The tests in Chapters 3 and 4 are meant to be applicable to actual datasets, and many researchers have investigated these applications using experiments, consumption surveys, and other sources of data. Naturally, there are complications that arise when one tries to carry out the tests we have described. We shall focus on the basic application of GARP (or SARP) to data on consumption expenditures. The difficulties in applying GARP can be summarized as follows:

First, GARP is an “all or nothing” notion. A dataset either falsifies the theory of a rational consumer or it does not. One may, however, want to distinguish a grayscale of degrees of violation of the theory. It is possible that some violations can be attributed to simple mistakes on the part of a fully rational consumer. We develop concepts along these lines in 5.1.

Second, the nature of budget sets introduces problems with the power of testing for GARP. When two observed budget sets are nested, then there are no choices that can indicate a violation of GARP (actually ofWARP in that case). More generally, any dataset in which budget sets have substantial overlap is biased towards the satisfaction of GARP. The problem of budget overlap is very real because often data contain more individual-level variation in expenditure levels than variation in relative prices. As we explain below (Section 5.2), these features cause budget sets to have substantial overlap.

Third, many studies do not track the identities of individual consumers.With such cross-sectional datasets, two observations (x1,p1) and (x2,p2) actually correspond to different individuals (or households), but they are identified as having the same preferences based on their observable characteristics. The procedure of identifying individuals based on their observable characteristics is called “matching” in statistics and econometrics. The basic problem is how to carry out this identification, or matching: when can we treat two individuals as the same for the purposes of revealed preference tests.

Moreover, certain cross-sectional datasets exacerbate the problem of power. The observations (x1,p1) and (x2,p2) in the data are of two different individuals (treated as the same agent) at similar points in time. Prices p1 and p2 are then bound to be similar, because even prices in different locations are similar at the same point in time.