Question 3 (b)

Linear demand and supply from two observations; consumer's and producer's surplus

9 marks

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

Question

Question 3 (b) · 9 marks

When the price of a smartphone averaged ₹5000, Zentra Co. Ltd. sold 30 units every week. When the price dropped to an average of ₹2000, 130 smartphones were sold every week. When the price was ₹5000, 240 smartphones were available per week for sale. When the price reached ₹2000, only 90 remained.

Determine the following:

The demand and supply functions, assuming both are linear.
The consumer's and producer's surplus.

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

assuming both are linearThe paper fixes this: use it as given, and say so in your answer.
DetermineFormula first, then the working, then the answer with its unit (₹, %, units).

Write it in this order

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

12345

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.

(i) Demand and supply functions

Demand passes through (p, x) = (5000, 30) and (2000, 130). Its slope is dx/dp = (130 − 30)/(2000 − 5000) = −1/30, so

x − 30 = −(1/30)(p − 5000) ⟹ x = (5900 − p)/30, i.e. p = 5900 − 30x.

Supply passes through (5000, 240) and (2000, 90). Its slope is dx/dp = (90 − 240)/(2000 − 5000) = 1/20, so

x − 240 = (1/20)(p − 5000) ⟹ x = (p − 200)/20, i.e. p = 200 + 20x.

Check: demand gives x = 30 at ₹5,000 and 130 at ₹2,000; supply gives 240 at ₹5,000 and 90 at ₹2,000.

Equilibrium. 5900 − 30x = 200 + 20x ⟹ 50x = 5700 ⟹ x₀ = 114 smartphones a week, p₀ = 200 + 20(114) = ₹2,480.

(ii) Consumer's and producer's surplus (perfect competition at the equilibrium, as the DU guideline sets these questions; definite integrals between 0 and x₀)

Consumer's surplus = ∫₀¹¹⁴ (5900 − 30x) dx − p₀x₀ = [5900x − 15x²]₀¹¹⁴ − 2480 × 114 = (6,72,600 − 1,94,940) − 2,82,720 = 4,77,660 − 2,82,720 = ₹1,94,940.

Producer's surplus = p₀x₀ − ∫₀¹¹⁴ (200 + 20x) dx = 2,82,720 − [200x + 10x²]₀¹¹⁴ = 2,82,720 − (22,800 + 1,29,960) = 2,82,720 − 1,52,760 = ₹1,29,960.

Consumers	₹
Amount spent, p₀x₀	2,82,720
Consumer's surplus	1,94,940
Total willingness to pay, ∫₀¹¹⁴ demand	4,77,660

Producers	₹
Minimum the producers would accept, ∫₀¹¹⁴ supply	1,52,760
Producer's surplus	1,29,960
Amount received, p₀x₀	2,82,720

Check by triangles (both curves are straight lines): CS = ½ × 114 × (5900 − 2480) = ₹1,94,940; PS = ½ × 114 × (2480 − 200) = ₹1,29,960.

Where the 9 marks sit

#	Component	Marks
1	Demand function	2
2	Supply function	2
3	Equilibrium	1
4	Consumer's surplus	2
5	Producer's surplus	2
	Total	9

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