The material contained herein is supplementary to the article named in the title and published in the American Journal of Agricultural Economics, Volume 89, Number 4, November 2007.Demand and Price Analysis,

Moschini, GianCarlo

Rizzi, Pier Luigi

English

Research Papers in Economics

 
 
 
AJAE Appendix: 
 
Deriving a Flexible Mixed Demand System: The Normalized Quadratic Model 
 
 
GianCarlo Moschini and Pier Luigi Rizzi * 
 
 
 
 
 
January 2007 
 
 
 
The material contained herein is supplementary to the article named in the title and published 
in the American Journal of Agricultural Economics (AJAE). 
 
* GianCarlo Moschini is professor and the Pioneer Chair in Science and Technology Policy, 
Department of Economics, Iowa State University. Pier Luigi Rizzi is professor, Dipartimento 
di Economia Politica, Università di Siena, Italy.  2 
Flexibility Properties of the Normalized Quadratic Mixed Demand System 
  For the functional specification proposed here to be “flexible” according to Diewert’s 
(1974) definition, it needs to be able to attain arbitrary values for the cost function and its first 
and second derivatives at a point  ,, p z u . For simplicity, let’s take this point to be unity for all 
variables involved (this is possible without loss of generality by appropriate choice of units of 
measurement). Also, because the  i a  coefficients are predetermined, we will assume 
(A.1) 
1
1
n
i
i
a  . 
Furthermore, note that, because utility is ordinal, arbitrary levels of C  and  u C  could be attained 
by any linear transformation of u. Finally, as in Diewert and Wales (1988), we concern ourselves 
only with the class of cost functions satisfying the money-metric scaling, i.e.,  , , 0 uu C p z u .  
  The arbitrary values that we need to consider, at the point  1 i pi ,  1 k zk , and 
1, u  are then: 
(A.2) 
1 1 1 1 1 1 1 1
11
, 1,...,
22
i
n m m m n n m m
p i i ij ik i k k ij ks
j k k k i j s k
C a i n  
(A.3) 
11
, 1,...,
k
mn
z k k ks jk
sj
C k m   
(A.4) 
1 1 1 1
, , 1,...,
ij
n n n n
p p ij j ir i jr i j rt
r r r t
C a a a a i j n  
(A.5)  , , 1,...,
ks z z ks C k s m 
(A.6) 
1
, 1,...   and   1,...,
ik
m
p z ik i k k ks
s
C a i n k m  
(A.7) 
1 1 1 1 1 1 1
11
, 1,...,
22
i
n m m n n m m
up i ij ik i k ij ks
j k k i j k s
C a i n   3 
(A.8) 
11
, 1,...,
k
mn
uz k ks jk
sj
C k m. 
  We can now ascertain the flexibility of our function. Arbitrary values for 
ks zz C  are 
possible as long as the  ks  parameters are unrestricted. Similarly, the free parameters  ij and 
ik  allow the form for achieving arbitrary values for 
ij pp C  and 
ik pz C , respectively. The 
parameter  k  allows arbitrary values for 
k uz C , and the parameter  i  allows arbitrary values for 
i up C . Finally, arbitrary values for 
k z C  are possible because the parameters  k  are unrestricted, 
and arbitrary values for 
i p C  are possible because of the free parameters  i . 
  Of course, the parameters just mentioned can be subject to some restrictions without 
destroying the flexibility of the model for the problem at hand. Specifically, be cause of 
symmetry of second derivatives, flexibility is still guaranteed if w e enforce the following 
restrictions: 
(A.9)  , , 1,..., ij ji i j n 
(A.10)  , , 1,..., ks sk k s m . 
Furthermore, so far we have neglected the restrictions implied by the fact that  ( , , ) C p z u  is 
homogeneous of degree one in  p . Thus, the Euler theorem implies that the first derivatives 
i p C  
are subject to one restriction. Hence, we can write 
(A.11) 
1
0
n
i
i
. 
Because each first derivative 
i p C  is homogeneous of degree zero in p, applying Euler’s theorem 
once more we find that second derivatives 
ij pp C are subject to n restrictions, second derivatives   4 
ik pz C are subject to m restrictions, and second derivatives 
i up C are subject to one restriction. 
These last three conditions allow, respectively, 
(A.12) 
1
0 , 1,...,
n
ij
j
in   
(A.13) 
1
0 , 1,...,
n
jk
j
km  
(A.14) 
1
1
n
i
i
. 
This concludes the illustration of the flexibility of the model presented in the  article, and 
provides a justification of the parametric restrictions of equations (21)-(27) in the text. ■ 
 
References 
See the corresponding AJAE article. 

AJAE Appendix: Deriving a Flexible Mixed Demand System: The Normalized Quadratic Model

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