Afriat and arbitrage

Alan Beggs

doi:10.1007/s40505-021-00208-w

Abstract

This paper presents a proof of Afriat’s (Int Econ Rev 8:67–77) theorem on revealed preference by using the idea that a rational consumer should not be vulnerable to arbitrage. The main mathematical tool is the separating hyperplane theorem.

Highlights

There are a number of proofs of Afriat (1967)’s famous theorem on revealed preference
It is based on the idea that a consumer should not be vulnerable to arbitrage or money pumps if her behavior is internally consistent
This paper shows that a separation argument can be used to prove another classical result, Afriat’s theorem

Summary

Introduction

There are a number of proofs of Afriat (1967)’s famous theorem on revealed preference. Echenique et al (2011) propose a measure of the seriousness of violations of the Generalized Axiom of Revealed Preference (GARP) by using a money pump argument. This idea can be used to motivate the proof in the current paper. In the example above, when GARP is violated the arbitrageur gains at each step of the trade (buying x j and selling xi to the consumer at prices pi and buying xi and selling x j at prices p j ) This is analogous to an asset which is profitable whatever the state (here the states are the observations i and j) and as such is a ‘free lunch’.

Results

Extensions

Discussion