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Towards a more general version of the almost ideal demand system
Economics Letters North-Holland
38 (1992) 305-308
305
Towards a more general version of the Almost Ideal Demand System * Pedro
Duarte
Neves
UtGersidade Cato’lica Portuguesa, Lisbon, Portugal Received Accepted
30 October 1991 4 January 1992
This note proposes a generalization of the Almost ideal Demand System. Two independent flexibility to the shape of the Engel curves and to the form of the price effects.
parameters
provide
more
1. Introduction The standard approach to model consumption behaviour proceeds by postulating functional forms representing the preferences of the consumer. From a theoretical point of view, this methodology suffers from the defect that one tends to impose unwanted restrictions on the parametric representation of preferences, in addition to the restrictions implied by utility maximization. The results will replicate properly the reality if and only if the postulated parametric forms provide good approximations to the unknown demand functions. In this way, the choice of functional forms is one of the most important issues in the empirical analysis of consumer behaviour. This note proposes a generalization of the Almost Ideal Demand System [Deaton and Muellbauer (1980)], one of the most commonly estimated demand systems (hereafter denoted as the AI model). The share equations implied by the AI model are linear functions of log prices and log real expenditure. This is unduly restrictive and it would be desirable to capture alternative shapes for these effects. The Box-Cox transformation [Box and Cox (196411 is a very convenient way to achieve this flexibility. Two Box-Cox parameters are added to the AI model: one determines the shape of the Engel curves; the other captures alternative forms of the (real-expenditure constant) price effects.
2. The representation Consider
of preferences
the following
V(P,m)=
indirect
m [ U(P)
utility
1
(-0)
function:
1 .--b(P)
(1)
’
Correspondence to: Pedro Duarte Neves, Universidade Cat6lica Portuguesa, Palma de Cima, 1600 Lisboa, Portugal. * The author thanks Anton Barten and Richard Blundell for helpful discussions. Financial support from the Commission the European Communities, under the SPES programme, is gratefully acknowledged. 0165.1765/92/$05.00
0 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
of
306
P. D. Necles / A more general version of the Almost Ideal Demand System
where m is the level of expenditure, p is the price vector, a(p) is a linear homogeneous price index, b(p) is a price index homogeneous of degree zero, and I ) denotes the Box-Cox transformation
x0- 1 XVI =
if
6
i In x
Of0
(2)
otherwise.
We select the following parameterizations
a(p)(“)= b(p)
a”
= exp(
Homogeneity
of
&Ykpy/21
+
CPk
A
CYkj=
2aj?
+ 0.5 C&kjpjr’2’pjh’2’,
(3)
ln pk).
and
a(p)
for a(p) and b(p):
b(p)
(4)
imposes the following restrictions:
k= l,...,n,
(5b)
j=l
(5c)
i A ZLYk>
CYkj=
,-..,n,
k
;s, =0.
(5d)
The demand functions implied by the representation wi =
of preferences
(1) are
(6)
c
Expression (6) shows that the shares are function of the two Box-Cox parameters, interesting to analyse special cases of this demand equation.
8 and A. It is
(i) The Almost Ideal model: 0 = A = 0
The Almost Ideal demand equations are the following: Wi=a,+
where the price index u(p)
=
In Pj+Pi
CYij
(7)
takes the form
a(p)
exp(a0
ln(m/a(p)),
+
cak h
pk +
o.5
cc Ykj In
pk
In
pj).
(8)
P. D. Neces /A
more general version of the Almost Ideal Demand System
307
Expression (7) shows that the AI model imposes the shares to be linear functions of log prices and the log of the real expenditure. (ii) Blundell, Browning and Meghir (1989): h = 0; 0 # 0
Allowing 0 to be different than zero leads to the demand functions m wi = ai + C yij In p, + pi [ a(l)> j
(0)
1
’
where a(p) has the same form as in (8). As we have seen in (i>, the case 0 = 0 implies the AI parameterization; if 0 = - 1 we have quasi-homothetic preferences, whilst 0 = + 1 implies quadratic Engel curves. If all the p’s are equal to zero we have homothetic preferences and 0 is not identified. (iii) The unrestricted model: A # 0; 19f 0
Allowing A to be different than zero, we are able to derive the following Marshallian demand functions:
c
(0)
yijpyp;/2
wi= ~-&/kjp;/2p;/2
+pi
[
a(;) 1
’
( lOa)
where the price index a(p) takes the form a(p)
Alternatively,
=
i
1 CCYkipkh
(11)
(lOa) could have been written as
wi = c yijp,?qj,?/2
m + p, [ a(p)
(01
1 ’
( 1Ob)
where jj, represents the relative price of the commodity i, using as deflator the price index (~~ YkjPih’2pkh’2)“2. A greed search procedure is a convenient way to investigate the empirical performance of alternative combinations of A and 0. Note that for model (iii) and for fixed values of A and 0 the total number of parameters to estimate is (n2 + 3n - 2)/2: n - 1 p’s, and n(n + 1)/2 y’s. For either the model (i) or the model (ii), and for fixed values of A (i.e. zero) and 8, the total number of parameters to estimate is also (n2 + 3n - 2)/2: a,,, n - 1 (Y’S,n - 1 /3’s and n(n - 1)/2 y’s. Thus, in order to have a comparable number of parameters in these models, Q, commonly interpreted as (the log of) the subsistence expenditure in the base year, should be estimated in the AI model, which is not usually the case in the empirical work.
3. Conclusions The usefulness of the Box-Cox transformations included in the indirect utility function (6) is clear. The parameter 8 captures alternative shapes of the Engel curves, without changing the form
308
P.D. Neues /A
more general version of the Almost Ideal Demand System
of the (real expenditure constant) price effects. The parameter A captures different forms of the (real expenditure constant) price effects, without changing the shape of the Engel curves.
References Box, G.E.P. and D.R. Cox, 1964, An analysis of transformations, Journal of the Royal Statistical Society, Series B, 26, no. 2, 211-252. Blundell, R.W., M.J. Browning and C. Meghir, 1989, A microeconometric model of intertemporal substitution and consumer demand, Discussion paper no. 89-11 (University College, London). Deaton, AS. and J. Muellbauer, 1980, Economics and consumer behaviour (Cambridge University Press, New York).
38 (1992) 305-308
305
Towards a more general version of the Almost Ideal Demand System * Pedro
Duarte
Neves
UtGersidade Cato’lica Portuguesa, Lisbon, Portugal Received Accepted
30 October 1991 4 January 1992
This note proposes a generalization of the Almost ideal Demand System. Two independent flexibility to the shape of the Engel curves and to the form of the price effects.
parameters
provide
more
1. Introduction The standard approach to model consumption behaviour proceeds by postulating functional forms representing the preferences of the consumer. From a theoretical point of view, this methodology suffers from the defect that one tends to impose unwanted restrictions on the parametric representation of preferences, in addition to the restrictions implied by utility maximization. The results will replicate properly the reality if and only if the postulated parametric forms provide good approximations to the unknown demand functions. In this way, the choice of functional forms is one of the most important issues in the empirical analysis of consumer behaviour. This note proposes a generalization of the Almost Ideal Demand System [Deaton and Muellbauer (1980)], one of the most commonly estimated demand systems (hereafter denoted as the AI model). The share equations implied by the AI model are linear functions of log prices and log real expenditure. This is unduly restrictive and it would be desirable to capture alternative shapes for these effects. The Box-Cox transformation [Box and Cox (196411 is a very convenient way to achieve this flexibility. Two Box-Cox parameters are added to the AI model: one determines the shape of the Engel curves; the other captures alternative forms of the (real-expenditure constant) price effects.
2. The representation Consider
of preferences
the following
V(P,m)=
indirect
m [ U(P)
utility
1
(-0)
function:
1 .--b(P)
(1)
’
Correspondence to: Pedro Duarte Neves, Universidade Cat6lica Portuguesa, Palma de Cima, 1600 Lisboa, Portugal. * The author thanks Anton Barten and Richard Blundell for helpful discussions. Financial support from the Commission the European Communities, under the SPES programme, is gratefully acknowledged. 0165.1765/92/$05.00
0 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
of
306
P. D. Necles / A more general version of the Almost Ideal Demand System
where m is the level of expenditure, p is the price vector, a(p) is a linear homogeneous price index, b(p) is a price index homogeneous of degree zero, and I ) denotes the Box-Cox transformation
x0- 1 XVI =
if
6
i In x
Of0
(2)
otherwise.
We select the following parameterizations
a(p)(“)= b(p)
a”
= exp(
Homogeneity
of
&Ykpy/21
+
CPk
A
CYkj=
2aj?
+ 0.5 C&kjpjr’2’pjh’2’,
(3)
ln pk).
and
a(p)
for a(p) and b(p):
b(p)
(4)
imposes the following restrictions:
k= l,...,n,
(5b)
j=l
(5c)
i A ZLYk>
CYkj=
,-..,n,
k
;s, =0.
(5d)
The demand functions implied by the representation wi =
of preferences
(1) are
(6)
c
Expression (6) shows that the shares are function of the two Box-Cox parameters, interesting to analyse special cases of this demand equation.
8 and A. It is
(i) The Almost Ideal model: 0 = A = 0
The Almost Ideal demand equations are the following: Wi=a,+
where the price index u(p)
=
In Pj+Pi
CYij
(7)
takes the form
a(p)
exp(a0
ln(m/a(p)),
+
cak h
pk +
o.5
cc Ykj In
pk
In
pj).
(8)
P. D. Neces /A
more general version of the Almost Ideal Demand System
307
Expression (7) shows that the AI model imposes the shares to be linear functions of log prices and the log of the real expenditure. (ii) Blundell, Browning and Meghir (1989): h = 0; 0 # 0
Allowing 0 to be different than zero leads to the demand functions m wi = ai + C yij In p, + pi [ a(l)> j
(0)
1
’
where a(p) has the same form as in (8). As we have seen in (i>, the case 0 = 0 implies the AI parameterization; if 0 = - 1 we have quasi-homothetic preferences, whilst 0 = + 1 implies quadratic Engel curves. If all the p’s are equal to zero we have homothetic preferences and 0 is not identified. (iii) The unrestricted model: A # 0; 19f 0
Allowing A to be different than zero, we are able to derive the following Marshallian demand functions:
c
(0)
yijpyp;/2
wi= ~-&/kjp;/2p;/2
+pi
[
a(;) 1
’
( lOa)
where the price index a(p) takes the form a(p)
Alternatively,
=
i
1 CCYkipkh
(11)
(lOa) could have been written as
wi = c yijp,?qj,?/2
m + p, [ a(p)
(01
1 ’
( 1Ob)
where jj, represents the relative price of the commodity i, using as deflator the price index (~~ YkjPih’2pkh’2)“2. A greed search procedure is a convenient way to investigate the empirical performance of alternative combinations of A and 0. Note that for model (iii) and for fixed values of A and 0 the total number of parameters to estimate is (n2 + 3n - 2)/2: n - 1 p’s, and n(n + 1)/2 y’s. For either the model (i) or the model (ii), and for fixed values of A (i.e. zero) and 8, the total number of parameters to estimate is also (n2 + 3n - 2)/2: a,,, n - 1 (Y’S,n - 1 /3’s and n(n - 1)/2 y’s. Thus, in order to have a comparable number of parameters in these models, Q, commonly interpreted as (the log of) the subsistence expenditure in the base year, should be estimated in the AI model, which is not usually the case in the empirical work.
3. Conclusions The usefulness of the Box-Cox transformations included in the indirect utility function (6) is clear. The parameter 8 captures alternative shapes of the Engel curves, without changing the form
308
P.D. Neues /A
more general version of the Almost Ideal Demand System
of the (real expenditure constant) price effects. The parameter A captures different forms of the (real expenditure constant) price effects, without changing the shape of the Engel curves.
References Box, G.E.P. and D.R. Cox, 1964, An analysis of transformations, Journal of the Royal Statistical Society, Series B, 26, no. 2, 211-252. Blundell, R.W., M.J. Browning and C. Meghir, 1989, A microeconometric model of intertemporal substitution and consumer demand, Discussion paper no. 89-11 (University College, London). Deaton, AS. and J. Muellbauer, 1980, Economics and consumer behaviour (Cambridge University Press, New York).