Labor Supply over Time

1. Labor Supply over Time 1 Hewi-Lin Chuang, Ph.D. 2010/05/6

2. 職場進化論：35而立作者：徐仁全出處：遠見雜誌 2010年4月號 ／ 第286期(HTTP://0RZ.TW/PSUVD) • 你今年35歲，單身？還是已婚？有沒有小孩？你今年35歲，失業中？還是一名基層職員？或已是小主管？你今年35歲，自己住？還是仍窩在爸媽的家裡？你今年35歲，曾經想過未來人生嗎？想像過65歲退休以後的光景嗎？ • 自古以來，孔子的「30而立」說，影響了世世代代的華人思想，認定30歲是人生成家立業的重要關卡。然而對照現在的台灣社會，對許多正處於30歲上下的年輕人而言，要在30歲達到成家立業的「而立」標準，似乎已是不可能的任務。

3. 職場進化論：35而立作者：徐仁全出處：遠見雜誌 2010年4月號 ／ 第286期 • 以立業來說，看看上達6萬人的流浪教師，或是去年突破12萬的公職報考人數，其中絕大多數都是30歲年紀的人。甚至30好幾、接近40歲，在職場上打滾了十餘年再轉公職者也大有人在。 • 成家更不易。就內政部的資料顯示，2008年男性成家的年齡為32.9歲，女性為29.5歲，皆較十年前平均結婚年齡延後了一到兩歲以上，主要因素包括經濟、個人自主、風氣轉變等，都是造成晚婚的原因，甚至生育下一代的意願也大大降低了。 • 成家難、立業也不易，讓此刻不少台灣的「30好幾」，普遍存在不自在、不知所措的窘態。因為30歲已難立，「30而立應該要改為35而立才對，」不少本次受訪者都這樣指出，應該多給年輕人幾年時間準備。35歲因此成為台灣年輕人的新人生關卡。

4. OUTLINE • Labor Supply over the Life Cycle • Retirement • Fertility • Labor Supply over the Business Cycle

5. 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.

10. 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.

11. 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:

12. 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

13. 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

14. 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: which is the same as

15. Wage Rate Hours of Work Age 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.

16. Hours of Work over the Life Cycle for Two Workers with Different Wage Paths Hours of Wage Rate Work Joe (if substitution effect dominates) Joe Jack Jack Joe (if income effect dominates) t * Age t Age * Joe’s wage exceeds Jack’s at every age. Although both Joe and Jack work more hours when the wage is high, Joe works more hours than Jack if the substitution effect dominates. If the income effect dominates, Joe works fewer hours than Jack.

17. 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 prime-working years, and low again for older workers.

18. 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.

19. 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.

20. 2. 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.

21. →The present vale of lifetime income for a worker who retires at age 60 equals: where Btgives the level of pension benefits received at age t.

22. 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.

23. 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. Consumption($) F PV80 P U1 U0 PV60 E 0 10 20 Years of Retirement

24. Determinants of the Retirement Age The worker’s retirement age depends on his wage and pension benefits. Consumption($) Consumption($) U1 G U0 F R F R H U1 P P U0 E 0 5 10 20 Years of Retirement 0 10 15 20 Years of Retirement a. Wage increase b. Increase in Pension Benefits

25. 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.

26. 3. 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!

27. 主要國家育齡婦女總生育率

30. 女性勞動參與情形

32. Average 1.7 100% 90% 1.6 80% 70% 1.5 60% 50% 1.4 40% 30% 1.3 20% Average Total Number of Children Born to Women 10% Proportion of Women with Live Birth 1.2 0% 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Figure 3. Trends of Fertility for Women in Taiwan - Women in the Labor Force

33. Average 2.1 100% 90% 2.0 80% 70% 1.9 60% 1.8 50% 40% 1.7 30% 20% Average Total Number of Children Born to Women 1.6 Proportion of Women with Live Birth 10% 1.5 0% 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Figure 4. Trends of Fertility for Women in Taiwan - Women Out of the Labor Force

34. Average 1.8 100% 90% 1.7 80% 70% 1.6 60% 50% 1.5 40% 30% 1.4 20% Average Total Number of Children Born to Women 10% Proportion of Women with Live Birth 1.3 0% 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Figure 5. Trends of Fertility for Women in Taiwan - EmployedWomen

35. Average 0.8 100% Average Total Number of Children Born to Women 90% 0.7 Proportion of Women with Live Birth 80% 0.6 70% 0.5 60% 0.4 50% 40% 0.3 30% 0.2 20% 0.1 10% 0 0% 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Figure 6. Trends of Fertility for Women in Taiwan - Unemployed Women

36. 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.

37. Goods 1/px P IC Number of Children 3 1/pn 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.

38. How Do Income and Price Influence the Household’s Fertility? Goods Goods 1/px R D Q P U1 U1 U0 U0 D 0 0 3 4 1 2 3 Number of Children a. Increase in Income b. Increase in Price of Children Number of Children

39. 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.

40. 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.

41. 4. 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.

42. 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.