Ch 18 Worksheet — Factor Markets

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

MPL is diminishing (almost immediately): the first worker adds 7 units, but each additional worker through the 4th adds only 6. From the 5th worker on, MPL falls further (3, 1, 0).

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.

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).

Question 1

If firms are competitive and profit-maximizing, the demand curve for labor is determined by:

the opportunity cost of workers' time the value of the marginal product of labor the value of the marginal product of capital the ratio of the marginal product of labor to the marginal product of capital

Hint Think about what the firm is actually willing to pay for one more worker. If a new worker adds $X of revenue, the firm is willing to pay up to $X in wages for her. What did we call that $X?

Explanation Answer: B. A competitive firm hires workers up to the point where the wage equals VMP_L. That means the firm's labor demand curve IS the VMP_L curve: at each wage W, it's willing to hire exactly the L that makes VMP_L = W. Because MPL is diminishing, this curve slopes downward. Option A describes labor supply, not demand.

Question 2

A bakery operating in competitive markets sells its output for $20 per cake and pays workers $10 per hour. To maximize profit, it should hire workers until the marginal product of labor is:

1/2 cake per hour 2 cakes per hour 10 cakes per hour 15 cakes per hour

Hint Profit-max rule: VMP_L = W, i.e., P × MPL = W. Solve for MPL.

Explanation Answer: A. Set P × MPL = W → 20 × MPL = 10 → MPL = 10/20 = 1/2 cake per hour. Common trap: option B (2 cakes/hour) confuses the ratio. The logic: each worker earning $10/hour must produce at least $10 worth of cakes (half a cake, since a cake sells for $20).

Question 3

Which of the following events will shift the labor supply curve to the right?

More parents leave the workforce to spend time raising children Great new video games are introduced, enhancing the value of leisure Relaxed immigration laws allow more workers to come in from abroad Government retirement benefits are increased, encouraging earlier retirement

Hint "Right" for labor supply means more workers are willing to work at every wage. Which of these events brings additional people into the labor market?

Explanation Answer: C. Immigration adds workers to the labor market at every wage — that's a rightward shift in labor supply. The new equilibrium has a lower wage and higher employment. Options A, B, and D all reduce the number of people willing to work at each wage, shifting labor supply to the left.

Question 4

A technological advance that increases the marginal product of labor shifts the labor-_________ curve to the _________.

demand ; left demand ; right supply ; left supply ; right

Hint Labor demand is VMP_L = P × MPL. If MPL rises, what happens to VMP_L? Which curve is that?

Explanation Answer: B — demand ; right. The labor demand curve is VMP_L = P × MPL. A technological advance raises MPL, so VMP_L rises at every level of employment — the demand curve shifts right. The new equilibrium has both a higher wage and higher employment. This is the textbook story for why rising productivity has gone hand in hand with rising real wages over the long run.

Question 5

In a competitive labor market equilibrium, each worker earns a wage equal to the value of the marginal product of labor.

Hint In equilibrium, firms hire up to the point where VMP_L = W. Rearrange that: what does each worker's wage equal?

Explanation True. Every competitive firm keeps hiring as long as VMP_L > W, and stops when VMP_L = W. So at any equilibrium wage, the last worker hired at each firm has VMP_L exactly equal to W. That's the central Ch 18 result: in a competitive factor market, each worker earns the value of her marginal contribution, and by symmetric logic each factor of production earns its VMP.

Question 6

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.

Question 7

An increase in the stock of capital — more or better tools per worker — tends to shift the labor demand curve to the right.

Hint A worker with better tools produces more output. What does that do to MPL? And therefore what does it do to VMP_L, which is the labor demand curve?

Explanation True. With more or better capital, each worker produces more output: MPL rises. Since labor demand is VMP_L = P × MPL, the labor demand curve shifts right. The new equilibrium features higher wages and more employment. (And the reverse is the Ch 18 hurricane example: destroying capital reduces MPL, shifting labor demand left and lowering wages.)

Question g

What must the market price of a box of pencils be, in dollars?

Price per box ($)

Hint At profit-max, the last worker's VMP_L = W. So P × MPL = W. You know W and MPL — solve for P.

Explanation At profit-max hiring: VMP_L = W, i.e., P × MPL = W.
Plug in: P × 30 = 150 → P = 150 / 30 = $5 per box.

This is the key Ch 18 reasoning in reverse: instead of going from P and MPL to VMP and labor choice, we know the wage and MPL and back out the output price. Every factor market problem reduces to this one identity — wage equals value of marginal product.

Practice Worksheet

Short Answer — Production Schedule & Labor Demand

Short Answer — Capital Demand at Smiling Cow Dairy

Multiple Choice

True or False

Closing Short Answer — Leadbelly Co.