Practice Worksheet

Chapter 18 — The Markets for the Factors of Production

Part 1

Short Answer — Production Schedule & Labor Demand

Given A perfectly competitive firm uses labor as its only variable input. Each unit of output sells for P = $10. The production function is shown in the table below.

Days of Labor (L)	Units of Output (Q)
0	0
1	7
2	13
3	19
4	25
5	28
6	29
7	29

Question a

Calculate the marginal product of labor (MPL) for the 1st, 2nd, 3rd, and 4th workers.

MPL, 1st worker

MPL, 2nd worker

MPL, 3rd worker

MPL, 4th worker

Hint MPL is the change in output when you add one more worker: MPL = ΔQ / ΔL. Since ΔL = 1 for each row, just subtract each row's output from the previous row's output.

Explanation MPL = ΔQ = Q(L) − Q(L−1).
1st worker: 7 − 0 = 7
2nd worker: 13 − 7 = 6
3rd worker: 19 − 13 = 6
4th worker: 25 − 19 = 6

Note on the flat region (workers 2–4 all add 6). Mankiw's "diminishing marginal product" is a weak rule: each extra worker adds no more than the previous, not strictly less. Workers 2, 3, and 4 all add 6, then it drops sharply (5→3, 6→1, 7→0). So the schedule is non-increasing throughout — consistent with diminishing returns even though it doesn't fall every single step. The scaffold's orchard example (100, 80, 60, 40, 20) is the strict version; this worksheet uses the more realistic weak version.

Question b

Output sells for $10/unit. Calculate the value of the marginal product of labor (VMP_L) for the 1st, 2nd, 3rd, and 4th workers.

VMP, 1st worker ($)

VMP, 2nd worker ($)

VMP, 3rd worker ($)

VMP, 4th worker ($)

Hint VMP_L = P × MPL. You computed each MPL in part (a); just multiply each by the output price P = $10.

Explanation VMP_L = P × MPL with P = $10:
1st worker: 10 × 7 = $70
2nd worker: 10 × 6 = $60
3rd worker: 10 × 6 = $60
4th worker: 10 × 6 = $60

VMP_L is the extra revenue the firm gets from hiring one more worker. Because MPL is diminishing, VMP_L is downward-sloping — and that's the firm's labor demand curve.

Question c

Suppose the daily wage is W = $50 per worker. How many workers does the firm hire to maximize profit?

Number of workers

Hint Rule: hire workers as long as VMP_L ≥ W. Extend your VMP table beyond the 4th worker (at P=$10, MPL for the 5th worker is 3, so VMP = $30). Count the workers whose VMP is at least $50.

Explanation Full VMP_L schedule at P = $10:
1st: $70, 2nd: $60, 3rd: $60, 4th: $60, 5th: $30, 6th: $10, 7th: $0.

The firm hires as long as VMP_L ≥ W = $50.
4th worker: VMP = $60 ≥ $50 → hire.
5th worker: VMP = $30 < $50 → do not hire.

Answer: 4 workers. At 4 workers, the firm is exactly at the profit-maximizing level — the next worker's VMP falls below her wage.

Question d

Now suppose the output price rises from $10 to $12 per unit. What is the new VMP_L of the 4th worker?

VMP, 4th worker ($)

Hint MPL doesn't change — that's about the physical production function. What changes is the price P. Recompute VMP_L = P × MPL for the 4th worker using the new price.

Explanation MPL of the 4th worker is still 6 (the production function didn't change). With the new price:
VMP_L = 12 × 6 = $72.

Every worker's VMP_L is now 20% higher than before, so the entire labor demand curve shifts right. Intuition: when the output sells for more, each worker is more valuable to hire. This is why higher output prices push wages up in a competitive labor market.

Part 2

Short Answer — Capital Demand at Smiling Cow Dairy

Given Smiling Cow Dairy can sell all the milk it wants for $4 per gallon, and it can rent all the milking robots it wants at a capital rental price of $100 per day. The dairy's production schedule (robots → gallons/day) is shown below.

Robots	Gallons of milk
0	0
1	50
2	85
3	115
4	140
5	150
6	155

Question e

Calculate the value of the marginal product (VMP_K) of the 3rd robot and the 4th robot.

VMP, 3rd robot ($)

VMP, 4th robot ($)

Hint Same recipe as labor, just swap in capital. MP_K of the 3rd robot = (gallons with 3 robots) − (gallons with 2 robots). Then VMP_K = price × MP_K.

Explanation 3rd robot: MP_K = 115 − 85 = 30 gallons/day. VMP_K = $4 × 30 = $120/day.
4th robot: MP_K = 140 − 115 = 25 gallons/day. VMP_K = $4 × 25 = $100/day.

Just like MPL, MP_K is diminishing — each extra robot helps a little less than the previous one.

Question f

The rental price of a robot is $100 per day. How many robots should the dairy rent to maximize profit?

Number of robots

Hint Rent robots as long as VMP_K ≥ rental price. Build the full VMP_K schedule for robots 1 through 6 at P = $4 and the rental = $100, then count how many robots clear the $100 bar.

Explanation Full schedule at P = $4:
1st robot: MP = 50, VMP = $200
2nd robot: MP = 35, VMP = $140
3rd robot: MP = 30, VMP = $120
4th robot: MP = 25, VMP = $100
5th robot: MP = 10, VMP = $40
6th robot: MP = 5, VMP = $20

Rent as long as VMP_K ≥ $100.
4th robot: VMP = $100 = rental → just worth it, rent it.
5th robot: VMP = $40 < $100 → don't rent.

Answer: 4 robots. The dairy hires capital by the same rule as labor: keep adding until marginal benefit (VMP) equals marginal cost (rental price).

Part 3

Multiple Choice

Question 1

Because a firm's demand for a factor of production is derived from its decision to supply a good in the market, the demand for a factor of production is called a:

direct demand supply demand derived demand indirect demand

Hint Would a bakery hire bakers if nobody wanted to buy bread? The demand for labor comes from (is derived from) the demand for the firm's output.

Explanation Answer: C. The textbook term is derived demand: firms don't want labor for its own sake, they want labor because it produces output that customers will buy. If demand for the output falls, demand for the labor that produces it falls too. This is the foundational concept of Ch 18.

Question 2

Value of marginal product is defined as the additional:

output a firm receives from hiring one more unit of a factor of production cost incurred from hiring one more unit of a factor of production profit earned from hiring one more unit of a factor of production revenue earned from hiring one more unit of a factor of production

Hint VMP = P × MPL. MPL is the extra output; multiplying by price converts it into extra _________.

Explanation Answer: D. VMP = P × MPL. Option A describes the marginal product (extra output), not the value of the marginal product. By multiplying MPL by the output price, we convert physical output into additional revenue — and that's VMP. This is the extra revenue the firm earns from hiring one more unit of a factor.

Question 3

To maximize profit, a competitive firm hires workers up to the point of intersection of:

the MPL curve and the wage the VMP curve and the wage the labor supply curve and the labor demand curve the production function and the wage

Hint The firm keeps hiring as long as the extra revenue from one more worker exceeds the wage. What curve measures that extra revenue?

Explanation Answer: B. The profit-maximizing hiring rule is VMP_L = W. The firm hires workers until the value of the marginal product equals the wage. Option A (MPL curve) is in physical units, not dollars — you can't compare units of output to a dollar wage. Option C describes the market equilibrium, not the individual firm's decision.

Figure 18-5 — Use for Questions 4, 5, and 6

Question 4

Refer to Figure 18-5. When the relevant labor supply curve is S1, and the labor market is in equilibrium, the:

VMP of labor to firms is W2 quantity of labor demanded exceeds the quantity of labor supplied quantity of labor supplied exceeds the quantity of labor demanded VMP of labor to firms is W1

Hint In equilibrium, the wage equals VMP_L. Where does the demand curve (which IS the VMP curve) intersect S1? What wage does that correspond to?

Explanation Answer: D. When the supply curve is S1, the equilibrium is where D intersects S1 — at wage W1. Since the demand curve IS the VMP_L curve, the VMP of labor at that equilibrium equals W1. Options B and C describe disequilibrium, and Option A confuses S2's equilibrium wage.

Question 5

Refer to Figure 18-5. Which of the following would shift the labor supply curve from S2 to S1?

an increase in immigration a technological advance a change in workers' attitudes toward the work-leisure tradeoff in favor of leisure an increase in the price of the firm's output

Hint S1 is to the left of S2, meaning fewer workers at every wage. What would make workers want to supply less labor? Think about what happens when people prefer more leisure.

Explanation Answer: C. Moving from S2 to S1 is a leftward shift — fewer workers at every wage. If workers' attitudes shift in favor of leisure, they're less willing to work at any given wage, reducing labor supply. Option A (immigration) would shift supply right, not left. Option B (tech advance) shifts labor demand. Option D (output price) also shifts demand.

Question 6

Refer to Figure 18-5. If the relevant labor supply curve is S2 and the current wage is W1, then the:

quantity of labor supplied exceeds the quantity of labor demanded quantity of labor demanded exceeds the quantity of labor supplied labor market is in equilibrium VMP of labor to firms is W2

Hint With S2, equilibrium would be at W2 (lower wage). W1 is above W2. At a wage above the equilibrium, is there a surplus or a shortage? Draw a horizontal line at W1 and see where it hits D vs. S2.

Explanation Answer: A. With the supply curve at S2, the equilibrium wage is W2. Since W1 is above W2, the current wage is higher than the market-clearing level. At wage W1, draw a horizontal line: it hits S2 to the right (workers supply a lot of labor at this high wage) and hits D to the left (firms demand less at a high wage). So Qs > Qd — there is a surplus of labor (unemployment). The wage will tend to fall toward W2 to restore equilibrium.

Part 4

True or False

Question 7

Technological advances can cause the labor demand curve to shift.

Hint Labor demand is VMP_L = P × MPL. If technology raises the amount each worker produces, what happens to MPL — and therefore to the VMP curve?

Explanation True. Technological advances raise the marginal product of labor (MPL). Since VMP_L = P × MPL, a higher MPL means a higher VMP at every level of employment — the labor demand curve shifts right. This is the textbook story for why productivity growth has gone hand in hand with rising real wages over time.

Question 8

The labor supply curve reflects how workers' decisions about the labor-leisure tradeoff respond to changes in the opportunity cost of leisure.

Hint When the wage rises, what is the cost of an extra hour of leisure? How does that affect the worker's choice between working and not working?

Explanation True. The opportunity cost of leisure is the wage you forgo by not working. When wages rise, leisure becomes more expensive, so workers tend to substitute toward more work (the substitution effect). The labor supply curve captures exactly this tradeoff: at higher wages, more people are willing to work more hours, which is why labor supply generally slopes upward.

Question 9

The demand for labor is a direct demand — firms hire workers because they value the workers themselves, not because of the output the workers produce.

Hint Would an apple orchard hire pickers if nobody was buying apples? What does Mankiw call this type of demand?

Explanation False. The demand for labor is a derived demand, not a direct demand. Firms hire workers because those workers produce output that the firm sells. If there's no demand for the output, there's no demand for the labor that produces it. This is also why the output price P appears in the labor demand curve VMP_L = P × MPL.

Part 5

Closer — Earnings, Human Capital & Minimum Wage

Question 10

The ownership of human capital:

makes it the same as other forms of capital is easily transferable between people is not easily transferable can be sold on the open market like physical capital

Hint You can sell a machine to someone else. Can you sell your education, skills, or experience to another person?

Explanation Answer: C. Human capital — education, training, skills, experience — is embodied in the person who acquired it. Unlike physical capital (machines, buildings), you can't sell or transfer your human capital to someone else. This is a key difference that makes human capital unique among factors of production.

Question 11

Economists who attempt to explain the increasing earnings gap between skilled and unskilled workers in recent decades have focused on:

minimum wage laws and unions discrimination and compensating differentials international trade and technology efficiency wages and signaling

Hint Think about what has changed in the economy over the last few decades. Which two forces have increased the demand for skilled workers relative to unskilled workers?

Explanation Answer: C. The two big forces are international trade and technology. Trade with lower-wage countries has reduced demand for unskilled labor in the U.S. (those jobs move abroad), while technology (especially computers and automation) has increased the productivity and demand for skilled workers. Together, they widen the gap between what skilled and unskilled workers earn.

Figure 19-1 — Use for Questions 12, 13, and 14

Question 12

Refer to Figure 19-1. If the minimum wage in this market is $6, then:

employment is 12 million employment is 13 million employment is 15.5 million there is no effect because the minimum wage is below equilibrium

Hint A minimum wage is a price floor. At $6, the wage is above the equilibrium of $5. Employment is determined by the short side of the market — the smaller of quantity demanded and quantity supplied.

Explanation Answer: A. At a $6 minimum wage (above the $5 equilibrium), firms only want to hire 12 million workers (reading off the demand curve at $6), even though 15.5 million workers want to work at that wage (the supply curve at $6). Actual employment equals the quantity demanded: 12 million. The surplus of labor (15.5M − 12M = 3.5M) represents unemployment caused by the binding price floor.

Question 13

Refer to Figure 19-1. Suppose the local labor market was in equilibrium but then the largest local employer changed compensation to $6 per hour to reduce worker turnover and increase productivity. This is an example of:

an efficiency wage a compensating differential a minimum wage a union wage premium

Hint The employer is voluntarily paying above the market wage. Why? To reduce turnover and boost productivity. What does Mankiw call a wage deliberately set above equilibrium to improve worker quality or effort?

Explanation Answer: A. An efficiency wage is a wage deliberately set above the equilibrium level to reduce turnover, attract better workers, and increase effort and productivity. The key feature is that the firm chooses to pay above market — unlike a minimum wage (imposed by law) or a union premium (negotiated by a union). Compensating differentials relate to job characteristics like danger or unpleasantness, not productivity incentives.

Question 14

Refer to Figure 19-1. What is the change in employment of having the minimum wage at $6 instead of the equilibrium wage of $5?

3.5 million jobs are lost no jobs are lost one million jobs are lost two million jobs are gained

Hint Employment at equilibrium ($5) is 13 million. Employment with the $6 minimum wage equals quantity demanded at $6, which is 12 million. What's the difference?

Explanation Answer: C. At the $5 equilibrium, employment is 13 million. With a $6 minimum wage, employment falls to 12 million (the quantity demanded at $6). The change is 13M − 12M = 1 million jobs lost. Note: 3.5 million is the size of the surplus (15.5M who want to work minus 12M hired), but the actual reduction from equilibrium employment is only 1 million.

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