Revealed preference theory started out as an exploration into the testable implications of neoclassical demand theory, and while it has expanded in many different directions, the analysis of rational demand is the most actively researched area in revealed preference theory. In this chapter, we present an exposition of the basic results in the revealed preference theory of rational demand.

We suppose here that we have observations on the purchasing decisions of a single consumer. The consumer makes a sequence of independent choices at different price vectors. The data consists of the consumer's choices, and we seek to understand the implications of rational consumption behavior for such data.

The material on rational demand is divided into three chapters. In Chapter 3 we discuss the basic results on weak and strong rationalization, including Afriat's Theorem, the main result in the revealed preference theory of rational demand. In Chapter 4 we turn to specific properties of demand functions; and in Chapter 5 to some of the practical issues that arise when applying the results of revealed preference theory to empirical research.

WEAK RATIONALIZATION

Consider an agent choosing a bundle of n goods to purchase. Consumption space is X ⊆ Rn+, meaning that the consumer chooses x ∈ X. We assume that for any x ∈ X and ε >0, there is ε’ with 0<ε’ <ε and x+ε’1 ∈ X; this means that it is possible to add more of every good to any bundle in X and still remain in X.

Given a preference relation on X, let d : Rn++ ×R+→2X be the demand correspondence associated to ; it is defined as

We refer to d as a demand function if d(p,m) is always a singleton.

A consumption dataset D is a collection (xk,pk), k = 1, …K, with K ≥ 1 an integer, xk ∈ X and pk ∈ Rn ++. For each k, xk is the consumption bundle purchased by the consumer at prices pk. We shall assume that the consumer exhausts all his income, so that the expenditure pk · xk is also the total income devoted to consumption at the time at which the purchases were made.