Question 3 OR (b)
Producer's surplus two ways for the supply law 64p = (x + 20)²
9 marks
- Time to write
- About 18 minutes
Question
Question 3 OR (b) · 9 marksIf the supply function for a particular commodity is 64 p = (x+20)² and the market price is ₹36, find the producers surplus in two ways:
Integrate with respect to x,
Integrate with respect to the price p.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- findFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
As printed, the marks marker is cut off at the page edge; it is read as (9).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Equilibrium quantity 1 mark 2 min
- 2 Producer's surplus by integrating in x 4 marks 7 min
- 3 Producer's surplus by integrating in p 4 marks 7 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Equilibrium at the market price. Supply: p = (x + 20)²/64. At p = 36: (x + 20)² = 2,304 ⟹ x + 20 = 48 ⟹ x₀ = 28. p₀x₀ = 36 × 28 = ₹1,008. (The lowest supply price, at x = 0, is 400/64 = ₹6.25.)
(i) Integrating with respect to x (definite integral between 0 and 28)
PS = p₀x₀ − ∫₀²⁸ (x + 20)²/64 dx = 1,008 − [(x + 20)³/192]₀²⁸ = 1,008 − (1,10,592 − 8,000)/192
= 1,008 − 1,02,592/192 = 1,008 − 534.33 = ₹473.67 (exactly 1,421/3).
(ii) Integrating with respect to p
From the supply law, x = 8√p − 20 (for p ≥ 6.25). The producer's surplus is the area between the price axis and the supply curve from the lowest supply price up to the market price:
PS = ∫ from 6.25 to 36 of (8√p − 20) dp = [(16/3)p^(3/2) − 20p] from 6.25 to 36
= [(16/3)(216) − 720] − [(16/3)(15.625) − 125] = (1,152 − 720) − (83.33 − 125) = 432 + 41.67 = ₹473.67.
| Method | Working | Producer's surplus (₹) |
|---|---|---|
| (i) With respect to x | 1,008 − 534.33 | 473.67 |
| (ii) With respect to p | 432 + 41.67 | 473.67 |
Both methods give the producer's surplus as ₹473.67.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Equilibrium quantity | 1 |
| 2 | Producer's surplus by integrating in x | 4 |
| 3 | Producer's surplus by integrating in p | 4 |
| Total | 9 |
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