Revealed Preference

In Chapter 6 we saw how we can use information about the consumer's preferences and budget constraint to determine his or her demand. In this chapter we reverse this process and show how we can use information about the consumer's demand to discover information about his or her preferences. Up until now, we were thinking about what preferences could tell us about people's behavior. But in real life, preferences are not directly observable: we have to discover people's preferences from observing their behavior. In this chapter we'll develop some tools to do this.

When we talk of determining people's preferences from observing their behavior, we have to assume that the preferences will remain unchanged while we observe the behavior. Over very long time spans, this is not very reasonable. But for the monthly or quarterly time spans that economists usually deal with, it seems unlikely that a particular consumer's tastes would change radically. Thus we will adopt a maintained hypothesis that the consumer's preferences are stable over the time period for which we observe his or her choice behavior.

7.1 The Idea of Revealed Preference

Before we begin this investigation, let's adopt the convention that in this chapter, the underlying preferences—whatever they may be—are known to be strictly convex. Thus there will be a unique demanded bundle at each budget. This assumption is not necessary for the theory of revealed preference, but the exposition will be simpler with it.

Consider Figure 7.1, where we have depicted a consumer's demanded bundle, (ei,X2)> and another arbitrary bundle, (1/1,2/2)» that is beneath the consumer's budget line. Suppose that we are willing to postulate that this consumer is an optimizing consumer of the sort we have been studying. What can we say about the consumer's preferences between these two bundles of goods?

Revealed preference. The bundle {x\, £2) that the consumer chooses is revealed preferred to the bundle , j/2), a bundle that he could have chosen.

Well, the bundle (3/1,2/2) is certainly an affordable purchase at the given budget—the consumer could have bought it if he or she wanted to, and would even have had money left over. Since (#1, £2) the optimal bundle, it must be better than anything else that the consumer could afford. Hence, in particular it must be better than (2/1,2/2)-

The same argument holds for any bundle on or underneath the budget line other than the demanded bundle. Since it could have been bought at the given budget but wasn't, then what was bought must be better. Here is where we use the assumption that there is a unique demanded bundle for each budget. If preferences are not strictly convex, so that indifference curves have flat spots, it may be that some bundles that are on the budget line might be just as good as the demanded bundle. This complication can be handled without too much difficulty, but it is easier to just assume it away.

In Figure 7.1 all of the bundles in the shaded area underneath the budget line are revealed worse than the demanded bundle (xi, X2). This is because they could have been chosen, but were rejected in favor of (xi, #2). We will now translate this geometric discussion of revealed preference into algebra.

Let x2) be the bundle purchased at prices {pi,p2) when the consumer has income m. What does it mean to say that (2/1,2/2) is affordable at those prices and income? It simply means that (2/1, y2) satisfies the budget constraint

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