Conditional Demands and Marginal Tax Reform
| Published date | 01 May 1997 |
| DOI | http://doi.org/10.1111/1468-0084.00062 |
| Date | 01 May 1997 |
| Author | David Madden |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 59, 2 (1997)
0305-9049
CONDITIONAL DEMANDS AND MARGINAL
TAX REFORM
David Madden*
I. INTRODUCTION
Traditional demand analysis usually assumes weak separability between
goods and leisure, despite casual observation to the contrary.1The
assumption of weak separability can be relaxed in two ways. Firstly, we
can regard goods and leisure (labour) as being endogenous or jointly
determined.2A second approach, which we concentrate on here, involves
the use of conditional demand functions.3In this approach leisure is
assumed to be fixed, and goods are demanded conditional on the quantity
of leisure being consumed. This conditional approach possesses a number
of advantages over the jointly determined approach. Firstly, the results
obtained from estimation are not dependent upon having the correct
specification for leisure demand, or equivalently, the correct model of
labour force participation or hours worked. Secondly, more flexible forms
for preferences for non-leisure goods may be used.
This second advantage is particularly relevant when we wish to test for
weak separability. This is because it avoids the trade-off present in jointly
determined models between exact tests for separability using quite
restrictive functional forms and approximate tests using less restrictive
forms.4The conditional approach allows us to test for weak separability
*I would like to thank Peter Neary, Rodney Thom and an anonymous referee for helpful
comments. I thank John Fitzgerald of the Economic and Social Research Institute and Fergal
O Brollchain of the Department of Finance for assistance with the data. I also gratefully
acknowledge financial support from the Foundation for Fiscal Studies and the HCM Network
on the Microeconometrics of Public Policy funded by grant 930225.
1For a recent example see the volume by Pollak and Wales (1992) and the references
therein. For examples in the Irish case, see Madden (1993).
2For examples of this approach see Abbott and Ashenfelter (1976), Barnett (1979) and
Blundell and Walker (1982). For examples in the Irish case, see Murphy and Thom (1987a)
and Madden (1995a).
3The earliest references here are Pollak (1969, 1971). For a recent application, see
Browning and Meghir (1991).
4For examples of the former where separability is tested for exactly, but with preferences
which are quasi-homothetic in full income, see Blundell and Walker (1982), Murphy and
Thom (1987a) and Madden (1995a). For an example of approximate testing for separability
see Barnett (1979). For an example of nonparametric testing which rejects separability see
Swofford and Whitney (1987). 237
© Blackwell Publishers Ltd, 1997. Published by Blackwell Publishers, 108 Cowley Road, Oxford
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exactly while at the same time using a flexible representation of prefer-
ences such as Deaton and Muellbauer’s Almost Ideal Demand System
(AIDS; Deaton and Muellbauer, 1980). Thus the possible misspecification
involved in assuming leisure demand to be fixed is balanced by our using
more flexible forms than those typically employed in jointly determined
models.5
Conditional demand functions may also be of use in analysing issues in
both optimal taxation and tax reform. Under certain conditions, weak
separability between goods and leisure may imply that uniform indirect
taxation is optimal.6Empirically, Ray (1986) has shown how optimally
estimated indirect tax rates are very sensitive to the functional forms
assumed for consumer preferences, while Ebrahimi and Heady (1988)
have shown their sensitivity to weak separability. Marginal tax reform
recommendations do not appear to show the same sensitivity to func-
tional form given the assumption of weak separability (Decoster and
Schokkaert, 1990; Madden, 1996). Madden (1995a) also shows that they
do not appear to show great sensitivity to the assumption of weak separa-
bility itself. However, this model uses aggregate time series data for the
estimation of a jointly determined unconditional commodity demand–
labour supply model thus implying relatively restrictive functional forms
(in effect generalizations of the Linear Expenditure System (LES)). Such
analysis using conditional demand functions instead allows us to examine
the sensitivity of marginal tax reform recommendations to weak separa-
bility in the context of more general preferences.7,8
In Section II we briefly discuss the theoretical results underlying the
use of conditional demand functions and in Section III we discuss
possible functional forms. Section IV compares demand responses from
conditional and unconditional systems and examines tests for weak
5An alternative, and, in principle, attractive, approach, which does not prejudge the issue
of whether labour supply is predetermined, would be to estimate matched pairs of rationed
and unrationed demands, generated from the same preferences. However, closed form repre-
sentations of preferences for such an approach with aggregate time-series data involve the
use of restrictive assumptions about either preferences and/or the form of rationing; see
Deaton and Muellbauer (1981), Murphy and Thom (1987b) and Madden (1994). For an
example with household data and more flexible functional forms, but using numerical
methods, see Kooreman and Kapteyn (1986).
6These conditions typically involve the presence of an optimal poll tax (grant) and an
optimal (linear or non-linear) direct tax. For a recent survey, see Stern (1990). However, the
form of separability in question (in particular whether weak or quasi separability is assumed)
can influence these results. See Deaton (1981) and Besley and Jewitt (1988).
7Since we do not specify any labour supply function, we can only analyse the sensitivity of
commodity tax reform recommendations to weak separability.
8We note that if labour supply is truly exogenous, then the optimal tax is a tax solely on
labour or equivalently uniform indirect taxation. However, it is probably fair to say that
optimally calculated indirect tax rates from preferences which have assumed weak separ-
ability have made this assumption with an econometric motivation, in order to utilize more
flexible functional forms. Also, even if we did wish to move towards uniform indirect
taxation, we may wish to do so in a piecemeal fashion, and marginal tax reform can indicate
the preferred direction.
© Blackwell Publishers 1997
238 BULLETIN
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