Source: Homework 6, Prof. McLeod, Penn State
Difficulty: Intermediate
Prerequisites: Understanding of production functions, the concept of diminishing marginal returns, and basic profit-maximisation logic.
Big picture: This topic covers how a competitive firm decides how many workers to hire when it can vary labour but not capital (the short run). The key insight is the value of the marginal product (VMPL) rule: keep hiring as long as the extra revenue from one more worker exceeds the wage. This is the demand side of the labour market, and it ties production theory directly to factor markets.
A profit-maximising firm hires workers until the value of the marginal product of labour (VMPL) equals the wage. VMPL is the extra output from one more worker, multiplied by the output price. When the wage is high, fewer workers are worth hiring; when the wage is low, more are.
Short-run production function
The relationship between the quantity of a variable input (labour) and the quantity of output, holding all other inputs (capital) fixed.
Think of it as: given the factory you already have, how much more stuff comes out as you add workers?
Marginal product of labour (MPL)
The additional output produced by hiring one more worker. Calculated as the change in output divided by the change in labour (here, one unit at a time from the table).
In simple terms, this is how much extra the next worker makes.
Diminishing marginal returns
The principle that each additional worker eventually adds less output than the one before, holding capital fixed. In the table, MPL starts at 25 and falls steadily.
Value of the marginal product of labour (VMPL)
MPL multiplied by the price of output. This is the dollar value of the extra output one more worker produces. The firm hires until VMPL = wage.
In simple terms, this is how much extra revenue the next worker brings in.
Hiring rule
Hire workers as long as VMPL is greater than or equal to the wage (w). Stop when the next worker's VMPL falls below w.
The production function gives output (q) for each level of labour (L = 0 to 10).
MPL = change in q when L increases by 1.
L=1: MPL = 25 – 0 = 25
L=2: MPL = 60 – 25 = 35
L=3: MPL = 100 – 60 = 40
L=4: MPL = 130 – 100 = 30
L=5: MPL = 150 – 130 = 20
L=6: MPL = 165 – 150 = 15
L=7: MPL = 170 – 165 = 5
L=8: MPL = 174 – 170 = 4
L=9: MPL = 177 – 174 = 3
L=10: MPL = 179 – 177 = 2
With output price Px = $6, VMPL = MPL x 6.
L=1: VMPL = 150
L=2: VMPL = 210
L=3: VMPL = 240
L=4: VMPL = 180
L=5: VMPL = 120
L=6: VMPL = 90
L=7: VMPL = 30
L=8: VMPL = 24
L=9: VMPL = 18
L=10: VMPL = 12
Hire as long as VMPL >= w (wage).
Wage = $110: Hire workers whose VMPL >= 110. Workers 1 through 5 qualify (VMPL of 150, 210, 240, 180, 120). Worker 6's VMPL is 90, which is below 110. Hire 5 workers.
Wage = $25: Hire workers whose VMPL >= 25. Workers 1 through 7 qualify (the 7th has VMPL = 30 >= 25). Worker 8's VMPL is 24, which is below 25. Hire 7 workers.
Wage = $15: Hire workers whose VMPL >= 15. Workers 1 through 9 qualify (the 9th has VMPL = 18 >= 15). Worker 10's VMPL is 12, which is below 15. Hire 9 workers.
MPL rises initially (from 25 to 35 to 40), then falls steadily from the 4th worker onwards. This is the classic pattern: early workers complement each other, then crowding sets in.
This is exactly the calculation a small business owner makes when deciding whether to bring on another employee. If a cafe can sell each coffee for $5 and a new barista would make 20 extra coffees per shift, that worker's VMPL is $100. If the wage is $90, hiring makes sense. If the wage is $120, it does not.
The VMPL framework also explains why wages differ across industries and skill levels: workers whose marginal product (in dollar terms) is higher command higher pay in competitive labour markets.
Students often confuse MPL with total output. MPL is the change in output from one more worker, not the output level.
A common mistake is forgetting to multiply MPL by the output price. The hiring decision uses VMPL (dollars), not MPL (units of output).
Students sometimes hire until VMPL exactly equals the wage. The rule is to hire whenever VMPL >= w, so a worker whose VMPL equals the wage should still be hired (the firm breaks even on that worker).
Some students assume MPL always declines. In this table, MPL rises for the first three workers before declining. Diminishing returns set in eventually, not necessarily from the first worker.
The standard exam format is a production table plus an output price. You will be asked to compute MPL, then VMPL, then determine hiring at a given wage. Practice the mechanical steps so they are fast.
Be ready to explain why the firm stops hiring where it does. The answer is always: the next worker's VMPL is less than the wage.
Know that the VMPL schedule is the firm's labour demand curve. A higher wage means fewer workers hired, which is a movement along this curve.
If the output price changes, the entire VMPL schedule shifts, which shifts labour demand.
True or false: A firm should hire a worker whose VMPL exactly equals the wage. (True, the firm breaks even on that worker.)
Fill in the blank: VMPL = MPL x ______. (Price of output.)
True or false: MPL always declines as you add more workers. (False, it can rise initially before diminishing returns set in.)
Fill in the blank: At a wage of $25, the firm hires ______ workers. (7.)
Q: What is the marginal product of the 5th worker?
A: MPL = 150 – 130 = 20 units.
Q: What is the VMPL of the 5th worker when Px = $6?
A: VMPL = 20 x $6 = $120.
Q: At a wage of $110, how many workers should the firm hire?
A: 5 workers. The 5th worker's VMPL is $120 (>= $110), and the 6th worker's VMPL is $90 (< $110).
Q: At a wage of $25, how many workers should the firm hire?
A: 7 workers. The 7th worker's VMPL is $30 (>= $25), and the 8th worker's VMPL is $24 (< $25).
Q: At a wage of $15, how many workers should the firm hire?
A: 9 workers. The 9th worker's VMPL is $18 (>= $15), and the 10th worker's VMPL is $12 (< $15).
Q: If the output price rose from $6 to $10, what would happen to the number of workers hired at a wage of $110?
A: Every VMPL would increase (multiplied by 10 instead of 6). More workers would have VMPL >= $110, so the firm would hire more workers. For example, the 6th worker's VMPL would become 15 x 10 = $150, and the 7th would become 5 x 10 = $50. The firm would now hire 6 workers.
This connects to the theory of the firm and cost curves. The production function here is the foundation for deriving short-run cost curves (total cost, marginal cost, average cost). If you understand diminishing MPL, you understand why marginal cost eventually rises.
The VMPL = w condition is the labour-market counterpart of the output-market condition P = MC. Both are profit-maximising rules for a price-taking firm.
This also links to labour economics and wage determination. In a competitive labour market, the equilibrium wage equals VMPL, which is why more productive workers earn more.
Short-run production function, marginal product of labour, MPL, value of marginal product, VMPL, MRP, labour demand, hiring rule, diminishing marginal returns, diminishing returns to labour, factor demand, input demand, profit maximisation, competitive firm hiring, wage and employment, ECON 102, intermediate microeconomics, Penn State, McLeod