Dynamic, or Life-Cycle Labor Supply
1. The setup
Dymo lives for T periods.
Let his utility in period
t
be given by the function:
)
,
(
t
t
L
C
U
(
1
)
where
C
is consumption and
L
is leisure. We’ll assume
U
has the usual properties:
it is
increasing in both
C
and
L
, and strictly concave.
Strict concavity implies negative (own)
second derivatives, but (as noted in the static case) does not restrict the sign of the cross-
partial derivative,
U
CL
.
Plausible stories can be told for both positive and negative
U
CL
’s
and both are consistent with well-behaved solutions to the problem.
1
If utility is
intertemporally separable and
ρ
is the intertemporal (subjective) discount factor, total
lifetime utility is:
T
t
t
t
t
L
C
U
W
1
)
1
(
)
,
(
(
2
)
Note that the
function
U
is not indexed by
t
; in this baseline case we are therefore not
allowing tastes for consumption or leisure to vary systematically with age (this is easy to
modify).
Dymo can work as many hours as he wants in each period of his life at fixed rate
of pay per hour. But the wage,
w
, Dymo can get isn’t the same in each period:
instead it
is indexed by
t
.
Suppose that Dymo knows, even at the start of his working life, what
w
will be in each period (this assumption is also fairly easily relaxed).
If Dymo also faces a
known stream of nonlabor income,
G
t
,
t
= 1, …
T
, we can write his income in period
t
as:
period
t
income =
+
G
t
(3)
)
1
(
t
t
L
w
(Note that we are normalizing the total amount of time available in each period to equal
one.)
In the above definition of income,
G
t
does
not
include interest income on money
that was saved earlier in life (the amounts of interest income Dymo actually ends up
earning in each period of his life will be
endogenous
outcomes of Dymo’s utility-
maximizing choice of a lifetime income and consumption plan).
G
t
as defined in (3)
includes only “exogenous” items of income such as inheritances, lottery winnings,
demogrants, etc.
(It is quite OK to think of all or most of the
G
’s as zero, but keeping
track of them helps in the interpretation of some results below).
1
If you can enjoy your spending more effectively when you have more leisure (e.g. a dollar yields more
marginal utils when spent on vacation in a beautiful place) then
U
CL
>0.
If your need to spend at the margin
is greater when you are working (e.g. you need to have better clothes, pay for commuting, eat out more,
and don’t have time to do your own home repairs), then
U
CL
<0.