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CHAPTER 5
Practical Issues in Revealed
Preference Analysis
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 of WARP 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


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72
Practical Issues in Revealed Preference Analysis
p′
p′
x′
x′
p
p
x
x
(x, p) and (x', p' ) violate GARP (in fact WARP).
(a)
(b)
A more “severe” violation of GARP.
Fig. 5.1 Two observations: (x, p) and (x′, p′).
bound to be similar, because even prices in different locations are similar at the
same point in time. The main source of variation in the data must then be in
expenditure levels pk · xk. As a consequence budget sets tend to be nested; the
power of testing for GARP is thus diminished.
5.1
MEASURES OF THE SEVERITY OF A VIOLATION OF GARP
The revealed preference tests we have seen are dichotomous: either the data
satisfy the test or they do not. But we are probably also interested in the degree
to which a test is violated. Speciﬁcally, suppose that a dataset violates GARP,
but that we somehow judge the violation to be mild. We may not be willing to
conclude that the agents involved behaved irrationally.
5.1.1
Afriat’s efﬁciency index
Afriat observes that if expenditures at each observation are “deﬂated” by some
number e ∈[0,1], then the violation of GARP will disappear. Afriat proposes
to measure the severity of a violation by how much expenditure needs to be
deﬂated for the data to satisfy GARP.
Formally, deﬁne a modiﬁed revealed preference relation Re by xk Re y iff
epk ·xk ≥pk ·y; deﬁne Pe similarly. For e ∈[0,1] small enough, the pair ⟨Re,Pe⟩
will be acyclic. Afriat’s efﬁciency index (AEI) is deﬁned as the supremum over
all the numbers e such that ⟨Re,Pe⟩is acyclic:
AEI = sup{e ∈[0,1] : ⟨Re,Pe⟩is acyclic.}
AEI is an intuitive measure of a violation of GARP. Consider, for example,
the violation in Figure 5.1. The violation represented in Figure 5.1(b) is more
severe than the one in 5.1(a). The difference is reﬂected in the AEI because
a large deﬂation of expenditure (a smaller e) is needed to account for the
violation.


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5.1 Measures of the severity of a violation of GARP
73
p′
x′
z
x
p
Fig. 5.2 Two violations of WARP: (x, p), (x′, p′) and (z, p), (x′, p′).
Afriat’s index is also called the critical cost efﬁciency index, or CCEI in the
literature.
5.1.2
Varian’s version of AEI
Varian modiﬁes Afriat’s index by allowing e to vary across the different price
vectors. Consider a vector θ = (ek)K
k=1 of numbers in [0,1], one for each
observation. Deﬁne the binary relation Rθ as xk Rθ xl if ekpk ·xk ≥pk ·xl. Deﬁne
the strict relation Pθ analogously. There is a set  of vectors θ such that the
corresponding ⟨Rθ,Pθ⟩satisﬁes GARP. Varian’s efﬁciency index (VEI) is the
closest distance of a vector θ to the unit vector (ek = 1 for all k), among those
θ for which ⟨Rθ,Pθ⟩is acyclic. Formally,
VEI = inf{∥1 −θ∥: ⟨Rθ,Pθ⟩is acyclic.}
5.1.3
The Money Pump Index
Both Afriat’s and Varian’s indices have a problem. Consider the example
in Figure 5.2. The AEI is the same for the two violations in the ﬁgure:
the data {(x,p),(x′,p′)} and the data {(z,p),(x′,p′)} have the same AEI. This
is counterintuitive. Arguably {(z,p),(x′,p′)} presents a worse violation than
{(x,p),(x′,p′)}. The source of the problem is that every violation of GARP
involves at least one cycle, and AEI (or VEI) try to break the cycle at its
“weakest link.” In the example in Figure 5.2, this requires deﬂating expenditure
to the point that x′ is on the budget set for x.
If one instead treats each “link” equally, then the problem goes away. Let
the sequence xk1,xk2,...,xkn deﬁne a violation of GARP, a cycle of ⟨⪰R,≻R⟩.


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Practical Issues in Revealed Preference Analysis
The money pump index (MPI) of this violation is deﬁned by
MPI{(xk1,pk1),...,(xkn,pkn)} =
n
l=1 pkl · (xkl −xkl+1)
n
l=1 pkl · xkl
(taking kn+1 = k1).
(5.1)
It is easy to see that, in Figure 5.2, {(z,p),(x′,p′)} has a higher level of MPI
than {(x,p),(x′,p′)}.
The MPI is named for the “money pump” that one can obtain from a
consumer who violates GARP. Arguably, what is wrong about the situation
in Figures 5.1 and 5.2 is that an outsider could take advantage of the consumer
that exhibits the behaviors in the ﬁgures. With data (x,p), (x′,p′) violating
WARP, for example, the outsider could trade the consumer x for x′ at prices
p, thereby getting an amount of money p · (x −x′) > 0, and then trade x′
back for x at prices p′, thereby getting an amount of money p′ · (x′ −x) > 0.
The total amount obtained by manipulating the consumer in such a way is
p·(x−x′)+p′ ·(x′ −x). The MPI simply expresses this magnitude as a fraction
of total expenditure.
Under certain assumptions, the MPI is the basis for a statistical test of
rational consumer behavior. If one assumes a source of statistical errors, one
can construct a critical region, so that rational consumer behavior is rejected
(beyond what can be explained as an error) if the MPI lies in this critical region.
5.2
POWER OF TESTING GARP
Failure to reject GARP can sometimes be caused by how budgets vary in
the data. In fact, certain datasets are problematic in the sense that it is
intrinsically hard to observe violations of GARP in them. Consider the example
in Figure 5.3(a). No matter what the consumer chooses at these budgets, there
will be no violation of GARP.
The ﬁgure is a stark example of a common phenomenon: when incomes vary
more than prices, it is harder to detect a violation of GARP. In cross-sectional
consumption data one has many observations of consumption chosen by
different households (individuals) at similar prices. The econometrician tries to
identify choices by different individuals as coming from the same (or similar)
preference relations, but since prices vary little, and individual incomes can
vary widely, GARP is easily satisﬁed. The situation is similar for aggregate
time-series data, meaning time-series data for a whole economy. Such data
often exhibit large year-on-year changes in aggregate expenditure (which
equals aggregate incomes), and comparatively small changes in relative prices.
One might argue that the issue of power does not apply to the realm of
revealed preference analysis. The theory only claims that agents’ behavior is
as if they were maximizing a utility function. So what if the budgets are not
set up in a way that might detect a violation of GARP? There is no sense in
which utility maximization could be a true model other than when behavior
satisﬁes GARP: this is in contrast to statistical models in which one afﬁrms


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5.2 Power of testing GARP
75
x1
x1
x2
e2
x2ˆ
x1ˆ
e1
x 2
Nested budget sets.
Choices adjusted by Engel curves violate WARP.
(a)
(b)
Fig. 5.3 Power of GARP.
the existence of an unknown true parameter. Having mentioned this point of
view, we shall nevertheless proceed with an exploration of the role of power in
revealed preference tests.
5.2.1
Bronars’ index
Recognizing the problems with the power of GARP, Stephen Bronars proposes
an index that tries to measure the power of using any particular collection
of budgets for testing GARP. His idea is: Given a collection of budget sets
B1,...,BK, simulate an “irrational” consumer choosing from these sets and see
how frequently the consumer violates GARP. Bronars assumes a consumer
who chooses a consumption bundle fully at random from the budget set.
The Bronars’ index of a collection of budget sets is the probability that
randomly chosen consumption bundles x1,...,xK violate GARP, where xk is
chosen uniformly at random on the boundary (budget line) of the budget Bk.
The index is implemented using Monte Carlo simulation. If there are L
goods, one method draws sk
1,...,sk
L at random and chooses xk ∈Bk such that the
budget share of good l is sk
l . This is done in such a way that the distribution over
the budget line is uniform. A second method targets the actual budget shares
in the data: If θk
l is the actual observed budget share of good l in observation k,
let sk
l = θk
l zk
l /
h θk
hzk
h, where the variables zk
1,...,zk
L are drawn uniformly and
independently from [0,1].
Bronars’ index is now routinely used in experimental studies of GARP.
When designing an experiment, researchers choose budgets so as to exhibit a
high Bronars’ index. An alternative measure of power, introduced by Andreoni,
Gillen and Miller, is based on having data from many agents choosing from
the same budget sets. One draws one choice at random, among the observed
choices, from each budget set. Then one tests if the resulting “bootstrapped”
choices pass GARP.


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Practical Issues in Revealed Preference Analysis
5.2.2
Engel curve correction
One solution to the problem of power is to use Engel curves to adjust the data.
Let d be a demand function. For given p, the Engel curve associated to p is the
function m →d(p,m).
Given is a consumption dataset D = {(xk,pk)}K
k=1. Suppose that we have
available an Engel curve ek : R+ →Rl
+ associated to price vector pk; that is,
ek(y) is the bundle demanded at prices pk when income is y (Engel curves ek
would be ﬁtted from the data). Note that y = pk · ek(y). Now we can correct
the problem by adjusting income so that the budget lines are no longer nested.
Figure 5.3(b) illustrates the idea. The choices and budgets from Figure 5.3(a)
exhibit no violation of GARP, but if we adjust the incomes for the two budget
lines (keeping the prices the same) to the two dashed lines, then the budget
sets are no longer nested and there is scope for a violation of GARP. Indeed,
at these hypothetical budgets, we can ﬁnd the choices consistent with Engel
curves e1 and e2. The choices induced by the Engel curves present a violation
of WARP.
The idea of using Engel curves was developed by Blundell, Browning, and
Crawford. They propose to consider a ﬁxed sequence of choices and prices
xk1,...,xkl,...xkL,
with a distinguished choice xkl. Transform the sequence as follows. For each
k, we let ek : R+ →Rn
+ be an Engel curve for prices pk. That is, ek satisﬁes the
following properties:
• For all k, ek(pk · xk) = xk.
• For all k and all y ≤y′, ek(y) ≤ek(y′).
• For all k and all y, pk · ek(y) = y.
So, these Engel curves need not come from preference maximization, but
importantly, the second criterion requires that the Engel curves are associated
with normal demand.
We proceed to describe the construction. For xkl−1, the term preceding xkl in
the sequence, we use the Engel curve for prices pkl−1 to ﬁnd a hypothetical
choice ˆxkl−1 on a budget line such that ˆxkl−1 is revealed preferred to xkl.
Speciﬁcally, let ˆmkl−1 = pkl−1 · xkl and ˆxkl−1 = ekl−1( ˆmkl−1). Given ˆxkl−1, we can
use the same idea to construct ˆxkl−2 from ˆxkl−1. That is, let ˆmkl−2 = pkl−2 · ˆxkl−1
and ˆxkl−2 = ekl−2( ˆmkl−2). Continuing in this fashion, we construct ˆxk1,..., ˆxkl−1
with the property that
ˆxk1 ⪰R ˆxk2 ⪰R ··· ⪰R ˆxkl−1 ⪰R xkl.
The construction of ˆxkl+1,..., ˆxkL is similar. Let ˆmkl+1 be the solution to the
equation
pkl · xkl = pkl · ekl+1( ˆmkl+1)
(such a solution exists and is unique under our assumptions on Engel curves;
namely, continuity is implied by normality). Let ˆxkl+1 = ekl+1( ˆmkl+1) and note


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5.3 An overview of empirical studies
77
that xkl ⪰R ˆxkl+1 by construction. Given, ˆxkl+1 and ˆmkl+1, let ˆmkl+2 be the
solution to the equation ˆmkl+1 = pkl+1 ·ekl+2( ˆmkl+2); and set ˆxkl+2 = ekl+2( ˆmkl+2).
Continuing in this fashion we obtain a sequence xk+l ⪰R ˆxkl+1 ⪰R ··· ⪰R ˆxkL.
The usefulness of the construction follows from the following proposition.
The construction never eliminates a violation of GARP. Thus every sequence
exhibiting a violation of GARP remains a violation after being adjusted by
Engel curves. In other words, transforming a sequence in this way can only
increase the chances of observing a violation of GARP.
Proposition 5.1
If xk1,...,xkL is a sequence with xk1 ⪰R ··· ⪰R xkL ≻R xk1 and
ˆxk1,..., ˆxkL is the Engel-curve adjusted sequence when xkL is the distinguished
element, then ˆxk1 ⪰R ··· ⪰R ˆxkL ≻R ˆxk1.
Proof. Let mkl = pkl · xkl. It is easy to see by induction that mkl ≥ˆmkl for
l = 1,...,L. By construction,
ˆmkL−1 = pkL−1 · xkL ≤pkL−1 · xkL−1 = mkL−1,
where the inequality follows from xkL−1 ⪰R xkL. Then, the normality of demand
implies that
xkL−1 = ekL−1(mkL−1) ≥ekL−1( ˆmkL−1) = ˆxkL−1,
so we obtain that
mkL−2 = pkL−2 · xkL−2 ≥pkL−2 · xkL−1 ≥pkL−2 · ˆxkL−1 = ˆmkL−2.
By continuing in this fashion, we show that mkl ≥ˆmkl for l = 1,...,L. Then,
xkL ≻R xk1 implies that
pkL · xkL > pkL · xk1 ≥pkL · ˆxk1,
as xk1 = ek1(mk1) ≥ek1( ˆmk1) = ˆxk1. Hence, xkL ≻R ˆxk1; so the fact that ˆxk1 ⪰R
··· ⪰R xkL establishes the result, using the fact that ˆxkL = xkL.
5.3
AN OVERVIEW OF EMPIRICAL STUDIES
We brieﬂy describe some of the best-known empirical studies of GARP,
classiﬁed by the kind of datasets that they use.
5.3.1
Panel data
Koo (1963) may be the ﬁrst empirical study of revealed preference in
consumption. He uses a panel dataset of consumption choices, and ﬁnds that
relatively few households (panelists) in his sample satisfy GARP fully. It
is interesting that such an early study uses panel data, a kind of data that
corresponds closely to the theory, and presents fewer complications than some
of the later uses of cross-sectional and time-series data. Koo introduces a
measure of the degree of satisfaction of GARP, based on the idea that some
subset of the observed choices could be consistent with GARP. His measure is
ﬂawed, however, as pointed out by Dobell (1965).


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Practical Issues in Revealed Preference Analysis
5.3.2
Cross-sectional data
Famulari (1995) tests for violations of GARP using a cross-sectional dataset
obtained from a consumer expenditure survey. She focuses on budget sets that
involve similar levels of expenditure, as a way of addressing the standard
problems of the power of GARP. She assumes that bundles with similar
costs, and where goods have similar expenditure shares, may be difﬁcult
to compare for consumers. Hence she focuses on comparing observations
where the consumption bundles have similar costs, but dissimilar expenditure
shares. Famulari also uses Afriat’s efﬁciency index to account for possible
measurement errors that could lead her to ﬁnd spurious violations of GARP.
Based on their economic and demographic characteristics, Famulari groups
her sample of some four thousand “consumer units” into 43 different types
of households (this methodology is called “matching” in statistics and
econometrics). She deﬁnes a rate of violation of GARP as follows: a pair of
observations xk and xl constitute a violation if xk stands in the transitive closure
of the revealed preference relation to xl (xk is indirectly revealed preferred to
xl), while xl is strictly revealed preferred to xk. Famulari’s rate of violation of
GARP is the number of pairs that induce a violation divided by the total number
of pairs in the data. Her results are that violation rates are quite small, and they
are attributable to measurement errors (using Afriat’s efﬁciency index).
Dowrick and Quiggin (1994) use a revealed preference approach to
cross-country welfare comparisons. As a ﬁrst step, they test for GARP
using cross-country aggregate consumption data. Speciﬁcally, they obtain one
observation (pk, xk) for each country in a sample of 60 countries (using data
from 1980). For country k, xk is a per-capita aggregate consumption bundle,
and pk is a country-speciﬁc price index. They treat the cross-country dataset as
a consumption dataset, the idea being that a common representative consumer
could exist across these countries. They use revealed preference comparisons
as a comparison in the standard of living. They ﬁnd almost no violations of
GARP (and that GARP is very close to WARP). The results are consistent with
the existence of seven categories of countries, such that the bundle consumed
by a high-category country is always revealed preferred to a lower-category
country; and such that most countries in the same category are not comparable
according to revealed preference.
The study of Blundell, Browning, and Crawford (2003) is also worth
mentioning here. They use cross-sectional data and introduce the technique
described in Section 5.2.2.
5.3.3
Time-series data
Varian (1982) studies yearly aggregate data on consumption in the period
1947–1978. A test for GARP in this case can be interpreted as a test for the
existence of an aggregate representative consumer. Varian ﬁnds no violation
of GARP. The test, however, has very low power for yearly aggregate data. In
the post-war period, the US aggregate income was increasing markedly year


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5.3 An overview of empirical studies
79
after year, while relative prices were relatively stable. This fact is emphasized
by Varian in his paper: an observation that prompted later researchers to be
concerned with the power of testing for GARP. Landsburg (1981) conducts a
similar study on UK consumption data, and also notes that increasing income is
problematic, attributing the observation to Gary Becker. See also Chalfant and
Alston (1988). Swofford and Whitney (1986, 1987) are early related studies on
the demand for ﬁnancial assets.
Browning (1989) is another well-known study using time-series data.
Instead of testing for GARP, Browning is interested in the maximization of
a utility function of a particular form: an additively time-separable utility
function, with no discounting and a per-period utility index that is constant
over time. The test for rationalizability by this kind of utility turns out to be
cyclic monotonicity: see Proposition 4.5.
Using aggregate time-series data from the UK and the US, Browning ﬁnds
that GARP can never be rejected, which is in line with Varian’s ﬁndings and
the comments on the power of GARP we made above. Cyclic monotonicity is
rejected, but there are fairly long periods of time for which cyclic monotonicity
holds. Given that the data satisfy GARP, we know that there are multipliers for
the prices for which the data with “adjusted prices” pass cyclic monotonicity
(see the discussion after Afriat’s axiom on Page 41). Browning proceeds to
calculate such multipliers, and uses them to inquire about the reasons behind
the falsiﬁcation of the additively separable model.
5.3.4
Experimental data
Battalio, Kagel, Winkler, Fischer, Basmann, and Krasner (1973) ran a ﬁeld
experiment in the female ward of a psychiatric hospital. Patients (who were
diagnosed psychotic) could exchange tokens for different consumption goods.
By varying the value of the tokens, the authors induce a variety of different
budget sets, and record the patients’ purchases. Battalio et al. found that many
patients satisﬁed GARP. In particular, if one allows for small measurement
errors in quantities, then almost all the psychiatric patients’ behavior is
consistent with GARP. The authors also look at the dynamic behavior of the
few patients who violate GARP, and argue that the patients reacted to price
changes in the correct direction, but that they failed to fully adjust to the new
prices. It is possible that if patients had had sufﬁcient time to get used to the
price change, then no violations would have been observed.
Sippel (1997) presented individuals with a menu of choices, determined
by standard budget sets. Subjects were according to the random decision
selection mechanism. Individuals were paid in consumption goods, and were
required to consume the goods at the experiment. Overall, subjects were found
to be inconsistent with classical demand theory. By setting an AEI of 95%,
most subjects were found to be consistent with the predictions of preference
maximization, but the power of the test also decreased substantially. The study
then chose perturbed demand close to actual demand, ﬁnding that the number


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Practical Issues in Revealed Preference Analysis
of violations changes little after perturbation, suggesting that inconsistency
cannot be due to error alone.
Andreoni and Miller (2002) are motivated by the standard experimental
ﬁnding that agents are too generous when asked to split a given amount of
money. In “dictator game” experiments, subjects are asked to split x dollars
between themselves and another subject, who is completely anonymous.
Andreoni and Miller run such dictator-game experiments, and test if the
observed altruistic behavior is utility-maximizing, by a utility that depends
on the monetary rewards received by both agents. They ﬁnd that 98% of
subjects make choices that are consistent with utility maximization. Andreoni
and Miller go further than most revealed-preference exercises in estimating a
parametric function of a utility function accounting for subject’s choices (about
half the subjects can be classiﬁed as using a linear, CES (constant elasticity of
substitution), or Leontief utility).
Harbaugh, Krause, and Berry (2001) perform an experiment using seven-
and eleven-year-old children, as well as college students. They test each of
these populations for compliance with GARP, after making them choose from
several budget sets. They ﬁnd a relatively small number of violations of GARP.
Seven-year-olds violate GARP much more than older children (but still exhibit
a relatively small number of violations). Eleven-year-old children behave close
to rationally, and there are few differences between college-age adults and
eleven-year-old children. It is interesting that the authors ﬁnd that violations
of GARP are largely uncorrelated with the results of a test that measures
mathematical ability in children.
Choi, Kariv, M¨uller, and Silverman (2014) run an experiment on choices
from budgets, and correlate the degree of consistency with GARP (as measured
by the Afriat efﬁciency index) with the demographic and socioeconomic
characteristics of the subjects. They work with a large sample of over
2,000 Dutch households. In their experiments, subjects are presented with 25
different budgets, each one chosen randomly. They ﬁnd an average AEI of .88,
and that almost half the population has an AEI above .95; so the population
is quite close to satisfying GARP. The meat of their study lies in correlating
AEI with the socioeconomic characteristics of the subjects. They ﬁnd that men
as well as highly educated and rich individuals score higher AEI than others.
They also ﬁnd that high AEI scores predict high levels of wealth: a standard
deviation increase in AEI predicts 15–19% higher household wealth.
5.4
CHAPTER REFERENCES
A basic reference to matching in statistics is Rubin (1973). Famulari (1995) is
an example of the use of matching in revealed preference analysis. Famulari
(2006) uses a similar methodology to study labor supply from the viewpoint
of revealed preference. She uses nonlinear budget sets to capture the effects
of taxes on labor supply. A different methodological approach is taken by


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5.4 Chapter references
81
Hoderlein and Stoye (2014), who focus on bounding the joint distribution of
the population of consumer characteristics, and by Kitamura and Stoye (2013),
who translate the problem into the framework analyzed in Chapter 7. Hoderlein
(2011) should also be mentioned: he focuses on testing the implications of
rationality for the Slutsky matrix of demand.
One issue we have not dealt with here is that consumption data usually
reﬂect the decisions of households, not of single agents. We shall consider
models of collective decision making in Chapter 9, 10, and 11, but the focus
on those chapters will not be on consumption data. In the consumption setting,
the issue is addressed by Cherchye, De Rock, and Vermeulen (2007, 2009),
based on the model of Chiappori (1988). See also Browning and Chiappori
(1998).
Afriat (1967) proposed Afriat’s efﬁciency index. Varian (1990) extended the
index as explained above, and gives it an interpretation in terms of consistency
with rationality and errors in measurement (see also Varian, 1985 and Epstein
and Yatchew, 1985). Afriat’s index is also called the critical cost efﬁciency
index, or CCEI in the literature. The critical cost efﬁciency index (CCEI)
terminology was introduced by Varian (1991), based on Afriat (1972) (a paper
on optimality in cost and production, see Chapter 6). It is used heavily in
applied work: see Choi, Fisman, Gale, and Kariv (2007) and Choi, Kariv,
M¨uller, and Silverman (2014).
The Money Pump Index is proposed by Echenique, Lee, and Shum (2011),
who show that the MPI has an interpretation as a statistical test. Smeulders,
Cherchye, De Rock, and Spieksma (2013) propose a variant of MPI that is
computationally easy. An alternative measure of a violation of GARP results
from computing the smallest set of observations one would need to delete
from the data in order for it to satisfy GARP: see Houtman and Maks (1985),
and more recently Dean and Martin (2013). An early related work in abstract
choice theory is Basu (1984).
We have omitted a discussion of Varian’s procedure for estimating demand
responses by using revealed preference inequalities: see Varian (1982),
Knoblauch (1992) and Blundell, Browning, and Crawford (2008). The
discussion in Section 5.2.2 on the Engel-curve corrections approach to the
problem of power is related to this procedure, and is taken from Blundell,
Browning, and Crawford (2003).
Bronars’ index was proposed in Bronars (1987). The model of random
choice as a model of irrational behavior was proposed by Becker (1962). It
is curious, however, that the point of Becker’s paper is that random behavior is
close to being rational. This point calls into question the idea of using random
choices as a measure of the power of GARP. The “bootstrap” approach to
power described in the text is due to Andreoni and Miller (2002).
There are alternative power indices formulated by Famulari (1995) and by
Andreoni, Gillen, and Harbaugh (2013). Andreoni, Gillen, and Harbaugh’s
test, in particular, rests on a clever “reversion” of AEI to measure how far an
observation that satisﬁes GARP is from not satisfying it. Beatty and Crawford


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Practical Issues in Revealed Preference Analysis
(2011) propose a measure of power that is related to Bronars’, but based on the
ideas in Selten (1991).
In addition to the evaluation of power based on a collection of budgets, it is
sensible to use the data on actual choices to get an idea for the power of GARP.
Andreoni, Gillen, and Harbaugh (2013) propose econometric methodologies
for carrying out such evaluations of power.
The selection of empirical papers in Section 5.3 is obviously arbitrary. It
is worth mentioning some papers that study the correlation between subjects’
pass rates for GARP, and the presence of factors that may impede subjects’
cognitive abilities. In particular, the paper of Burghart, Glimcher, and Lazzaro
(2013) ﬁnds that subjects in experiments who have consumed substantial
amounts of alcohol still pass GARP. The paper by Castillo, Dickinson, and
Petrie (2014) in contrast compares subjects who are sleepy with those who
are fully alert: again pass rates for GARP are the same across treated and
non-treated subjects. It is important to note that both papers do ﬁnd an effect
of the treatment on agents’ risk-taking behavior.
