The Basic Static Labor Supply Model
Consider a single individual with a utility function
U
(
y
,
ℓ
) where
y
is income and
ℓ
is leisure.
Both
y
and
ℓ
are “goods”, i.e. the consumer prefers more of each:
U
1
> 0;
U
2
> 0.
Suppose this person has nonlabor income of
G
, and can work as many hours,
h
, as she wishes at a wage of
w
per hour.
Total time available for the only two possible activities, work (
h
) and leisure (
ℓ
) is T.
If she allocates her time between work and leisure to maximize her utility, what can we say about her
decisions, and about how these decisions will respond to changes in the exogenous parameters,
w
and
G
?
Using the constraints
h +
ℓ
= T and
y
=
wh
+
G
, this problem can be written as a singlevariable
maximization problem without constraints:
Choose
h
to maximize:
U
(
wh
+
G
,
T
–
h
).
(1)
Firstorder conditions for a maximum are:
0
)
,
(
,
2
1
h
T
G
wh
U
h
T
G
wh
wU
(2)
or simply:
)
(
)
(
1
2
income
MU
leisure
MU
U
U
w
(3)
Diagrammatically, (3) represents the familiar tangency condition between the slope of the budget constraint
(
w
) and the slope of an indifference curve (U
2
/U
1
), shown for individual #1 below:
y
G
G
“Tangency” Solution
“Corner” Solution
y
l
l
# hours worked
# hours
at home
chooses not to
work at all
Individual #1
Individual #2
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Of course, it is also possible (e.g. when
G
is high and
w
low) for the solution to be at a “corner” where it is
optimal not to work at all, as shown for individual #2.
We could show this formally using KuhnTucker
conditions in the maximization but the intuition is perfectly clear from the diagram.
To understand the comparativestatic predictions of the model, assume an interior solution (as for
individual #1), and totally differentiate equation (2), yielding:
0
)
(
)
(
)
(
21
11
22
21
12
11
21
11
1
dG
U
wU
dh
U
w
U
U
w
U
w
dw
h
U
h
wU
U
, or simply:
0
CdG
Bdh
Adw
(4)
To help interpret this, note that, by the secondorder condition for a maximum (
U
as defined in equation 1
must be concave in
h
),
B
< 0.
(btw the SOC also require
U
11
< 0 and
U
22
<0).
Note that
w
and
G
are
exogenous and
h
endogenous, so we can define ceterisparibus thought experiments where we introduce
small changes in
w
and
G
one at a time, and consider their consequences for the utilitymaximizing level of
h
:
Effects of nonlabor income on labor supply:
pos
U
wU
B
C
dw
holding
dG
dh
21
11
)
0
(
,
(5)
which can be greater, less than, or equal to zero.
It will be negative for sure if
U
21
>0, i.e. if the utility
function is such that additional income always raises the marginal utility of leisure.
It will also be negative
as long as
U
21
is not “too” negative, i.e. as long as additional income does not reduce the marginal utility of
leisure “too much”.
From here on we will
define
leisure as a
normal good
if
the partial derivative defined
by (5) is negative.
(This is the standard definition:
a good is normal if its optimal consumption increases
when the budget constraint shifts outward without changing slope).
Of course, if (5) is positive, we will
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 Fall '09
 Kuhn
 Utility, Labor Supply

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