How does a product price change alter a hiring calculation?
Your task: Plot the step schedule of marginal revenue product and wage. Shift only the output price and identify the changed hiring range and the break-even tie.
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.
Derived demand
Demand for an input arising from demand for the output it helps produce.
Assumption
A condition accepted for a model or argument; it needs justification before applying the result elsewhere.
01 · TRY
How does a product price change alter a hiring calculation?
Plot the step schedule of marginal revenue product and wage. Shift only the output price and identify the changed hiring range and the break-even tie.
Plot the step schedule of marginal revenue product and wage. Shift only the output price and identify the changed hiring range and the break-even tie. Use the supplied records and show one intermediate step.
Make a provisional claim
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04 · UNDERSTAND
Connect evidence and mechanism
Derive labor demand from additional output
Question and evidence
How does a product price change alter a hiring calculation?
A fictional price-taking workshop’s first three workers add 5, 4 and 2 units daily. The product price is 3 credits per unit and the wage is 10 per worker. Other costs do not change with these hires.
This packet uses fictional data to isolate a mechanism. It is not an empirical estimate.
Mechanism and worked example
Marginal revenue product links additional input to additional revenue. In this simple competitive case it equals marginal physical product times output price. Hiring depends on the product market as well as wages. A sole or influential employer can face a different marginal hiring-cost relationship from a wage-taking firm.
At price 3, revenue products are 15,12,6. The first two exceed wage 10; the third does not. At price 2 they become 10,8,4, so the first hire only breaks even and later hires reduce profit.
Complication
A lower product price of 2 credits leaves physical productivity unchanged but reduces the revenue from it. The model excludes training, team complementarities, employment protection, discrimination and monopsony power.
Your investigation
Plot the step schedule of marginal revenue product and wage. Shift only the output price and identify the changed hiring range and the break-even tie.
Revise your answer to “How does a product price change alter a hiring calculation?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.
Vocabulary
Derived demand: Demand for an input arising from demand for the output it helps produce.
Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.
Independent extension
Independently extend the investigation “Derive labor demand from additional output” 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.
Inspect the worked graph
Points are discrete observations or marginal-unit values; no continuous interpolation is asserted.
Series
Worker number
Revenue product / wage (credits)
Revenue product at price 3
1
15
Revenue product at price 3
2
12
Revenue product at price 3
3
6
Wage
1
10
Wage
2
10
Wage
3
10
Choose hours on a continuous input graph
A separate original fictional workshop uses L labor-hours per day, not a count of people. Output is Q=αL−L² units/day, where α is the selected technology coefficient. The coefficients carry the units needed for this stipulated relationship. L can vary continuously from 0 to 6; fixed equipment limits the day to six hours. Other inputs stay fixed. This does not interpolate the earlier three-worker schedule.
The workshop takes output price p (credits/output unit) and wage w (credits/labor-hour) as given. Its marginal product is α−2L output units per extra hour. Multiplying by p gives marginal revenue product MRP=p(α−2L), in credits per extra hour. Compare that with w. Market wages are not determined by this one workshop.
Predict: sketch MRP and wage, choose hours, and explain which line changes if the product price falls. Then test your choice; distinguish a movement caused by wage from a shift caused by price or technology.
Fixed hours 0–6 and marginal credits/hour 0–32. Solid navy MRP; dashed red wage. Circle marks trial MRP, square the maximizing choice's MRP.
Trial: L=3, Q=27, MRP=12 versus wage 8. Revenue 54 − wage bill 24 = net contribution 30 credits/day. This is before other fixed costs, not total economic profit.
Maximizing feasible hours: 4 (interior). Output=32, revenue=64, wage bill=32, net contribution=32. Your trial forgoes 2 credits/day compared with this choice.
For an interior choice solve p(α−2L)=w: L=(α−w/p)/2=4. Then respect 0≤L≤6. At zero, MRP can be below wage; at the capacity boundary it can exceed wage. Do not force equality outside the feasible set. Falling MRP makes the net-contribution curve strictly concave here, so the feasible maximum is unique—even when a boundary has marginal equality.
Numerical hourly schedule
L hours
Output
Marginal product
MRP
Revenue
Wage bill
Net
0
0
12
24
0
0
0
1
11
10
20
22
8
14
2
20
8
16
40
16
24
3
27
6
12
54
24
30
4
32
4
8
64
32
32
5
35
2
4
70
40
30
6
36
0
0
72
48
24
Rows sample a continuous model. A whole extra hour changes output by α−2L−1, while marginal product α−2L is the instantaneous slope. They are not interchangeable.
Transfer: price, technology and a binding limit
Explain why lower p changes revenue product without changing the physical production function. Then set α=16, p=2, w=4: is the intersection feasible? Justify the boundary using both the graph and foregone revenue net of wage. Finally set α=12, p=1, w=32 and explain why zero hours can maximize contribution. These are one-firm comparative cases, not a forecast of total employment.
Reasoning guide and optional algebra
Initially L=4, Q=32 and net=32. Lower price 1 gives L=2, Q=20 and net=4; keeping L=4 instead gives net=0. Changed technology 16 with price2/wage4 would intersect at L=7, outside capacity: choose L=6, Q=60, net=96; MRP8 exceeds wage4. At technology12/price1/wage32 choose zero; the computed intersection −10 is infeasible.
Completing the square gives pQ−wL = −p[L−(α−w/p)/2]² + p[(α−w/p)/2]². Thus the feasible point nearest the unconstrained peak maximizes contribution. The guided graph does not require calculus. Product-market power would require marginal revenue rather than p; wage-setting power would require marginal hiring cost rather than the fixed wage line. Production, pay and a person's intrinsic worth are distinct.
Method reference and scope
OpenStax Microeconomics 3e section 14.1 supplies context for marginal product, price-taking revenue product and wage-taking hiring. Equations and examples here are original, with no observed data or copied source prose. The technology change is stipulated, not evidence of an intervention's effect. Homogeneous hours, fixed complementary inputs and no adjustment costs are simplifying assumptions.
Copy the continuous hiring report
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Worked example
At price 3, revenue products are 15,12,6. The first two exceed wage 10; the third does not. At price 2 they become 10,8,4, so the first hire only breaks even and later hires reduce profit.
