Optimization, question 10

Topic: Optimization
Difficulty: Standard
Suggested time: 5 min
Marks: 3
Theory: UBC Math 100 §8 CLP-1 §3.5Key results

A shop sells $q$ phone cases a week when it charges $p$ dollars each. The demand and the price elasticity of demand $E$ are given below. At what price is demand unit elastic $(E = 1)$ , and should the shop raise or lower its current price of $12 to increase its weekly revenue?

q = 768 - p^{2}, \qquad E(p) = -\frac{p}{q}\,\frac{dq}{dp}

Unit elastic at $16. At $12, E = 13/6 > 1, so demand is elastic: lowering the price increases revenue.
Unit elastic at $16. At $12, E = 6/13 < 1, so demand is inelastic: raising the price increases revenue.
Unit elastic at $16. At $12, E = 6/13 < 1, so demand is inelastic: lowering the price increases revenue.
Unit elastic at 8√6 ≈ $19.60. At $12, E = 3/13 < 1, so demand is inelastic: raising the price increases revenue.

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Theory UBC Math 100 §8 CLP-1 §3.5Key results

Worked solution

Strategy

Compute the elasticity from the demand curve, solve E = 1 for the unit-elastic price, then use R′(p) = q(1 − E) to read off which way revenue moves at the current price.

Working

$E(p) = -\frac{p}{768 - p^{2}}\,(-2p) = \frac{2p^{2}}{768 - p^{2}}$
Differentiate the demand and substitute.
$E = 1 \iff 2p^{2} = 768 - p^{2} \iff p^{2} = 256 \iff p = 16$
Only the positive root is a price.
$E(12) = \frac{2(144)}{768 - 144} = \frac{288}{624} = \frac{6}{13} < 1$
Inelastic: a 1 % rise in price loses only about 0.46 % of sales.
$R'(p) = q + p\,\frac{dq}{dp} = q\,(1 - E), \qquad R'(12) = 624\cdot\frac{7}{13} = 336 > 0$
Revenue rises with the price at $12, so raise it; revenue is largest at $16, where E = 1.

Answer

p = \$16 \text{ (unit elastic)}; \quad E(12) = \frac{6}{13} < 1, \text{ so raise the price}

Check

Revenue p(768 − p²) grows from 12 · 624 = $7488 at $12 to 13 · 599 = $7787 at $13, and peaks at 16 · 512 = $8192 at $16.

Where this goes wrong

Reading inelastic demand as a reason to cut the price. When E < 1, sales fall by a smaller percentage than the price rises, so revenue goes up with the price; only elastic demand, E > 1, rewards a price cut.

Leans on

The product rule, from Differentiation rules

Try a similar question

A food truck sells 400 burritos a week at $11 each, and estimates that every $1 cut in price would sell 40 more a week. Each burrito costs $3 to make, and the truck has fixed costs of $1200 a week. Assuming demand is linear in price, find the weekly sales and the price that maximise weekly profit.

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Optimization practice question 10