Empirical life cycle models of labour supply and - Statistisk sentralbyrå

Empirical Life Cycle Models Social and Economic Studies 91
Section 3.5.1 discusses the estimation procedure in more details.
3.4 A Particular Empirical Specification
In what follows we assume that the monotonic transformation function G is
given by
ufk — 1
(127) G=
and that (within-period) preferences are of the Box-Cox type
(128)
(Z ik-{- zor - 1 fik
Lw • - 1
Uk=ik + Rk mt k
where subscript i denotes household and 0, a, 7 and w are unknown coefficients.
If e = 1, the coefficients 7 and w determine the intertemporal substitution
elasticities 1/(7 - 1) and 1/(w - 1), which measure the percentage change in the
consumption of leisure in any two periods in response to a percentage change in
the relative wage rate for those periods, cf. Heckman and MaCurdy (1980). The
interpretation of a is less straightforward, since /3 0, implying that Z is not
an ordinary physical good.
If the within-period utilities are to be strictly concave, a, 7 and w must all be
less than one. Since the cash flow related to the purchases of durables and
non-durables, as well as the parameter 8p, are allowed to be negative, Z can be
negative. If zo is zero and Z is negative, the (Box-Cox) utility of Z is undefined;
in order to avoid this problem, we introduce the parameter zo. Notice also that
since sp is unknown, the smallest value of Yt -F 6pDt_ 1 can not be observed, and
we (must) treat zo as a parameter to be estimated.
It is also assumed that the person- and age-specific modifiers of tastes, rik and
can be related to a vector of exogenous and observable consumer
characteristics, Xik and Bik, and unmeasured characteristics, Eik and
according to Fik = exp(XikOx -F eik) and Qik = exp(BikOb riik). The error
terms are needed since the econometrician cannot observe all components that
influence preferences.
The first order conditions for consumption and female and male leisure become
(129) U21-1- (Zit -I- 4) 1 = AitPt,
(130) uPt—' rit qi-t 1 = Aitmfitf it
and
(131) QitLrn-it1 = Aitimmit amit.
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