In investigating linear approximations to the Quadratic Almost Ideal Demand System (QUAIDS), Matsuda (2006) proposed a composite variable to approximate a price aggregator function. This paper provides an axiomatic characterization of this composite price variable.QUAIDS, Linear approximation, Price aggregator, Characterization

Amita Majumder

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
A characterization of the composite price variable to approximate a price 
aggregator function in the Quadratic Almost Ideal Demand System 
  
 
 
Amita Majumder  
Indian Statistical Institute 
Abstract 
In investigating linear approximations to the Quadratic Almost Ideal Demand System (QUAIDS), Matsuda (2006) 
proposed a composite variable to approximate a price aggregator function. This paper provides an axiomatic 
characterization of this composite price variable.
The author thanks Professor Satya R. Chakravarty for his valuable suggestions. The usual disclaimer applies. 
Citation: Amita Majumder, (2009) ''A characterization of the composite price variable to approximate a price aggregator function in the 
Quadratic Almost Ideal Demand System'', Economics Bulletin, Vol. 29 no.3 pp. 2050-2054. 
Submitted: Jul 22 2009.   Published: August 20, 2009. 
 
       1
1. Introduction 
 
In recent times the Quadratic Almost Ideal Demand System (QUAIDS) proposed by 
Banks et al. (1997) has been the most extensively used model in the analysis of consumer 
behaviour because of its versatility and flexibility (see Matsuda 2006 and the references 
therein). As Matsuda (2006) points out, observed price and expenditure data are often 
non-stationary, and application of nonlinear systems to such data is difficult. In order to 
be able to properly handle the popular QUAIDS in the context of non-stationary data, 
Matsuda (2006) proposed a composite variable to approximate one of the price 
aggregator functions. In this paper we provide an axiomatic characterization of this 
composite price variable. 
 
 
2.  QUAIDS and Linear Approximation 
 
 
The cost function underlying QUAIDS is of the form 
 
)
) ( ) ln / 1 (
) (
exp( ). ( ) , (
p u
p b
p a p u C
λ −
= ,                                                                        (1)  
 
where p is the price vector of n commodities,  ) p ( a is a homogeneous function of degree 
one in prices, given by  ∑∑ ∑ + + =
== =
n
i
n
i
n
j
j i ij i i p ln p ln p ln ) p ( a ln
11 1
0 2
1
δ α α , with  0 = ∑
i
i α , 
0 = ∑ = ∑
j
ij
i
ij δ δ ;  ) p ( b and  ) (p λ  are homogeneous functions of degree zero in prices, 
given by  ∏ =
=
n
i
i
i p ) p ( b
1
β , with  0 = ∑
i
i β ;  ∑ =
=
n
i
i i p ln ) p (
1
λ λ , with  0 = ∑
i
i λ  and u denotes 
the level of utility.  
 
By Shephard’s lemma, the budget share functions corresponding to the cost function 
“(1)” are of the form 
 
2 )
) p ( a
y
(ln
) p ( b ) p ( a
y
ln p ln w
i
i j
j
ij i i
λ
β δ α + + ∑ + = ,  i=1,2,…,n,             (2)                    
 
where y denotes nominal  income and i denotes item of expenditure.  
 
Linear approximations to the QUAIDS require that both a(p) and b(p) be replaced by 
composite variables, which are free of unknown parameters. Various approximations to 
a(p) in the context of the Almost Ideal demand System (AIDS) of Deaton and Muellbauer 
(1980) have been suggested in the literature (see, for example, Deaton and Muellbauer   2
1980, Moschini 1995)
1.  In order to approximate b(p), Matsuda (2006) proposed a 
composite variable, 
z P , given by the following equation: 
 
  
z P ln =  0
1
0
1
1
i
i
i
n
i
i p
p
ln ) w w ( − ∑
=
,                               (3) 
 
where  n  is  the number of commodities in two time periods t=0 (base period) and 
1(current period),  ) p ,..., p , p ( p
t
n
t t t
2 1 =  and  ) q ,..., q , q ( q
t
n
t t t
2 1 =  are the vectors of prices  
and quantities, respectively, at period t,  and 
∑
=
=
n
i
t
i
t
i
t
i
t
i t
i
q p
q p
w
1
 is the budget share of 
commodity i at period t, i=1,2,…,n.
2  Evidently, 1
1
= ∑
=
n
i
t
i w , for t=0,1. 
In what follows, we provide an axiomatic characterization of 
z P ln .
3  
 
 
3. Characterization  of 
z P  
 
Consider the class of aggregated price relatives  
0
1
1 0
1
0 0 1 1
i
i
i i
n
i
i p
p
ln ) w , w ( m ) q , p , q , p ( P ln ∑ =
=
, 
where  + → × R ] , [ ] , [ : mi 1 0 1 0,  0 0 0 = ) , ( mi  and  i m is continuous. 
 
Let us look at the following two properties of  ) q , p , q , p ( P
0 0 1 1 . 
 
Property 1:  ) q , p , q , p ( P
0 0 1 1  is homogeneous of degree zero separately in base and 
current period prices. 
 
  That  is,  ) q , p , q , p ( P
0 0 1 1 λ  =  ) q , p , q , p ( P
0 0 1 1  for any scalar  0 > λ . 
and        ) q , p , q , p ( P
0 0 1 1 λ  =  ) q , p , q , p ( P
0 0 1 1  for any scalar  0 > λ . 
 
This implies that   0
1 0
1
= ∑
=
) w , w ( m i i
n
i
i . 
 
                                                 
1 Of these, the  rnqvist o T & &  Index has been axiomatically characterized by Balk and Diewert (2001). 
 
2 Note that the approximation is a function of  . , , ,
1 0 1 0
i i i i p p w w  If the base period prices are set to 1, then 
z P  is a function of current period prices only. 
 
3 It may, however, be pointed out that no optimization relating 
z P  to b(p) is attempted here. The linear 
approximation, which is purely for practical convenience (Matsuda 2006), is the starting point.    3
Property 2:  ) q , p , q , p ( P
0 0 1 1  is symmetric with respect to base and current period prices. 
  That  is,  ) q , p , q , p ( P
0 0 1 1  =  ) q , p , q , p ( P
1 1 0 0 .              
 
This implies that   
1
0
0 1
1
0
1
1 0
1 i
i
i i
n
i
i
i
i
i i
n
i
i p
p
ln ) w , w ( m
p
p
ln ) w , w ( m ∑ = ∑
= =
           (4) 
  Or,     
0
1
0 1
1
0
1
1 0
1 i
i
i i
n
i
i
i
i
i i
n
i
i p
p
ln ) w , w ( m
p
p
ln ) w , w ( m ∑ − = ∑
= =
.             (5) 
 
Theorem:  Property 1 and Property 2 hold together if and only if     
                                 ) w w .( c ) w , w ( m i i i i i
0 1 1 0 − = , where c is a positive constant. 
  
Proof: 
Since “(5)” is true for all n, it is in particular true for n=1. This is turn implies that   
            ) w , w ( m ) w , w ( m
0
1
1
1 1
1
1
0
1 1 − = . 
 
Using this successively in “(5)” for n = 2,3,…, we can show that 
                ) w , w ( m ) w , w ( m i i i i i i
0 1 1 0 − =    for all i. 
            Or,            0
0 1 1 0 = + ) w , w ( m ) w , w ( m i i i i i i  for all i. 
 
Now, this is of the form F(x,y)+F(y,x)=0, which is a special case of the general form  
F(x,y)+F(y,z)=F(x,z), with z=x,  
where F is continuous. 
 
The solution of this functional equation is given by 
     F(x,y) = g(y)-g(x)                   [see  l e Acz ′ 1966, page 223, Theorem 1], 
where g is continuous. 
 
Therefore, the solution in our case is  ) w ( g ) w ( g ) w , w ( m i i i i i
0 1 1 0 − =  for all i. 
 
Observe from Property 1 that  0
1 0
1
= ∑
=
) w , w ( m i i
n
i
i . Hence,  0
1
0 1 = ∑ −
=
n
i
i i )) w ( g ) w ( g ( . 
Now, assume that  , ) w ( g ) w ( g
n
i
i
n
i
i θ = ∑ = ∑
= = 1
0
1
1  say. Note that for any 
k
wi
1
= , i=1,2,…,k, 
k
) w ( g i
θ
= .  Since g is continuous, it now follows that the form of g is given by  
g(x) = c.x, where c is a positive constant. 
 
Hence, 
0
1
1
0 1
0
1
1 0
1 i
i
n
i
i i
i
i
i i
n
i
i p
p
ln ) w w ( . c
p
p
ln ) w , w ( m ∑ − = ∑
= =
.   4
It is straightforward to verify that 
0
1
0 1
i
i
i i p
p
ln ) w w ( − ∑  satisfies the two properties 
mentioned above. This completes the proof of the theorem.                                                
 
Thus, the price aggregator  ) q , p , q , p ( P ln
0 0 1 1  is a unique (up to a scalar multiple) 
member of the class of the price aggregators with  ) q , p , q , p ( P
0 0 1 1 satisfying the 
property of homogeneity of degree zero in current period prices and symmetry with 
respect to base and current period prices. 
  
As already observed earlier, the approximation uses data from a pair of time periods. So, 
for practical application to data sets covering many time periods, one time period may be 
chosen as base (time period 0 in the formula). 
  
 
References 
 
l e Acz ′ , J. (1966) Lectures on Functional Equations and Their Applications, Academic 
Press. 
 
Balk, B. M. and W. E. Diewert (2001)  “A Characterization of the Tornqvist Price Index”   
Economics Letters 72, 279-281. 
 
Banks, J., R. Blundell and A. Lewbel (1997) “Quadratic Engel Curves and Consumer  
Demand” The Review of Economics and Statistics 79, 527-539. 
 
Deaton, A. S. and J. Muellbauer (1980) “An Almost Ideal Demand System” American  
Economic Review 70, 312-326. 
 
Matsuda, T. (2006) “Linear Approximations to the Quadratic Almost Ideal Demand  
System” Empirical Economics 31, 663-675. 
 
Moschini, G. (1995) “Units of Measurement and the Stone Index in Demand System  
Estimation” American Journal of Agricultural Economics 77, 63-68. 
 

Lectures on Functional Equations and Their Applications,

Linear Approximations to the Quadratic Almost Ideal Demand System”

Units of Measurement and the Stone Index in Demand System Estimation”

A characterization of the composite price variable to approximate a price aggregator function in the Quadratic Almost Ideal Demand System

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A characterization of the composite price variable to approximate a price aggregator function in the Quadratic Almost Ideal Demand System

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