derivation of an individual firm’s demand for labour using marginal revenue product

Labour Market – Deriving a Firm’s Demand for Labour (MRP Approach)

Objective

To show how an individual firm derives its demand for labour using the marginal‑revenue‑product (MRP) method, to identify the factors that shift both labour‑demand and labour‑supply curves, and to evaluate the impact of key government policies on the labour‑market equilibrium.

1. Key Concepts

• Marginal Physical Product of Labour (MP_L): extra output from one additional unit of labour, holding other inputs constant.
• Marginal Revenue (MR): extra revenue from selling one more unit of output.
• Marginal Revenue Product of Labour (MRP_L): extra revenue generated by an additional worker.
  Formula: \(MRP_{L}=MR \times MP_{L}\)
• Derived (or Derived) Demand for Labour: the demand for labour that follows from the demand for the product the firm produces.
• Profit‑maximising rule: hire labour up to the point where \[MRP_{L}=W\] (where \(W\) is the market wage rate).

2. Derivation of the Individual Firm’s Labour‑Demand Curve

1. Total‑Revenue (TR) function (product market) \[TR = P \times Q\] – \(P\) = price of output (constant under perfect competition).
   – \(Q\) = total output.
2. Short‑run production function (labour is the only variable input) \[Q = f(L)\]
3. Differentiate TR with respect to labour \[\frac{dTR}{dL}=P\frac{dQ}{dL}=P \times MP_{L}\] When the firm is a price‑taker, this derivative is the MRP.
4. Imperfect competition (price‑setting firm) Output price falls as output rises, so we replace \(P\) by marginal revenue: \[MR = P\Bigl(1-\frac{1}{|E_{d}|}\Bigr)\] where \(|E_{d}|\) is the absolute value of the price elasticity of demand for the product. Hence, \[MRP_{L}=MR \times MP_{L}\] Because \(MR
5. Profit‑maximising condition \[MRP_{L}=W\] The firm hires the last worker as long as the extra revenue from that worker is at least as large as the wage cost.

3. The Individual Firm’s Labour‑Demand Curve

• Downward‑sloping because of the law of diminishing marginal returns (MP_L falls as L rises).
• Vertical position depends on the product price (or MR) and on technology (which determines MP_L).

4. Factors that Shift the Labour‑Demand Curve (Syllabus 8.3.1‑8.3.4)

All factors affect the MRP curve by changing either \(MP_{L}\) or \(MR\). The shift is vertical because the wage line is horizontal.

Factor	Effect on MRPL	Direction of shift	Illustrative example
Product price (or MR)	Multiplies MPL	Upward if price rises; downward if price falls	World wheat price rises → wheat farm’s labour demand increases.
Technology / productivity	Changes the shape of the MPL schedule	Upward shift if technology raises MPL	New assembly robot raises output per worker in a car plant.
Output tax (or subsidy) on the product	Reduces (or raises) MR by the tax amount \(t\) (or subsidy \(s\))	Downward shift with a tax; upward shift with a subsidy	Excise tax on cigarettes lowers the cigarette‑maker’s labour demand.
Number of firms in the industry	Aggregated demand changes, but each firm’s derived‑demand curve is unchanged	Market‑demand curve shifts; individual firm’s curve stays the same	Entry of new smartphone manufacturers expands total industry labour demand.
Expectations about future price or demand	Firms may hire more (or fewer) workers today if they expect higher (or lower) future MR	Upward shift if optimistic; downward if pessimistic	Anticipated rise in oil price leads a refinery to increase current staffing.

5. Numerical Example (Profit‑maximising Rule)

Assume a perfectly competitive product market with price \(P=£10\). The firm’s MP_L schedule is:

L (workers)	MPL (units)	MRPL=P×MPL (£)
1	15	150
2	12	120
3	9	90
4	6	60
5	3	30

If the prevailing wage is £80, the firm hires while MRP ≥ W:

• 1st worker: MRP = £150 > £80 → hire.
• 2nd worker: MRP = £120 > £80 → hire.
• 3rd worker: MRP = £90 > £80 → hire.
• 4th worker: MRP = £60 < £80 → stop.

Optimal employment = 3 workers.

6. Graphical Representation (Suggested Diagram)

• Horizontal axis: Quantity of labour (L).
• Vertical axis: £ per unit of labour (W) and MRP_L.
• Plot the downward‑sloping MRP_L curve.
• Draw a horizontal line at the market wage (W). The intersection gives the profit‑maximising employment.
• Show how a rise in product price or an improvement in technology shifts the MRP curve upward, increasing equilibrium L.

7. Labour‑Supply Side (Syllabus 8.3.5‑8.3.6)

7.1. Individual Labour‑Supply Curve

• Usually upward sloping: higher wages induce workers to supply more hours or attract new workers.
• Derived from the trade‑off between leisure and work (utility maximisation).

7.2. Factors that Shift Labour Supply

Factor	Effect on Supply Curve	Direction of Shift	Illustrative example
Population growth / immigration	More potential workers	Rightward (increase)	Net migration of skilled engineers expands the supply of engineering labour.
Changes in preferences for leisure	Workers value leisure more or less	Leftward if leisure becomes more attractive	Longer school terms reduce the number of part‑time teenage workers.
Education and training	Raises skill levels; may increase or decrease supply of a specific type of labour	Rightward for skilled labour if training expands the pool	Government‑funded IT courses increase the supply of qualified programmers.
Taxes on labour income (pay‑roll tax)	Reduces the net wage received	Leftward (supply falls)	Higher income‑tax rates discourage extra work hours.
Wage subsidies / tax credits	Increase the effective wage received by workers	Rightward (supply rises)	Child‑care vouchers make it easier for parents to work.
Occupational licensing	Restricts entry to an occupation	Leftward (supply falls)	Medical licensing limits the number of practising doctors.
Union bargaining power	Can raise the reservation wage for members	Rightward shift of the *effective* supply curve (higher wage for a given L) or creation of a wage floor	National teachers’ union negotiates higher minimum pay.
Minimum‑wage‑induced labour‑supply effect	If a statutory minimum exceeds the market wage, some workers may exit the labour force (e.g., discouraged workers)	Leftward for low‑skill workers	High minimum wage reduces part‑time employment among teenagers.

7.3. Interaction of Labour Demand and Supply

• Perfectly competitive labour market: equilibrium wage (\(W^{*}\)) and employment (\(L^{*}\)) are where the derived‑demand curve intersects the upward‑sloping supply curve.
• Any shift in either curve changes the equilibrium:
  • Demand shift (e.g., higher product price) → higher \(W^{*}\) and higher \(L^{*}\).
  • Supply shift (e.g., immigration) → lower \(W^{*}\) and higher \(L^{*}\).

8. Wage Determination in Different Labour‑Market Structures (Syllabus 8.3.7‑8.3.10)

8.1. Perfect Competition

• Firms are wage takers; each hires up to MRP\(_L\)=W.
• Market wage is set by the intersection of the industry‑wide derived‑demand curve and the aggregate labour‑supply curve.

8.2. Monopsony (single large employer)

• Labour supply to the firm is upward sloping; the firm faces a Marginal Factor Cost (MFC) that lies above the supply curve: \[MFC = \frac{d(W \times L)}{dL}= W + \frac{dW}{dL}L\]
• Profit maximisation: hire where \(\text{MRP}_{L}=MFC\). Because MFC > W, a monopsonist hires fewer workers and pays a lower wage than in a competitive market.
• Graphically, the equilibrium is at the intersection of MRP\(_L\) and MFC, with the wage determined by the supply curve at that level of employment.

8.3. Role of Trade Unions

• Unions can shift the effective labour‑supply curve upward (higher reservation wage) or negotiate a wage floor above the competitive equilibrium.
• Result: higher wage, lower employment (ceteris paribus), similar to a minimum‑wage effect.

9. Transfer Earnings and Economic Rent (Syllabus 8.3.11‑8.3.12)

• Transfer earnings: the minimum amount a worker must receive to stay in the current occupation; equal to the wage that would be earned in the next best alternative use of labour.
• Economic rent: any earnings above transfer earnings. In a diagram, rent is the area between the actual wage line and the transfer‑earnings line, for the quantity of labour employed.

These concepts help explain why wages may be above the marginal product of labour in the presence of scarcity, licensing, or strong bargaining power.

10. Government Intervention and Its Effect on Labour‑Market Equilibrium (Syllabus 8.3.13‑8.3.14)

10.1. Minimum Wage

• Statutory floor \(W_{\text{min}}\) above the competitive wage.
• If \(W_{\text{min}} > W^{*}\): the horizontal wage line intersects the derived‑demand curve left of the original equilibrium → unemployment = L^{*}_{\text{competitive}} - L_{\text{post‑min‑wage}}\).

10.2. Wage Subsidies / Tax Credits

• Government pays part of the wage, lowering the firm’s effective cost to \(W_{\text{eff}} = W - s\) (where \(s\) is the subsidy per worker).
• Effect on the diagram: the labour‑demand curve shifts rightward (or the wage line shifts downwards) → higher employment, lower effective wage for workers.

10.3. Output Taxes / Subsidies

• Tax \(t\) on each unit of output reduces marginal revenue: \(MR = P - t\).
  → MRP\(_L\) shifts downward → lower labour demand.
• Subsidy \(s\) raises marginal revenue: \(MR = P + s\).
  → MRP\(_L\) shifts upward → higher labour demand.

10.4. Training and Education Policies

• Public investment raises workers’ productivity → raises MP\(_L\) for firms.
• Result: outward shift of the derived‑demand curve (higher labour demand at every wage).

10.5. Combined Diagram (MRP + Wage Line + Minimum‑Wage Floor)

Draw the MRP\(_L\) curve, the competitive wage line (intersection = \((W^{*},L^{*})\)), and a horizontal line at \(W_{\text{min}}\) above \(W^{*}\). The new equilibrium is where \(W_{\text{min}}\) meets MRP\(_L\); the horizontal distance between the two employment points shows the magnitude of involuntary unemployment created by the policy.

11. Evaluation of Government Policies (Syllabus 8.3.15‑8.3.16)

Policy	Potential Benefits	Potential Costs / Drawbacks	Overall Effectiveness (A‑Level perspective)
Minimum Wage	Raises living standards for low‑paid workers; reduces in‑work poverty.	Creates unemployment if set above equilibrium; may encourage informal employment or automation.	Effective for redistribution but can distort labour allocation; impact depends on how far the floor is above the competitive wage.
Wage Subsidy (employer‑paid)	Encourages firms to hire more workers; can target specific groups (youth, long‑term unemployed).	Fiscal cost to government; may be temporary – employment can fall when subsidy ends; risk of “subsidy‑induced” inefficiency.	Generally more efficient than a minimum wage because it lowers the firm’s cost without creating a wage floor for all workers.
Output Tax (e.g., excise)	Internalises negative externalities (e.g., pollution, health costs).	Reduces MRP → lower employment in taxed industry; may raise price for consumers.	Useful for welfare‑maximisation when externalities are present; employment effects are a side‑effect.
Output Subsidy	Boosts production in strategic sectors; raises MRP → higher employment.	Fiscal burden; can lead to over‑production if not well‑targeted.	Effective if the goal is to expand a specific industry, but must be weighed against budgetary constraints.
Training / Education Programs	Increases MP\(_L\) → outward shift of labour‑demand; improves long‑run productivity.	Up‑front public expenditure; benefits accrue over time; risk of mismatch if training does not match employer needs.	Highly effective for structural unemployment; complements other policies.
Occupational Licensing (as a policy tool)	Protects consumers; ensures quality standards.	Restricts entry → leftward supply shift; can create artificial scarcity and higher wages without productivity gains.	Beneficial for safety‑critical occupations but may reduce overall welfare if licensing is overly restrictive.

12. Summary

• The firm’s demand for labour is a derived demand obtained from the marginal‑revenue‑product:
  \(MRP_{L}=MR \times MP_{L}\).
• Profit maximisation requires hiring labour up to the point where MRP_L = W.
• Labour‑demand curves slope downwards because MP_L falls as more workers are employed.
• Key demand‑shifters: product price/MR, technology, output taxes or subsidies, number of firms, and expectations.
• Labour‑supply curves are generally upward sloping; they shift with population changes, preferences, education, taxes, subsidies, occupational licensing, union power, and minimum‑wage‑induced labour‑supply effects.
• Wage determination differs across market structures: perfect competition (MRP = W), monopsony (MRP = MFC), and environments with strong union bargaining.
• Transfer earnings represent the minimum wage a worker would accept; any earnings above this are economic rent.
• Government policies (minimum wage, wage subsidies, output taxes/subsidies, training, licensing) alter either the demand or supply side, creating new equilibrium wages and employment levels. Their effectiveness depends on the size of the shift, fiscal cost, and possible side‑effects such as unemployment or market distortions.
• Understanding the MRP derivation equips students to analyse real‑world events—price changes, technological innovation, or policy decisions—and predict their impact on the labour market.