Eliciting the just-noticeable difference
Pawel Dziewulski∗
June 2016
Abstract
In this paper we provide the testable implications for the model of consumer
choice with just-noticeable differences. A preference relation admits such a representation whenever there is a utility function u and a constant δ such that bundle x
is preferred to y if and only if u(x) > u(y)+δ. Equivalently, we say that the relation
is a semiorder. We introduce a necessary and sufficient condition under which a
finite set of observations can be rationalised with the above model. Specifically,
our restriction weakens the well-known generalised axiom of revealed preference, or
GARP for short. In addition, we argue that the condition allows to determine an
informative and computationally efficient measure of violations of GARP.
Keywords: just-noticeable difference, revealed preference, Afriat’s theorem, generalised budget sets, Afriat’s efficiency index, money-pump index
JEL Classification: C14, C60, C61, D11, D12
∗
Department of Economics, University of Oxford, Manor Road Building, Manor Road, Oxford OX1
3UQ, United Kingdom. Email: [email protected].
1
1
Introduction
Suppose we observe a consumer making a choice from ℓ goods, with a typical observation t
consisting of a bundle xt ∈ Rℓ+ selected from a budget set Bt ⊆ Rℓ+ .1 Denote a finite
{
}
set of observations by O = (xt , Bt ) t∈T . Stemming from the work by Afriat (1967),
a significant part of the economic literature was devoted to the problem of the testable
restrictions for the utility maximisation model.2 The objective of this research was to
provide necessary and sufficient conditions on set O under which there existed a locally
non-satiated function u : Rℓ+ → R such that, for all t ∈ T ,
u(xt ) ≥ u(y), for all y ∈ Bt .
Following Forges and Minelli (2009), the empirical content of this model is summarised
by the generalised axiom of revealed preference, henceforth GARP.3 The condition requires
{
}S
that, for any sequence (xs , Bs ) s=1 of observations in O such that xs+1 ∈ Bs , for all
s = 1, . . . , S − 1, and x1 ∈ BS , all the bundles belong to the upper boundaries of the
respective sets. Equivalently, the restriction implies the existence of a locally non-satiated
weak order ⪰ over bundles in Rℓ+ such that xt ⪰ y, for all y ∈ Bt and t ∈ T .
Applying the notion of weak orders to consumer preference has been criticised in the
literature, yet most properties of this model are widely accepted. This includes transitivity
of the strict counterpart of such an order. However, transitivity of indifferences seems to
be more questionable. Referring to the classical example in Luce (1956), even though an
agent might be indifferent between a cup of coffee with n and n + 1 grains of sugar, for
any natural n, it seems unlikely that the same decision maker would have no preference
at all over any amount of sugar in the coffee. Citing Armstrong (1950, p. 122):
“The nontransitiveness of indifferences must be recognised and explained on
any theory of choice, and the only explanation that seems to work is based on
imperfect powers of discrimination of the human mind whereby inequalities
become recognisable only when of sufficient magnitude.”
The above claim is supported by the evidence from psychophysics. According to the
well-known Weber-Fechner law, people can not discriminate between very close objects,
1
We consider consumer choices over generalised budget sets Bt , as in Forges and Minelli (2009).
A survey summarising those results can be found in Varian (2006). For an exhaustive lecture on the
revealed preference analysis, see the book by Echenique and Chambers (2016).
3
Forges and Minelli (2009) generalise the original result in Afriat (1967) to non-linear budget sets Bt .
2
2
and only when the difference in their physical traits, like mass or temperature, exceeds
a certain just-noticeable difference a distinction in their perception emerges. Moreover,
the law postulates the existence of a so-called Weber’s constant λ∗ ≥ 1 such that the
individual notices the difference between two objects only if the ratio of their respective
traits exceeds λ∗ or falls below 1/λ∗ . In particular, in a one-dimensional case this would
imply a representation: x ≻ y if and only if log x−log y = log(x/y) > δ, where δ = log λ∗ .
We employ this idea to the problem of consumer choice.
In order to capture the above phenomenon, one is required to model consumer preferences in terms of semiorders introduced in Luce (1956), rather than weak orders. Roughly
speaking, a semiorder is defined as a transitive binary relation with non-transitive indifferences.4 In particular, under some regularity conditions, any semiorder ⪰ admits a
representation in terms of just-noticeable differences.5 That is, there is a utility function
u and a constant δ ≥ 0 such that x ≻ y if and only if u(x) > u(y) + δ, while x ∼ y if
and only if |u(x) − u(y)| ≤ δ. Specifically, whenever the agent chooses bundle x over y, it
must be that u(x) ≥ u(y) − δ. Otherwise, we would have u(y) > u(x) + δ, which would
contradict that x is preferred to y.
{
}
Given the above definition, a finite set of observations O = (xt , Bt ) t∈T is rational-
isable in the just-noticeable sense if there is a utility function u : Rℓ+ → R and a positive
number δ such that, for all observations (xt , Bt ) in O, we obtain
u(xt ) ≥ u(y) − δ, for all y ∈ Bt .
(1)
Alternatively, we say that pair (u, δ) rationalises set O in the just-noticeable sense.
In general, there are no testable restrictions for the above model. That is, any set of
observations O can be rationalised in the just-noticeable sense by some function u and
a number δ ≥ 0. This is true for two reasons. First of all, for any bounded function u,
it is always possible to choose an arbitrarily large number δ for which condition (1) is
satisfied. However, the model is not testable even for a particular value of δ. In fact,
as long as δ > 0, for any bounded function u : Rℓ+ → R there exists a sufficiently small
[
]
number β > 0 such that β u(y) − u(xt ) ≤ δ, for all y ∈ Bt and (xt , Bt ) ∈ O. Therefore,
pair (βu, δ) would rationalise set O in the just-noticeable sense. Hence, unless δ = 0, in
which case the model is reduced to the standard utility maximisation problem, there are
no observable implications for the above representation.
4
A formal definition of a semiorder can be found in Luce (1956). See also Beja and Gilboa (1992) and
Gilboa and Lapson (1995) for a further discussion and references.
5
See Beja and Gilboa (1992) for details.
3
The above claim suggests that, in order to determine any restrictions of the justnoticeable model of consumer choice, we need to restrict our attention to a narrower
class of utility functions that may rationalise O. In particular, our previous argument
suggests that it is crucial to impose assumptions on differences of function u. In this
paper, we provide a necessary and sufficient condition under which a set of observations
O is rationalisable in the just-noticeable sense by a pair (u, δ), where δ ≥ 0 and the utility
function u : Rℓ+ → R is such that, for any non-zero x ∈ Rℓ+ and λ ≥ 1, we have
u(λx) − u(x) ≥ log λ.6
(2)
As we show in Section 2, the above restriction on the class of admissible utility functions is sufficient to determine testable implication of the model in question. That is,
given δ ≥ 0, not every set of observations O can be rationalised in a just-noticeable
sense, with a utility function satisfying condition (2). Second of all, any such function
u is locally non-satiated. In particular, as δ tends to 0 the observable restrictions of
the just-noticeable rationalisation of consumer choice converge to the implications of the
utility maximisation model, hence, GARP.
We find this class of utility functions to be a natural choice for a study of justnoticeable differences. Our framework admits as a special case the one-dimensional model
implied by the Weber-Fechner law.7 At the same time, it provides an intuitive interpretation of δ. Notice that, whenever λ > exp δ, then u(λx) − u(x) ≥ log λ > δ. Therefore,
the value of λ∗ := exp δ can be interpreted as the just-noticeable difference. Clearly, once
we scale an arbitrary non-zero bundle x ∈ Rℓ+ by λ∗ , then the agent has to prefer λ∗ x to
x in the just-noticeable sense. Hence, it must be that u(λ∗ x) ≥ u(x) + δ.
In Section 2 we introduce a restriction called δ-generalised axiom of revealed preference, or δ-GARP for short, that is defined relatively to a scalar δ ≥ 0. Specifically, our
condition generalises GARP. In fact, a set of observations obeys the latter property if and
only if it satisfies δ-GARP for δ = 0. Therefore, GARP is a special case of our axiom. In
the Main Theorem we claim that a set of observations O obeys δ-GARP for some δ ≥ 0
if and only if it is rationalisable in the just-noticeable sense by a pair (u, δ), where the
utility function u satisfies condition (2). Hence, our condition exhausts all the observable
implications of the model in question. Moreover, the results presented in this paper allow
6
Whenever function u is differentiable, this is equivalent to the derivative of function u at x to be
greater than 1/∥x∥ along any ray rooted at the origin and passing through x.
7
Clearly, in this case, we have log(λx) − log x = log λ, for any non-zero x ∈ R+ and λ ≥ 1.
4
to determine the least value of δ for which set O obeys δ-GARP via linear programming
methods. Thus, our method is easy to apply in empirical studies.
It is reasonable to argue that, even though semiorders are more accurate whilst describing consumer behaviour, weak orders may still provide a good approximation of
reality and substantially simplify the analysis. However, from the empirical point of
view, verifying whether a set of observations is rationalisable by the utility maximisation
model, or equivalently: if it obeys GARP, might be problematic because of the deterministic nature of the test; a data set either obeys GARP or not. In practice, it would be
useful to evaluate the degree to which a consumer violates this condition.
We address the above issue is Section 3. Suppose that δ ≥ 0 is the lowest scalar for
which a set of observations obeys δ-GARP. Therefore, number λ∗ = exp δ constitutes the
corresponding just-noticeable difference. We argue that the inverse of the just-noticeable
difference 1/λ∗ = exp(−δ) constitutes an informative measure of revealed preference
violations. Moreover, as δ can be determined via linear programming methods, evaluating
this index is computationally efficient. We compare this measure with two other existing
concepts: Afriat’s efficiency index discussed in Afriat (1967) or Varian (1990) and the
money-pump index introduced in Echenique, Lee, and Shum (2011). In particular, we
show that 1/λ∗ provides an alternative response to the critique of Afriat’s efficiency
index expressed by Echenique, Lee, and Shum. At the same time, the low computational
burden and appealing economic interpretation of our measure in terms of just-noticeable
differences makes it attractive for empirical applications.
2
Revealed just-noticeable differences
In this section we present a necessary and sufficient condition under which a set of observations O can be rationalised in the just-noticeable sense by a pair (u, δ), for a specific
value of δ ≥ 0 and a utility function u that satisfies condition (2). Before we proceed with
our discussion, we need to introduce some useful notation and one preliminary result.
Suppose that, for any observation (xt , Bt ) in O, set Bt is compact and downward
comprehensive, i.e., for any z ∈ Rℓ+ , if y ∈ Bt and z ≤ y then z ∈ Bt . Moreover, we
assume that there is some y ∈ Bt such that y ≫ 0. Denote the upper bound of set Bt by
{
}
∂Btu := y ∈ Bt : if z ≫ y then z ̸∈ Bt .
Let λy ∈ Bt \ ∂Btu , for any y ∈ ∂Btu and λ ∈ [0, 1). That is, for any vector y ∈ Rℓ+ ,
5
{
}
ray λy : λ ≥ 0 intersects ∂Btu at most once. Define the gauge function γt : Rℓ+ → R
{
}
of set Bt by γt (y) := inf λ > 0 : y ∈ λBt . Roughly speaking, function γt assigns to a
vector y ∈ Rℓ+ the lowest scalar λ such that y belongs to λBt . Given our assumptions,
the following result from Forges and Minelli (2009) holds true.
Lemma 1. For all t ∈ T , function γt is continuous, homogeneous of degree one, and
{
}
y ≫ z implies γt (y) > γt (z). Moreover, we have Bt = y ∈ Rℓ+ : γt (y) ≤ 1 .
For any t ∈ T , let function gt : Rℓ+ → R be given by gt (y) := log γt (y), where γt
is defined as previously. By Lemma 1, we conclude that gt is continuous, it satisfies
gt (λy) = log λ + gt (y) for any non-zero vector y ∈ R+ and λ > 0, while y ≫ z implies
{
}
gt (y) > gt (z). Moreover, we have Bt = y ∈ R+ : gt (y) ≤ 0 . Finally, we assume that
xt ̸= 0, for all t ∈ T .8 In particular, this implies that gt (xs ) is finite, for all t, s ∈ T .9
{
}S
Axiom 1 (δ-Generalised axiom of revealed preference). For any sequence (xs , Bs ) s=1
in O such that gs (xs+1 ) ≤ 0, for s = 1, . . . , S − 1, and gS (x1 ) ≤ 0, we have
g1 (x2 ) + g2 (x3 ) + . . . + gS−1 (xS ) + gS (x1 ) ≥ −Sδ.
In the remainder of this paper we refer to the above condition as δ-GARP. To better
{
}S
understand this property, take any sequence (xs , Bs ) s=1 specified as in the thesis of
the axiom. Fix an arbitrary index s = 1, . . . , S − 1. Since gs (xs+1 ) ≤ 0, by construction
of function gs , we know that xs+1 ∈ Bs . Moreover, given the assumptions imposed on
set Bs , for all s = 1, . . . , S − 1, there exists some number λs ≤ 1 such that the bundle
xs+1 is an element of the upper boundary of set λs Bs . Similarly, there is some λS for
which x1 is in the upper boundary of λS BS . Therefore, by definition, it must be that
gs (xs+1 ) = log λs , for all s = 1, . . . , S − 1, and gS (x1 ) = log λS . In particular, this allows
to restate the inequality introduced in the above axiom as
√
S
λ1 λ2 · · · λS−1 λS ≥ exp(−δ).
Hence, for any such a sequence, the geometric average of the magnitudes λs can not
exceed the exponent of −δ. Recall that, at the same time the latter value is the inverse
of the just-noticeable difference λ∗ = exp δ. Finally, observe that the condition has to be
{
}
satisfied for every one-element sequence (xt , Bt ) . Hence, for any t ∈ T , it requires that
8
The above assumption is not without loss of generality. However, it substantially simplifies our
analysis. Moreover, we find it {to be insignificant from the
} empirical point of view.
9
Linear budget sets Bt = y ∈ Rℓ+ : pt · y ≤ pt · xt , for some pt ∈ Rℓ++ , satisfy all of the above
assumptions. Moreover, in such a case, we have gt (y) = log λ, where λ = pt · y/pt · xt .
6
Good 2
b
xt
Bt
exp(−δ)Bt
Good 1
{
}
Figure 1: A singleton set O = (xt , Bt ) obeys δ-GARP. However, since xt does not belong to
the upper boundary of the set Bt , the observation (xt , Bt ) can not be rationalised by maximisation of a locally non-satiated utility function.
gt (xt ) = log λt ≥ −δ, or equivalently: λt ≥ exp(−δ). This implies that, whenever δ > 0,
the axiom admits observations (xt , Bt ) for which choice xt does not belong to the upper
boundary of set Bt . For example, see Figure 1.
A closer look at Axiom 1 reveals three additional properties of this condition. First
of all, for any set of observations O there exists a sufficiently large number δ ≥ 0, for
{
}S
which the set obeys δ-GARP. Clearly, since any sequence (xs , Bs ) s=1 specified as in
the thesis of the axiom is finite, so is the sum g1 (x2 ) + g2 (x3 ) + . . . + gS−1 (xS ) + gs (x1 ).
Hence, it is always possible to find some δ ≥ 0, for which the restriction is satisfied. The
same observation implies that any data set admits the least δ ≥ 0 for which our condition
holds. Namely, such a number is equal to the greatest average of values −gs (xs+1 ), for
s = 1, . . . , S −1, and −gS (x1 ) over all the sequences specified as in the thesis of the axiom.
Since any such average is finite, the minimal δ exists. Finally, it is straightforward to
show that if set O obeys δ-GARP for some δ ≥ 0, then it also satisfies the condition
for any δ ′ ≥ δ. Therefore, the focus of the researcher should be restricted solely on the
minimal value of the constant δ.
Our condition is weaker than the generalised axiom of revealed preference, introduced
7
Good 2
B1
9
B
10 1
xb 2
xb 1
B2
4
B
5 2
Good 1
{
}
Figure 2: Set O = (x1 , B1 ), (x2 , B2 ) violates GARP. In fact, it fails to satisfy the weak axiom
√
9
of revealed preference. However, it obeys δ-GARP for any δ ≥ − log 45 10
.
in Forges and Minelli (2009).10 Given our notation, a data set O obeys GARP if for any
{
}S
sequence (xs , Bs ) s=1 in O such that gs (xs+1 ) ≤ 0, for s = 1, . . . , S − 1, and gS (x1 ) ≤ 0
all the inequalities are binding. Therefore, set O obeys GARP only if it satisfies δ-GARP
for all δ ≥ 0. Conversely, the data set satisfies δ-GARP for δ = 0 only if it obeys GARP.
Hence, the latter axiom is a special case of the former. In particular, there are data sets
that violate GARP, but satisfy Axiom 1 for some δ > 0. For instance, see Figure 2.
Main Theorem. Set O obeys δ-GARP for some δ ≥ 0 if and only if it is rationalisable
in the just-noticeable sense by a pair (u, δ), where function u satisfies condition (2).
Given that the argument supporting this theorem highlights both the properties as
well as applicability of Axiom 1, we find it convenient to prove the result via three lemmas
that follow. First, we show that δ-GARP is a necessary condition for a set of observations
to be rationalisable in the just-noticeable sense, with a utility u satisfying condition (2).
Lemma T1. Whenever set O is rationalisable in the just-noticeable sense by a pair (u, δ),
for some δ ≥ 0 and a function u that satisfies condition (2), then it obeys δ-GARP.
10
Hence, it also generalises the original version of the axiom presented in Afriat (1967).
8
Proof. Take a function u satisfying condition (2) and a number δ ≥ 0 such that the
pair (u, δ) rationalises set O in the just-noticeable sense. Consider an arbitrary sequence
{
}S
(xs , Bs ) s=1 in O such that gs (xs+1 ) ≤ 0, for all s = 1, . . . , S − 1, and gS (x1 ) ≤ 0.
Fix any s = 1, . . . , S − 1. By construction of function gs , it must be that xs+1 ∈ Bs .
Moreover, there is some λs ≥ 1 such that λs xs+1 is in the upper boundary of set Bs , or
equivalently: gs (λs xs+1 ) = 0. This implies that gs (xs+1 ) = − log λs . By condition (2),
u(λs xs+1 ) ≥ u(xs ) + log λs = u(xs ) − gs (xs+1 ).
In addition, given that the pair (u, δ) rationalises set O in the just-noticeable sense, while
element λs xs+1 belongs to Bs , we conclude that
u(xs ) ≥ u(λs xs+1 ) − δ ≥ u(xs+1 ) − gs (xs+1 ) − δ.
Analogously, we show that u(xS ) ≥ u(x1 )−gS (x1 )−δ. In order to complete our argument,
observe that the above inequalities imply
u(x1 ) ≥ u(x2 ) − g1 (x2 ) − δ ≥ u(x3 ) − g1 (x2 ) − g2 (x3 ) − 2δ
≥ . . . ≥ u(x1 ) − g1 (x2 ) − g2 (x3 ) − . . . − gS (x1 ) − Sδ,
which can be satisfied only if g1 (x2 ) + g2 (x3 ) + · · · + gS (x1 ) ≥ −Sδ. Hence, the set of
observations O must obey δ-GARP. This concludes our proof.
A careful read of the proof of Lemma T1 provides a deeper understanding of the
revealed preference relation that is induced by the set of observations. Suppose there
{
}S+1
is a sequence (xs , Bs ) s=1 in O such that gs (xs+1 ) ≤ 0, for all s = 1, . . . , S. By the
argument presented in the above proof, the set is rationalisable in our sense only if
u(x1 ) ≥ u(xS+1 ) −
S
∑
gs (xs+1 ) − Sδ.
s=1
This implies that, whenever we have
∑S
s=1
gs (xs+1 ) ≤ −Sδ, the agent must assign a
higher utility to bundle x1 than xS+1 , i.e., u(x1 ) ≥ u(xS+1 ). In such a case, we say
that the bundle x1 is revealed preferred to xS .11 Moreover, the relation must be strict if
∑S
∑S
s=1 gs (xs+1 ) = −Sδ.
s=1 gs (xs+1 ) < −Sδ. Therefore, we may have x1 = xS+1 only if
Otherwise, we would have u(x1 ) > u(xs+1 ) = u(x1 ), yielding a contradiction. Hence,
11
Notice that, bundle x1 is revealed preferred to xS with respect to the weak order induced by utility
function u. However, it need not be revealed preferred relatively to the semiorder generated by the model
of just-noticeable consumer choice. That is, it need not be true that u(x1 ) ≥ u(xS+1 ) + δ.
9
the condition stated in Axiom 1 excludes the existence of any sequence within the set of
observations, that would induce a strict cycle of the revealed preference relation.
In the next lemma we show that δ-GARP implies the existence of a solution to a
particular system of linear inequalities. In the remainder of our discussion we shall use
any such solution to construct a utility function that rationalises the set of observations
in the just-noticeable sense. The linear system specified in the lemma shows a very close
resemblance to the so-called Afriat’s inequalities. However, it is distinct.
Lemma T2. Whenever set O obeys δ-GARP, there exist numbers ϕt , µt , and νt , t ∈ T ,
such that ϕs − δ ≤ ϕt + µt gt (xs ), if gt (xs ) ≤ 0, and ϕs − δ ≤ ϕt + νt gt (xs ), if gt (xs ) > 0,
for all t, s ∈ T , where additionally, we have µt , νt ≥ 1, for all t ∈ T .
We prove this result by applying a variation of Farkas’ Lemma. In particular, we show
that the above system of linear inequalities has no solution only if set O fails to satisfy
δ-GARP. As our argument is rather extensive, we present it in the Appendix.
Lemma T2 plays a significant role in the applicability of the Main Theorem. In
particular, as the existence of a solution to the system of inequalities is necessary for
O to obey δ-GARP, the lemma provides a method of verifying the axiom via linear
programming methods. In fact, in the remainder of this section we argue that determining
a solution to these inequalities is also sufficient for δ-GARP to hold.
The linear system presented in the thesis of Lemma T2 differs in two important aspect
from the so-called Afriat’s inequalities analysed in Forges and Minelli (2009). First of
all, our system is conditioned on the value of coefficient δ, which by assumption is equal
to zero in the aforementioned paper. Moreover, Forges and Minelli specify their system
such that µt = νt , for all t ∈ T . As we show in our next result, this difference directly
affects the way we construct a utility function u that rationalises the set of observations.
Lemma T3. Suppose there exist numbers specified in Lemma T2. There is a continuous
function u : Rℓ+ → R satisfying condition (2) such that (u, δ) rationalises O in the justnoticeable sense. Moreover, for any y, z ∈ Rℓ+ , y ≫ z implies u(y) > u(z).
Proof. Take any numbers ϕt , µt , and νt , for t ∈ T , that solve the system of inequalities
presented in Lemma T2. For any t ∈ T , define function vt : Rℓ+ → R by

 ϕ + µ g (y) if g (y) ≤ 0,
t
t t
t
vt (y) :=
 ϕt + νt gt (y) otherwise.
10
By continuity and monotonicity of gt , any such function is continuous, while y ≫ z
implies vt (y) > vt (z), for any y, z ∈ Rℓ+ . Next, we show that it satisfies condition (2).
First, take a non-zero y ∈ Rℓ+ and λ ≥ 1 such that gt (y) ≤ 0 and gt (λy) ≤ 0. Then,
[
]
vt (λy) − vt (y) = µt gt (λy) − gt (y) = µt log λ ≥ log λ,
since gt (λy) = log λ + gt (y) and µt ≥ 1. Analogously, we can show that the condition
holds for any y and λ such that gt (y) > 0 and gt (λy) > 0. Finally, take any y and λ with
gt (y) ≤ 0 and gt (λy) > 0. By continuity of gt , there is some κ ≥ 1 such that gt (κy) = 0.
Moreover, it must be that λ > κ. In particular, by our previous argument, we obtain
vt (λy) − vt (y) =
[
] [
]
vt (λy) − vt (κy) + vt (κy) − vt (y) ≥ νt g(λy) + log κ
[
( )]
= νt log λ/κ + gt κy + log κ ≥ log λ/κ + log κ = log λ.
Therefore, function vt obeys condition (2), for any t ∈ T . Moreover, by construction of
numbers ϕt , µt , and νt , for t ∈ T , we have ϕt − δ ≤ vs (xt ), for any t, s ∈ T . Clearly, this
is equivalent to ϕt − δ ≤ mins∈T vs (xt ), for all t ∈ T .
Define the utility function u : Rℓ+ → R by u(y) := mint∈T vt (y). Given the argument
presented above, it is continuous, while y ≫ z implies u(y) > u(z), for all y, z ∈ Rℓ+ .
To show that the function satisfies property (2), take a non-zero y ∈ Rℓ+ and λ ≥ 1. By
construction, there are some indices t, s ∈ T such that u(λy) = vt (λy) and u(y) = vs (y).
Given that function vt satisfies property (2), for any t ∈ T , we have
u(λy) − u(y) = vt (λy) − vs (y) ≥ vt (λy) − vt (y) ≥ log λ.
Hence, function u satisfies condition (2). It remains to show that (u, δ) rationalises O in
the just-noticeable sense. Take any t ∈ T and y ∈ Bt . Since gt (y) ≤ 0, we obtain
u(y) − δ ≤ vt (y) − δ = ϕt + µt gt (y) − δ ≤ ϕt − δ ≤ min vs (xt ) = u(xt ),
s∈T
which concludes our argument.
Lemmas T1–T3 establish a fourfold equivalence. First of all, the three results imply that δ-GARP is a necessary and sufficient condition for a set of observations to be
rationalisable in the just-noticeable sense by a pair (u, δ), where function u obeys condition (2). Moreover, the axiom is equivalent to the existence of a solution to the system of
linear inequalities specified in Lemma T2. Third, existence of such a solution is necessary
and sufficient for the set of observations to be rationalisable in the just-noticeable sense
11
by a pair (u, δ), where function u is continuous, monotone in the particular sense, and
satisfies condition (2). Finally, this establishes the fourth equivalence, according to which
a data set O is rationalisable by a utility function u satisfying condition (2) if and only
if it is rationalisable by a utility function that is additionally continuous and increasing
in the sense specified in Lemma T3. Hence, given our framework, neither continuity nor
monotonicity are testable properties of the utility function u.12
The second equivalence mentioned above provides conditions that allow to verify Axiom 1 via linear programming methods, for a specific value of δ. However, as it was
pointed out in one of the preceding paragraphs, it is sufficient for a researcher to focus solely on the minimal value of δ for which δ-GARP is satisfied. Notice that, given
Lemma T2, such a value can be determined via a linear optimisation problem. In particular, it suffices to find the lowest value of δ subject to the existence of a solution to
the system of linear inequalities specified in the thesis of the lemma. Since such problems are generally solvable, our result provides a convenient way of determining the least
just-noticeable difference λ∗ = exp δ. Moreover, recall that its inverse 1/λ∗ = exp(−δ) is
equal to the least geometric average of scalars λs , defined earlier in this section, over all
sequences in O that may potentially violate GARP. We use this fact extensively in the
next section, where we argue that the inverse of the just-noticeable difference constitutes
an informative measure of revealed preference violations.
3
Measures of revealed preference violations
There is an extensive economic literature on evaluating the severity of revealed preference
violations. For an extensive discussion, see the survey by Varian (2006) or Chapter 5.1
in Echenique and Chambers (2016). In this section we explore some properties of an
alternative measure: the inverse of the just-noticeable difference. Formally, suppose that
δ ≥ 0 is the minimal number for which a set of observations O obeys Axiom 1. By the
discussion provided in the previous section, we know that such a number exists. Moreover,
by the Main Theorem, there is a function u satisfying condition (2) such that O can be
rationalised in the just-noticeable sense by the pair (u, δ). In particular, for any non-zero
bundle x ∈ Rℓ+ , we have u(λ∗ x) ≥ u(x) + δ, where λ∗ := exp δ is the just-noticeable
It is easy to verify that, whenever set Rℓ+ \ Bt is convex, for all t ∈ T , then function u defined in the
proof of Lemma T3 is additionally quasiconcave. Therefore, within this narrower class of experiments,
quasiconcavity of a utility function is not a testable property.
12
12
difference. That is, once we scale the bundle x by λ∗ , the agent is able to “distinguish
between” options x and λ∗ x, and prefers the latter to the former.
We argue that the inverse of the just-noticeable difference 1/λ∗ = exp(−δ) constitutes
a relevant measure of the extent to which a data set O violates GARP. Suppose that the
{
}S
set O admits a sequence (xs , Bs ) s=1 such that gs (xs+1 ) ≤ 0, for all s = 1, . . . , S − 1,
and gS (x1 ) ≤ 0. By construction, there are some positive numbers λs ≤ 1, for all
s = 1, . . . , S, such that gs (xs+1 ) = log λs , for s = 1, . . . , S − 1, and gS (x1 ) = log λS .
Recall that, number λs is the minimal value for which bundle xs+1 belongs to λs Bs , for
s = 1, . . . , S − 1. Analogously, λS is the lowest scalar such that x1 ∈ λS BS . Alternatively,
the magnitude λs may be interpreted as the relative “distance” of bundle xs+1 from the
upper boundary of set Bs , where λs = 1 whenever the vector belongs to ∂Bsu .
Recall from the previous section that the inverse of the just-noticeable difference 1/λ∗
{
}S
is equal to the least geometric average of such numbers λs , over all sequences (xs , Bs ) s=1
defined as above. In particular, the measure takes values between 0 and 1, while 1 is
attained if and only if the set of observations O obeys GARP. In fact 1/λ∗ = 1 if and
only if λs = 1, for all s = 1, . . . , S, which is equivalent to the generalised axiom. In order
to provide a better context, we discuss the remaining properties of our index by comparing
it to two other measures of revealed preference violations existing in the literature.
3.1
Afriat’s efficiency index and just-noticeable differences
First, we compare the inverse of the just noticeable difference to a generalised version
of the well-known Afriat’s efficiency index, proposed originally in Afriat (1967).13 A set
of observations O obeys GARP for an efficiency index α ∈ [0, 1], if for any sequence
{
}S
(xs , Bs ) s=1 in O such that xs+1 ∈ αBs , for all s = 1, . . . , S − 1, and x1 ∈ αBS
all the vectors belong to the upper bound of their respective sets. Equivalently, using
the notation introduced in the previous section, the condition requires that whenever
gs (xs+1 ) ≤ log α, for each s = 1, . . . , S − 1, and gS (x1 ) ≤ log α, then all the inequalities
are binding. Afriat’s efficiency index is then defined as the supremum over all efficiency
indices α for which the set O obeys GARP.
Similarly to Axiom 1, Afriat’s efficiency index imposes a weaker restriction on the
set of observations than GARP. In particular, a data set O obeys the generalised axiom
if and only if α = 1. Before discussing any distinctions between this measure and the
13
See also Varian (1990) for a modified version of the measure.
13
inverse of the just-noticeable difference, we present the following observation.
Proposition 1. Whenever set O obeys δ-GARP for some δ ≥ 0, then it satisfies GARP
for any efficiency index α ≤ exp(−δ).
Proof. Fix any δ ≥ 0. We show that, whenever set O obeys δ-GARP, then is satisfies
{
}S
GARP for efficiency index α = exp(−δ). Take any sequence (xs , Bs ) s=1 such that
gs (xs+t ) ≤ −δ, for all s = 1, . . . , S − 1, and gS (x1 ) ≤ −δ. It suffices to show that all the
above inequalities are binding. Clearly, we have gs (xs+t ) ≤ 0, for all s = 1, . . . , S − 1, and
gS (x1 ) ≤ 0. Hence, given that O obeys δ-GARP, we obtain
−Sδ ≥ g1 (x2 ) + g2 (x3 ) + . . . + gS−1 (xS ) + gS (x1 ) ≥ −Sδ,
which holds only if gs (xs+1 ) = −δ, for all s = 1, . . . , S − 1, and gS (x1 ) = −δ. Hence, set
O obeys GARP for α = exp(−δ). Obviously, it satisfies the axiom for any α′ ≤ α.
Proposition 1 implies that the inverse of the just-noticeable difference is always (weakly)
lower than Afriat’s efficiency index. However, the opposite implication need not hold. In
other words, there are observation sets that obey GARP for some α, but fail to satisfy δ-GARP for δ = − log α. For example, recall the data set form Figure 2. Observe
that the corresponding Afriat’s efficiency index is equal to 9/10, while the inverse of the
√
just-noticeable difference is 36/50 < 0.85.
The last observation is implied by one crucial property of Afriat’s efficiency index. As
it was noted in Echenique, Lee, and Shum (2011), the above measure seeks to “break” a
violation of GARP at its weakest link. In particular, the measure is independent of other
observations, as long as the weakest link remains unchanged. Consider the example in
Figure 3. Clearly, both sets O and O′ violate GARP. The inverse of the just-noticeable
difference corresponding to the latter set is lower than the one evaluated for the former.
This would suggest that the violation in O′ is more severe than in O, as it would require
a larger value of δ to rationalise set O′ in the just-noticeable sense. At the same time,
Afriat’s efficiency index does not discriminate between these two cases.
{
}S
More generally, consider a set of observations O and its sequence (xs , Bs ) s=1 such
that gs (xs+1 ) ≤ 0, for s = 1, . . . , S−1, and gS (x1 ) ≤ 0. Suppose that gs (xs+1 ) = log λs , for
s = 1, . . . , S − 1, and gS (x1 ) = log λS . A closer look at the definition of Afriat’s efficiency
index and the above example reveals that the measure is equivalent to the minimum over
values maxs=1,...,S λs , among all sequences in O constructed as above. Hence, unlike the
14
Good 2
Good 2
B2′
B2
xb ′1
xb 1
4
B
5 2
4 ′
B
5 2
xb 2
4
B
5 1
B1
7
B′
10 1
B1′
xb ′2
Good 1
Good 1
{
}
{
}
Figure 3: Sets O = (x1 , B1 ), (x2 , B2 ) and O′ = (x′1 , B1′ ), (x′2 , B2′ ) violate GARP. At the
same time, Afriat’s efficiency index is equal to 4/5 is either case. However, the inverse of the
√
just-noticeable difference is equal to 4/5 for O, but 28/50 for O′ .
inverse of the just-noticeable difference, Afriat’s efficiency index is independent of the
distribution of observations in O, as long as the above minmax value is unchanged.
Below we revisit an example studied in Echenique, Lee, and Shum (2011), which
highlights the extent to which the same violation of revealed preference can be evaluated
differently by the two measures discussed in this subsection.
{
}3
Example 1. Take any ϵ > 0 and consider a set of observations O = (xt , Bt ) t=1 , where
(
)
x1 = (ϵ2 , 0), x2 = ϵ2 /(1 + ϵ2 ), ϵ2 /(1 + ϵ2 ) , and x3 = (0, 1), while
{
}
Bt := y ∈ R2+ : pt · y ≤ pt · xt ,
for all t = 1, 2, 3, where p1 = (1/ϵ, ϵ), p2 = (ϵ, 1/ϵ), and p3 = (1, 1). Notice that g1 (x2 ) = 0,
g2 (x3 ) = 0, while g3 (x1 ) = 2 log ϵ. Therefore, the inverse of the just-noticeable difference
√
3
must be less than ϵ2 . In particular, as ϵ tends to 0, so does our measure. At the same
time, Afriat’s efficiency index is equal to 1, irrespectively of ϵ.
3.2
Money-pump index and just-noticeable differences
Echenique, Lee, and Shum (2011) introduce an alternative measure of revealed preference
violations, called the money-pump index. As the above notion refers to consumer choice
problems over linear budget sets, in the remainder of this section we restrict our attention
15
{
}
to observation sets O = (xt , Bt ) t∈T such that, for all t ∈ T , we have
{
}
Bt := y ∈ Rℓ+ : pt · y ≤ pt · xt ,
for some vector pt ∈ Rℓ++ . Recall that, in such a case, an observation set O obeys GARP
{
}S
if for any sequence (xs , Bs ) s=1 such that ps · xs ≥ ps · xs+1 , for all s = 1, . . . , S − 1, and
pS · xS ≥ pS · x1 all the inequalities are binding.
{
}
Take a sequence S = (xs , Bs ) s∈S such that ps · xs ≥ ps · xs+1 , for all s = 1, . . . , S − 1,
and pS · xS ≥ pS · x1 . The money pump index mS corresponding to S is given by
mS =
p1 · x2 + p2 · x3 + . . . + pS−1 · xS + pS · x1 14
.
∑S
p
·
x
s
s
s=1
In order to obtain a single measure of revealed preference violations, Echenique, Lee, and
Shum propose to use the mean or the median of money-pump indices, over all sequences
S of set O satisfying the above property.15 Specifically, the index is equal to 1 if and only
if the set of observations obeys GARP. See Echenique, Lee, and Shum (2011) for details.
The money-pump index has a very intuitive interpretation. Consider a sequence S in
set O, specified as previously. Suppose there is a devious “arbitrageur” who is aware of
the choices made by the consumer. In particular, by buying bundle xs+1 from the agent
at prices ps and reselling it at ps+1 , for s = 1, . . . , S − 1, while acquiring bundle x1 at
prices pS and selling it at p1 , the arbitrageur could make a positive profit equal to
p1 · (x1 − x2 ) + p2 · (x2 − x3 ) + . . . + pS · (xS − x1 ).
We refer to this value as the money-pump cost associated with the sequence S. Therefore,
(1−mS ) is equal to the money-pump cost normalised with respect to the total expenditure
∑S
s=1 ps · xs , induced by the sequence.
In order to show the distinction between the inverse of the just-noticeable difference
{
}
and the money-pump index, take any sequence S = (xs , Bs ) s∈S in O, specified as in
the preceding paragraphs. Moreover, denote λs := ps · xs+1 /ps · xs , for s = 1, . . . , S − 1,
and λS := pS · x1 /pS · xS . It is straightforward to verify that gs (xs+1 ) = log λs , for
s = 1, . . . , S − 1 and gS (x1 ) = log λS . This implies that, for any δ for which the set obeys
√
δ-GARP, it must be that S λ1 λ2 . . . λS−1 λS ≥ exp(−δ). Specifically, this has to hold
for the lowest value of δ, for which exp(−δ) is the inverse of the just-noticeable difference.
The original definition of the money-pump index for sequence S is equivalent to (1 − mS ). We
modified the notion in order to make it comparable with other measures discussed in this paper.
15
Alternatively, Smeulders, Cherchye, De Rock, and Spieksma (2013) propose to use the least or the
greatest value of the index, over all such sequences.
14
16
Let as := ps · xs /
∑S
pt · xt be the share of budget ps · xs in the total expenditure
∑
corresponding to sequence S, for s = 1, . . . , S. Clearly, we have Ss=1 as = 1. Moreover,
t=1
the money-pump index associated with S is equivalent to
mS =
S
∑
as λs .
s=1
Therefore, the measure is given by the average of ratios λs , weighted by the budget
shares as . In particular, unlike the inverse of the just-noticeable difference, for the same
scalars λs , where s = 1, . . . , S, the money-pump index might take different values, depending on the corresponding weights as . This is to say that, a low value of λs affects
the index less if the corresponding budget ratio as is small. Clearly, this is not the case
in for the measure proposed in our paper, as it is independent of the budget shares.
Finally, as pointed out in Remark 1 in Echenique, Lee, and Shum (2011), the computational burden of evaluating the above measure is significant. However, by the result in
Smeulders, Cherchye, De Rock, and Spieksma (2013), once we restrict our attention to
the maximal or the minimal values of the money-pump index, rather than the mean or
the median, the measure can be determined in a polynomial time. Hence, conditional of
the approach, the complexity of computing the index is at least as high as determining
the value of the inverse of the just-noticeable difference.
A
Appendix
We begin this section by introducing a variation of the well-known Farkas’ Lemma. See
Gale (1989, Theorem 2.7, p. 46) for the proof of the following result.
Theorem A.1 (Farkas’ Lemma). For any k × ℓ real matrix A and vector b ∈ Rk exactly
one of the following alternatives is true.
(i) There is some x ∈ Rℓ such that A · x ≥ b.
(ii) There is some π ∈ Rk+ such that π · A = 0 and π · b > 0.
The above result is applied directly to the proof of Lemma T2 that follows.
Proof of Lemma T2. Let ϵt ∈ {0, 1}|T | be a vector in which the t’th coordinate is equal
to 1 and all the remaining ones are 0, for t ∈ T . First, we show that whenever set O
17
obeys δ-GARP, there exist vectors ϕ, µ̂, and ν̂ in R|T | such that, for all t, s ∈ T , we have
(ϵt − ϵs ) · ϕ + gt (xs )ϵt · µ̂
(ϵt − ϵs ) · ϕ
≥ −gt (xs ) − δ, if gt (xs ) ≤ 0
+ gt (xs )ϵt · ν̂ ≥ −gt (xs ) − δ, if gt (xs ) > 0
ϵt · µ̂
≥
0,
ϵt · ν̂ ≥
0.
In particular, there are |T |2 inequalities of the first two types, and 2|T | of the last two.
We prove the above claim by contradiction. Suppose there is no solution to the above
|T |2
system of linear inequalities. By Theorem A.1, there exist vectors π ∈ R+
and ρ,
|T |
σ ∈ R+ such that, for all t ∈ T , we have
∑
∑
s∈T
πts =
∑
πrt ,
r∈T
πts gt (xs ) = −ρt , where gt (xs ) ≤ 0, for all s ∈ T,
s∈T
∑
πts gt (xs ) = −σt , where gt (xs ) > 0, for all s ∈ T,
s∈T
while
∑
∑
t∈T
s∈T
[
]
πts − gt (xs ) − δ > 0. Clearly, the final inequality implies that πts > 0,
for some t, s ∈ T . However, the penultimate set of restrictions can be satisfied only if
πts = 0, for all t, s ∈ T such that gt (xs ) > 0, and σ = 0.
Define vectors π̂ := π, ρ̂ := ρ, and σ̂ := σ. Obviously, they satisfy the set of restrictions
∑
∑
specified above. Take any a, b ∈ T such that π̂ab > 0. Since s∈T π̂bs = s∈T π̂sb , there
is some c ∈ T such that π̂bc > 0.16 Continuing this way, we are able to construct a
{
}
cycle C = (a, b), (b, c), . . . , (z, a) such that π̂ts > 0, for all (t, s) ∈ C. Without loss of
generality, we may assume that every index in the cycle appears exactly twice.17 By the
previous observation, it must be that gt (xs ) ≤ 0, for all (t, s) ∈ C. In particular, since
observation set O obeys δ-GARP, this implies that ga (xb ) + gb (xc ) + . . . + gz (xa ) ≥ −|C|δ,
[
]
∑
or equivalently: (t,s)∈C − gt (xs ) − δ ≤ 0.
{
}
Define π̂ := min π̂ts : (t, s) ∈ C , which is strictly positive. Moreover, construct a
|T |2
new vector π ∈ R+
such that πts := π̂ts − π̂, for all (t, s) ∈ C, and πts := π̂ts otherwise.
|T |
Similarly, define a vector ρ ∈ R+ such that ρt := ρ̂t + π̂gt (xs ), for all (t, s) ∈ C, and
ρt := ρ̂t otherwise. Finally, let σ = 0. It is straightforward to verify that the newly
16
In particular, it might be that a = b = c, in which case we determine a one-element cycle.
Otherwise, there would have to be another cycle within the cycle C. In such a case, we could simply
address the reminder of our argument to the smaller cycle.
17
18
constructed vectors satisfy the first three conditions specified above. Moreover, we have
0 <
≤
∑∑
[
]
π̂ts − gt (xs ) − δ
t∈T s∈T
∑ [
[
]
]
π̂ts − gt (xs ) − δ − π̂
− gt (xs ) − δ
∑∑
t∈T s∈T
=
∑∑
[
]
(t,s)∈C
πts − gt (xs ) − δ .
t∈T s∈T
Therefore, the forth condition also holds. Additionally, we have π < π̂ and ρ ≤ ρ̂. This
implies that, by repeating the procedure, we eventually find positive vectors π, ρ, and
σ that satisfy the restrictions specified above, while all non-zero entries of π are equal.
{
}
However, this would imply that there exists a cycle C := (a, b), (b, c), . . . , (z, a) such
[
]
∑
that gt (xs ) ≤ 0, for all (t, s) ∈ C, and (t,s)∈C − gt (xs ) − δ > 0, or equivalently
ga (xb ) + gb (xc ) + . . . + gz (xa ) < −|C|δ
which would contradict that set O obeys δ-GARP. Therefore, we conclude that there
|T |
exist vectors ϕ, µ̂, and ν̂ in R+ that satisfy the system of inequalities introduced at the
beginning. In order to complete the proof, define µ and ν such that µt := µ̂t + 1 and
νt := ν̂t + 1, for all t ∈ T . It is straightforward to show that numbers ϕt , µt , and νt ,
t ∈ T , satisfy the inequalities presented in the thesis of Lemma T2.
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20