Random Choice and Market Demand
Javier A. Birchenall∗
University of California at Santa Barbara
September 10, 2016
Abstract
This paper presents comparative statics predictions for a statistical model
of choice in which consumption is randomly chosen subject to a linear budget
constraint. Market demands are shown to satisfy all the properties of the behavioral model of choice, including symmetry and negative semidefiniteness of the
Slutsky matrix. The findings characterize market demands for random choice
behaviors that abandon all behavioral postulates of rational choice theory but
also for realistic individual choice behaviors representable by choice probabilities. The findings also serve to demonstrate the inconsistency of statistical tests
of revealed preferences that rely on rationality indices.
Keywords : random choice, market behavior, demand theory
JEL classification: D01; D11; C02
Communications to: Javier A. Birchenall
Department of Economics, 2127 North Hall
University of California, Santa Barbara CA 93106
Phone/fax: (805) 893-5275
[email protected]
∗
I am grateful to audiences at several locations for comments and helpful suggestions. I am specially
grateful to Geir Asheim, Ted Bergstrom, Gary Charness, Luis Corchón, John Duffy, Martin Dufwenberg,
Enrique Fatas, Raya Feldman, Eric Fisher, Rod Garratt, Zack Grossman, John Hartman, David Hinkley,
Marek Kapicka, Tee Kilenthong, Natalia Kovrijnykh, John Ledyard, Steve LeRoy, Gustavo Ponce, ChengZhong Qin, Warren Sanderson, Perry Shapiro, Joel Sobel, Glen Weyl, Eduardo Zambrano, and Giulio
Zanella for useful conversations. The usual disclaimer applies.
1
Introduction
Modern economic theory is founded on the assumption of rational individual behavior. In
the behavioral model of choice, consumer demand functions are derived from a rational
preference relationship and maximizing behavior. Demand functions, however, are not
exclusive to the behavioral model of choice. Gary Becker examined a statistical model of
choice that dispenses with any kind of preferences and maximizing behavior. In Becker [9],
individual consumption is randomly (i.e., “irrationally”) chosen subject to a linear budget
set. Yet, compensated price changes force individuals to, on average, satisfy the Law of
Demand. Becker’s [9] example, while an alternative to the behavioral model of choice, is
impractical and very restrictive: it relies on two commodities, random choices drawn from
a uniform distribution, and non-satiated demands.1 It is not obvious, for instance, that
one would retain any predictive power without these assumptions or that these predictions
would apply to random choice procedures of interest to behavioral scientists.2
This paper presents comparative statics predictions for a statistical model of choice
based on a general probability distribution function, an arbitrary number of commodities,
and for demands that lie in the interior or in the boundary of the budget set. I demonstrate
that market demands satisfy the compensated Law of Demand quite generally. Indeed, I
show that market demand functions satisfy all the properties of the behavioral model of
choice including symmetry and negative semidefiniteness of the Slutsky matrix. This last
property is the defining feature of the behavioral model. Market demands in the statistical
model of choice are therefore shown to be indistinguishable from those derived from a
behavioral model of choice.
1
Hildenbrand ([36], p. 34) remarked “I do not know of any successful alternative for modelling the
dependence of [demand] on [prices]. There is, of course, the well-known example of Becker.” He them
proceeds to point out some of the crucial limitations of Becker [9]; particularly, the use of uniform distributions. Varian ([63], p. 105) similarly remarked “there seem to be few alternative hypotheses other than
Becker’s that can be applied using the same sorts of data used for revealed preference analysis.”
2
The uniform distribution is special; a famous example is Caplin and Spulber [14] where monetary
neutrality follows entirely due to their use of uniform distributions. Integrability in two-commodities is
also special as it is easily resolved; see Katzner ([38], Theorem 4.1-2). Finally, nonsatiation, e.g., “more
is preferred to less,” is responsible for downward sloping indifference curves, for the direction of increasing utility in the indifference map, and for homogeneity and adding up conditions in behavioral models.
Demsetz ([20], p. 489) was indeed agnostic about Becker’s [9] findings: “as imaginative and informative as
[Becker’s] arguments are, they cannot be sustained in full without appealing to rationality.”
1
I illustrate the theory’s value-added using a number of applications. I place no restrictions on the choice probabilities beyond those implied by probability theory. Thus the
comparative statics derived here characterize, not just a number of “irrational” behaviors,
but any individual choice behavior representable by a continuous probability distribution
over the commodity space. Decision theorists have in effect formulated numerous realistic (i.e., descriptive) random choice models based on axiomatic rules or choice heuristics.
Naturally their focus is on individual choice behavior not on market behavior; see, e.g.,
Gonzalez-Vallejo [32], Luce [43], Roe et al. [54], and Tversky ([59], [60]). This paper finds
that these realistic individual choice behaviors share the market predictions of the standard
behavioral model of choice.3
The statistical model of choice has been primarily used to derive the statistical power
for revealed preference tests that rely on rationality indices; see, e.g., Varian [62] and
Echenique et al. [22].4 (Apesteguia and Ballester [5] and Dean and Martin [19] recently
derived additional rationality indices.) Quantifying statistical power requires an alternative
hypothesis in case the behavioral postulates are incorrect. The statistical model of choice
provides an alternative hypothesis to measure statistical power. Power calculations have
employed the uniform distribution, the logistic distribution, and random perturbations from
observed choices; see, e.g., Bronars [12], Choi et al. [17], and Andreoni et al. [4].5 This
paper shows that revealed preference tests are inconsistent and that their inconsistency is
not sensitive to the choice of distribution function. For instance, tests based on the amount
of money that could be pumped out of non-rational individuals have no power to detect
3
A recent literature in economics studies sophisticated random choices but also at the individual level
and not at the market level; see, e.g., Fudenberg et al. [28], Gul and Pesendorfer [34], and Manzini and
Mariotti [46]. An early exception that focuses on market behavior is Mossin’s [53] characterization of mean
demands in Luce’s [43] model of choice.
4
Tests of individual rationality abound in economics and they are likely to become more prevalent
as detailed individual purchase data becomes more accessible. Revealed preference tests have even been
applied to nonstandard populations such as animal subjects (Kagel et al. [37] and Chen et al. [16]),
psychiatric patients (Cox [18]), children (Harbaugh et al. [35]), and individuals under the influence of
alcohol (Burghart et al. [13]) to name a few.
5
Varian ([61], [62], and [63]) provide an authoritative overview of revealed preference theory. Bronars [12]
first applied Becker’s [9] model to derive statistical power. Choi et al. [17] considered boundedly rational
individuals that follow Luce’s [43] choice axioms. Andreoni et al. [4] considered numerous alternatives.
Beatty and Crawford [8] discuss a separate concern based on nonstatistical aspects of rejectability in
revealed preference tests. Essentially, if budget sets do not intersect, violations of rationality cannot be
detected even in deterministic settings. This concern will not be present here.
2
individual rationality.
Additional applications of statistical models of choice are very specialized. Grandmont
[33], in the spirit of Becker [9], derived through aggregation market demands of the CobbDouglas type; see also Hildenbrand [36] and Kneip [40]. These demands are consistent with
positive income effects and hence ensure equilibrium uniqueness in general equilibrium. I
characterize income effects here and show that positive income effects are not general
to the statistical model of choice. Gode and Sunder ([29], [30]), in another application,
used a double auction mechanism to document high allocative efficiency of markets under
statistical choices; see also Duffy [21]. This literature relies extensively but unnecessarily
on the uniform distribution used by Becker [9]. Market interactions turn out to serve
as a partial substitute for individual rationality in far more general settings than those
previously analyzed.
This paper also substantiates some of Becker’s [9] remarks about rational economic
behavior. Statistical choices, for example, make some of the criticisms of the behavioral assumptions of economics less forceful. This paper shows that the consistency of preferences
and maximizing behavior, often refuted by psychologists and experimental economists, are
unnecessary for demand theory as its main predictions can be derived using a completely
different route. The findings also attest to the independence between positive and normative assessments in economics. In the behavioral model, positive and normative assessments
are always intertwined leading to a number of difficulties in the presence of unstable preferences or inconsistent behavior. Choices in a statistical model have no normative content;
welfare can be measured (i.e., by the consumer’s surplus) but it has no significance or practical value. Finally, the findings demonstrate by example that the intensive understanding
of individual decision-making does not necessarily hold the key to the understanding of aggregate behavior. Statistical choices prompt all sorts of violations of individual rationality;
even in a general stochastic rationality sense. Aggregate behavior, however, is consistent
with the existence of a rational “representative agent.”
A statistical model of choice is more general than random preference models (i.e., models
in which individuals maximize randomly chosen preferences). To guarantee that choices can
be rationalized by a random preference model, observed choice probabilities must satisfy
3
the axiom of revealed stochastic preference; see McFadden and Richter [50].6 The choice
procedure here is entirely statistical so it could, in principle, dispense with any behavioral
assumption. In fact, random choices here are not necessarily stochastically rational.7
The paper unfolds as follows. Section 2 generalizes Becker’s [9] example; Section 3
presents the general theorems; Section 4 lists a few examples of random choice models;
Section 5 discusses the power of revealed preference tests; and Section 6 concludes.
2
A simple example
I first generalize the two-commodity example of Becker [9]. The two commodities are
denoted 1 and 2 . Prices  = (1  2 ) and income  are given and are assumed to be
strictly positive. There is a continuum of individuals and their choices must lie in a linear
budget set
©
ª
( ) ≡ (1  2 ) ∈ R2+ : 0 ≤ 1 1 + 2 2 ≤  .
(1)
The model of choice is statistical: 1 is a random variable with a well-behaved distribution function  (1 ) defined over 1 ∈ R+ ; the density function is (1 )  0 for all
1 ∈ R+ . For technical reasons (i.e., differentiability), I assume that all random variables
are absolutely continuous and that their first and second moments are finite. Analytical
problems concerning results that hold on sets of measure zero are neglected throughout
and all subsets are assumed non-empty and measurable.
Throughout the paper, I also maintain the requirement of consumer sovereignty: I
assume that  (1 ) is invariant with respect to ( ). This assumption is the analog of
assuming that tastes are invariant to the economic environment. As in the behavioral
6
The literature that studies random preference models places a special emphasis on econometric applications; see, e.g., Anderson et al. [3]. These models typically rely on an indirect utility function that
varies with prices and income. Individuals are subject to optimization errors in the form of a random
utility component. This paper is direct with respect to prices and income, and there is no need to assume
consistent preferences or maximizing behavior at any stage of the decision process. “Irrational” choices
tough admit a random utility representation; see Section 3.
7
McFadden and Richter [50] provides a definite treatment of revealed stochastic preference theory; see
also Bandyopadhyay et al. [7], Falmagne [25], Lewbel [42], and McFadden [49]. In the approach taken
here, preferences or maximizing behavior are not needed at any stage of the decision process. I also focus
on market demands and not on observed choice probabilities, which are motivated by the discreteness of
applied random preference models.
4
model of choice, in which preferences are given and independent of the budget set, the
distribution function  (1 ) is given and does not vary with ( ). Obviously, observed
choices will depend on ( ) because the budget constraint limits choices.
Indeed, not all statistical choices are feasible. Due to (1), the maximum quantity of 1
that can be bought is X max
≡ 1 . Choices of 1 are thus subject to 0 ≤ 1 ≤ X max
1
1 .
Mean demand for 1 is given by a right-truncated mean formula:
̄1 ( ) ≡ E [1 | 0 ≤ 1 ≤
X max
1 ]
=
Z
 max
1
1
0
(1 )
1 ,
 (X max
1 )
(2)
max
where  (X max
1 ) ≡ Pr{1 : 0 ≤ 1 ≤ X 1 }.
As in Becker [9], once 1 is chosen, the remaining income is spent in 2 , i.e., 2 satisfies
2 2 =  − 1 1 as long as 0 ≤ 1 ≤ X max
1 . Its mean demand is
̄2 ( ) =
 − ̄1 ( )1
.
2
(3)
A random choice procedure consistent with interior demands is discussed later on.
The “Law of Demand.” It is obvious but neither a Hicksian compensation nor standard duality results are possible here because demands do not rely on utility functions.
The analysis is carried out in terms of Slutsky compensated demands.
I first show that the compensated Law of Demand holds,
¯
 ̄1 ( )¯¯
 0,
1 ¯
(4)
where ·| denotes the standard Slutsky compensation,  = ̄1 ( )1 .8
The relevant derivative needed to evaluate (4) is:
8
¯
¯ ¶
µ
¯
 ̄1 ( )¯¯
X max
E [1 | 0 ≤ 1 ≤ X max
1 ]
1 ¯
.
=
max
¯
1
X 1
1 ¯

(5)
When 1 changes to 1 +1 mean demand changes to ̄1 (1 +1  2   +), where  = ̄1 ( )1
is the Slutsky compensation. Every individual is a set of measure zero so individual demands are not
well defined. Income compensations, however, can be thought out as given to every feasible realization of
demand, i.e., the truncated mean of 1 1 is ̄1 ( )1 . This means that individuals are not compensated
so that everyone can afford mean demands but so that individuals can afford their original bundle.
5
The first term represents the effect of changes in X max
on mean demands and the second
1
the effect of a compensated change in 1 on X max
1 . This first term satisfies
 (X max
E [1 | 0 ≤ 1 ≤ X max
1 ]
1 )
=
− ̄1 ( ))  0,
(X max
1
max
X 1
 (X max
)
1
(6)
by a standard application of Leibniz’s rule for differentiation under the integral sign in (2).
The second term in (5) satisfies
¯
¯
X max
− ̄1 ( )
X max
1 ¯
1
=
−
 0.
1 ¯
1
(7)
Expressions (5), (6) and (7) yield the following proposition:
Proposition 1 Assume that individuals randomly choose 1 and determine 2 as a residual from (1). For any well-behaved continuous probability distribution function  (1 ), the
compensated Law of Demand holds for ̄1 ( ) and ̄2 ( ).
The (compensated) Law of Demand holds due to a statistical fact and an economic fact.
First, the truncated mean ̄1 ( ) is less than the truncation point X max
1 , and its value
increases as the truncation point increases; see (6). Second, even after taking into account
the income compensation, the maximum amount of 1 that can be purchased declines as
1 increases; see (7).
Figure 1 illustrates Proposition 1. As the figure shows, some values of 1 are not feasible
max
so (1 ) must be truncated at X max
1 ; that is, the dark-shaded area to the right of X 1
is not feasible at ( ). When 1 increases, and income is compensated, the truncation
max 00
00
point changes to X max
 X max
is no longer
1
1 ; the light-shaded area to the right of X 1
feasible. A compensated increase in 1 thus reduces X max
and the mean demand declines
1
from ̄1 ≡ ̄1 (1  2  ) to ̄001 ≡ ̄1 (01  2  0 ).9
9
The only generalizations of Becker [9] I have been able to find are in Sanderson ([55], [56]). These
papers noted that an increase in 1 expands the choice set in a first-order stochastic sense. Sanderson
([55], [56]), however, restrict the distribution of consumption opportunities in ways that are not actually
needed. Since the relevant choice distribution is  (1 ) (X max
), the orderings assumed in Sanderson
1
0
([55], [56]) are always obtained here:  (1 ) (X max
)
stochastically
dominates  (1 ) (X max
) in the
1
1
max 0
first degree sense in [0 X 1 ].
6
2
 2 6
0
e
e
e
2
e
Z
eZ e
e
e Z
e
r
002 e
|
Z 
Z
e
e
|
 e
e Z
Z e

e  Z
|
(1 )
er
Z
e
2 

Z
e
 e
 Z |

 e
 eZ

|

 e|Z
 e 

e
| ZZ |

e 
e


Z
e


 e | e |Z

Z

e

 e


Z
e|
e|Z
001
1
0
Xmax
1
00
Xmax
1
-
1
Xmax
1
Figure 1: Mean demand for 1 , ̄1 ( ), drawn from (1 ). Initial choices are truncated by
0
Xmax
= 1 . For uncompensated changes, the truncation point is Xmax
. For compensated
1
1
max 00
changes, the truncation point is X1 .
Some remarks. (i) Proposition 1 does not depend on particular assumptions about
 (1 ). The uniform distribution used by Becker [9] yields
1
̄1 ( ) =
(1 )
Z
0
1
1 1 =
1
,
2 1
(8)
and ̄2 ( ) = (12)(2 ); these mean demands are exactly the same as the ones derived
by Becker [9]. These demands also coincide with those of a rational “representative agent”
who maximizes a Cobb-Douglas utility function. (Theorem 2 below provides integrability
results that generalize this case.)
(ii) Uncompensated price effects satisfy the standard Slutsky equation,
¯
¶
µ
 ̄1 ( )
 ̄1 ( )¯¯
 ̄1 ( )
,
=
− ̄1 ( )
1
1 ¯

7
(9)
with income effects given by
 ̄1 ( )
E [1 | 0 ≤ 1 ≤ X max
1 ] 1
=
 0.
max

X 1
1
(10)
As Figure 1 suggests, 1 is on average a normal good. Income effects, however, are not
necessarily positive for ̄2 ( ), or in general. A sufficient condition for ̄2 ( ) to be a
normal good is that  (1 ) is log-concave; see Appendix A. This condition is implicit in
Becker [9] since the uniform distribution is log-concave.
(iii) Proposition 1 examines mean demands but other aggregators also satisfy the Law
−1
of Demand. Particularly, 1 ’s median demand med
( (X max
1 )2) satisfies
1 ( ) ≡ 
med
1  (X max
1 ( )
1 )
=
 0,
max
med
X 1
2  (1 ( ))
(11)
which, as in (5), is enough to verify the Law of Demand. The Law of Demand also holds
for the mid-range aggregator since this statistic treats  (1 ) as a uniform distribution.
Similar results cannot be established for the mode, but the mode is not meaningful here.
(The mode does not generally depend on X max
1 .)
(iv) When 1 increases, the fraction of individual demands  (̄1 ( )) −  (0) agrees
00
with the compensated Law of Demand but the fraction  (X max
) −  (̄1 ( )) violates
1
it, as demands increase for these individuals; see Figure 1. These violations have been used
to measure the statistical power of revealed preference tests and the economic significance
of “irrational” behavior. I discuss this application in detail in Section 5.
(v) I focused on 1 but the Law of Demand also holds, on average, for 2 ; see Appendix A. Indeed, (2) and (3) satisfy all the properties of the behavioral model of choice.
Homogeneity of degree zero in ( ) follows because X max
is a defined in real terms; see
1
(2). By construction, ̄2 ( ) is also homogeneous and adding up conditions are satisfied;
see (3). Mean demands are symmetric with respect to cross-price changes so they can
be rationalized, e.g., integrated. Integrability, however, is expected because this example
considers two commodities and non-interior demands; see, e.g., Mas-Colell et al. ([47], p.
36 and Exercise 2.F.15) and Katzner ([38], Theorem 4.1-2).
8
3
A general framework
This section deals with generalizations of the previous example. Further generalizations
that consider several new dual (i.e., “irrational”) representations of demands in the behavioral model of choice, a derivation of an aggregate supply curve under random choice
behavior, and numerical simulations that examine how ‘large’ the economy needs to be in
order to observe consistent results are discussed in an Appendix not for publication.
Statistical choices will now take place over   1 commodities using a general probability distribution function. Mean demands will also be allowed in the interior of the budget
set.
Let  ≡ (1       ) denote the commodity vector and let  ≡ (1       ) be the
corresponding price vector. The budget set generalizes (1),
©
ª
( ) ≡  ∈ R+ : 0 ≤  ·  ≤  .
(12)
The budget constraint (12) may hold as an inequality depending on whether one or
more commodities are randomly chosen or determined as a residual. I will separately study
the case when the “last” commodity  is chosen randomly and when it is determined as
a residual.
As before, all random variables are absolutely continuous with finite first and second
moments. Statistical choices are derived from a well-behaved probability distribution function  (1       ) ≡ Pr(1 ≤ 1       ≤  ) with density (1       ). The support
of these functions is given by the commodity space R+ . (This is also the support for the
behavioral model of choice; see Mas-Colell et al. ([47], p. 18).)
To ensure that (12) is satisfied for every realization of ,  (1       ) must be truncated according to
 (X
max
)=
Z
0
 max
1

Z
 max
 (1 −1 )
 (1       ) 1     ,
(13)
0
max
where X max ≡ (X max
1      X  (1      −1 )) is the vector of maximum feasible consump-
max
tions and  (X max ) ≡  (X max
1   X  (1      −1 )). This truncation takes place over
9
the -dimensional budget hyperplane associated with (12). In particular, the vector X max
satisfies  X max
=  for  = 1 and

(1      −1 ) = −1 [X max
 X max

−1 (1   −2 ) − −1 ] , for  = 2     .
(14)
(1      −1 ) is a subset of −1 X max
This vector is ordered:  X max

−1 (1   −2 ) because a
positive consumption for commodities    limits the maximum amount that can be consumed of commodity , i.e., if  = X max
(1      −1 ) then X max

 (1      −1 ) = 0 for
all  = +1     . For instance, the “last” commodity  satisfies  X max
 (1      −1 ) =
P−1
 − =1
  .
Mean demands. Mean demand ̄ ( ) ≡ E( |0 ≤  ·  ≤ ) satisfies
̄ ( ) =
Z
0
 max
1

Z
 max
 (1 −1 )

0
(1       )
1     .
 (X max )
(15)
The vector of mean demands is denoted by ̄( ) ≡ (̄1 ( )     ̄ ( )).
Notice that mean demands are homogeneous of degree zero in ( ). Homogeneity is
easy to verify because ( ) influence ̄( ) only through X max and a proportional change
in  and  leave X max unchanged; see (14). Moreover, if commodity  is residual, then
 = X max
 (1      −1 ) for any realization of 1      −1 . In this case, the relevant
distribution function is  (1      X max
 (1      −1 )) and individual and mean demands
add up to income. Notice also that the order of integration is in general irrelevant. The
“last” commodity is special, but only when it is residually determined.10
I characterize mean demands next. All derivations are in Appendix A; they are not
difficult but tedious as they rely on repeated differentiations under the integral sign.
Let  ≡ (1       ) and let () denote the vector  when  is excluded, i.e., () ≡
(1      −1 ). Likewise, let  ≡ (1       ) be the vector of differential changes and
let () denote this vector when  is excluded, i.e., () ≡ (1      −1 ). Finally,
let max (() ) denote the density () when X max
 (() ) takes the place of  . That is,
10
Under interior demands, for example, the order of integration in (13) and (15) can be changed without
consequence due to Fubini’s Theorem; see, e.g., Fikhtengol’ts ([26], Vol. II, Section 344). (This is generally
true for the first  − 1 goods.) This change would be somewhat analogous to changing the order in which
first order conditions are obtained in the behavioral model.
10
max (() ) ≡  (()  X max
 (() )), which is only a function of () . This density function
appears repeatedly in the comparative statics in this section.
(a) Own-price effects. As derived in Appendix A,
¯
Z  max
Z  max
2
1
−1 ((−1) ) [ − ̄ ( )]  max (
 ̄ ( )¯¯
() )



=
−



max () .
¯


 (X )
0
0

(16)
A compensated own-price change is proportional to the negative of the conditional
variance of  . Slightly abusing notation in the definition of variance, i.e., ignoring the
conditioning terms and proportionality factors, let  [ ] ≡ E[( − ̄ ( ))2 |() ≤
max
X max
()   = X  ]. Then (16) can be written as
¯
 ̄ ( )¯¯
= − [ ].
 ¯
(17)
Thus, for any well-behaved probability distribution function  (), a compensated increase
in  reduces ̄ ( ).11
Figure 2 illustrates the own-price effect for ̄1 ( ) when  = 2 and demands are
0
interior. A compensated change in 1 removes the triangular area ( X max
 X max
1
1 ) and
max 0
adds the area ( X max
(0)) to the budget set. This ‘pivoting’ of the budget set
2 (0) X 2
max
is integrated along X max
2 (1 ) using  (1  X 2 (1 )). (In general, changes are integrated
max
along X max
 (() ) using  (() ).) As in Section 2, the compensated Law of Demand holds
because the area that is removed once prices increase favors high values of 1 whereas the
area that is added favors low values.
(b) Income effects. Income changes satisfy
 ̄ ( )
=

Z
0
 max
1

Z
 max
−1 ((−1) )
0
[ − ̄ ( )] max (() )
() .

 (X max )
(18)
To determine the sign of (18) one needs to make additional assumptions on the distribution function  (). In particular, the sign of the previous expression depends on the
11
Mean demand ̄ ( ) when  is not residual needs a minor adaptation; see (A5) in Appendix A.
11
2
6
00
Xmax
(0)
2
e
e
e
max
X2 (0) Q e
Q e
Q
Qe
Qe 
00  r
2
r
 Qe
Q

eQQ

e rQ
2 

e
 QQ

e

Q
 e
Q

Q

e
Q

Q

e


Q
e
Q
e

Q

001
00
Xmax
1
1
-
1
Xmax
1
Figure 2: The Law of Demand for interior demands and  = 2. (The density (1  2 )
is not represented.) To determine own-price changes, (1  Xmax
2 (1 )) must be integrated
max
along the “thick” boundary X2 (1 ).
association between  and  . One needs to sign
Z
 max
1

0
Z
 max
−1 ((−1) )
0
 max (() )()
Z
≷ ̄ ( )
0
 max
1

Z
 max
−1 ((−1) )
max (() )() .
0
This expression compares ̄ ( ), the mean value of  , to E[ |() ≤ X max
()   =
X max
 ], which is the mean value of  when  takes its highest possible value. “Positive
association” in the likelihood ratio sense implies that a high value of  likely yields a high
value for  .12 Thus, if  and  are positively associated, income effects in (18) will be
positive; if  and  are negatively associated, income effects in (18) will be negative.
Figures 3 and 4 illustrate this dependence for  = 2 and interior demands. Figure 3 considers positively associated variables. At income , mean demands (̄1  ̄2 ) are determined
12
Formally,  (1  2 ) is said to be positively likelihood ratio dependent if  (01  02 ) (1  2 ) ≥
for 01  1 and 02  2 . Thus it is more likely to observe that 1 and 2 take
larger values together and smaller values together than any mixture of these; see, e.g., Figure 3. The
multivariate analog requires  (1       ) to satisfy a positive likelihood ratio dependence for every pair
(   ) when the  − 2 remaining variables are fixed; see Shaked and Shanthikumar [57]. The relationship
with log-concavity is also discussed by these authors.
 (01  2 ) (1  02 ),
12
2
0
Xmax
(0) 6
2
e
e
e
e
max
X2 (0)
e
e
e
e
e
(1  2 ) = 
e
e
e
e ¡
e  ¡
ª
e
e
e





e
e


e
e
0 r
2
 e
e
e

e
e
2 r 
e

e
 
e
 
e
e
 
e
e
e
1 01
Xmax
1
-
1
0
Xmax
1
Figure 3: Income effects for 1 are positive when 1 and 2 are “positively associated.”
(The density (1  2 ) is represented by its contour.)
excluding the shaded area. At 0  , the shaded area becomes relevant to determine
demands. Since this area favors high values of 1 , its mean demand increases to ̄01 . Negative income effects are depicted in Figure 4. If 1 and 2 are negatively associated, the
new (shaded) area added as income increases favors low values of 1 and this lowers mean
values.
(c) The Slutsky matrix. Let S( ) denote the Slutsky matrix of substitution effects.
I next show that the Slutsky matrix is symmetric and negative semidefinite.
A compensated change in  on commodity  yields
¯
Z  max
Z  max
1
−1 ((−1) ) [ − ̄ ( )][ − ̄ ( )]  max (
 ̄ ( )¯¯
() )





=
−



max () ,
¯


 (X )
0
0

(19)
which can be written as the -th entry of a well-behaved variance-covariance matrix.
Several of the properties of the Slutsky matrix will follow from this representation.
More specifically, let Σ be the variance-covariance matrix of (()  X max
 (() )) about
̄( ). A typical element of Σ is of the form Σ = [   ] ≡ E[( − ̄ ( ))( −
13
2
0
Xmax
(0) 6
2
e
e
e
e
max
X2 (0)
e
e  e








e e (1  2 ) = 
e e¡
ª
e ¡
 e
0 e





2
r e ee
e 
e
2 r e
e
 
e
e
 
e
e
 
e
 
e
e
 
e
 
e
e
 
e
e
e
01
1
Xmax
1
-
1
0
Xmax
1
Figure 4: Income effects for 1 are negative when 1 and 2 are “negatively associated.”
(The density (1  2 ) is represented by its contour.)
max
̄ ( ))|() ≤ X max
()   = X  ], which is equivalent to (19) up to a proportionality
factor,
¯
 ̄ ( )¯¯
S ( ) ≡
= −[   ],
 ¯
(20)
for   = 1     − 1, and with  = X max
 (() ) when   = ; see Appendix A.
The Slutsky matrix can be written as a matrix of second order moments, S( ) = −Σ.
Since the variance-covariance matrix Σ is symmetric and positive semidefinite; see, e.g.,
Fisz ([27], pp. 89-90), the Slutsky matrix S( ) is symmetric and negative semidefinite.
Moreover, if  is randomly determined and not selected as a residual, then () −̄() ( )
would be linearly independent from X max
 (() ) − ̄ ( ) making all terms in Σ linearly
independent. In this case, Σ would be symmetric and positive definite; see, e.g., Fisz
([27], Theorem 3.6.6). Thus, when demands are interior, ( ) will be symmetric and
negative definite instead of just semidefinite. The distinction between positive definiteness
and positive semidefiniteness in S( ) is not a curiosity; it will be important for demand
integrability.
Theorem 1 summarizes the key implications I have presented so far:
14
Theorem 1
(a) Assume that individuals randomly choose  subject to (12). For any well-behaved
continuous probability distribution function  (), mean demands ̄( ) are: linearly
homogeneous in ( ); interior,  · ̄( )  ; and their Slutsky matrix S( ) is
symmetric and negative definite.
(b) Assume that individuals randomly choose () subject to (12) and determine the
“last” commodity  as a residual. For any well-behaved continuous probability distribution function  (), mean demands ̄( ) are: linearly homogeneous in ( );
add up to income,  · ̄( ) = ; and their Slutsky matrix S( ) is symmetric and
negative semidefinite.
Economically, the difference between (a) and (b) in Theorem 1 is that if the “last” commodity is a residual, (individual and mean) demands would add up to income so  would
be redundant. This redundancy of the “last” commodity is of the usual kind: knowledge
of () and (12) is enough to determine individual and mean demands. In the behavioral
model, homogeneity and adding up also imply that the negative semidefiniteness of the Slutsky matrix cannot be extended to negative definiteness; see, e.g., Mas-Colell et al. ([47],
Proposition 2.F.3). In the statistical model, the linear dependence implicit when the “last”
commodity is a residual makes  a linear combination of () , i.e.,  = X max
 (() ).
This gives rise to positive definite but singular variance-covariance matrix; see, e.g., Fisz
([27], Theorem 3.6.6).
The intuition for the symmetry and negative semidefiniteness of the Slutsky matrix is
transparent in (20). This transparency should not detract from the fact that symmetry
and negative semidefiniteness are striking findings. For once, negative semidefiniteness
establishes the compensated Law of Demand under general conditions including interior
demands; see Mas-Colell et al. ([47], pp. 34-35) for uses of S( ) in the behavioral model
of choice. More importantly, symmetry and negative semidefiniteness in the Slutsky matrix
S( ) are exhaustive properties of the behavioral model of choice; see, e.g., Mas-Colell
et al. ([47], pp. 75-76). Mean demands ̄( ) can therefore be rationalized as being the
result of the maximization of some utility function:
15
Theorem 2
(a) Under the conditions of Theorem 1(a), there exists some continuous, non-decreasing,
→ R such that ̄( ) and ̄0 ( ) ≡
and quasi-concave utility function  : R+1
+
 −  · ̄( ) are the unique solution to max0 ≥0 {( 0 ) :  ·  + 0 = }.
(b) Under the conditions of Theorem 1(b), there exist some continuous, non-decreasing,
and quasi-concave utility function  : R+ → R such that ̄( ) is the unique solution
to max≥0 {() :  ·  = }.
Theorem 2 shows that mean demands can be rationalized even when they are interior.
To do so, Theorem 2(a) treats ̄( ) as an incomplete demand system. Epstein [24]
characterized integrability in incomplete systems; see also LaFrance and Hanemann [41].
Incomplete systems are more economically and mathematically complex than complete
systems because one must make economic assumptions about how to complete the system.
(Theorem 2(b) relies on a standard complete demand system; see Katzner ([38], Chap. 4).)
Suppose for illustration that incompleteness is due to unobserved commodities. In this
case, one has to limit the way observed prices of unobserved commodities influence observed
demands. A “residual” commodity 0 with 0 = 1 is the simplest way to complete the
system. (Some remarks about the interpretation of 0 in the present context are provided
below.) Incomplete demands also require a symmetric and negative definite Slutsky matrix.
This mathematical requirement cannot be weakened for semidefiniteness; see Epstein ([24],
Example 1).
Some remarks. (i) Random choices clearly violate individual rationality. On average,
however, behavior is consistent with a rational “representative agent.” The utility function
that rationalizes ̄( ) even has a dual random representation. (An Appendix not for
publication elaborates on this kind of duality principle. In particular, I show that any
consumer demand function in the behavioral model of choice can be represented as the
outcome of random choice behavior.) In case (b) of the previous theorems, for example,
E[|0 ≤  ·  = ] = argmax {() : 0 ≤  ·  = )}.
∈R
+
16
(21)
The utility function () and its associated welfare measures (i.e., the consumer’s surplus)
have no content in the statistical model of choice. The process by which choices are revealed,
even the choices themselves, provide no basis for normative statements. The difficulty
with normative assessments is not the standard problem that the welfare function of the
“representative agent” is in conflict with the preferences of the disaggregated individuals,
as in the traditional treatment of the normative “representative agent” discussed, e.g.,
in Mas-Colell et al. ([47], Section 4D). The issue here is more fundamental. Theorem
2 demonstrates that well-behaved preferences would emerge even when individual choices
have no preferential basis and there is no maximizing behavior. As the dual representation
(21) shows, well-behaved preferences can be recovered from purely random choice data.13 As
Mas-Colell et al. ([47], pp. 121-122) note, however, “[t]he moral of all this is clear: The
existence of preferences that explain behavior is not enough to attach them any welfare
significance. For the latter, it is also necessary that these preferences exist for the right
reasons.”
(ii) Positive statements about price and income effects have empirical content. They are,
however, unable to discriminate between the statistical and the behavioral model of choice.
At the market level, both models are equally consistent with, for example, a symmetric
and negative semidefinite Slutsky matrix. Even at the individual level it will be difficult to
discriminate between a statistical model of choice and a behavioral model. The behavioral
model of choice is often assessed by applying revealed preference tests to individual choice
data. To make these tests operational, the literature typically adds “noise” to individual
demands. The behavioral model of choice is deterministic so ‘small’ violations of behavioral
assumptions are traditionally attributed to a stochastic component in the decision process
(i.e., mental errors) or to measurement error in prices or individual choices. In statistical
tests of the behavioral assumptions, individual demands are assumed to have a “noise”
component; see, e.g., Choi et al. [17], Echenique et al. [22], Lewbel [42], and Varian [62].
In the statistical model of choice, individual demands do have a “rational” and a “noise”
component. Deviations from “rational” behavior are in fact bounded. Let   denote  =
13
This point was discussed by Becker [9] in the context of an ‘as if’ justification for rationality and
irrationality in economics. Blundell et al. ([11], p. 211) also mentioned this point in passing in their
analysis of demand integrability.
17
1      independent sample realizations of an individual’s vector of demands. The sample
P

average for the individual demand for commodity  is defined by ̃ ≡  −1 
=1  .
Corollary 1 Individual demands in the statistical model of choice have a “rational-plusnoise” representation
  = ̄( ) +   ,
(22)
with   ≡   − ̄( ) and E[  ] = 0. Moreover, given   0 and for  = 1  , individual
demands satisfy
¯
o  [ ]
n¯
¯ 
¯

Pr ¯̃ − ̄ ( )¯   
.
2

(23)
(iii) Individual choices can also be represented by “observed” choice probabilities, in
the sense of McFadden and Richter [50]. Let  denote a subset of ( ). The probability
that choices lie in  is
1
(|( )) =
 (X max )
Z
 ().
(24)

“Observed” choice probabilities are the starting point of stochastic revealed preference
theory. Stochastic revealed preference theory seeks to rationalize observed choice probabilities using a random preference model (see, e.g., footnote 7). Individual choices here
are not based on random preferences or maximizing behavior. “Observed” choices, for
instance, are not required to satisfy any form of stochastic transitivity.14
Hence, the
current statistical model of choice is “irrational” not just in the standard deterministic
sense, but also in the more general stochastic sense. Still, Theorem 2 and Corollary 1
imply that individual choices in the statistical model admit a random utility representation
14
There are several notions of stochastic transitivity. Consider  ,  , and  under the same feasibility
conditions and as binary choices, i.e., a choice between one unit of commodity  and one unit of  .
The proportion of individuals for whom chance selects alternative  out of alternatives  and  is   ≡
( |{   }) = Pr{   };  and  are analogously defined. If   ≥ 12 and   ≥ 12,
choice probabilities satisfy: weak stochastic transitivity if  ≥ 12, moderate stochastic transitivity if
  ≥ min{    }, and strong stochastic transitivity if   ≥ max{    }; see, e.g., Tversky [59].
Binary choice probabilities can be ordered following basic probability theory; see Appendix A. For instance,
  ≥   and   ≥   if and only if Pr{     } ≥ Pr{     } and Pr{   
 } ≥ Pr{     }, respectively. Violations of stochastic transitivity can only be ruled out by
restricting these probabilities. There is no simple interpretation for these restrictions but none of these
conditions is needed in Theorems 1 or 2.
18
(  ) = (̄( )) +   , with   ≡ (̄( ) +   ) − (̄( )) and   ≡   − ̄( ).
Purely random choice data can therefore be behaviorally represented as a random utility
model in which an individual’s utility (  ) results from random perturbations   to a
well-behaved deterministic indirect utility function (̄( )).
(iv) Finally, the “residual” commodity 0 needed for integrability in Theorem 2(a) is
not too restrictive. If the number of commodities  is large, the “residual” commodity
0 ≡  −  ·  will become insignificant. (Trivially, by Markov’s inequality, Pr{0  } 
̄0 ( ), so lim→∞ Pr{0  } = 0.) Moreover, since 0 = 1, 0 could be interpreted
as an individual’s cash balance; it could also be interpreted as wasted money or as savings
in an intertemporal context. Tests of the behavioral assumptions in economics effectively
analyze incomplete demand systems since they are typically carried out in terms of total
expenditure and not in terms of income. This means that existing tests could not assign
any particular meaning to 0 . As trivial as it might sound, I’m not aware of empirical
assessments of non-satiation. Non-satiation is not actually tested as part of conventional
revealed preference tests. Empirical analyses typically assume or impose non-satiation to
avoid trivial rationalizations of the data; see, e.g., Varian ([61], p. 969).
4
Applications to random choice behaviors
The only restrictions needed in Theorems 1 and 2 are those implied by probability theory.
One can therefore showcase the applicability of these theorems by considering statistical
models of choice that restrict choice probabilities. This section lists a few random choice
models formulated for individual decision-making and provides some remarks about the
applicability of the previous theorems.
Preferential choices. I first consider some classical preferential choice models in which
 () is derived from behavioral postulates or choice heuristics. I make the dependence on
the choice set explicit and use continuous probability distributions.15
(a) Luce’s choice axiom and Tversky’s elimination model. Let  (|) be the probabil15
Continuous choice models can be derived as the infinitesimal limit of discrete choice models; see, e.g.,
Ben-Akiva and Watanatada ([10], p. 327) and McFadden ([48], pp. 311-312).
19
ity that an alternative from  is chosen when the set of available alternatives is . In Luce
[43], choice probabilities satisfy the independence from irrelevant alternatives: for  ⊆ 
and  0 ⊆  such that  ⊆  0 ,  (|) =  (| 0 ) ( 0 |). This assumption yields a
continuous logit formulation
Z
exp{()}
 (|) = Z
,
(25)
exp{()}

where () is a direct utility function associated with reflexive, transitive, and complete
preferences.16 The parameter  accommodates extremes in the rationality spectrum; the
larger  is, the greater the degree of “irrationality.” The limit of  (|) as  → ∞ is the
uniform distribution used by Becker [9], and when  → 0, the choice problem becomes a
deterministic utility maximization problem; see Anderson et al. ([3], p. 42).
Tversky [60] considered a more general choice process in which alternatives are sequentially eliminated. Individuals first choose a subset  ⊆  with probability ∫ ( 0 )0
and then choose alternatives within this subset, e.g., choices are of the form  (|) =
R
 (|0 )( 0 )0 , where  (|0 ) represents the probability of selecting  given 0 ,

and ( 0 ) is a weighting scheme that determines the probability of choosing 0 in ,
with ∫ ( 0 )0 = 1. Luce [43], for instance, is a special case in Tversky [60]. Moreover, Luce [43] and Tversky [60] can be represented as random preference models; see, e.g.,
Anderson et al. ([3], Chap. 2).
(b) Choice control models. Even more general probabilistic choice behaviors can be
seen as outcomes of individual randomization that maximize a utility function that faces a
cost of implementing the choice, i.e., a desire for randomization. This probabilistic choice
structure has been studied by Machina [44] and more recently by Fudenberg et al. [28].
Let () denote a choice function that specifies the probability of choosing a commodity
bundle  ∈  ⊆ R+ . Let () be a direct utility function and, as in Fudenberg et al.
([28], p. 2371), let (()) denote “a convex perturbation function that may reward the
16
Random preference models often assume that there is an indirect utility function that depends on prices
an income, and that individuals randomly choose among these indirect utilities. The utility functions used
in this section are all direct so they are independent of the economic environment.
20
agent for randomizing.” Individual choices, which can be seen as randomly selecting among
deterministic bundles ranked according to (), satisfy
∗
 (|) ≡ argmax
()≥0
½Z

[()() − (())] :
Z

¾
() = 1 ,
(26)
where the feasibility restriction is implied by the probabilistic nature of choice. The presence
of a convex cost (()) means that individuals may find random choice preferable to
R
deterministic choices. In (26), for  ⊆ , choice probabilities  (|) =  ∗ (|) are
behaviorally constructed.
(c) Difference models. Gonzalez-Vallejo [32] and Roe et al. [54] are examples of multialternative preferential choice models consistent with violations of stochastic rationality. These models are also more general than Luce [43] and Tversky [60]. (Tversky [59]
also used proportional difference models to study stochastic intransitivity.) For instance,
individual choices in these models are based on pure preferences, inattention, informational updates, and a comparative evaluation of the different alternatives. In a simplified version of Roe et al. [54], for example, individual choices satisfy  (|) = Φ( :
{() · () ≥ max[() · () :  ∈ ]}), where () is an attention weight for the at-
tributes  = (1       ) and () =  1 (1 ) +    +  ( ) is a pure preference component
perturbed by a stochastic error; see, e.g., Roe et al. ([54], Eq. 1b). In Gonzalez-Vallejo
[32], individual choices also have a preferential basis, but the comparative evaluation is
based on proportional differences between alternative attributes. Difference models are
not based on primitive axioms. Behavioral scientists postulate these models as descriptive
representations of actual choices. These models, for instance, are consistent with violations
of stochastic regularity and stochastic transitivity along the lines discussed by remark (iii)
in Section 3.
(d) Satisficing. Consumption alternatives in the previous examples are evaluated based
on maximizing behavior. Simon [58] advocated satisficing as a comparative principle of
individual choice. In Simon ([58], p. 252), a bundle  is satisfactory provided that () ≥ ,
where  is a given aspiration level. In a random choice formulation, individuals would be
satisfied with a bundle whose utility value exceeds , e.g.,  (|) = Φ( : {() ≥  ∈ }).
21
“Irrational” choices. The previous examples have a preferential basis. Individual
choices can also be based on ‘pure’ statistical models of choice in which  () lacks behavioral foundations. These “irrational” choices typically arise when preferential choices have
limited value. Individuals sometimes act impulsively, instinctively, and/or emotionally due
to limited decision time or attention.17 Preferences may also be incomplete or individuals
might find desirable to follow random choices out of fairness or due to strategic considerations; see, e.g., Moore [52] for a classical application in which individual choices require an
“impersonal and relatively uncontrolled process.” Subtle forms of random choice behavior,
some still common today (i.e., organized religion, astrology, superstition, divination, etc.),
appear to resolve situations that exhibit indecisiveness; see, e.g., Elster [23] for a discussion
of random choice in the context of limits of rational behavior.18 As noted by Agranov and
Ortoleva ([1], p. 2), random choice might also arise from “preferences originating from the
desire to reduce regret, incomplete preferences, difficulty to judge one’s true risk aversion,
or other forms of Non-Expected Utility.” Indeed, Agranov and Ortoleva [1] found, in an
experimental setting, that random choice is deliberate especially when subjects are dealing
with ‘hard’ (i.e., complex) questions.
Some remarks. (i) The previous list of examples is obviously non-exhaustive. These
examples, however, are more general than Becker [9]; they, for instance, rely on descriptive
assumptions that explicitly take into account inconsistent preferences, optimization errors,
and bounded rationality. In conjunction with Theorem 1, these examples show how broad
the domain of market demand theory can be and how unnecessary it is to rely on the
‘pure’ behavioral model of choice (i.e., on individual rationality) or the ‘pure’ statistical
(i.e., “irrational”) model of choice to derive its main testable predictions.
17
These instances motivated Becker [9] and an empirical demand literature on nonhuman or nonstandard
populations (see footnote 4). McGrath ([51], p. 433), for example, observed male shoppers in gift shops
on Christmas Eve tended “to make large, rapid, spontaneous, and often random purchases.” Becker [9]
considered “impulsive good deciders” and Chant [15] “impulsive money deciders” but the difference is
irrelevant here since individuals are price-takers.
18
As noted by Aumann ([6], p. 446): “[c]ertain decisions that our individual is asked to make might
involve highly hypothetical situations, which he will never face in real life; he might feel that he cannot reach
an ‘honest’ decision in such cases. Other decision problems might be extremely complex, too complex for
intuitive ‘insight,’ and our individual might prefer to make no decision at all in these problems.” Indifference
and indecisiveness though generally have different testable implications in the behavioral model; see, e.g.,
Mandler [45] and the subsequent literature.
22
(ii) In conjunction with Theorem 2, these examples also show that a rational “representative agent” carries through for a large class of individual behaviors. The emergence
of rationality is useful to study markets in which individuals systematically depart from
conventional behavioral assumptions. Gode and Sunder [29], for instance, documented high
allocative efficiency in a market mechanism (i.e., a double auction) with “zero-intelligence”
(ZI) traders characterized by uniformly distributed bids and asks. Gode and Sunder ([29],
p. 121), however, explicitly acknowledge that “[t]hese ZI traders are not intended as descriptive models of individual behavior.” The preferential choice examples listed above are
descriptive models of individual behavior. The allocative efficiency of markets in which
these descriptive behaviors are present has not been systematically studied. Market interactions, however, should also serve as a partial substitute for individual rationality under
more realistic individual choice behaviors.
(iii) The preferential choice examples listed above recognize that individual choices are
context-dependent. The choice set , for instance, is an argument of market demands,
as in ̄( |). The fact that individual choices depend on the choice context is not
inconsistent with demand theory or with the existence of a rational “representative agent.”
Market demands have been characterized holding other things constant and assuming that
the budget set constrains choice, i.e., that ( ) ⊂ . The analysis of market demands
derived from  (|) complements psychological and experimental research that formalizes
how individual choices depend on the choice context. This paper does not study the
relationship between market demands and the set of alternatives  or between the utility
function of the “representative agent” (|), the choice set , and the primitive individual
utility function (). These questions are beyond the scope of this paper.
5
Statistical power
As corollaries to Theorem 1, this section discusses the statistical power and economic
significance of tests of the weak axiom of revealed preferences, the most famous testable
prediction of the behavioral model of choice. A bundle  is weakly revealed preferred to
0 (i.e., satisfies WARP) if whenever  ·  ≥  · 0 it is false that 0 · 0  0 · ; see, e.g.,
23
Mas-Colell et al. ([47], Chap. 2) and Echenique et al. ([22], p. 1206). My emphasis here is
on WARP, the simplest possible setting for statistical tests of rational behavior. I present
some remarks about the generalized axiom of revealed preference (GARP) later on.
Tests of revealed preferences measure statistical power against particular alternative
hypotheses. As Echenique et al. ([22], p. 1220) notes, it is difficult “to find an acceptable
alternative benchmark to rationality under which to measure power.” As an alternative
hypothesis, the statistical model of choice requires parametric assumptions on  (); see
Bronars [12], Choi et al. [17], and Andreoni et al. [4] for specific examples. This section shows that revealed preference tests are inconsistent, and that their inconsistency is
insensitive to the choice of the distribution function under the alternative.
I consider compensated price changes and formulate the null and the alternative hypotheses as:19
0 : Choices are drawn from a behavioral model of choice in which no ‘true’ violation of revealed preferences exists but there is (additive) measurement error to the
consumer’s choices.
 : Choices are drawn from a statistical model of choice according to a distribution
function  ().
Tests based on the size of the violations. The economic significance of violations
of rationality can be measured in terms of the amount of money that can be ‘pumped out’
of non-rational individuals; see Echenique et al. [22].
Before describing the money pump statistic, consider a more general approach. Given
a pair of consumption bundles of the form ( ) and (0   0 ), the statistic  =  · ( −
 0 ) + 0 · ( 0 − ) measures the potential profits or losses of a devious “arbitrager” that
exploits violations of rationality. In particular, the positive part + ≡ max{0  } and the
negative part − ≡ max{− 0} denote the money pump cost and the money drain cost,
respectively. Obviously,  = + − − so an arbitrager either makes money out of these
19
Tests are uninformative if budget sets do not intersect, even with deterministic data; see Beatty and
Crawford [8]. Intersecting budget sets is a standard assumption; see, e.g., the equal marginal utility of
income (EMUI) assumption in Echenique et al. [22]. Andreoni et al. [4] discuss this point in great detail.
24
transactions when +  0 (and there is a violation of the weak axiom) or loses money
when −  0 (and the weak axiom is satisfied). Echenique et al. [22] focused on the money
pump but I will describe the asymptotic properties of tests based on  and + . Since mean
demands are well-behaved in the statistical model of choice, the power of tests based on
these statistics agree almost completely.
Consider first a test based on  . (The case of + is presented below.) Under the null,
 = ( − 0 ) · ( ( ) −  (0  0 )) + ( − 0 ) · ( − 0 ), where  ( ) and  (0  0 )
represent the unobserved ‘true’ choices and  and 0 are distributed according to Φ( 2 ).
The appropriate test is one-sided and rejects the null if the  -statistic exceeds a critical
test-value. The distribution of the  -statistic under the null cannot be calculated without
additional assumptions on the ‘true’ choices. As first explored by Varian [61], under the null
of no ‘true’ violations, there is an upper bound statistic given by  = ( − 0 ) · ( − 0 ), with
 ≥  and with  normally distributed with  2 ≡ 2 · k − 0 k2 ·  2 . Given a significance
level  and a value for  2 , one can calculate a critical test-value   0 from the normal
distribution according to Pr{   |0 } = Φ( : {   }) = . The probability of a
Type I error is thus no greater than , i.e., Pr{   |0 } ≤ .
The power function of tests based on the  -statistic is Pr{   | }. Power based
on the critical values from the -statistic cannot exceed the power of a test based on the
 -statistic, i.e., Pr{   | } ≥ Pr{   | }. To characterize asymptotic power,
let {     :   = 1     } denote the sample data. The   observations are obtained at
( ) and the   observations at (0  0 ). Let   ≡  (     ) = ( − 0 ) · (  −   ) for
 6=  denote the sample  -statistic for the ( )-pair. The  -statistic can be computed for
a total of ( − 1)2 pairwise sample comparisons. The sample average of the  -statistic
P

is ̃  ≡  −1 
()6=  , or
̃

=
−1

X
()6=
( − 0 ) · (  −   ).
(27)
The behavior of the  -statistic under the alternative is standard: ̃  converges (weakly)

to ̄ ≡ ( − 0 ) · (̄( ) − ̄(0  0 )) ≤ 0 (i.e., ̃  → ̄ when  → ∞), where the inequality
25
follows from Theorem 1. Moreover, given  2 ≡ 2 · k − 0 k2 · Σ, with Σ as the variance-
covariance matrix of  and  [̃  ] =  2 , Chebyshev’s inequality bounds the power
function:
Pr{̃    | } 
 2
.
 2 + ( − ̄ )
(28)
This upper bound depends on the first two moments of the  -statistic under the alternative
(̄  2 ): the more “rational” the statistical model, the less powerful the test would be, i.e.,
lower values of ̄ and/or  2 (a small price change k − 0 k and/or a small variance in )
yield lower statistical power.
Tests based on the money pump statistic + are related to the previous more general
test statistic. In particular, the + -statistic under the null is + = max{0 ( − 0 ) ·
( ( ) −  (0  0 )) + ( − 0 ) · ( − 0 )} and the critical values for the upper bound
statistic + ≡ max{0 } are given by + = max{0 ( − 0 ) · ( − 0 )}. The critical values
are defined analogously as Pr{+  + |0 } = Φ(+ : {+  + }) = , which relies on a
truncated normal distribution.
The power function of tests based on the money pump + -statistic is Pr{+  + | }.
As expected, statistical power for the  and the + statistics decreases with the size of
the critical area  and + ; see, e.g., (28). Also, for any finite sample of size , the
power function of tests based on the money pump statistic + cannot be smaller than the
power of tests based on the  -statistic, i.e., Pr{̃+  + | } ≥ Pr{̃   + | } since
̃+ = ̃  + ̃− by definition. In the limit as  → ∞, however, the  -statistic and even the
money pump statistic + lose all their ability to detect whether or not individuals behave
rationally:
Corollary 2 For any well-behaved distribution function  (), the asymptotic power of
revealed preference tests that rely on the  -statistic or the money pump + -statistic is zero.
That is, lim→∞ Pr{̃    | } = lim →∞ Pr{̃+  + | } = 0.
Tests based on the  -statistic or the money pump cost + are inconsistent. A statistical
test is consistent if its power against an alternative hypothesis tends to one as  → ∞.
(This is a standard statistical definition and a minimal requirement in hypothesis testing.)
The inconsistency of these tests is not due the use of a particular distribution function
26
under the alternative hypothesis  . Asymptotic power is actually not sensitive to the
distribution function  (). This means that at any level of significance , there is no wellbehaved distribution function  () that generates a consistent test of individual rationality
based on the  - or the + -statistics.
Tests based on the number of violations. Statistical power has also been computed
for tests based on the number of revealed preference violations; see Bronars [12]. Assume
without loss of any generality that 0  . Consider the bundles ( ) and (0   0 ) but
study them in relation to ̄( ): the test involves pairwise comparisons between  and
̄( ), and between  0 and ̄( ). There are two kinds of violations: for some individuals
0 ·  0 might exceed 0 · ̄( ) thus revealing a violation of  0 being weakly preferred to
̄( ), while for some others  · ̄( ) might exceed  ·  thus revealing a violation of
̄( ) over .20
The sample distributions of each of these kinds of violations are

Π̃
+ { } ≡
#{ · (  − ̄( ))}+
#{− · (  − ̄( ))}+

, and Π̃
,
{
}
≡
−


(29)
so the sample distribution of a violation of either the first or the second kind (or of both)







is Π̃ {(     )} = Π̃
+ { } + Π̃− { } − Π̃+ { }Π̃− { }.
Under the null, the probability of observing a ‘true’ violation should be zero. The
appropriate test is also one-sided and rejects the null if the number of observed violations
exceeds some critical test-value. Under the null, Π+ { 0 |0 } = Φ(0 : { (0  0 ) + 0 
̄( )}) and Π− {|0 } = Φ( : {̄( )   ( ) + }). The probability of a Type I
error can be determined (or at least bounded) as before so I assume that Π{(  0 )|0 } ≤
, where  is some significance level. The power function of the test is Π{(  0 )| }.
Corollary 3 For a given well-behaved distribution function  (), the asymptotic power of
revealed preference tests that rely on the Π-statistic is
lim Π̃ {(  0 )| } = 1 −
 →∞
 (̄( ))[ (X max ) −  (̄( ))]
.
 (X max 00 ) (X max )
20
(30)
In the case of the two goods depicted in Figure 1, these violations lie in the familiar segments [2  ̄1 ]
00
and [̄1  X max
]. An indifference curve tangent to these segments would have to be concave to the origin;
1
an inconsistency with the weak axiom.
27
Tests based on the number of violations are also inconsistent. Asymptotic power
Π{(  0 )| } equals one if and only if  (̄( )) = 0 or  (̄( )) =  (X max ). If mean
demands are interior in the statistical model of choice, asymptotic power will necessarily
be less than one. As in the case of the  - and the + -statistics, at any level of significance
, there is no well-behaved distribution function  () that generates a consistent test of
individual rationality based on the Π-statistic.
In contrast to tests based on the  - and + -statistics, asymptotic power for tests based
on the Π-statistic is sensitive to the distribution function in the alternative hypothesis.
On one hand, power under a (truncated) symmetric distribution about ̄( ) is larger
than power under asymmetric distributions, all else equal. Under a symmetric distribution, asymptotic power is 1 −  (X max )[4 (X max 00 )]; power would be further enhanced
if  (X max ) does not exceed  (X max 00 ) by a large margin. Asymptotic power would be
at most 34 if  (X max ) =  (X max 00 ). This upper bound might still represent non-trivial
power. Asymptotic power, on the other hand, would equal zero if  (X max 00 ) =  (X max )4
and  (̄( )) =  (X max )2. In this case, tests are also completely uninformative.
Some remarks. (i) Corollary 2 analyzes the average money pump cost. Echenique et
al. [22] also studied the median money pump cost. I have not studied median demands
since there is no general way to define the median for the multivariate case. (One approach
is to generalize the notion of univariate conditional median using marginal distributions,
as in (11).) Median demands in Section 2 also satisfy the compensated Law of Demand.
Thus the asymptotic power of tests of revealed preferences based on the median behavior
of the  and + -statistics is also likely to be small and possibly zero.
(ii) I have examined cycles of length two and not the more general cycles associated
with the generalized axiom of revealed preferences (GARP). A data set satisfies GARP if
for each pair of bundles   and   with   = 1      if  ·   ≥  ·   then it is false
that  ·     ·   . Given a sample sequence of  observations of the form 1 , 2 , ...,  ,
P P

one can compute a generalization of the  - and + -statistics as T̃  =  −1 
=1
=1  ·
(  −  +1 ), with  +1 =  1 , and T̃+ = max{0 T̃  }.21 For any finite sample of size
21
Power calculations for tests of GARP in terms of the number of violations cannot be derived in closedform, even for the uniform distribution used by Becker [9]. The literature relies on Monte Carlo simulations
to measure power in these more general cases; see, e.g., Bronars ([12], p. 695).
28
, the power of tests based on the generalized statistics T̃  and T̃+ cannot be smaller
than the power of tests based on the  and + -statistic since longer cycles allow for more

revealed preference violations. Under the alternative hypothesis, however, T̃  → T̄ ≡
P



+1
 +1 )) with ̄(+1  +1 ) = ̄(1  1 ), so one should not
=1  · (̄(   ) − ̄(
see any arbitrage opportunity as mean demands ̄(   ) satisfy GARP, i.e., they are the
outcome of maximizing a quasi-concave utility function (Theorem 2). Even the generalized
money pump T̃+ -statistic will have a difficult time identifying individual rational behavior
since the T+ -statistic has no statistical power, i.e., lim →∞ Pr{T̃+  0| } = 0.
(iii) Although I have focused on particular test statistics, it is possible to construct
additional statistical tests of the behavioral axioms using rationality indices. Afriat’s efficiency index measures the maximum “margin of error” across violations of the weak axiom
(Echenique et al. [22] and Varian [62]), the minimum cost index measures the monetary
cost of breaking all revealed preference violations in a given data set (see Dean and Martin
[19]), and the swaps indices measure the number of “choice swaps” required to rationalize
choice inconsistencies in a given data set (Apesteguia and Ballester [5]). Even the Slutsky
matrix can be used to assess individual rationality (Aguiar and Serrano [2]). These and
similar rationality indices are unlikely to yield consistent statistical tests. For a statistical
test of individual rationality to be consistent, the test statistic should be able to perfectly
discriminate between the statistical and the behavioral model of choice when the number
of observations is unlimited. That is, a consistent statistical test of individual rationality requires that, for a given choice of  () in  , individuals in the statistical model of
choice eventually make only “irrational” choices, i.e., all random choices should violate the
behavioral axioms. This statistical requirement clearly contradicts Theorem 1.
(iv) The “rational-plus-noise” representation of individual demands in (22) implies that
the null and alternative hypotheses are economically indistinguishable. This means that the
revealed preference tests formulated here are inherently unrevealing. There are, however,
some important statistical issues with the previous tests:
First, consistency in hypothesis testing is defined in a large-sample limit. In some
simple numerical examples available in an Appendix not for publication, however, I find
that random choice behavior leads to well-behaved demand functions for realistic sample
29
sizes as well. Therefore, the inconsistency discussed here is not only of theoretical interest,
but it is a practical concern for tests of revealed preferences and for tests of individual
rationality.22
Second, revealed preference theory is non-statistical: economic theory does not impose
any meaningful restriction on the stochastic behavior of individual demands under the null
hypothesis 0 . This is clearly a very important analytical gap and the most serious limitation for statistical testing of the behavioral model of choice. One can assume that the
behavioral model is misspecified: prices may be the ones mismeasured (as in Echenique
et al. [22]) or the measurement error may be multiplicative; or there may be a stochastic
component in the decision process instead of measurement error (see remark (ii) after Theorem 2). These alternative specifications change the meaning of ‘small’ failures under the
null hypothesis but they are not likely to reverse the test’s inability to detect individual
rationality since these additional specifications do not affect the alternative hypothesis  .
Third, the null and the alternative hypotheses are not mutually exclusive because 0
and  are not standard statistical hypotheses, i.e., these hypotheses are not nested. (This
is why statistical power is not bounded by size, as in standard statistical hypothesis testing.) In general, standard hypothesis testing might not be appropriate for testing revealed
preference theory. In standard hypothesis testing, for example, only two hypotheses are
considered and they are treated asymmetrically, i.e., rejection of the behavioral model of
choice does not necessarily imply that one should accept the statistical model of choice.
One might consider separate families of hypotheses, as in classical non-nested tests, or an
alternative approach based on model selection. These approaches are beyond the scope
of the present paper. Non-nested hypothesis testing and model selection, however, could
confront observed choice data using several behavioral and statistical models.23
22
Echenique et al. ([22], p. 1220) noted that their test procedure had “low power” in terms of the
number of violations of GARP: the number of violations under in statistical choices drawn from a uniform
distribution (as in Bronars [12]) is “much lower than the number of GARP violations observed in the actual
choice data.”
23
It seems possible, for example, to use Vuong [64] non-nested tests to formulate the null hypothesis that
the behavioral and the statistical models of choice are indistinguishable against the alternative that one
model of choice is closer to the ‘true’ data generating process.
30
6
Some final remarks
This paper has rigorously derived a well-behaved consumer demand system from purely
random choice behavior. This paper’s findings, colloquially and loosely, represent the
economic equivalent of the Shakespearean Monkey Theorem. The interpretation of these
findings is open and depends on whether one views the glass as half full or half empty.
On one hand, the findings are reassuring. Psychologists and experimental economists
have raised countless objections against the consistency of preferences and maximizing
behavior. These objections, as valid as they are, are not very forceful since all the testable
predictions of the behavioral model of choice can be reached by an alternative route that
completely abandons these assumptions. In this defense of the behavioral model, which
confirms and extends the one advanced by Becker [9], behavioral objections are largely
misguided because the predictive power of demand theory depends primarily on how budget
sets change and not on the psychological or neurological process of reasoning involved in an
individual’s decision making. In effect, the comparative statics derived in this paper imply
that a large class of realistic random choice behaviors rooted in decision theory and aware
of systematic departures from individual rationality have the same market predictions of
the standard behavioral model of choice.
On the other hand, the fact that purely random choices are as predictive as sophisticated rational choices is troublesome; especially for empirical studies of revealed preferences
and welfare, and for statistical tests of the behavioral postulates of rational choice theory.
The essence of revealed preference theory is to deduce properties of an individual’s preferences from observed choices. But statistical tests of revealed preferences are inherently
unrevealing since even a naive (i.e., statistical) model of choice will pass these tests with
flying colors. Moreover, and this is a special difficulty for demand integrability and welfare
analyses, successfully recovering an individual’s preferences from observed choice data does
not necessarily represent a meaningful exercise. Observed choices could, in principle, be
fully rational yet completely lack normative content. The present findings, in short, largely
diminish the economists’ widely held confidence in the behavioral model of choice.
31
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7
Appendix A: Omitted proofs and derivations
Omitted derivations from the simple example. Consider first the Slutsky equation
for 1 . Notice that the Slutsky compensation at income  is  = ̄1 ( )1 . The Slutsky
equation (9) follows from the definition. For completeness, notice that
¯
max
¯
X max
1
¯ = X 1 + ̄1 ( ) .
1 ¯
1
1
The Slutsky equation (9) is a consequence of this expression as well as (5) and (10).
Next consider the mean demand for 2 . Some steps in this case are analogous to the
proof of Proposition 1. Consider first the own-price effect. Using equation (3), one has:
with
µ
¯
¯ ¶
¯
µ
 ̄1 ( ) ¯¯ 1 ̄1 ( )1
 ̄2 ( ) ¯¯
(2 ) ¯¯
−
=
−
,
2 ¯
2 ¯
2 ¯ 2
22
¯ ¶
¯
̄1 ( )1  ̄1 ( ) ¯¯
E [1 | 0 ≤ 1 ≤ X max
(2 ) ¯¯
1 ] ̄2 ( )
=
,
=
, and
max
2
¯
¯
2 
2
2
X 1
1

¯
µ
¶
̄2 ( )
E [1 | 0 ≤ 1 ≤ X max
 ̄2 ( )¯¯
1 ]
0.
=−
2 ¯
X max
2
1
The Slutsky equation for 2 follows from simple rearrangements. For instance, notice that
the uncompensated own-price effect is  ̄2 ( )2 = −̄2 ( )2 which rules out Giffen
goods. Income effects, however, can be negative for ̄2 ( ). That is,
 ̄2 ( )
=

¶
µ
1
E [1 | 0 ≤ 1 ≤ X max
1 ]
.
1−
max
X 1
2
(A1)
Corollary 4 Under the conditions of Proposition 1, suppose  (1 ) is a log-concave distribution. Then, ̄2 ( ) is a normal good.
A1
Proof. The proof relies on (A1) and the following fact about log-concave distributions:
0≤
E [1 | 0 ≤ 1 ≤ X max
1 ]
≤1,
max
X 1
see Goldberger ([31], Appendix A).
Moreover, notice that |S( )| can be written as
|S( )| = ̄2 ( )
µ
E [1 | 0 ≤ 1 ≤
X max
1
¯
¯
¶
¯
X max
1 ] ¯
−2 21
¯
¯ 11
¯
¯
11 ¯
¯ ,
¯
−12 ¯
with |S( )| = 0. Symmetry and negative semidefiniteness always hold in the special case
of two-commodities. Integrability follows trivially from these properties; see Katzner ([38],
Theorem 4.1-2).
Omitted derivations from the general framework. In order to simplify the derivations in this section, I next present an auxiliary Lemma that will be used repeatedly. The
Lemma serves to simplify the calculations of price and income effects. The central implication of the following Lemma is that in order to determine the response in mean demands to
changes in prices and income, one only needs to evaluate the changes in the most interior
integral. Economically, this result makes sense: recall that  X max
 (() ) =  − () · () ,
which is the only maximum feasible consumption that depends on the entire price vector
 and income .
Lemma 1 Let ( ) ≡
Z
0
 max
1

Z
 max
 (() )
(), where X max is given in (14). Then,
0
"Z max
#¯
¯
Z  max
¯
  (() )
1
( )¯¯

¯
=



()
 ¯ () ,
¯
 ¯


0
0

¯ ¶
µ
Z  max
Z  max
1
−1 ((−1) )
¯
X max
 (() )¯
max
=

(()  X  (() ))
¯ () ,

0
0

for  = 1     .
Proof. The proof is a repeated application of Leibniz’s rule for differentiation under the
integral sign. The case of  = 2 and 1 = 2 = 1 is available in Khuri ([39], pp. 307-308).
A2
For the general proof, it is enough to consider changes in 1 since this price enters in
all limits of integration. All other price changes can be seen as special cases.
#
"Z max
¯
¯
Z  max
 2 (1 )
 (1 −1 )
¯
( )¯¯
X max
1 ¯
=



(






)




1

1

1 ¯
1 ¯
0
0
1 = max
1
"Z max
#¯
Z  max
Z  max
¯
 2 (1 )
(1 −1 )
1


¯
+

(1       )2     ¯ 1 .
¯
1 0
0
0

max
The first term is zero since X max
2 (X 1 ) = ( − 1 (1 ))2 = 0. Further, as noted
max
max
in the text, since X max
= 0 for  ≥ 2 also as there would be no
2 (X 1 ) = 0, then X 
income left for the consumption of these commodities.
The second term becomes
Z
 max
1
0
Z
+
0
⎧"
⎨ Z
⎩
 max
(1 2 )
3

0
Z  max
 max
(1 )
1
2
0
Z
 max
 (1 −1 )
()2    
⎫
¯ ⎬
¯ ⎭ 1
¯
X max
2 (1 )¯
1

#¯
Z  max
¯
 max
(1 2 )
3
 (1 −1 )
¯

()3     ¯ 1 2 ,
¯
0
0
0

1
#
"Z
2 = max
(1 )
2

max
max
max
whose first component is also zero as 3 X max
3 (1  X 2 (1 )) = 2 (X 2 (1 ) − X 2 (1 )) =
0. Thus, as before, the first component evaluates a definite integral over a degenerate
interval and this equals zero. The only relevant component is the second one which also
needs to be evaluated using Leibniz’s rule.
By the way the limits of integration are defined, the derivative operator moves toward
high values of . For instance, the  step of the sequence of integrals is given by
¯
( )¯¯
= 0 +    + 0 (  − 1 times)
1 ¯
"Z max
#¯
Z  max
Z  max
¯
  (1 −1 )
1
 (1 −1 )

¯


()     ¯ 1    −1 ,
+
¯

1
0
0
0

where the evaluation of the integral with the upper limit of integration X max
+1 (1       )
evaluated at  = X max
(1      −1 ) will also equal zero. Moreover, X max
= 0 for all


 ≥ .
A3
The last term in the sequence is
"Z max
#¯
¯
Z  max
¯
  (1 −1 )
1
( )¯¯

¯
=



()
 ¯ 1    −1 ,
¯
¯
1
1 0
0


which under Leibniz’s rule simply becomes
¯
¯ ¶
µ
Z  max
Z  max
1
−1 (1 −2 )
¯
X max
( )¯¯
 (() )¯
max
=



(

X
(
))
()
()

¯ () .
1 ¯
1
0
0

Proof of Theorem 1. The proof was started in the text and is completed here.
To obtain (16), consider (15) and write this expression as ̄ ( ) =  ( ) (X max ),
with
 ( ) ≡
 (X
max
Z
 max
1

 max
 (() )
  () , and
0
0
)=
Z
Z
 max
1
0

Z
 max
 (() )
 () ,
0
as the numerator and the denominator respectively.
Using Lemma 1, notice that
¯
¯ ¶
µ
Z  max
Z  max
1
 (() )
¯
X max
 ( )¯¯
 (() )¯
max
=




(

X
(
))

()
()

¯ () ,
 ¯

0
0

¯
¯ ¶
µ
Z  max
Z  max
1
 (() )
¯
X max
 (X max )¯¯
 (() )¯
max
=

(()  X  (() ))
¯
¯ () .


0
0


Further, notice that the quotient rule implies
¯
¯ µ
¯ µ
¶
¶
1
 ( )
 (X max )¯¯
 ( )¯¯
 ̄ ( )¯¯
,
−
=
¯
 ¯
 ¯  (X max )

 (X max )2

which can be written in simple terms as
¯
¯
¯ ¸
¶∙
µ
 ( )¯¯
1
 (X max )¯¯
 ̄ ( )¯¯
.
=
− ̄ ( )
¯
 ¯
 (X max )
 ¯


A4
This last expression, upon substitution, implies
¯
¯ ¶
µ max
Z  max
Z  max
1
−1 ((−1) )
¯
 (() ) X max
 ̄ ( )¯¯
 (() )¯
=



{
−
̄
(
)}


¯ () .
 ¯
 (X max )

0
0

(A2)
where, as in the text, max (() ) ≡ (()  X max
 (() )).
A compensated change in  only needs to be evaluated in terms of its effect on
X max
 (() ). This result implies that price effects are channeled through changes in the
maximum feasible consumption. Recall that feasibility implies
X max
 (1      −1 ) =
´
X−1
1 ³
−
  ,
=1

(A3)
and that income is compensated according to  = ̄ ( ) . Then,
¯
∙µ
¶
¸
¯
X max

̄ ( ) − 
1
 (() ) ¯
−  =
=
.
¯





(A4)
Finally, substitution of (A4) into (A2) yields (16).
For commodity , (16) implies that the own price effect is given by
¯
µ
¶
Z  max
Z  max
1
−1 ((−1) ) ©
ª2 max (() ) 1
 ̄ ( )¯¯
max
() .
=−

X  (() ) − ̄ ( )
 ¯
 (X max ) 
0
0
(A5)
The main difference is that  has been substituted by X max
 (() ), which is a (hyper)
surface along () .24
To obtain income effects (18), one simply needs to evaluate the appropriate derivative
of X max
 (() ). For instance
 ( )
=

Z
0
 max
1

Z
 max
 (() )
 (()  X max
 (() ))
0
µ
¶
X max
 (() )
() ,

and similarly for the denominator. Once these expressions are substituted back into the
quotient rule result, one obtains (18).
24
Notice that in the example of Section 2, the own-price effect (5) can be written as  ̄1 ( )1 | =
−{X max
− ̄1 ( )}2  (X max
) (X max
)1 , which is the analog of (A5).
1
1
1
A5
Finally, the cross-partial effects (A6) and (A7) also follow using the appropriate derivative of X max
 (() ). For instance
¯
¯ ¶
µ
Z  max
Z  max
max
1
 (() )
¯
X
(
)
 ( )¯¯
()

max
¯ () ,
=




(

X
(
))

()
()

¯
 ¯


0
0

and similarly for the denominator.
These calculations yield
¯
¯ ¶
µ
Z  max
Z  max
1
−1 ((−1) )
¯
(() ) X max
 ̄ ( )¯¯
 (() )¯
=



{
−
̄
(
)}


¯ () ,
 ¯
 (X max )

0
0

(A6)
where the compensated change in X max
 (() ) is the same as the one obtained in (A4).
Next consider the response of ̄ ( ) to a compensated change in  . Following Lemma
1, and repeating the steps just taken, one can show that:
¯
¯ ¶
µ
Z  max
Z  max
1
−1 ((−1) )
¯
(() ) X max
 ̄ ( )¯¯
 (() )¯
=



{
−
̄
(
)}


max
¯ () ,
 ¯
 (X )

0
0

(A7)
¯
¯
where X max
 (() )  = (̄ ( ) −  ) . Simple substitutions establish symmetry.
The only property that deserves some further comment is the negative semidefiniteness
of S( ), which is analogous to the positive semidefiniteness of Σ. I mentioned before
that S ( ) can be written as a covariance term given by
[   ] =
Z
 max
1

Z
 max
−1 ((−1) )
0
0
{ − ̄ ( )} { − ̄ ( )} ˆ(() )() ,
where ˆ(() ) is a simple correction since the density (()  X max
 (() )) does not integrate
to one. That is, the normalization ˆ(() ) ≡ (()  X max
 (() ))( ) with
( ) ≡
1
 (X
max
)
ÃZ
0
 max
1

Z
 max
−1 ((−1) )
0
(()  X max
 (() ))()
!
,
ˆ () ) as a proper density in order to define the matrix of second moments.
serves to treat (
Notice that the term ( ) is a positive constant and hence it does not influence any of
A6
the conclusions of the analysis.
To complete the proof, one only needs to consider standard properties of the variancecovariance matrix. Consider, for instance, a vector  and the non-negative quadratic form
[ · ( − ̄( ))]2 =  · ( − ̄( ))( − ̄( )) ·  . The conditional expectation (20)
satisfies E[ · ( − ̄( ))]2 =  · Σ , which only takes non-negative values. Thus Σ is
positive semidefinite which is a well-known property of variance-covariance matrices; see
Fisz ([27], pp. 89-90).
To verify the conditions for positive definiteness, one only needs to check that the
“last” commodity is linearly independent from the remaining  − 1 commodities because
these other commodities have a joint density, i.e., their distribution is non-degenerate;
see Fisz ([27], p. 90). Notice that X max
 (() ) is a linear function of () . That is, for
a given realization of demands () , the maximum feasible consumption X max
 (() ) is
linear in () , i.e.,  X max
 (() ) =  − () · () for all possible realizations of () ;
see (14). Notice, however, that  is not necessarilly equal to X max
 (() ). In particular,
 = X max
 (() ) for all possible realizations of () only when  is selected as a residual.
In such case, ̄ ( ) will be linearly related to ̄() ( ) since  ̄ ( ) =  − () ·
̄() ( ). In other words, if  is a residual, then Σ is positive semidefinite but not
positive definite since  · Σ = 0 holds for some  6= 0.25
Preliminaries for integrability conditions. Katzner ([38], Chap. 4) discusses in
detail the conditions for integrability in a complete demand system. The integrability of
incomplete demand systems is based on Epstein [24]. In terms of notation, let ( ) denote
the expenditure function: ( ) ≡ min≥0 { ·  : () ≥ }. Thus,
( )
=  ( ) ,

(A8)
for  = 1      and with  ( ) as the Hicksian demand for , i.e.,  ( ) = ̄ ( ( )).
Consider the case of  = 2. Then, X max
(1 ) − ̄2 ( ) = ( − 1 1 )2 − ̄2 ( ). Assume 2 =
2
for all possible realizations of 1 . In this case, 2 is determined as a residual which is the
example of Section 2. This implies that ̄2 ( ) = ( − 1 ̄1 ( ))2 ; see, e.g., (3). Then X max
(1 ) −
2
̄2 ( ) becomes −(1 2 )(1 − ̄1 ( )) which is a linear function of (1 − ̄1 ( )). In this case the
variance-covariance matrix will be positive semidefinite but not positive definite.
25
X max
(1 )
2
A7
Proof of Theorem 2. Consider first part (b). Symmetry in S( ) is necessary and
sufficient for the existence of a solution for the partial differential equation system (A8);
see, e.g., Epstein [24]. A negative semidefinite S( ) is necessary and sufficient for the
solution of the previous system to be concave in . As Theorem 1 noted, these requirements
are met if  is determined as a residual. (The utility function can then be recovered from
the expenditure function.)
For part (a), integrability in incomplete systems requires a negative definite Slutsky
matrix. As Theorem 1(a) noted, interior demands satisfy this condition here. The constant
of integration in the previous system cannot be uniquely determined in an incomplete
system so one must make assumptions about how to complete the system. Assuming the
existence of 0 with 0 = 1 accomplishes this in the simplest possible way. In general, let
̄0 ( 0  ) denote the vector of demands for  commodities that complete the demand
system and let 0 represent their corresponding vector of prices. If ̄( 0  ) = ̄( ) and
̄0 ( 0  ) = ̄0 (0  ), then there is a utility function  : R+
→ R such that ̄( )
+
and ̄0 (0  ) are the solution to max0 ≥0 {( Ψ(0 )) :  ·  + 0 · 0 = }, with Ψ(0 )
linearly homogeneous. Epstein [24] and LaFrance and Hanemann [41] provide additional
remarks about the integrability of incomplete demand systems.
Proof of Corollary 1. Representing individual demands as a mean-plus-noise random
variable   = ̄( ) +   is standard. Expression (23) in Corollary 1 is simply Chebyshev
inequality.
Derivation of binary choice probabilities. Statistical choices can be ordered as
follows:
  ≡ Pr{   } = Pr{     } + Pr{     } + Pr{     } ,
  ≡ Pr{   } = Pr{     } + Pr{     } + Pr{     } ,
  ≡ Pr{   } = Pr{     } + Pr{     } + Pr{     } .
Then,   −   = Pr{     } − Pr{     } and   −   = Pr{   
 } − Pr{     } which are the statements used in footnote 14 in the text.
A8
Proof of Corollary 2.
The proof of Corollary 2 follows from the one-sided Cheby-
shev inequality (28). Further, lim →∞ Pr{̃+  + | } ≡ lim→∞ Pr{max{0 ̃  } 
+ | }. The limit for the + -statistic follows from the continuous mapping theorem, i.e.,
lim→∞ Pr{max{0 ̃  }  + | } = max{0 Pr{lim→∞ ̃  }  + | } = 0. (The max
function is continuous although not differentiable.)
Proof of Corollary 3. Under  , the power components in (29) converge (strongly)
to
0
lim Π̃
+ { | } = 1 −
→∞
 (̄( ))
 (̄( )) −  (0)

, (A9)
max 00 , and lim Π̃− {| } =
→∞
 (X
)
 (X max )
where X max 00 corresponds to a budget set (0  0 ). The rest of the proof follows from
simple rearrangements.
A9
8
Appendix B: Additional results [NOT FOR PUB-
LICATION]
Random (“dual”) representations. Theorem 2 show that mean demands that arise
from individuals randomly choosing their consumptions can be represented as demands
that arise from the maximization of some utility function. This sub-section provides a
partial examination of the “converse” problem and provides two random or “irrational”
representations of behavioral demands. The first representation assumes that “irrational”
demands are on average feasible whereas the second is individually feasible but it restricts
the support of the distribution.
Consider a given consumer demand function  ( ) for commodity  with ( ) ≡
(1 ( )      ( )). Demands ( ) are homogeneous in ( ) and satisfy ·( ) =
. These are the basic properties of demand functions in the behavioral model of choice.
(i) Average feasibility. This feasibility condition is the same as the one used by Katzner
[38] to discuss errors and shocks to individual demand functions. In particular, as in
Katzner ([38], p. 161), “if [the consumer] were to choose from the same budget set many
times, ‘on average’ he would choose the utility maximizing bundle.”
Lemma 2 Let z  ≡  and z ≡ (z 1      z  ) denote the set of extreme points of the
budget set (12). Then, ( ) = (z).
Proof. This is just a consequence of homogeneity: demands satisfy  (1        ) =
 (1        1).
The goal in the first representation is to find a distribution function  ∗ () such that
 (z) =
Z
0
1

Z
0


 ∗ ()
 ,
 ∗ (z)
(B1)
for all  = 1      where  ∗ (z) ≡  ∗ (z 1      z  ) and  ∗ () is the joint density of  ∗ ().
Notice that (B1) integrates random individual demands over a rectangular area instead of
B1
over the triangular area given by (12).
The derivations needed to construct the first representation rely on an inversion formula
for the right-truncated mean. Write (B1) as
∗
 (z) (z) =
Z
0


µ
¶
 ∗ (z 1   z −1    z +1      z  )
 ,

which on differentiation with respect to z  yields ( (z)z  ) ∗ (z)+ (z)( ∗ (z) ) =
z  ( ∗ (z) ). Let   (z) ≡  (z)z  denote the derivative function of  (z). A con-
venient way to write the previous expression is  ln  ∗ (z) =  (z)[z  −  (z)]. Integration yields
½ Z
 (z ()   ) = Φ (z () ) exp −
+∞
∗

µ
¶ ¾
 (z ()  )
 ,
 −  (z ()  )
(B2)
where  ∗ (z ()   ) ≡  ∗ (z 1   z −1    z +1      z  ) and similarly for  (z ()  ) and   (z ()  ),
and with Φ (z () ) = Φ (z 1   z −1  z +1      z  ) determined such that  ∗ () is a distribu-
tion function. This distribution function  ∗ () can be found by solving the  simultaneous
equations (B2).
(ii) Individual feasibility. A second representation can be constructed using results from
convex analysis. Every point in a convex and compact set of finite dimension can be written
as a convex combination of the extreme points of the set; see, e.g., Phelps ([2], p. 1). Let 
be a compact convex subset of R and let z denote the extreme points of . Every point
 ∈  can be written as a convex combination of z:

X
=
 z  ,
=1
where  ≥ 0 and
P
=1
 = 1. There is a probabilistic interpretation based on Minkowski’s
integral representation; see Phelps [2]:
Theorem 3 (Minkowski) Let  be a compact convex subset of R . Then, for every point
 ∈ , there exists a probability measure  concentrated in the extreme points of  such
R
that  =  (). If  is a simplex,  is unique.
B2
Proof. Let   be the Dirac measure on the point z  , i.e., for every Borel set B,
P
  (B) = 1 if z  ∈ B and zero otherwise. Then, () = =1  = 1 and  is a measure
R
R
P
P
with support {z 1      z  }. Therefore,  () = =1    () = =1  z  = .
Uniqueness can be established generally for the case of a simplex; see Phelps ([2], Chap.
10).
To apply the previous representation it is enough to use Lemma 2 and to notice that the
budget set (12) is convex and compact; see Mas-Colell et al. ([47], p. 22). This means that
it is possible to construct a probability measure concentrated in z that is always feasible
and represents any behavioral demand function (z) as the mean demand from a random
choice procedure that selects bundles on the budget hyperplane.
Supply and market equilibrium. In this sub-section, I present a simple partial
equilibrium analysis in competitive markets to outline the nature of results that random
choice may yield in market situations. The purpose of this section is to show that the nice
properties of random choice also apply to the analysis of supply curves.26
Consider the market of commodity 1 . The demand side is determined by Proposition
1 and the supply side can be derived based on maximizing behavior or as the outcome of
randomization. Let  denote the amount of output produced of this commodity. For a
given price , profits are  −  −  (), where  and  () denote the fixed cost and
the variable cost function respectively. Factor prices are held constant and costs are wellbehaved. Let Qmin () denote the minimum scale (i.e., the “shut down” point) determined
by  =  (Qmin ())Qmin (). Notice that Qmin () is positive along increasing average
costs.
Suppose that firms randomly select production levels  using a density function ().
Each randomly selected production plan must satisfy  ≥ Qmin (). Let ̄() denote the
mean supply curve. This curve results from the truncation of unfeasible production plans:
min
̄() ≡ E[| ≥ Q
()] =
26
Z
+∞
 ()
min ()
1 −  (Qmin ())
,
(B3)
These discussions are motivated in part by Kirzner’s [1] earlier criticisms to models of irrational behavior. Essentially, Kirzner [1] suggested that leaving out the supply side of the market considerably weakens
any conclusion one can draw about random choices.
B3
min
where 1 −  (Q
()) ≡
Z
+∞
(). Thus any production plan in [Qmin () +∞) may
min ()
be selected depending on () but any production plan in [0 Qmin ()) must be discarded.
The comparative statics properties with respect to prices satisfy:
¶
µ
 ̄()
Qmin ()
(Qmin ())
min
= [̄() − Q ()]
0.


1 −  (Qmin ())
(B4)
The first two terms in (B4) are positive due to the statistical properties of the lefttruncated mean formula and have interpretations that are analogous to those given in
Section 2. The last term contains all the economic information needed to determine the
slope of the supply curve under randomization. As I noted before, this term is positive if
average costs are increasing and so in this case the mean supply curve will be positively
sloped even in the absence of maximizing behavior. For instance, under profit maximization, firms will choose to produce at a level  ∗ () such that  =  ( ∗ ) as long as
 ∗ ≥ Qmin (); see, e.g., Mas-Colell et al. ([47], Section 5.D).
Proposition 1 and (B4) also imply that there is a single equilibrium in the market
for commodity 1 which can be identified as the intersection point of the mean demand
and supply curves. This finding confronts the criticism of Kirzner [1] and it implies that
changes in underlying market conditions affect equilibrium outcomes in the expected way
even under “irrational behavior.”
Simulation exercises. A key feature in all previous derivations is that mean demand
curves are defined from the aggregation of individual choices. In here, I explore some
simple simulation exercises whose purpose is to determine how ‘large’ the economy needs
to be in order to observe consistent results. I consider a simple uniform distribution and a
multivariate log-normal distribution.
(i) Uniform distribution. Assume two goods and non-interior choices. This is the
setting used in the simple example of Section 2. Assume also that 2 and  are constant
throughout. At a fixed price level 1 let 1 ,  = 1      be an i.i.d. sequence of uniform
random variables on [0 1 ]. Each  represents an individual realization of demand for
B4
good 1 . The mean demand is:
̃1
=
−1

X
1 ,
=1
which, by the strong law of large numbers, satisfies ̃1 → ̄1 ( ). Moreover, when
prices 1 change, one can trace an uncompensated demand curve whose elasticity should
be ε = −1.
In the following simulations  = 1 and 1 varies from 1 to 2. The results consider two
incremental steps. The first is 0005 and the second 005. This means that each individual 
has 200 realizations of demand in the first case and 20 realizations in the second case. Mean
demand is computed for different values of  that range from  = 1 to  = 1 000. In each
sample, and for each value of , a log-log linear regression estimates the uncompensated
elasticity of the demand curve. The number of cross-samples is 500.
Table B1. Simulation results for uniform distribution.
Number of individuals aggregated
 =1
5
10
25
50
100
500
1 000
A. Grid size for 1 of 200 sample points
ε̂
−1.0086 −1.0095 −1.0061 −1.0005 −1.0011 −1.0015 −1.0007 −1.0001
std.err.
(0.349)
(0.100)
(0.067)
(0.041)
(0.029)
(0.020)
(0.009)
(0.006)
std.dev. [0.345]
[0.102]
[0.067]
[0.040]
[0.029]
[0.020]
[0.008]
[0.006]
2
R
0.0396
0.3343
0.5242
0.7423
0.8538
0.9219
0.9834
0.9917
[0.027]
[0.054]
[0.043]
[0.026]
[0.014]
[0.008]
[0.001]
[0.0008]
B. Grid size for 1 of 20 sample points

ε̂
−1.101 −0.9901 −0.9966 −0.9977 −0.9992 −1.0013 −0.9998 −1.0002
std.err.
(0.954)
(0.276)
(0.188)
(0.118)
(0.083)
(0.058)
(0.026)
(0.018)
std.dev. [1.090]
[0.297]
[0.208]
[0.125]
[0.085]
[0.061]
[0.027]
[0.018]
2
R
0.0562
0.3586
0.5497
0.7633
0.8672
0.9316
0.9857
0.9928
[0.109]
[0.160]
[0.146]
[0.075]
[0.043]
[0.022]
[0.004]
[0.002]
Note: The number of cross-samples is 500. The elasticities are estimated using a linear fit to
log[̃1 ] and log[1 ]. The average value of the standard errors across samples is in parentheses.
The cross-sample standard deviation for the estimate of the elasticity of demand and the R2 is in
brackets.

Table B1 shows that ‘individual’ demands are on average negatively sloped with an
B5
elasticity consistent with the predicted pattern. The estimates of the elasticity, however,
are unreliable when only one individual realization is considered and the sample variation
in prices is small. In fact, the elasticity cannot be statistically distinguished from zero
in this case. Further, the standard deviation across and between samples is 1.090 and
0.954 respectively. Both estimates suggest that some individuals have a positively-sloped
demand curve. Finally, the goodness of fit from the R2 is, on average, about 5 percent.
Thus, overall, individual demands are not consistently determined.
Consider the two cases in which  = 5. In these cases, the goodness of fit increase to
over 30 percent and the estimates of the elasticity of demand become (statistically) close
to ε̂ = −1. When the sample variation in prices is 200, and  = 10, the across sample
standard deviation for the elasticity of demand is about 0.06. The goodness of fit and
the statistical significance of the estimates also suggest that average demands are precisely
estimated. When only 20 sample points for 1 are considered, a similar conclusion follows
if the aggregation takes place for 50 individuals. As expected, with  = 500 or  = 1000,
the goodness of fit is well over 98 percent and the elasticity is precisely estimated up to
three digits.
(ii) Multivariate log-normal distribution. Consider next three goods and non-interior
choices. Prices are 1 , 2 , and 3 . Prices 2 = 1 and 3 = 1, and  = 3 are constant
throughout. 1 varies from 1 to 2. Let (1  2  3 ),  = 1      be an i.i.d. sequence
of log-normal random variables on ( ). Each  represents an individual realization
of demand for the three goods 123 . The draws are from a log-normal distribution with
mean [−05 −05 −05], variances [2 2 2] and with positive pairwise correlations between
all variables of 05.
In the following simulations 1 varies in incremental steps of 005, so there are 20
sample points. Mean demand is computed for different values of  that range from  = 1
to  = 10 000. In each sample, and for each value of , a log-log linear regression
estimates the compensated elasticity of the demand curve, ε . This elasticity should be
negative, although there are no specific numerical values suggested by the theory. Again,
the number of cross-samples is 500.
Table B2 shows again that ‘individual’ demands are negatively sloped but unreliable.
B6
As the number of individuals aggregated increases, mean demands become better behaved
especially when more than 100 individuals are considered. The goodness of fit from the
R2 increases and the precision of the estimated elasticity of demand also increases. The
compensated elasticity is negative, as predicted by Theorem 1.
Table B2. Simulation results for multivariate log-normal distribution.
Number of individuals aggregated

ε̂
std.err.
std.dev.
R2
 =1
10
100
1 000
10 000
−0.0965 −0.2325 −0.2598 −0.2524 −0.2541
(0.723)
(0.363)
(0.107) (0.0384) (0.022)
[2.301]
[0.614]
[0.184]
[0.056]
[0.018]
0.0813
0.0869
0.222
0.656
0.858
[0.162]
[0.152]
[0.213]
[0.134]
[0.036]
Note: The number of cross-samples is 500. The elasticities are estimated using a linear fit
regression. The average value of the standard errors across samples is in parentheses. The crosssample standard deviation for the estimate of the elasticity of demand and the R2 is in brackets.
In conclusion, Tables B1 and B2 show that consistent mean demand curves do not
require an excessively large number of individuals aggregated; this is specially true with
enough price variation. That is, while the theoretical analyses rely on a large number of
individuals, numerical approximations with fewer individuals still give support to the main
theoretical conclusions of the analysis.
References
[1] Kirzner, I.M. (1962) “Rational Action and Economic Theory,” Journal of Political Economy, Vol. 70(4), 380-385.
[2] Phelps, R.P. (2001) Lectures on Choquet’s Theorem, Berlin: Springer.
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