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Price Elasticity of Demand and Supply: Question 5

Syllabus 2.6

Structured 9 marks

A gym raises the price of its monthly membership from $40 to $50. Before the price rise, 2000 people held a monthly membership. After the price rise, the number of members falls to 1700.

(a) Calculate the price elasticity of demand (PED) for gym memberships, showing your working, and state whether demand is price elastic, price inelastic or unitary. [3]

(b) Calculate the gym's total monthly revenue from memberships before the price rise and after the price rise. [2]

(c) Using your answers to (a) and (b), explain the relationship between price elasticity of demand and the change in a firm's total revenue when it raises its price. [4]

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Worked solution

Part (a): Calculating and classifying PED

Percentage change in price, using the original price of $40 as the base: %ΔP=50−4040×100=+25%\%\Delta P = \frac{50-40}{40}\times100 = +25\%

Percentage change in quantity demanded, using the original number of members (2000) as the base: %ΔQd=1700−20002000×100=−3002000×100=−15%\%\Delta Q_d = \frac{1700-2000}{2000}\times100 = \frac{-300}{2000}\times100 = -15\%

PED=−1525=−0.6PED = \frac{-15}{25} = -0.6

Ignoring the sign, the size of PED is 0.60.6. Since 0.6<10.6 < 1, demand for gym memberships is price inelastic.

Part (b): Total revenue before and after the price rise

Total revenue is price multiplied by the quantity sold at that price.

Before the price rise: 40×2000=80 00040 \times 2000 = 80\,000 which is $80,000 per month.

After the price rise: 50×1700=85 00050 \times 1700 = 85\,000 which is $85,000 per month.

Part (c): The relationship between PED and total revenue

Total revenue rose from $80,000 to $85,000 when the gym raised its price. This is exactly what the theory of price elasticity of demand predicts, because demand is price inelastic: the percentage fall in quantity demanded (15%) is smaller than the percentage rise in price (25%). Since the extra revenue earned on each remaining membership (from the 25% price rise) outweighs the revenue lost from members who cancel (the 15% fall in quantity), total revenue increases overall.

The general rule is:

  • If demand is price inelastic (∣PED∣<1|PED| < 1), a rise in price increases total revenue, because quantity falls proportionately less than price rises, as shown here.
  • If demand is price elastic (∣PED∣>1|PED| > 1), a rise in price decreases total revenue instead, because quantity would fall proportionately more than price rises, so the loss of revenue from lower sales would outweigh the gain from the higher price per membership.

This is why firms find PED so useful when deciding whether to raise or lower prices: it lets them predict, before changing the price, whether the change will raise or lower their total revenue.

Final answers

  • (a) PED=−0.6PED = -0.6. Demand is price inelastic
  • (b) Total revenue was $80,000 before the price rise and $85,000 after the price rise
  • (c) Because demand is price inelastic, the price rise increased total revenue; a price rise would instead decrease total revenue if demand were price elastic