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Proving Adam Smith: The Nobel Prizing Winning Work of Kenneth Arrow and Gerard Debreu

In 1954, two economists -Kenneth Arrow (American) and Gerard Debreu (French) published a paper in the prestigious economics journal, Econometrica with the forbidding title “Existence of an Equilibrium for a Competitive Economy.” Arrow and Debreu went on to win the Nobel Prize in Economics for their development of their theory, one which offered a rigorous, mathematical proof of the correctness of Adam Smith’s “invisible hand” conjecture. Here is the conjecture in Smith’s own words, taken from his famous book, The Wealth of Nations published in 1776:

“Every individual .. neither intends to promote the public interest, nor knows how much he is promoting it … he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention.”

The invisible hand is Smith’s metaphor for the unseen forces that guides a free market economy to the best interest of society even though every man or woman acts purely out of self-interest. Best social interest does not mean that economic gains are shared equally between individuals; rather it means that through the price mechanism, goods that are desired by society are brought to the market because there are economic incentives to do so and at the “right” price if free trade can take place unimpeded.

It is a nice idea, but how to prove that this rather counter-intuitive conjecture? Also, what conditions must exist for markets to function unimpeded to deliver the predicted social benefits? This was the challenge that Arrow and Debreu set for themselves.

Instead of observing how real markets work on a case-by-case basis (which would be tedious and ad-hoc), they decided to go for a general proof of whether the invisible hand really works. More formally, they wanted to know precisely whether there exists a state of general economic equilibrium whereby price signals coordinate demand and supply so well that there is neither too little or too much of any private good or service. The task was a tall order, and the only way to prove such a general statement was to use mathematics, the way theoretical physicists use math to predict the existence of abstract stuff like gravitational waves or black holes.

In what follows, I will sketch the key mathematical idea used by Arrow and Debreu in their proof which is both elegant and powerful. You can either stop here or continue reading the sections below (requires some familiarity with real analysis).

The Mathematics of General Economic Equilibrium

The math that Arrow and Debreu used was state-of-the-art at the time; it goes by the name of fixed-Point theorems. The particular fixed-point theorem featured in their 1954 paper is called the Brouwer Fixed-point Theorem, named after the Dutch mathematician, L.E.J. Brouwer (1881-1966) who was a major figure in the field of topology.

Topology is the branch of mathematics that studies the properties of geometric shapes that are preserved through deformations, twisting, and stretching (but not tearing). In more colorful terms, topology has been described as “rubber sheet geometry” or “doughnut geometry”. I will briefly state Brouwer’s famous fixed-point theorem for the one-dimension case, then give some remarks about the more difficult n-dimensional case. 

For the one-dimension case, Brouwer’s fixed-point theorem can be stated as follows:

If {\displaystyle \psi}  is a continuous mapping of the interval 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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