On Gale's Contribution in Revealed Preference Theory

TL;DR

This paper critically examines Gale's 1960 work on demand functions, identifying a proof gap, and provides three complete proofs to confirm Gale's original claim about demand functions satisfying the weak axiom.

Contribution

It clarifies Gale's original proof, identifies a gap, and offers three complete proofs, including a modern one using Shephard's lemma.

Findings

Gale's candidate demand function satisfies the weak axiom.

Gale's original proof had a gap but his claim remains valid.

Three complete proofs confirm Gale's claim.

Abstract

We investigate Gale's important paper published in 1960. This paper contains an example of a candidate of the demand function that satisfies the weak axiom of revealed preference and that is doubtful that it is a demand function of some weak order. We examine this paper and first scrutinize what Gale proved. Then we identify a gap in Gale's proof and show that he failed to show that this candidate of the demand function is not a demand function. Next, we present three complete proofs of Gale's claim. First, we construct a proof that was constructible in 1960 by a fact that Gale himself demonstrated. Second, we construct a modern and simple proof using Shephard's lemma. Third, we construct a proof that follows the direction that Gale originally conceived. Our conclusion is as follows: although, in 1960, Gale was not able to prove that the candidate of the demand function that he constructed is not a demand function, he substantially proved it, and therefore it is fair to say that the credit for finding a candidate of the demand function that satisfies the weak axiom but is not a demand function is attributed to Gale.