## Thermodynamics and Economic Equilibrium

I’m having another round of studying thermodynamics, and I’m running into more interesting leads than I can keep up with. Like this paper:

• Eric Smith and Duncan K. Foley, Classical thermodynamics and economic general equilibrium theory, Journal of Economic Dynamics and Control 32 (2008) 7–65.

I’ve always been curious about the connection between economics and thermodynamics, but I know too little about economics to make this easy to explore. There are people who work on subjects called thermoeconomics and econophysics, but classical economists consider them ‘heterodox’. While I don’t trust classical economists to be right about things, I should probably learn more classical economics before I jump into the fray.

Still, the introduction of this paper is intriguing:

The relation between economic and physical (particularly thermodynamic) concepts of equilibrium has been a topic of recurrent interest throughout the development of neoclassical economic theory. As systems for defining equilibria, proving their existence, and computing their properties, neoclassical economics (Mas-Collel et al., 1995; Varian, 1992) and classical thermodynamics (Fermi, 1956) undeniably have numerous formal and methodological similarities. Both fields seek to describe system phenomena in terms of solutions to constrained optimization problems. Both rely on dual representations of interacting subsystems: the state of each subsystem is represented by pairs of variables, one variable from each pair characterizing the subsystem’s content, and the other characterizing the way it interacts with other subsystems. In physics the content variables are quantities like asubsystem’s total energy or the volume in space it occupies; in economics they area mounts of various commodities held by agents. In physics the interaction variables are quantities like temperature and pressure that can be measured on the system boundaries; in economics they are prices that can be measured by an agent’s willingness to trade one commodity for another.

In thermodynamics these pairs are called conjugate variables. The ‘content variables’ are usually called extensive and the ‘interaction variables’ are usually called intensive. A vector space with conjugate pairs of variables as coordinates is a symplectic vector space, and I’ve written about how these show up in the category-theoretic approach to open systems:

• John Baez, A compositional framework for passive linear networks, Azimuth, 28 April 2015.

Continuing on:

The significance attached to these similarities has changed considerably, however, in the time from the first mathematical formulation of utility (Walras, 1909) to the full axiomatization of general equilibrium theory (Debreu, 1987). Léon Walras appears (Mirowski, 1989) to have conceptualized economic equilibrium as a balance of the gradients of utilities, more for the sake of similarity to the concept of force balance in mechanics, than to account for any observations about the outcomes of trade. Fisher (1892) (a student of J. Willard Gibbs) attempted to update Walrasian metaphors from mechanics to thermodynamics, but retained Walras’s program of seeking an explicit parallelism between physics and economics.

This Fisher is not the geneticist and statistician Ronald Fisher who came up with Fisher’s fundamental theorem. It’s the author of this thesis:

• Irving Fisher, Mathematical Investigations in the Theory of Value and Prices, Ph.D. thesis, Yale University, 1892.

Continuing on with Smith and Foley’s paper:

As mathematical economics has become more sophisticated (Debreu, 1987) the naive parallelism of Walras and Fisher has progressively been abandoned, and with it the sense that it matters whether neoclassical economics resembles any branch of physics. The cardinalization of utility that Walras thought of as a counterpart to energy has been discarded, apparently removing the possibility of comparing utility with any empirically measurable quantity. A long history of logically inconsistent (or simply unproductive) analogy making (see Section 7.2) has further caused the topic of parallels to fall out of favor. Samuelson (1960) summarizes well the current view among many economists, at the end of one of the few methodologically sound analyses of the parallel roles of dual representation in economics and physics:

The formal mathematical analogy between classical thermodynamics and mathematic economic systems has now been explored. This does not warrant the commonly met attempt to find more exact analogies of physical magnitudes—such as entropy or energy—in the economic realm. Why should there be laws like the first or second laws of thermodynamics holding in the economic realm? Why should ‘utility’ be literally identified with entropy, energy, or anything else? Why should a failure to make such a successful identification lead anyone to overlook or deny the mathematical isomorphism that does exist between minimum systems that arise in different disciplines?

The view that neoclassical economics is now mathematically mature, and that it is mere coincidence and no longer relevant whether it overlaps with any body of physical theory, is reflected in the complete omission of the topic of parallels from contemporary graduate texts (Mas-Collel et al., 1995). We argue here that, despite its long history of discussion, there are important insights still to be gleaned from considering the relation of neoclassical economics to classical thermodynamics. The new results concerning this relation we present here have significant implications, both for the interpretation of economic theory and for econometrics. The most important point of this paper (more important than the establishment of formal parallels between thermodynamics and utility economics) is that economics, because it does not recognize an equation of state or define prices intrinsically in terms of equilibrium, lacks the close relation between measurement and theory physical thermodynamics enjoys.

Luckily, the paper seems to be serious about explaining economics to those who know thermodynamics (and maybe vice versa). So, I will now read the rest of the paper—or at least skim it.

One interesting simple point seems to be this: there’s an analogy between entropy maximization and utility maximization, but it’s limited by the following difference.

In classical thermodynamics the total entropy of a closed system made of subsystems is the sum of the entropies of the parts. While the second law forbids the system from moving to a state to a state of lower total entropy, the entropies of some parts can decrease.

By contrast, in classical economics the total utility of a collection of agents is an unimportant quantity: what matters is the utility of each individual agent. The reason is that we assume the agents will voluntarily move from one state to another only if the utility of each agent separately increases. Furthermore, if we believe we can reparametrize the utility of each agent without changing anything, it makes no sense to add utilities.

(On the other hand, some utilitarian ethicists seem to believe it makes sense to add utilities and try to maximize the total. I imagine that libertarians would consider this ‘totalitarian’ approach morally unacceptable. I’m even less eager to enter discussions of the foundations of ethics than of economics, but it’s interesting how the question of whether a quantity can or ‘should’ be totaled up and then maximized plays a role in this debate.)

### 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.

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:
$\Phi(t,p,q,\cdots)$
and
$0 = a_1 \partial_t S+ a_2 \partial_p S+ a_3 \partial_q S+a_4 \partial_{qt} S+\cdots$
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:

$0=\gamma \Phi \partial_{TT}\Phi-(\gamma-1)\partial_T\Phi \partial_T \Phi$

• 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
$\Phi^{\alpha-1}=0$
or
$0=\frac{\alpha}{2} \partial_T \Phi \partial_T \Phi+\Phi \partial_{TT} \Phi$
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:

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:

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.

8. Jesse C. McKeown says:

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

• John Baez says:

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

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