Chapter 8 -- Workers, Wages and Unemployment

Problem 1. Earnings and Education For recent data on earnings and educational attainment, go to:

Census Bureau

from which the following data is taken:

Education	Median income 2000		Median income 2004
	Male	Female	Male	Female
Less than 9th grade	$14,149	$8,404	na	na
9-12; no HS diploma	$18,953	$9,995	$ 21,760	$ 13,280
HS diploma	$27,666	$15,120	$ 31,183	$ 19,821
Some college, no degree	$33,039	$20,181	$ 37,883	$ 25,235
Associate degree	$37,953	$23,270	na	na
Bachelor's degree	$49,178	$30,489	$52,242	35,195
Master's degree	$59,376	$40,246	na	na
Master's or doctorate degree	na	na	$ 68,239	$46,004
Doctorate degree	$71,738	$48,885

Problem 2. Bob's Bicycle Factory

Note: the text definition of the value of the marginal product of labor is not quite exact enough for this problem. The text definition is simply VMPL = Price times MPL. The full definition is that:

In this problem, we must note that the non-labor cost per bike is $100, so that the price minus non-labor cost is $30 per bike.

It is straightfoward to construct the following table, noting that

Workers	Bikes	MPL	VMPL (P = $130)	VMPL (P = $140)
1	10	10	$300	$400
2	18	8	$240	$320
3	24	6	$180	$240
4	28	4	$120	$160
5	30	2	$60	$80
6	31	1	$30	$40

For the case in which productivity increases by 50 percent:

Workers	Bikes	MPL	VMPL (P = $130)
1	15	15	$450
2	27	12	$360
3	36	9	$270
4	42	6	$180
5	45	3	$90
6	46.5	1.5	$45

Note that the VMPL is the price that employers are willing to pay the last worker hired ; since the simple version of the marginal productivity theory of wages does not regard workers as different in skills, there is no reason for the employer to pay any more to any other worker.

Hence the VPML values just calculated represent the labor demand curve .

Problem 3. Lightbulb factory

In the Bright Idea Lightbulb factory, the marginal product of workers is given by the equation

where N is the number of workers employed. Lightbulbs sell for $2, and since there are no non-labor costs, we can write the value of the marginal product equation as:

Since the VMPL equation is the firm's demand curve for labor:

• If the wage is $20 an hour, the firm will find the number of workers to hire by solving the following equation for N:
  $20 = 60 - 2N

  which yields N = 20 .
  

• If the wage were $30 an hour, the equation would be:
  $30 = 60 - 2N

  which yields N = 15
  

• The demand for labor will be represented graphically on a plot with X-axis representing N and Y-axis representing the VMPL (or the wage which employers are willing to pay) as a straight line with intercept 60 (no workers would be hired at $60) on the Y-axis and intercept 30 (30 workers would be hired at $0) on the X-axis.
  
• If the price of the product were $3 rather than $2, the VMPL equation would be:
  VMPL = 90 - 3N

  and the line representing this demand for labor would start at $90 on the Y-axis, but would have the same X-axis intercept. (Once the marginal product of labor hits zero, there is no reason to hire labor even if it's free).
  

• If the supply of workers in the town is 20 workers, no more and no less, the labor supply curve is vertical, and we can find the market wage by solving the labor demand curve (assuming price = $2) wage = 60 - 2N

  or

  N = 30 - 1/2 wage

  along with the labor supply curve:

  N = 20

  The solution is most easily arrived at by substituting N = 20 into the first form of the labor demand curve, giving
  

  wage = 60 - 2 (20) = $20

  You should be able to solve for Price = $3; the equilibrium wage will be $30.

Problem 4. Labor as a derived demand: the auto assembly line

The point of the questions in this problem is that the demand for labor depends on the demand for the product being produced. The demand for labor is therefore derived from the demand for the product, and hence

• If demand for the car increases, the price of autos will go up, and hence the demand for labor will increase.
• If consumers switch to mass transit because of a rise in the price of gas, the demand for autos will decline, the price of autos will fall, and hence the demand for labor will decrease.
• If workers find other job opportunities more attractive than factory work, the supply of labor will shift up and to the left (decrease), and hence the wage will rise.
• If new assembly line methods increase the productivity of labor, the MPL and hence (assuming the same price) the VMPL (which is the demand for labor) will increase. Unless the greater number of cars on the market as a result drives the price of cars down very sharply, wages will increase.
• The introduction of robots is likely to result in a decrease of the demand for unskilled labor (robots and unskilled labor are substitutes ), but may well lead to an increase in the demand for engineers and robotics technicians (robots and skilled labor may be complements .
• Unionization will not by itself shift the demand for labor -- unless greater employee satisfaction leads to increased productivity (Richard Freeman, in his book What do Unions Do? , suggests that this is a real possibility).
  Unionization will not shift the supply of labor directly, since no one can be required to be a member of a union before applying for a job. Unions may however try to persuade the state legislatures or Congress to pass measures (certification requirements, immigration restrictions) which reduce the supply of labor and so raise the equilibrium wage .
  Unions may negogiate contracts which provide for above-equilibrium wages and so lead to unemployment.

Problem 5. The supply of labor

Note that in the following, supply = supply curve , and the questions can be rephrased "Does this shift the supply curve?"

• If the retirement age increases, the supply of labor will shift down and to the right (increase)
• If productivity increases, the demand for labor increases, but nothing will happen to supply . When wages rise, the quantity of labor supplied will increase , but the supply curve will still remain fixed.
• With a military draft, the supply of labor to civilian occupations will decrease. The supply curve will shift up and to the left.
• More children mean a decrease in the supply of labor in the short-run, since some people will withdraw from the labor force to become full-time parents; but it will lead to an increase in the labor force when the children themselves enter the work force.
• More generous social security benefits will give an incentive to retire earlier, and will decrease the supply of labor.

Problem 6. Skilled and unskilled workers

In this problem, we must derive two VMPL equations, and use them to find wages for skilled and unskilled workers. We use VMPLS to denote the value of the marginal product of skilled workers, and VMPLU to denote the value of the marginal product of unskilled workers; Ns to denote the number of skilled workers, and Nu to denote the number of unskilled workers.
The equations themselves are simply the price of the output times the given MPL equations:

and

The specific questions asked in this problem are:

• The wage of each group if there are 100 skilled and 50 unskilled workers.

  Since the value of the MPL is equal to the wage in equilibrium, we have:

  Ws = 600 - 3 Ns = 600 - 3 (100) = $ 300

  and

  Wu = 300 - 3 Nu = 300 - 3 (50) = $ 150

• The wage of skilled workers given the introduction of new equipment which raises their marginal productivity to
  MPLS = 300 - Ns

  or the value of their marginal product to:

  VMPLS = 900 - 3 Ns

  Simply substitute into this the number of skilled workers to get

  Ws = 900 - 3 Ns = 900 - 3 (100) = $600

• The result if the growing wage differential motivates unskilled workers to acquire skills. The problem states that a wage differential of $300 or more will provide sufficient motivation; to solve it, we need to rewrite the equilibrium condition in terms of the wage differential.
  Ws - Wu = $300

  Using the VMPL equations in place of Ws and Wu, this means:

  (900 - 3 Ns) - (300 - 3 Nu) = $ 300

  or after a little algebra:

  Ns - Nu = 100

  There will be 100 more skilled workers than unskilled workers in long-run equilbrium in this case.

  We also know that Ns + Nu = 150, and we can use this to find that in long run equilbrium,

  Ns = 125 and Nu = 25

  This in turn allows us to calculate wages with the VMPLS equations from the last questions:

  Ws = 900 - 3 (125) = $ 525

  and

  Wu = 300 - 3 (25) = $ 225

Problem 7. Trade and wages

Key assumptions of the problem:

• The economy produces two goods, sweaters and dresses.
• Workers are not mobile between sectors.
• Wages will adjust to maintain full employment in each sector.

Data:

• MPLs = 20 - Ns ,
  where MPLs = Marginal product of labor in the sweater industry.
  and Ns = number of workers in the sweater industry.
• MPLd = 30 - Ns ,
  where MPLd is the marginal product of workers in the dress industry,
  and Nd = number of workers in the dress industry.
• Ns = 14 and Nd = 26 .

The above information is to be used in the following problems:

• Initially Ps = $40 and Pd = $60 .
  What are wages in the sweater industry (Ws) and the dress industry (Wd) ?

  At these prices,
  

  Ws = VMPLs = 40 (20 - Ns) = 800 - 40 Ns = 800 - 40 (14) = $ 240

  and

  Wd = VMPLd = 60 (30 - Nd) = 1800 - 60 Nd = 1800 - 60 (26) = $ 240

• When the price of sweaters rises to $50 and the price of dresses falls to $50 as a result of international trade, what happens to wages?
  The answer can be found by simply substituting the new prices in the VMPL equations we just used:
  Ws = VMPLs = 50 (20 - Ns) = 1000 - 50 Ns = 1000 - 50 (14) = $ 300
  Wd = VMPLd = 50 (30 - Nd) = 1500 - 50 Nd = 1500 - 50 (26) = $ 200

  The wages of sweater workers increase; those of dress workers decrease. Trade results in losses for some workers, and gains for others.

• What happens if workers are free to move between the sweater industry and the dress industry?
  This is a problem similar to problem 6, where workers moved to the other industry by acquiring appropriate training. As in that problem, we must first formulate an appropriate equilibrium condition -- a good one is simply that for an equilibrium in which workers are happy where they are, wages in the two industries must be equal.

  This may be written as the equation

  Ws = Wd

  or substituting the formula for VMPLs and VMPLd,

  50 (20 - Ns) = 50 (30 - Nd)

  From which we can find after some algebra

  Nd - Ns = 10

  Since it is also true that
  Nd + Ns = 26 + 14 = 40 the equilbrium values of Ns and Nd will be
  Ns = 15 and Nd = 25 and more importantly, wages will be
  

  Ws = 1000 - 50 Ns = 1000 - 50 (15) = $ 250

  Note that wages in the sweater industry and in the dress industry have both risen above their pre-trade level.

Problem 8. Frictional, structural or cyclical?

The classification of the types of unemployment described in this problem is simple:

• Ted, the displaced worker , is structurally unemployed .
• Alice, laid off because of a recession is cyclically unemployed
• Lance, who works only during busy seasons, is structurally unemployed . (Some texts have a separate classification for "seasonally unemployed", which to me fits this case more appropriately)
• Gwen, who quit and moved, and has not yet found a new job, is frictionally unemployed .
• Tao, the new college grad who has been looking for a job, is also frictionally unemployed.
• Karen (like Ted) is a displaced worker, but this time the displacement resulted from the bad luck of her company rather than the structural problems of the steel injury. She has other opportunities, but is taking her time to search for the best one, so she is frictionally unemployed.

Problem 9. Minimum wage and disequilibrium

Demand and supply of workers are given by the following equations:

• Equilibrium is found by setting Ns = Nd, or
  400 - 2 w = 240 + 2w

  From which we find (adding 2w to each side, and subtracting 240 from each side:

  4w = 160

  or wage = $40.
  At this wage, Nd = 400 - 2 (40) = 320 and Ns = 240 + 2 (40) = 320.

• If the minimum wage were set at $50, the amount of labor demanded would be less than the amount supplied, and we would have unemployment.
  Nd = 400 - 2 (50) = 300
  Ns = 240 + 2 (50) = 340

  Employment would be 300 (the smaller of the quantity demanded and the quantity supplied is the amount actually exchanged on the market -- in microeconomics, we called this the short-side rule ) and 40 workers would be unemployed. Note however that in the previous problem the 320 workers who were unemployed were making $40 each, or a total of 320 (40) = $12,800. In this case the 300 workers who are employed are making $50 each, a total of 300 (50) = $ 15,000. The minimum wage may lead to unemployment, but does not necessarily lead to less money going to workers as a whole.

Problem 10. Duration of unemployment

The statistics on unemployment below are from the Bureau of Labor Statistics web site , specifically from the latest Employment Situation News Release . Table A-6 in the January 2002 report gives the percentage distribution of the duration of unemployment -- that is, what percentage of the unemployed were unemployed for less than 5 weeks, from 5 to 15 weeks, and for more than 15 weeks. The data for January 2001 -- a very good month, when the recession of 2001 had not yet started -- should be contrasted with the data for January 2002, with the recession underway.

The January 6,2005 report gives data for December 2004 and 2005.
Note that the percentage unemployed less than 5 weeks has been increased, but long run unemployment is relatively more of a problem now than it was during the recession.

Duration	Jan.2001	Jan.2002	Dec.2004	Dec.2005
Less than 5 weeks	46.6	36.8	35.7	37.2
5 to 14 weeks	31.8	31.3	28.2	30.2
More than 15 weeks	21.6	29.9	36.1	32.6