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S5 Economics paper 3: cost functions and scale
S5 Economics paper 3: cost functions and scale
Ten items derive average and marginal cost from a cubic total cost function, locate the minimum, then handle a lump-sum tax and minimum efficient scale.
Direct answers Items 01 to 03 derive the cost functions and locate the minimum, 04 to 06 compute marginal cost, judge the range and compare firm sizes, 07 to 09 cover the average to marginal ratio, firm numbers and a lump-sum tax, and 10 corrects a confusion of minima.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 25 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S5 Economics Long-run Cost and Economies of Scale |
| Level | S5 |
02 Questions (10 items)
- FoundationA firm has long-run total cost Q^3 – 30Q^2 + 400Q. Find the average cost function.
- FoundationFor the same firm, at what output is average cost lowest?
- FoundationFor the same firm, find marginal cost at 15 units.
- IntermediateFor the same firm, find marginal cost at 20 units.
- IntermediateFor the same firm, output rises from 15 to 20 units and average cost from $175 to $200. Is this range economies or diseconomies of scale?
- IntermediateOne firm makes 30 units at average cost $400; three firms each make 15 units at $175. By what percentage is the single firm dearer?
- ChallengeFor the same firm, average cost is $200 and marginal cost $100 at 10 units. Find the ratio of average cost to marginal cost.
- ChallengeMarket demand is 100 units and minimum efficient scale is 15 units. How many firms can reach minimum cost?
- ChallengeThe government charges a fixed annual tax of $300. Near the minimum point, how does marginal cost change?
- Error redoA student takes the lowest point of marginal cost (10 units) to be the minimum of long-run average cost. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | AC = Q^2 – 30Q + 400. | Average cost is long-run total cost divided by output. |
| 2 | 15 units. | Setting the derivative of average cost to zero gives Q = 15. |
| 3 | $175. | At the minimum of average cost, marginal cost equals average cost. |
| 4 | $400 (3 x 400 – 1,200 + 400). | Marginal cost is 3Q^2 – 60Q + 400. |
| 5 | Diseconomies of scale. | Rising average cost means diseconomies of scale. |
| 6 | About 128.6 per cent ((400 – 175) / 175). | Diseconomies make concentrating output in one plant dearer. |
| 7 | 2. | A ratio above one means average cost is still falling. |
| 8 | About 6.7 (100 / 15), so at most six in practice. | A whole number of firms cannot exceed 6.7, so six at most. |
| 9 | It stays the same; a lump-sum tax does not change the cost of one more unit. | A lump-sum tax raises average cost only, not marginal cost. |
| 10 | Marginal cost is lowest at 10 units, but average cost is lowest at 15 units. | The two curves bottom out at different outputs and cannot be interchanged. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
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