Cost-minimising choice of factor inputs

7.5 Types of Cost, Revenue and Profit

1. Cost Classifications

Understanding how costs behave is the first step in deciding the most efficient combination of factor inputs.

Short‑run Cost Curves (U‑shaped)

• AFC continuously falls because the same fixed cost is spread over a larger output.
• AVC falls at low output (increasing marginal returns) and then rises (diminishing marginal returns).
• ATC = AFC + AVC – therefore ATC is U‑shaped.
• MC (short‑run) cuts ATC at its minimum; this is the point where the cost of an extra unit equals the average cost of all units produced.

Long‑run Cost Curves

• LRAC is drawn by taking the lowest possible ATC for each level of output – the classic “envelope” diagram.
• Three sources of economies of scale that cause LRAC to fall:
  1. Technical – more efficient plant and equipment.
  2. Managerial – specialisation of labour and management.
  3. Marketing – bulk buying, wider distribution.
• Diseconomies of scale arise when LRAC rises (e.g., coordination problems, excessive bureaucracy).
• The output at which LRAC is at its lowest is the Minimum Efficient Scale (MES) – the smallest output at which a firm can enjoy all possible economies of scale.

Diagram – SRAC vs. LRAC (envelope)

2. Revenue Classifications

Example – Perfectly Competitive Firm

Market price P = £5 and the firm sells Q = 200 units:

• TR = 5 × 200 = £1 000
• AR = TR / Q = £5 (equal to P)
• Because the demand curve is horizontal, MR = £5 as well.

3. Profit Types and Their Conditions

4. Short‑run vs. Long‑run Production Decisions

• Short run: at least one factor (usually capital, K) is fixed. The firm can only vary the variable input (labour, L).

  Cost‑minimising rule (with K fixed):

  \(\displaystyle \frac{MP_L}{w} \;=\; \frac{MP_K}{r}\)

  where \(MP_L\) and \(MP_K\) are marginal products, \(w\) the wage rate, and \(r\) the rental rate of capital.
• Long run: both L and K are variable. The firm chooses the combination that satisfies the isoquant–isocost tangency condition:

  \(\displaystyle \frac{MP_L}{MP_K} \;=\; \frac{w}{r}\)

  and places the isoquant on the lowest possible isocost line, thereby achieving the minimum LRAC for the chosen output.

5. Cost‑Minimising Input Mix (Isoquant–Isocost Analysis) – Optional/Advanced

Key concepts

• Isoquant: curve showing all (L, K) combinations that produce a given output Q.
• Isocost line: all (L, K) combinations that cost the same total amount C, given by

  \(C = wL + rK\)

  where \(w\) is the wage rate and \(r\) the rental rate of capital.
• Rate of Technical Substitution (RTS): the slope of an isoquant, \(-\dfrac{MP_L}{MP_K}\). It shows how many units of K can be replaced by one more unit of L while keeping output constant.

Step‑by‑step procedure

1. Determine the required output level \(Q\).
2. Draw the corresponding isoquant.
3. Write the isocost equation \(C = wL + rK\) and sketch several isocost lines (higher‑cost lines lie farther from the origin).
4. Identify the **lowest** isocost that still touches the isoquant – the tangency point gives the cost‑minimising input mix.
5. Check the tangency condition: \(\displaystyle \frac{MP_L}{MP_K} = \frac{w}{r}\) (or equivalently \(\frac{RTS}{w/r}=1\)).
6. Use the output condition (the production function) together with the isocost equation to solve for the exact quantities of L and K.

Diagram – Isoquant–Isocost (placeholder)

6. Linking Cost‑Minimisation to Profit Maximisation

• Profit maximisation requires MR = MC. The MC curve is derived from the cost‑minimising input choices at each output level.
• Short‑run decision: choose the output where MR meets the short‑run MC.
  • If price P < AVC, the firm shuts down (produces Q = 0) because it cannot cover its variable costs.
  • If P ≥ AVC, produce where MR = MC (or P = MC in perfect competition).
• Long‑run decision: choose the output where MR meets LRMC.
  • In perfect competition MR = P, so the firm produces where P = LRMC = LRAC – the normal‑profit equilibrium.
  • In monopoly MR < P; the firm produces where MR = LRMC and then charges the price given by the demand curve.

Key Takeaways

• Costs are classified as fixed, variable, short‑run total, average (AFC, AVC, ATC) and marginal (MC); each has a characteristic shape in the short‑run and long‑run.
• Long‑run average cost (LRAC) reflects economies and diseconomies of scale; the Minimum Efficient Scale (MES) is the output at which LRAC is lowest.
• Revenue is measured as total, average and marginal. Only under perfect competition do we have AR = MR = P.
• Profit types:
  • Normal profit ⇔ P = ATC (long‑run equilibrium).
  • Super‑normal profit ⇔ P > ATC (short‑run) or P > LRAC (long‑run with barriers).
  • Loss ⇔ P < AVC (short‑run shutdown) or P < ATC (long‑run exit).
• The cost‑minimising input mix is found where the isoquant is tangent to the lowest feasible isocost line, i.e. where the rate of technical substitution equals the ratio of input prices.
• Short‑run constraints limit adjustment to the variable input; the long‑run allows full flexibility, enabling the firm to operate at the lowest possible average cost for any chosen output.