Why can hiring one more worker raise the wage bill by more than their wage?
Your task: Plot wage and discrete marginal hiring cost separately, mark the four-worker decision and report wage 6. In the explanation’s wage-floor workbench, compare floors 0, 7, 10 and 11; preserve tied choices and explain the fifth worker’s extra cost. Compare the wage-taking contract and record the model limits.
Read counts with their units. A percentage divides a part by its named whole and multiplies by 100. A model assumption is not an observed fact.
Monopsony
A market setting where a buyer has influence over the price paid for an input.
Assumption
A condition accepted for a model or argument; it needs justification before applying the result elsewhere.
01 · TRY
Why can hiring one more worker raise the wage bill by more than their wage?
Plot wage and discrete marginal hiring cost separately, mark the four-worker decision and report wage 6. In the explanation’s wage-floor workbench, compare floors 0, 7, 10 and 11; preserve tied choices and explain the fifth worker’s extra cost. Compare the wage-taking contract and record the model limits.
Plot wage and discrete marginal hiring cost separately, mark the four-worker decision and report wage 6. In the explanation’s wage-floor workbench, compare floors 0, 7, 10 and 11; preserve tied choices and explain the fifth worker’s extra cost. Compare the wage-taking contract and record the model limits. Use the supplied records and show one intermediate step.
Make a provisional claim
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Inspect evidence and test an explanation
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04 · UNDERSTAND
Connect evidence and mechanism
Distinguish the wage from marginal hiring cost
Question and record
Why can hiring one more worker raise the wage bill by more than their wage?
A fictional employer must pay the same wage to all workers. To attract L workers it pays wage w=2+L. Each worker adds 10 credits of revenue. Consider L=0 through 5, and no other variable costs.
The quantities and graphs are teaching assumptions, not observed policy outcomes.
Mechanism and worked calculation
A monopsonistic employer faces an upward-sloping labor supply relationship. Hiring more at a uniform wage raises pay on earlier workers as well, so marginal factor cost can exceed wage. Minimum-wage effects depend on the relevant range and market structure; neither universal job loss nor universal job gain follows without a model.
Total wage bills for L=0–5 are 0,3,8,15,24,35. Incremental costs are3,5,7,9,11. Revenue gains are10 each, so four workers maximize profit among the listed choices:40−24=16 versus50−35=15.
Experiment: can a higher minimum wage increase hiring?
Test a wage floor against the whole wage bill
Fictional one-period experiment: choose 0–5 workers, pay everyone the same wage, and earn 10 credits per worker. There are no other costs. Before changing the floor, predict employment and the fifth worker’s extra cost. A floor is a minimum permitted wage, not a requirement to hire.
For any positive worker count L, wage = max(floor, 2 + L): use whichever number is larger. Wage bill = L × wage. Incremental cost is the change from the preceding row, including raises for workers already hired.
All feasible hiring choices; credits for one common period
Workers
Wage per worker
Wage bill
Extra cost
Revenue
Profit
Maximum?
0
No workers
0
Not applicable
0
0
No
1
3
3
3
10
7
No
2
4
8
5
20
12
No
3
5
15
7
30
15
No
4
6
24
9
40
16
Yes
5
7
35
11
50
15
No
Maximum profit: 16 credits. Maximizing worker counts: 4.
Compare floors 0 and 7: explain why the fifth worker’s extra cost changes although their wage remains 7. Then try 10 and 11. Record the floor, wage bills, maximizing choices and your explanation in the source notebook in the Experiment stage. Controls are temporary, reset on reload, and are not included in saved notebook captures.
At a floor of 7, four workers cost 28 and five cost 35. The fifth worker now adds 7, not 11, to the bill: the first four already receive 7 under this floor. Profit rises from 12 at four workers to 15 at five. Relative to the original optimum, employment rises from four to five while maximum profit falls from 16 to 15. The employer optimizes under a new constraint; this is not a free gain for every participant.
The five-worker cap matters: choosing five proves an optimum among the allowed choices, not that an unrestricted firm would stop there. At a floor of 10, every allowed choice, including zero hires, earns zero profit. At 11, every positive employment choice loses money and zero hires uniquely maximizes profit. Do not silently pick the highest employment in a tie.
Compare the employer’s market position
A separate wage-taking employer can hire up to five workers at an externally fixed wage and also earns 10 credits per worker. At wage 6 its profits for zero through five workers are 0, 4, 8, 12, 16 and 20. At wage 7 they are 0, 3, 6, 9, 12 and 15. Both select five: there are no raises on earlier hires triggered by adding a worker. The employer faces a different supply contract, so these are not two predictions about the same labor market. Neither example supplies a complete competitive market equilibrium or an empirical employment estimate.
OpenStax’s labor-market explanation distinguishes wage-taking from an employer facing upward-sloping labor supply. Our numbers and cap are invented for this investigation. To evaluate an actual policy, investigate enforcement, hours, job quality, worker movement, product demand and a credible comparison group; a simulated profit maximum cannot establish those effects.
Transfer: change productivity, not just the label
Before opening the feedback, use paper or a source annotation. Keep wage = 2 + L, the five-worker cap and uniform pay, but reduce revenue to 8 credits per worker. Build profit rows for no binding floor and for a floor of 6. Predict both maximizing choices and explain the change in the fourth worker’s incremental cost. State one missing fact needed before using this result in a policy brief.
Check the lower-productivity hiring case
Without a binding floor, profits for zero through five workers are 0, 5, 8, 9, 8 and 5: three workers maximize profit. With a floor of 6, bills are 0, 6, 12, 18, 24 and 35, so profits are 0, 2, 4, 6, 8 and 5: four workers maximize profit. The fourth worker’s incremental cost falls from 9 to 6 because the first three already receive 6. This supports a conditional model explanation, not a measured effect. If you read this feedback first, label your response as guided practice.
Challenge the interpretation
A mistaken calculation hires until wage equals revenue product while ignoring raises paid to existing workers. In a wage-taking comparison, wage is fixed independently of the employer’s hiring, so marginal hiring cost equals wage.
Your evidence task
Plot wage and discrete marginal hiring cost separately, mark the four-worker decision and report wage 6. In the explanation’s wage-floor workbench, compare floors 0, 7, 10 and 11; preserve tied choices and explain the fifth worker’s extra cost. Compare the wage-taking contract and record the model limits.
Revise your answer to “Why can hiring one more worker raise the wage bill by more than their wage?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.
Vocabulary
Monopsony: A market setting where a buyer has influence over the price paid for an input.
Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.
Independent extension
Independently extend the investigation “Distinguish the wage from marginal hiring cost” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.
Worked graph: original no-floor example
This static graph shows the original no-floor example; it does not change with the experiment controls. Discrete points are not interpolated. Table labels provide a non-color way to identify every series. Axes begin at zero.
Series
Worker number
Credits per additional worker
Wage
1
3
Wage
2
4
Wage
3
5
Wage
4
6
Wage
5
7
Marginal hiring cost
1
3
Marginal hiring cost
2
5
Marginal hiring cost
3
7
Marginal hiring cost
4
9
Marginal hiring cost
5
11
Revenue product
1
10
Revenue product
2
10
Revenue product
3
10
Revenue product
4
10
Revenue product
5
10
Worked example
Total wage bills for L=0–5 are 0,3,8,15,24,35. Incremental costs are3,5,7,9,11. Revenue gains are10 each, so four workers maximize profit among the listed choices:40−24=16 versus50−35=15.
