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On the estimation of 'an implicitly additive demand
system'
John A. L. Cranfield , Paul V. Preckel , James S. Eales & Thomas W. Hertel
Published online: 04 Oct 2010.
To cite this article: John A. L. Cranfield , Paul V. Preckel , James S. Eales & Thomas W. Hertel (2000)
On the estimation of 'an implicitly additive demand system', Applied Economics, 32:15, 1907-1915, DOI:
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±
Applied Economics, 2000, 32, 1907 1915
On the estimation of `an implicitly additive
demand system’
JOHN A . L. CR A NFIELD*, PA UL V. PR ECKEL , JA MES S.
{
EA LES and THOMA S W. HER TEL
{
{
Department of Agricultural Economics and Farm Management, University of
Manitoba, W innipeg Manitoba, Canada, R3T 2N2 and Department of Agricultural
{
Economics, Purdue University, W est L afayette, Indiana, 47907-1145, USA
Motivated by a lack of Engel ¯ exibility in commonly used demand systems, Rimmer
and Powell developed a new demand system. This system, referred to AIDADS, is
implicitly, directly additive, possesses marginal budget shares and thus Engel elasti-
cities, that vary nonlinearly with expenditure such that predicted budget shares are
[
]
restricted to the 0,1 interval. Due to these attractive Engel properties, AIDADS
represents a signi®cant contribution to the literature on demand analysis. This paper
presents an alternative estimation procedure to the one used by Rimmer and Powell
and examines its properties via a case study. The proposed approach avoids a linear
approximation employed in Rimmer and Powell’s estimation framework. Based on a
small Monte Carlo study, it appears that the approach produces more accurate
estimates of the parameters and Engel elasticities.
I. INTR ODUCTION
odological limitation in the estimation procedure used by
Rimmer and Powell, and suggests a preferred alternative.
The AIDADS functional form is highly non-linear and
implicit in the level of utility attained by the consumer. As
a result, estimation is a di cult task. Rimmer and Powell
use a linear approximation to the change in utility across
observations to implement their estimation framework. It
is found that this approximation results in parameter esti-
mates that are sensitive to the ordering of the data. For this
reason an alternative estimation framework was developed
which results in estimates that are (i) insensitive to the
ordering of the data, and (ii) that appear to produce
more accurate estimates of Engel elasticities. A comparison
using a six good demand system estimated with cross-
sectional data from the 64 countries contained in the
1985 International Comparisons Project (United Nations,
1992) data set is used to illustrate the two di€ erent
approaches. In the next section, the AIDADS speci®cation
is reviewed, along with Rimmer and Powell’s estimation
strategy. The proposed estimation strategy is then pre-
sented, with results from the strategy compared to those
In a recent paper, Rimmer and Powell (1996) propose an
innovative implicitly, directly additive demand system. The
motivation for developing this demand system, referred to
as an implicitly, directly additive demand system (or
AIDADS), stems from a lack of Engel ¯ exibility in com-
monly used demand systems. In particular, some forms
(e.g. the Linear Expenditure System) have constant mar-
ginal budget shares over the range of expenditure, while
others (e.g. Deaton and Muellbauer’s (1980) Almost Ideal
Demand System, or Working’s (1943) model) vary linearly
with the log of expenditure, but do not constrain budget
shares to lie within the unit interval. AIDADS possess
marginal budget shares that vary non-linearly with expen-
diture, such that predicted budget shares are restricted to
the unit interval. This makes AIDADS particularly well
suited to estimation of consumer demand when real income
varies dramatically. Due to these attractive Engel proper-
ties, AIDADS represents a signi®cant contribution to the
literature on demand analysis. This note points out a meth-
*Corresponding author: E-mail: John_Cran®eld@Umanitoba.CA
1907
#
±
±
/
Applied Economics ISSN 0003 6846 print ISSN 1466 4283 online
2000 Taylor & Francis Ltd
//
/
http: www.tandf.co.uk journals

1908
J. A. L . Cran®eld et al.
from Rimmer and Powell’s procedure. Next, a Monte
Carlo experiment is presented that examines the e€ ective-
ness of the proposed approach. The paper concludes with
some comments on computational issues.
is the marginal budget share for the ith good and wik is the
budget share for the ith good in the kth observation.
Flexible Engel elasticities derive from the fact that when
ˆ
, the Engel elasticities are a function of utility, which
-
i
¬
i
depends on Mk.
However, estimation of Equation 2 requires treating the
uk as parameters. Rimmer and Powell’s approach is to
estimate u1 and the change in utility between subsequent
observations based on a linear approximation.1 This
change is then used to compute the level of utility for the
next observation, as follows:
II. R EVIEW OF A IDA DS
Following Hanoch (1975), Rimmer and Powell (1996)
assume an implicit, directly additive utility function of
P
n
ˆ
ˆ
the form
U x
u
;
k
1 for all k, where i
ˆ
1
n
;. . . ;
…
†
i
ik
ˆ
i
1
n
indexes the commodities, k
1
T indexes observa-
;. . . ;
X
ˆ
ˆ
^
x
jk
u
Cjk
8k
2
T
3
… †
;. . . ;
x
u
D
D
tions,
is a twice-di€ erentiable, monotonic func-
…
Ui
;
†
k
ik
k
ˆ
j
1
tion, xik is consumption of the ith good in the kth
2
observation and uk < is the level of utility attained
where:
from the consumption bundle, . Fur-
x
x
x
2k ;. . . ;
nk
…
;
†
1k
u
pjk
x
thermore,
is assumed to satisfy the requisite con-
…
;
†
Ui
cavity requirements.
Rimmer and Powell then specify the following function
ik
k
ˆ
Cjk
¡
£
M ¡ p0
…
k®†
k
"
#
1
¡
n
x
u
for
:
…
Ui
;
†
X
ik
k
exp u
…
†
k
¡
u
ln x
¡
¡ 1
…- i ’ik… k†† …
®i†
ik
xik
¡
1 ‡exp u
®
…
†
k
i
ˆ
i
1
ˆ
U x
u
u ln
k
8i k
1
… †
…
i
;
†
’ik…
†
;
ik
k
…A exp u †
…
†
,
k
ˆ
=
where
u
‡
G u
1 ‡G u
and A
ˆ
ˆ
’ik…
†
‰¬i - i … k†Š ‰
… k†Š ¬i ;- i ;®i
8k
2
T j
;
1
n
4
… †
;. . . ;
;. . . ;
k
are parameters, and G u is a positive, monotonic, twice
…
†
k
di€ erentiable function. In addition, 0 µ
1 for all i,
¬i ;- i <
Mk‡1 ¡ p 0
Mk ¡ p 0
P
P
®
®
k
n
n
k‡1
ˆ
ˆ
and
1. The ®rst order conditions for
¬
i
-
i
x^
º
¡
u
k
’ik…
†
D
ˆ
ˆ
i
1
i 1
jk
…
†
pjk‡1
pjk
utility maximization, subject to a linear budget constraint,
can be solved to obtain the AIDADS demands:
ˆ
ˆ
ˆ
8k
uk‡1
2
T j
;
1
n
5
… †
;. . . ;
;. . . ;
u
pik
’ik…
†
k
ˆ
xik
‡
M ¡ p0 8i k
2
… †
®
®
k
;
…
i
k
ˆ
u ‡
k
u
k
8k
2
T
6
… †
;. . . ;
D
where Mk is per capita expenditure, pik is the price of the
ith good in the kth observation, pk is a vector of prices with
is the ®rst di€ erence operator across observations, and
D
^
xjk are ®tted values of xjk. These four equations are key to
Rimmer and Powell’s estimation of AIDADS.2
If the ordering of the data changes such that the value of
Mk or Mk‡1 (as well as pk and pk‡1) changes, the approx-
^
components p , and is a vector with components . In
®
®
ik
i
this context,
sumption level for the ith good. If
Equation 2 becomes Stone’s (1954) Linear Expenditure
can be thought of as the subsistence con-
®
i
ˆ
for all i then
-
i
¬
i
imation of
x
±
changes, as does u and hence uk‡1. Since
D
D
jk
k
System (LES).
Equations 3 6 are constraints in Rimmer and Powell’s
Flexible Engel properties are a key feature of the
approach, and represent a recursive system (in the order
ˆ
AIDADS model. Assuming G u
exp u Rimmer and
…
†
… †
4, 5, 3, 6), errors in approximating u , and hence u
will a€ ect the estimates of the other parameters. To mini-
mize the consequences of linearization errors in computing
,
k
k
D
k
k‡1
Powell show that
tained in the
u
behaves logistically and is con-
’ik…
†
k
interval (Rimmer and Powell, 1996:
1615). Engel elasticities in AIDADS are de®ned as
‰¬i ;- iŠ
u , Rimmer and Powell order the data by increasing per
D
k
ˆ
=
wik, where:
²
ik
Áik
capita expenditure. Nevertheless in cases where the
bracketed term in Equation 5 is large, the linearization
errors can still be large. In the next section a reformulation
of this estimation problem is introduced that eliminates the
linear approximation, and results in estimates that are
independent of the ordering of the data.
ˆ
u
¡
¡
Áik ’ik…
†
…- i ¬i†
k
"
#
1
¡
2
n
X
1 ‡exp u
… k††
…
£
¡
ln x
¡
¡
-
…
¬
®
j
…
j
j
jk
… †
exp u
…
†
k
ˆ
j
1
1 The appendix shows the derivation of Rimmer and Powell’s linear approximation, with reference to the equations used in their 1992
paper.
2 In Rimmer and Powell’s formulation, the value A from Equation 1 drops out of the model, but is recoverable as an implicit parameter
^
of the model through u1.

An implicitly additive demand system
1909
III. A PR OPOSED ESTIMA TION STR A TEGY
Appropriate choice of starting values also helps reduce
the computational burden. Starting values are chosen to be
consistent with a LES speci®cation. When the proposed
Rimmer and Powell are followed and a maximum likeli-
hood estimation framework is used. Ignoring constants
that are independent of the unknown parameters, the con-
centrated log-likelihood function is expressed as:
framework is used, and i are started at the mean budget
¬
i
-
share for the ith good across the sample, while subsistence
parameters are set at half the minimum observed consump-
ˆ
tion level in the sample (i.e.
0 5 £ min
x
). Utility,
®
:
k f ikg
uk, is started at a value consistent with Stone-Geary pref-
erences evaluated at the starting values for and
i
j
j
†
0 5 £ ln W
7
:
…
… †
P
T
k
1
¡
ˆ
where wij
T
"ik"jk is the (i, j )th element of the
,
®
i
¬i ;- i
ˆ
matrix W, "ik is the e1rror term for the ith equation in the
and observed consumption levels. Fitted consumption
levels, xik, are chosen to satisfy Equation 9 evaluated at
the starting values for and uk. Starting values for
are chosen to satisfy Equation 8, while that for is set
equal to one.
The same starting values and bounds for
employed when Rimmer and Powell’s approach is used for
estimation. The same upper and lower bounds are also
used for uk. However, the starting values for uk
±
kth observation (see Greene, 1993: 495 6). Equation (7) is
^
minimized with respect to the unknown values, subject to a
de®ning equation for the error term, expressed for numer-
ical convenience in share form as:
¬i ;- i ;®i
"
µ
ik
and are
¬i ;- i
®
pik
Mk
i
ˆ
^
"
x
¡ xik
8i k
8
… †
…
†
;
ik
ik
A de®ning equation for x^ is also included as a constraint,
ik
k
1
> †
…
and is expressed as:
are derived from Rimmer and Powell’s linear approxima-
±
tion in Equations 3 6, while the starting value for u1 is the
‡
exp u
k
¬
-
… †
i
i
M ¡ p0
8i k
9
ˆ
x^
‡
®
®
k
;
… †
…
ik
i
k
same as that used in the proposed framework. The same
bounds on xik are used when estimating with Rimmer and
Powell’s approach, while starting values for xik are consis-
tent with the starting values of uk. The optimization prob-
lems were programmed in the General Algebraic Modelling
System (GAMS) Release 2.25 (Brooke et al. 1992), and
solved using the MINOS solver. Next, the AIDADS
model is estimated using Rimmer and Powell’s approach
and the proposed framework.
1 ‡exp u
…
and
†
k
^
Constraints on
take the form of
¬
ˆ
-
i
i
^
P
P
n
n
ˆ
1 and bounds given below. Finally,
¬
-
i
i
ˆ
ˆ
i 1
i
1
the de®ning equations for the implicit utility function are
included as constraints:
n
X
‡
exp u
k
¬
-
… †
i
i
ˆ
ln x
…
¡
¡ u
8k
10
†
®i†
µ
…
ik
k
1 ‡exp u
…
†
k
ˆ
1
i
ˆ
where
1 ‡ln A . It is the inclusion of these de®ning
… †
µ
equations for utilities that allows us to avoid the approx-
imation employed by Rimmer and Powell.
IV. A IDA DS ESTIMA TION : A COMPAR ISON
USING THE 1985 ICP DA TA
Upper and lower bounds on all choice variables are
included to reduce the size of the feasible set, and to main-
tain consistency with the AIDADS model. The bounds are
In this section, the e€ ect of Rimmer and Powell’s linear
2
2
2
de®ned as follows:
0 0 8 ,
Š
0 0 8 ,
0 0 99£
2
¡12 30 . Reducing
‰
approximation of
x^ is explored by estimating the
D
ik
¬
‰ ; : Š - i ‰ ; : Š ®i ‰ ; :
i
2
mink
x
f
, x^
ikgŠ
0 01 100 , and u
‰ :
AIDADS model using their approach and the proposed
framework. As mentioned, the 1985 International
Comparisons Project (ICP) data, aggregated to a six
good level, are used. The six goods are food (FOOD),
beverages and tobacco (BEVRTOBC), clothing and foot-
wear (CLTHFOOT), gross rent and fuel (RENTFUEL),
household furnishings and operations (HFURNOPS),
and other expenditure (OTHEREXP). The ICP data set
allows one to obtain price (stated in international dollars)
and quantity data for 113 goods in 64 countries that span
the development spectrum.3 Table 1 shows summary
statistics for the budget shares, prices and per capita expen-
diture data used to estimate the model.
;
;
Š
ik
k
the size of the feasible set tends to reduce the computa-
tional burden of ®nding an optimal solution. The precise
choice of bounds depends on the data used. In particular,
^
x , and u are data dependent. If
ik k
upper bounds on
¬i ;- i ;
one of these upper bounds is active, it should be relaxed
until no longer binding, or reaches a level dictated by
theory to be an active constraint. Upper bounds on
®
i
are such that
x
¡
is strictly positive, thus avoiding
…
®i†
ik
over¯ ows (or unde®ned results) in taking the logarithm
of the level of discretionary consumption in Equation 10.
Lower bounds on
and are based on the theoretical
®
i
¬i ;- i
restrictions of the model and are not changed if binding.
^
Lower bounds on xik and uk help to limit the search to the
Table 2 presents parameter estimates corresponding to
the proposed framework and Rimmer and Powell’s
approach. The ®rst panel in Table 2 shows estimates
2
range of values expected. Finally,
<, which implies
µ
2
A
<
.
‡‡
3 These countries range from Ethiopia, with the lowest level of per capita expenditure ($89), to the USA, with the highest level of per
capita expenditure ($12 018).

1910
J. A. L . Cran®eld et al.
Table 1. Means and standard deviations of 1985 ICP budget shares, prices, and per capita expenditure
Beverages
and tobacco
Clothing
and footwear
Gross rent
and fuel
Household furnishing
and operations
Food
Other expenditure
Budget sharesa
0.313
(0.143)
0.053
(0.027)
0.079
(0.027)
0.139
(0.049)
0.067
(0.026)
0.349
(0.125)
Prices
0.742
(0.315)
0.816
(0.495)
0.742
(0.378)
0.867
(0.381)
0.807
(0.424)
0.741
(0.423)
Per capita expenditure (in hundreds of international dollars)
25.750
(28.621)
Note: a Means are at the top of each cell, while standard deviations are shown in parentheses.
Table 2. Parameter estimates from AIDADS speci®cation using 1985 ICP data
Beverages
and tobacco
Clothing
and footwear
Gross rent
and fuel
Household furnishing
and operations
Other
expenditure
Food
Randomly ordered data
Proposed framework
¬
-
®
0.459
0.000*
0.589
1.908
0.066
0.036
0.048
0.095
0.053
0.099
0.086
0.229
0.086
0.076
0.218
0.618
0.235
0.064
0.029**
µ
Rimmer and Powell’s framework
¬
-
®
0.181
0.051
0.052
0.059
0.072
0.079
0.103
0.162
0.139
0.026
0.069
0.071
0.029**
0.465
0.329
0.000*
0.329
0.640**
7.593
µ
Di€ erence between estimates from Rimmer and Powell and proposed strategies, as a percentage of the latter
7
7
7
7
9.411
¬
-
®
60.589
1
22.884
46.077
22.180
24.282
50.976
4.526
89.336
39.384
69.721
113.143
7
7
7
46.736
11.119
0.000
7
8.588
100.000
µ
297.942
Sequentially ordered data
Proposed framework
¬
-
®
0.459
0.000*
0.590
1.908
0.066
0.036
0.048
0.095
0.053
0.099
0.086
0.229
0.086
0.076
0.218
0.618
0.236
0.064
0.029**
µ
Rimmer and Powell’s framework
¬
-
®
0.356
0.000*
0.640**
15.207
0.059
0.036
0.047
0.086
0.052
0.097
0.114
0.232
0.064
0.072
0.314
0.614
0.000*
0.066
0.029**
µ
Di€ erence between estimates from Rimmer and Powell and proposed strategies, as a percentage of the latter
7
7
7
7
7
7
¬
-
®
22.521
0.000
8.588
10.116
0.936
3.338
9.657
1.987
2.008
32.511
1.177
25.842
5.638
2.990
0.000
43.835
7
0.632
7
7
7
100.000
µ
696.918
Notes: *denotes an active lower bound; ** denotes an active upper bound.
obtained using randomly ordered data, while the second
Compared across randomly ordered and expenditure-
panel shows estimates obtained using data ordered by
increasing per capita expenditure. Di€ erences between esti-
mates from Rimmer and Powell’s and the approach are
expressed as a percentage of the latter.
ordered data sets, three points emerge. First, the proposed
approach results in estimates of AIDADS that are inde-
pendent of the ordering of the data. Second, Rimmer and
Powell’s approach results in estimates of AIDADS that

An implicitly additive demand system
1911
Table 3. Engel elasticities evaluated at the means of the data, and percentage di€ erencea using estimates from Rimmer and Powell’s
approach and the proposed framework
Beverages
and tobacco
Clothing
and footwear
Gross rent
and fuel
Household furnishing
and operations
Other
expenditure
Food
Proposed Framework
b
²
i
0.671
0.934
0.916
1.010
1.179
1.047
1.244
0.937
±
Rimmer and Powell randomly ordered data
²
i
1.057
0.985
(5.44)
0.999
15.24)
1.064
(1.59)
7
7
7
24.67)
(57.68)
(10.30)
(
(
(
(
±
Rimmer and Powell sequentially ordered data
²
i
1.111
(65.69)
1.102
(17.95)
1.079
(17.79)
0.838
28.92)
1.074
(2.59)
0.917
26.29)
7
Notes: a Percentage di€ erences, which are shown in parentheses, are computed as the di€ erence between elasticities based on estimates
from Rimmer and Powell’s strategy and the proposed approach expressed as a percentage of the latter. The values are not exact due to
rounding.
b
² denotes Engel elasticity for the ith good.
i
di€ er across the di€ erently ordered data sets. Finally,
compared across estimation strategies, ordering the data
by increasing per capita expenditure lessens the percentage
ordering of the data. To address this point, a Monte Carlo
experiment is used to demonstrate the e cacy of the
approach.
di€ erences in the estimates of
and , lessens or does
-
i
¬
i
not change percentage di€ erences in the estimates of
, but increases the percentage di€ erence in the estimate
®
V. A SSESSING THE PR OPOSED
PR OCEDUR E : A MONTE CA R LO
EX PER IMENT
i
4
of
.
µ
While di€ erences in the parameter estimates are of inter-
est, the behavioural characteristics of the demand system
are of greater concern. Table 3 presents Engel elasticities,
computed at the means of the data with parameter esti-
mates shown in Table 2. The main result is that substantial
di€ erences exist between Engel elasticities from this pro-
posed approach and those from Rimmer and Powell’s
approach. Note, however, that when the data are ordered
by expenditure, the magnitudes of the percentage di€ er-
ences actually increase (in absolute value) for all goods.
In summary then, the proposed framework produces
identical parameter estimates regardless of the ordering
of the data. However, Rimmer and Powell’s approach pro-
duces estimates that depend on the ordering of the data.
This results in di€ erent Engel elasticity estimates across
di€ erent orderings of the data. While sensitivity of
Rimmer and Powell’s approach to data order is bother-
some, it still may produce estimates which are closer to
the underlying data generating process (DGP) with some
To evaluate the proposed estimation framework, a small
Monte Carlo experiment is conducted. The experiment
consists of 100 replications. In each replication, pseudo-
random consumption data are generated by adding a ran-
dom error to the ®tted consumption levels from the pro-
posed estimation strategy (which have been ordered by
increasing per capita expenditure). The errors are drawn
from a multivariate normal distribution with zero mean
vector and covariance matrix de®ned by the estimated W
matrix in Equation 7. The new values of xik, and the ICP
price and per capita expenditure data (ordered by increas-
ing expenditure) are then used to estimate AIDADS with
the proposed strategy and Rimmer and Powell’s
approach. 5
Table 4 shows the parameter values assumed for the
DGP of the Monte Carlo experiment, and the means of
the parameter estimates across replications for each estima-
4 While Rimmer and Powell’s estimation strategy circumvents direct estimation of A in Equation 1, A is, in fact, implicitly estimated. To
see this, note that u1 is an estimated parameter in Rimmer and Powell’s approach. Consequently, one can use u1 and the de®ning
^
equation of utility for the ®rst observation to obtain a value of µ as follows:
n
X
‡
¬
- i exp…u1†
i
^
^ ˆ
µ
^
ln…xi1 ¡®i† ¡ u1
‡
1
exp…u1†
ˆ
i
1
Since µ is estimated directly in our proposed framework, its value can be compared to the implied values in Rimmer and Powell’s
approach.
5 Some of the drawn error terms are such that the corresponding pseudo-random consumption levels are negative. In observations where
this occurs, the vector of error terms for that observation is discarded and new vector of error terms is drawn.

1912
J. A. L . Cran®eld et al.
Table 4. Data generating process (DGP) parameter values and means of the Monte Carlo parameter estimates using sequentially ordered
data
Beverages
and tobacco
Clothing
and footwear
Gross rent
and fuel
Household furnishing
and operations
Other
expenditure
Food
DGP parameter values
¬
-
®
0.459
0.000
0.590
1.908
0.066
0.036
0.048
0.095
0.053
0.099
0.086
0.229
0.086
0.076
0.064
0.029
0.218
0.618
0.235
µ
Proposed framework
¬
-
®
0.463
0.006
0.564
1.870
0.066
0.095
0.086
0.076
0.214
0.036
0.053
0.227
0.064
0.614
0.045**
0.092**
0.082**
0.027**
0.217**
µ
Rimmer and Powell’s framework
¬
-
®
0.438**
0.217
0.646**
5.238
0.066
0.049
0.050**
0.097
0.073
0.099**
0.103
0.161
0.066**
0.074
0.070
0.029**
0.223
0.431
0.076**
µ
Note: ** Denotes an active upper bound was encountered in at least one replication.
tion approach.6 A striking observation from Table 4 is the
percentage di€ erence values greater than the percentage
di€ erence values from Rimmer and Powell’s approach.
Again, for the assumed error structure, DGP parameter
values, price and expenditure data, our approach appears
^
large di€ erence in mean of the estimates of FOOD. The
-
mean estimate of
in Rimmer and Powell’s approach
-
FOOD
is 0.217, but 0.006 in the proposed framework. The corre-
sponding DGP parameter value is 0. Even though the mean
to estimate
,
and better, but neither approach esti-
- µ
i
¬
i
of the estimates of
in this approach are not equal to
mates very well.
®
i
-
FOOD
zero, it is far closer to the DGP parameter value than the
The means of the Engel elasticities (evaluated at the
means of the data in each replication) are shown in Table
6, along with the RMSPE and percentage di€ erence from
the elasticities computed using the DGP parameter values
at the means. The estimated Engel elasticities from this
approach have lower RMSPE than elasticities from
Rimmer and Powell’s approach. In addition, the percen-
tage di€ erence values are smaller using this approach than
Rimmer and Powell’s. Thus, it appears that this approach
to estimation produces parameter estimates that are closer
to the DGP parameter values, and that the behavioural
properties of the system estimated with this approach
better re¯ ect the underlying preference structure when
compared to Rimmer and Powell’s approach.
mean estimate of
approach.
from Rimmer and Powell’s
-
FOOD
Table 5 shows root mean square percentage error
(RMSPE) and percentage di€ erence values between esti-
mates from each approach and the DGP parameter values.
^
With the exception of FOOD, the RMSPE values for the
-
and are larger when Rimmer and Powell’s approach
¬i ;- i
µ
is employed.7 The RMSPE values for estimates for the
®
i
suggest that neither approach estimates them well.8
^
Except for FOOD, the percentage di€ erences between the
-
means of the estimates of
and and the DGP par-
µ
¬i ;- i
ameter values tend to be smaller (in absolute value) with
this approach than with Rimmer and Powell’s. However,
three of the mean estimates of
from our approach have
®
i
6 To obtain the estimates reported in Table 4, the upper bounds for utility were set to 25, which are inactive constraints. In two
replications, estimates for ¬FOOD from Rimmer and Powell’s approach are at their upper bounds. Given this small number of active
bounds, these upper bounds were not changed. With Rimmer and Powell’s approach, upper bounds on ®i are active in three replications
for CLTHFOOT and OTHEREXP, in 12 replications for BEVRTOBC, in 16 replications for HFURNOPS, in 19 replication for
RENTFUEL, and in 24 replications for FOOD. With this approach, upper bounds on estimates of ®i are binding once for CLTHFOOT,
in nine replications for HFURNOPS, in 11 replications for BEVRTOBC, in 12 replications for OTHEREXP, and in 36 replications for
RENTFUEL.
7 Since the true value of - FOOD is zero, the RMSPE equals in®nity since the squared error is expressed as a percentage of the true value
(i.e., zero).
8 While not reported, the standard deviation for the implied value of µ from Rimmer and Powell’s approach is 6.34, while the estimate of
µ from our approach is 0.14. Based on these values and RMSPE values, it appears the mean value of µ is in error and that the estimate of
µ exhibits greater variability when using Rimmer and Powell’s approach.

An implicitly additive demand system
1913
Table 5. RMSPE and percentage di€ erencea of Monte Carlo experiment parameter estimates from true valuesb,c
Beverages
and tobacco
Clothing
and footwear
Gross rent
and fuel
Household furnishing
and operations
Other
expenditure
Food
Proposed framework
¬
-
®
6.15
4.67
(0.46)
5.95
4.18
(0.21)
4.30
16.32
(0.00d )
3.74
6.12
0.79)
4.23
(0.47)
77.79
6.27)
12.58
7
7
(0.89)
(
(
(
(
(
1.83)
2.09
e
1
(
)
(0.00d)
31.79
(1.33)
25.44
(
(
1.13)
7
7
0.74)
1
15.10
37.79
52.93
7
7
6.02)
7
6.80)
7
7
7
6.73)
(
4.41)
7.82
(
(
5.34)
µ
7
(
2.05)
Rimmer and Powell
¬
-
®
32.05
55.87
(0.92)
41.06
(39.49)
35.69
(4.56)
44.50
(2.11)
39.16
(37.83)
28.16
(0.31)
119.86
(20.16)
30.85
58.84
3.28)
9.19
52.84
(1.65)
31.36
30.36)
81.57
67.06)
7
7
(
4.59)
(
(
e
1
7
7
7
(
)
(
(
29.86)
50.82
24.16)
(8.74)
80.37
(
(
1
19.95
(9.49)
375.52
7
7
0.35)
µ
(174.82)
Notes: a The percentage di€ erence values are not exact due to rounding.
b RMSPE is computed as:
=
1 2
2
100
Y rs ¡ Y ra
X
1
100
ˆ
RMSPE
¤ 100
yra
…
†
…
†
ˆ
r
1
where r 1;. . . ;100 is an index of replications, Y r is the parameter estimate and Y ra is the DGP parameter value.
s
ˆ
c The percentage di€ erence between the mean of the estimate and the DGP parameter value, expressed as a percentage of the latter, is
shown in parentheses.
d Di€ erences in the mean and DGP parameter value were zero to three decimal places.
e Since the DGP parameter value for this parameter is zero any deviation is divided by 0, which results in a RMSPE of , likewise for the
1
percentage di€ erence terms.
Table 6. Engel elasticities based on assumed DGP parameter values, mean of the Monte Carlo Engel elasticities (evaluated at the means of
the data in each replication), and RMSPE and percentage di€ erencesa from the DGP Engel elasticities, expressed as a percentage of the
latter
Beverages
and tobacco
Clothing
and footwear
Gross rent
and fuel
Household furnishing Other
Food
0.619
and operations
expenditure
DGP value
0.909
0.893
1.232
1.039
1.306
Proposed framework
²
i
0.648
5.42
0.924
2.72
(1.61)
0.91
2.32
(1.89)
1.191
3.89
3.39)
1.043
2.05
(0.43)
1.258
3.86
7
3.65)
RMSPEb
7
(4.70)
(
(
Rimmer and Powell’s framework
²
0.898
50.09
1.019
13.50
(12.02)
1.001
13.54
(12.07)
1.007
20.14
18.31)
1.069
4.78
(2.91)
1.069
19.65
18.11)
i
RMSPE
7
7
(45.08)
(
(
Notes: a Percentage di€ erence from the DGP parameter value is shown in parentheses.
b Denotes root mean squared percentage error.
VI. SUMMA R Y A ND CONCLUSIONS
ties. Assuming an implicitly, directly additive utility func-
tion, they derived a demand system that is characterized by
marginal budget shares, and hence Engel elasticities, that
vary non-linearly with real per capita expenditure. Given
these ¯ exible Engel properties, this new functional form,
Recently, Rimmer and Powell (1996) made a major con-
tribution in the ®eld of demand analysis by developing a
new functional form which possesses ¯ exible Engel proper-

1914
J. A. L . Cran®eld et al.
referred to as AIDADS, will likely play an important role
in future analysis of consumer demand. However, non-
linearities make estimation of AIDADS di cult. To sim-
plify estimation, Rimmer and Powell used an approxima-
tion that introduces linearization errors in estimation of the
system. This paper proposes an alternative estimation
methodology that avoids this approximation.
Results based on a multi-country, cross-sectional data
set illustrates the extent to which Rimmer and Powell’s
estimation strategy produces estimates that are di€ erent
from the proposed approach. These di€ erences appear
not only in the parameter estimates but also in the Engel
elasticities. Results from a Monte Carlo experiment pro-
vide further evidence regarding the ability of the proposed
approach to recover the unknown parameters, and thus the
Engel elasticities. Speci®cally, the percentage di€ erences
between Engel elasticities computed using estimates from
the approach and the DGP Engel elasticities are less than
5% in absolute value. In contrast, ®ve of six Engel elasti-
cities from Rimmer and Powell’s approach have percentage
di€ erences greater than 10% in absolute value.
A number of computational issues deserve mention at
this juncture. In particular, given that AIDADS is a highly
nonlinear model it is important to begin the nonlinear esti-
mation algorithm at appropriate starting values. Based on
experience, it is essential to begin at feasible starting values.
In addition, it is also important to place reasonable bounds
on parameters to be estimated. Finally, it is recommended
that one use appropriate algorithms with active set strate-
gies to facilitate identi®cation of binding inequality con-
straints (Gill et al., 1981).
In summary, the formulation presented here provides an
alternative means to operationalize the estimation of
AIDADS. Based on limited testing, the procedure appears
to have some advantages over earlier estimation
approaches. Further work in alternative empirical settings
is needed to con®rm these somewhat promising ®ndings. It
is hoped that this work will facilitate widespread adoption
of Rimmer and Powell’s AIDADS functional form for
analysis of Engel relationships in demand.
R EFER ENCES
Brooke, A., Kendrick, D. and Meeraus, A. (1992) GAMS : A
±
User’s Guide
Release 2.25, The Scienti®c Press, San
Francisco.
Deaton, A. and Muellbauer, J. (1980) An almost ideal demand
±
system, American Economic Review, 70, 312 26.
Gill, P. E., Murray, W. and Wright, M. H. (1981) Practical
Optimization , Academic Press Ltd, London.
Greene, W. H. (1993) Econometric Analysis (2nd edn), Macmillan
Publishing Company, New York.
Hanoch, G. (1975) Production or demand models with direct or
±
indirect implicit additivity, Econometrica, 43, 395 419.
Rimmer, M. T. and Powell, A. A. (1992) An implicitly directly
additive demand system: estimates for Australia. Impact
Project Preliminary Working Paper No. Op-73, Monash
University, Clayton, Victoria, Australia.
Rimmer, M. T. and Powell A. A. (1996) An implicitly additive
±
demand system, Applied Economics, 28, 1613 22.
Stone, R. (1954) Linear expenditure system and demand analysis:
an application to the British pattern of demand, Economic
±
Journal, 64, 511 32
Working, H. (1943) Statistical laws of family expenditure, Journal
±
of the American Statistical Association, 38, 43 56.
United Nations (1992) Handbook of the International Comparisons
±
Programme Series F, No. 62, United Nations, New York,
NY.
A PPENDIX
Derivation of Rimmer and Powell’s L inear Approximation
This appendix outlines the linear approximation employed
by Rimmer and Powell. In Rimmer and Powell (1992),
Equation 3.3.2 equates the change in utility as follows:
n
X
ˆ
ˆ
^
x
i
½
u
Ci
8
1
T ¡ 1
½
;. . . ;
D
D
½
½
ˆ
i
1
^
x
i
Since Ci and
depend on u , the above function
D
½
½
½
cannot be evaluated directly. Consequently, Rimmer and
Powell (1992) totally di€ erentiate the AIDADS demands
(i.e. Equation 2 in this paper) and evaluate the total di€ er-
ential at u1 :
M1 ¡ p 01
M1 ¡ p 0
®
®
1
ˆ
^
dxj1
u d
¡
’j1…
†
1
… † … †
pj1
pj1
u
@’j1…
†
du1
A CKNOWLEDGEMENTS
1
ˆ
£
8j
1
n
. . .
u
@
The authors acknowledge the helpful comments of
Maureen Rimmer, Alan Powell, Yves Surry, Channing
Arndt, Ken Foster and an anonymous referee. Bettina
Aten kindly provided the data for this study. Partial
®nancial support of the United States Department of
1
Discretizing this total di€ erential results in:
M2 ¡ p02
M1 ¡ p0
®
®
1
ˆ
^
^
^
x
xj2 ¡ xj1
º
u
¡
’j1…
†
D
j1
1
…
†
pj2
pj1
±
Agriculture National Research Initiative Grant # 97-
M1 ¡ p 0
u
@’j1…
†
®
1
1
35400-4752 and the Purdue Research Foundation is grate-
fully acknowledged.
‡
u ¡ u
…
†
2
1
… †
pj1
u
1
@
ˆ
8j
1
n
;. . . ; :

An implicitly additive demand system
1915
P
n
ˆ
These changes are likely to be smallest when the M are
sequenced in either increasing or decreasing order.
Substituting the above into Equation 3.3.1 of Rimmer
and that
u
u
0 results in Equation 3.3.10
1 @’jk… k† @
k
½
ˆ
in Rimmer iand Powell (1992), which is used to approxi-
ˆ
mate
u
2
n . Note that changing the order of
the data changes the interval over which the approximation
…½
;. . . ; †
D
½
ˆ
and
‰…- i ’ik… k††
Powell
(1992),
and
also
u
u
@’jk… k† @
k
ˆ
¡
u
exp u
1 ‡exp u
evaluated at k 1,
is made (i.e.,
changes).
Mk‡1 ¡ p0k‡1
pjk‡1¡ Mk ¡ p 0k
pjk
=
…
k†Š ‰
…
k†Š
®
®
noting that Cj1 M1 ¡ p 01
pj1 is independent of j,
®