With the advent of the almost ideal demand system (AIDS) of Deaton and Muellbauer, the estimation of consumer demand functions revolves around specifications that use flexible functional forms of the indirect utility function. This dual approach has put on the backburner the traditional primal approach because the direct utility function exists only in a latent state. The lack of explicit, analytical invertibility of either system, however, is an indication that focusing exclusively on the dual side of the consumer problem is equivalent to disregard potentially important and independent information residing with the primal side. This paper suggests that efficient estimates (in the sense of using all the available information) of the demand functions require the joint estimation of all the primal and dual relations. The specification of this objective assumes that risk-neutral households maximize their expected utility subject to their expected budget constraint. This theoretical framework leads to a two-step procedure that produces consistent and efficient estimates of the model’s parameters. The generality of the approach proposed here can handle also the frequently encountered case when all the sample units face the same observed commodity prices. Finally, we present a general solution of the nonlinear errors-in-variables problem with a novel estimation procedure that avoids the pitfalls of the traditional approach.Consumer demand functions, primal, dual, nonlinear errors-in-variables, Demand and Price Analysis, Research Methods/ Statistical Methods, D0,

Paris, Quirino

English

Research Papers in Economics

Department of Agricultural and Resource Economics
University of California, Davis
Estimation of Consumer Demand Functions
When the Observed Prices  
Are the Same for All Sample Units
by
Quirino Paris
Working Paper No. 03-004
September 2003
 Copyright @ 2003 by Quirino Paris 
 All Rights Reserved.  
Readers may make verbatim copies of this document for non-commercial  
purposes by any means, provided that this copyright notice appears on all such copies.
Giannini Foundation of Agricultural Economics1
Estimation of Consumer Demand Functions
When the Observed Prices Are the Same for All Sample Units
Quirino Paris
University of California, Davis
September 2003
Abstract
With the advent of the almost ideal demand system (AIDS) of Deaton and Muellbauer,
the estimation of consumer demand functions revolves around specifications that use
flexible functional forms of the indirect utility function.  This dual approach has put on
the backburner the traditional primal approach because the direct utility function exists
only in a latent state. The lack of explicit, analytical invertibility of either system,
however, is an indication that focusing exclusively on the dual side of the consumer
problem is equivalent to disregard potentially important and independent information
residing with the primal side.  This paper suggests that efficient estimates (in the sense of
using all the available information) of the demand functions require the joint estimation
of all the primal and dual relations. The specification of this objective assumes that risk-
neutral households maximize their expected utility subject to their expected budget
constraint. This theoretical framework leads to a two-step procedure that produces
consistent and efficient estimates of the model’s parameters. The generality of the
approach proposed here can handle also the frequently encountered case when all the
sample units face the same observed commodity prices. Finally, we present a general
solution of the nonlinear errors-in-variables problem with a novel estimation procedure
that avoids the pitfalls of the traditional approach.
JEL Classification: D0
Keywords: Consumer demand functions, primal, dual, nonlinear errors-in-variables
Quirino Paris is a member of the Giannini Foundation of Agricultural Economics.2
Estimation of Consumer Demand Functions
When the Observed Prices Are the Same for All Sample Units
Introduction
Statistical surveys of consumer behavior rarely gather price information that is specific
for each individual sample unit. How are, then, demand functions estimated in the
absence of any price variability? The answer varies from the use of time series and panel
data  to  construction  of  “unit  value”  prices.  Indeed,  this  statistical  reality  is  the
consequence of the following conundrum of consumer theory. Theory suggests that
observed  commodity  prices  should  be  identical  for  households  that  make  their
consumption decisions in the same market environment, but econometric estimation of
consumer demand functions requires a significant variability of prices without which the
duality approach for their estimation is infeasible.
Another unsatisfactory aspect of the traditional approach to demand functions’
estimation consists in the absence of any use of the primal information represented by the
first-order-necessary conditions for utility maximization.   With the advent of flexible
functional  forms  such  as  the  almost  ideal  demand  system  (AIDS) of  Deaton  and
Muellbauer, the primal relations contribute independent information which exists only in
a latent state. Hence, efficient estimators of the demand functions require the utilization
of the complete system of primal and dual relations.
This paper contributes three main results.  First, we propose the joint and novel
estimation of primal and dual relations in order to obtain efficient estimates of systems of
demand functions. Second, we present a theoretical and empirical framework that admits3
the estimation of demand functions also when the observed commodity prices are the
same for all the sample units. Third, we offer a general solution of the nonlinear errors-
in-variables problem that results from the specification of the empirical framework
outlined above.
These results are obtained on the basis of plausible and general assumptions: The
typical household is risk neutral and its objective is the maximization of the expected
utility functions subject to an expected budget constraint. In other words, the household
unit makes its consumption decisions through a planning process that uses expected
information.  The expectation process is specific to each household.
When the econometrician desires to measure commodity quantities and prices for
the purpose of inferring the demand functions that were used by the sample households,
measurement errors occur.   We assume an additive error structure for each piece of
sample information, that is, commodity quantities, prices and the household’s disposable
money income.
The resulting nonlinear errors-in-variables model will be estimated using a novel
approach. Traditionally, errors-in-variables specifications include unobservable quantities
that are replaced by their observable counterparts. This replacement step, however,
induces undesired properties on the estimator with inconsistent estimates and other
difficulties. In order to avoid these undesirable properties, we will not replace the latent
expected quantities and prices. A two-phase procedure will first generate consistent
estimates  of  those  expected  quantities  and  prices  which,  then,  will  be  used  as
instrumental variables to produce efficient estimates of the model’s parameters.4
The Theoretical Framework
We assume that risk-neutral households maximize their expected utility subject to an
expected budget constraint. Let 
† 
p
e be the 
† 
(J ¥1) vector of expected prices and 
† 
y
e the
expected disposable income available to the risk-neutral household that solves the
following problem
(1)
† 
U
*(p
e,y
e)=
def
max
x {U
e(x)  s.t.  y
e = ¢  p 
ex}
where 
† 
x is a 
† 
(J ¥1) vector of commodity quantities. The first-order-necessary conditions
are stated as
(2)
† 
∂L
∂x
=Ux
e(x)- lp
e = 0
∂L
∂l
= y
e - ¢  p 
ex= 0
where 
† 
L is the Lagrangean function and 
† 
l is the Lagrange multiplier.  After solving
equations (2), the vector of commodity quantities 
† 
x will become the vector of expected
quantities 
† 
x
e.  Similarly, the Lagrange multiplier 
† 
l will become the expected Lagrange
multiplier 
† 
l
e which is equivalent to the marginal utility of money income.   We assume
an interior solution of equations (2) that will generate commodity demand functions with
values
(3)
† 
x
e = d
e(p
e,y
e).
In case the first-order-necessary conditions (2) do not admit an explicit, analytical
solution, the demand functions (3) exist via the duality principle.  The indirect utility
function is obtained by inserting the optimal quantities (3) into the direct utility
(4)
† 
U
*(p
e,y
e)=U
e(x
e)
with the marginal utility of money income defined as 
† 
U
y
e
* (p
e,y
e)= l
e.5
The above development corresponds to the textbook discussion of consumer
theory. The econometric specification of the model requires the specification of the error
structure that accompanies any statistical measurement.
A Measurable Model of Consumer Demand
We assume that all the commodity quantities and prices and the disposable income are
measured with errors. That is, the econometrician observes 
† 
x, p and y that bear an
additive relation with their expected counterparts, that is 
† 
x = x
e +e, p= p
e +n  and
† 
y = y
e +n0. The vector of errors 
† 
¢  e = ( ¢  e , ¢  n ,n0) is distributed as 
† 
e ~ N(0,S). We further
assume that the errors are independently distributed across sample units.
The measurable econometric model of consumer behavior can thus be assembled
using all the primal and dual relations and the postulated error structure as follows:
Primal relations
(5)
† 
p =U
x
e
e (x
e)/U
y
e
* (p
e,y
e)+ n
(6)
† 
y = ¢  p 
ex
e +n0
Dual relations
(7)
† 
x = d
e(p
e,y
e)+e
Error structure
(8)
† 
p = p
e + n
(9)
† 
y = y
e +n0
(10)
† 
x = x
e +e.
Equations (5) represent the latent first-order-necessary conditions involving commodity
prices and the marginal utilities of goods. These relations are traditionally disregarded6
during  the  estimation  of  demand  functions  (7).  However,  in  the  case  of  model
specifications involving flexible functional forms, relations (5) convey information
independent  from  the  demand  functions  (equations  (7)).   The  pursuit  of  efficient
estimates, therefore, requires the joint estimation of all the primal and dual relations of
the consumer model.
Econometric Model
We assume a sample of N household with index 
† 
n =1,...,N. The estimation procedure
proposed here is articulated in two-phases. The Phase I objective consists in obtaining
consistent estimates of all the individual households’ expected quantities and prices.
These estimates, then, will be used as instrumental variables during Phase II to obtain
efficient estimates of the model’s parameters.
The Phase I specification assumes the form of a nonlinear errors-in-variables
model to be estimated by nonlinear least-squares. Specifically, let 
† 
b = (b
U
e,b
U
e,bx) be the
vector of parameters that constitute the final goal of the estimation process.  Then, the
Phase I estimation problem is stated as follows:
(11)
† 
min
b,p jn
e ,x jn
e ,yn
e,en
¢  e  nen
n=1
N
Â
subject to
(12)
† 
p =U
x
e,n
e (xn
e,b
U
e)/U
y
e,n
* (pn
e,yn
e,b
U
*)+ nn
(13)
† 
yn = ¢  p  n
exn
e +n0n
(14)
† 
xn = dn
e(pn
e,yn
e,bx)+en
(15)
† 
p = pn
e + nn7
(16)
† 
yn = yn
e +n0n
(17)
† 
xn = xn
e +en
(18)
† 
pjn
e n jn
n=1
N
Â = 0,   j =1,...N
(19)
† 
yn
enn
n=1
N
Â = 0
(20)
† 
x jn
e e jn
n=1
N
Â = 0,   j =1,...N
The sample index was removed from the observed vector of prices in order to indicate
that all the sample units face the same commodity prices.  The solution of the Phase I
model is not a trivial pursuit. For example, with N=100 and J=5, the number of
constraints is (2J+1)(2N+1)+1=2211. In the case of first-order-conditions that admit an
explicit solution the number of constraints is reduced considerably because either the
primal or the dual constraints are redundant. With a utility specification corresponding to
the flexible AIDS model, all the primal and dual constraints are required for obtaining
efficient estimates. The number of expected quantities and prices, expected money
income, parameters and errors is somewhat larger than 2211.  The Phase I problem can
be solved with commercially available computer applications for nonlinear programming
such as GAMS (Brooke et al.)
Constraints  (19)-(20)  are  the  orthogonality  conditions  for  the  expected
unobservables. The estimates of these quantities and prices 
† 
ˆ  p  n
e,ˆ  y  n
e,ˆ  x  n
e will be used as
instrumental variables in the Phase II estimation problem. Let 
† 
ˆ  S 
-1 be the estimated
covariance matrix computed from the estimated residuals of Phase I. Then, the Phase II
estimation problem is the following nonlinear seemingly unrelated (NSUR) equation
model:8
(21)
† 
min
b,en
¢  e  n ˆ  S 
-1en
n=1
N
Â
subject to
(22)
† 
p =U
x
e,n
e (ˆ  x  n
e,b
U
e)/U
y
e,n
* (ˆ  p  n
e,ˆ  y  n
e,b
U
*)+ nn
(23)
† 
yn = ¢  ˆ  p  n
eˆ  x  n
e +n0n
(24)
† 
xn = dn
e(ˆ  p  n
e,ˆ  y  n
e,bx)+en.
The Phase II problem can be solved with conventional econometric packages such
as SHAZAM (Whistler et al.).   The consistency of the above nonlinear least-squares
estimator has been discussed by Davidson and MacKinnon (1993) in their Theorem 5.1.
Empirical applications
We present two empirical applications of the methodology developed in the previous
sections.  The first example deals with a Cobb-Douglas utility function that admits an
explicit solution of the first-order-necessary conditions. In this case, either the primal or
the dual set of relations is redundant for the Phase I problem.   It is not redundant,
however, for the Phase II estimation problem, as each component of the error structure in
the NSUR specification conveys independent information.
The second example deals with a flexible functional form of the indirect utility
function such as the AIDS specification.  In this case, the direct utility function exists
only in a latent form and its econometric specification will be given as a second-order
Taylor series expansion. Thus, this primal specification of the direct utility function
conveys information that is independent from the dual set of relations and all the primal
and dual relations, therefore, are required in the Phase I problem.9
Cobb-Douglas Utility Function
The direct utility takes on the follow specification:
(25)
† 
U
e(x)= b j log(x j
j=1
J
Â -g j)
where 
† 
x j > g j ≥ 0  and  the  parameter 
† 
g j  represents  a  subsistence  level  of  the
corresponding good.
The Phase II econometric model takes on the following specification:
† 
min
b,en
¢  e  n ˆ  S 
-1en
n=1
N
Â
subject to
Primal relations
(26)
† 
pj =U
x j
e,n
e (ˆ  x jn
e ,b
U
e)/U
y
e,n
* (ˆ  p jn
e ,ˆ  y  n
e,b
U
*)+n jn
    =
b j
bk k=1
J Â
(ˆ  y  n
e - ˆ  p  kn
e gk) k=1
J Â
(ˆ  x  jn
e -g j)
+n jn
(27)
† 
yn = ˆ  p  jn
e ˆ  x  jn
e
j-1
J
Â +n0n
Dual relations
(28)
† 
x jn = d jn
e (ˆ  p jn
e ,ˆ  y  n
e,bx j)+e jn
      =g j +
b j
bk k Â
(ˆ  y  n
e - ˆ  p  kn
e gk) k=1
J Â
ˆ  p  jn
e +e jn
The Cobb-Douglas model of consumer behavior admits an explicit solution of the first-
order-necessary  conditions.  We  re-emphasize,  however,  that,  during  the  Phase  II
estimation relations (26) and (28) convey independent information through different
errors and their distributions. They are both required, therefore, if one wishes to obtain
efficient estimates of the model’s parameters.10
AIDS Indirect Utility Function
The AIDS model is traditionally specified as the following expenditure function:
(29)
† 
logE
e(p
e,U
e)= (1-U
e)log[a(p
e)]+U
e log[b(p
e)]= logy
e
where 
† 
a(p
e) and b(p
e) are price indexes defined (by Deaton and Muellbauer, 1980) as
(30)
† 
log[a(p
e)]= a0 + ak
k=1
J
Â log pk
e + gkj
j
J
Â
k
J
Â log pk
e log pj
e /2
(31)
† 
log[b(p
e)]= log[b(p
e)]+b0 (pk
e)
bk
k=1
J
’ .
The AIDS indirect utility function, therefore, can be written as
(32)
† 
U
*(p
e,y
e)=
logy
e - log[a(p
e)]
b0 (pk
e)
bk
k
’
with the marginal utility of money income given by
(33)
† 
U
y
e
* (p
e,y
e)=1/y
eb0 (pk
e)
bk
k
’ = l
e.
The Phase II estimation problem of the AIDS model can then be stated as follows:
† 
min
b,en
¢  e  n ˆ  S 
-1en
n=1
N
Â
subject to
Primal relations
(34)
† 
pj =U
x j
e,n
e (ˆ  x jn
e ,b
U
e)/U
y
e,n
* (ˆ  p jn
e ,ˆ  y  n
e,b
U
*)+n jn
     =
First derivative of 2nd order 
Taylor series of Un
e(ˆ  x 
e)
È 
Î 
Í 
˘ 
˚ 
˙ ˆ  y  n
eb0 ( ˆ  p  kn
e )
bk
k=1
J
’ È 
Î  Í 
˘ 
˚  ˙ +n jn
(35)
† 
yn = ˆ  p  jn
e ˆ  x  jn
e
j-1
J
Â +n0n
Dual relations11
(36)
† 
x jn =
a j ˆ  y  n
e
ˆ  p  jn
e +
ˆ  y  n
e
ˆ  p  jn
e gkj
k=1
J
Â log ˆ  p  kn
e +
b j ˆ  y  n
e
ˆ  p  jn
e log[ˆ  y  n
e /a(ˆ  p  n
e)]+e jn.
The  Taylor  series  expansion  of  the  latent  direct  utility  function  provides  the
complementary  primal  information  for  obtaining  efficient  estimates  of  the  AIDS
parameters.
Conclusion
We have tackled threes neglected aspects of the econometric estimation of consumer
demand functions. First, we have proposed a theoretical framework based upon the
assumption that households are risk-neutral and maximize their expected utility subject to
an expected budget constraint. We then assumed that all the sample information is
measured with an additive error.
Secondly, we suggested that efficient estimates of the consumer demand functions
require the joint estimation of primal and dual relations. As a byproduct of this general
framework we obtain the possibility of estimating demand functions also when all the
sample households face the same observed commodity prices. This statistical reality
corresponds to a rather common event associated with many national surveys and
censuses of consumer behavior.
Third, the resulting nonlinear errors-in-variables problem was solved using a two-
phase  approach  that  produces  consistent  and  efficient  estimates  of  the  model’s
parameters.  This novel approach avoids the pitfalls of the traditional method for handling
errors-in-variables models where the latent, unobservable information is replaced by its
observable counterpart.   This replacement step induces undesirable properties on the12
estimator such as inconsistent estimates.   The methodology presented in this paper is
rather general and can be applied to several other micro-econometric contexts.13
References
Brooke, A., K., D. Kendrick and A. Meeraus. GAMS, User Guide. Boyd and
Fraser Publishing Company. Danvers, MA, 1988.
Davidson, R. and J. G. MacKinnon. Estimation and Inference in Econometrics.
Oxford University Press, New York, N.Y., 1993.
Deaton, A and J. Muellbauer. “An Almost Ideal Demand System,” American Economic
Review 70(3), June 1980, 313-326.
Stone, J. The measurement of Consumers’ Behaviour in the United Kingdom, Cambridge
University Press, Cambridge, 1953.
Whistler, D., K. White, S.D. Wong, and D. Bates. SHAZAM, The Econometric
Computer  Program, User’s Reference Manual, Version 9. Northwest
Econometrics Ltd. Vancouver, B.C. Canada, 2001.

Estimation of Consumer Demand Functions When the Observed Prices Are the Same for All Sample Units

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