Statistical Equilibrium In Economics

(Notes added by Enrico Scalas on 19.08.2009)

  1. What is the common notion of equilibrium in economics?

The concept of equilibrium referred to in General Equilibrium Theory is taken from Physics. It coincides with mechanical equilibrium.

When looking for mechanical equilibrium one minimizes a potential function
subject to boundary conditions, in order to find equilibrium positions;
when looking for standard (micro)economic equilibrium, one maximizes
a utility function subject to budget constraints (this is the consumer side, in other words: demand) and maximizes the profit subject to cost constraints (this is producer side, in other words, supply); then one equates supply and demand, and finds
equilibrium quantities and prices. In both cases, the mathematical tool is optimization with constraints using the method of Lagrange multipliers.

Walras and Pareto explicitly inspired their pioneering work on General Equilibrium Theory to Physics and mechanical equilibrium. This was made clear by Ingrao and Israel (1990).

  1. What is statistical equilibrium?

Statistical equilibrium is another notion of equilibrium in Physics. It was defined by Maxwell and Boltzmann in their early work on the theory of gases, trying to reconcile mechanics and thermodynamics. In order to better understand this notion, it is useful to make use of a Markovianist approach to statistical equilibrium as discussed by Oliver Penrose (the brother of Roger Penrose) in his 1970 book. By the way, a similar
approach was promoted by Richard von Mises (the brother of Ludwig von Mises) in a book reprinted in 1945 (actually the book was written by
R. von Mises before WWII).

A finite Markov chain is a stochastic process defined as a sequence of random variables
$X_1,\ldots,X_n$ on the same probability space that assume values in a finite set $S$, known as the state space. For a Markov chain, the predictive probability
$\mathbb{P}(X_n = x_n|X_{n-1}=x_{n-1},\ldots,X_1=x_1)$ has the following simple form

(1)
\begin{align} \mathbb{P}(X_n = x_n|X_{n-1}=x_{n-1},\ldots,X_1=x_1) = \mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1}). \end{align}

In other words, the predictive probability does not depend on al the past states, but only on the last state occupied by the chain. As a consequence of the multiplication theorem, one gets that the finite-dimensional distribution $\mathbb{P}(X_1 = x_1,\ldots,X_n=x_n)$ is given by

(2)
\begin{align} \mathbb{P}(X_1=x_1,\ldots,X_n=x_n) = \nonumber \end{align}
(3)
\begin{align} \mathbb{P}(X_n =x_n|X_{n-1}=x_{n-1}) \cdots \mathbb{P}(X_2=x_2|X_1=x_1) \mathbb{P}(X_1=x_1); \end{align}

As a consequence of Kolmogorov's representation theorem, this means that a Markov chain is fully characterized by the knowledge of the functions $\mathbb{P}(X_m=x_m|X_{m-1}=x_{m-1})$, also known as transition probabilities and $\mathbb{P}(X_1=x_1)$, also known as initial probability
distribution
. If the transition probabilities do not depend on the index $m$, but only on the initial and on the final state, than the Markov chain is called homogeneous. In the following, only homogeneous Markov chains will be considered. For the sake of simplicity, it is useful to introduce the notation

(4)
\begin{align} P(x,y) = \mathbb{P}(X_m = y|X_{m-1}=x) \end{align}

for the transition probability and

(5)
\begin{align} \pi(x) = \mathbb{P}(X_1 = x) \end{align}

for the initial probability distribution. Note that $P(x,y)$ is nothing else than
a matrix in the finite case under scrutiny, with the property that

(6)
\begin{align} \sum_{y \in S} P(x,y) = 1; \end{align}

in other words the rows of the matrix sum up to 1 as a consequence of the addition axiom. Such matrices are called stochastic matrices (to be distinguished from
random matrices which are matrices with random entries). Note that the initial distribution can be written as a row vector, so that one can obtain the marginal
distribution of the random variable $X_n$ as

(7)
\begin{align} \mathbb{P}(X_n = y) = \sum_{x \in S} \pi(x) P^n (x,y). \end{align}

Now suppose there is a distribution $p(x)$ satisfying the equation

(8)
\begin{align} \mathbb{P}(X_n=y) = p(y) = \sum_{x \in S} p(x) P (x,y), \end{align}

then $p(x)$ is called a stationary distribution or invariant measure. If the chain starts with states distributed according to $p(x)$, this distribution does not change as time goes by. Note that the states are jumping from one to another one, but the probability of finding the system in a specific state does not change. This is exactly the idea of statistical equilibrium put forward by Ludwig Boltzmann.

However, more can and should be said. First of all, the stationary distribution may not exist; secondly the chain usually starts from a specific state, so that the initial distribution is a vector full of 0's and with a single 1 in the initial state. The latter state of affairs can be represented by a Kronecker delta $\pi(x) = \delta(x,x_0)$, where $x_0$ is the specific initial state. Usually, this is not a stationary distribution and the convergence of the chain to the stationary distribution is not granted at all.
Fortunately, it turns out that under some rather mild conditions:

  • the stationary distribution exists and is unique;
  • the chain always converges to the stationary distribution irrespective of its initial distribution.

It is indeed sufficient to consider a finite chain that is aperiodic and irreducible or ergodic. If $x$ is a state of the Markov chain, its period $d_x$is defined as the greatest common divisor of the set $\{n \geq 1: \, P^n (x,x) > 0\}$. A chain is irreducible if any state leads to any other state with finite probability and in a finite number of steps. If a state $x$ leads to a state $y$, then one can
prove that also $y$ leads to $x$ and the set of communicating states have a common period. In an irreducible chain all the states communicate and they have a common period $d$. The chain is aperiodic if $d=1$.

If the finite Markov chain is irreducible and aperiodic, then it has a unique stationary distribution $p(x)$ and

(9)
\begin{align} p(y) = \lim_{n \to \infty} P^n(x,y) \end{align}

irrespective of the initial state $x$. This means that, after a transient period, the distribution of chain states reaches a stationary distribution, which can then be interpreted as an equilibrium distribution in the statistical sense.

  1. Why and where statistical equilibrium may be useful in economics?

There are several possible domains of application of the concept of statistical equilibrium in Economics. Incidentally, note that many agent-based models used in Economic theory are intrinsically Markov chains (or Markovian processes). Therefore the ideas discussed above naturally apply. Up to now, with my coworkers I have used these concepts:

  • to discuss some toy models for the distribution of wealth (not of income!) as in Scalas et al. (2006) and in Garibaldi et al. (2007).
  • to study a simple agent-based model of financial market with heterogeneous knowledge as in Toth et al. (2007).
  • to generalize a sectoral productivity model originally due to Aoki and Yoshikawa, in Scalas and Garibaldi (2009).

In Scalas et al. (2006), Garibaldi et al. (2007), and Scalas and Garibaldi (2009), we promote the use of a finitary approach to combinatorial stochastic processes which will be the main topic of my presentation in Reykjavik.

References

B. Ingrao and G. Israel (1990), The Invisible Hand. Economic Equilibrium in the History of Science,
MIT Press, Cambridge MA.

O. Penrose (1970), Foundations of Statistical Mechanics: A Deductive Treatment, Dover, NY.

R. von Mises (1945), Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik, Rosenberg, NY.

E. Scalas, U. Garibaldi and S. Donadio (2006), Statistical equilibrium in simple exchange games I - Methods of solution and application to the Bennati-Dragulescu-Yakovenko (BDY) game, European Physical Journal B, 53(2), 267-272.

U. Garibaldi, E. Scalas and P. Viarengo (2007), Statistical equilibrium in simple exchange games II. The redistribution game, European Physical Journal B, 60(2), 241-246.

B. Toth, E. Scalas, J. Huber and M. Kirchler (2007), The value of information in a multi-agent market model - The luck of the uninformed, European Physical Journal B, 55(1), 115-120.

E. Scalas and U. Garibaldi (2009), A Dynamic Probabilistic Version of the Aoki–Yoshikawa Sectoral Productivity Model, Economics, The Open-Access, Open-Assessment E-Journal, 3, 2009-15.
http://www.economics-ejournal.org/economics/journalarticles/2009-15.

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