For any given number of working hours the budget constraint is
2
y s wh q m
4
Ž .
where m is non-labour income. Ž
. At a given point in time, an individual receives n possibly zero job offers,
each of them consisting of a wage w, 0 - w - ` and a number of working hours 4
h g h , . . . ,h . Furthermore, an individual can always choose not to work. The
1 m
alternative which yields the highest level of utility will be chosen. An individual will be observed to be non-working if the utility level of not working exceeds the
utility level of all of the n job offers.
3. Maximum likelihood estimation
The model parameters will be estimated by maximum likelihood. For working Ž
. individuals, let the observed wage–hours pair be denoted by
w , h , l
g
l
4 0,1, . . . , m , and u denotes utility at 0 hours. In Appendix A, it is shown that the
likelihood contribution of a working individual, given the random preferences e, is
X
l w , h N e s lexp yl 1 y P w ,h
N e k w ,h x ,s
p
4
Ž .
Ž .
Ž .
l l
l Õ
l
if u h ,w h
q m u , zero otherwise 5
Ž .
Ž .
l l
Ž . Ž
The function k . denotes the log-normal wage offer density function. P w , h N
l
. e is the probability that the observed wage–hours combination has a utility level
at least as high as that of any other possible combination of wages and hours.
3
Let Ž .
g e denote the wage at which the individual is indifferent between the observed
l
Ž .
combination w , h and working hours of h , at random preferences e, i.e.
l l
Ž Ž .
. Ž
. Ž
. u h , g e h
q m s u h ,w h
q m . Then P w , h N e can be written as
l l
l l
l l
m
P w , h N e [
p P w F g e N e
6
Ž .
Ž . Ž .
Ž .
Ý
l l
l
l s1
This is the probability that any other combination of wages and hours h , l s
l
Ž . 1, . . . , m, has a wage at most as high as its indifference wage g e . If wages
l
follow the log-normal distribution, the probability can be expressed in terms of the Ž .
standard normal distribution function F . : ln g
e y h
X
x
Ž .
l l
P w F g e N e s F
if g e 0
7
Ž . Ž .
Ž .
Ž .
l l
ž
s
Õ
s 0 if g
e F 0
8
Ž . Ž .
l
2
Non-linearity in hours arises if the wage rate w depends on hours.
3
Derivations and further details can be found in Appendix A.
For non-working individuals, the conditional likelihood function is l h
s 0 N e s exp yl 1 y P 0 N e 9
4
Ž .
Ž .
Ž .
Ž .
Ž . with P 0
N e as in Eq. 6 , evaluated at 0 hours, being the probability that the utility of not working is at least as high as the utility level of an arbitrary job offer.
After integrating over random preferences, the likelihood contribution for workers is
1 e
4
l w , h s
l w , h N e
f d e,l
g 1, . . . ,m , 0 - w - `
Ž .
Ž .
H
l l
ž
s s
B e
e
10
Ž .
Ž . f . being the standard normal density function, and B being the region of
random preferences in which utility of not-working is lower than the utility of the observed wage–hours combination:
B [ e N u F u h
,w h q m
11
4
Ž .
Ž .
l l
For non-workers, the likelihood contribution is 1
e l h
s 0 s exp yl 1 y P 0 N e f
d e 12
4
Ž .
Ž .
Ž .
H
ž
s s
e e
4. Empirical results
