An ‘economics proof’ of the supporting hyperplane theorem

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Abstract

Based centrally on the economic concept of a cost function, an ‘economics proof’ (by induction) is given of the supporting hyperplane theorem.

Introduction

Separation theorems for convex sets are a basic mathematical tool that find enormously widespread use throughout economics. As has been recognized in the literature, all of the standard separating hyperplane theorems are readily derivable from the supporting hyperplane version of the theorem. (The most difficult case to prove is where the two convex sets are ‘kissing’ along their common boundary, which is essentially the case treated here of the supporting hyperplane theorem for a closed convex set.) In a sense, we are really talking about one unified ‘family’ of separating (or supporting) hyperplane theorems. So far as I have ever seen, the standard proofs of the family of separating (or supporting) hyperplane theorems found in the literature consist of either formal abstract mathematics1 or, if motivated at all, are geometrically motivated.2 Although the separating hyperplane theorems are ultimately loaded with economic content, I have never encountered a proof that showed clearly and directly within the proof itself the basic connection between the ‘economics of convexity’ and the ‘mathematics of convexity’.

This paper contains a proof of one of the main separating hyperplane theorems that starts with a rigorously stated, but economically intuitive, economic proposition about cost functions, and derives from it the proof of the supporting hyperplane theorem. The proof is by induction on the number of ‘inputs’ as arguments of a concave production function. The proof is constructive in the sense that it indicates precisely how to construct the n-plus-first price (the ‘output’ price), given the previous n prices (the ‘input’ prices). The proof is also economically intuitive in the sense of being based on a fundamental economic idea about the possibility of being able to decentralize (production) decisions in a convex technology. Some variant of this basic idea finds application almost everywhere throughout economics, being, e.g., the subject of the celebrated first essay (‘Allocation of Resources and the Price System’) in Koopmans (1957).

Of course, the separating hyperplane theorem itself has been stated and proved, in its many versions, a long time ago. The only novelty of this paper is that the proof, which is by induction, is based upon economic concepts. The proof is taken from lecture notes to a graduate course I teach on ‘optimization for economists’. Students studying formal economics, in this course or elsewhere, may wish (or even need) to see a rigorous proof of a separating hyperplane theorem. In this context, I then asked myself the following question. Why not expose such students to a proof that in itself reveals the close connection between the economics and the mathematics of convexity — by being based on the language and ideas of economics directly? In the hopes that such a proof may perhaps be found to have some modest interest, pedagogic or otherwise, I offer it here to a wider audience than that of my own classroom.

Section snippets

Strategy of proof

Because the goal of this proof is to emphasize economic content and a connection with economics, before diving into the details I want first to give an overview of the strategy.

Let y stand for a single ‘output’, while x represents a n-dimensional vector of ‘inputs’. The convex (n+1)-dimensional closed ‘production set’ is denoted S. Thus, the ‘output–input’ combination (y, x) is feasible if and only if(y, x)∈S.

Let (y∗, x*) represent a boundary point of S. In economics the interesting boundary

Three lemmas for the initial case n=1

We will assume without proof the following three intuitive lemmas, which constitute the initial induction step.

Lemma 1

If A⊂E1 is a closed convex set on the one-dimensional line, then A has one of the following three possible forms: (−∞,b1], [b2,∞), [b3,b4], where b1, b2, b3, b4 represent (finite) boundary points of A and b3≤b4.

Lemma 2 (one-dimensional version of separating or supporting hyperplane theorem)

If A is a closed convex set on the one-dimensional line, with boundary b, then there exists a scalar γ≠0 such thatx∈A⇒γx≤γb.

Lemma 3

Let f(y) be a convex function defined over the closed

The supporting hyperplane theorem

Theorem

Let S⊂Em be a closed convex set. Let z* be any boundary point of S. Then there exists a m-dimensional vector P≠0 such thatz∈S⇒P·z<=P·z∗.

Proof

By Lemma 2, the theorem is true for m=1. Now suppose it is true for m=n, with n a arbitrary positive integer. Then we must prove that the theorem is also true for m=n+1.

Let z be a (n+1)-dimensional vector, let x be a n-dimensional vector, and let y be a one-dimensional scalar. Think of z being decomposed into y and x as follows:z=(y, x).

Define parametrically

Conclusion

All of the other separating hyperplane results are derivable from (8). Thus, in effect, a proof by induction has been provided here of the family of separating hyperplane theorems. The proof uses directly the language and intuition of one of the most powerful concepts in economic theory: the idea that with convexity there always exist shadow prices that support efficient production outcomes via decentralized profit maximization. This proof is one more piece of evidence that the mathematical and

References (3)

  • G. Debreu

    Separation theorems for convex sets

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