Question 3 OR (b)

Consumer's and producer's surplus with quadratic demand and supply

9 marks

Time to write: About 18 minutes
Topic: Integration: elasticities and surplus

Question

Question 3 OR (b) · 9 marks

The demand and supply laws of a commodity are p_d = 62 - x² and p_s = 8+ (x²/2). Find the consumer surplus and produce surplus at the equilibrium price.

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

18 minfor 9 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 question asks for the "produce surplus"; it is read as the producer surplus.

Write it in this order

Read the data; note exactly what is asked 1 min
1 Equilibrium 2 marks 4 min
2 Consumer's surplus 3½ marks 6 min
3 Producer's surplus 3½ marks 6 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

123

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. p_d = 62 − x², p_s = 8 + x²/2 (the question's produce surplus is read as producer's surplus, as the paper note says).

62 − x² = 8 + x²/2 ⟹ 1.5x² = 54 ⟹ x² = 36 ⟹ x₀ = 6; p₀ = 62 − 36 = ₹26.

Consumer's surplus (perfect competition, as the DU guideline sets these questions; definite integrals):

CS = ∫₀⁶ (62 − x²) dx − p₀x₀ = [62x − x³/3]₀⁶ − 26 × 6 = (372 − 72) − 156 = 300 − 156 = ₹144.

Producer's surplus:

PS = p₀x₀ − ∫₀⁶ (8 + x²/2) dx = 156 − [8x + x³/6]₀⁶ = 156 − (48 + 36) = 156 − 84 = ₹72.

Consumers	₹	Producers	₹
Amount spent, p₀x₀	156	Minimum acceptable, ∫₀⁶ p_s dx	84
Consumer's surplus	144	Producer's surplus	72
Willingness to pay, ∫₀⁶ p_d dx	300	Amount received, p₀x₀	156

At equilibrium x = 6 and p = ₹26: consumer's surplus ₹144, producer's surplus ₹72.

Where the 9 marks sit

#	Component	Marks
1	Equilibrium	2
2	Consumer's surplus	3.5
3	Producer's surplus	3.5
	Total	9

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