Short-run labour demand

Short-run labour demand is the number of workers a firm hires at each wage when its capital is fixed. A competitive firm hires until the value of the marginal product of labour equals the wage, $VMP_E = p \times MP_E = w$, on the falling part of $VMP_E$ and only if revenue per worker covers the wage. Diminishing returns make the resulting demand curve slope down.

Labour demand is a derived demand

Firms hire workers because consumers want the goods and services those workers produce. The demand for labour is therefore a derived demand: it comes from the demand for the firm’s output. If households buy fewer new houses, builders need fewer framers and electricians even though nothing about those workers has changed.

This separates labour demand from the other curves in the course. Consumer demand comes from utility, and labour supply comes from a choice between consumption and leisure. Labour demand comes from profit: what workers produce and what that output sells for.

The production function

The firm’s technology is its production function,

$$q = f(E, K),$$

the most output $q$ it can produce with $E$ workers (or employee-hours) and $K$ units of capital. Workers are assumed identical, so who is hired does not matter.

The marginal product of labour is the extra output from one more worker with capital held constant:

$$MP_E = \frac{\Delta q}{\Delta E} \quad (K \text{ fixed}).$$

The marginal product of capital, $MP_K = \Delta q / \Delta K$, is defined the same way with labour fixed. Both are assumed non-negative. The average product of labour is output per worker, $AP_E = q / E$.

With a few workers, one more lets the team divide tasks, so $MP_E$ rises: these are gains from specialization. With equipment fixed, additional workers eventually crowd each other, and each adds less than the one before. That is the law of diminishing returns: adding more of one input to a fixed amount of another eventually lowers the variable input’s marginal product.

Workers hired (E)

6.0

Output (q)

28.8

Marginal product (MP_E)

8.40

Average product (AP_E)

4.80

Open in Graph Lab

Figure 1. Total product (TP) with capital fixed. The tangent's slope is $MP_E$ and the slope of the ray from the origin is $AP_E$. The tangent flattens after 10 workers as diminishing returns set in, and at 15 workers the tangent and the ray coincide, so $MP_E = AP_E$.

Workers hired (E)

12.0

Marginal product (MP_E)

9.60

Average product (AP_E)

7.20

MP_E minus AP_E

2.40

Open in Graph Lab

Figure 2. $MP_E$ peaks at 10 workers and reaches zero at 20, where total product peaks. $AP_E$ rises while $MP_E$ is above it and peaks where the two curves cross, at 15 workers.

The link between the two curves is arithmetic. A worker who produces more than the current average raises the average; one who produces less lowers it. So $AP_E$ rises while $MP_E > AP_E$, falls while $MP_E < AP_E$, and peaks where they cross.

Key idea

Diminishing returns comes from a fixed input. It does not mean later hires are less able (workers are identical), and it does not mean total output falls: output rises as long as $MP_E$ is positive.

Profit maximization with capital fixed

The firm maximizes profit,

$$\pi = pq - wE - rK = p\,f(E, K) - wE - rK,$$

where $p$ is the output price, $w$ the wage and $r$ the price of a unit of capital. Total revenue is $pq$; total cost is the wage bill $wE$ plus the capital bill $rK$. The firm is a price taker in every market: it cannot move $p$, and it can hire any number of workers at $w$.

In the short run capital is fixed at $K_0$, so $rK_0$ is a fixed cost. Output changes only through $E$, and only the wage bill responds. Two dollar measures drive the decision:

• The value of the marginal product, $VMP_E = p \times MP_E$: extra revenue from one more worker.
• The value of the average product, $VAP_E = p \times AP_E$: revenue per worker.

The hiring rule

Hire another worker whenever the revenue they add is at least the wage, and stop where the two are equal:

$$VMP_E = p \times MP_E = w.$$

This is the first-order condition of $\max_E \; p\,f(E, K_0) - wE - rK_0$. Two side conditions complete the rule:

1. $VMP_E$ must be falling at the chosen point. Where $VMP_E$ is still rising, the next worker adds more than this one, so equality there is a profit minimum.
2. $VAP_E$ must be at least $w$ (equivalently, $VAP_E \ge VMP_E$ at the optimum). If revenue per worker is below the wage, every worker loses money on average and the firm hires no one in the short run. The capital bill is paid either way, so it does not enter this choice.

Wage (w)

$32

Output price (p)

$5.00

Workers hired (E^*)

16.0

VAP_E where VMP_E = w on the falling part

$37.33

Revenue minus wage bill, per hour

$85.33

Open in Graph Lab

Figure 3. The hiring decision at an output price of $5. At a $32 wage the wage line crosses $VMP_E$ at 4 and 16 workers; the firm hires 16, where $VMP_E$ is falling. The shaded rectangle is revenue minus the wage bill. At any wage above $37.50, the peak of $VAP_E$, the firm hires no one.

The stopping rule

The same answer follows from thinking about output. One more worker costs $w$ and adds $MP_E$ units, so the marginal cost of a unit of output is

$$MC = \frac{w}{MP_E}.$$

A competitive firm expands output until $MC = p$, since price is its marginal revenue. Setting $w / MP_E = p$ and multiplying by $MP_E$ gives $w = p \times MP_E = VMP_E$, the hiring rule again. Because labour is the only variable input, a rule for when to stop producing is a rule for when to stop hiring: choosing output $q^*$ fixes employment $E^*$. This is the stopping rule. Diminishing returns make it work, since a falling $MP_E$ makes $w / MP_E$ rise until it meets the price.

Wage (w)

$32

Output price (p)

$5.00

Output (q^*)

119.5

Workers needed (E^*)

16.0

Minimum AVC

$4.27

Open in Graph Lab

Figure 4. The stopping rule for the firm in Figure 3. Marginal cost $w/MP_E$ falls while $MP_E$ rises, then rises with diminishing returns; at a $32 wage and a $5 price the firm produces 119.5 units, where the price line meets the rising part of marginal cost, using the same 16 workers. The price must also be at least the minimum of average variable cost $w/AP_E$, here $4.27, which is the output version of $VAP_E \ge w$.

The short-run demand curve

The short-run labour demand curve gives employment at each wage with capital fixed. Each wage selects one point on the falling part of $VMP_E$ (Figure 3), so the demand curve is the downward-sloping part of $VMP_E$ below the peak of $VAP_E$. Above that peak the firm hires no one.

The curve slopes down because of diminishing returns: a lower wage makes it worth hiring workers with a lower marginal product, and with capital fixed, extra workers have a lower marginal product.

• A higher wage moves the firm along its demand curve. In Figure 3, a rise in the wage from $32 to $36 cuts employment from 16 to about 15.3 workers.
• A higher output price shifts the curve. Because $VMP_E = p \times MP_E$, a higher $p$ raises $VMP_E$ at every $E$. At a $32 wage, a rise in the price from $5 to $6 raises employment from 16 to about 16.8.

Anything that raises $MP_E$ at every level of employment, such as better equipment, also shifts the curve up.

On the exam

Draw $VMP_E$ rising then falling, with $VAP_E$ peaking where $VMP_E$ cuts it from above, and a horizontal wage line. Mark $E^*$ on the falling part of $VMP_E$, label the stretch of $VMP_E$ below the peak of $VAP_E$ as the demand curve, and state both side conditions.

Firm and industry demand

Industry demand starts as the horizontal sum of firm demands: at each wage, add up what every firm would hire. That sum holds the output price fixed, and the price does not stay fixed. When the wage falls, every firm hires and produces more at once, industry output rises, the output price falls, and each firm’s $VMP_E = p \times MP_E$ shifts down, taking back part of the extra hiring.

The industry demand curve is therefore steeper (less elastic) than the sum of firm demand curves at a constant price. The difference is the price feedback: one small firm cannot move its output price, but a whole industry can.

Market wage (w)

$14.00

Employment at B if the price stayed at p₀ (thousands)

204.1

Industry employment at C (thousands)

142.9

Change in the output price

−16.3%

Open in Graph Lab

Figure 5. A wage cut from $20 to $14 for a whole industry. At a constant output price $p_0$, firms would move from A to B along $D(p_0)$. Because all firms expand, the price falls to $p_1$, each firm's demand shifts down to $D(p_1)$, and the industry ends at C. The industry demand curve through A and C is steeper than $D(p_0)$.

The feedback requires the industry’s own output to move its price. Producers selling into a much larger world market at a price they cannot influence get little of it, so their industry demand is close to the simple sum.

The short-run elasticity of labour demand, $\delta_{SR} = \%\Delta E_{SR} / \%\Delta w$, measures how responsive employment is. It is negative; demand is elastic when $\delta_{SR}$ is less than $-1$ and inelastic when it lies between $-1$ and $0$. Estimates are in elasticity of labour demand.

Worked example

A bakery has a fixed oven. Bread sells for $4 a unit and the wage is $30 an hour.

Bakers ($E$)	Output ($q$)	$MP_E$	$AP_E$	$VMP_E$	$VAP_E$
0	0
1	10	10	10.0	$40	$40.00
2	30	20	15.0	$80	$60.00
3	54	24	18.0	$96	$72.00
4	74	20	18.5	$80	$74.00
5	88	14	17.6	$56	$70.40
6	98	10	16.3	$40	$65.33
7	104	6	14.9	$24	$59.43
8	106	2	13.3	$8	$53.00

1. Marginal products. Each $MP_E$ is the change in output from the row above: the third baker adds $54 - 30 = 24$ units. $MP_E$ rises for three bakers (specialization), then falls (diminishing returns).

2. Dollar values. Multiply by the price: the sixth baker’s $VMP_E$ is $4 \times 10 = 40$ dollars, and with six bakers $VAP_E$ is $4 \times 98 / 6 = 65.33$ dollars.

3. Hiring rule. On the falling part of $VMP_E$, the sixth baker adds $40, more than the $30 wage; the seventh adds $24, less. Hire 6. The first baker’s $VMP_E$ of $40 also exceeds the wage, but $VMP_E$ is still rising there, so stopping at one would be wrong.

4. Revenue per worker. At $E = 6$, $VAP_E$ is $65.33, above the $30 wage. Revenue is $4 \times 98 = 392$ dollars an hour against a wage bill of $30 \times 6 = 180$ dollars. Revenue minus the wage bill is $212, compared with $202 at 5 bakers and $206 at 7.

5. Stopping rule. The sixth baker produces at a marginal cost of $w / MP_E = 30 / 10 = 3$ dollars a unit, below the $4 price; the seventh would cost $30 / 6 = 5$ dollars, above it. Stop at 6.

6. Demand curve. At a $20 wage the seventh baker ($24) is worth hiring and the eighth ($8) is not, so employment rises to 7: a movement along the curve. With the wage at $30 and bread at $6, the seventh baker’s $VMP_E$ is $6 \times 6 = 36$ dollars, so the bakery again hires 7: a shift of the curve.

Exercise: the same bakery at a wage of $78

On the falling part of $VMP_E$, the fourth baker adds $80 and the fifth $56, so the hiring rule alone says 4. But with four bakers $VAP_E$ is $74, below the $78 wage: revenue of $4 \times 74 = 296$ dollars does not cover a wage bill of $4 \times 78 = 312$ dollars. The highest $VAP_E$ in the table is $74, so no level of employment covers the wage. The bakery hires no one.

Evidence: an output price collapse

A fall in the output price lowers $VMP_E$ at every level of employment, so hiring should fall even though workers are unchanged. The monthly average price of West Texas Intermediate crude, a benchmark for North American oil, fell from $105.79 a barrel in June 2014 to $30.32 in February 2016, a drop of about 70 percent.

Crude oil price: West Texas Intermediate

Monthly average spot price, Cushing, Oklahoma

Figure 6. Monthly average price of West Texas Intermediate crude oil, 2010 to 2019. A lower output price shifts every producer's $VMP_E = p \times MP_E$ curve down.Source: FRED, Crude Oil Prices: West Texas Intermediate (WTI) - Cushing, Oklahoma (MCOILWTICO), U.S. Energy Information Administration. Retrieved 2026-10-23.

Show the data and notes

Monthly averages of daily spot prices, nominal dollars (not adjusted for inflation).

Period	WTI
Jan 2010	$78.33
Feb 2010	$76.39
Mar 2010	$81.20
Apr 2010	$84.29
May 2010	$73.74
Jun 2010	$75.34
Jul 2010	$76.32
Aug 2010	$76.60
Sep 2010	$75.24
Oct 2010	$81.89
Nov 2010	$84.25
Dec 2010	$89.15
Jan 2011	$89.17
Feb 2011	$88.58
Mar 2011	$102.86
Apr 2011	$109.53
May 2011	$100.90
Jun 2011	$96.26
Jul 2011	$97.30
Aug 2011	$86.33
Sep 2011	$85.52
Oct 2011	$86.32
Nov 2011	$97.16
Dec 2011	$98.56
Jan 2012	$100.27
Feb 2012	$102.20
Mar 2012	$106.16
Apr 2012	$103.32
May 2012	$94.66
Jun 2012	$82.30
Jul 2012	$87.90
Aug 2012	$94.13
Sep 2012	$94.51
Oct 2012	$89.49
Nov 2012	$86.53
Dec 2012	$87.86
Jan 2013	$94.76
Feb 2013	$95.31
Mar 2013	$92.94
Apr 2013	$92.02
May 2013	$94.51
Jun 2013	$95.77
Jul 2013	$104.67
Aug 2013	$106.57
Sep 2013	$106.29
Oct 2013	$100.54
Nov 2013	$93.86
Dec 2013	$97.63
Jan 2014	$94.62
Feb 2014	$100.82
Mar 2014	$100.80
Apr 2014	$102.07
May 2014	$102.18
Jun 2014	$105.79
Jul 2014	$103.59
Aug 2014	$96.54
Sep 2014	$93.21
Oct 2014	$84.40
Nov 2014	$75.79
Dec 2014	$59.29
Jan 2015	$47.22
Feb 2015	$50.58
Mar 2015	$47.82
Apr 2015	$54.45
May 2015	$59.27
Jun 2015	$59.82
Jul 2015	$50.90
Aug 2015	$42.87
Sep 2015	$45.48
Oct 2015	$46.22
Nov 2015	$42.44
Dec 2015	$37.19
Jan 2016	$31.68
Feb 2016	$30.32
Mar 2016	$37.55
Apr 2016	$40.75
May 2016	$46.71
Jun 2016	$48.76
Jul 2016	$44.65
Aug 2016	$44.72
Sep 2016	$45.18
Oct 2016	$49.78
Nov 2016	$45.66
Dec 2016	$51.97
Jan 2017	$52.50
Feb 2017	$53.47
Mar 2017	$49.33
Apr 2017	$51.06
May 2017	$48.48
Jun 2017	$45.18
Jul 2017	$46.63
Aug 2017	$48.04
Sep 2017	$49.82
Oct 2017	$51.58
Nov 2017	$56.64
Dec 2017	$57.88
Jan 2018	$63.70
Feb 2018	$62.23
Mar 2018	$62.73
Apr 2018	$66.25
May 2018	$69.98
Jun 2018	$67.87
Jul 2018	$70.98
Aug 2018	$68.06
Sep 2018	$70.23
Oct 2018	$70.75
Nov 2018	$56.96
Dec 2018	$49.52
Jan 2019	$51.38
Feb 2019	$54.95
Mar 2019	$58.15
Apr 2019	$63.86
May 2019	$60.83
Jun 2019	$54.66
Jul 2019	$57.35
Aug 2019	$54.81
Sep 2019	$56.95
Oct 2019	$53.96
Nov 2019	$57.03
Dec 2019	$59.88

Statistics Canada reports that employment in mining, quarrying, and oil and gas extraction fell by 25,400 jobs, or 19.4 percent, in the 12 months to January 2016, and that payroll employment across Alberta fell by 105,500 (5.1 percent) between January 2015 and June 2016 as the province adjusted to lower oil prices.¹ Part of that decline came over a horizon long enough for firms to change their capital as well, which the long-run model covers. The direction is the one the hiring rule predicts.

Common mistakes

Traps on exams

• Stopping on the rising part of $VMP_E$. The wage line often crosses $VMP_E$ twice. The first crossing is a profit minimum.
• Skipping the $VAP_E$ check. If revenue per worker is below the wage at the crossing, the firm hires no one. Capital costs are fixed in the short run and do not enter the decision.
• Misreading diminishing returns. It comes from the fixed input, not from later hires being worse, and total output still rises while $MP_E$ is positive.
• Treating a price change like a wage change. A wage change is a movement along the demand curve; an output price change shifts it.
• Summing firm demands and stopping there. Industry demand is steeper because the output price falls when all firms expand together.

Check yourself

1.

A competitive firm with fixed capital chooses how many workers to hire. Which conditions hold at its profit-maximizing employment when it hires a positive number of workers? Select all that apply.

Select all that apply.

1$VMP_E = w$2$VMP_E$ is falling as employment rises3$VAP_E \ge w$4$MP_E = AP_E$5$VMP_E$ is at its maximum

Check answer

2.

At a firm's current wage, the wage line crosses the falling part of its $VMP_E$ curve at 12 workers. At 12 workers the value of the average product is below the wage. What should the firm do in the short run?

1Hire no workers, because revenue per worker does not cover the wage2Hire 12 workers, because $VMP_E = w$ there3Hire more than 12 workers to spread its fixed capital costs4Hire fewer than 12 workers, where $VMP_E$ is above the wage

Check answer

3.

Compared with the horizontal sum of firms' labour demand curves drawn at a constant output price, the industry demand curve for labour is:

1Steeper, because when all firms expand together the output price falls and each firm's VMP falls2Flatter, because adding many firms together makes demand more responsive3Identical, because the industry is just the sum of its firms4Steeper, because firms collude to hold down employment

Check answer

Practise more questions on this topic

Sources

1. Fields, A. and Gellatly, G. (2016). The impact of lower earnings in Alberta on earnings growth at the national level. Economic Insights, Statistics Canada catalogue no. 11-626-X, November 18, 2016. ↩