Author: Eiko

Time: 2025-01-21 10:08:30 - 2025-01-21 10:08:30 (UTC)

Thermal Economics

Introduction

Not about the economics of heat, it is a theory of macroeconomics based on mathematics of thermal-dynamics and statistical mechanics.

References:

• Theory: arXiv 2412.00886

• Simulations, tests: arXiv 2410.20497

Theory paper has some inconsistencies, probably need some rewrite, treat it as unfinished work but still useful. Actually teaching something is a good way to learn it, including your own work.

Basic : thermal macroeconomics based on plausible axioms without specifying the macroeconomics. Does not require the notion of rational agents, it deduces an entropy function which governs the allowed transitions. We gonna have a pre-order and the entropy function realizes the pre-order.

It leads to notions like economic temperature, values of goods, relations between partial derivatives etc. (derivatives in the math sense, not the financial sense).

The other half we will talk about thermal-microeconomics. We are gonna to have stochastic models for microeconomics and the point is derive the entropy function for the resulting macroeconomics.

What is the scope of the story? So far we are only doing “exchange economics”, this means we have finitely many agents, finitely many types of goods which are infinitely divisible and durable. The agents exchange on encounter according to a stochastic process with unique attracting stationary probability distribution.

And require extendability, we can scale any economy by any positive factor and get one with same properties except for size. (a bit dubious assumption, but for simplicity now)

State space: A set $\mathrm{\Gamma }$ of possible equilibrium states. Assume

• A0 : An isolated system goes to an equilibrium state as $t\to \mathrm{\infty }$, meaning an attracting stationary probability on to micro description.

  eg. basic system: $N$ agents, money $M$, vector $G$ of amounts of goods, set $n\ge 1$ other types of good, no internal barriers. Assume unique equilibrium for each $(M,G)\in {\mathbb{R}}_{+}^{n+1}$ so $\mathrm{\Gamma }\subset {\mathbb{R}}_{+}^{n+1}$.

• Accessibility, state $Y$ of a system is accessible from $X$ then $X\le Y$ if an external trader with unlimited $M,G$ and access to some other economic systems can move the state from $X$ to $Y$ with arbitrarily small net change in the other systems but allowing arbitrary change in the traders money and goods.

• A2. transitive, we assume $X\le Y,Y\le Z$ then $X\le Z$.

Then $\le$ is a pre-order (reflexive, transitive, no anti-symmetry).

• Say $Y$ is reversibly accessible from $X$, $X\sim Y$ if $X\le Y$ and $Y\le X$. This is an equivalence relation.

  write $X<Y$ if $X\le Y$ but not $Y\le X$, (equivalently, not $X\sim Y$).

Given two economies $A,B$ in states $X_A,Y_B$, we denote the state as a product by $(X_A,Y_B)\in {\mathrm{\Gamma }}_A\times {\mathrm{\Gamma }}_B$.

• A3. $X_A\le X_A^{\prime }$ and $Y_B\le Y_B^{\prime }$ then $(X_A,Y_B)\le (X_A^{\prime },Y_B^{\prime })$.

• Scaling, A4. For any economy $A$ and $\lambda >0$, there exists a scaled version $\lambda A$ and if $X\le Y$ for $A$ then $\lambda X\le \lambda Y$ for $\lambda A$. (we’d like to go beyond it, develop non-extensive macroeconomics).

• A5. Any system can be subdivided in arbitrary ratio $\lambda :1-\lambda$ by cutting connections and state $X\sim (\lambda X,(1-\lambda )X)$.

• A6. If $(X,\epsilon Z_0)\le (Y,\epsilon Z_1)$ for a sequence of $\epsilon \to 0$, then $X\le Y$.

Comparison Hypothesis

Say CH holds for a state space $\mathrm{\Gamma }$ if for all $X,Y\in \mathrm{\Gamma }$, $X\le Y$ or $Y\le X$ (including both), ($X,Y$ are comparable).

Definition. A multiple scaled copy of $\mathrm{\Gamma }$ is ${\lambda }_1\times \dots {\lambda }_m\mathrm{\Gamma }$ for some ${\lambda }_i>0$.

Theorem[LY]. If $\le$ satisfies A1-6 on $\mathrm{\Gamma }$ and CH on each MSC, then there exists an entropy function $S:\mathrm{\Gamma }\to \mathbb{R}$ such that if $\sum {\lambda }_i=\sum {\lambda }_j^{\prime }$ for two ms-copy then

$$({\lambda }_1{X}_1,\dots )\le ({\lambda }_1^{\prime }{X}_1^{\prime },\dots )$$

iff

$$\sum {\lambda }_{i}S(X_i)\le \sum {\lambda }_j^{\prime }S(X_j^{\prime })$$

$S$ is unique up to affine transformation.

The construction of $S$ is to choose two reference states $X_0<X_1$ For $X_0\le X\le X_1$, let $S(X)=sup\{\lambda :((1-\lambda )X_0,\lambda X_1)\le X\}$.

Theorem 2[LY]. If CH holds for all product systems formed from scaled copies of a collection of systems ${\mathrm{\Gamma }}_k$ and for each ${\mathrm{\Gamma }}_k$ we have an entropy function $S_k$ as per theorem $1$, then there exists $a_k>0$ such that

$$S(X)=a_k{S}_k(X)$$

for $X\in {\mathrm{\Gamma }}_k$ extended to all systems by $S(\lambda X)=\lambda S(X)$ and

$$S(X,Y)=S(X)+S(Y)$$

satisfying $X\le Y$ iff $S(X)\le S(Y)$ for $X,Y$ in the same product system.

So we need to justify CH for product systems.

Simple System

A simple system is one whose state space $\mathrm{\Gamma }$ is an open convex subset of ${\mathbb{R}}^{n+1}$ with $n\ge 1$, with one distinguished coordinate called money, rest called goods coordinates.

e.g. a basic exchange economy, also an exchange economy with internal barriers to some types of goods.

Then for $X,Y$ states of a simple system,

• A7. For $t\in [0,1]$, $(tX,(1-t)Y)\le tX+(1-t)Y$.

  because the trader can just connect the two parts.

In particular, the forward sector $A_X=\{Y\in \mathrm{\Gamma }:X\le Y\}$ is automatically a convex set.

• Assume A8. for $X\in \mathrm{\Gamma }$ there is a $Y\in \mathrm{\Gamma }$ with $X<Y$.

  This follows (for example) if we assume money is desirable at macro level, then we can just take $Y=X+(M,0)\Rightarrow X<Y$. (i.e. economy won’t give it away)

  and if trader is not a $M$

• A9. Assume $A_X$ has unique support plane at $X$ and it is locally Lipschitz function of $X$.

• A10. The boundary of $A_X$, $\partial A_X$ is connected.

Theorem 3[LY]. If $X,Y\in \mathrm{\Gamma }$ simple then $X\le Y$ or $Y\le X$. $X\sim Y$ iff $Y\in \partial A_X$.

Financial Join

The financial join of two simple economies $A,B$ is the joint economy where money is allowed to flow between the two distinguished parts but nothing else. We call this financial contact.

The financial join is a simple system. Money might flow without goods flowing, people in $A$ send money to people in $B$, or one person owns assets in both economies.

Denote by $\theta (X_A,Y_B)$ the financial join of $A,B$ in states $X_A,Y_B$ after coming to equilibrium. It is specified by the total distinguished money and the individual amounts of all other goods in each part. We deduce

• A11. $(X_A,Y_B)\le \theta (X_A,Y_B)$.

• A12. There is $X_A^{\prime },Y_B^{\prime }$ such that $\theta (X_A,Y_B)\sim (X_A^{\prime },Y_B^{\prime })$. (financial cut)

  Say $X_A,Y_B$ are in financial equilibrium if $(X_A,Y_B)\sim \theta (X_A,Y_B)$, write $X_A\equiv Y_B$. It is symmetric, reflexive (A13, but this is deducible), transitive (A14, this is probably not true, but it is assumed for now).

  So financial equilibrium is an equivalence relation, the Zeroth Law of Thermo-macroeconomics.

Accessibility and Financial Equilibrium

Assume A14, for each state $X$ of a simple system $A$, there exists $X_0,X_1$ with $X_0\equiv X_1$ and $X_0<X<X_1$.

Example: Cabb-Douglas Economy

Consists of $N$ agents, some money $M>0$, one good $G>0$ and exponents $\alpha ,\eta >0$. Each agent has ‘utility’ $u(g,m)=g^{\alpha -1}{m}^{\eta -1}$ for possessing amounts $g$ of good and $m$ of money.

Suppose agents $i,j$ encounter independently at a rate $k_{ij}\ge 0$ (symmetric), forming a connected undirected symmetric graph.

On encounter, they pool their belongings and redistribute between them with probability density $\propto u(g_i^{\prime },m_i^{\prime })u(g_j^{\prime },m_j^{\prime })$. Conditional on $g_i+g_j=g_i^{\prime }+g_j^{\prime }$ and $m_i+m_j=m_i^{\prime }+m_j^{\prime }$, independent of previous transitions, this is a Markov Process, and it is reversible with respect to the probability density

$$\rho (g,m)=\frac{1}{Z(G,M)}\prod _{i}u(g_i,m_i)$$

conditional on $\sum g_i=G,\sum m_i=M$, product of two simplices.

$$Z(G,M)=\frac{G^{N\alpha -1}\mathrm{\Gamma }(\alpha {)}^N}{\mathrm{\Gamma }(N\alpha )}\frac{M^{N\eta -1}\mathrm{\Gamma }(\eta {)}^N}{\mathrm{\Gamma }(N\eta )}$$

so $\rho$ is a stationary distribution. Under the $k$ connected assumption, it is attracting [to be addressed later]. So state space is ${\mathbb{R}}_{+}^2$ labelled by $(G,M)$.