The Café That Raised Its Price and Lost More Than It Gained
A suburban café sells 4,000 coffees a month at $4.50. The owner raises the price to $5.00 to cover higher wages. Monthly sales fall to 3,600 cups. Total costs were $14,000 a month before the change and rise to $14,800 after it (higher wages plus slightly lower input usage). The owner's accountant argues the price rise was a mistake because revenue fell. A neighbouring café kept its price at $4.50 and reported a small rise in customers. The owner also notices that sales of muffins, which are often bought with coffee, fell by 15% over the same period, while sales of a cheaper slice rose. The café has spare seating capacity in the afternoons and a fixed lease running for two more years.
1. Calculate the price elasticity of demand for coffee over this price change and use it to judge whether the owner's revenue fell. Show your method and interpret the coefficient.
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Method (midpoint formula):
- Percentage change in quantity = (3,600 − 4,000) ÷ [(4,000 + 3,600) ÷ 2] = −400 ÷ 3,800 = −10.5%
- Percentage change in price = (5.00 − 4.50) ÷ [(4.50 + 5.00) ÷ 2] = 0.50 ÷ 4.75 = +10.5%
- PED = −10.5 ÷ 10.5 = −1.0
Interpretation: demand is unit elastic at this point. Revenue before = 4,000 × $4.50 = $18,000. Revenue after = 3,600 × $5.00 = $18,000. Revenue is unchanged, so the accountant's claim that revenue 'fell' is wrong - but so is the owner's implicit hope that revenue would rise.
Judgement: because the coefficient is close to 1, the café sits near the boundary between elastic and inelastic demand. A small error in the estimates (for example if sales actually fell to 3,500) would push PED above 1 and revenue down. The exam point is that the revenue effect depends on elasticity, not on the direction of the price change alone, and that a unit-elastic result gives the owner no margin for error.
2. Using the cost and revenue data, assess whether the price rise improved the café's profit. Then explain what the fall in muffin sales and the rise in slice sales suggest about the relationship between the two goods.
Read the worked answer
Profit calculation:
- Before: revenue $18,000 − costs $14,000 = $4,000 profit
- After: revenue $18,000 − costs $14,800 = $3,200 profit
Profit fell by $800. The price rise covered part of the wage increase through the price channel but the volume loss removed the revenue growth that would have absorbed the rest. Note that the cost figures include the wage rise, so the comparison is fair.
Complement and substitute relationships: coffee and muffins are complements - they are consumed together, so a rise in the price of coffee reduced the quantity demanded of muffins even though the muffin price did not change (cross elasticity is negative). Coffee and the cheaper slice are substitutes within the café's own range: as the coffee visit became more expensive, some customers switched to the lower-priced item (positive cross elasticity).
Judgement: the owner should evaluate the price rise on joint revenue, not coffee revenue alone. If muffin margins are high, the lost muffin sales make the price rise clearly value-destroying. A better response might be to hold the coffee price and lift afternoon sales into the spare capacity, since the lease is fixed and marginal cost per cup is low.