Transcript Lecture 3 Labor Supply over Time

Lecture 3
Labor Supply over Time
We
make labor supply decisions continuously
over the life cycle, and our current decisions
influence economic opportunities in the future
and are obviously influenced by the decisions
that we made in the past.
It
is evident that we allocate our time in
different ways at different stages of our life cycle.
1. Labor Supply over the Life Cycle
Because consumption and leisure decisions are
made over the entire working life, workers can
“trade” some leisure time today in return for
additional consumption tomorrow.
We will generally find it optimal to concentrate
on work activities in those years when the
wage is high, and to concentrate on leisure
activities in those years when the wage is low.

Present Value
The present value of a payment of, say, y dollars next
year is given by:
where r is the rate of interest. The quantity PV tells us
how much needs to be invested today in order to
have y dollars next year. By the same token, the
present value of y dollars received t years from now
equals:

Lifetime Utility
It is important to “discount” the utility received in the
future in much the same way that we discount future
earnings in calculating present values. We will,
therefore, write down the two-year life’s worker’s
lifetime utility function as:
Lifetime utility
Where, for simplicity, we assume that the worker uses
the same rate of interest r to discount both future
utilities and future earnings. From the definition of
lifetime utility, it follows that:
Marginal utility of hour of leisure in first year
Marginal utility of hour of leisure in second year

Hours of Work over Time
The optimal allocation of resources between any two activities
requires that the last dollar spent on each commodity generates
the same marginal utility. Applying this condition to our model
of labor supply over the life cycle implies that:
witch is the same as
Wage Rate
Hours of Work
t*
Age
A person will work few hours in those periods of the
life cycle when the wage is low and will work many
hours in those periods when the wage is high.

Labor Force Participation over Time
The labor force participation decision depends on a comparison of
the reservation wage to the market wage. In each year of the life
cycle, therefore, the worker will compare the reservation wage to
the market wage.
→The worker’s participation decision in each year can be summarized as:
Work in the first year if
Work in the second year if
The person is more likely to enter the labor market in periods
when the wage is high. As a result, participation rates are likely
to be low for young workers, high for workers in their primeworking years, and low again for older workers.

The Intertemporal Substitution Hypothesis
A particular worker will time his leisure activities so
that he enters the labor market and works more hours
in those periods of the life cycle when the wage is
high. The notion that people substitute their time over
the life cycle so as to take advantage of changes in the
price of leisure is called the intertemporal
substitution hypothesis.
In the one period model, an increase in the wage
expands the worker’s opportunity set, and hence
creates an income effect for more leisure hours. In the
life-cycle model, the wage increase we are
considering is the wage increase associated with the
process of aging for a given worker, which we call an
evolutionary wage change.
The change in hours of work over time,
therefore, is a response to the change in the price
of leisure, not to an expansion in the lifetime
opportunity set.
2. Labor Supply over the Business Cycle
The added worker effect suggests that “secondary” workers
who are currently out of the labor market (such as young
persons or mothers with small children) are affected by the
recession because the main breadwinner becomes
unemployed or faces a wage cut. As a result, family income
falls and secondary workers get a job to make up the loss.
→The added worker effect thus implies that the labor force
participation rate of secondary workers has a
countercyclical trend (that is, it moves in a direction
opposite to the business cycle); it rises during recessions
and falls during expansions.

The discouraged worker effect argues that
many unemployed workers lose their hopes of
finding a job during a recession. As a result of
the discouraged worker effect, the labor force
participation rate has a pro-cyclical trend; it
falls during recessions and increases during
expansions.
3. Retirement

To simplify our discussion of the retirement decision,
we assume that workers do not participate in the labor
market after they retire. Suppose a male worker has
just turned 60 years old and that his life expectancy is
20 more years. He can choose to retire at age 60 and
collect employer- and government-provided pension
benefits for the remainder of his life.
→The present vale of lifetime income for a
worker who retires at age 60 equals:
where Bt gives the level of pension benefits received
at age t.
As an alternative, the worker can choose to remain in
the work force until he turns 80 years old (so that he
never retires). The present value of his income stream
would then equal the discounted sum of labor
earnings, or:
where Wt gives the worker’s labor earnings at age t.
Consumption($)
PV80 F
P
U1
U0
PV60
0
10
E
20
The worker can also choose to
retire at any age between 60
and 80. He would then receive
labor earnings while employed
and collect his pension from
the time of retirement until age
80. By calculating the present
value of the lifetime income
associated with each retirement
age, we can derive the
worker’s “budget line” FE.
Years of
Retirement
Determinants of the Retirement Age
The worker’s retirement age depends on his wage and
pension benefits.
Consumption($)
Consumption($)
U1
G
F
R
U0
F
R
P
U1
P
H
U0
0
5
10
E
20
a. Wage increase
Years of 0
Retirement
10
15
20
Years of
Retirement
b. Increase in Pension Benefits
A wage increase generates both income and
substitution effects. The high-wage worker has a larger
opportunity set and would like to consume more leisure,
so that he will want to retire earlier. At the same time,
the wage increase raises the price of retirement, so that
the worker will want to delay retirement. As drawn, the
substitution effect dominates and the high-wage worker
cuts the duration of his retirement from 10 to 5 years.
A more generous pension plan rotates the budget line
around point F, from FE to FH. The increase in pension
benefits generates both income and substitution effects.
Both of these effects, however, work in the same
direction.
4. Fertility

The economic analysis of fertility dates back to the
famous (and some would say infamous) Essay on the
Principle of Population written by the Reverend
Thomas Malthus in 1798. Malthus’ pessimistic views
on the long-run prospects for the human species
earned economics its hard-to-shake nickname of
being the “dismal science.”

The Malthusian model of fertility clearly failed to
predict what actually happened to fertility behavior in
modern economies. As per-capita income rose,
fertility rates did not rise; they declined!

How Many Children Should the Household Have?
Suppose that the household cares both about the number of
children that it has and about the goods that it consumes. This
household’s utility function can then be written as:
where N is the number of children in the household and X
denotes all other goods.
The household’s consumption activities, however, are constrained
by its income, I. The budget constraint is given by:
where pn is the price of having an additional child, and px is the
price of other goods.
Goods
1/px
P
IC
3
1/pn
Number of Children
The household maximizes utility by choosing
point P, where the indifference curve is tangent to
the budget line. As drawn, the household wishes to
have three children.
How Do Income and Price Influence the
Household’s Fertility?
Goods
Goods
1/px
R
D
Q
P
U0
0
3
U1
U1
4
a. Increase in Income
D
0
Number of
Children
1 2 3
U0
Number of
Children
b. Increase in Price of Children
Assuming that children are a normal good, the
increase in household income increases the demand for
children, from three to four. This income effect yields
precisely the positive correlation between incomes and
fertility that Malthus had in mind.
Income effects, however, are only half the story.
After all, the desired number of children also depends
on their price. An increase in the direct costs of having
children rotates the budget line inward. Initially, the
household is at point P and desires three children. After
the price of children goes up, the household moves to
point R and the household only wishes to have one
child.
Do Households Really Look at Economic
Variables When Determining Their Fertility?
The negative correlation between the price of
children and the demand for children helps us
understand why Malthus failed to predict what
actually happened to fertility as countries became
wealthier.
Our simple model of the fertility decision has been
extended in a number of important ways. Many of
these extensions are based on the sensible hypothesis
that households get utility not only from the number
of children they have, but also from the “quality” of
the children.
→The analysis of quantity and quality decisions in
fertility has led to the discovery of an empirically
important quality-quantity interaction.
International Differences in Female Labor Force Participation Rate (women aged 15-64)
Country
Australia
Canada
France
Germany
Greece
Ireland
Italy
Japan
Korea, South
Mexico
New Zealand
Portugal
Spain
Sweden
Taiwan
Turkey
United Kingdom
United States
1980
52.7%
57.8
54.4
52.8
33.0
36.3
39.6
54.8
---33.7
44.6
54.3
32.2
74.1
39.3
---58.3
59.7
1990
62.9%
67.8
57.6
57.4
43.6
38.9
45.9
60.4
51.3
23.6
62.9
62.9
41.2
80.5
44.5
36.7
65.5
68.8
1999
64.4%
69.6
60.8
62.8
49.0
54.9
46.0
63.8
53.1
42.1
67.7
66.8
48.9
74.6
46.0
34.0
67.5
71.7