Fixed and variable costs
Fixed costs do not change with output, variable costs rise with it. Together they give the cost function C(x) = fixed + variable × x.
Formulas
- Total cost = fixed + variable × x
- Cost per unit = total cost / x
- High-low: variable = (C₂ − C₁) / (x₂ − x₁), then fixed = C₁ − variable × x₁
Your turn
Three problems on this topic. Amounts with or without thousands separators, a point for pence.
Task
These total costs were recorded in two months. Split them into fixed and variable costs with the high-low method.
| Month | Output | Total cost |
|---|---|---|
| low | 4,700 | £57,200.00 |
| high | 7,300 | £72,800.00 |
Variable cost per unit in £
Fixed costs in £
Amounts like 12,500 or 12500.50; percentages as a number (12.5). Round to the penny or 2 decimal places.
Worked example
A firm's cost function is C(x) = 26000 + 36.5x. Work out total cost and cost per unit at 3,100 units.
Worked solution
- 1.C = fixed + variable × x = £26,000.00 + £36.50 × 3,100 = £139,150.00
- 2.Cost per unit = £139,150.00 / 3,100 = £44.89
- 3.Cost per unit falls as quantity rises because fixed costs are spread over more units.
Typical mistake: Giving the variable cost per unit as the cost per unit; the fixed cost share is missing.
These total costs were recorded in two months. Split them into fixed and variable costs with the high-low method.
| Month | Output | Total cost |
|---|---|---|
| low | 5,400 | £122,200.00 |
| high | 6,800 | £143,900.00 |
Worked solution
- 1.Variable cost = change in cost / change in output = £21,700.00 / 1,400 = £15.50
- 2.Fixed = C₁ − variable × x₁ = £122,200.00 − £15.50 × 5,400 = £38,500.00
Typical mistake: Taking C₂ / x₂ as the variable cost; it still contains fixed costs.
Frequently asked questions
Why does cost per unit fall with output?
Fixed costs are spread over more units.