marginal revenue product (MRP) theory: definition and calculation of marginal revenue product

Labour Market – Derived Demand and the Marginal Revenue Product (MRP) Theory (Cambridge A‑Level Economics 9708)

1. Derived demand for labour

• Labour is a derived factor of production: firms demand labour only because it helps them produce a good or service that can be sold.
• The demand for labour therefore follows the demand for the firm’s output.
  • If the product‑market demand curve shifts, the demand for all the inputs (including labour) shifts in the same direction.
  • The shape of the labour‑demand curve mirrors the shape of the product‑demand curve – a more elastic product demand produces a flatter labour‑demand (MRP) curve.
• In a perfectly competitive product market the firm’s labour‑demand curve is the Marginal Revenue Product (MRP) curve.

2. Factors that affect the demand for labour (shifts of the MRP curve)

Factor	How it shifts the labour‑demand (MRP) curve
Product price (or change in product‑market demand)	Higher price → higher marginal revenue (MR) → higher MRP → right‑shift.
Technology / productivity of capital	More productive capital raises the marginal product of labour (MPL) → higher MRP → right‑shift.
Price of complementary inputs (e.g., energy, raw materials)	Higher input costs reduce MPL (less output per worker) → lower MRP → left‑shift.
Input prices of other factors (e.g., cost of capital)	Higher non‑labour input costs lower MPL → lower MRP → left‑shift.
Taxes or subsidies on output	Output tax reduces MR, cutting MRP (left‑shift); output subsidy raises MR, raising MRP (right‑shift).
Expectations about future product price or demand	Positive expectations raise expected MR → firms may hire more now → right‑shift.
Changes in the quantity of other factors of production	More capital can increase MPL (right‑shift) or, if it substitutes for labour, reduce MPL (left‑shift).
Labour‑productivity improvements from training / education	Higher MPL raises MRP → right‑shift of the labour‑demand curve.
Government regulation that alters production technology	Regulations that improve (or hinder) technology affect MPL and therefore shift MRP.

3. Movements vs. shifts of the labour‑demand curve

• Movement along the MRP curve: caused only by a change in the wage rate (W).
  • Higher W → firm hires fewer workers (move up the curve).
  • Lower W → firm hires more workers (move down the curve).
• Shift of the MRP curve: caused by any of the factors listed above (price, technology, input costs, taxes, expectations, etc.). The whole curve moves right (increase in demand) or left (decrease in demand).

4. MRP theory – definition, calculation and algebraic example

4.1 Definition

\[ \text{MRP}_L \;=\; \frac{\Delta \text{TR}}{\Delta L}\;=\;\text{MP}_L \times \text{MR} \]

• ΔTR = change in total revenue when one more worker is hired.
• ΔL = change in labour (normally one worker).
• MP_L = marginal product of labour – extra output from the additional worker.
• MR = marginal revenue from selling that extra output.
  • In a perfectly competitive product market, MR = P (price).
  • In an imperfectly competitive market, MR < P and is calculated from the demand curve.

4.2 Step‑by‑step procedure to obtain the MRP curve

1. Derive the marginal product of labour (MP_L) for each additional worker.
2. Determine the marginal revenue (MR) of the product:
   • Perfect competition: MR = P (constant).
   • Imperfect competition: MR = P \[1 + (1/ε)\] where ε is the price elasticity of demand (or use the formula MR = P \[1 - (1/|ε|)\] for a linear demand curve).
3. Calculate MRP = MP_L × MR for each worker.
4. Plot MRP (vertical axis) against the number of workers (horizontal axis). The resulting downward‑sloping curve is the firm’s labour‑demand curve.

4.3 Numerical example – perfect competition

Labour (workers)	Total output (units)	Marginal product of labour (MPL)	Price (P) = MR (£)	MRP = MPL × P (£)
1	20	20	5	100
2	45	25	5	125
3	65	20	5	100
4	80	15	5	75
5	90	10	5	50

The MRP curve falls as MP_L diminishes – a clear illustration of the law of diminishing marginal product.

4.4 Algebraic example – imperfectly competitive product market

Suppose the firm faces a linear downward‑sloping demand curve:
\(P = 20 - 0.5Q\)

When the firm produces 30 units, the price is £5.5. The marginal revenue is:

\[ \text{MR} = \frac{d(TR)}{dQ}= \frac{d(PQ)}{dQ}= P + Q\frac{dP}{dQ}= (20-0.5Q) + Q(-0.5)=20- Q \]

At Q = 30, MR = 20 – 30 = –£10 (negative, indicating we are past the profit‑maximising output). Assume the firm is at Q = 20, then MR = 20 – 20 = £0, so the relevant range for the example is Q = 10:

\[ \text{At } Q = 10,\; P = 20-0.5(10)=15,\; MR = 20-10 = £10 \]

If the marginal product of the 4th worker is 12 units, the MRP of that worker is:

\[ \text{MRP}_4 = MP_4 \times MR = 12 \times 10 = £120 \]

This shows how, when MR ≠ P, the MRP is lower than the simple MP × P calculation used under perfect competition.

5. Labour‑market equilibrium (demand = supply)

• Firms hire workers up to the point where MRP = wage (W). This is the profit‑maximising rule.
• In a competitive labour market the equilibrium wage is where a horizontal wage line intersects the downward‑sloping MRP (labour‑demand) curve.
• The labour‑supply curve is upward‑sloping because:
  • Higher wages increase the opportunity cost of not working (more people are willing to supply labour).
  • Each worker has a reservation wage – the minimum wage they are willing to accept. As wages rise, more workers’ reservation wages are met, shifting them into the labour force.
• If the actual wage is above the MRP for a given worker, the firm will not employ that worker, creating involuntary unemployment.

6. Factors that affect the supply of labour (shifts of the labour‑supply curve)

Factor	Effect on the labour‑supply curve
Population size & demographics	More working‑age people → right‑shift; ageing population → left‑shift.
Alternative employment opportunities / unemployment benefits	Better alternatives or generous benefits lower the willingness to work at a given wage → left‑shift.
Education, training and skill acquisition	Higher skill levels raise the reservation wage and the marginal productivity of workers → right‑shift.
Cultural / social attitudes towards work (e.g., gender roles)	Changes in attitudes can increase or decrease labour‑force participation, shifting the curve.
Taxes on labour income (e.g., income tax) and subsidies	Higher income tax lowers net wage → left‑shift; tax credits or subsidies raise net wage → right‑shift.
Non‑wage benefits (health insurance, childcare, pensions)	Improved benefits raise the effective remuneration → right‑shift.
Immigration policies	Opening borders adds to the labour pool → right‑shift; restrictions reduce supply → left‑shift.
Labour‑force participation decisions (part‑time vs. full‑time, retirement age)	Changes in the willingness to work more or fewer hours shift the supply curve.

7. Government intervention and its impact on the labour market

• Minimum wage – a legal floor above the equilibrium wage.
  • If set below the equilibrium, it has no effect.
  • If set above the equilibrium, the wage line lies to the left of the MRP curve → unemployment (workers whose MRP < minimum wage).
• Wage subsidies / tax credits – lower the effective cost of labour.
  • Shift the labour‑demand curve right (higher MRP) and/or shift the supply curve right (higher net wage), increasing employment.
• Output taxes / subsidies – affect marginal revenue.
  • Output tax reduces MR → lowers MRP → left‑shift of labour‑demand.
  • Output subsidy raises MR → raises MRP → right‑shift.
• Training and education programmes – raise workers’ productivity.
  • Higher MP_L raises MRP → right‑shift of the labour‑demand curve.
• Immigration controls – directly affect the labour‑supply curve.
  • Relaxed controls increase supply (right‑shift); tighter controls reduce supply (left‑shift).
• Regulation of working conditions – can affect both supply (through willingness to work) and demand (through production costs).

8. Suggested diagrams for exam answers

1. Labour‑demand (MRP) curve with a horizontal wage line – shows the equilibrium wage (W) and equilibrium employment (L*).
2. Full labour‑market diagram – simultaneous labour‑demand (MRP) and labour‑supply curves; illustrate the effect of a minimum wage (horizontal line above equilibrium) and the resulting unemployment gap.
3. Shift diagram – demonstrate a right‑shift of the MRP curve after a rise in product price, a training programme, or an output subsidy.
4. Imperfectly competitive product market – draw the product‑demand curve, derive the MR curve, and then show how the MR curve (not the price line) is used to calculate MRP.

9. Key points to remember

• Labour demand is derived from product‑market demand; the MRP curve represents this derived demand.
• MRP = MP_L × MR (or MP_L × P when the product market is perfectly competitive).
• Firms hire workers up to the point where MRP = wage (W).
• Changes in product price, technology, input costs, taxes, subsidies, expectations, and training shift the labour‑demand curve.
• The labour‑supply curve is upward‑sloping because higher wages attract more workers; the reservation wage explains why workers will not supply labour below a certain price.
• Factors that shift labour‑supply include population, alternative opportunities, education, cultural attitudes, taxes, non‑wage benefits, immigration policy, and participation decisions.
• Government policies can affect either the demand side (minimum wage, output taxes/subsidies, training) or the supply side (taxes on income, immigration controls, benefit reforms).
• Distinguish clearly between movements along a curve (caused by a change in the wage rate) and shifts of the whole curve (caused by any other factor).