Demand Systems With Unobserved Heterogeneity andUnrestricted Engel Curves

by

Arthur Lewbel and Krishna Pendakur*

Abstract

We propose a class of �exible utility derived consumer demand systems, called "Pseudo-

Marshallian" demands, that can have the following properties: 1. Budget shares are

nearly linear in parameters. 2. Engel curve shapes for each good are almost com-

pletely unrestricted and can have any rank. 3. Like the Almost Ideal Demand System

(AID), the model has budget shares that are nearly linear in log prices, but, unlike

the AID system, our model is not restricted to linearity in log de�ated total expendi-

tures, e.g., it can be a polynomial or spline of any order in de�ated expenditures and

in other observed heterogeneity characteristics. 4. Error terms correspond to unob-

served preference heterogeneity. 5. Demand functions are estimated using ordinary

GMM, and, like the AID system, an approximate model can be estimated by linear

regression methods.

1 Introduction

Recent empirical nonparametric and semiparametric work with large consumer ex-

penditure data sets reveals income expansion paths (Engel curves) with signi�cant

curvature and variation across goods. For example, some goods have Engel curves

that are close to linear or quadratic, while others are more S shaped (see, e.g. Banks,

*Department of Economics, Boston College, USA and Department of Economics, Simon FraserUniversity, CANADA. This version is June, 2005. We would like to thank David Donaldson forsuggesting Theorem 2, and Stefan Sperlich.

1

Blundell and Lewbel 1997 and Blundell, Chen and Kristensen 2003). Typical para-

metric demand models cannot encompass this variety of shapes. We propose a class

of �exible cost (expenditure) functions characterizing utillity, and associated budget

share demand functions, that have the following properties.

1. Budget share demand functions are nearly linear in parameters.

2. Budget share Engel curves for each good are almost completely unrestricted,

that is, for one price regime, Engel curves can have almost any functional form, and

the resulting demand system can have any rank up to J � 1, where J is the numberof goods.

3. Like Deaton and Muellbauer�s (1980) Almost Ideal Demand (AID) System ,

our models have budget shares that, up to an income de�ator, are linear or close to

linear in log prices. However, unlike the AID system, our models are not restricted to

linearity in log de�ated total expenditures y, e.g., they can be polynomials or splines

of any order in y.

4. Error terms in our model can interpreted as unobserved preference hetero-

geneity parameters. This unobserved preference heterogeneity is locally coherent and

invertible.

5. Observable parameters that a¤ect tastes, such as demographic variables, can

a¤ect demands in the same way as the unobserved heterogeneity, or more generally

can also appear in the model interacting in general ways with y, e.g., the demand

functions can be polynomials or splines of any order in both y and in demographic

variables.

6. The demand functions can be estimated using ordinary nonlinear instrumental

variables via GMM. Like the AID, approximate versions of our models can be esti-

mated by linear regression methods. We �nd almost no empirical di¤erence between

the exact GMM and approximate linear regression estimates.

The methodology begins by specifying a cost (expenditure) function and uses

Shephard�s lemma to obtain Hicks demands. Then, instead of obtaining Marshallian

demands from Hicks demands by solving for utility in terms of total expenditure, we

solve for demands as functions of log prices and log de�ated (or real) total expendi-

tures y, where the de�ator is similar to a Stone index. We call the results Pseudo-

Marshallian demand functions. These Pseudo-Marshallian demands circumvent the

2

di¢ culty of �nding simple analytic expressions for indirect utility or Marshallian de-

mands. A similar idea is Browning�s (2001) �M-demands,�which expresses demand

functions in terms of prices and the quantity of one good, instead of in terms of

prices and total expenditures, though in our case the goal is obtaining simple �exible

functional forms rather than coping with missing or mismeasured total expenditures.

De�ne a J�vector of logged prices p, a J�vector of budget shares w, and loggedtotal expenditure x. Assume an individual chooses w by maximizing a regular utility

function subject to a linear budget constraint. We wish to explicitly include both

observable and unobservable sources of preference heterogeneity in the model, so let

z be an L� vector of observable demographic (or other) characteristics that a¤ect

preferences, and let " be a J�vector of unobserved preference characteristics (tasteparameters). De�ne the log cost or expenditure function x = C(p; u; z; ") which

equals the minimum log-expenditure required for an individual with characteristics

z; " to attain utility level u when facing log prices p.

Two models serve to illustrate this paper�s methodology. The �rst is

C(p; u; z; ") = u+ p0a+ p0Bs(u; z) + (p0Ap)1=2 + p0"

where a is a J�vector of constants with 10ja = 1, A is a J � J symmetric matrixof constants with 10jA = 00J , B is a J � L matrix of constants with 10jB = 00L, and

s(u; z) is an L�vector of arbitrary functions such as polynomials, splines, logistictransforms of polynomials (to remain bounded), or almost any other functional form

we choose. In our empirical application, we let s(u; z) = (u�1, u, u2,..., u5, z, uz)0.

By Shephard�s lemma, this cost function has Hicksian (compensated) budget shares

w = a + Bs(u; z) + Ap(p0Ap)�1=2 + " and it can be readily checked from these

formulas that C(p; u; z; ") = u+ p0w, which means that

w = a+Bs(y; z) +Ap(p0Ap)�1=2 + "

where

y = x� p0w.

Also, in the base time period where p is the zero vector, we get Engel curves w =

3

a+Bs(x; z) + ", so Engel curves in this price regime may have almost any shape.

This model is an example of what we call Pseudo-Marshallian demand functions,

meaning that budget shares are expressed in terms of log de�ated expenditures y

instead of log nominal expenditures x. Unlike Deaton and Muellbauer�s (1980) AID

system, here the logged Stone index p0w is not an approximation, but is instead the

exact, correct logged de�ator for x. This model does not have a closed form expression

for indirect utility or ordinary Marshallian demand function except for very special

choices of s(x; z), but it can still be readily estimated because it expresses budget

shares as functions of observables p, z, and y.

The key is de�ning a cost function that makes y ordinally equivalent to u, thereby

allowing us to replace u with y in the Hicksian demand functions. The resulting model

has budget shares that are �exible in prices, additive in random utility parameters

(the errors terms "), and has Engel curves that are almost unrestricted functions,

such as polynomials of arbitrary degree, in total expenditures and in other observable

characteristics.

A second example is the closely related speci�cation

C(p; u; z; ") = u+ p0a+ p0Bs(u; z) + p0Ap=2 + p0"

This cost function has the simpler Hicksian budget shares w = a+Bs(u; z)+Ap+"

and for this model we obtain C(p; u; z; ") = u+p0w+p0Ap=2, which means that we

have Pseudo-Marshallian budget share demand functions

w = a+Bs(y; z) +Ap+ "

where now

y = x� p0(w �Ap=2).

Log de�ated expenditures y are more complicated in this model, depending on A

instead of only being a function of observed variables as in the previous model. How-

ever, in this model the Pseudo-Marshallian demand functions are completely linear

in parameters given y, while still maintaining �exibility in prices, additive random

utility parameter errors, and arbitrary Engel curve functional forms.

4

We provide general theorems showing the restrictions on cost functions required

to construct pseudo-Marshallian demands. We then show that, with approximations

similar to those used for AID system estimation, approximate versions of these mod-

els are fully linear, and can be estimated by least squares regression. We show that

instrumental variable and GMM estimators can be used to deal with the endogeneity

that results from having a regressor y that depends on w, though we also �nd that

accounting for this endogeneity has very little e¤ect empirically. We provide exten-

sions and additional mathematical properties of pseudo Marshallian demands, which

are relevant for evaluating these models and for other applications of our general

methodology.

2 Tricks With Hicks

Ignore preference heterogeneity for now, that is, consider z and " as �xed. We will

reintroduce them again later. De�ne preferences to be regular if they can be repre-

sented by a log cost function C(p; u) that satis�es: exp[C(p; u)] is concave, increas-

ing, di¤erentiable and homogeneous of degree one in (unlogged) prices exp(p) and

is monotonically increasing and di¤erentiable in u. In most cases, explicit functions

we provide for C(p; u) will not be globally regular, that is, they may only be regular

over some limited range of values of p; u, in which case the reported results will only

be valid over these limited ranges. Shephard�s lemma relates Hicksian (compensated)

budget shares to regular cost functions by

w = !(p; u) = rpC(p; u):

The Stone de�ator is de�ned as the price index equal to the share-weighted geo-

metric mean of prices, that is, as exp(p0w). De�ne log stone-de�ated expenditure, y,

as y = x � p0w. If y = u (or more generally if y is ordinally equivalent to u) then

real expenditures, which hold utility constant when prices change, are equal to Stone

de�ated expenditures. De�ne the Stone de�ator to be exact if u = y = x � p0w, orequivalently if

u = x� p0w = C(p; u)� p0[rpC(p; u)]: (1)

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Exactness of the Stone de�ator implies w = !(p; x � p0w), which transforms theHicksian demands into a Pseudo-Marshallian demand system, which is based solely

on observables. Let 1J denote the J�vector of ones.

Theorem 1: Assume C(p; u) is regular. De�ne d(p; u) � C(p; u)�u. The Stonede�ator is exact for C(p; u) if and only if d(�p+1J�; u) = �d(p; u)+� for any scalars

� and �.

Proofs are in the Appendix. Theorem 1 can be equivalently stated as the require-

ment that ed be linearly homogeneous in ep, which follows from ordinary cost function

homogeneity, and also that d be linearly homogeneous p, which makes u = y. Both

homogeneity conditions must hold for the Stone index to be the exact de�ator of a

regular cost function.

Proofs are in the Appendix. By ordinality, instead of de�ning exactness by y = u,

we could have de�ned it by y = h(u) for any strictly monotonically increasing function

h, and Theorem 1 would then hold with d(p; u) de�ned by C(p; u) � h(u) There isno gain in generality from doing so, because by ordinality, C(p; u) � h(u) has thesame indi¤erence curves, and hence the same Marshallian and Pseudo-Marshallian,

demand functions, as C(p; h�1(u)) � u = eC(p; u) � u, and in this representationof the cost function (i.e., in this cardinalization of utility) we would have y = u as

before. We will for convenience write cost functions in the cardinalization that makes

u = x� p0w instead of h(u) = x� p0w.The following Corollary and theorem illustrates the restrictiveness of the con-

straint that d(�p+ 1J�; u) = �d(p+ 1J�; u), as required for the Stone de�ator.

Corollary 1: Assume the Stone de�ator is exact for C(p; u). Then Hicksiandemands !(p; u) = rpC(p; u) are homogeneous of degree zero in p and in exp(p).

Cost function regularity implies that Hicksian demands do not change when all

prices are scaled by a constant factor. Corollary 1 shows that Stone index exactness

requires that Hicksian demands also not change when all prices undergo the same

power transformation, for example, when all prices are squared. This also rules out

Hicksian demands that are linear in p.

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Theorem 2: Consider the case where J = 2. Then d(�p1 + �; �p2 + �; u) =

�d(p1; p2; u)+� for any scalars � and � if and only if d(p1; p2; u) = m(u)p1+[1�m(u)]p2for some function m(u).

Theorem 2 shows that, with J = 2 goods, the only cost functions C(p; u) that have

an exact Stone de�ator are linear in p, which we may write as C(p; u) = u+ p0m(u)

for any J�vector of functionsm(u) that satisfym(u)01J = 1. This has an exact Stonede�ator for any J , but also has Hicksian budget shares m(u) that are independent of

p.

Systems having more than J = 2 goods can be nonlinear in p. One example is

C(p; u) = u+ p0m(u) + [p0M(u)p]1=2

where m(u) is as above andM(u) is a J by J symmetric matrix-valued function of u

withM(u)1J = 0J . The �rst model described in the introduction is a special case of

this example. Note that, as the above example shows, we could add a term of the formhp0 eApi1=2 u to the log cost function of the �rst model in the introduction, which wouldthen add the term eApy(p0 eAp)�1=2 to the Pseudo-Marshallian demands, allowing for�exible interactions between prices and real Stone de�ated total expenditures in the

demand functions.

Another example is

C(p; u) = u+ p0m(u) + [p0M(u)p] =p0n(u)

where m(u) and M(u) are as above and n(u) is a J�vector function of u withn(u)01J = 0. The functions m(u), M(u), and n(u) can depend in arbitrary ways

on u, with cost function regularity only imposing inequality constraints on these

functions.

3 Obtaining Flexible Linearity by Generalizing the De�ator

In the models of the previous section, Pseudo-Marshallian shares require the unattrac-

tive feature of not changing when prices are squared, and require nonlinear terms like

7

(p0Ap)1=2 in the cost function to have any relative price e¤ects. We can avoid these

drawbacks by generalizing the form of the permitted de�ator, thereby relaxing the ho-

mogeneity restriction required by Stone de�ator exactness. Speci�cally, now consider

log real total expenditures de�ned by

y = x� p0w � t(p) (2)

for some function t(p). Exactness for the more general de�ator p0w + t(p) requires

y = u with y given by equation (2), which implies

u = x� p0w � t(p) = C(p; u)� p0[rpC(p; u)]� t(p). (3)

Theorem 3: Assume C(p; u) is regular. Again let d(p; u) � C(p; u)� u. Thereexists a function t(p) that makes the de�ator p0w+t(p) exact for C(p; u) if and only if

d(p+1J�; u) = d(p; u)+� andrud(�p; u) = �rud(p; u) for any scalars � and �. Also,

if the de�ator p0w+ t(p) is exact for some cost function, then t(p+ 1J�) = t(p) + �.

Theorem 3 describes the general conditions required for exactness of de�ators

of the form p0w + t(p). Comparing Theorems 3 and 1 shows that the di¤erence

between this exactness and Stone de�ator exactness is that the latter requires linear

homogeneity of d(p; u), while the former only needs linear homogeneity of rud(p; u).

The following theorem provides a convenient way to construct such cost functions.

Theorem 4: Assume the function C(p; u) satis�es the homogeneity conditionsrequired to be a log cost function with an exact Stone de�ator, as given by Theorem

1. Assume C(p; u) = C(p; u)+T (p) is a regular log cost function. Then the de�ator

p0w + t(p) is exact for C(p; u), where t(p) = T (p)� p0 [rpT (p; u)].

It follows from Theorem 4 that we may take any log cost function described in the

previous section, and add to it a function T (p) that satis�es T (p+ 1J�) = T (p) + �,

to obtain a log C(p; u) that satis�es the necessary price homogeneities for exactness

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of p0w + t(p). In particular, this means that C(p; u) = u + p0m(u) + T (p) will

satisfy the necessary price homogeneities for exactness of p0w + t(p) where t(p) =

T (p) � p0 [rpT (p; u)], given any function T (p) having the property that exp [T (p)]

is homogeneous of degree zero in exp(p). An example of particular interest is

C(p; u) = u+ p0m(u) + p0Ap=2

where m(u)01J = 1, A is symmetric, and A1J = 0J . This has Hicks demands

!(p; u) =m(u)+Ap and, by Theorem 3, has Pseudo-Marshallian demandsw =m(y)+

Ap where y = x�p0 (w �Ap=2). This is the model of Pendakur and Sperlich (2005),although they estimate its ordinary Marshallian demands, leaving m nonparametric

and numerically solving for u. The second model described in the introduction has

this form. By allowing the de�ator to depend on Ap=2, we can have prices appear

linearly and �exibly in the Pseudo-Marshallian demand functions.

4 Preference Heterogeneity

We wish to explicitly include preference heterogeneity in the model, both from ob-

servable sources z and unobservable sources ". A simple way to include these in the

models without interfering with required price homogeneities is to include them in

the functions m(u). Example models provided in the previous sections then have the

general form

C(p; u; z; ") = u+ p0m(u; z; ") + (p0Ap)1=2 (4)

or

C(p; u; z; ") = u+ p0m(u; z; ") + p0Ap=2

where m(u; z; ")01J = 1. Equation (4) has Stone index exactness, while the other

equation has exactness of the de�ator p0 (w �Ap=2). Note that u may also be im-plicitly a function of z; ", since by ordinality, replacing u with h(u; z; ") for any func-

tion h that is strictly monotonically increasing in its �rst element leaves Marshallian

demands unchanged.

9

A convenient general form for m(u; z; ") is

m(u; z; ") = a+Bs(u; z) + "

where s(u; z) are polynomials, splines, or other known functions of u and z. This then

yields the models described in the introduction. These models are

C(p; u; z; ") = u+ p0a+ p0Bs(u; z) + (p0Ap)1=2 + p0" (5)

which has a Stone index de�ator and Pseudo-Marshallian demands

w = a+Bs(y; z) +Ap(p0Ap)�1=2 + ", y = x� p0w (6)

and

C(p; u; z; ") = u+ p0a+ p0Bs(u; z) + p0Ap+ p0" (7)

which has a more general de�ator, but linear in parameters Pseudo-Marshallian de-

mands

w = a+Bs(y; z) +Ap+ ", y = x� p0w � p0Ap=2. (8)

The complete set of properties required for de�ator exactness and cost func-

tion regularity of these models are a01J = 1, A1J = 0J, 10JB = 0K, "01J = 0,

p0Brus(u; z) > �1, and concavity of exp[C(p; u)] (a su¢ cient condition for concav-ity is that A be negative semide�nite). These constraints are assumed to hold for

every value that x;p; z; " can take on, and hence every value that u;p; z; " can take

on.

5 Models For Estimation

The model we estimate has cost function

C(p; u; z; ") = u+p0�a+Ap=2 + b�1u

�1 + b1u+ b2u2 + :::+ b5u

5 +Cz+Dzu+ "�

(9)

where each bk is a J�vector of constants with b0k1J = 0, and C and D are J by L

matrices of parameters with 10JC = 0L and 10JD = 0L. This model has log real total

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expenditures y = x� p0w � p0Ap=2 and pseudo-Marshallian budget shares

w = a+Ap+ b�1u�1 + b1y + b2y

2 + :::+ b5y5 +Cz+Dzy + ". (10)

This model is the special case of equations (7) and (8) that has s(y; z) = (y�1; y; y2; :::; yK ; z0; z0y)0

and B = (b�1 : b1 : ::: : b5 : C : D). Typical elements of z would include house-

hold size, age, and composition. This model has Engel curves that are high order

polynomials. Unlike Marshallian demands, our pseudo-Marshallian demands can be

polynomials of any degree K without being bound by Gorman (1981) and Lewbel

(1991) type rank restrictions. Note, however, that monotonicity of the cost function

does place inequality constraints on s(y; z), which restrict the range of possible pa-

rameter values and the range of values of p; y; z; " for which these demand functions

satisfy regularity. Instead of polynomials, we could alternatively have taken s(y; z) to

be splines, or bounded functions such as trigonometric functions or logistic transfor-

mations of polynomials, which would automatically bound estimated budget shares.

In our model the D parameters allow for interactions between y and z. Demand

functions having interactions of z and y with p are also possible, e.g., by adding the

termhp0 eApi1=2 u to the log cost function. Other possible extensions are discussed in

the Appendix.

6 Approximate Fully Linear Model

The demand functions (8) are linear in parameters except for the term p0Ap in the

de�ator. A similar nonlinearity appears in Deaton and Muellbauer�s (1980) AID

system and Banks, Blundell, and Lewbel�s (1997) QUAID system, and can be dealt

with in an analogous way by replacing it with an observable approximation.

In the models given by equations (7) and (8), consider approximating log real

expenditure y with nominal total expenditures de�ated by an average Stone price

index, that is, replace y = x� p0w � p0Ap=2 with ey de�ned byey = x� p0w (11)

11

for some set of budget shares w. Then equation (8) becomes

w � a+Ap+Bs(ey; z) + ".and, in particular, our budget share demands (10) become

w � a+Ap+ b�1ey�1 + b1ey + :::+ bKeyK +Cz+Dzey + ". (12)

For K = 1 and D zero, this is identical to the popular approximate Almost Ideal

Demand System (AID). Note, however, that our exact model with K = 1 di¤ers from

the exact AID, in that the exact AID has a de�ator for x is that is quadratic in p,

while our model�s exact de�ator includes both a quadratic in p and the Stone index.

Unlike the AID system, our model permits s(ey; z) to have any functional form, suchas a polynomial of any order. A special case is the model w � a+Ap+ b1ey+b2ey2+Cz+ ", which is the model estimated by Blundell, Pashardes, and Weber (1993).

Their motivation for this model was by analogy with the Almost Ideal, but if this

model was really to be Marshallian with a, b1, and b2 constant then, as they show,

utility maximization would require linear rank restrictions on these coe¢ cients, which

they did not impose. The above approximate model therefore provides a rationale for

the model they actually estimated.

The approximate model (12) can be estimated by linear regression methods, with

linear cross-equation symmetry restrictions on theA coe¢ cients. A natural choice for

w is the sample average of budget shares across consumers. A better approximation to

y would be to let w be each consumer�s own w, so each consumer has their own Stone

index de�ator based on their own budget shares. As discussed later, this introduces

endogeneity, but dealing with the endogeneity makes the estimator be just linear three

stage least squares with linear cross-equation constraints on the A coe¢ cients.

In our later empirical application, we estimate the approximate model withw = w

using seemingly unrelated regressions ignoring endogeneity, and we estimate the exact

model accounding for endogeneity. We �nd empirically that these approximate model

estimates did not di¤er much from the exact model estimates.

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7 Applying the Model

Consider evaluating the cost for each individual of a price change from p0 to p1. A

consumer surplus measure for this price change is the log cost of living index, which

for our general model (4) is given by

C(p1; u; z; ")� C(p0; u; z; ") = (p1 � p0)0 [a+m(u; z; ")] + p01Ap1=2� p00Ap0=2.

If C(p0; u; z; ") is the cost function of a household that has budget shares w, then

this expression simpli�es to

C(p1; u; z; ")� C(p0; u; z; ") = (p1 � p0)0 (w �Ap0) + p01Ap1=2� p00Ap0=2.= (p1 � p0)0w + (p1 � p0)0A(p1 � p0)=2:

The �rst term in this cost of living index is (p1�p0)0w the share-weighted geomet-ric mean of the price changes. Such indices are commonly used on the grounds that

they are appropriate for small price changes and they allow for arbitrary unobserved

preference heterogeneity across households. In our model, the presence of the second

term, which depends upon A, allows us to explicitly model substitution e¤ects, and

so consider large price changes, while also accounting for the behavioural importance

of both observed and unobserved heterogeneity.

De�ne semielasticities to be derivatives of budget shares with respect to log prices

p, log real total expenditures y, log nominal total expenditures x, and demographic

characteristics (or other observed taste shifters) z. The semielasticity of a budget

share can be converted into an ordinary elasticitiy by dividing by that budget share.

We present semielasticities because they are easier to present algebraically. Our gen-

eral model (4), and the speci�cations (7) and (9) have Hicksian demands

!(p; u; z; ") = a+Ap+m(u; z; ")

= a+Ap+Bs(u; z) + "

= a+Ap+ b1u+ b2u2 + :::+ bKu

K +Cz+Dzy + "

13

so the Hicksian price semielasticities are

rp0!(p; u; z; ") = A.

These are equivalently the price semielasticities holding real expenditures y �xed, and

the real total expenditures semielasticities are

ry!(p; y; z; ") = rym(y; z; ") = Brys(y; z)

= b1 + 2b2y + :::+KbKyK�1 +Dz.

Marshallian elasticities can be calculated by writing the general form of the de-

mand functions as

w = a+Ap+m(x� p0w � p0Ap=2; z; ") (13)

and taking the total derivative of w with respect to p and x. Details regarding these

and other elasticity calculations are given in the Appendix.

One could also numerically solve x = C(p; u; z; ") for u, with C given by equations

(4), and substitute the result into the Hicksian demand functions, to obtain Marshal-

lian demands. The appendix also provides partial characterizations and closed form

approximations of these Marshallian demands.

8 Estimation

We are estimating demand systems with J goods, so as usual we can drop the last

equation (the J�th good) from the system and just do estimation on the remaining

system of J � 1 equations. The parameters of the the J�th good are then recoverablefrom the adding up constraint that budget shares sum to one. Assume this is done

in all of the following discussion.

The approximately linear model of equations (12) and (11) is the simplest one

to estimate. Assuming " is homoskedastic and uncorrelated with p and s(ey; z), thismodel �ts the form of ordinary SUR (Seemingly Unrelated Regressions). Without im-

posing symmetry of the A matrix, estimating each equation separately by ordinary

14

least squares is consistent and equivalent to SUR. Symmetry of the A matrix imposes

linear cross equation equality constraints on the coe¢ cients. The SUR estimator is

asymptotically equivalent to maximum likelihood assuming " is normal. This estima-

tor remains consistent if " is heteroskedastic, but in that case it will be more e¢ cient

to use the Generalized Method of Moments (GMM), based on the moments de�ned

by p and s(ey; z) uncorrelated " and by " having mean zero. SUR estimates from the

linear approximate model can be used to obtain good parameter starting values for

exact model estimation. If we let ey = x� p0w, then endogeneity of w means that "

will not be uncorrelated with s(ey; z). However, in our empirical application we �ndthat the bias from ignoring this endogeneity is small, probably because variation in

the endogeneous portion of y; that is, variation in the p0" component of p0w, is small

relative to variation in x and the other components of y.

The exact model of equation (8) is

w = a+Ap+Bs [x� p0w � p0Ap=2; z] + " (14)

and the version we estimate has matrix B and functions s described by equation

(10). A complication for estimation of these models is that w, which is endogenous,

appears on the right side of this expression. One possible estimator is to ignore

this endogeneity and simply estimate the system of equations (14) by nonlinear least

squares.

To formally account for endogeneity, nonlinearity, and heteroskedasticity, we use

an instrumental variables estimator. Let r be an M�vector of observable variablesthat are uncorrelated with ", which will be used as instruments for estimation. If

E(" j x;p; z) = 0J then r can include any bounded functions of p, z, and x. However,if unobserved heterogeneity is correlated with some observed characteristics such as x

or elements of z, then those elements should be excluded from r. Let r = (r1,...,rM)0.

It then follows from E(" j r) = 0J that

E [(w � a�Ap�Bs (x� p0w � p0Ap=2; z)) r`] = 0J , ` = 1,...,M (15)

If " is homoskedastic, then one possible estimator based on these moment conditions is

nonlinear three stage least squares. Otherwise, allowing for generally heteroskedastic

15

", parameters may be estimated by applying GMM to this set of moment conditions.

By adding up, the moments associated with the J�th good are super�uous, so with

M instruments we will have (J � 1)M moments, which is a rather large number of

moments. The matrix A is constrained to be symmetric.

Writing this system linearly asw = a+Ap+Bs(y; z)+" suggests that good instru-

ments r should be highly correlated with s(y; z) and p. We assume E(" j p; x; z) = 0Jand take r to be s(y; z) and p, using y = x� p0

�w �Ap=2

�where w is the average

budget shares across consumers in our sample, and A is the estimated value of A

based on estimation of the linear approximate model. Speci�cally, the instruments

we use for the model (10) are therefore 1, y�1, y, y2,...,y5, z, yz, and p. Note that

use of w and possible inconsistency of A in the construction of y only a¤ects the

quality of the instruments r and hence e¢ ciency of the GMM estimation, but does

cause inconsistency of the GMM, because r remains uncorrelated with " given any

choice of these parameters w and A. Cross terms such as xp could also be included

as additional instruments. Reasonable starting values for the GMM estimator may

be obtained from the linear approximate model.

Since we have assumed E(" j p; x; z) = 0J , we could interpret Bs(y; z) as a sieveapproximation to a general unknown vector of smooth functionsm(y; z), where s(y; z)

are elements of a spanning basis. Ai and Chen (2003) provide associated rates, optimal

instrument construction, and limiting distribution theory for this GMM estimator,

which can attain the semiparametric e¢ ciency bound for parameters such as A.

The parametric GMM estimator can be readily modi�ed to deal with possible

measurement error or endogeneity in some regressors, by suitably modifying r. For

example, if log total-expenditure x su¤ers from measurement error then one might

use functions of income (eg, log income) instead of functions of x in r to instrument

the problem. Similarly, if unobserved preference heterogeneity is correlated with

some observed taste shifters (i.e., elements of z), then those may be excluded from

the instrument list and replaced by, e.g, nonlinear functions x and of the remaining

elements of z. However, in these cases one would need to take care in interpreting the

resulting model residuals b", because they will then contain both unobserved preferenceheterogeneity and measurement error. With panel data one might separate these two

e¤ects by modeling the unobserved preference heterogeneity using standard random

16

or �xed e¤ects methods.

These estimators do not impose the inequality constraints implied by regularity

of the cost function, or equivalently by utility maximization. If the model is correctly

speci�ed then imposing these constraints would not be binding asymptotically. These

inequalities can be checked using the elasticity calculations, e.g., the estimated Slutsky

matrix can be checked for negative semide�niteness.

9 Empirical Work

The data used in this paper come from the following public use sources: (1) the Family

Expenditure Surveys 1969, 1974, 1978, 1982, 1984, 1986, 1990, 1992 and 1996; (2)

the Surveys of Household Spending 1997, 1998 and 1999; and (3) Browning and

Thomas (1999), with updates and extensions to rental prices from Pendakur (2002).

Price and expenditure data are available for 12 years in 5 regions (Atlantic, Quebec,

Ontario, Prairies and British Columbia) yielding 60 distinct price vectors. Prices are

normalised so that the price vector facing residents of Ontario in 1986 is (1; :::; 1),

that is, these observations de�ne the base price vector.

Table 1 gives summary statistics for 19276 observations of rental-tenure house-

holds located in cities with at least 30,000 residents. The empirical analysis uses

annual expenditure in nine expenditure categories: food-in, food-out, rent, cloth-

ing, household operation, household furnishing & equipment, private transportation

operation, public transportation and personal care. Personal care is the left-out

equation, yielding eight expenditure share equations which depend on 9 log-prices

and log-expenditure. These expenditure categories account for about three quarters

of the current consumption of the households in the sample.

We estimate expenditure-share equations of the form (10) corresponding to the

cost function (9), where z is a the vector of 7 demographic characteristics reported

in Table 1. For our speci�cation of m, we use powers of y in [�1; :::; 5] and allow forinteractions between z and y, so that

w = a+Ap+ b�1y�1 + b1y + b2y

2 + :::+ b5y5 +Cz+Dzy + ":

To estimate, we need instruments orthogonal the behavioural heterogeneity parame-

17

ters " to deal with endogeneity of log real-expenditure y. As discussed in the previous

section, we use (y)k as an instrument for yk, where y = x� p0�w �Ap=2

�, w is the

average budget shares across consumers in our sample, and A is a pre-estimate of

A. This instrument y ranges from 7.68 to 11.57, and has somewhat lower vari-

ance than log expenditure. To mitigate numerical di¢ culties associated with our

high-order polynomial in y, we subtract 7.5 from log-expenditure (and thus from log

real-expenditure and its instrument as well).

When symmetry is not imposed on A, the model is exactly identi�ed, and when

symmetry is imposed on A, then the model has (J � 1) (J � 2) =2 = 28 overidentify-ing moments. With symmetry constraints, the model has 188 parameters. The model

is estimated by the GMM, with starting values obtained by seemingly unrelated re-

gressions estimation of the linear approximate model. We obtained convergence in

�ve iterations, due to the near linearity of the model. We obtain a test statistic for

overidentifying restrictions of 1836, distributed under the null as a �228. Since the

overidentifying restrictions are symmetry restrictions, this is a joint test of instru-

ment validity and Slutsky symmetry. It is likely that this rejection of symmetry

is due to not in part to constraining A to be constant across all consumers. This

constraint could be relaxed by letting A depend upon z.

Tables 2-4 in the appendix show estimated coe¢ cients for our model. The left

side of each table gives GMM estimates for the exact model, and the right side of

the table gives estimates for the linear approximate model (estimated using SUR)

which ignores both the endogeneity of y and the appearance of p0Ap=2 in y. Table

2 gives parameters describing the shape of the Engel curve, a and B. Table 3 gives

parameters describing demographic e¤ects, C and D. Table 4 gives parameters

describing price e¤ects, A. A summary of own price e¤ects is:

18

j wj Ajj

food-in 0.22 -0.47

food-out 0.06 -0.45

rent 0.33 0.097

clothing 0.10 0.029

operation 0.08 -0.033

furnish/equip 0.05 0.046

private trans 0.09 0.023

public trans 0.03 -0.002

Note that the Hicks elastiticity of the quantity of good j with respect to its own price

is (Ajj=wj)� 1.Turning to income e¤ects, Figures 1 and 2 show estimated budget share equations

from GMM estimates for 8 of the 9 budget shares for a household where z = 0, a

single childless adult aged 40. Each budget share function is evaluated at the base

price vector p = 0J (that faced by residents of Ontario in 1986) at each percentile of

the distribution of y for single childless adults. In each curve, there are 99 dots rep-

resenting 99 percentiles. At di¤erent price vectors, the expenditure-share equations

over y are shifted vertically by Ap, and so have the same shape over log real expendi-

ture y. For households with di¤erent demographic characteristics, expenditure-share

equations are shifted vertically by Cz and have a linear e¤ect Dzy added to them.

The estimated expenditure-share equation for Food at Home is almost linear,

which is consistent with many other parametric and nonparametric studies that �ne

nearly linear food shares. The Rent expenditure-share equation exhibits substantial

nonlinearity beyond what a quadratic model could capture. The private transporta-

tion operation equation is approximately S-shaped, and the clothing equation appears

more like two linear pieces joined with a curve. Figure 2 suggests two in�ection points

for the public transportation equation and one in�ection point at very low expenditure

for the food out equation with near linearity above the median..

Figures 3 and 4 overlay a quadratic version of our model for selected expenditure

shares. This quadratic version is given by

w = a+Ap+ b1y1 + b2y

2 +Cz+Dzy + ";

19

and, in comparison with our baseline model, lacks both the term in the reciprocal

of log real-expenditure, and higher-order terms in log real-expenditure. This model

di¤ers from the Quadratic Almost Ideal (QAI) model of Banks, Blundell, and Lewbel

(1997) in that it has a di¤erent de�ator for total expenditures, and does not require

the coe¢ cient of squared log real-expenditures to depend on prices. Our speci�cation

also allows for more general demographic e¤ects than is typical in applied QAI models,

which usually only have terms similar to C, but not D.

Here, we see that non-quadratic features may be important in a few expenditure-

share equations. At the lower end of the expenditure distribution, the rent share

is mapped poorly by the quadratic model, overstating rent shares by as much as

5 percentage points for the poorest households. Similarly, the expenditure share of

poorer households on private transportation operation is understated by the quadratic

model, and indeed is predicted to be negative for the very poorest households. The

private transportation operation equation is also over-estimated for the second quintile

of the expenditure distribution by as much as 3 percentage points.

Turning to Figure 4, we see some small e¤ects at the bottom of the expendi-

ture distribution: public transportation shares are overstated and food out shares

overstated by the quadratic model.

Figures 5 and 6 overlay SUR estimates of the approximate model

w � a+Ap+ b�1ey�1 + b1ey + b2ey2 + :::+ b5ey5 +Cz+Dzey + "where approximate log real-expenditure ey = x � p0w is treated as though it were

exogenous and exact. The message of these �gures is that the SUR does a remarkably

good job of getting the shape of expenditure-share equations right. The bias due

to the endogeneity of approximate log real expenditure is small and to measurement

error resulting from the exclusion of p0Ap=2 in that approximation is pretty small.

10 Conclusions

We have provided a utility derived consumer demand system that is as �exible in

price responses and as close to linear in parameters as the Almost Ideal Demand

System, but also allows for virtually unrestricted �exibility in Engel curve shapes for

20

each good. Error terms in the model correspond to coherent and invertible unobserved

preference heterogeneity parameters. Demand functions are estimated by GMM, and,

like the AID system, an approximate model can be estimated by linear regression. We

empirically estimate the model allowing Engel curves to be high order polynomials,

and �nd some signi�cant departures from linear or quadratic demands.

The model is an application of a new general methodology we propose for con-

structing demand systems, which we call pseudo-Marshallian demands. These are

essentially the Hicks demands associated with cost functions having the property that

utility can be represented by total expenditures de�ated by an easily constructed price

de�ator, such as a variant of a Stone index.

References

Ai, Chunrong and Xiaohong Chen, 2003, E¢ cient Estimation of Models with

Conditional Moment Restrictions Containing Unknown Functions, Economet-

rica, 71(6), pp 1795-1944.

Banks, Blundell, R. and A. Lewbel. 1997. Quadratic Engel curves and consumer

demand, Review of Economic studies 79, 527�539.

Beckert, W. and R. Blundell, 2004, Invertibility of Nonparametric Stochastic

Demand Functions, Birkbeck Working Papers in Economics and Finance No.

406.

Blackorby, C., and D. Donaldson. 1993, "Adult-Equivalence Scales and the

Economic Implementation of Interpersonal Comparisons of Well-Being." Social

Choice and Welfare 10, 335-61.

Blundell, Richard, Xiaohong Chen and Dennis Kristensen, 2003, Semi-

Nonparametric IV Estimation of Shape-Invariant Engel Curves, Centre for Mi-

crodata Methods and Practice, working paper CWP15/03.

Blundell, R., A. Duncan and K. Pendakur. 1998. "Semiparametric Estimation

of Consumer Demand", Journal of Applied Econometrics 13, pp 435-461.

21

Blundell, R. , and A. Lewbel. "The Information Content of Equivalence Scales."

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Blundell, R. W., P. Pashardes, and G. Weber (1995), �What do we learn about

Consumer Demand Patterns from Micro Data?�, American Economic Review,

vol. 83, 570-597.

Brown, D. and R. Matzkin, 1995, "Estimation of a Random Utility Model From

Data on Consumer Demand,"

Browning, Martin and Irena Thomas, 1999, Prices for the FAMEX, unpublished

working paper, Department of Economics, McMaster University.

Browning, Martin, 2001, Modelling Commodity Demands and Labour Supply

with M-Demands, unpublished working paper, Institute of Economics, Univer-

sity of Copenhagen, 21 pages.

Deaton, A. S., and J. Muellbauer (1980), �An Almost Ideal Demand System,�

American Economic Review, vol. 70, 312-326.

Gorman, W. M. (1981), "Some Engel Curves," in Essays in the Theory and

Measurement of Consumer Behaviour in Honor of Sir Richard Stone, ed. by

Angus Deaton, Cambridge: Cambridge University Press.

Lewbel, A. "Household Equivalence Scales and Welfare Comparisons." Journal

of Public Economics 39 (1989): 377-91.

Lewbel, A. (1991), �The Rank of Demand Systems: Theory and Nonparametric

Estimation,�Econometrica, 59, 711-730.

Pendakur, K. 1999. "Estimates and Tests of Base-Independent Equivalence

Scales", Journal of Econometrics 88(1), pp 1-40.

Pendakur, Krishna. 2002. "Taking Prices Seriously in the Measurement of In-

equality", Journal of Public Economics, October 2002, 86(1), pp 47-69.

22

Pendakur, Krishna and Stefan Sperlich, 2005, Semiparametric Estimation of

Consumer Demand Systems with Almost Partially Linear Price E¤ects, unpub-

lished working paper, Department of Economics, Simon Fraser University.

11 Appendix

This appendix provides proofs, extensions, and additional mathematical properties of

our models, which are relevant for evaluating these models and for other applications

of our general methodology.

11.1 Proofs

Proof of Theorem 1: The condition that d(�p+1J�; u) = �d(p; u)+� is equivalentto d(p; u) being linearly homogeneous in p and exp[d(p; u)] linearly homogeneous

in exp(p). The latter condition is required by cost function regularity. For the

former, given equation (1) we have C(p; u) � u = p0 [rpC(p; u)]. This is equivalent

to d(p; u) = p0 [rpd(p; u)], which by the Euler Theorem shows that d(p; u) � u islinearly homogeneous in p of degree one.

Proof of Corollary 1: Homogeneity of degree zero in exp(p) follows from or-

dinary cost function regularity, and homogeneity of degree zero in p follows from

equalling the derivative of a linearly homogeneous function.

Proof of Theorem 2: Since d(�p1 + �; �p2 + �; u) = �d(p1; p2; u) + � holds

for any � and �, let � = 1=(p2 � p1) and � = �p1=(p2 � p1) to obtain d(0; 1; u) =[d(p1; p2; u)= (p2 � p1)]� [p1= (p2 � p1)]. The theorem holds withm(u) = 1�d(0; 1; u).

Proof of Theorem 3: The condition that d(p+1J�; u) = d(p; u)+� is equivalentto exp[d(p; u)] linearly homogeneous in exp(p), which is required by cost function

regularity. Given equation (3) we have C(p; u) � u � t(p) = p0 [rpC(p; u)]. This is

equivalent to d(p; u)� t(p) = p0 [rpd(p; u)]. Taking the derivative of this expression

with respect to u gives

rud(p; u) = p0 [rprud(p; u)] (16)

23

which by the Euler Theorem shows that rud(p; u) is linearly homogeneous in p,

that is, rud(�p; u) = �rud(p; u). In the other direction, if rud(p; u) is linearly

homogeneous then equation (16) holds. Integrate this equation with respect to u

(de�ning t(p) be the constant of integration) and continue following the above steps

in reverse to obtain equation (3). Given exactness, Equation (3) implies t(p) =

u�C(p; u)+p0!(p; u). Ordinary homogeneity of cost functions and Marshallian de-mands requires that exp [u� C(p; u) + p0!(p; u)] be linearly homogeneous in exp(p),so t(p+ 1J�) = t(p) + �.

Proof of Theorem 4: C(p; u) = C(p; u) + T (p), so

C(p; u)� p0[rpC(p; u)] = C(p; u)� p0[rpC(p; u)] + T (p)� p0 [rpT (p; u)]

= u+ T (p)� p0 [rpT (p; u)]

where the second equality follows from Stone de�ator exactness for C(p; u) from

Theorem 1, and this expression by (3) implies exactness of the de�ator p0w+T (p)�p0 [rpT (p; u)].

Proof of Theorem 5: Let R be a scalar valued function of rpC(p; u) and p

that is strictly monotonic in u. Since R(p;!(p; u)) is strictly monotonic in u, it can

be inverted to obtain u = S(p; R), and we may de�ne a function g by g(w;p) =

C(p; S(p; R(w;p)))� S(p; R(w;p)).

11.2 Extensions

Instead of Stone index or related de�ators, we could more generally de�ne de�ated

log total expenditures by

y = x� g(w;p) (17)

for any de�ator function g. Exactness of the de�ator g(w;p) would then require that

y = u, and therefore that

C(p; u) = u+ g(rpC(p; u);p) (18)

24

The resulting Pseudo-Marshallian demand functions would then be

w = ! [p; x� g(w;p)] . (19)

where !(p; u) = rpC(p; u). At this level of generality, there exists a de�ator g(w;p)

that is exact for almost any log cost function C(p; u), as the following Theorem shows.

Theorem 5: Assume C(p; u) is regular. A su¢ cient condition for existence of afunction g(w;p) that is an exact de�ator for C(p; u) is the existence of some scalar

valued function of rpC(p; u) and p that is strictly monotonic in u.

Theorem 5 provides an extremely weak su¢ cient condition for existence of g,

and hence existence of pseudo-Marshallian demands. For example, this condition

is satis�ed if any good or combination of goods has a budget share that is strictly

increasing or strictly decreasing in total expenditures (and hence in utility). Moreover,

this condition is su¢ cient but not necessary, e.g., if preferences are homothetic, then

C(p; u) = u+t(p) for some function t and we may let g(w;p) = �t(p). The only kindsof situations where g might fail to exist is when all budget shares are independent of

u for some (but not all) ranges of values of u.

To give an example of Theorem 5, suppose that the Hicksian budget share function

for good 1, w1 = !1(p; u), is invertible in u, so u = !�11 (p; w1). Then a de�ator g

that makes u = y is g(w;p) = C(p; !�11 (p; w1))� !�11 (p; w1).The function g satisfying equation (18) is not unique, e.g., it can be freely trans-

lated by an ordinal transformation of u, and in the previous example, di¤erent in-

vertible budget shares could be used to obtain di¤erent expressions for g. For appli-

cations, the main point of Pseudo-Marshallian demands is not existence, but rather

convenience for demand estimation, which is why this paper focused on examples

where g has simple closed forms, like the log Stone de�ator.

One potentially interesting generalization of the cost function (4), is the general

class of models

x = C(p; u; z; ") = u+ c(p; z; ")0 em(u; z; ") + d(p; z; ") (20)

25

where a is a J�vector of constants with a01J = 1, em(u; z; ") and c(p; z; ") areJ�vector valued functions, d(p; z; ") is a scalar valued function, 1+ru em(u)0c(p; z; ") >0, em(u)01J = 1, exp[c(p; z; ")] is di¤erentiable and homogeneous of degree one in

prices, exp [d(p; z; ")] is homogeneous of degree zero in prices, rpc(p; z; ") is nonsin-

gular, and exp[C(p; u; z; ")] is concave. By Shephard�s lemma the Hicksian budget

shares for this class are

w = !(p; u; z; ") = rpc(p; z; ")0 em(u; z; ") +rpd(p; z; ") (21)

Solving this equation for em(u; z; ") and substituting the result into equation (20)gives the Pseudo-Marshallian demand functions

w = rpc(p; z; ")0 em(y; z; ") +rpd(p; z; ")

where real log total expenditures y are given by

y = x� d(p; z; ") + c(p; z; ")0 [rpc(p; z; ")0]�1[w � d(p; z; ")]

Assuming d(p; z; ") and c(p; z; ") are independent of " and em(y; z; ") is additive in aconstant vector a and in " gives the model

x = C(p; u; z; ") = u+ c(p; z)0 [a+m(u; z) + "] + d(p; z) (22)

which has real log expenditures and pseudo Marshallian demand functions

y = x� d(p; z) + c(p; z)0 [rpc(p; z)0]�1[w � d(p; z)] (23)

w = rpc(p; z)0 [a+m(y; z)] +rpd(p; z) + e". (24)

This makes y depend only on observables, and has Pseudo-Marshallian demand func-

tions that are additive in errors e" = rpc(p; z)0". These errors are conditionally

mean zero if " is conditionally mean zero, so this general model can still be esti-

mated by GMM given parameterizations of the functions c(p; z), d(p; z), andm(y; z).

Our estimated models are the special case of equation (22) where c(p; z) = p,

26

d(p; z) = p0Ap=2, and m(y; z) = Bs(y; z). The generalizations above show how

further interactions between p, z, and y can be introduced in our model, if empiri-

cally required.

11.3 Coherency and Invertibility

Coherency of a structural model is de�ned as the property that, for each value of

the exogenous variables and errors, there exists a unique corresponding value of the

endogenous variables, and invertibility is the property that a unique value of errors is

associated with each value of the endogenous and exogenous variables. In the demand

system context (see, e.g., Brown and Matzkin (1995) or Beckert and Blundell (2004)),

coherency requires that a unique value of budget shares w be associated with each

possible value of p; x; z; ", and invertibility requires that a unique value of errors "

be associated with each possible value of p; x; z;w.

Consider the class of cost functions (22). Invertibility of this class (and hence of

our estimated models) is established by observing that, by equation (24),

" = [rpc(p; z)0]�1[w �rpd(p; z)]� a�m(y; z)

where y is given by equation (23). Coherency follows from C(p; u; z; ") having all

of the properties of a regular cost function, which ensures existence of Marshallian

demands (treating " as preference parameters), even though we do not have a closed

form analytic expression for them. This coherency is local in the sense that it requires

joint restrictions on variables, errors and parameters to attain the assumed regularity

of C.

11.4 Closure Under Unit Scaling

A desirable feature of demand models is that they be closed under unit scaling, that

is that a change in the units that goods are measured in only changes the values of

the parameters or functions that de�ne the model. The general model (22) is closed

under unit scaling. To see this, observe that changing the units of measurement is

equivalent to replacing p with p+k for some J�vector of constants k. This changes

27

the cost function to

C(p; u; z; ") = u+ c(p+ k; z)0 [a+m(u; z) + "] + d(p+ k; z)

= u+ c�(p; z)0 [a+m(u; z) + "] + d�(p; z)

so changing the units just changes the functions c(p; z) and d(p; z) but leaves the

general form of the model unchanged.

The general form of the model we estimated, equation (4) is also closed under unit

scaling, since

C(p; u; z; ") = u+ (p+ k)0 [a+A(p+ k)=2 +m(u; z; ")]

= u+ k0m(u; z; ") + k0(a+Ak=2) + p0 [(a+Ak) +Ap=2 +m(u; z; ")]

= u� + p0 [a� +Ap=2 +m�(u�; z; ")]

where a� = a+Ak, u� = h(u; z; ") = u+k0m(u; z; ")+k0(a+Ak=2), andm�(u�; z; ") =

m(h�1(u�; z; "); z; "), where h�1 denotes the inverse of h(u; z; ") with respect to its

�rst element, and invertibility of h(u; z; ") follows from strict monotonicity of the

cost function with respect to u. This reinforces the point made earlier that u may

also be implicitly a function of z; ", since by ordinality, replacing u with h(u; z; ")

for any function h that is strictly monotonically increasing in its �rst element leaves

Marshallian demands unchanged.

Although the general form of our model is closed under unit scaling, speci�c

parametric functional forms for m might not be, since the functional form for m�

need not be the same as the functional form for m (though they are the same when

m is linear in its arguments). When m� does not have the same functional form

as m, closure under unit scaling can be attained by introducing k as an additional

parameter vector to be estimated. The presence of a implies that, without loss of

generality, we may take k01J = 0. In our empirical application, attempts to estimate

a parameter k were always either completely insigni�cant or failed to converge.

28

11.5 Additional Elasticity Calculations

This section provides elasticity calculations in addition to those provided in the section

on applying the model. Marshallian semielasticities are obtained by taking total

derivatives and solving. With

w = a+Ap+m [x� p0(w �Ap=2); z; "] ,

Marshallian total expenditure semielasticities are

rxw(p; x; z; ") = ry!(p; y; z; ") [rx(x� p0(w(p; x; z; ")�Ap=2))]= rym(y; z; ") [1� p0rxw(p; x; z; ")]

= rym(y; z; ")� [rym(y; z; ")]p0rxw(p; x; z; ")

= [IJ +rym(y; z; ")p0]�1rym(y; z; ")

where IJ is the J by J identity matrix. Marshallian price e¤ects are most easily

obtained from the above Hicksian price e¤ects and Marshallian total expenditure

e¤ects by the Slutsky matrix

rp0w(p; x; z; ") = rp0!(p; u; z; ")� [rxw(p; x; z; ")]!(p; y; z; ")0.

From the general model, Hicksian demographic semielasticities are

rz0!(p; u; z; ") = rz0m(u; z; ") = Brz0s(u; z).

These are equivalently the demographic semielasticities holding real expenditures y

�xed. Marshallian demographic semielasticities are obtained by

rz0w(p; x; z; ") = rz0m(y; z; ")�rym(y; z; ")p0rz0w(p; x; z; ")

= [IJ +rym(y; z; ")p0]�1rz0m(y; z; ").

The derivative of log nominal expenditures x with respect to utility u is

ruC(p; u; z; ") = 1 + p0 [rum(u; z; ")] = 1 + p

0 [rym(y; z; ")]

29

and the derivative of the indirect utility function with respect to x equals 1=ruC(p; u; z; ").

Some of the above elasticity expressions depend on " either directly or through w.

Given estimated model parameters, " for each consumer can be estimated as the model

residuals (the di¤erence between �tted and observed w). If each consumer is only

observed onece, the empirical distribution of the estimated residuals " can be used

as an estimate of the population distribution of unobserved preference heterogeneity.

Empirical moments or quantiles of " are generally consistent estimates of population

moments or quantiles. If the same consumers are observed more than once (panel

data), then each observation of the same consumer will likely yield a di¤erent ", which

can either be interpreted as time varying preferences, or the mean of the estimated

" for each individual could be modeled as a �xed preference e¤ect, with remaining

variation attributed to measurement error.

Similarly, in any price regime, features of the distribution of y = x�p0 (w �Ap=2)may be estimated by calculating the corresponding features of the distribution esti-

mated y across consumers observed in that price regime. For example, to estimate the

mean and variance of the average elasticities of budget shares with respect to x, based

on the general model (4) and the linear in coe¢ cients speci�cation (??), we may takethe sample mean and variance of each element of [IJ + [Brys(yi; zi)]p

0]�1 [Brys(yi; zi)]

divided by the corresponding element of wi across all consumers i that faced prices

p, where yi = xi�p0 (wi �Ap=2), and parameters are replaced with their estimatedvalues. Moreover, since yi is ordinally equivalent to ui, social welfare calculations can

be based directly estimates of each consumer�s yi.

If Marshallian demand functions are of direct interest, evaluated at points other

than those observed in the sample using the above formulas, they can be obtained

numerically by, e.g., numerically solving x = C(p; u; z; ") for u, with C given by equa-

tion (4), and substituting the result into the Hicksian demand functions. However, as

the above equations show, this will often not be necessary for evaluating the e¤ects

on demand or welfare of price, expenditure, or demographic changes.

11.6 Shape Invariance

A property of some demand functions that is relevant for the construction of equiv-

alence scales, and has been found to at least approximately hold empirically in some

30

consumer demand applications, is shape invariance. See, e.g., Blundell and Lewbel

(1991), and Blundell, Duncan and Pendakur (1998), Pendakur (1999), and Blundell,

Chen, and Kristensen, (2003). In our notation, shape invariance can be de�ned as

Marshallian Engel curves of the form

w = h [x� t0(z; ")] + t1(z; ")

for some function t0 and some J�vector valued functions h and t. In a graph ofengel curves t1(z; ") is a vector of vertical shifts of each budget share and t0(z; ") is

a horizontal shift of x. Utility derived demand functions possess shape invariance if

and only if

C(p; u; z; ") = h0[p; h(u; z; ")] + h1(p; z; ") (25)

for some functions h0, h1, and h2. Shape invariance is the only observable implica-

tion of equivalence scale exactness or independence of base of equivalence scales (see

Lewbel 1989, Blackorby and Donaldson 1993, and Pendakur 1999), which is the same

as equation (25) with the restriction that h(u; z; ") = h(u) for some function h.

A special case of the class of cost functions (22) is

x = C(p; u; z; ") = u+ c(p)0 [a+m0(u) +m1(z) + "] + d(p; z) (26)

which satis�es the above de�nition of shape invariance. The cost function (9) is

a special case of equation (26) when D is a matrix of zeroes. In that case, the

Marshallian Engel curves of this model simplify to w = h [x� �0(Cz+ ")] +Cz+ "for some J�vector valued function h. This shows that our model with zeroD providesa functional form for vertical shifts t1(z; "), and restricts the horizontal shift t0(z; ")

to equal a weighted sum of the vertical shifts. Nonzero D in our model violates shape

invariance by providing more �exibility with respect to z than simple horizontal and

vertical shifts.

An alternative model speci�cation that also provides interactions between y and

z while satisfying shape invariance is the cost function

C(p; u; z; ") = u+ p0�a+Ap=2 + b1(u� dz) + :::+ bK(u� dz)K +Cz+ "

�(27)

31

where d is a J�vector of constants with d01J = 0. This again has y = x � p0w �p0Ap=2 and has pseudo-Marshallian budget shares

w = a+Ap+ b1(y � dz) + :::+ bK(y � dz)K +Cz+ ". (28)

The cost function (27) is a special case of equation (26), and so is shape invariant.

This model has Marshallian Engel curves of the form w = h(x� dz��0") +Cz+ "for some J�vector valued function h, and so has vertical shifts t1(z; ") = Cz+ ",

and horizontal shifts dz � �0". We empirically estimated this model, but do notreport results because this model yielded very similar Engel curves and elasticities,

but had multiple local minima and other numerical problems that were likely due to

the nonlinearity dz: This alternative model might be prefered in applications involving

equivalence scale calculations, where independence of base is desired.

11.7 Indirect Utility and Marshallian Demands

Our model does not have an explicit analytic expression for indirect utility or for

Marshallian demands, except under special cases, but this section provides some par-

tial characterizations of these functions. Implicitly solving the cost function equation

(9) for u gives indirect utility

u = V [x� p0(Cz+ ");p0Dz;p]

where the function V is a root of a K 0th order polynomial which depends upon the

parameters a, A, b1; :::bK . This can be substituted into the associated Hicksian

demand functions w = a+Ap+b1u+ :::+bKuK +Cz+Dzu+" to get Marshallian

demands. Similarly, solving the cost function (27) for u gives indirect utility

u� = v[x� dz� p0(Cz+ ");p] (29)

where v is also a root of a K 0th order polynomial, and u� = u�dz is an unobservablemonotonic transformation of utility. Applying Roy�s identity to equation (29) shows

32

that Marshallian demands for this model have the form

w = h[x� dz� p0(Cz+ ");p] +Cz+ " (30)

where the J�vector valued function h is the negative of the ratio of the derivativesof v with respect to it�s second and �rst arguments.

The following approximation may provide additional insight into the structure

of Marshallian demands. Substituting y = x � p0w � p0Ap=2 into (8) and Taylorexpanding the result around the point p = 0J gives, with the general cost function

(7), w � a+Ap+Bs(x; z) + ", and substituting this expression into (??) yields

y � x� p0 (a+Ap=2 +Bs(x; z) + ") : (31)

Finally, substituting this approximate expression for y into the Pseudo-Marshallian

demand function (8) gives approximate Marshallian demands

w � a+Ap+Bs [x� p0 (a+Ap=2 +Bs(x; z) + ") ; z] + "

These expressions show that while our prefered model speci�cation has no interaction

terms between p and z,y, there do exist interactions between p and z,x.

11.8 Engel Curves

Evaluating equation (8) at some arbitrary price vector p = p gives Pseudo-Marshallian

Engel curves

w = �+Bs(y; z) + " (32)

with

y = x� � �0w (33)

where � = a+Ap, � = p, and = ��0A�=2. The Pseudo-Marshallian Engel curvesfor models (9) and (27) are therefore w = �+ b1y+ :::+ bKyK +Cz+Dzy+ " and

w = �+ b1(y � dz) + :::+ bK(y � dz)K +Cz+ " with y = x� � �0w. Based on

33

equation (31), approximate Marshallian Engel curves are given by equation (32) with

y � x� � �0 [�+Bs(x; z) + "] (34)

By equation (30), the exact Marshallian Engel curves for the (27) model have the

shape invariant form

w = h(x� dz� �0") +Cz+ "

where d = d� �0C, for some J�vector valued function h.

11.9 Data and Parameter Estimates

Table 1: The Data Min Max Mean Std Dev

expenditure shares food-in 0.00 0.81 0.22 0.09

food-out 0.00 0.64 0.06 0.10

rent 0.01 0.95 0.33 0.06

clothing 0.00 0.59 0.10 0.06

operation 0.00 0.64 0.08 0.05

furnish/equip 0.00 0.65 0.05 0.06

private trans 0.00 0.59 0.09 0.08

public trans 0.00 0.45 0.03 0.04

log-expenditure 6.58 11.61 9.38 0.64

childless single adult dummy 0 1 0.36 0.48

childless adult couple dummy 0 1 0.22 0.41

log household size no children 0 1.95 0.21 0.37

w/ children 0 2.40 0.46 0.64

single parent dummy 0 1 0.06 0.23

age of head (less 40) no children -15 24 1.11 9.73

w/ children -15 24 -1.42 5.05

34

Table 2: Engel Curve Parameters, GMM and

SUR

GMM-exact SUR-approximate

Commodity Parameter Estimate Std Err Estimate Std Err

Food at Home a 0.344 0.751 0.303 0.177

Food at Home y1 0.052 1.340 0.135 0.346

Food at Home y2 -0.243 1.141 -0.319 0.321

Food at Home y3 0.137 0.499 0.174 0.153

Food at Home y4 -0.033 0.108 -0.041 0.036

Food at Home y5 0.003 0.009 0.004 0.003

Food at Home yinv -0.016 0.151 -0.015 0.033

Food Out a 0.251 0.135 0.210 0.135

Food Out y1 -0.416 0.270 -0.323 0.265

Food Out y2 0.385 0.253 0.299 0.246

Food Out y3 -0.159 0.121 -0.122 0.117

Food Out y4 0.033 0.028 0.025 0.027

Food Out y5 -0.003 0.003 -0.002 0.003

Food Out yinv -0.037 0.024 -0.029 0.025

Rent a -0.441 0.786 -0.372 0.231

Rent y1 1.679 1.411 1.540 0.453

Rent y2 -1.215 1.208 -1.077 0.421

Rent y3 0.383 0.532 0.312 0.200

Rent y4 -0.057 0.116 -0.039 0.047

Rent y5 0.003 0.010 0.001 0.004

Rent yinv 0.124 0.158 0.109 0.043

Clothing a 0.168 0.290 0.169 0.107

Clothing y1 -0.095 0.498 -0.120 0.209

Clothing y2 -0.002 0.416 0.041 0.195

Clothing y3 0.029 0.181 0.001 0.093

Clothing y4 -0.010 0.039 -0.002 0.022

35

Clothing y5 0.001 0.003 0.000 0.002

Clothing yinv -0.007 0.063 -0.008 0.020

HH Operation a -0.044 0.114 -0.076 0.127

HH Operation y1 0.124 0.236 0.214 0.248

HH Operation y2 -0.154 0.233 -0.253 0.230

HH Operation y3 0.095 0.117 0.147 0.110

HH Operation y4 -0.024 0.029 -0.038 0.026

HH Operation y5 0.002 0.003 0.004 0.002

HH Operation yinv 0.007 0.020 0.010 0.023

HH Furn/Eq a -0.129 0.131 -0.058 0.129

HH Furn/Eq y1 0.320 0.263 0.123 0.253

HH Furn/Eq y2 -0.298 0.246 -0.098 0.235

HH Furn/Eq y3 0.147 0.117 0.054 0.112

HH Furn/Eq y4 -0.031 0.027 -0.011 0.026

HH Furn/Eq y5 0.002 0.002 0.001 0.002

HH Furn/Eq yinv 0.034 0.023 0.038 0.024

Priv Trans a 0.747 0.226 0.617 0.180

Priv Trans y1 -1.560 0.401 -1.255 0.353

Priv Trans y2 1.411 0.346 1.092 0.328

Priv Trans y3 -0.567 0.155 -0.407 0.156

Priv Trans y4 0.104 0.034 0.067 0.037

Priv Trans y5 -0.007 0.003 -0.004 0.003

Priv Trans yinv -0.110 0.047 -0.093 0.033

Pub Trans a -0.018 0.084 0.072 0.094

Pub Trans y1 0.088 0.171 -0.092 0.185

Pub Trans y2 -0.045 0.160 0.124 0.171

Pub Trans y3 0.000 0.076 -0.081 0.082

Pub Trans y4 0.005 0.018 0.024 0.019

Pub Trans y5 -0.001 0.002 -0.003 0.002

Pub Trans yinv 0.013 0.014 -0.001 0.017

36

Table 3: Demographic E¤ects, GMM and SUR

GMM-exact SUR-

approximate

Commodity Parameter Estimate Std Err Estimate Std Err

Food at Home childless couple 0.009 0.014 0.013 0.011

Food at Home head age (kids) 0.002 0.001 0.001 0.001

Food at Home head age (no kids) 0.003 0.000 0.003 0.000

Food at Home log size (kids) 0.220 0.010 0.214 0.007

Food at Home log size (no kids) 0.200 0.018 0.199 0.014

Food at Home single parent -0.015 0.013 -0.003 0.011

Food at Home y1*childless couple -0.009 0.006 -0.010 0.005

Food at Home y1*head age (kids) 0.000 0.000 0.000 0.000

Food at Home y1*head age (no kids) -0.001 0.000 -0.001 0.000

Food at Home y1*log size (kids) -0.047 0.004 -0.046 0.003

Food at Home y1*log size (no kids) -0.040 0.007 -0.041 0.006

Food at Home y1*single parent 0.000 0.006 -0.005 0.005

Food Out childless couple -0.016 0.008 -0.017 0.009

Food Out head age (kids) 0.000 0.000 0.000 0.001

Food Out head age (no kids) -0.001 0.000 -0.001 0.000

Food Out log size (kids) -0.018 0.005 -0.023 0.006

Food Out log size (no kids) -0.035 0.010 -0.038 0.011

Food Out single parent 0.006 0.006 0.001 0.008

Food Out y1*childless couple 0.005 0.004 0.005 0.004

Food Out y1*head age (kids) 0.000 0.000 0.000 0.000

Food Out y1*head age (no kids) 0.000 0.000 0.000 0.000

Food Out y1*log size (kids) -0.011 0.003 -0.008 0.003

Food Out y1*log size (no kids) 0.000 0.005 0.003 0.005

Food Out y1*single parent -0.009 0.003 -0.007 0.004

Rent childless couple 0.017 0.017 0.019 0.015

Rent head age (kids) -0.001 0.001 -0.001 0.001

37

Rent head age (no kids) -0.001 0.000 -0.001 0.000

Rent log size (kids) -0.196 0.011 -0.174 0.010

Rent log size (no kids) -0.188 0.020 -0.183 0.019

Rent single parent 0.016 0.018 0.000 0.014

Rent y1*childless couple 0.000 0.007 -0.001 0.006

Rent y1*head age (kids) 0.000 0.000 0.000 0.000

Rent y1*head age (no kids) 0.001 0.000 0.001 0.000

Rent y1*log size (kids) 0.073 0.005 0.063 0.004

Rent y1*log size (no kids) 0.063 0.008 0.059 0.008

Rent y1*single parent 0.008 0.008 0.015 0.007

Clothing childless couple 0.005 0.006 0.004 0.007

Clothing head age (kids) 0.003 0.000 0.003 0.000

Clothing head age (no kids) 0.000 0.000 0.000 0.000

Clothing log size (kids) 0.002 0.005 0.000 0.005

Clothing log size (no kids) 0.061 0.007 0.061 0.009

Clothing single parent -0.018 0.008 -0.014 0.007

Clothing y1*childless couple -0.003 0.002 -0.003 0.003

Clothing y1*head age (kids) -0.002 0.000 -0.002 0.000

Clothing y1*head age (no kids) 0.000 0.000 0.000 0.000

Clothing y1*log size (kids) 0.001 0.002 0.001 0.002

Clothing y1*log size (no kids) -0.029 0.003 -0.029 0.004

Clothing y1*single parent 0.025 0.004 0.023 0.003

HH Operation childless couple -0.008 0.009 -0.013 0.008

HH Operation head age (kids) 0.000 0.000 0.000 0.000

HH Operation head age (no kids) 0.001 0.000 0.001 0.000

HH Operation log size (kids) 0.043 0.006 0.034 0.005

HH Operation log size (no kids) 0.038 0.011 0.034 0.010

HH Operation single parent -0.008 0.008 -0.008 0.008

HH Operation y1*childless couple 0.005 0.004 0.007 0.004

HH Operation y1*head age (kids) 0.000 0.000 0.000 0.000

38

HH Operation y1*head age (no kids) 0.000 0.000 0.000 0.000

HH Operation y1*log size (kids) -0.026 0.003 -0.021 0.002

HH Operation y1*log size (no kids) -0.026 0.005 -0.023 0.004

HH Operation y1*single parent 0.002 0.004 0.003 0.004

HH Furn/Eq childless couple -0.024 0.009 -0.030 0.008

HH Furn/Eq head age (kids) -0.001 0.001 -0.001 0.000

HH Furn/Eq head age (no kids) 0.000 0.000 0.000 0.000

HH Furn/Eq log size (kids) 0.018 0.006 0.015 0.005

HH Furn/Eq log size (no kids) 0.020 0.011 0.023 0.010

HH Furn/Eq single parent 0.032 0.008 0.028 0.008

HH Furn/Eq y1*childless couple 0.007 0.004 0.011 0.004

HH Furn/Eq y1*head age (kids) 0.001 0.000 0.000 0.000

HH Furn/Eq y1*head age (no kids) 0.000 0.000 0.000 0.000

HH Furn/Eq y1*log size (kids) -0.013 0.003 -0.011 0.002

HH Furn/Eq y1*log size (no kids) -0.011 0.005 -0.011 0.004

HH Furn/Eq y1*single parent -0.015 0.004 -0.013 0.004

Priv Trans childless couple 0.024 0.012 0.035 0.012

Priv Trans head age (kids) -0.001 0.001 -0.001 0.001

Priv Trans head age (no kids) -0.001 0.000 0.000 0.000

Priv Trans log size (kids) -0.074 0.007 -0.069 0.008

Priv Trans log size (no kids) -0.115 0.016 -0.118 0.014

Priv Trans single parent 0.001 0.009 0.008 0.011

Priv Trans y1*childless couple -0.008 0.005 -0.012 0.005

Priv Trans y1*head age (kids) 0.000 0.000 0.000 0.000

Priv Trans y1*head age (no kids) 0.000 0.000 0.000 0.000

Priv Trans y1*log size (kids) 0.029 0.003 0.026 0.003

Priv Trans y1*log size (no kids) 0.053 0.007 0.052 0.006

Priv Trans y1*single parent -0.016 0.005 -0.020 0.005

Pub Trans childless couple -0.006 0.006 -0.010 0.006

Pub Trans head age (kids) 0.000 0.000 0.000 0.000

39

Pub Trans head age (no kids) 0.000 0.000 -0.001 0.000

Pub Trans log size (kids) -0.002 0.004 -0.003 0.004

Pub Trans log size (no kids) 0.012 0.007 0.016 0.008

Pub Trans single parent -0.016 0.005 -0.017 0.006

Pub Trans y1*childless couple 0.001 0.003 0.003 0.003

Pub Trans y1*head age (kids) 0.000 0.000 0.000 0.000

Pub Trans y1*head age (no kids) 0.000 0.000 0.000 0.000

Pub Trans y1*log size (kids) -0.004 0.002 -0.003 0.002

Pub Trans y1*log size (no kids) -0.007 0.003 -0.008 0.003

Pub Trans y1*single parent 0.007 0.002 0.007 0.003

Table 3: Price E¤ects, GMM and SUR

GMM-exact SUR-

approximate

Commodity Price Estimate Std Err Estimate Std Err

Food at Home Food at Home -0.047 0.010 0.027 0.002

Food Out Food at Home 0.037 0.007 0.008 0.007

Food Out Food Out -0.045 0.006 -0.036 0.010

Rent Food at Home -0.072 0.004 -0.017 0.003

Rent Food Out 0.042 0.003 0.045 0.002

Rent Rent 0.097 0.005 0.109 0.005

Clothing Clothing 0.029 0.007 0.034 0.008

Clothing Food at Home 0.012 0.007 0.002 0.004

Clothing Food Out 0.009 0.005 -0.033 0.006

Clothing Rent 0.003 0.002 -0.029 0.003

HH Operation Clothing -0.045 0.006 0.014 0.005

HH Operation Food at Home 0.058 0.006 -0.042 0.006

HH Operation Food Out 0.005 0.005 -0.049 0.004

HH Operation HH Operation -0.033 0.007 0.007 0.005

HH Operation Rent -0.046 0.003 0.007 0.003

40

HH Furn/Eq Clothing 0.004 0.004 0.027 0.007

HH Furn/Eq Food at Home 0.022 0.005 0.015 0.005

HH Furn/Eq Food Out -0.049 0.004 0.052 0.006

HH Furn/Eq HH Furn/Eq 0.046 0.004 0.027 0.004

HH Furn/Eq HH Operation 0.042 0.005 -0.052 0.003

HH Furn/Eq Rent -0.033 0.002 -0.075 0.004

Priv Trans Clothing -0.027 0.002 -0.008 0.002

Priv Trans Food at Home 0.026 0.004 0.042 0.003

Priv Trans Food Out 0.004 0.003 0.052 0.004

Priv Trans HH Furn/Eq -0.004 0.002 -0.002 0.002

Priv Trans HH Operation 0.000 0.003 0.001 0.002

Priv Trans Priv Trans 0.023 0.004 0.042 0.005

Priv Trans Rent -0.024 0.003 -0.035 0.008

Pub Trans Clothing 0.025 0.002 -0.006 0.003

Pub Trans Food at Home -0.053 0.003 -0.022 0.002

Pub Trans Food Out -0.007 0.002 0.020 0.004

Pub Trans HH Furn/Eq 0.002 0.002 0.007 0.002

Pub Trans HH Operation -0.020 0.002 -0.005 0.002

Pub Trans Priv Trans 0.007 0.002 0.000 0.002

Pub Trans Pub Trans -0.002 0.002 -0.037 0.002

Pub Trans Rent 0.046 0.002 -0.045 0.003

41