Thermodynamics and Economic Equilibrium




Still, the introduction of this paper is intriguing:




Continuing on:




Continuing on with Smith and Foley’s paper:









19 Responses to Thermodynamics and Economic Equilibrium

  1. Cal Abel says:

    The thermodynamic relationships are derivable directly from the axioms of game theory. The derivation is actually fairly straight forward. The 0th, 2nd, and 3rd laws fall out as a direct consequence of this approach. It also formally includes uncertainty in choice under uncertainty models through the entropy functional. I haven’t been able to get the first law (conservation of utility) so waiting on publishing the results.

  2. Toby Bartels says:

    While I agree that you should learn classical economics, it is surely the neoclassical economists who you heard criticizing econophysics and all that. The classical economists were from the 18th and 19th centuries, predating both Marx and the marginal revolution.

    • John Baez says:

      Okay, I’m so behind the curve I’d need to learn the classical stuff before I can figure out what makes the neoclassical stuff different.

      • Toby Bartels says:

        I think that the neoclassical economists would tell you not to bother with the classical stuff at all, but I could be mistaken.

        • Go Careful says:

          They would. Classical economics, which I would say runs from Smith (1776) through Ricardo, Marx, Mill and George (1880), was heading in the ‘wrong’ direction. JB Clark (1886) and Alfred Marshall (1890) invented neoclassical economics to turn things in the ‘right’ direction. ‘Wrong’ and ‘right’ as defined by the robber barons of the day that is. Yes, you are entering the domain of politics.

  3. Doug Summers Stay says:

    This analogy between economics and physics reminds me of the idea behind “psychohistory” in Asimov’s Foundation novels. He imagined that only in a galactic civilization, with trillions of individuals, would a statistical approach to the flow of history make sense.

    • John Baez says:

      I read it a long time ago. I forgot the idea that a enormous civilization would allow the application of statistical techniques that would fail for smaller numbers of people. I don’t really believe it, but it’s interesting. Asimov was a chemist.

    • Glyn Adgie says:

      I think recent history shows that particular individuals are not just statistics. What about influential politicians and businessmen? I am thinking of people like Donald Trump and Jeff Bezos. Also, please note what Karl Popper said about theories of history, which he called “historicism”. A great deal of history is influenced by powerful people doing what they feel like, which is not always rational.

  4. domenico says:

    I am thinking that thermodynamics could work like an analogy, but that a statistical mechanics differential equation could work better in econophysics.
    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

    0=a_0 s+a_1 \frac{\partial s}{\partial v}+a_2 \frac{\partial s}{\partial p}+a_3 \frac{\partial^2 s}{\partial p \partial v}+  cdots

    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.

    • domenico says:

      I am thinking that the temporal dynamics of an economic system could be modeled by:
      0 = a_1 \partial_t S+ a_2 \partial_p S+ a_3 \partial_q S+a_4 \partial_{qt} S+\cdots
      when the system is in equilibrium
      0 = a_2 \partial_p S+ a_3 \partial_q S+\cdots
      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 0=\partial_t S=\partial_{qt}S=\cdots (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 \Phi=(T-T_c)^{\gamma} is 0=\gamma \Phi \partial_{TT} \Phi - (\gamma-1) \partial_T \Phi 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.

        • domenico says:

          It seem that exist a Lagrangian in a critical point
          {\cal L}=\Phi^{\alpha}\partial_T \Phi \partial_T \Phi
          the Euler-Lagrange equation is
          0=-\alpha \Phi^{\alpha-1}\partial_T \Phi \partial_T \Phi-2\Phi^{\alpha} \partial_{TT} \Phi
          that has two solution
          0=\frac{\alpha}{2} \partial_T \Phi \partial_T \Phi+\Phi \partial_{TT} \Phi
          then if there is this econophysics Lagrangian, then there are critical index, and surely phases. I write this result in vixra, where I write immediately results that look beautiful.

  5. Giampiero Campa says:

    Very nice find! I am definitely going to read it too.

    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.

  6. Allen Knutson says:

    If economics can be set up in symplectic coordinates, is there an economic notion of “completely integrable system”?

    • John Baez says:

      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.

  7. Alex says:

    With regard to your last paragraph, I think this is a misconception, albeit a pretty common one. “Utility” was originally introduced (I believe) by Bentham to refer to pleasure and pain, clearly psychological entities. Early economists adopted the idea as a scientific postulate that individual maximise utility — Edgeworth even called his book “Mathematical Psychics”! However although the term “utility maximisation” is still used in economics, it has been totally stripped of all psychological connotation. The terminology is confusing because “utility” as used by philosophers continues to have psychological connotations. Utility in economics is now understood to be just a way of representing a rank-ordering (preference relation), so that outcomes with higher utility are preferred by the agent to those with a lower utility. The advantage of using a utility function is that you can use techniques of calculus, like Lagrange multipliers. But the utility function is fictitious, it’s the preference relation that’s “real” and behaviour only depends on the latter. Even if individuals did maximise the pleasure minus pain, the actual amounts of pleasure and pain would be economically irrelevant, only the implied rank-ordering. 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?

    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!

    • John Baez says:

      Thanks! By the way, about this:


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


      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.

  8. Jesse C. McKeown says:

    I’m impressed that Fisher’s thesis was able to cite publications nearly a century into the future!



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