Question 2 OR (b)
Four partial elasticities of demand and the nature of the two commodities
9 marks
- Time to write
- About 18 minutes
Question
Question 2 OR (b) · 9 marksThe demand functions for two commodities are:
Q₁ = 1800 + 300/(p₁+2) - 40 p₂ and Q₂ = 1800 + 450/(p₁+5) - 80 p₂.
Calculate all four partial elasticities of demand at p₁ = 4 and p₂ = 2.
Comment on the nature of commodities.
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.
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Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Quantities and partial derivatives 3 marks 5 min
- 2 Four partial elasticities 4 marks 7 min
- 3 Nature of commodities 2 marks 4 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
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Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Q₁ = 1800 + 300/(p₁ + 2) − 40p₂; Q₂ = 1800 + 450/(p₁ + 5) − 80p₂. (The four partial elasticities and the nature of commodities are the Unit 3 partial-derivative topic named in the DU guideline.)
Q₁ = 1800 + 300/6 − 80 = 1,770; Q₂ = 1800 + 450/9 − 160 = 1,690.
Partial derivatives (quotient and power rules):
∂Q₁/∂p₁ = −300/(p₁ + 2)² = −300/36 = −8.3333; ∂Q₁/∂p₂ = −40; ∂Q₂/∂p₁ = −450/(p₁ + 5)² = −450/81 = −5.5556; ∂Q₂/∂p₂ = −80.
(i) The four partial elasticities
Own-price elasticities are written as positive numbers, η₁₁ = −(p₁/Q₁)(∂Q₁/∂p₁) and η₂₂ = −(p₂/Q₂)(∂Q₂/∂p₂); cross elasticities keep their sign, η₁₂ = (p₂/Q₁)(∂Q₁/∂p₂) and η₂₁ = (p₁/Q₂)(∂Q₂/∂p₁).
| Elasticity | Working | Value |
|---|---|---|
| η₁₁ (own price, good 1) | 8.3333 × 4/1,770 | 0.0188 |
| η₁₂ (good 1 w.r.t. p₂) | −40 × 2/1,770 | −0.0452 |
| η₂₁ (good 2 w.r.t. p₁) | −5.5556 × 4/1,690 | −0.0131 |
| η₂₂ (own price, good 2) | 80 × 2/1,690 | 0.0947 |
(ii) Nature of the commodities
Both cross elasticities are negative: a rise in the price of either good reduces the demand for the other. The two goods are complements (jointly demanded). Both own-price elasticities are far below 1, so each good's demand is highly inelastic at these prices.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Quantities and partial derivatives | 3 |
| 2 | Four partial elasticities | 4 |
| 3 | Nature of commodities | 2 |
| Total | 9 |
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