Emerging Trends in The Social and Behavioral Sciences

Models of Revealed Preference

Item

Title

Models of Revealed Preference

Author

Adams, Abi

Crawford, Ian

Research Area

Cognition and Emotions

Topic

Decision Making

Abstract

Revealed preference theory is concerned with what we can learn about the process by which economic agents make decisions using simply the features of the world that we observe: choices. Different economic models place different restrictions on these choices. The revealed preference literature derives these restrictions and then puts them to use. Research effort has recently been extended far beyond the axiomatic characterization of neoclassical models of choice to consider data‐consistency and preference‐recoverability for a wide class of models. This essay places these recent developments in context, giving a brief introduction to the revealed preference approach before elaborating on recent research that has dramatically extended the domain and ambition of the discipline. We address the widening of the scope of revealed preference theory to new classes of models, which have introduced novel techniques, and challenges, to the discipline, before charting emerging trends within the areas of identification and power, which have arisen as revealed preference has emerged as an empirical method.

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extracted text

Models of Revealed Preference
ABI ADAMS and IAN CRAWFORD

Abstract
Revealed preference theory is concerned with what we can learn about the process
by which economic agents make decisions using simply the features of the world that
we observe: choices. Different economic models place different restrictions on these
choices. The revealed preference literature derives these restrictions and then puts
them to use. Research effort has recently been extended far beyond the axiomatic
characterization of neoclassical models of choice to consider data-consistency and
preference-recoverability for a wide class of models. This essay places these recent
developments in context, giving a brief introduction to the revealed preference
approach before elaborating on recent research that has dramatically extended the
domain and ambition of the discipline. We address the widening of the scope of
revealed preference theory to new classes of models, which have introduced novel
techniques, and challenges, to the discipline, before charting emerging trends within
the areas of identification and power, which have arisen as revealed preference has
emerged as an empirical method.

MODELS OF REVEALED PREFERENCE
One could object to the presence of an article entitled “Models of Revealed
Preference” in a project entitled “Emerging Trends.” The approach has a long
and distinguished history, dating back to Samuelson and an initial citation
date of 1938. Varian describes it as “one of the most influential ideas in
economics” (2006, p. 99). The essential idea is very simple. Economic models
of individual choice are based on the premise that people are endowed
with coherent preferences. Yet, preferences are not directly observable; all
we can observe are the choices people make. Revealed preference theory is
concerned with what we can learn about the process by which economic
agents make decisions using the features of the world that we observe,
namely, choices. Different economic models place different restrictions on
how individuals’ make their choices. The revealed preference literature
derives these restrictions and puts them to use.
The constructive application of revealed preference techniques is a novel
development within the literature and has prompted the emergence of the
Emerging Trends in the Social and Behavioral Sciences. Edited by Robert Scott and Stephen Kosslyn.
© 2015 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

discipline as an empirical method. Research effort has now extended far
beyond the axiomatic characterization of theoretical models of choice to
address questions of data-consistency and preference-recoverability for a
wide class of models. A revealed preference approach offers the prospect
of eliminating the gap between the economic theorist and the applied
economist who deals with data, providing a purer way to assess the empirical performance of a theory and complementing the growing sentiment that
the identification of a model’s parameters is not an “all-or-nothing” affair.
This essay places these developments in context, giving a brief introduction
to the revealed preference approach before elaborating on recent research
that has dramatically extended the domain and ambition of the discipline.
We first address the widening of the scope of revealed preference theory. The
approach has now been extended to new classes of models, which has introduced novel techniques, and challenges, to the discipline. We then consider
developments and trends within the areas of identification and power that
have arisen as revealed preference has emerged as an empirical method.
THE ESSENTIAL IDEA
General models, which offer a simplified version of an interesting aspect of
reality, have a central place in the social sciences. We live in a complicated
world. Thus, there can be great power in the elimination of irrelevant detail.
An appeal to Occam’s razor is generally apt in our setting: all things equal, a
simpler explanation should be preferred to a more complex one.
However, there is a trade-off between accuracy and parsimony. We wish
to adopt the simplest model that remains capable of illuminating the situation under examination. This desire leads the social scientist to ask: Can a
model accurately account for the aspect of reality under consideration? As
Sen remarks: “The primary concern here is not with the relation of postulated models to the real economic world but with the accuracy of answers to
well-defined questions posed with preselected assumptions which severely
constrain the nature of the models” (1977, p. 322). Addressing this concern
requires an empirical test of the theory: is the model consistent, or not, with
data we have on the situation observed?
Yet, many models deal with motivations and mechanisms that we cannot
acquire data on. This is certainly true of rational consumer choice theory,
the birthplace of the revealed preference approach. The utility maximization
model, at the heart of this theory, simply states that individuals choose the
course of action that makes them “best off.” The notion of “best off” is formalized via the specification of a “utility function,” a numerical representation
of an individual’s preferences over different goods and actions. Formally, the

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utility maximization model posits that individuals decide how much of each
good to buy in order to maximise their utility function.
To fix ideas, imagine a world in which there are only two goods: whisky
and bread. An individual’s utility function assigns a number to any bundle
of whisky and bread that we can imagine, allowing us to rank these bundles
according to an individual’s relative preference. For example, let the amount
of whisky that an individual consumes be given by W, and the amount of
bread that an individual consumes be given by B. The utility function, which
describes their total amount of happiness, takes the form:
Utility(W, B) = f (W, B)
For example,
Utility(W, B) = W + B
In this example, an individual is indifferent over the split of a bundle
between whisky and widgets; she just wants more of both goods.
Although the utility maximization model is a simple way to model choice
behavior, it is hard to test. We do not observe an individual’s utility function
and, therefore, the choice that actually makes them best off. Given this, how
are we to determine if rational choice theory can account for the behavior we
observe in the marketplace?
Standard approaches start by placing structure on the aspects of an
economic model that go unspecified by theory and are unobserved in reality,
i.e. the precise structure of the utility function. Therefore, following this
approach, we would assume that preferences can be described by some
specific utility function (as in the example above) and then determine if the
choices we observe in reality best advance these preferences.
The problem with this approach is that it is not purely a test of economic
theory. It is a test of the theory plus a test of the hypothesis that the structure
of the utility function takes the assumed form. To highlight the problem with
this approach, imagine that we found that observed choices differed from
those dictated by the hypothesized utility function. We would not be justified
in discarding rational choice theory on this basis. The observed inconsistency
could arise from the fact that preferences in reality do not coincide with the
utility function we have assumed, rather than arising from any irrationality. Continuing to try different specifications for the utility function is not
guaranteed to help either. There are an infinite number of ways to formulate preferences and thus, following this approach, we are unable to reject
the model in a finite number of steps (see Popper, 1959 and Miller, 1996 for a
deeper discussion).
Revealed preference theory offers an alternative method to assess how well
a model performs when confronted with reality. The approach derives the

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

restrictions on the observable features of the world that are logically implied
by the model alone. No appeal to any specific utility function is necessarily involved. If reality respects these conditions, one can conclude that it is
within a model’s ability to account for what we see. However, if the revealed
preference restrictions are violated by observations on the world, one is justified in concluding that the model does not provide an adequate description
of reality. The approach thus provides us with “effective” tests of theory.
FOUNDATIONS OF REVEALED PREFERENCE
Samuelson (1938, 1948) was the first to consider the restrictions that rationality alone imposes on observed choice behavior, concluding that: “if an individual selects batch one over batch two, he does not at the same time select
two over one” (1938, p. 7). If at some point in time, an individual selects batch
one over two, then they reveal their preference for one over two. Therefore,
to be consistent with pure rational choice theory, that same individual cannot reveal a preference for two over one at a later date. Therefore, regardless
of what form an individual’s utility function does take, the model imposes
a requirement of consistency on choice behavior and can be rejected if this
requirement is not upheld in reality.
The foundational revealed preference literature largely concerned itself
with the utility maximization model and the refinement of Samuelson’s
conditions. Samuelson’s Weak Axiom of Revealed Preference, introduced
intuitively above, was extended by Houthakker (1950) to impose consistency
on choices that reveal bundles to be indirectly revealed preferred to one another.
For example, if a rational individual chooses batch X over Y, but Y over Z,
then they must choose X over Z when presented with this choice. However,
few attempts were made to implement the derived tests despite their being
motivated by a quest for empirical falsifiability. In the words of Mas-Colell,
revealed preference represented as “foundational and purely theoretical a
subject as one can find” (1982, p. 72).
This changed with the translation (Diewert, 1973) of a ground breaking,
but “virtually impenetrable” (Pollak, 1990, p. 142) paper by Afriat (1967),
which opened the door to greater emphasis on implementation and identification within the literature. Although the question of consistency between
a model and reality is an important one, there is often a desire to go beyond
this. Typically, we seek knowledge of the structure of a model’s unobserved
components and want to predict behavior in new situations. For example,
imagine we find that an individual’s choices satisfy the revealed preference
conditions implied by rational choice theory. It is then interesting to ask what
kind of preferences this individual holds. Afriat (1967) developed a method to
reconstruct preferences rationalizing observed choices given finite data. He

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showed how simple linear programming techniques could be used to both
implement revealed preference tests of the theory and then recover individual preferences.
Once the value of Afriat’s approach was recognized and made intelligible
by Diewert (1973),1 and further extended and refined by Varian (1982, 1983)
the stage was set for the literature to move beyond revealed preference to
“preferences revealed” and to the prediction of behavior.
Varian’s (1982) contribution was central to this movement, providing a
clear exposition of how a revealed preference methodology offers a viable
alternative to statistical methods for applied demand analysis. However,
recoverability and prediction using a revealed preference methodology
rarely leads to a single answer. Unlike methods that place prior structure on
the unobserved components of a model, model features are “set identified.”
For example, given a change in prices, revealed preference methods will not
yield a unique prediction for the new choice, but will identify ranges within
which the new choices must lie to remain consistent with rationality. As we
will explore in further in this review, achieving tighter identification while
maintaining a revealed preference approach is a key theme in modern work.
In summary, revealed preference theory has the potential to develop
into a powerful tool and offers a way of reducing the distance between
econometrician and theorist. The approach allows us to assess whether the
simplifications a model makes are good ones by addressing the consistency
of theory and reality. However, it can facilitate much more than this, providing a methodology to recover unobservable features of models and predict
action in new environments.
MODERN THEMES AND EMERGING TRENDS
This section considers modern research and where we believe the future of
the discipline lies. We structure our discussion thematically as the literature
has expanded along a number of dimensions. We begin by considering the
extension of the methodology to more complex models before addressing
developments that have occurred as revealed preference has emerged as an
empirical method.
SCOPE
The scope of the literature has extended far beyond the canonical utility maximization model. Chambers, Echenique, and Shmaya (2012) prove that there
1. This essay was born out of a referee report that he wrote concerning a paper that Afriat sent to the
Journal of Political Economy. Diewert recognized the immense value of the paper but recommended that it
be revised to make it more understandable. Afriat did not resubmit the paper to that journal.

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exist revealed preference necessary and sufficient conditions, defined only in
terms of observables, for any model that can be expressed as a series of universal statements. Thus, in principle, there awaits a huge class of models to be
given a revealed preference treatment. However, deriving the conditions for
many models requires the use of mathematical techniques that are not part
of the mainstream economist’s toolkit, potentially slowing the expansion of
the methodology.
A large portion of early research energy was devoted to characterizing
the restrictions implied by different preference structures, reflected in the
general form of the utility function. The structure of utility functions is
enormously important for both theoretical and applied attempts to model
consumer behavior, with implications for topics as apparently diverse as
choice under uncertainty and intertemporal choice2 and the measurement
of inflation.3 Varian (1983) introduced revealed preference characterizations
for some key preference structures, which made it possible for researchers
to widen the span of revealed preference methods to a number of important
areas. For example, building on Varian’s work, Browning (1989) provided a
revealed preference analysis of an intertemporal model of consumer choice,
whereas Green and Srivastava (1986) considered models of choice under
uncertainty.
The derivation of the testable restrictions implied by different hypotheses
regarding the “separability” of goods in the utility function has led to a
number of interesting developments. Separability assumptions partition
the choice space into subsets that the individual evaluates independently.
Separability is a matter of degree. At one end of the scale is the characteristics
model of Gorman (1956) and Lancaster (1966), under which preferences
are defined with respect to characteristics rather than goods, i.e. we have
a desire for energy (a characteristic of a good) and so buy different food
items to satisfy this. Blow, Browning, and Crawford (2008) were the first
to give the model a revealed preference characterization and, in so doing,
extended the scope and ambition of the discipline. The notion of latent
separability was introduced in Gorman (1968) and thoroughly developed
in Blundell and Robin (2000). It too is an important functional structure as
it underlies Samuelson’s (1956) account of the representative consumer and
was characterized by Cherchye, Crawford, De Rock, and Vermeulen (2012)
in their revealed preference exploration of aggregation.
Recent years have seen an extension of the revealed preference methodology to the analysis of group behavior. The key issue here is that we typically
only observe the aggregate outcome of a group’s decision-making process.
2. Expected utility theory and the discounted utility model are both based on additive separability.
3. The existence and uniqueness of subcost of living indices depends on weak separability and homotheticity.

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Given that the aggregate behavior of a group of rational consumers is not
necessarily rational (see Kirman, 1992 for example), the idea that it might be
possible to test collective models using revealed preference techniques seems
incredible. However, Cherchye, De Rock, and Vermeulen (2007) have shown
that it is possible and have derived a number of subtle results in this area. The
extension of the revealed preference methodology to collective models has
opened the door to an extensive research programme that aims at opening
the “black box” of the family. A wide variety of models, which make varying assumptions about the degree of cooperation and commitment between
family members, have now been given the revealed preference treatment
(Cherchye et al., 2007, 2012; Cherchye, De Rock, & Vermeulen, 2009; Adams,
Cherchye, De Rock, & Verriest, 2014).
Testing the conditions associated with these models has introduced new
techniques to the discipline. Typically, the revealed preference restrictions
for this class of models are nonlinear and, therefore, difficult to test. A key
contribution of Cherchye et al. (2007) was to show how to reformulate the
revealed preference restrictions as a mixed integer linear programme (MILP)
that can be solved using standard, robust techniques. Given a MILP formulation of the problem, one searches directly for each family member’s revealed
preference relation. This is a powerful recharacterization of the problem and
the introduction of these new programming techniques will make it easier to
operationalize tests for other complex models in the future.
A further research agenda, which has significantly widened the scope of
revealed preference theory, is that considering the stochastic demand function. All the work cited above assumes that each consumer has a single set
of preferences. However, simple introspection suggests that this is typically
not the case. Individual preferences are often thought to embody a random
component. Alternatively, an absence of information on the context of choice
can cause observed choices to appear somewhat random. Bandyopadhyay,
Dasgupta, and Pattanaik (1999) and McFadden (2005) derive revealed preference characterizations of models of stochastic demand. These results are
important and their value has not yet been fully utilised by the empirically
focused strain to the literature. Given the concern with distributions of preferences, this work provides a coherent way to extend revealed preference
results to a population setting and facilitate greater use of cross-section data
sets. Many of these extended revealed stochastic preference problems have
not yet been studied but this area is likely to prove something of a treasure
trove to economic researchers.
The gradual widening of the scope of revealed preference analysis has
not yet taken in behavioral economics to any great extent. This promises to
be an important area of research in the future. Behavioral economics has
recently become a prominent subfield and moves somewhat away from the

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

neoclassical tradition of treating people as always-rational decision makers.
The approach focuses on models that have more plausible sociological and
psychological foundations to behavior, allowing for people to be influenced
by others, make mistakes, and regret their choices. Behavioral economics
promises much in terms of its potential to help us understand behavior that
would otherwise prove resistant to straightforward explanation by standard
rational choice models.
However, whether the important classes of models in behavioral economics
will succumb to a revealed preference analysis is currently an open question. In many respects, the models do not seem likely to provide firm enough
predictions, nor the necessary stability in behavior, that revealed preference
methods typically make use of. If it can be shown that these models are
not falsifiable using revealed preference arguments then it may undermine
them—it being all very well to claim that such models can fit the data but
quite another thing if it turns out that it is impossible for them to do otherwise. Alternatively, if they do turn out to be rejectable using revealed preference methods, it is likely that this will speed their acceptance amongst
traditionalist nonconverts.
IDENTIFICATION
As should now be clear, a wide variety of economic models have been
given a revealed preference treatment, and their consistency with real-life
decision-making behavior ascertained. Yet, revealed preference techniques
do not just provide tools to address whether a model can rationalize choices.
They also provide a methodology to recover the aspects of a model that go
unobserved in reality and allow us to predict behavior in new circumstances.
Recoverability has been addressed in a variety of contexts. Varian (1982)
yet again provides the key foundational reference for the utility maximization model, providing a detailed exposition of how to recover a consumer’s
utility function and predict their theory-consistent demands at new budget
regimes. Revealed preference techniques have also been used to throw light
on the power relations presiding within the household and the differences
in preferences that exist within the family unit. Cherchye et al. (2008) identify bounds on the “sharing rule,” a parameter that summarizes the split of
resources within household, whereas Adams et al. (2014) identify minimum
differences in the levels of patience of different family members.
However, the methodology typically identifies ranges within which an
answer must lie if behavior is to be characterized by a particular model.
For example, a revealed preference analysis is only able to predict that, in
reaction to an increase in the price of widgets by 10%, an individual will
alter their behavior to demand somewhere in the range of, say, 5 and 10

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widgets. The potential response is not uniquely identified; we are not able
to give a unique prediction of, for example, 8 widgets.
A key problem for researchers is that survey data typically imply uninformative bounds on predicted responses and other model features. The informational content of consumption data is removed by income variation as
this negates any notion of “selecting over” from the problem at hand. Recent
research has explored ways of integrating more standard estimation techniques to the revealed preference methodology, in an attempt to get the best
of both worlds: the functional form agnosticism of revealed preference and
the tighter identification of traditional, statistical techniques.
Blundell, Browning, and Crawford (2003, 2008) exploit information on how
choices vary with income to tighten the revealed preference bounds on predicted demand responses. Using their method, one first uses more traditional methods to estimate Engel curves, functions describing how choices
change with income while keeping everything else constant, to, in effect,
remove the impact of income variation from choice behavior. One then performs the usual revealed preference analysis on these “corrected” demands.
Their method is further extended by Blundell, Browning, Cherchye, Crawford, De Rock, and Vermeulen (2012) to exploit the information that can be
gleaned from the transitivity of preferences. Cherchye, De Rock, Lewbel, and
Vermeulen (2012) also combine revealed preference and statistical methods
in order to identify household sharing rules from cross-section data. The
results presented in these works signal the enormous benefits that can flow
from blending statistical econometric methods and revealed preference techniques. The bounds on demand responses calculated by Blundell et al. (2003,
2008, 2012) are significantly tighter, and therefore, much more useful, than
those generated by a pure revealed preference approach and, without prior
estimation of the household demand function, Cherchye et al. (2012) would
not even have been able to proceed with a revealed preference analysis of
the family choice problem. We consider the emergence of the trend toward
greater integration of these separate branches of econometrics an important
one and expect this to be an active research topic in the coming years.
We also expect a greater engagement with literature from information theory, statistical physics, and engineering. A range of techniques exist in these
fields in order to select a unique answer from the feasible solution set. Infusing revealed preference with these techniques could help further narrow the
distance between it and mainstream econometrics by facilitating the point
identification of model features. Exactly what form the knowledge crossovers
between these disciplines will take is unclear but further research in this area
offers substantial promise.

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POWER
The emergence of revealed preference as an empirical method has prompted
discussion over how one should interpret the results gained from the application of the methodology to real-world data. Revealed preference conditions provide very “crisp” tests of optimizing models: if the data satisfy the
revealed preference restrictions then the data are consistent with the model
and when they do not satisfy the restrictions they are not. However, it can be a
little difficult to know what to make of the results of these sorts of tests. One
particularly important question is whether the optimizing model in question is not, in fact, the “true” behavioral model: can we be sure that revealed
preference tests are sensitive enough to detect this? In statistical hypothesis testing, this question concerns the “power” of the test: the probability of
rejecting the null hypothesis when the null is not true.
Revealed preference tests are typically nonstochastic and the statistical
notion of power is not applicable. Yet, there remains a need to consider the
sensitivity of the testing procedure to nonmodel-consistent data-generating
processes. Just as is the case with statistical power calculations, the results
will depend on the alternative hypothesis considered. The difficulty is
that there are many alternatives to rational choice models and no obvious
benchmark. The question of the right alternative model was considered by
Becker (1962), and in particular, he focused on the alternative hypothesis
that consumer make random choices subject to their budget constraints. This
idea was applied to revealed preference tests by Bronars (1987) who asked
how likely it would be for a revealed preference test to detect a negative
result when the true data-generating hypothesis was random behavior
with a uniform probability distribution. More recent contributions (notably
Andreoni, Gillen, & Harbaugh, 2012) consider more data-driven alternatives
to random choice with uniform probability—they use the empirical distribution of choices to allow for a more realistic alternative hypothesis. Work
in this area is ongoing, but the leading approach remains Bronars’ (1987)
method. It seems that the field has yet to settle on an agreed procedure for
evaluating the sensitivity of revealed preference methods in this manner
due mainly to the difficulty of finding a compelling, nonrational benchmark
for the alternative data-generating process.
Given this, researchers have developed different approaches that do not
require an alternative model of behavior. Andreoni et al. (2012), for example,
also do this. One of the most interesting and intuitively appealing among
these is what they term the Afriat Power Index. Given a dataset in which
no revealed preference violations are detected, this measures how much the
consumer’s budget would have to be adjusted in order to induce a violation.

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If the required adjustment is small, then the test is considered to be sensitive,
if it is high then it is not.
A third type of approach, recently applied by Beatty and Crawford (2011),
derives from experimental game theory and is due to Selten (1991). Under
a revealed preference characterization, economic models typically delineate
well-defined sets of choices that are consistent with the model of interest. It is
then useful to consider the size of the theory-consistent set of behaviors relative to the size of the set of all possible behaviors. The essential idea is that
if the set of observations explainable by the model is large relative to the set
of possible behaviors, then simply noting that many of the observed choices
are theory-consistent is not a very demanding requirement; they could hardly
have done otherwise. Thus, “fit,” the proportion of the sample which passes
the relevant test, is not a sufficient basis for ranking the empirical performance of alternative theories. If this were the case, then a meaningless theory
that ruled nothing out could not be outperformed. A better approach is to
consider the trade-off between the pass rate and a measure of how sensitive
the test is. We can measure the size of the theory-consistent set relative to the
possibility set for the model of interest. The relative area of the empty set is
zero and the relative area of all outcomes is one. We can also measure the
proportion of the data that satisfies the restrictions of the model of interest
(the “hit rate”). Selten (1991) argues that a good measure of the predictive
success of a theory trades off these elements: the size, as it were, of the target
against the number of times the data manage to hit it. In fact, he provides an
axiomatic argument that the trade-off between these two elements should be
the simple difference measure4 : hit rate minus the size of the target.
Axiomatisations other than Selten’s would produce different forms for the
measure of the predictive success of a theory and are yet to be explored. However, the basic idea that the measure should combine both the pass rate and
some measure of sensitivity remains and appears to be an important and
promising area for further work.
CONCLUSION
Revealed preference has undergone something of a resurgence in recent
years and is now established as an empirical discipline. This essay has
charted the extension of the discipline into new classes of models and
has described recent developments in the areas of identification, power,
and inference, which have occurred as revealed preference has emerged
as an empirical method. Among those discussed here, we consider the
integration of revealed preference and traditional statistical techniques to
4. See also Beatty and Crawford (2011) for an application of this to revealed preference test

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

be the key emerging trend to be on the look out for. Research in this area
has the potential to open up a variety of new avenues and prompt exciting
developments in the areas of identification and inference.
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ABI ADAMS SHORT BIOGRAPHY
Abi Adams is a Junior Research Fellow at Merton College, University of
Oxford and a Research Associate of the Institute for Fiscal Studies. She works
on revealed preference analysis of individual and household behaviour.
IAN CRAWFORD SHORT BIOGRAPHY
Ian Crawford is a Professor of Economics at the University of Oxford and a
Research Fellow of the Institute of Fiscal Studies, where he was previously
Deputy Director. He works on nonparametric/revealed preference analysis
of consumer behavior.
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