Random Choice and Market Demand

Javier A. Birchenall∗

University of California at Santa Barbara

September 10, 2020

Abstract

I present a model of choice with no preferential basis or optimizing behavior,

a statistical model of choice. Individual consumption is randomly chosen while

respecting a linear budget set. I show that mean (i.e., market) demands satisfy

all the properties of classical demand theory, including symmetry and negative

semidefiniteness of the Slutsky matrix. The derived comparative statics serve to

introduce a new form of (random) duality to demand theory, characterize market

demands for general probabilistic choice behaviors, and examine the consistency

and statistical power to detect rational behavior in revealed preference tests.

Keywords: demand theory, random choice behavior, aggregation

JEL classification: D01; D11; E03

Communications to: Javier A. Birchenall

Department of Economics, 2127 North Hall

University of California, Santa Barbara CA 93106

Phone/fax: (805) 893-5275

Javier.Birchenall@ucsb.edu

∗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, Cheng-

Zhong Qin, Warren Sanderson, Perry Shapiro, Joel Sobel, Glen Weyl, Eduardo Zambrano, and Giulio

Zanella for useful conversations. The usual disclaimer applies.

1

1 Introduction

Classical demand theory is deterministic. To account for random variation in actual choices,

or to test the implications of revealed preference theory using statistical methods, econo-

mists typically view consumers as boundedly rational. A consumer’s deterministic choices,

or the process by which their choices are made, are often augmented by a random element.

Examples include additive error terms to individual demand functions or additive random

preference shocks to deterministic utility functions. This paper takes these bounded ra-

tionality arguments to the extreme. I study consumer demands derived from a statistical

model of choice where individual choices are entirely made at random.

A statistical model of choice has no preferential basis and there is no optimizing be-

havior at any stage of the decision process. I consider a continuum of individuals that

select their consumption over a linear budget set by drawing random choices from a con-

tinuous probability density function. This choice behavior is an extreme form of bounded

rationality. Individuals are actually “irrational” in the sense used by Becker [7]. Becker

[7] demonstrated by example that compensated price changes sometimes force “irrational”

individuals to, on average, satisfy the Law of Demand. Characterizing a mean (i.e., market)

demand function derived from a statistical model of choice was an important contribution

of Becker [7].1 Becker’s [7] example, however, is impractical and restrictive. It relies on

random choices drawn from a uniform distribution, two commodities only, and non-satiated

individual choices.2

In this paper, individual choices are drawn from a general probability density function, an

arbitrary number of commodities, and for demands that lie in the interior or the boundary

of the budget set when the “last” commodity is left as a residual. I show that market

1Hildenbrand ([32], 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 also

pointed out some limitations in Becker [7]; particularly, the use of uniform distributions. Varian ([59], p.

105) 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.”2The uniform distribution is special. Some aggregation properties in Caplin and Spulber [11] follow

entirely because of their use of uniform distributions. Using two-commodities is also special for demand

integrability; see Katzner ([35], Theorem 4.1-2). Finally, non-satiation is responsible for downward sloping

indifference curves and adding up in classical demand theory. Demsetz ([16], p. 489) was agnostic: “as

imaginative and informative as [Becker’s] arguments are, they cannot be sustained in full without appealing

to rationality.”

1

demands satisfy the compensated Law of Demand quite generally. Indeed, I show that

market demand functions in a statistical model satisfy all the properties of classical demand

theory, including symmetry and negative semi-definiteness of the Slutsky matrix (Theorems

1 and 3). This last property is the defining feature of classical demand theory. Mean

demands in a statistical model of choice are thus indistinguishable from classical demands.

Mean demands can be rationalized as being the result of utility maximization, even when

interior (Theorems 2 and 4).

The main theorems yield several insights. First, mean demand integrability implies that

classical demand functions can be derived either from randomizing or optimizing behavior

(i.e., utility maximization or expenditure minimization). The paper hence introduces a

new (random) dual representation to classical demand theory. While the characterization is

partial, I make headway into the formulation of random duality in demand theory. Second, a

statistical perspective on choice behavior yields comparative statics that characterize a wide

range of probabilistic choice models. It is possible, for instance, to incorporate restrictions

based on behavioral postulates or choice heuristics. An example listed here is Luce’s [42]

independence from irrelevant alternatives, but many other examples based on preferential

and ‘pure’ (i.e., non-preferential) random choice behaviors are possible.3

Third, statistical models of choice have been used to derive statistical power to detect

rational behavior in revealed preference tests; see, e.g., Varian ([57], [58], and [59]), Bronars

[9], and Echenique et al. [18]. In statistical testing, consistency is generally viewed as a

necessary condition. It ensures that hypotheses can be distinguished in sufficiently large

samples. The statistical model of choice provides an alternative hypothesis to revealed

preference theory.4 It also provides power calculations. Power calculations have used the

uniform distribution, the logistic distribution, and random perturbations from observed

3The model does not differentiate between preferential and ‘pure’ (i.e., non-preferential) random choice

behavior. Traditional decision theory considers ‘pure’ randomizing behavior useful when preferential choices

have limited value; see, e.g., Elster [20] and Moore [49]. Recent laboratory and field experiments that

showcase the value of randomizing behavior in individual decision-making settings include Agranov and

Ortoleva [1] and Levitt [40].4Revealed preference tests abound including ”irrational” (i.e., non-standard) populations such as animal

subjects (Kagel et al. [34] and Chen et al. [12]), psychiatric patients (Cox [14]), children (Harbaugh et al.

[31]), and individuals under the influence of alcohol (Burghart et al. [10]) to name a few.

2

choices; see, e.g., Bronars [9], Choi et al. [13], and Andreoni et al. [4].5 I find here that re-

vealed preference tests and tests based on the amount of money pumped out of non-rational

individuals have low (asymptotic) power. Asymptotic power is generally independent of the

distribution function used in the alternative hypothesis (Corollaries 3 and 4).

Many other literatures have made use of statistical models of choice. In the spirit of

Becker [7], Grandmont [29] derived through aggregation market demands of the Cobb-

Douglas type (see also Hildenbrand [32] and Kneip [37]). Their positive income effects

ensure equilibrium uniqueness. I show that positive income effects are not general to the

statistical model of choice. In Becker [7], positive income effects follow due to the log-

concavity of the uniform distribution (Corollary 5). I provide a general characterization of

income effects (Corollary 2). Gode and Sunder ([25], [26]), in another application, used a

double auction to study the allocative efficiency of market mechanisms for zero-intelligence

traders, also inspired by Becker [7]; see Duffy [17] for a discussion of agent-based modeling

and decision-making in economic experiments. This paper suggests that the literature’s

reliance on a uniform distribution to model zero-intelligence traders is unnecessary.

2 Simple Examples

This section presents simple examples to illustrate the intuition behind the general theorems.

I consider an economy with two infinitely divisible commodities, 1 and 2. Their prices

= (1 2) and income are given and are assumed to be strictly positive.

There is a continuum of individuals whose choices must respect a linear budget set

() ≡ ©(12) ∈ R2+ : 0 ≤ 11 + 22 ≤ ª. (1)

The only choice is 1. Once 1 is chosen, the remaining income is spent in 2. (Simul-

taneous choices are discussed below.) The model of choice is statistical : 1 is selected at

random by draws from a distribution function (1) with density (1).

5Varian ([57], [58], and [59]) provides an overview of revealed preference theory. Bronars [9] first applied

Becker’s [7] model to derive statistical power. Choi et al. [13] considered boundedly rational individuals that

follow Luce’s [42] choice axioms; see also Andreoni et al. [4] for many alternatives. Beatty and Crawford

[6] discuss some nonstatistical aspects in revealed preference tests.

3

I assume that (1) is absolutely continuous and that first and second moments are

finite. Measure zero events are neglected throughout and all subsets are assumed non-

empty and measurable. To make sure that some choices are unfeasible, I assume that the

support of (1) is sufficiently large (i.e., possibly infinite). I also assume that (1) is

invariant with respect to ( ). This is the analog of assuming that tastes are invariant to

the economic environment. As in classical demand theory, in which preferences are given

and independent of the budget set, the density function (1) is given and does not vary

with (). Throughout the paper, I maintain these “well-behaved” requirements for all

other densities from where choices are drawn.

The Law of Demand. The maximum quantity of 1 that individuals can afford

in (1) is Xmax1 ≡ 1. Therefore, statistical choices must lie in the restricted choice set

0 ≤ 1 ≤Xmax1 . Let (Xmax

1 ) ≡ Pr{1 : 0 ≤ 1 ≤Xmax1 }. The mean demand for 1 is:

1() ≡ E [1| 0 ≤ 1 ≤Xmax1 ] =

Z max1

0

1(1)

(Xmax1 )

1. (2)

Given1,2 satisfies 22 = −11 so its mean demand is 2() = (−11())2.Let ·| denote the standard Slutsky compensation, = 1( )1.

6

Proposition 1 Suppose that individuals randomly choose 1 from (1) using a well-behaved

probability density function (1), and determine 2 as a residual. Then, the compensated

own price effects are negative for 1() and 2( ), i.e.,

1()

1

¯

0, (3)

and the Slutsky matrix of substitution effects is symmetric and negative semidefinite.

The proof, as other technical results, is in the Appendix. The relevant derivative for (2)

is:1( )

1

¯

=E [1| 0 ≤ 1 ≤Xmax

1 ]

Xmax1

µXmax

1

1

¯

¶, (4)

6When 1 changes, mean demand changes to 1(1 + 1 2 + ), where = 1()1 is the

Slutsky compensation. Income compensations are not so that everyone can afford mean demands but so that

individuals can, on average, afford their original bundle. Obviously, a Hicksian compensation or standard

duality results are not possible here because demands do not rely on utility functions.

4

so (3) follows as each term in (4) is given by:

E [1| 0 ≤ 1 ≤Xmax1 ]

Xmax1

=(Xmax

1 )

(Xmax1 )

(Xmax1 − 1( )) 0, and (5)

Xmax1

1

¯

= −Xmax1 − 1()

1 0. (6)

In Proposition 1, the (compensated) Law of Demand for 1() holds due to a statis-

tical fact and an economic fact. First, 1(), a right-truncated mean, is lower than the

truncation point Xmax1 and its value increases as the truncation point increases; see (5).

Second, even after taking into account the income compensation, the maximum amount of

1 that can be purchased declines as 1 increases; see (6).

Figure 1 illustrates Proposition 1. Values of 1 that exceed Xmax1 are not feasible so

(1) is truncated atXmax1 , i.e., the dark-shaded area to the right ofXmax

1 is not feasible at

(). When 1 increases, the maximum quantity of 1 that individuals can afford declines

to Xmax 01 . Once income is compensated, the truncation point changes to Xmax

100 Xmax

1 .

A compensated increase in 1 thus reduces Xmax1 and the mean demand declines from

1 ≡ 1(1 2 ) to 001 ≡ 1(

01 2

0).7

Some remarks. Proposition 1 does not depend on functional form assumptions on

(1). Becker [7] assumed a uniform distribution with (1) = 1+ and support [0 +], so

1() =1

(1)

Z 1

0

11 =1

2

1, (7)

and 2() = (12)(2). These demands coincide with those of a symmetric Cobb-

Douglas utility. (A power density (1) = (+)(1

+)−1 yields non-symmetric Cobb-

Douglas demands for 6= 1.)Proposition 1 examines mean demands but other aggregates also satisfy the Law of

Demand. For example, let med1 ( ) ≡ −1( (Xmax1 )2) denote 1’s median demand.

Since med1 ()Xmax1 = (12)(Xmax

1 )(med1 ( )) 0, median demands increase

7The only generalizations of Becker [7] I have been able to find are in Sanderson ([52], [53]). They assume

that an increase in 1 expands the choice set in a first-order stochastic sense. Since the relevant choice

distribution is (1) (Xmax1 ), the orderings assumed in Sanderson ([52], [53]) are always obtained here:

(1) (Xmax1

0) stochastically dominates (1) (Xmax1 ) in the first degree sense in [0Xmax

10].

5

6

-

(1)

1

2

Xmax1Xmax

100Xmax

101

2

001

002

2

02

eeeeeeeeeeeeeee

eeeeeeeeeeeeeee

eeeeeeeeeeeeeee

ZZZZZZZZZZZZZZZZZZZZ

ZZZZZZZZZZZZZZZZZZZZ

ZZZZZZZZZZZZZZZZZZZZ

eeeeeeeeeeeeeeeeeee||||

||||||||

|

rr

Figure 1: Mean demand for 1, drawn from (1). Initial choices are truncated by Xmax1 .

As prices change, the truncation point is Xmax 01 . For compensated changes, the truncation

point is Xmax 001 .

with Xmax1 hence they satisfy the Law of Demand.

The Law of Demand holds on average but not for every realization. When 1 increases,

the fraction of individual demands given by (1())− (0) satisfies the Law of Demandbut a fraction (Xmax 00

1 ) − (1()) violates it since demands increase for these indi-

viduals; see Figure 1. These violations have been used to measure the statistical power of

revealed preference tests. I return to this application of statistical choices in Section 5.

As in any demand function, uncompensated price effects satisfy the standard Slutsky

equation,1()

1=

1( )

1

¯

− 1()1( )

, (8)

with income effects given by 1() = E [1| 0 ≤ 1 ≤Xmax1 ] Xmax

1 (11) 0.

Hence mean demand 1() is normal (Figure 1), but positive income effects for 2()

are not general. A sufficient condition for 2( ) to be a normal good is that (1) is log-

concave; see Appendix. This condition is implicit in Becker [7] as the uniform distribution

is log-concave.

6

001

002

QQQQQQQQQQQQQQQQQQQQ

QQQQQQQQQQQQQQQQQQQQ

QQQQQQQQQQQQQQQQQQQQ

eeeeeeeeeeeeeeeee

6

- 1

2

Xmax1Xmax

1001

2

Xmax2 (0)

Xmax2

00(0)

(1 2) =

¡¡ª

rr

Figure 2: The Law of Demand for interior demands and = 2 when 1 is stochastically

decreasing in 2.

Mean demands are homogeneous (of degree zero) in (), add up to income, and

symmetric with respect to compensated cross-price changes. Means demands can even be

rationalized as the outcome of optimizing behavior. Integrability, however, is special under

two commodities because there is no room for non-transitive behavior; see, e.g., Mas-Colell

et al. ([45], p. 36 and Exercise 2.F.15) and Katzner ([35], Theorem 4.1-2). I provide general

integrability results in Section 3.

Interior demands. Suppose now that individuals choose 1 and 2 simultaneously

with draws from a distribution function (1 2) with density (1 2). The maximum

feasible consumption of 1 is still Xmax1 ≡ 1. The maximum feasible consumption of

2 depends on 1, as in 2Xmax2 (1) = − 11 or 2X

max2 (1) = 1[X

max1 − 1], for

1 ≤ Xmax1 . Its bounds are Xmax

2 (0) = 2 and Xmax2 (Xmax

1 ) = 0. Hence, for feasibility,

one now needs to consider choices drawn from

(Xmax1 Xmax

2 (1)) ≡ Pr{(12) : 0 ≤ 11 + 22 ≤ }

=

Z max1

0

Z max2 (1)

0

(1 2) 21. (9)

7

Mean demands 1( ) and 2( ) rely on the truncated distribution (9). Adding up

is violated (a.s.), but 1( ) and 2() satisfy all other properties of classical demand

theory. Figure 2 illustrates the Law of Demand for interior demands using a contour of

(1 2). A compensated increase in 1 ‘pivots’ the budget line and removes the dark-

shaded triangular area along (Xmax 001 Xmax

1 ). The area along (Xmax2 (0)Xmax 00

2 (0)) is

added. The Law of Demand holds because the area that is removed once 1 increases favors

high values of 1 whereas the area that is added favors low values of 1.

Figure 2 assumes that 1 and2 are “negatively associated.” It is possible to see that as

income increases, the feasible set favors more low values of 1. This yields inferior demands

for 1() and normal demands for 2(). If 1 and 2, are “positively associated,”

both goods would be normal. I formalize these observations later on. I was unable to find

conditions to make both demands inferior, an impossibility for classical demand theory.

3 General Theorems

This section will deal with generalizations of the previous examples. Let ≡ (1 )

denote the commodity vector and let ≡ (1 ) be the corresponding price vector.

The budget set,

() ≡ © ∈ R+ : 0 ≤ · ≤

ª, (10)

may hold as an inequality depending on whether one or more commodities are randomly

chosen or determined as a residual. I first study a system of mean demands where the “last”

commodity is chosen randomly. The corresponding analysis for the case when is

determined as a residual, which is less general, follows.

3.1 Interior demands

Choices are simultaneously drawn from (1 ) ≡ Pr(1 ≤ 1 ≤ ) with

density (1 ). To ensure that individual choices respect (10), (1 ) are drawn

from

(Xmax) =

Z max1

0

Z max (1−1)

0

(1 ) 1, (11)

8

where Xmax ≡ (Xmax1 Xmax

(1 −1)) is the vector of maximum feasible consump-

tions.8 The vector Xmax satisfies 1Xmax1 = and

Xmax (1 −1) = −1[X

max−1 (1 −2)− −1] , for = 2 . (12)

This vector is ordered: Xmax (1 −1) is a subset of −1X

max−1 (1 −2) be-

cause a positive consumption for commodities ≤ limits the maximum consumption of

commodities . If = Xmax (1 −1), positive consumption for commodities

becomes unfeasible as Xmax (1 −1) = 0 for = + 1 ; see, e.g., (12).

The upper bound for feasible consumptions is Xmax (1 = 0 −1 = 0) = for all

= 1 . These upper bounds are the vertices of the budget hyperplane.

Compensated price effects. The mean demand for commodity is given by

() ≡ E[|0 ≤ · ≤ ], or

() =

Z max1

0

Z max (1−1)

0

(1 )

(Xmax) 1, (13)

with () ≡ (1( ) ()) denoting the demand system.These demands are the truncated means of the random realizations of individual choices.

Mean demands depend on the () because the maximum feasible consumptionsXmax de-

pend on (). I have omitted the dependence ofXmax on ( ) for notational convenience.

The vector of maximum feasible consumption, however, is homogeneous (of degree zero) in

(). As a function of ( ), max( ) = max() for all ( ) and 0. Notice

also that in (11) and (13), the order of integration is irrelevant. Changing it amounts to a

relabeling of the different commodities.9

Compensated price changes are summarized by the × Slutsky matrix of substitution

effects, whose ()-entry is

S() ≡ ()

¯

. (14)

8For a uniform density, (Xmax) is the volume of a simplex, (Xmax) = (!)−1Q

; see Ellis [19].9The order of integration in (11) and (13) can be changed without consequences due to Fubini’s Theorem;

see, e.g., Fikhtengol’ts ([22], Vol. II, Section 344). This change would be somewhat analogous to changing

the order in which first-order conditions are obtained in the classical model.

9

The following theorem characterizes the resulting demand system:

Theorem 1 Suppose that individuals randomly choose subject to (10) using a well-

behaved probability density function (). Then, mean demands () are: interior,

· () ; homogeneous (of degree zero) in (); and the Slutsky matrix S()is symmetric and negative definite.

To simplify notation, let ≡ (1 ) and let () denote the vector when

is excluded, i.e., () ≡ (1 −1). Likewise, let ≡ (1 ) be the vector ofdifferential 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, max(()) ≡ (()X

max (())), which is only a function of ().

This density is central in all the comparative statics of this paper. The term Xmax (())

plays a central role because it is the only term that depends on the prices of all commodities,

i.e., all changes in ( ) are channeled through Xmax (()).

Proof (overview). Mean demands are (a.s.) interior since exhausting the budget

set is a zero-probability event. Homogeneity follows since Xmax is homogeneous in (),

as previously discussed. Symmetry and negative definiteness in the Slutsky matrix follow

from differential changes to (13). The relevant derivatives are based on repeated differen-

tiation under the integral sign so they are collected in the Appendix. I show there that a

compensated change in on commodity yields

S( ) = −Z max

1

0

Z max−1((−1))

0

[ − ( )][ − ()]

max(())

(Xmax)(). (15)

This term can be written as the second-order moment of linearly independent random

variables. For instance, let Σ be the × variance-covariance matrix of the random vari-

ables (() − ()( )Xmax (())− ()). By definition, then, Σ = [ ] ≡

E[( − ())( − ())| ≤Xmax] so one has

S() = −[], (16)

for = 1 − 1, and with = Xmax (()) when = . Since the variance-

10

covariance matrix Σ is symmetric and positive definite; see, e.g., Fisz ([23], Theorem 3.6.6),

the Slutsky matrix () = −Σ is symmetric and negative definite.Individual and mean demands are interior, but ( ) is homogeneous in (). As in

classical demand theory, this is a feature of the budget set (10). Symmetry and negative

definiteness are technical properties that have an intuitive interpretation. Individual choices

are random variables with a well-behaved variance-covariance matrix. The Slutsky matrix

S( ) is the negative of an × positive definite variance-covariance matrix. Hence

S( ) is symmetric and negative definite.Integrability. There are no preferences or optimizing behavior behind the demand

system (). Mean demands are actually interior so they would violate non-satiation.

Still, as the following theorem shows, () can be rationalized as being the result of the

maximization of some utility function:

Theorem 2 Under the conditions of Theorem 1, there exists some continuous, non-decreasing,

and quasi-concave utility function : R+1+ → R such that ( ) and 0( ) ≡ − ·

() are the unique solution to max0≥0 {( 0) : · + 0 = }.

Proof (overview). The system () is an incomplete demand system whose Slutsky

matrix S( ) satisfies the integrability conditions in Epstein [21] (see also LaFrance andHanemann [39]).

Incomplete demand systems can be rationalized if S() permits so. Theorem 2 con-

siders a “residual” commodity 0 with 0 = 1. The role of 0 is to exhaust the budget set

in order to determine a constant of integration for the expenditure function associated with

( 0). Not all incomplete demand systems can be rationalized. For integrability, S()must be symmetric and negative definite, as in Theorem 1. Negative definiteness in S()cannot be weakened for semidefiniteness; see Epstein ([21], Example 1).

3.2 Non-interior demands

As in Becker [7], and the simple example depicted in Figure 1, assume now that individuals

choose () commodities randomly but the “last” commodity is a residual. The density

11

function is (1 −1), which is only a function of () arguments. Mean demands

()() are given by (13) and, by construction, () = ( − () · ()()).

Theorem 3 Suppose that individuals randomly choose () subject to (10) using a well-

behaved probability density function (()). The “last” commodity is determined as a

residual. Then, mean demands (): add up to income, · () = ; are homoge-

neous (of degree zero) in (); and the Slutsky matrix S() is symmetric and negativesemidefinite.

The proof is an adaptation of the proof of Theorem 1. In contrast to Theorem 1, the

Slutsky matrix S( ) is now symmetric and negative semidefinite instead of just symmetricand negative definite. This is so because the “last” commodity is redundant. Knowing

the individual realizations of demands for commodities () and the budget set (10) is

enough to determine individual and mean demands for the “last” commodity .10 Since

the random variables()−()( ) andXmax (())−() in (16) are linearly related,

their variance-covariance matrix Σ is positive definite but singular; see, e.g., Fisz ([23],

Theorem 3.6.6).11 As Σ is positive semidefinite, the Slutsky matrix ( ) is symmetric

but only negative semidefinite.

As before, there is no optimizing behavior at any stage, but ( ) can rationalized :

Theorem 4 Under the conditions of Theorem 3, there exist some continuous, non-decreasing,

and quasi-concave utility function : R+ → R such that ( ) is the unique solution to

max≥0 {() : · = }.

Interior and non-interior demands ( ) derived from randomizing behavior can be

rationalized as the outcome of maximizing behavior. A “residual” commodity 0 in the

incomplete system may seem restrictive but rational behavior is typically tested in terms of

expenditure and not in terms of income. Since it is not generally possible to study all the

goods relevant for an individual, empirical tests of revealed preferences effectively analyze

10In classical demand theory, homogeneity and adding up also imply that the negative semidefiniteness

cannot be extended to negative definiteness; see, e.g., Mas-Colell et al. ([45], Proposition 2.F.3).11For example, when = 2 and2 =X

max2 (1), as in the non-interior demands in Section 2,X

max2 (1)−

2() becomes −(12)(1 − 1()) which is a linear in (1 − 1()).

12

incomplete demand systems.12 Of course, if only prices and demands are observed, but

not income , one can define 0 so that 0 ≡ · ( 0).

4 Some Extensions

This section discusses some simple extensions of the previous findings.

Bounded rationality “errors.” Applied consumer demand studies and revealed

preference tests typically augment deterministic individual demands with small “errors.”

These errors are seen as a random component in the decision process (i.e., mental errors)

or as additive measurement error in prices or individual choices; see, e.g., Choi et al. [13],

Echenique et al. [18], Lewbel [41], and Varian [58].

Individual demands here do have a “rational” and a “error” component. Deviations from

“rational” behavior are also bounded. Let denote = 1 independent sample real-

izations of an individual’s vector of demands. I assume that the realizations hold ( ) fixed.

The sample average for the individual demand for is defined by ≡ −1

X

=1

.

Corollary 1 Individual demands for in the statistical model of choice have a “rational-

plus-error” representation

= ( ) +

, (17)

with ≡

− ( ) and E[

] = 0. Moreover, for a given 0, individual demands

satisfy

Prn¯

− ( )¯ o

[]

2. (18)

Proof. Expression (17) is a standard construction and (18) is Chebyshev inequality.

By construction, a sample of individual demands generates a random variable that de-

viates from a “rational” mean demand by a purely stochastic “error” term. Since mean

demands can be rationalized by a well-behaved utility function, individual choices in the sta-

tistical model also admit a random utility representation given by () = (())+ ,

12Applied work relies on separability, composite goods, or an incomplete system of demands. Conventional

revealed preference tests, for example, do not test non-satiation. It is typically assumed to avoid trivial

data rationalizations; see, e.g., Varian ([57], p. 969).

13

where ≡ (() + ) − (()) and ≡ − () with E[] = E[()] −(E[]) 0. Corollary 1 thus shows that purely random choice data can be represented

as a rational-plus-error demand function, or as random utility model in which an individ-

ual’s utility () results from perturbations to a deterministic indirect utility function

(()) or from perturbations to a deterministic demand function ().

Income effects. As in classical demand theory, no special restrictions on () are

needed in Theorems 1 and 3. Income effects, however, cannot be unambiguously signed.

Income effects depend on the association between commodity and the “last” com-

modity . The random variables ( ) are said to be “positively associated” if one

is more likely to observe and taking larger (or smaller) values together than any

mixture of these. For 0 and 0 , while the remaining − 2 variables are fixed,“positive association” restricts () according to (0

0)( ) (0 )(

0);

see Shaked and Shanthikumar ([54], Section 9D).

Corollary 2 Under the conditions of Theorem 1, suppose that and are “positively

(resp. negatively) associated” random variables. Then, income effects for () are posi-

tive (resp. negative).

Proof (partial). For , with = 1 − 1, mean demand () satisfies:

()

=

Z max1

0

Z max−1((−1))

0

[ − ()]

max(())

(Xmax)(). (19)

This expression compares E[|() ≤ Xmax() = Xmax

(())], the mean value of

when takes its highest possible value, with ( ), or E[| ≤ Xmax]. For “posi-

tively associated” random variables, E[|() ≤Xmax() =X

max (())]− () 0,

yielding positive income effects. For = , expression (19) replaces byXmax (()), which

always exceeds ( ). Income effects for are therefore positive.

Income effects depend on the relationship between and . To understand this

relationship, recall the example of random choices over a single good in Figure 1. Income

effects are positive for 1 because 1 is trivially associated with 1. In the two-good case

of Figure 2, income effects for 1 are negative because 1 is “negatively associated” with

14

2. Income effects are positive for 2 since 2 is the “last” commodity.

The “last” commodity is central because changes in are channeled throughXmax (()).

Expression (12) implies that Xmax (()) = −

X−1=1

(), which is increasing in

. As the (hyper) surface Xmax (()) shifts out, the feasible values of expand. When

and are “positively associated,” and the budget set expands, the relevant density

added to determine mean demands for favors high values of , so () increases. For

“negatively associated” variables, the added area favors high values of but low values

of . In such case, () would be an inferior good. As before, it is not possible for all

goods to be inferior. Income effects for the “last” commodity are always positive.

A preferential random choice example. Other than having a sufficiently large

support, the restrictions required from () are in general implied by probability theory. The

previous theorems thus apply to choice probabilities restricted through behavioral postulates

or choice heuristics.

A well-known choice restriction is Luce’s [42] independence from irrelevant alternatives

(IIA). As an illustration, let (|) be the probability that an alternative subset is chosenwhen the set of available alternatives is and IIA holds. Since (|) = (| 0) ( 0|),for ⊆ 0 ⊆ , one has a continuous logit formulation with

(|) =

Z

exp{()}Z

exp{(0)}0, and (|) = exp{()}Z

exp{(0)}0, (20)

where () is a direct utility function associated with reflexive, transitive, and complete

preferences over ∈ .13 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 [7], and when → 0, the choice problem

becomes a deterministic utility maximization problem; see Anderson et al. ([3], p. 42).

Using (|), it is possible to derive a system of mean demands of the form ( |).These mean demands, holding the set of alternatives constant, satisfy the general theorems

13Continuous choice models can be derived as the infinitesimal limit of discrete choice models; see, e.g.,

Ben-Akiva and Watanatada ([8], p. 327) and McFadden ([46], pp. 311-312). In location theory, continuous

choices are unavoidable.

15

presented before. They, therefore, follow standard comparative statics for compensated price

changes and can be rationalized. This is a more general characterization than Mossin’s [50]

early study of the market demand behavior in Luce [42].14

Non-preferential random choice behaviors also satisfy the previous theorems.15 ‘Pure’

random choice may arise out of fairness or strategic considerations; see, e.g., Moore [49]

for a classical discussion on the use of an “impersonal and relatively uncontrolled process”

to coordinate social actions. ‘Pure’ random choices may also arise when preferential or

rational choices have limited value; see, e.g., Elster [20]. Agranov and Ortoleva ([1], p.

2) study random choice behavior in laboratory experiments. They argue for deliberate

randomization, especially when subjects are dealing with ‘hard’ (i.e., complex) questions.

Levitt [40] reports happiness changes from following actual random choice behavior in a

field experiment.16

Random duality. Mean demands obtained from the statistical model of choice admit

a random dual representation:

E[|0 ≤ · ≤ ] = argmax∈R+

{() : 0 ≤ · ≤ )}. (21)

The demand system () can hence be derived either from randomizing or optimizing

behavior (i.e., utility maximization or expenditure minimization).

Expression (21) does not imply that any classical demand function can be represented

as the mean outcome of random choice behavior. For instance, a random choice procedure

will (a.s.) lead to interior individual and mean demands. Non-interior demands based on a

14Luce’s [42] formulation, and the more general elimination process studied by Tversky [56], can be rep-

resented as random utility models; see, e.g., Anderson et al. ([3], Chap. 2). Tversky [55], Gonzalez-Vallejo

[28], and Roe et al. [51] are additional examples of preferential choice models based on the comparative eval-

uation of the different alternatives. A recent literature in economics studies sophisticated random choices.

A very small set of examples is Fudenberg et al. [24], Gul and Pesendorfer [30], Machina [43], and Manzini

and Mariotti [44]. These papers focus on individual and not on market behavior.15Individual choices are often represented by “observed” choice probabilities. This is the starting point

of the stochastic revealed preference theory developed by McFadden and Richter [47]. The probability that

individual choices lie in ⊂ () is () = () (Xmax). “Observed” choices here are not required

to satisfy any form of stochastic transitivity. The current statistical model of choice is “irrational” not only

in the standard deterministic sense, but also in the more general stochastic sense.16Anecdotal and ethnographic evidence also exists. McGrath ([48], p. 433) observed male shoppers in

gift shops on Christmas Eve tended “to make large, rapid, spontaneous, and often random purchases.”

16

residual “last” commodity also restrict cross-price effects. In the = 2 example in Section

2, non-interior demands 1( ) cannot vary with 2, as 2 is not part of Xmax1 .

To establish random duality starting with a continuously differentiable classical demand

system ∗( ) ≡ (∗1() ∗( )), one requires a density ∗() that satisfiesZ max1

0

Z max (())

0

[ − ∗()] ∗() = 0, for = 1 . (22)

Expression (22) is an orthogonality condition. The density ∗() must exist and be

independent of (). It should also integrate to one over a (sufficiently large) support.

In order to induce a classical demand system, the density ∗() must be such that mean

demands drawn from ∗() agree with ∗() along a ( )-expansion path.

Since () is a truncated mean, random duality requires characterizing a probability

distribution function in terms of its truncated mean, as in Kotz and Shanbhag [38]. By

homogeneity, ∗() can be written as a function of the extreme points of the budget

set (10). These points are Xmax1 = 1 and X

max (1 = 0 −1 = 0) = for

= 2 . Classical demands ∗() can then be written as ∗() ≡ ∗(1 ) with

≡ . As changes, ∗() change and so does the maximum feasible consumptions

that determine (). (The dependence ofXmax on is implicit to avoid cluttered notation.)

Differentiate (22) for commodity with respect to , with 6= . Then, the relevant

density ∗() can be constructed using the inversion formula for truncated means, as in

Z max1

0

Z max−1((−1))

0

Z max (())

0

[ − ∗()]∗() =

∗()

∗(). (23)

This generates a system of partial differential equations. By simultaneously tracing the

-paths, one can trace a density ∗() that induces ∗(). (For the “last” commodity,

the relevant term is of the form [Xmax ((); ) − ∗()]. If demands are non-interior,

Xmax (()) = ∗() for all() so the “last” equation is redundant.) Unfortunately, because

each affects multiple terms in Xmax, it is not generally easy to solve the system (23),

even when = 2. I describe the = 2 case under interior demands in the Appendix, but

the complete characterization of random duality requires a dedicated paper.

17

Even though mean demands can be rationalized, utility functions and welfare measures

have no content in a statistical model of choice. There is no conflict between the “represen-

tative agent” and the disaggregated individuals, as in Mas-Colell et al. ([45], Section 4D). In

here, there is no preferential or behavioral basis for normative assessments. As Mas-Colell

et al. ([45], pp. 121-122) note “[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.”

5 Statistical Power and Rational Choice

Instead of starting from a choice behavior, revealed preference theory confronts observed

individual choices in order to infer an individual’s preferences. In this section, I apply the

general theorems to measure the statistical power to detect rational behavior in revealed

preference tests. 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., Mas-Colell et al. ([45], Chap.2). I consider compensated price changes to ensure testability.17

Revealed preference theory is non-statistical. Thus I formulate the null and the alterna-

tive hypotheses as:

0 : Choices are derived from a deterministic demand function in which no ‘true’

violation of WARP exists but there are (additive) errors to the consumer’s choices.

: Choices are drawn from a statistical model of choice according to a distribution

function ().

The weak axiom is the simplest possible setting for testing of rational economic behavior.

Demands that satisfyWARP do not require symmetry in cross-priced effects and hence allow

for non-transitive behavior. Consistency with utility maximization requires a generalized

axiom of revealed preference (GARP). I present some remarks about GARP later on.

17Tests are uninformative if budget sets do not intersect; see Beatty and Crawford [6]. Intersecting

budget sets is a standard assumption; see, e.g., the equal marginal utility of income (EMUI) assumption in

Echenique et al. [18]. Testability is discussed in detail by Andreoni et al. [4].

18

Tests on the size of the violations. Given a pair of consumption bundles of the

form () and (0 0), the statistic = · (− 0)+0 · ( 0−) measures the potentialprofits or losses of an “arbitrager” that exploits violations of rationality. The positive part

+ ≡ max{0 } and the negative part − ≡ max{− 0} denote the money pump and themoney drain, respectively. The significance of violations of rationality can be measured by

the amount of money ‘pumped out’ of individuals that violate WARP; see Echenique et al.

[18]. Because the asymptotic properties of the money pump + agree almost completely

with those of the -statistic, I first consider tests based on .

Under 0, = ( − 0) · (∗( ) − ∗(0 0)) + ( − 0) · ( − 0), where ∗( ) and

∗(0 0) represent the unobserved ‘true’ choices and and 0 are distributed according to

N ( 2). The test is one-sided and rejects 0 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 [57], under the null of no ‘true’

violations and normality, there is an upper bound statistic given by = (− 0) · (− 0),

with ≥ and with normally distributed with 2 ≡ 2·k− 0k2 ·2 . Given a significancelevel and a value for 2, one can calculate a critical test-value 0 from the normal

distribution according to Pr{ |0} = N ( : { }) = . For the size of the test,

one has Pr{ |0} ≤ .

Define the power of tests based on the -statistic as Pr{ |}. To characterizeasymptotic power, let { : = 1 } denote the sample data. The obser-

vations are obtained at ( ) and the observations at (0 0). Let ≡ ( ) =

( − 0) · ( − ) denote the sample -statistic for the ( )-pair. The -statistic can

be computed for a total of ( − 1)2 pairwise sample comparisons. I consider a simple

matched sample average of the -statistic, ≡ −1X=

, or

= −1X

=(− 0) · ( −). (24)

The -statistic under is standard: converges (weakly) to ≡ (−0) · (()−

(0 0)) ≤ 0 (i.e., → when → ∞), where the inequality follows from Theorems

19

1 and 3. Given 2 ≡ 2 · k− 0k2 · Σ, with Σ as the variance-covariance matrix of and

[ ] = 2 , Chebyshev’s inequality bounds the power function:

Pr{ |} 22 +( − )

. (25)

This upper bound depends on the first two moments of the -statistic under the alternative:

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− 0k and/or a small variance in ) yield

lower statistical power.

The “money pump” statistic. Tests based on the money pump are related to the

-statistic. Under the null, the +-statistic is + = max{0 (−0) · (∗()−∗(0 0))+

(− 0) · (− 0)} and the critical values for + ≡ max{0 } are given by + = max{0 (−0) · ( − 0)}. The critical values are defined analogously as Pr{+ +

|0} = N (+ :{+ +

}) = , which relies on a truncated normal distribution. Define the power of

tests based on the money pump as Pr{+ + |}. Statistical power for the and the

+ statistics decreases with the size of the critical area and + ; see, e.g., (25). Also, the

power function of tests based on + cannot be smaller than the power of tests based on the

-statistic, i.e., Pr{+ +

|} ≥ Pr{ + |} since

+ = + − by definition.

In the limit as →∞, however, both tests behave similarly:

Corollary 3 For a given and well-behaved distribution function () in , the asymptotic

power of revealed preference tests based on the -statistic and the money pump +-statistic

satisfy lim→∞Pr{ |} = lim→∞ Pr{+ +

|} = 0.

Proof. The proof follows from the one-sided Chebyshev inequality (25). Further, notice

that lim→∞ Pr{+ +

|} ≡ lim→∞ Pr{max{0 } + |}. Hence the limit for

the power function of the +-statistic follows from the continuous mapping theorem, i.e.,

lim→∞ Pr{max{0 } + |} = max{0Pr{lim→∞ } +

|} = 0. (The maxfunction is continuous although not differentiable.)

The -statistic and the money pump are inconsistent. A statistical test is consistent if

its power against an alternative hypothesis tends to one as → ∞. (This is a standard

20

requirement in hypothesis testing.) The inconsistency of these tests is not due the use

of a particular distribution function under the alternative hypothesis . At any level of

significance , there is no well-behaved distribution function () that generates a consistent

test of individual rationality based on the - or the +-statistics.

Tests on the number of violations. Tests can also be based on the number of

revealed preference violations. 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: 0· 0 might exceed 0·() thus revealing a violationof 0 being weakly preferred to (), or · () might exceed · thus revealing a

violation of ( ) over . In the case of = 2 in Figure 1, these violations lie in the

segments [2 1] and [1Xmax 001 ]. An indifference curve tangent to these segments would

be concave to the origin leading to an inconsistency with the weak axiom.

The sample distributions of these kinds of violations depend on their frequency, as in

Π+{} ≡ #{ · (

− ( ))}+

, and Π−{} ≡ #{− · (

− ())}+

. (26)

The sample distribution of a violation of either the first or the second kind (or of both) is

given by

Π{()} = Π+{}+ Π

−{}− Π+{}Π

−{}.

The test statistic Π under 0 must be such that 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} = N (0 : {∗(0 0) + 0 ( )}), andΠ−{|0} = N ( : {() ∗() + }).

The size of the test can be determined (or at least bounded) as before so thatΠ{( 0)|0} ≤, where is the significance level. Define the asymptotic power of the test asΠ{( 0)|},or as the value of the test statistic Π under .

21

Corollary 4 For a given and well-behaved distribution function () in , the asymptotic

power of revealed preference tests based on the Π-statistic satisfies

Π{( 0)|} = 1− (( ))[ (Xmax)− (( )) + (0)]

(Xmax 00) (Xmax). (27)

Proof. Under , the power components in (26) converge (strongly) to

lim→∞

Π+{ 0|} = 1− (( ))

(Xmax 00), and lim

→∞Π−{|} = (( ))− (0)

(Xmax), (28)

where Xmax 00 corresponds to a budget set (0 0). Since the power function is given by

Π{( 0)|} ≡ lim→∞ Π{( 0)|}, (27) follows from simple rearrangements.

Tests based on the number of violations are also inconsistent. The asymptotic power for a

test based on Π{( 0)|} equals one if and only if (( )) = 0 or (())− (0) = (Xmax). These conditions essentially require () to be insensitive to price changes. As

in Corollary 5, 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.

Asymptotic power for the Π-statistic, however, is sensitive to the distribution function in

the alternative. To maximize power, (()) = [ (Xmax) + (0)]2, so at most

Π{( 0)|} = 1− [ (Xmax) + (0)]2

4 (Xmax 00) (Xmax),

with the highest value given by (Xmax) = 1 and (0) = 0. Power would increase further

if (Xmax 00) is close to (Xmax). As (Xmax 00) → 1, power approaches 3/4. In contrast,

if (Xmax 00) = 14, asymptotic power would be zero.

Some remarks. A test of individual rationality is consistent if it can, with unlimited

observations, perfectly discriminate between demands that satisfy WARP and demands

based on the statistical model of choice. For a given probability distribution function ()

in , the statistical model of choice should eventually make only choices that violate the

weak axiom.18 This contradicts Theorems 1 and 3, and Corollary 1.

18Consistency in hypothesis testing is defined in a large-sample limit. In numerical examples available in

an Appendix not for publication, I find that random choice behavior leads to well-behaved demand functions

for realistic sample sizes. Inconsistency seesm to be a practical concern.

22

The previous tests are problematic. The null and the alternative hypotheses are not

mutually exclusive. The reason is that 0 and are not standard statistical hypotheses,

i.e., 0 and are not nested. (This is why statistical power is not bounded by size, as in

standard statistical hypothesis testing.) The issue is that economic theory does not impose

any meaningful restriction on the stochastic behavior of individual demands under 0. One

can assume that prices may be mismeasured, that measurement error is multiplicative, or

that there is a stochastic component in the decision process. These specifications change the

meaning of ‘small’ failures under the null but they do not affect power under the alternative

hypothesis.

The literature has considered additional rationality indices.19 Afriat’s efficiency index

measures the maximum “margin of error” across violations of the weak axiom (Echenique et

al. [18] and Varian [58]), theminimum cost index measures the monetary cost of breaking all

revealed preference violations in a given data set (see Dean and Martin [15]), 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 deterministic

indices can be used to formulate statistical tests of rational behavior but it seems unlikely

that they will yield consistency seeing that the statistical model delivers, on average, well-

behaved demand functions.

I 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 , one

can generalize the - and +-statistics as T = −1P

=1

P

=1 · ( − +1), with

+1 = 1 , and T + = max{0 T }. For a finite sample of size , the power of tests based

on the generalized statistics T and T + cannot be smaller than the power of tests based

on the and +-statistic since longer cycles allow for more revealed preference violations.

19Corollary 3 analyzes the average money pump cost. Echenique et al. [18] also studied the median

money pump cost. I have not studied median demands in general. Median demands in Section 2 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 zero.

23

Under the alternative hypothesis, however, mean demands ( ) satisfy GARP since

they are the outcome of maximizing a quasi-concave utility function according to Theorems

2 and 4. Asymptotic statistical power is thus likely to be small for T and T + . Power

calculations for tests of GARP in terms of the number of violations cannot be derived in

closed-form.20

6 Some Final Remarks

This paper derived a well-behaved consumer demand system from a statistical model of

choice with no preferential basis or optimizing behavior. The findings, colloquially and

loosely, represent the economic equivalent of the proverbial typing monkeys. The interpre-

tation 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 numerous objections against the consistency of preferences and maximizing

behavior. These objections are valid but not very forceful in a market context. The testable

predictions of classical demand theory can be reached by an alternative route that abandons

all behavioral assumptions. In this defense of the classical model, which confirms and

extends the one advanced by Becker [7], pure behavioral objections are 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 can be applied to realistic

random choice behaviors rooted in decision theory and aware of systematic departures from

individual rationality.

On the other hand, the fact that random choices are as predictive as sophisticated ra-

tional choices is troublesome; especially for empirical 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 pref-

erences are unrevealing since even a naïve (i.e., statistical) model of choice will pass these

20The literature uses simulations to measure power in general cases; see, e.g., Bronars ([9], p. 695).

24

tests with flying colors. Moreover, and this is a difficulty for demand integrability and wel-

fare analyses, successfully recovering an individual’s preferences from observed choice data

does not necessarily represent a meaningful exercise. Observed choices could, in principle,

be rational yet lack normative content. The findings, in short, diminish the economists’

confidence on the rational choice paradigm.

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29

7 Appendix A: Omitted proofs and derivations

Proof Proposition 1 (completed). Mean demands for 1 were studied in the text.

Consider the mean demand for 2. Some steps in this case are analogous to the partial

proof in the text. Consider first the own-price effect. By definition,

2()

2

¯

=

µ(2)

2

¯

¶− 1( )

2

¯

1

2− 1()1

22,

with

µ(2)

2

¯

¶=

1( )1

22,1()

2

¯

=E [1| 0 ≤ 1 ≤Xmax

1 ]

Xmax1

2( )

1, so

2()

2

¯

= −E [1| 0 ≤ 1 ≤Xmax1 ]

Xmax1

µ2( )

2

¶ 0.

Moreover, the Slutsky matrix is such that |S()| can be written as

|S( )| = 2()

µE [1| 0 ≤ 1 ≤Xmax

1 ]

Xmax1

¶ ¯ −221 1111 −12

¯,

with |S( )| = 0. Symmetry and negative semidefiniteness hold here. Integrability followsfrom these properties; see Katzner ([35], Theorem 4.1-2).

For the Slutsky equation for 1, notice that the Slutsky compensation at income is

= 1( )1. The Slutsky equation (8) follows by definition. For completeness, notice

thatXmax

1

1

¯

=Xmax

1

1+

1( )

1.

The Slutsky equation (8) is a consequence of this expression as well as of (4). The Slutsky

equation for 2 follows from simple rearrangements. For instance, notice that the uncom-

pensated own-price effect is 2()2 = −2( )2 which rules out Giffen goods.Income effects, however, can be negative for 2(). That is,

2()

=

µ1− E [1| 0 ≤ 1 ≤Xmax

1 ]

Xmax1

¶1

2. (A1)

Corollary 5 Under the conditions of Proposition 1, suppose (1) is a log-concave distri-

bution. Then, 2() is a normal good.

Proof. The proof relies on (A1) and the following fact about log-concave distributions

0 ≤ E [1| 0 ≤ 1 ≤Xmax1 ] Xmax

1 ≤ 1; see Goldberger ([27], Appendix A).

Omitted derivations from the general theorems. In order to simplify the deriva-

tions in this section, I first present an auxiliary Lemma that will be used repeatedly. The

Lemma serves to simplify the calculations of price and income effects. The central implica-

tion 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

A1

integral. Economically, this result makes sense: recall that Xmax (()) = − () · (),

which is the only maximum feasible consumption that depends on the entire price vector

and income . Hence, any price change will invariably alter Xmax (()).

Lemma 1 Let ( ) ≡Z max

1

0

Z max (())

0

(), where Xmax is given by (12). Then,

( )

¯

=

Z max1

0

"Z max (())

0

()

#¯¯

(),

=

Z max1

0

Z max−1((−1))

0

(()Xmax (()))

µXmax

(())

¯

¶(),

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 ([36], pp. 307-308).)

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.

()

1

¯

=

"Z max2 (1)

0

Z max (1−1)

0

(1 )1

#1=

max1

Xmax1

1

¯

+

Z max1

0

1

"Z max2 (1)

0

Z max (1−1)

0

(1 )2

#¯¯

1.

The first term is zero since Xmax2 (Xmax

1 ) = (− 1(1))2 = 0. Further, as noted in

the text, sinceXmax2 (Xmax

1 ) = 0, thenXmax = 0 for ≥ 2 also as there would be no income

left for the consumption of these commodities.

The second term becomesZ max1

0

⎧⎨⎩"Z max

3 (12)

0

Z max (1−1)

0

()2

#2=

max2 (1)

Xmax2 (1)

1

¯

⎫⎬⎭ 1

+

Z max1

0

Z max2 (1)

0

1

"Z max3 (12)

0

Z max (1−1)

0

()3

#¯¯

12,

whose first component is also zero as 3Xmax3 (1X

max2 (1)) = 2(X

max2 (1)−Xmax

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

A2

high values of . For instance, the -th step of the sequence of integrals is given by

( )

1

¯

= 0 + + 0 ( − 1 times)

+

Z max1

0

1

"Z max (1−1)

0

Z max (1−1)

0

()

#¯¯

1−1,

where the evaluation of the integral with the upper limit of integration Xmax+1 (1 )

evaluated at =Xmax (1 −1) will also equal zero. Moreover, X

max = 0 for all ≥ .

The last term in the sequence is

( )

1

¯

=

Z max1

0

1

"Z max (1−1)

0

()

#¯¯

1−1,

which under Leibniz’s rule simply becomes

()

1

¯

=

Z max1

0

Z max−1(1−2)

0

(()Xmax (()))

µXmax

(())

1

¯

¶().

Proof of Theorems 1 and 3 (completed). The proof was started in the text and is

completed here. I focus on the case of interior demands but provide the needed adaptations

for the case of a residual commodity at the end of the proof.

Consider (13) and write this expression as () = ( ) (Xmax), with

( ) ≡Z max

1

0

Z max (())

0

(), and

(Xmax) =

Z max1

0

Z max (())

0

(),

as the numerator and the denominator respectively.

Using Lemma 1, notice that

()

¯

=

Z max1

0

Z max (())

0

(()Xmax (()))

µXmax

(())

¯

¶(),

for all 6= and with Xmax (()) taking the place of for = . Likewise

(Xmax)

¯

=

Z max1

0

Z max (())

0

(()Xmax (()))

µXmax

(())

¯

¶(),

A3

for all . The quotient rule implies

( )

¯

=

µ1

(Xmax)

¶ ∙()

¯

− () (Xmax)

¯

¸.

This last expression, upon substitution, yields

()

¯

=

Z max1

0

Z max−1((−1))

0

{ − ( )}µmax(())

(Xmax)

Xmax (())

¯

¶().

(A2)

where, as in the text, max(()) ≡ (()Xmax (())).

A compensated own-price change in only needs to be evaluated in terms of its effect

on Xmax (()). This result implies that price effects are channeled through changes in the

maximum feasible consumption. Recall that feasibility implies

Xmax (1 −1) =

−X−1

=1

µ

¶, (A3)

and that income is compensated according to = ( ). Then,

Xmax (())

¯

=1

∙µ

¶−

¸=

()−

. (A4)

Finally, substitution of (A4) into (A2) yields the own-price effect. For commodity , the

own price effect is given by

()

¯

= −Z max

1

0

Z max−1((−1))

0

©Xmax

(())− ( )ª2µmax(())

(Xmax)

1

¶().

(A5)

The main difference is that has been substituted by Xmax (()), which is a (hyper)

surface along (). Expression (A5) is consistent with the ones in the simple examples.21

The cross-partial effects follow from the appropriate derivative of Xmax (()). For in-

stance

()

¯

=

Z max1

0

Z max (())

0

(()Xmax (()))

µXmax

(())

¯

¶(),

and similarly for the denominator. Following Lemma 1, and repeating the steps just taken,

one can show that:

()

¯

=

Z max1

0

Z max−1((−1))

0

{ − ( )}µmax(())

(Xmax)

Xmax (())

¯

¶(),

(A6)

where Xmax (())

¯= (()− ). Simple substitutions yield (15).

21Notice that in the example of Section 2, the own-price effect (4) can be written as 1()1| =−{Xmax

1 − 1()}2(Xmax1 ) (Xmax

1 )1, which is the analog of (A5).

A4

The only property that deserves some further comment is the negative (semi)definiteness

of S(), which is analogous to the positive (semi)definiteness of Σ. The term S()can be written as a covariance term given by

[ ] =

Z max1

0

Z max−1((−1))

0

{ − ( )} { − ()} (())(),

where (()) is proportional to (()Xmax (())). Consider a vector and the non-

negative quadratic form [ · ( − ( ))]2 = · ( − ( ))( − ()) · . Theconditional expectation (16) satisfies E[ · ( − ( ))]2 = · Σ , which only takesnon-negative values. This is a well-known property of variance-covariance matrices; see Fisz

([23], 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 theseother commodities have a joint density, i.e., their distribution is non-degenerate; see Fisz

([23], p. 90). Notice that Xmax (()) is a linear function of (). That is, for a given

realization of demands (), the maximum feasible consumption Xmax (()) is linear in

(), i.e., Xmax (()) = −()·() for all possible realizations of(); see (12). Notice,

however, that is not necessarily equal to Xmax (()). 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 ( ) = − () · ()( ). If is a residual, then Σ is positive semidefinite but not positive definite since · Σ = 0

holds for some 6= 0. In other words, the -dimensional distribution has a degenerated

character as the probability is not distributed over the whole -dimensional space, but is

concentrated on − 1 dimensions.Preliminaries and remarks for integrability conditions. In terms of notation,

let ( ) denote the expenditure function: ( ) ≡ min≥0{ · : () ≥ }. Thus,( )

= ∗( ) , (A7)

for = 1 and with ∗( ) as the Hicksian demand for , i.e., ∗( ) = ∗( ( )).

Integrability requires a boundedness condition in terms of the partial derivative of mean

demands with respect to income; see Katzner [35]. Boundedness is assumed to hold here

due to finite first and second moments.

Proof of Theorems 2 and 4. Symmetry and a negative definite Slutsky matrix

S( ) is necessary and sufficient for the existence of a solution for the partial differentialequation system (A7); see, e.g., Epstein [21]. These conditions hold here (Theorem 1). The

constant of integration in (A7) cannot be uniquely determined in an incomplete demand

system. (In a complete system, the budget constraint serves to determine the constant of

integration; see, e.g., Katzner ([35], Chap. 4).) In an incomplete demand system, 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. See the remarks below for additional

ways to complete the system.

A5

If the “last” commodity is a residual, () becomes a standard demand system.

A negative semidefinite S() is necessary and sufficient for the solution of the previoussystem to be fully determined. These requirements hold here (Theorem 3). The utility

function can then be recovered from the expenditure function; see, e.g., Katzner ([35],

Chap. 4).

In general, an incomplete system can be completed in many ways. Let 0( 0 ) de-

note 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 max0≥0 {(Ψ(0)) : · + 0 · 0 = }, with Ψ(0) linearly homo-

geneous. Epstein [21] and LaFrance and Hanemann [39] provide additional remarks about

the integrability of incomplete demand systems.

Income effects. Before completing the proof of Corollary 2, I need to define the

notion of “positive association.” The variables random and are said to be “positively

associated” in the positive likelihood ratio dependence sense of Shaked and Shanthikumar

([54], Section 9D) or the Total Positivity order TP2 in Joe ([33], Chapter 1), if

(0 0)( ) (0 )(

0), (A8)

for 0 and 0 , when the − 2 remaining variables are fixed. Positive association

means that one is more likely to observe that and take larger values together and

smaller values together than any mixture of these. Figure 2 depicts the case of negatively

associated variables.

Proof of Corollary 2. To derive income effects (19), one simply needs to evaluate the

appropriate derivative of Xmax (()). Using the notation of the previous proofs,

()

=

Z max1

0

Z max (())

0

(()Xmax (()))

µXmax

(())

¶(),

and similarly for the denominator. Once these expressions are substituted back into the

quotient rule, one obtains (19).

As noted in the text, expression (19) can be written as E[|() ≤ Xmax() =

Xmax ] − E[|() ≤ Xmax

() ≤ Xmax ]. To characterize this difference, I focus on a

pair of commodities as this implies the general case. Let Φ( ) denote the conditional

probability distribution for a feasible value of when = , as in

Φ( ) ≡ Pr{ | ≤Xmax = } =

Z

0

( )Z max

0

(0 )0, (A9)

and define Φ( 0) for

0 similarly. Since is positive, E[| ≤ Xmax

=

] =Xmax −

Z max

0

Φ( ).

A6

Notice that E[| ≤ Xmax = 0] E[| ≤ Xmax

= ] requires

Φ( ) Φ( 0), which, from (A9), corresponds toZ

0

Z max

0

[( )(0 0)− ( 0)(

0 )]0 0.

By assumption, this expression holds when and are “positively associated” in (A8).

A7

8 Appendix B: Additional results [NOTFORPUB-LICATION]

Extended random (“dual”) representations. Mean demands that arise from individ-

uals randomly choosing their consumptions can be represented as demands that arise from

the maximization of some utility function. This sub-section provides a partial examina-

tion of the “converse” problem for two random representations. The first representation

assumes that 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 ∗() arehomogeneous 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

[35] to discuss errors and shocks to individual demand functions. In particular, as in Katzner

([35], p. 161), “if [the consumer] were to choose from the same budget set many times, ‘on

average’ he would choose the utility maximizing bundle.” The goal in the first representation

is to find a distribution function ∗() such that

∗() =Z 1

0

Z

0

∗() ∗()

, (B1)

for all = 1 where ∗() ≡ ∗(1 ) and ∗() is the joint density of ∗().Notice that (B1) integrates random individual demands over a rectangular area instead of

over the triangular area given by (10).

As in the text, the derivations needed to construct the first representation rely on an

inversion formula for the right-truncated mean. Write (B1) as

∗(z)∗() =

Z

0

µ ∗(1 −1 +1 )

¶,

which on differentiation with respect to yields (∗())

∗() + ∗()(∗()) =

(∗()). As in the text, let (z) ≡ [∗()][ − ∗()]. A convenient way to

write the previous expression is ln ∗() = (). Integration yields

∗(() ) = exp

½−Z +∞

(() )

¾, (B2)

where ∗(() ) ≡ ∗(1 −1 +1 ), with the ‘constant’ of integration

determined such that ∗() is a distribution function over its support. This distributionfunction ∗() 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

B1

as a convex combination of the extreme points of the set; see, e.g., Phelps ([1], p. 1). Let

be a compact convex subset of R and let denote the extreme points of . Every point

∈ can be written as a convex combination of z:

=X

=1,

where ≥ 0 andP

=1 = 1. There is a probabilistic interpretation based on Minkowski’s

integral representation; see Phelps [1]:

Theorem 5 (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

that =R(). If is a simplex, is unique.

Proof. Let be the Dirac measure on the point , i.e., for every Borel set B, (B) =1 if ∈ B and zero otherwise. Then, () =

P

=1 = 1 and is a measure with support

{1 }. Therefore,R() =

P

=1

R() =

P

=1 = . Uniqueness can

be established generally for the case of a simplex; see Phelps ([1], Chap. 10).

To apply the previous representation it is enough to notice that the budget set (10) is

convex and compact; see Mas-Colell et al. ([45], p. 22). This means that it is possible to

construct a probability measure concentrated in that is always feasible and represents any

behavioral demand function ∗() as the mean demand from a random choice procedure

that selects bundles on the budget hyperplane. An alternative way to interpret the previous

theorem is by making reference to the coarea integral formula applied to the level set of

(10).

Random random (“dual”) representation for = 2. Consider as an example with

interior demands. Standard differentiation in (22) for ∗2(), in terms of 2, yieldsZ max1 ()

0

½[Xmax

2 (1; )− ∗2()]∗(1X

max2 (1; ))

Xmax2 (1; )

2

−∗2()

2

Z max2 (1;)

0

∗()2

)1 = 0.

One can rewrite the previous term asZ max1 ()

0

"

2log

Z max2 (1;)

0

∗()2 − 2(1; )

#1 = 0, (B3)

for 2(1; ) ≡ (Xmax2 (1; )− ∗2())

−1∗2(), so upon integrationZ max2 (1;12)

0

∗(1 2)2 = exp

½−Z ∞

2

2(1; 1 )

¾,

for a ‘constant’ of integration . By varying 2 it is possible to trace the density along the

2 dimension, while holding 1 constant.

B2

Now consider ∗1(). Using integration by parts, expression (22) can be written as

[Xmax1 ()− ∗1()]

∗(Xmax1 ()Xmax

2 (1; )) =

Z max1 ()

0

∗(1Xmax2 (1; ))1.

Differentiation with respect to 1 yields

0 = 1()−

1log ∗(Xmax

1 ()Xmax2 (1; ))

− 1

[Xmax1 ()− ∗1()]

1

∗(Xmax1 ()Xmax

2 (1; ))

Z max1 ()

0

1 ∗(1X

max2 (1; ))1,

for

1() ≡ ∗1()1

1

Xmax1 ()− ∗1()

.

This equation is more complex than (B3) due to the fact that as 1 varies, bothXmax1 ()

and Xmax2 (1; ) change. If there is a solution, one can then trace the cumulative density

∗() over its support as the extreme points (1 2) vary. The constant of integration mustbe then such that the density integrates to one over its support.

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 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 0005 and the second 005. 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 shows that ‘individual’ demands are on average negatively sloped with an

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

B3

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.

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

ε −1.0086 −1.0095 −1.0061 −1.0005 −1.0011 −1.0015 −1.0007 −1.0001std.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]

R2 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

ε −1.101 −0.9901 −0.9966 −0.9977 −0.9992 −1.0013 −0.9998 −1.0002std.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]

R2 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.

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 ε = −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 123. The draws are from a log-normal distribution with

mean [−05−05−05], variances [2 2 2] and with positive pairwise correlations betweenall variables of 05.

In the following simulations 1 varies in incremental steps of 005, 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,

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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.

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.

Table B2. Simulation results for multivariate log-normal distribution.

Number of individuals aggregated

= 1 10 100 1 000 10 000

ε −0.0965 −0.2325 −0.2598 −0.2524 −0.2541std.err. (0.723) (0.363) (0.107) (0.0384) (0.022)

std.dev. [2.301] [0.614] [0.184] [0.056] [0.018]

R2 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 cross-

sample 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] Phelps, R.P. (2001) Lectures on Choquet’s Theorem, Berlin: Springer.

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