Microeconomics guide, Unit 5, step 21 of 28Previous step Next step

Step 21: Factor Demand and the Least-Cost Rule

By the EconTutorials team. 10-minute read with 6 practice questions.

SyllabusAP Micro 5.1–5.4 · Cambridge 8.3

In 30 seconds

Demand for a factor (labour, capital, land) is derived demand: firms want workers because they want the goods those workers make.
Marginal revenue product $ MRP = MP \times MR $. A firm hires a factor up to the point where $ MRP = MRC $; the MRP curve is the firm’s demand curve for that factor.
Least-cost rule: $ \frac{MP_L}{w} = \frac{MP_K}{r} $. Profit-maximising rule: $ \frac{MRP_L}{MRC_L} = \frac{MRP_K}{MRC_K} = 1 $.
A monopsony faces an MRC curve above the labour supply curve, so it hires fewer workers at a lower wage than a competitive market.

Video lesson coming soonA short animated lesson for this step will appear here.

Derived demand: why firms hire

A bakery hires a baker because customers want bread. If bread sales collapse, it needs fewer bakers. Demand for the worker comes from demand for the product.

Key termDerived demand: the demand for a factor of production that comes from the demand for the good or service it helps to produce.

In factor markets firms are the buyers. The price of labour is the wage ($ w $); the price of renting capital is the rental rate ($ r $).

Marginal product, MRP and MRC

For each extra worker the firm asks: does this worker bring in more money than they cost?

Marginal product (MP): the extra output from one more worker. Because of diminishing marginal returns, MP eventually falls.
Marginal revenue product (MRP): the extra revenue from one more worker.
Marginal resource cost (MRC), also called marginal factor cost (MFC): the extra cost of hiring one more worker. In a competitive labour market the firm is a wage taker, so $ MRC = w $.

\[ MRP = MP \times MR \qquad \text{Hire until } MRP = MRC \]

If the firm sells its product in a perfectly competitive market, $ MR = P $, so $ MRP = MP \times P $. If it is a monopolist, MR falls as it sells more, so MRP falls even faster.

Worked example

For example, a café sells sandwiches in a competitive market at $5 each and can hire workers at $40 per day. How many workers should it hire?

Workers	Total product	MP	MRP ($ MP \times \$5 $)
1	12	12	$60
2	22	10	$50
3	30	8	$40
4	36	6	$30
5	40	4	$20

Step 1MP is the change in total product: e.g. third worker $ 30 – 22 = 8 $.

Step 2MRP $ = MP \times P $: third worker $ 8 \times 5 = \$40 $.

Step 3Compare with $ MRC = w = \$40 $. Workers 1–3 have $ MRP \geq 40 $; worker 4 adds $30 but costs $40.

The café hires 3 workers, where $ MRP = MRC = \$40 $. If the wage fell to $30 it would hire 4. So the MRP curve is the firm’s labour demand curve.

Common mistakeUsing total product instead of marginal product, or forgetting to multiply by MR. MRP is about the extra worker, measured in money, not in units of output.

What shifts factor demand and factor supply

A change in the wage moves the firm along its MRP curve. Anything that changes MRP at every wage shifts the whole curve:

Product price (demand for the product). If sandwiches sell for more, each worker’s MRP rises, so labour demand shifts right.
Productivity. Better training or technology that helps workers raises MP, so MRP and labour demand rise.
Prices of other inputs. If capital and labour are substitutes in production (self-checkout machines and cashiers), cheaper machines may reduce demand for cashiers (the substitution effect), although lower costs can also raise output and hiring (the output effect); the net result depends on which is stronger. If they are complements in production (lorries and drivers), cheaper lorries mean more lorries and more demand for drivers.

Labour supply to a market shifts with population and migration, wages in other occupations, non-wage features of the job (safety, hours, prestige), and entry requirements such as qualifications or licences.

DiagramMarket for labour: wage on the vertical axis, quantity of labour on the horizontal. Draw a downward-sloping demand curve labelled $ D_L = MRP $ and an upward-sloping supply curve $ S_L $. Equilibrium gives $ W^* $ and $ L^* $. A rise in the product’s price shifts $ D_L $ right: the wage and employment both rise. For a single competitive firm, draw a horizontal line at $ W^* $ labelled $ S = MRC $; the firm hires where it meets MRP.

The least-cost rule

Most firms use more than one input. A delivery business uses drivers (labour) and vans (capital). Compare the extra output you get from the last dollar spent on each input. If the last dollar on drivers adds more output than the last dollar on vans, you should shift spending towards drivers. You stop shifting when the last dollar gives the same output whichever input it buys.

\[ \frac{MP_L}{w} = \frac{MP_K}{r} \]

This is the least-cost rule: any given output is produced at the lowest possible cost when the marginal product per dollar is equal for all inputs.

Worked example

For example, the wage is $20 and the rental rate of a van is $50 per day. The last worker adds 30 deliveries and the last van adds 60. Is the firm minimising cost?

Step 1$ \frac{MP_L}{w} = \frac{30}{20} = 1.5 $ deliveries per dollar.

Step 2$ \frac{MP_K}{r} = \frac{60}{50} = 1.2 $ deliveries per dollar.

Step 3$ 1.5 > 1.2 $: moving $1 from vans to workers gains 1.5 deliveries and loses 1.2, a net gain of 0.3 at the same cost.

The firm should use more labour and less capital. As it does, $ MP_L $ falls and $ MP_K $ rises (diminishing returns) until the two ratios are equal.

Least cost is not the same as maximum profit

The least-cost rule tells you the cheapest mix, but not how much to produce. Profit is maximised when each input is hired up to the point where its MRP equals its MRC:

\[ \frac{MRP_L}{MRC_L} = \frac{MRP_K}{MRC_K} = 1 \]

In competitive factor markets, $ MRC_L = w $ and $ MRC_K = r $. A combination that satisfies this rule also satisfies the least-cost rule, but not the other way round.

Worked example

For example, a firm sells output at $2. The wage is $8 and the rental rate is $12. Find the least-cost combinations and the profit-maximising one.

Units	$ MP_L $	$ MP_L/w $	$ MRP_L $	$ MP_K $	$ MP_K/r $	$ MRP_K $
1	10	1.25	$20	15	1.25	$30
2	8	1.00	$16	12	1.00	$24
3	6	0.75	$12	9	0.75	$18
4	4	0.50	$8	6	0.50	$12

Step 1Least-cost combinations have equal ratios: (1L, 1K), (2L, 2K), (3L, 3K) and (4L, 4K) all qualify.

Step 2Profit maximum: labour where $ MRP_L = w $: $ 8 = 8 $ at 4 workers. Capital where $ MRP_K = r $: $ 12 = 12 $ at 4 units.

Step 3Check: $ \frac{MRP_L}{w} = \frac{8}{8} = 1 $ and $ \frac{MRP_K}{r} = \frac{12}{12} = 1 $.

Only 4 workers and 4 units of capital maximise profit. Notice that there $ \frac{MP_L}{w} = \frac{MP_K}{r} = 0.5 = \frac{1}{P} $.

Exam tipAP Micro questions on Unit 5 reward three moves: compute MP and MRP correctly from a table, state the hiring rule “hire until MRP = MRC” (not “MRP = MP” or “wage = price”), and, for two inputs, say which input to increase by comparing $ MP/\text{price} $. If both $ MRP/MRC $ ratios are above 1, say “hire more of both”. Cambridge 9708: link MRP to the labour demand curve.

Monopsony: one big employer

A monopsony is a market with a single buyer of a factor, such as one large employer in a small town. To hire one more worker, it must raise the wage, and (with no wage discrimination) it must pay the higher wage to everyone already employed. So the MRC of the extra worker is above the wage: the MRC curve lies above the labour supply curve.

Worked example

For example, a mine is the only employer in town.

Workers	Wage needed (supply)	Total labour cost	MRC	MRP
1	$10	$10	$10	$28
2	$12	$24	$14	$24
3	$14	$42	$18	$20
4	$16	$64	$22	$16

Step 1MRC = change in total cost: e.g. fourth worker $ 64 – 42 = 22 $.

Step 2Monopsony hires while $ MRP \geq MRC $: worker 3 ($ 20 \geq 18 $) yes; worker 4 ($ 16 < 22 $) no. So 3 workers, paid the supply wage of $14.

Step 3Competitive market: wage = MRP where supply meets demand, at 4 workers and $16.

The monopsony hires fewer workers (3 not 4) and pays a lower wage ($14 not $16).

DiagramWage on the vertical axis, labour on the horizontal. Draw upward-sloping supply $ S_L $, a steeper MRC curve above it, and downward-sloping MRP ($ D_L $). The monopsony hires $ L_m $ where MRC meets MRP, then drops down to the supply curve to find the wage $ W_m $. The competitive outcome, $ W_c $ and $ L_c $, is where $ S_L $ crosses MRP: higher wage, more workers.

Practise: 6 questions

Try each one on paper first, then check the worked answer.

EasyQ1. A worker’s MP is 8 units and the product sells for $5 in a competitive market. What is the MRP? Should the firm hire at a wage of $35? Show worked answer

$ MRP = 8 \times 5 = \$40 $. Since $ 40 > 35 $, yes: the worker adds more revenue than cost.

EasyQ2. Explain why the demand for nurses is a derived demand. Show worked answer

Hospitals hire nurses because patients need care. If demand for health care rises (for example, an ageing population), the demand for nurses rises too.

MediumQ3. Total product for 1 to 5 workers is 20, 36, 48, 56, 60. Output sells for $3. How many workers are hired at a wage of $36? At $24? Show worked answer

MP: 20, 16, 12, 8, 4. MRP (×$3): $60, $48, $36, $24, $12.
At $36: hire 3 (MRP of the 3rd = $36).
At $24: hire 4 (MRP of the 4th = $24).
A lower wage moves the firm down its MRP (labour demand) curve.

MediumQ4. A monopolist’s total product for 1 to 4 workers is 10, 18, 24, 28, and to sell these amounts it must charge $10, $9, $8, $7. The wage is $30. Find MRP and the number hired. Show worked answer

Total revenue: $ 10 \times 10 = 100 $, $ 18 \times 9 = 162 $, $ 24 \times 8 = 192 $, $ 28 \times 7 = 196 $.
MRP = change in TR: $100, $62, $30, $4.
Hire 3 workers (MRP of the 3rd = $30 = wage). MRP falls faster than for a competitive seller because both MP and MR fall.

MediumQ5. $ MP_L = 40 $, $ w = \$10 $, $ MP_K = 90 $, $ r = \$30 $, and output sells for $0.50. (a) Is the firm using the least-cost mix? (b) Is it maximising profit? Show worked answer

(a) $ \frac{40}{10} = 4 $ and $ \frac{90}{30} = 3 $. Not equal: use more labour and less capital.
(b) $ MRP_L = 40 \times 0.5 = 20 $, so $ \frac{MRP_L}{w} = 2 $. $ MRP_K = 90 \times 0.5 = 45 $, so $ \frac{MRP_K}{r} = 1.5 $. Both exceed 1, so it is not maximising profit: it should hire more of both inputs.

ExamQ6. A monopsony employer faces these supply wages for 1 to 5 workers: $6, $8, $10, $12, $14. MRP for workers 1 to 5 is $24, $20, $16, $12, $8. (a) Calculate MRC. (b) Find employment and wage. (c) Compare with a competitive market. (d) A minimum wage of $12 is set. What happens to employment? [6 marks] Show worked answer

(a) Total cost: 6, 16, 30, 48, 70. MRC: $6, $10, $14, $18, $22 [1].
(b) Hire while MRP ≥ MRC: worker 3 (16 ≥ 14) yes, worker 4 (12 < 18) no. 3 workers [1] at the supply wage of $10 [1].
(c) Competitive: supply wage = MRP at 4 workers, $12. The monopsony hires fewer workers at a lower wage [1].
(d) With a $12 minimum, MRC is $12 for workers 1–4 (they all accept $12). Worker 4: MRP $12 = MRC $12, so hire 4 [1]. Employment rises from 3 to 4: a well-set minimum wage can raise both pay and jobs under monopsony [1].

Check yourself

1. A competitive firm hires labour up to the point where:

Each worker is worth hiring while the extra revenue (MRP) covers the extra cost (MRC, the wage).

2. $ MP_L/w = 3 $ and $ MP_K/r = 2 $. To cut costs the firm should:

The last dollar on labour buys more output, so shift spending to labour.

3. Compared with a competitive labour market, a monopsony:

MRC lies above supply, so the monopsony stops hiring sooner and pays the lower supply wage.

Your score: 0 out of 3

The short version

Factor demand is derived demand; the MRP curve ($ MP \times MR $) is the firm’s demand curve for a factor.
Hire until $ MRP = MRC $; in a competitive labour market $ MRC = w $.
Labour demand shifts with product price, productivity, and the prices of substitute or complementary inputs.
Least cost: $ MP_L/w = MP_K/r $. Maximum profit: $ MRP_L/MRC_L = MRP_K/MRC_K = 1 $.
A monopsony’s MRC lies above supply, so it hires fewer workers at a lower wage.

Step 21: Factor Demand and the Least-Cost Rule

Derived demand: why firms hire

Marginal product, MRP and MRC

What shifts factor demand and factor supply

The least-cost rule

Least cost is not the same as maximum profit

Monopsony: one big employer

Practise: 6 questions

Check yourself

The short version

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