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The Economics of a Billion Interactions

Economics isn’t physics although it strives to emulate it. That doesn’t mean that economists have an easier job than physicists or that there is no beauty in economic theories. Consider the task economists have set themselves to. How do you prove that an equilibrium exist in a market economy with millions of transactions taking place daily across sellers and buyers who are only interested in maximizing their own self interests? Remarkably, the best minds in economics have shown formally that such an equilibrium exist even though intuitively, we expect chaos rather than order. How did they do it?

The answer – the subject of today’s post – is that they resorted to clever mathematics, in a way that mirrors the use of mathematics in physics to model gravity and relativity.

To demonstrate the existence of general equilibrium in a market economy, the conceptual tool that economists utilized is known as fixed point theorems. I first came across fixed-point theorems in my senior undergraduate days but went away clueless as to its relevance for economics until much later. There are actually a family of fixed point theorems, but for this post, I will discuss only one of them: the famous (to economists) Brouwer fixed-point theorem, named after its discover, the Dutch mathematician L.E.J. Brouwer (1881 – 1966).

L.E.J. Brouwer was a major figure in the field of topology, the branch of mathematics that studies the properties of geometric shapes that are preserved through deformations, twisting, and stretching (but not tears). In more colorful terms, topology has been described as “rubber sheet geometry”.

L.E.J. Brouwer (1881 – 1966)

Forty years after Brouwer published his theorem, two economists: Kenneth Arrow (American, 1921 – 2017) and Gerard Debreu (French, 1921 – 2004) realized that it was the tool they needed to prove the existence of an equilibrium in a decentralized economy [1]. Their elegant proof formally vindicates Adam Smith’s penetrating insight that the price mechanism or “invisible hand” is a powerful way to align the economic interests of individuals and the collective economy. For their achievements, Arrow and Debreu were awarded the Nobel Prize in Economic Science in 1972 and 1983 respectively.

To understand the relevance of Brouwer’s fixed point theorem in demonstrating economic general equilibrium, I will first discuss Brouwer’s fixed-point theorem for the “simple” one-dimension case. After that, I will make some brief remarks about the more difficult n-dimensional case used by Arrow and Debreu in their proof. In the interest of readers who have “math phobia”, I will keep the discussion to be as informal as I possibly can.

Brouwer’s Fixed Point Theorem: The One-Dimensional Case

For the one-dimension case, Brouwer’s fixed-point theorem can be stated as follows (some very basic knowledge of real analysis is assumed here):

If {\displaystyle \psi} is a continuous mapping of the interval $latex I = [-1,1] $ to itself, then there exists at least one point, {\displaystyle \ x_0} in I such that {\displaystyle \psi(x_0) = x_0.}

A picture helps (see figure below). Since \psi is a continuous mapping from I to itself, it is a continuous function defined over the interval from -1 to 1, inclusive. From the figure, we see that the graph of \psi intersects the diagonal line of the square with side I at a specific point, P.  Denote the x-coordinate of this point by x_0 . Then, we have {\displaystyle \psi(x_0) = x_0}, indicating that x_0 is a fixed point of \psi.

Side note: to get a rigorous proof for the one-dimension case, we need something called the intermediate value theorem applied to continuous functions. Applying this theorem to the continuous function \phi(x) = x , we see that there is a point, x_0 in the interval I such that \phi(x_0) = x_0, and hence, \phi(x_0) = x_0 . 

Things become more abstract and harder in higher dimensions. For the two-dimension case, we need to deal with the concept of a disk, the region in a plane bounded by a circle. We say that a disk is closed if it contains the circle that constitutes its boundary and open it if does not. The following is Brouwer’s fixed-point theorem for two dimensions.

Brouwer’s Fixed-Point Theorem (Two Dimensions)

Let B be the closed disk of radius 1 and let \psi be a continuous mapping from B to itself. Then there exists a point, P_0 in B such that \psi(P_0) = P_0 .

The theorem now sounds more freakish. Moreover, is not possible to provide a graphical proof like we did earlier for one dimension. This is because the graph of a continuous mapping from the closed disk to itself can’t be drawn (it lives in a four-dimensional space!). An analogy helps. Picture a circular tray which is completely filled with sand. If you shake the tray slightly but continuously, the sand will also move continuously from one location to another. If we represent each grain of sand as a point, P of the closed disk, then we can consider the new location of P to be \psi(P) . Brouwer’s theorem guarantees that no matter how you shake the tray, there exists at least one grain of sand which does not move from its original location. In the context of the economy, each grain of sand represents an equilibrium. Arrow and Debreu completed their proof by showing that there exists one and only one “grain of sand”. Thus, not only does equilibrium exists, it is unique.

The intuition for the two-dimensional case carries over to the n dimensional case. This general version can be stated as follows:

Let B be the unit ball in \mathcal R^n defined by \displaystyle {B = \{ (x_1, x_2, ...,x_n)| {x_1}^2 + {x_2}^2 + ... + {x_n}^2 \leq 1 \} } . Let \psi be a continuous mapping from B to itself. Then, there exists a point, P_0 in B such that \psi(P_0) = P_0 .

This general version was proved independently in 1910 by Hadamard and Brouwer himself, and is the version that Arrow and Debreu used in their seminal paper.

General equilibrium theory is one of the high points of economic theorizing. It showed the surprising (and beautiful) connection between an esoteric branch of mathematics and the invisible hand of a market economy. In mathematics itself, fixed point theorems are used to show the existence of solutions of differential equations, which in turn find applications in many branches of science and engineering [see 2] 

Notes:

[1] Arrow, Kenneth. J. and Gerard Debreu (1954). “Existence of an equilibrium for a competitive economy”. Econometrica. 22 (3): 265–290. doi:10.2307/1907353.

[2] See for example, Eberhard Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-Point Theorems, Springer-Verlag, 1986.

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