Labor Supply over TimeLabor Supply over Time

We make labor supply decisions continuously We make labor supply decisions continuously over the life cycle, and our current decisions over the life cycle, and our current decisions influence economic opportunities in the future influence economic opportunities in the future and are obviously influenced by the decisions and are obviously influenced by the decisions that we made in the past.that we made in the past.

It is evident that we allocate our time in It is evident that we allocate our time in different ways at different stages of our life different ways at different stages of our life cycle.cycle.

age range 總 男 女15-19 9.8 9.8 9.8120-24 52.88 48.86 56.3825-29 84.68 88.68 80.7930-34 84.37 94.14 74.835-39 83.17 95.32 71.1840-44 81.42 94.03 68.7945-49 75.57 90.8 60.3950-54 65.57 83.25 48.0955-59 49.64 68.15 31.5860-64 32.04 46.32 18.48>65 8.13 11.95 4.45

年齡組別勞動力參與率96 Ave., 2007年平均

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

勞動參與率(%)

15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 >65

年齡組別

2007 年 年齡組別平均勞動力參與率

總男女

1. Labor Supply over the Life Cycle1. Labor Supply over the Life Cycle

Because consumption and leisure decisions are Because consumption and leisure decisions are made over the entire working life, workers can made over the entire working life, workers can “trade” some leisure time today in return for “trade” some leisure time today in return for additional consumption tomorrow.additional consumption tomorrow.

We will generally find it optimal to We will generally find it optimal to concentrate on work activities in those years concentrate on work activities in those years when the wage is high, and to concentrate on when the wage is high, and to concentrate on leisure activities in those years when the wage leisure activities in those years when the wage is low.is low.

Present ValuePresent Value

The present value of a payment of, say, The present value of a payment of, say, yy dollars next dollars next year is given by:year is given by:

where where rr is the rate of interest. The quantity PV tells us is the rate of interest. The quantity PV tells us how much needs to be invested today in order to how much needs to be invested today in order to have have yy dollars next year. By the same token, the dollars next year. By the same token, the present value of present value of yy dollars received dollars received tt years from now years from now equals:equals:

Lifetime UtilityLifetime Utility

It is important to “discount” the utility received in the It is important to “discount” the utility received in the future in much the same way that we discount future future in much the same way that we discount future earnings in calculating present values. We will, earnings in calculating present values. We will, therefore, write down the two-year life’s worker’s therefore, write down the two-year life’s worker’s lifetime utility function as:lifetime utility function as:

Lifetime utilityLifetime utility

Where, for simplicity, we assume that the worker uses Where, for simplicity, we assume that the worker uses the same rate of interest the same rate of interest rr to discount both future to discount both future utilities and future earnings. From the definition of utilities and future earnings. From the definition of lifetime utility, it follows that:lifetime utility, it follows that:

Marginal utility of hour of leisure in first yearMarginal utility of hour of leisure in first year

Marginal utility of hour of leisure in second yearMarginal utility of hour of leisure in second year

Hours of Work over TimeHours of Work over Time

The optimal allocation of resources between any two activities The optimal allocation of resources between any two activities requires that the last dollar spent on each commodity generates requires that the last dollar spent on each commodity generates the same marginal utility. Applying this condition to our the same marginal utility. Applying this condition to our model of labor supply over the life cycle implies that:model of labor supply over the life cycle implies that:

witch is the same aswitch is the same as

Age

Wage Rate

Hours of Work

t*

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 TimeLabor Force Participation over Time The labor force participation decision depends on a comparison The labor force participation decision depends on a comparison

of the reservation wage to the market wage. In each year of the of the reservation wage to the market wage. In each year of the life cycle, therefore, the worker will compare the reservation life cycle, therefore, the worker will compare the reservation wage to the market wage.wage to the market wage.

→→The worker’s participation decision in each year can be summarized as:The worker’s participation decision in each year can be summarized as:

Work in the first year if Work in the first year if

Work in the second year ifWork in the second year if

The person is more likely to enter the labor market in periods The person is more likely to enter the labor market in periods when the wage is high. As a result, participation rates are likely when the wage is high. As a result, participation rates are likely to be low for young workers, high for workers in their prime-to be low for young workers, high for workers in their prime-working years, and low again for older workers.working years, and low again for older workers.

The Intertemporal Substitution HypothesisThe Intertemporal Substitution Hypothesis

A particular worker will time his leisure activities so tA particular worker will time his leisure activities so that he enters the labor market and works more hours ihat he enters the labor market and works more hours in those periods of the life cycle when the wage is hign those periods of the life cycle when the wage is high. The notion that people substitute their time over thh. The notion that people substitute their time over the life cycle so as to take advantage of changes in the pe life cycle so as to take advantage of changes in the price of leisure is called the rice of leisure is called the intertemporal substitutiointertemporal substitution hypothesis.n hypothesis.

In the one period model, an increase in the wage In the one period model, an increase in the wage expands the worker’s opportunity set, and hence expands the worker’s opportunity set, and hence creates an income effect for more leisure hours. In the creates an income effect for more leisure hours. In the life-cycle model, the wage increase we are life-cycle model, the wage increase we are considering is the wage increase associated with the considering is the wage increase associated with the process of aging for a given worker, which we call an process of aging for a given worker, which we call an evolutionary wage change.evolutionary wage change.

The change in hours of work over time, The change in hours of work over time, therefore, is a response to the change in the price therefore, is a response to the change in the price of leisure, not to an expansion in the lifetime of leisure, not to an expansion in the lifetime opportunity set.opportunity set.

2.2. Labor Supply over the Business CycleLabor Supply over the Business Cycle

The The added worker effectadded worker effect suggests that “secondary” workers suggests that “secondary” workers who are currently out of the labor market (such as young who are currently out of the labor market (such as young persons or mothers with small children) are affected by the persons or mothers with small children) are affected by the recession because the main breadwinner becomes recession because the main breadwinner becomes unemployed or faces a wage cut. As a result, family income unemployed or faces a wage cut. As a result, family income falls and secondary workers get a job to make up the loss. falls and secondary workers get a job to make up the loss.

→→The added worker effect thus implies that the labor force The added worker effect thus implies that the labor force participation rate of secondary workers has a participation rate of secondary workers has a countercyclical trend (that is, it moves in a direction countercyclical trend (that is, it moves in a direction opposite to the business cycle); it rises during recessions opposite to the business cycle); it rises during recessions and falls during expansions.and falls during expansions.

The The discouraged worker effectdiscouraged worker effect argues that argues that many unemployed workers lose their hopes of many unemployed workers lose their hopes of finding a job during a recession. As a result of finding a job during a recession. As a result of the discouraged worker effect, the labor force the discouraged worker effect, the labor force participation rate has a pro-cyclical trend; it participation rate has a pro-cyclical trend; it falls during recessions and increases during falls during recessions and increases during expansions.expansions.

3. Retirement3. Retirement

To simplify our discussion of the retirement decision, To simplify our discussion of the retirement decision, we assume that workers do not participate in the labor we assume that workers do not participate in the labor market after they retire. Suppose a male worker has market after they retire. Suppose a male worker has just turned 60 years old and that his life expectancy is just turned 60 years old and that his life expectancy is 20 more years. He can choose to retire at age 60 and 20 more years. He can choose to retire at age 60 and collect employer- and government-provided pension collect employer- and government-provided pension benefits for the remainder of his life.benefits for the remainder of his life.

→→The present vale of lifetime income for a worker wThe present vale of lifetime income for a worker who retires at age 60 equals:ho retires at age 60 equals:

where where BBtt gives the level of pension benefits receivedgives the level of pension benefits received

at age at age tt..

As an alternative, the worker can choose to remain in As an alternative, the worker can choose to remain in the work force until he turns 80 years old (so that he the work force until he turns 80 years old (so that he never retires). The present value of his income stream never retires). The present value of his income stream would then equal the discounted sum of labor would then equal the discounted sum of labor earnings, or:earnings, or:

where where WWtt gives the worker’s labor earnings at age gives the worker’s labor earnings at age tt..

Consumption($)

Years of Retirement

0

U1

U0

PV80

PV60

P

10 20E

F

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.

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.

Determinants of the Retirement AgeDeterminants of the Retirement Age

The worker’s retirement age depends on his wage and The worker’s retirement age depends on his wage and pension benefits.pension benefits.

Consumption($)

Years of Retirement

0

a. Wage increase

E

G

FU1

U0

R

P

5 10 20

Consumption($)

Years of Retirement

0 10 15 20

F

b. Increase in Pension Benefits

U0

U1

PR

H

A wage increase generates both income and A wage increase generates both income and substitution effects. The high-wage worker has a larger substitution effects. The high-wage worker has a larger opportunity set and would like to consume more leisure, opportunity set and would like to consume more leisure, so that he will want to retire earlier. At the same time, so that he will want to retire earlier. At the same time, the wage increase raises the price of retirement, so that the wage increase raises the price of retirement, so that the worker will want to delay retirement. As drawn, the the worker will want to delay retirement. As drawn, the substitution effect dominates and the high-wage worker substitution effect dominates and the high-wage worker cuts the duration of his retirement from 10 to 5 years.cuts the duration of his retirement from 10 to 5 years.

A more generous pension plan rotates the budget line A more generous pension plan rotates the budget line around point F, from FE to FH. The increase in pension around point F, from FE to FH. The increase in pension benefits generates both income and substitution effects. benefits generates both income and substitution effects. Both of these effects, however, work in the same Both of these effects, however, work in the same direction.direction.

4. Fertility4. Fertility The economic analysis of fertility dates back to the The economic analysis of fertility dates back to the

famous (and some would say infamous) famous (and some would say infamous) Essay on the Essay on the Principle of PopulationPrinciple of Population written by the Reverend written by the Reverend Thomas Malthus in 1798. Malthus’ pessimistic views Thomas Malthus in 1798. Malthus’ pessimistic views on the long-run prospects for the human species on the long-run prospects for the human species earned economics its hard-to-shake nickname of earned economics its hard-to-shake nickname of being the “dismal science.”being the “dismal science.”

The Malthusian model of fertility clearly failed to The Malthusian model of fertility clearly failed to predict what actually happened to fertility behavior in predict what actually happened to fertility behavior in modern economies. As per-capita income rose, modern economies. As per-capita income rose, fertility rates did not rise; they declined!fertility rates did not rise; they declined!

Figure 1. Total Fertility Rate, 1961~2005

4.82

3.71

5.59

2.84

3.09

2.70

2.72 2.52

2.672.46

2.32

2.171.88

1.68

1.701.86

1.68 1.72 1.76

1.73 1.76

1.78

1.75

1.77

1.47

1.56

1.68

1.40

1.34 1.18

1.101.24

2.061.81

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

1961 1971 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

year

Total

Fer

tility

Rate

%

Figure 2. Female Labor Participation Rates, 1961~2005

35.45

35.8139.19

44.03 45.82

48.12

0

10

20

30

40

50

60

1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003year

Fem

ale

Par

tici

pati

on R

ate

(%)

Figure 3. Trends of Fertility for Women in Taiwan -Women in the Labor Force

1.2

1.3

1.4

1.5

1.6

1.7

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year

Average

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Average Total Number of Children Born to Women

Proportion of Women with Live Birth

Figure 4. Trends of Fertility for Women in Taiwan -

Women Out of the Labor Force

1.5

1.6

1.7

1.8

1.9

2.0

2.1

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year

Average

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Average Total Number of Children Born to Women

Proportion of Women with Live Birth

Figure 5. Trends of Fertility for Women in Taiwan -Employed Women

1.3

1.4

1.5

1.6

1.7

1.8

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year

Average

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Average Total Number of Children Born to Women

Proportion of Women with Live Birth

Figure 6. Trends of Fertility for Women in Taiwan -

Unemployed Women

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year

Average

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Average Total Number of Children Born to Women

Proportion of Women with Live Birth

How Many Children Should the Household Have?How Many Children Should the Household Have?

Suppose that the household cares both about the number of childrSuppose that the household cares both about the number of children that it has and about the goods that it consumes. This houseen that it has and about the goods that it consumes. This household’s utility function can then be written as:hold’s utility function can then be written as:

where where NN is the number of children in the household and is the number of children in the household and XX denot denotes all other goods. es all other goods.

The household’s consumption activities, however, are constrained The household’s consumption activities, however, are constrained by its income,by its income, I I. The budget constraint is given by: . The budget constraint is given by:

where where ppnn is the price of having an additional child, and is the price of having an additional child, and ppxx is the is the

price of other goods.price of other goods.

Goods

Number of Children

P

IC

1/px

1/pn3

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 How Do Income and Price Influence the Household’s Fertility?Household’s Fertility?

Goods

Number of Children

0

U0

U1P

R

3 4

Goods

Number of Children

0

U0

U1

1/px

1 2 3

D

D

Q

a. Increase in Income b. Increase in Price of Children

Assuming that children are a normal good, the Assuming that children are a normal good, the increase in household income increases the demand for increase in household income increases the demand for children, from three to four. This income effect yields children, from three to four. This income effect yields precisely the positive correlation between incomes and precisely the positive correlation between incomes and fertility that Malthus had in mind.fertility that Malthus had in mind.

Income effects, however, are only half the story. Income effects, however, are only half the story. After all, the desired number of children also depends After all, the desired number of children also depends on their price. An increase in the direct costs of having on their price. An increase in the direct costs of having children rotates the budget line inward. Initially, the children rotates the budget line inward. Initially, the household is at point P and desires three children. After household is at point P and desires three children. After the price of children goes up, the household moves to the price of children goes up, the household moves to point R and the household only wishes to have one point R and the household only wishes to have one child.child.

Do Households Really Look at Economic Do Households Really Look at Economic Variables When Determining Their Fertility?Variables When Determining Their Fertility? The negative correlation between the price of The negative correlation between the price of

children and the demand for children helps us children and the demand for children helps us understand why Malthus failed to predict what understand why Malthus failed to predict what actually happened to fertility as countries became actually happened to fertility as countries became wealthier.wealthier.

Our simple model of the fertility decision has been Our simple model of the fertility decision has been extended in a number of important ways. Many of extended in a number of important ways. Many of these extensions are based on the sensible hypothesis these extensions are based on the sensible hypothesis that households get utility not only from the number that households get utility not only from the number of children they have, but also from the “quality” of of children they have, but also from the “quality” of the children.the children.

→ →The analysis of quantity and quality decisions in The analysis of quantity and quality decisions in fertility has led to the discovery of an empirically fertility has led to the discovery of an empirically important important quality-quantity interaction.quality-quantity interaction.

Country Country

Australia Australia

Canada Canada

FranceFrance

GermanyGermany

GreeceGreece

IrelandIreland

ItalyItaly

JapanJapan

Korea, SouthKorea, South

MexicoMexico

New ZealandNew Zealand

PortugalPortugal

SpainSpain

SwedenSweden

TaiwanTaiwan

TurkeyTurkey

United KingdomUnited Kingdom

United StatesUnited States

19801980

52.7%52.7%

57.857.8

54.454.4

52.852.8

33.033.0

36.336.3

39.639.6

54.854.8

--------

33.733.7

44.644.6

54.354.3

32.232.2

74.174.1

39.339.3

--------

58.358.3

59.759.7

19901990

62.9%62.9%

67.867.8

57.657.6

57.457.4

43.643.6

38.938.9

45.945.9

60.460.4

51.351.3

23.623.6

62.962.9

62.962.9

41.241.2

80.580.5

44.544.5

36.736.7

65.565.5

68.868.8

19991999

64.4%64.4%

69.669.6

60.860.8

62.862.8

49.049.0

54.954.9

46.046.0

63.863.8

53.153.1

42.142.1

67.767.7

66.866.8

48.948.9

74.674.6

46.046.0

34.034.0

67.567.5

71.771.7

International Differences in Female Labor Force Participation Rate (women aged 15-64)