In this paper, we build on Ryan and Wales (1998) and Moschini (1999) and impose curvature conditions locally on the generalized Leontief model, introduced by Diewert (1974). In doing so, we exploit the Hessian matrix of second order derivatives of the reciprocal indirect utility function, unlike Ryan and Wales (1998) and Moschini (1999) who exploit the Slutsky matrix.

Apostolos Serletis

Asghar Shahmoradi

English

Research Papers in Economics

 
Department of Economics 
  Discussion Paper 2005-17 
 
 
 
A Note on Imposing Local Curvature in 
Generalized Leontief Models 
 
 
 
Apostolos Serletis 
University of Calgary 
 and 
Asghar Shahmoradi 
University of Calgary 
 
 
 
 
 
Department of Economics 
University of Calgary 
Calgary, Alberta, Canada 
T2N 1N4 
 
This paper can be downloaded without charge from 
http://www.econ.ucalgary.ca/research/research.htm  A Note on Imposing Local Curvature in
Generalized Leontief Models￿
Apostolos Serletisy and Asghar Shahmoradi
Department of Economics
University of Calgary
Calgary, Alberta, T2N 1N4,
Canada
September 11, 2005
Abstract
In this paper, we build on Ryan and Wales (1998) and Moschini
(1999) and impose curvature conditions locally on the generalized
Leontief model, introduced by Diewert (1974). In doing so, we ex-
ploit the Hessian matrix of second order derivatives of the reciprocal
indirect utility function, unlike Ryan and Wales (1998) and Moschini
(1999) who exploit the Slutsky matrix.
JEL classi￿cation: C3; C13; C51.
Keywords: Regularity; Slutsky matrix; Negative semide￿nite.
￿Serletis gratefully acknowledges ￿nancial support from the Social Sciences and Hu-
manities Research Council of Canada.
yCorresponding author. Phone: (403) 220-4092; Fax: (403) 282-5262; E-mail: Ser-
letis@ucalgary.ca; Web: http://econ.ucalgary.ca/serletis.htm.
11 Introduction
As Moschini (1999, p. 487) puts it ￿[a] critical implication of consumer-
demand theory is that the expenditure function is concave in prices, which
implies that the so-called Slutsky matrix is negative semide￿nite. Yet this
curvature property is often not maintained in empirical applications, and
estimated demand functions frequently fail to satisfy the implications of a
concave expenditure function.￿For this reason, in the recent literature there
is an attempt to treat the curvature property as a maintained hypothesis and
built it into the models being estimated, very much like the homogeneity in
prices and symmetry properties of neoclassical consumer theory.
In particular, Ryan and Wales (1998) suggest a relatively simple pro-
cedure for imposing local curvature conditions. Their procedure applies to
those locally ￿ exible demand systems for which, at the point of approxima-
tion, the n ￿ n Slutsky matrix S can be written as
S = B + C (1)
where B is an n ￿ n symmetric matrix, containing the same number of
independent elements as the Slutsky matrix, and C is an n￿n matrix whose
elements are functions of the other parameters of the system. Curvature
requires the Slutsky matrix to be negative semide￿nite. Ryan and Wales
(1998) draw on related work by Lau (1978) and Diewert and Wales (1987)
and impose curvature by replacing S in (1) with ￿KK0, where K is an
n ￿ n lower triangular matrix, so that ￿KK0 is by construction a negative
semide￿nite matrix. Then solving explicitly for B in terms of K and C
yields
B = ￿KK
0 ￿ C;
meaning that the models can be reparameterized by estimating the parame-
ters in K and C instead of the parameters in B and C. That is, we can
replace the elements of B in the estimating equations by the elements of
K and the other parameters of the model, thus ensuring that S is negative
semide￿nite at the point of approximation, which could be any data point.
Ryan and Wales (1998) applied their procedure to three locally ￿ exible
functional forms ￿ the almost ideal demand system of Deaton and Muell-
bauer (1980), the normalized quadratic of Diewert and Wales (1988), and the
linear translog. Moreover, Moschini (1999) suggested a possible reparame-
terization of the basic translog of Christensen, Jorgenson, and Lau (1975) to
2overcome some problems noted by Ryan and Wales (1998) and also imposed
curvature conditions locally in the basic translog.
In this paper, we build on Ryan and Wales (1998) and Moschini (1999)
and impose curvature conditions locally on the generalized Leontief model,
introduced by Diewert (1974). In doing so, we exploit the Hessian matrix
of second order derivatives of the reciprocal indirect utility function, unlike
Ryan and Wales (1998) and Moschini (1999) who exploit the Slutsky matrix.
The rest of the paper is organized as follows. Section 2 brie￿ y discusses the
generalized Leontief demand system while Section 3 discusses our procedure
for imposing local curvature conditions. The ￿nal section provides a brief
conclusion.
2 The Generalized Leontief Model
The generalized Leontief (GL) reciprocal indirect utility function, introduced
by Diewert (1974), is
h(v) = ￿0 +
n X
i=1
￿iv
1=2
i +
1
2
n X
i=1
n X
j=1
￿ijv
1=2
i v
1=2
j (2)
where B ￿ [￿ij] is an n ￿ n symmetric matrix of parameters and ￿0 and ￿i
are other parameters, for a total of (n2 + 3n + 2)=2 parameters.
Using Roy￿ s identity the share equations of the GL demand system can
be writtten as
si =
￿iv
1=2
i +
n P
j=1
￿ijv
1=2
i v
1=2
j
n P
j=1
￿jv
1=2
j +
n P
k=1
n P
m=1
￿kmv
1=2
k v
1=2
m
; i = 1;￿ ￿ ￿;n (3)
Because the share equations are homogenous of degree zero in the para-
meters, one can follow Barnett and Lee (1985) and impose the following
normalization in estimation
2
n X
i=1
￿i +
n X
i=1
n X
j=1
￿ij = 1 (4)
33 Local Curvature for GL Models
Curvature of the reciprocal indirect utility function requires that the Hessian
matrix is negative semide￿nite. We can therefore impose local curvature
(that is, at the reference point) by evaluating the Hessian terms of (2) at
￿￿
i = 1 8i, as follows
Hij = ￿￿ij
 
￿i +
n X
j=1;j6=i
￿ij
!
+ (1 ￿ ￿ij)￿ij
where ￿ij = 1 if i = j and 0 otherwise. By replacing H by ￿KK0, where K
is an n ￿ n lower triangular matrix and K0 its transpose, the above can be
written as
￿(KK
0)ij = ￿￿ij
 
￿i +
n X
j=1;j6=i
￿ij
!
+ (1 ￿ ￿ij)￿ij (5)
Solving for the ai and ￿ij terms as a function of the (KK0)ij terms we can
get the restrictions that ensure the negative semide￿niteness of the Hessian
matrix (without destroying the ￿ exibility properties of (2), since the number
of free parameters remains the same). In particular, when i 6= j (5) yields
￿ij = ￿(KK
0)ij (6)
and for i = j it yields
(KK
0)ii = ai +
n X
j=1;j6=i
￿ij
Substituting ￿ij from (6) in the above equation
(KK
0)ii = ai ￿
n X
j=1;j6=i
(KK
0)ij
or
ai = (KK
0)ii +
n X
j=1;j6=i
(KK
0)ij
4which after some rearrangement yields
ai =
n X
j=1
(KK
0)ij (7)
As an example, for the case of three goods (6) and (7) imply the following
six restrictions
￿12 = ￿k11k21
￿13 = ￿k11k31
￿23 = ￿(k21k31 + k22k32)
a1 = k
2
11 + k11k21 + k11k31
a2 = k
2
21 + k
2
22 + k11k21 + k21k31 + k22k32
a3 = k
2
31 + k
2
32 + k
2
33 + k11k31 + k21k31 + k22k32
where the kij terms are the elements of the replacement matrix K.
4 Conclusion
We have illustrated a simple procedure for imposing curvature conditions
locally in generalized Leontief models, building on earlier work in this area
by Ryan and Wales (1998) and Moschini (1999). The procedure is rooted in
the work by Lau (1978) and Diewert and Wales (1987), but unlike the Ryan
and Wales (1998) and Moschini (1999) procedures that exploits the Slutsky
matrix, our procedure exploits the Hessian matrix of second order derivatives
of the reciprocal indirect utility function.
5References
[1] Barnett, W.A. and Y.W.Lee. ￿The Global Properties of the Min￿ ex Lau-
rent, Generalized Leontief, and Translog Flexible Functional Forms.￿
Econometrica 53 (1985), 1421-1437. Reprinted in W.A. Barnett and J.
Binner (eds), Functional Structure and Approximation in Econometrics.
Elsevier, Amsterdam (2004), Chapter 4, pp. 79-97.
[2] Christensen, L.R., D.W. Jorgenson, and L.J. Lau. ￿Transcendental Log-
arithmic Utility Functions.￿American Economic Review 65 (1975), 367-
383.
[3] Deaton, A. and J. N. Muellbauer. ￿An Almost Ideal Demand System.￿
American Economic Review 70 (1980), 312-326.
[4] Diewert, W. Erwin. ￿Applications of Duality Theory.￿ In Frontiers
of Quantitative Economics (Vol. 2), eds. M.D. Intriligator and D.A.
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[5] Diewert, W. Erwin and Terence J. Wales. ￿Normalized Quadratic Systems
of Consumer Demand Functions.￿ Journal of Business and Economic
Statistics 6 (1988), 303-312.
[6] Diewert, W. Erwin and Terence J. Wales. ￿Flexible Functional Forms and
Global Curvature Conditions.￿Econometrica 55 (1987), 43-68.
[7] Lau, L.J. ￿Testing and Imposing Monotonicity, Convexity, and Quasi-
Convexity Constraints.￿In Production Economics: A Dual Approach to
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sterdam: North Holland, (1978), p. 409-453.
[8] Moschini, Giancarlo. ￿Imposing Local Curvature in Flexible Demand Sys-
tems.￿Journal of Business and Economic Statistics 17 (1999), 487-490.
[9] Ryan, David L. and Terence J. Wales. ￿A Simple Method for Imposing
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of Business and Economic Statistics 16 (1998), 331-338.
6

A Note on Imposing Local Curvature in Generalized Leontief Models

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A Note on Imposing Local Curvature in Generalized Leontief Models

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