a feeling for what it means! So that’s a good puzzle to add to my list. I’m thinking about this stuff again these days.

]]>I have been reading similar things lately as I am looking for ways to approach macroeconomics from a dynamical systems perspective (I had some preliminary success too, but that’s maybe a post for another time).

In my high-level understanding, originally (before the 70s) models were based on simple (mostly static) equations describing an equilibrium among markets (e.g. goods, money, currency and labor markets), using aggregate prices as variables.

There was later a push to add “microfoundations” to these models, which aimed to explicitly derive the relationship among aggregate quantities as a result of a number of agents seeking to maximize their utility. This was mostly a theoretical effort, and I might be wrong but I don’t think there was enough attention to reproducing observed real-world behavior. Or at least I couldn’t find many textbooks where these models are introduced and in which real-world economic data play an important part (but it is possible that I haven’t looked hard enough).

Today there seem to be two main classes of macroeconomics models. On one hand the ones based on first principles, and derived from the microfoundations effort, like DSGE (Dynamic Stochastic General Equilibrium) which have really hard time with prediction. On the other hand there are more empirical models, which can do some forecasting but are more towards the “black box” end of the spectrum (that is not so much based on first principles), and therefore they are not good at explaining things. There is a nice short blog post from Olivier Blanchard that explains the current situation well, IMO.

It seems that there is a growing dissatisfaction in the field with the whole microfoundations effort and DSGE models. According to some, these models attempt “to derive macroeconomics from the wrong end—that of the individual rather than the economy”, and are therefore doomed to failure.

Furthermore, the Sonnenschein–Mantel–Debreu theorem means (roughly speaking) that when you aggregate the behavior of individual rational agents you basically can end up with any demand function you want. So if you pick some structure for the demand function, like the simple ones assumed in textbooks for example, you really have very low chance of picking the “right one”, based only on theoretical assumptions. Only data (and lots of it) can help here, I think.

Again i am not an economist and these are just mostly my perceptions from outside, i can’t claim any kind of deep understanding here. Nevertheless, it would be interesting to read more viewpoints.

]]>If there are N>>1 agents (share ownership, customers,etc.), then it is possible a statistical mean of M economics quantity (share values, prize of goods,etc.) as a function of L continuous economic variables: so that could be possible to write the first and second partial derivatives (experimental evaluated) of the economic quantity in the continuous variables.

It should be possible (with some mathematical tricks) to write the multiple linear regression using M(1+2L+L(L-1)/2+…) parameters

a surface in the derivative space that intersect the experimental points with minimum error; if there are product of derivatives the number of terms grow exponentially.

The economic (physical) law obtained could describes the dynamics of the economic statistical system, and the laws are a system of M differential equations with L variables; there should be phase transitions, critical points, coalescence and entropy in economic laws.