Issuu

Type:

Solution Manual

Resource:

Labor Economics

Edition:

8th Edition

Author(s):

George Borjas

CHAPTER 2 2-1. The table below reports the unemployment rate, labor force participation rate, and (working-age) population for the United States in January 2008, 2011, and 2016. Using the data, answer the following questions. a. What was the size of the labor force at the start of each year? b. How many people were officially unemployed at the start of each year? c. What about these numbers may cause some concern even though the unemployment rate to start 2016 was a notch below the unemployment rate in 2008 as the economy was entering the Great Recession? 2008 5.0% 66.2% 234m

Unemployment Rate Labor Force Participation Rate Working-age Population

2011 9.1% 64.2% 238m

2016 4.9% 62.7% 251m

Parts (a) and (b) require some mathematical manipulations. As the labor force participation rate equals the LF / P, the size of the labor force (LF) each year is simply the working age population (P) times the labor force participation rate. Once the size of the labor force (LF) is know, the number of unemployed individuals is calculated by multiplying the labor force by the unemployment rate. For 2008, for example, LF = 234m × 0.662 = 154.9 million, and therefore U = 154.9m × 0.05 = 7.745 million. The numbers for the other years are found similarly. The answers are contained in the following table: 2008 154.9m 7.745m

Labor Force (LF) Unemployed Population (U)

2011 152.8m 13.9m

2016 157.4m 7.713m

As for part (c), these numbers are concerning despite the unemployment rate returning to something less than 5%. The concern is that the labor force participation rate has fallen drastically. According to the above numbers, the labor force consisted of 157.4 million individuals in 2016. Of these, only 4.9% were unemployed. Put differently, 95.9% × 157.4 = 150.9 million individuals were employed. Had the labor force participation rate remained at 66.2% in 2016, the labor force would have consisted of 251m × 0.662 = 166.2 million individuals. Of these, only 150.9 million were employed. Taking into account these individuals who have left the labor force (i.e., the hidden unemployed), the unemployment rate in 2016 could have been as high as (166.2 – 150.9) / 166.2 = 9.2%.

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2-2. 2-2. Charlie and Larry both face the same budget line for consumption and leisure. At every possible consumption-leisure bundle on the budget line, Charlie always requires marginally more leisure than does Larry in order to be equally happy when asked to forego a dollar of consumption. Using a standard budget line, graph several indifference curves and the optimal consumption-leisure bundle for both people. Which person optimally chooses more consumption? Which feature of indifference curves guarantees this result? Because Charlie requires receiving more leisure than Larry when giving up consumption, Charlie’s indifference curves are flatter relative to Larry’s. This feature – shallower or flatter indifference curves – results in that person (Charlie) optimally choosing more of the Y-axis good. Similarly, Larry’s indifference curves are steeper relative to Charlie’s, because Larry does not need to receive as much leisure when giving up consumption. This feature – steeper indifference curves – results in that person (Larry) optimally choosing more of the X-axis good. These ideas are all incorporated in the graph below where the solid lines represent an assortment of indifference curves for Charlie while the dashed lines represent an assortment of indifference curves for Larry.

Consumption

Charlie’s ICs

Larry’s ICs

C*Charlie

C*Larry

L*Charlie

L*Larry

Leisure

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2-3. Tom earns $15 per hour for up to 40 hours of work each week and $30 per hour for every hour in excess of 40. Tom also faces a 20 percent tax rate, pays $4 per hour in child care expenses for each hour he works, and receives $80 in child support payments each week. There are 110 (non-sleeping) hours in the week. Graph Tom’s weekly budget line. • • •

If Tom does not work, he leisures for 110 hours and consumes $80. For all hours Tom works up to his first 40, his after-tax and after-child care wage equals (80 percent of $15) – $4 = $8 per hour. Thus, if he works for 40 hours, he will be able to leisure for 70 hours and consume $80 + $8(40) = $400. For all hours Tom works over 40, his after-tax and after-child care wage equals (80 percent of $30) – $4 = $20. Thus, if he works for 110 hours (70 hours at the overtime wage), he will not leisure at all, but he will consume $80 + $8(40) + $20(70) = $1,800.

Tom’s weekly budget line is pictured below. Dollars of Consumption $1,400

$400 $80 70

110

Hours of Leisure

2-4. Cindy gains utility from consumption C and leisure L. The most leisure she can consume in any given week is 110 hours. Her utility function is U(C, L) = C  L. This functional form implies that Cindy’s marginal rate of substitution is C / L. Cindy receives $660 each week from her great-grandmother–regardless of how much Cindy works. What is Cindy’s reservation wage? The reservation wage is the MRS when not working at all. Thus, wRES = MRS at maximum leisure equals C/L = $660/110 = $6.00.

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2-5. Currently a firm pays 10% of each employee’s salary into a retirement account, regardless of whether the employee also contributes to the account. The firm is considering changing this system to a 10% match meaning that the firm will match the employee’s contribution into the account up to 10% of each employee’s salary. Some people at the firm think this change will lead employees to save more and therefore be more able to afford to retire at a younger age, while others believe this change will lead employees to have less retirement savings and therefore be less able to afford to retire. Explain why either point of view could be correct. Either point of view may be correct. The first assumes that the new matching system will encourage workers to save at least 10% of their salary into the retirement account, because it is matched. In essence, each dollar of personal savings receives an automatic and immediate 100% return. Alternatively, if the workers feel that they simply cannot save for retirement, then the change to a matching system may result in fewer dollars saved for retirement as the workers save very little (say 2%) and the firm then only matches the 2%. With this example, a worker’s retirement account is receiving 4% of his or her salary each year compared to the 10% it received before the change. Clearly, the matching system provides fewer funds for retirement if the workers are not “savers” during their worklife. 2-6. Shelly’s preferences for consumption and leisure can be expressed as U(C, L) = (C – 100)  (L – 40). This utility function implies that Shelly’s marginal utility of leisure is C – 100 and her marginal utility of consumption is L – 40. There are 110 hours in the week available to split between work and leisure. Shelly earns $10 per hour after taxes. She also receives $320 worth of assistance benefits each week regardless of how much she works. (a) Graph Shelly’s budget line. If Shelly does not work, she leisures for 110 hours and consumes $320. If she does not leisure at all, she consumes $320 + $10(110) = $1,420. Shelly’s weekly budget line, therefore, is: Dollars of Consumption $1,420

$320 110 Hours of Leisure

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(b) What is Shelly’s marginal rate of substitution when L = 100 and she is on her budget line? If Shelly leisures for 100 hours, she works for 10 hours and consumes $320 + $10(10) = $420. Thus, her MRS when doing this is:

MRS =

MU L C − 100 420 − 100 320 = = = = $5.33 . MUc L − 40 100 − 40 60

(c) What is Shelly’s reservation wage? The reservation wage is defined as the MRS when working no hours. When working no hours, Shelly leisures for 110 hours and consumes $320. Thus,

wRES =

MU L C − 100 320 − 100 220 = = = = $3.14 . MUc L − 40 110 − 40 70

(d) Find Shelly’s optimal amount of consumption and leisure. Her optimal mix of consumption and leisure is found by setting her MRS equal to her wage and solving for hours of leisure given the budget line: C = 320 + 10(110 – L). w = MRS 10 = 10 =

C − 100 L − 40

320 + 10(110 − L) − 100 L − 40

10 L − 400 = 1320 − 10 L L = 86.

Thus, Shelly will choose to leisure 86 hours, work 24 hours, and consume $320 + $10(24) = $560 each week.

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2-7. Explain why receiving a cash grant from the government can entice some workers to stop working (and entices no one to start working) while the earned income tax credit can entice some people who otherwise would not work to start working (and entices no one to stop working). A lump sum transfer is associated with an income effect but not a substitution effect, because it doesn’t affect the wage rate. Thus, if leisure is a normal good, a lump sum transfer will likely cause workers to work fewer hours (and certainly not cause them to work more hours) while possibly enticing some workers to exit the labor force all together. On the other hand, the Earned Income Tax Credit raises the effective wage of low-income workers by 40 percent (at least for the poorest workers). Thus, someone who had not been working faces a wage that is 40 percent higher than it otherwise was. This increase may be enough to encourage the person to start working. For example, if a worker’s reservation wage is $10.00 per hour but the only job she can find pays $8.00 per hour, she will not work. Under the earned income tax credit, however, the worker views this same job as paying $11.20 per hour, which exceeds her reservation wage. Furthermore, the EITC cannot encourage a worker to exit the labor force, as the benefits of the EITC are received only by workers.

2-8. In 1999, 4,860 TANF recipients were asked how many hours they worked in the previous week. In 2000, 4,392 of these recipients were again subject to the same TANF rules and were again asked their hours of work during the previous week. The remaining 468 individuals were randomly assigned to a “Negative Income Tax” (NIT) experiment which gave out financial incentives for welfare recipients to work and were subject to its rules. Like the other group, they were asked about their hours of work during the previous week. The data from the experiment are contained in the table below.

Number Of Recipients TANF NIT Total

4,392 468 4,860

Number of Recipients Who Worked At Some Time in the Survey Week

Total Hours Of Work By All Recipients in the Survey Week

1999

2000

1999

2000

1,217 131 1,348

1,568 213 1,781

15,578 1,638 17,216

20,698 2,535 23,233

(a) What effect did the NIT experiment have on the employment rate of public assistance recipients? Develop a standard difference-in-differences table to support your answer. Employment Rate TANF NIT

1999 27.7% 28.0%

2000 35.7% 45.5%

Diff 8.0% 17.5%

Diff-in-Diff 9.5%

The NIT increased the probability of employment by 9.5 percentage points. Note that the percent numbers are found by dividing the “Number of Recipient” columns (2nd and 3rd columns of the original table) by the Number of Recipients column (1st column of the original original). 6 ©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

(b) What effect did the NIT experiment have on the weekly hours worked of public assistance recipients who worked positive hours during the survey week? Develop a standard difference-in-differences table to support your answer. Weekly Hours Worked Per Working Person TANF NIT

1999 12.8 12.5

2000 13.2 11.9

Diff 0.4 -0.6

Diff-in-Diff -1.0

The NIT decreased weekly hours worked, of those working, by 1 hour. Note that the average weekly hours of work per persons is found by dividing the “Total Hours of Work” columns (4th and 5th columns of the original table) by the Number of Recipients column (1st column of the original table).

2-9. Consider two workers with identical preferences, Phil and Bill. Both workers have the same life cycle wage path in that they face the same wage at every age, and they know what their future wages will be. Leisure and consumption are both normal goods. (a) Compare the life cycle path of hours of work between the two workers if Bill receives a one-time, unexpected inheritance at the age of 35. Because the workers have the same life cycle wage path and the same preferences, they will have the same life cycle path of hours of work up to the unexpected event. An inheritance provides an income effect for Bill with no substitution effect, and thus, he will work fewer hours (or at least not more hours) than Phil from the age of 35 forward. See the following graph.

Hours Worked

Life Cycle Path of Hours Worked After Age 35: Before Age 35: Phil

Bill

Bill and Phil

Age

(b) Compare the life cycle path of hours of work between the two workers if Bill had always known he would receive (and, in fact, does receive) a one-time inheritance at the age of 35. In this case, because the inheritance is fully anticipated, and because it offers the same income effect with no substitution effect, Bill will work fewer hours (or at least not more hours) than Phil over their entire work lives. See the following graph. 7 ©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

Hours Worked

Life Cycle Path of Hours Worked

Phil

Bill

Age

2-10. Under current law, most Social Security recipients do not pay federal or state income taxes on their Social Security benefits. Suppose the government proposes to tax these benefits at the same rate as other types of income. What is the impact of the proposed tax on the optimal retirement age? Suppose social security benefits are the only pension benefits available to a retiree. The tax, therefore, can be interpreted as a cut in pension benefits. The cut in pension benefits shifts the budget line from FH to FE in the figure below, shifting the worker from point P to point R. (Note that FE and FH are both downward sloping, indicating that total retirement consumption is greater the later in life one retires meaning that one has fewer years of retirement.) This shift from FH to FE generates both income and substitution effects. Both of these effects, however, work in the same direction. First, the tax reduces the retiree’s wealth, reducing her demand for leisure, and leading her to retire later (the income effect). At the same time, the tax reduces the “wage” that retirees receive when retired, effectively increasing (in relative terms) the wage they earn while working and generating a substitution effect that leads to more work hours, thus further delaying retirement. Under normal conditions, therefore, a tax on pension benefits will increase the optimal retirement age (i.e., workers will delay retirement and have fewer years of retirement).

Consumption During Retirement F

P R H U0

E 20

Years of Retirement

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2-11. A worker plans to retire at the age of 65, at which time he will start collecting his retirement benefits. Then there is a sudden change in the forecast of inflation when the worker is 63 years old. In particular, inflation is now predicted to be higher than it had been expected so that the average price level of market goods and wages is now expected to be higher. What effect does this announcement have on the person’s preferred retirement age: (a) if retirement benefits are fully adjusted for inflation? There will be no effect on the person’s retirement decision if retirement benefits are fully adjusted for inflation as nothing changes in the person’s calculations in real terms: the relative magnitudes of prices, wages and retirement benefits are the same with or without inflation. The person faces the same choice set, so his decision does not change. (b) if retirement benefits are not fully adjusted for inflation? If retirement benefits are not adjusted for inflation, the purchasing power of retirement benefits falls. If the person does not retire, he can enjoy the same consumption as he would without inflation as wages are assumed to fully adjust for inflation. If he retires at 65, his benefits are worth less in real terms (they can buy him less consumption) with inflation than without, so he cannot afford the same consumption path as before. Hence, his choice set over the years of retirement and consumption lies below the original (pre-inflation) choice set except at one point—where he does not retire at all. Thus, as long as leisure (i.e., years of retirement) and consumption are normal goods, the income and substitution effects both lead to the individual retiring later in life.

2-12. Presently, there is a minimum and maximum social security benefit paid to retirees. Between these two bounds, a retiree’s benefit level depends on how much she contributed to the system over her work life. Suppose Social Security was changed so that everyone aged 65 or older was paid $12,000 per year regardless of how much she earned over her working life or whether she continued to work after the age of 65. How would this likely affect the number of hours worked by retirees? Labor force participation is likely greatest for those retirees whose social security income is low (below $12,000 per year). Thus, the change in benefits offers these retirees a pure (positive) income effect. These retirees should reduce their hours worked if not leave the labor force all together after the age of 65. In contrast, the policy change offers all retirees who would have earned more than $12,000 per month a pure (negative) income effect. These retirees will become more likely to work, or, if already working, more likely to work more hours after the age of 65.

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2-13. Over the last 100 years, real household income and standards of living have increased substantially in the United States. At the same time, the total fertility rate, the average number of children born to a woman during her lifetime, has fallen in the United States from about three children per woman in the early twentieth century to about two children per woman in the early twenty-first century. Does this suggest that children are inferior goods? The conventional wisdom (and empirical evidence) suggests that children are normal goods. Economically, children are a lot more expensive today than they were 100 years ago (consider education, housing, clothing, entertainment expenses, etc.). Children also produce less for the household in the 21st century than they did 100 years ago. There is also a biology/evolution argument is that infant mortality rates have fallen dramatically over the last 100 years, so a woman needs to have fewer children to be more confident that some of her children will reach adulthood. This argues against children being an inferior good as the “good” in question can be thought of as the number of offspring who live long enough to procreate.

2-14. Consider a person who can work up to 80 hours each week at a pre-tax wage of $20 per hour but faces a constant 20% payroll tax. Under these conditions, the worker maximizes her utility by choosing to work 50 hours each week. The government proposes a negative income tax whereby everyone is given $300 each week and anyone can supplement her income further by working. To pay for the negative income tax, the payroll tax rate will be increased to 50%. (a) On a single graph, draw the worker’s original budget line and her budget line under the negative income tax. Under the original scenario, let I be total weekly income, L be hours of leisure, and H be hours worked. The worker’s after-tax wage rate is 80% of $20 which equals $16 per hour. Thus, when the worker works all 80 hours in the week, she earns $16 x 80 = $1,280 and her budget line is described by I = 1280 – 16L. Notice that when L = 80, the worker earns $0. And when L = 30, the worker earns $16 × 50 = $800. Under the negative income tax, the worker is given $300 each week, but now her after-tax wage rate is 50% of $20 which equals $10 per hour. In this case, when the worker works all 80 hours in the week, she earns $10 × 80 + $300 = $1,100 and her budget line is properly described by I = 1100 – 10L. Notice that when L = 80, the worker receives $300. And when L = 30, the worker receives $300 + $10 x 50 = $800. The two budget lines for both scenarios are graphed on the next page.

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Weekly Budget Lines Weekly $1,280 Income $1,100

Original Scenario

$800

Negative Income Tax $300

Hours of Leisure

(b) Show that the worker will choose to work fewer hours if the negative income tax is adopted. To answer this question, one needs to find where the budget lines intersect. Setting the budget lines equal and solving for L reveals that the budget lines intersect at L = 30. Thus, the indifference curve that is tangent to the original budget line at L = 30 must not be tangent to the budget line under the negative income tax (because L = 30 was the optimal choice without the negative income tax). In particular, the worker’s original indifference curve must be below the new budget line to the right of L = 30. Therefore, when faced with the negative income tax, the worker will move in that direction, which requires her to increase L (hours of leisure) and concurrently decrease H (hours of work). (c) Will the worker’s utility be greater under the negative income tax? In this particular case, the worker’s utility will increase under the negative income tax because she could have continued to leisure 30 hours each week and receive $800 (which was her outcome before the negative income tax) but instead the worker decides to leisure more (and consume less). This change in behavior must increase her utility.

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2-15. The absolute value of the slope of the consumption-leisure budget line is the after-tax wage, w. Suppose some workers earn w for up to 40 hours of work each week, and then earn 2w for any hours worked thereafter (called overtime). Other workers earn w for up to 40 hours of work each week, and then only earn 0.5w thereafter as working more than 40 hours requires getting a second job which pays an hourly wage less than their primary job. Both types of workers experience a “kink” in their consumption-leisure budget line. (a) Graph in general terms the budget line for each type of worker. Weekly Budget Lines Weekly Income The wage increases (to 2w) for hours of work in excess of 40.

The wage falls (to 0.5w) for hours of work in excess of 40.

Both face a wage of w for 40 hours

T - 40

Hours of Leisure

(b) Which type of worker is likely to work up to the point of the kink, and which type of worker is likely to choose a consumption-leisure bundle far away from the kink? The worker who experiences a decrease in her wage after working 40 hours is more likely to work exactly 40 hours as the marginal benefit of working experiences a negative jump down at this point. In contrast, the worker who experiences an overtime premium after working 40 hours is more likely to not work exactly 40 hours. Because of the overtime premium, once the worker hits 40 hours of work, the worker experiences a positive jump up in the marginal benefit of working. Put differently, this worker may opt to only work 20 or 30 hours, but if she finds herself having worked 40 hours because the T – 40th hour of leisure was not as valuable as w, then it is very likely that she will also find that the T – 41st hour of leisure is not as valuable as 2w, and therefore she works the 41st hour (and possibly quite more).

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CHAPTER 3 3-1. Suppose there are two inputs in the production function, labor and capital, and these two inputs are perfect substitutes. The existing technology permits 1 machine to do the work of 3 workers. The firm wants to produce 100 units of output. Suppose the price of capital is $750 per machine per week. What combination of inputs will the firm use if the weekly salary of each worker is $300? What combination of inputs will the firm use if the weekly salary of each worker is $225? What is the elasticity of labor demand as the wage falls from $300 to $225? Because labor and capital are perfect substitutes, the isoquant for producing 100 units of output (in bold in the figure below) is linear and the firm will use only labor or only capital, depending on which is relatively cheaper in producing 100 units of output. The (absolute value of the) slope of the isoquant (MPE / MPK) is 1/3 because 1 machine does the work of 3 workers. When the wage is $300, the slope of the isocost is 300/750. The isocost curve, therefore, is steeper than the isoquant, and the firm only hires capital (at point A). To calculate this in a different way, one machine does the work of three workers. The one machine costs $750; the three workers cost $300 × 3 = $900. Clearly the firm should hire only machines. When the weekly wage is $225, the isoquant is steeper than the isocost and the firm hires only labor (at point B). To calculate this in a different way, one machine does the work of three workers. The one machine costs $750; the three workers cost $225 × 3 = $675. Clearly the firm should hire only workers.

Capital Smallest isocost line that achieves 100 units of output when the wage is $225 per week: w / r = 225 / 750 < 1/3.

Output = 100 Units Isoquant A Smallest isocost line that achieves 100 units of output when the wage is $300 per week: w / r = 300 / 750 > 1/3.

Labor

The elasticity of labor demand is defined as the percentage change in labor divided by the percentage change in the wage. Because the demand for labor goes from 0 to a positive quantity when the wage drops to $225, the (absolute value of the) elasticity of labor demand is infinity.

3-2. Figure 3-18 in the text shows the ratio of the federal minimum wage to the average hourly manufacturing wage. (a) Describe how this ratio has changed from the 1950s to the 1990s. What might have caused this apparent shift in fundamental economic behavior in the United States? The ratio of the federal minimum wage to the average hourly manufacturing wage averaged about 0.5 from 1950 to 1968. That is, a manufacturing job paid about 2 times as much as a minimum wage job. Since 1968, however, the ratio has steadily fallen. In the last quarter of the 20th century, the ratio approached 0.3, meaning that a manufacturing job paid about 3 times as much as a minimum wage job. Consider three (of the many possible) explanations for this behavior. First, although the nominal minimum wage has risen over time, the real minimum wage fell quit a bit from 1968 to 1990. Second, the U.S. economy has lost a lot of manufacturing jobs recently, ostensibly due to globalization. Economic theory suggests that the jobs most likely to be lost are those with the least productive workers (relative to foreign workers), which likely correlates to the lowest paid manufacturing jobs. Thus, when the lowest paid manufacturing jobs are lost, the average manufacturing wage will increase, which in turn increases the ratio presented in Figure 3-18. Third, one might point toward the return to skills. Most low-wage, minimum wage jobs require very little skill (and some of these jobs are becoming more and more automated). Manufacturing jobs, however, still require skill in terms of using high-tech equipment and machines. (b) This ratio fell steadily from 1968 to 1974 and again from 1980 to 1990, but the underlying dynamics of the minimum wage and the average manufacturing wage were different during the two time periods. Explain. First, notice that over both of these periods that the federal minimum wage was unchanged. The earlier period (1968-1974) was a time of stagflation, war, and an oil price shock. High levels of inflation led to higher nominal wages (for those who had jobs), thus lowering the ratio of the minimum wage to the average manufacturing wage. That is, the decreasing ratio was largely a product of an eroding real minimum wage during an inflationary period. The later period (1980-1990) is generally characterized as a time of economic growth and prosperity. Thus, while inflation was largely in check, wages were increasing substantially as a result of productivity growth. This productivity growth (and in turn wage growth) resulted in a lower ratio. (c) What has been happening to the ratio of the federal minimum wage (nominal) to the average hourly manufacturing wage from 1990 to today? More or less, it has been holding steady between 0.3 and 0.4, depending on the timing of legislative increases in the minimum wage.

3-3. Firm would hire 20,000 workers if the wage rate is $12 but will hire 10,000 workers if the wage rate is $15. Firm B will hire 30,000 workers if the wage is $20 but will hire 33,000 workers if the wage is $15. The workers in which firm are more likely to organize and form a union? The union will be more likely to attract the workers’ support when the elasticity of labor demand (in absolute value) is small. The elasticity of labor demand facing firm A is given by:

A =

%E (20,000 − 10,000) 20,000 = = −2 . %w (12 − 15) 12

The elasticity of labor demand facing firm B is given by:

B =

%E (33,000 − 30,000) 33,000 =  −0.45 %w (15 − 20) 15

Workers at Firm B, therefore, are more likely to organize as |-0.45| < |-2|.

3-4. Consider a firm for which production depends on two normal inputs, labor and capital, with prices w and r, respectively. Initially the firm faces market prices of w = 6 and r = 4. These prices then shift to w = 4 and r = 2. (a) In which direction will the substitution effect change the firm’s employment and capital stock? Prior to the price shift, the absolute value of the slope of the isocost line (w/r) was 1.5. After the price shift, the slope is 2. In other words, labor has become relatively more expensive than capital. As a result, there will be a substitution away from labor and towards capital (the substitution effect). (b) In which direction will the scale effect change the firm’s employment and capital stock? Because both prices fall, the marginal cost of production falls, and the firm will want to expand. The scale effect, therefore, increases the demand for both labor and capital as both are normal inputs. (c) Can we say conclusively whether the firm will use more or less labor? More or less capital? The firm will certainly use more capital as the substitution and scale effects reinforce each other in the direction of using more capital. The change in labor hired, however, will depend on whether the substitution or the scale effect dominates for labor.

3-5. What happens to employment in a competitive firm that experiences a technology shock such that at every level of employment its output is 200 units/hour greater than before? Because output increases by the same amount at every level of employment, the marginal product of labor does not change (and hence, the value of the marginal product of labor does not change). Therefore, as the value of the marginal product of labor will equal the wage rate at the same level of employment as before, the level of employment will not change.

3-6. Consider each of the following, and explain why it is or is not a valid instrument for estimating labor supply elasticity and/or labor demand elasticity in the United States. (1) Variation in state income tax rates. (2) Variation in state corporate tax rates. (3) Changes in federal income tax rates over time. A valid instrumental variable for estimating labor supply elasticity must be something that shifts labor demand but not labor supply. Similarly, a valid instrumental variable for estimating labor demand elasticity must be something that shifts labor supply but not labor demand. For each of the options above: (1) State income taxes affect labor supply (because income taxes affect the wage), and therefore variation in state income taxes would be a valid instrument for estimating the elasticity of labor demand but not for estimating labor supply. (2) State corporate taxes affect labor demand (because corporate taxes affect profits and the value of marginal product of labor), and therefore variation in state corporate taxes would be a valid instrument for estimating the elasticity of labor supply but not for estimating labor demand. (3) Changes in the federal income tax rate over time is not a valid instrument for estimating either elasticity. The goal is to estimate either elasticity for the United States. Therefore, one needs something that changes within the United States for different workers or firms. One cannot use a nation-wide variable to do this. Moreover, using time as the variation is dangerous as it requires assuming everything else is constant over time, which likely is not the case.

3-7. Suppose a firm purchases labor in a competitive labor market and sells its product in a competitive product market. The firm’s elasticity of demand for labor is −0.4. Suppose the wage increases by 5 percent. What will happen to the amount of labor hired by the firm? What will happen to the marginal productivity of the last worker hired by the firm? Given the estimates of the elasticity of labor demand and the change in the wage, we have that

=

%E %E = −0.4 => = −0.4 => %E = −2% . %w 5%

Thus, the firm hires 2 percent fewer workers when wages increase by 5%. Furthermore, because fewer workers are hired, under normal conditions the marginal productivity of the last worker hired will increase. (More formally, because the labor market is competitive, the marginal worker is paid the value of his marginal product. As the product market is competitive, we also know that the output price does not change so that the marginal productivity of the marginal worker increases by 5 percent as well.) 3-8. A firm’s technology requires it to combine 5 person-hours of labor with 3 machinehours to produce 1 unit of output. The firm has 15 machines in place when the wage rate rises from $10 per hour to $20 per hour. What is the firm’s short-run elasticity of labor demand? Unless the firm shuts down (i.e., goes out of business in the short run), it will combine 25 persons with the 15 machines it has in place regardless of the wage rate. Therefore, employment will not change in response to the movement of the wage rate, and the short-run elasticity of labor demand is zero.

3-9. In a particular industry, labor supply is ES = 10 + w and labor demand is ED = 40 − 4w, where E is the level of employment and w is the hourly wage. (a) What is the equilibrium wage and employment if the labor market is competitive? What is the unemployment rate? In equilibrium, the quantity of labor supplied equals the quantity of labor demanded, so that ES = ED. This implies that 10 + w = 40 – 4w. The wage rate that equates supply and demand is $6. When the wage is $6, 16 persons are employed. There is no unemployment because the number of persons looking for work equals the number of persons employers are willing to hire at the going wage rate of $6 per hour. (b) Suppose the government sets a minimum hourly wage of $8. How many workers would lose their jobs? How many additional workers would want a job at the minimum wage? What is the unemployment rate? If employers must pay an hourly wage of $8, employers would only want to hire ED = 40 – 4(8) = 8 workers, while ES = 10 + 8 = 18 persons would like to work. Thus, 8 workers lose their job following the minimum wage as 16 workers used to be employed but now only 8 are; and 2 additional people enter the labor force following the minimum wage as 16 workers used to want a job but now 18 do. Under the minimum wage, the unemployment rate would be 10/18, or 55.6 percent.

3-10. Suppose the hourly wage is $10 and the price of each unit of capital is $25. The price of output is constant at $50 per unit. The production function is f(E,K) = E½K ½, so that the marginal product of labor is MPE = (½)(K/E) ½ . If the current capital stock is fixed at 1,600 units, how much labor should the firm employ in the short run? How much profit will the firm earn? The firm’s labor demand curve is its value of marginal product curve, VMPE, which equals the marginal productivity of labor, MPE, times the marginal revenue of the firm’s product. But as price is fixed at $50, MR = 50. Thus, we have: 1600 1,000 1 . VMPE = MPE  MR =     50 = E 2 E

Now, by setting VMPE = w and solving for E, we find that the optimal number of workers for the firm to hire is 10,000 workers (i.e., 1000/sqrt(E) = 10 solves as E = 10,000). The firm then makes (1,600)½(10,000)½ = 4,000 units of output and earns a profit of π = 4,000($50) – 1,600($25) – 10,000 ($10) = $60,000.

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Solution Manual for Labor Economics, 8th Edition George Borjas