Download An analytical solution to determine Equivalence Scales from single

An analytical solution to determine Equivalence Scales from
single equation models
Mauro Maltagliati
Dipartimento di statistica “G.Parenti”
Viale Morgagni, 59 - 50134 Florence Italy
[email protected]
Introduction
There are basically two ways to estimate Equivalence Scales (ES) from households’ budget
data: single equation models, namely Engel and Rothbarth models, and complete demand systems
(for example LES, Translog, AIDS, GAIDS). The ES is an index that measures the ratio between the
expenditures of the two households at the same level of utility, i.e. at the same level of food share.
The aim of this work is to determine an analytical solution for the ES starting from a
quadratic single equation model. We will consider only the Engel model, but the methodology we
will use can be easily extended to the Rothbarth case.
1. The Engel model, linear case
The principle of the Engel model is that two households have the same level of utility if the
shares of food expenditure (food shares) are the same for both the households:
(1)
wc=wr⇔Uc=Ur
Where wc and wr are, respectively, the food shares of a generic household c, and the reference
one r. Uc and Ur are the utilities for the two households.In single equation models, is generally used
a demographically extended version of the Working-Leser Engel curve:
(2)
w=a0+a1lnx
where lnx is the logarithm of total expenditure and a0 and a1 are parameters.
If we consider, for simplicity, only the households with exactly two adults, we can
differentiate the households only by the number of children (NC).
(3)
w=a0+a1lnx+kNC
If we take, as the reference household, the one without children, we have:
(4)
wc=a0+a1lnxc+kNC
(5)
wr=a0+a1lnxr
If we define:
(6)
z=ln(ES)=ln(xc/xr)=lnxc-lnxr
As it is well known, by equalising the two food shares an subtracting (4) from (5) we can
easily obtain:
(7)
z=−kNC/a1
and hence:
(8)
(−kNC/a1)
ES=e
1
The obtained ES does not depend on the level of total expenditure that means, using
Blackorby and Donaldson terminology, mantain the assumption of Equivalent Scale Exactness
(ESE).
2. The quadratic case
Several authors (see, for example, Banks and Lewbel, 1992) have stressed the importance of
inserting a quadratic term of the log-expenditure in the Engel curve of the kind of (2), obtaining, for
the two households:
(9)
wc=a0+a1lnxc+a2(lnxc)2+ktNC
(10)
wr=a0+a1lnxr+a2(lnxr)2
Usually (see, among the others, Lancaster and Ray, 1996) a numeric iterative method is used
to obtain ES from (9) and (10) equations, but an analytical way is possible too. If we equalise the
two food shares, and then subtract (9) from (10), we obtain, by operating the transformation (6):
(11)
a1z+a2[(z+lnxr)2–(lnxr)2]+kNC=0
Two solutions of the ES can now be obtained:
(12)
{–(2a2lnxr+a1) ±[(2a2lnxr+a1)2−4a2kNC] 0,5}/2a2
ES=e
In the paper is proved that only one of them is allowable, and finally we have:
(13)
{–(2a2lnxr+a1) – [(2a2lnxr+a1)2−4a2kNC] 0,5}/2a2
ES=e
Conclusion
It is evident that (13) is not ESE, because lnxr appears in equation (13). This means that we
have a different ES for every level of expenditure we choose for the reference household. The test of
significance for a2 can thus be used to reject (or accept) the ESE assumption. The main advantage of
the procedure is that study of the parameters of (13) can immediately show if the ES is increasing
(a2 positive) or decreasing (a2 negative) with the growth of lnxr, without resorting to different
numerical solutions of the (9) and (10).
REFERENCES
Banks, J. Blundell, R. e Lewbel, A. (1992), “Quadratic Engel curves and consumer demand”,
Institute for Fiscal Studies, Working Paper W92/8.
Blackorby, C. e Donaldson, D., (1994), “Measuring the Cost of Children: A Theoretical
Framework”, in R. Blundell, I. Preson e I. Walker, eds., The Measurement of Household Welfare,
CUP, Cambridge.
Lancaster, G. e Ray, R. (1996), “Comparison of Alternative Methods of Estimating Household
Equivalence Scales: The Australian Evidence on Pooled Time Series of Unit Record Data”,
Discussion Paper, 1996-11 of Department of Economics, University of Tasmania.
RESUME
Dans cette note on propose une méthode alternative d’estimation les echelle d’equivalence pour le
modéle d’Engel quadratique
2