Why a price cut raises total revenue when demand is elastic and lowers it when demand is inelastic, with the revenue rectangles. Through Debenhams.
Over Christmas 2017, the department store Debenhams cut prices hard to pull in shoppers. In January 2018 it issued a profit warning: the discounts had not attracted enough extra customers. It warned twice more that year, went into administration in 2019 and again in 2020, and closed its last stores in May 2021.
When does cutting your price bring in more money, and when does it just give money away?
Sources: Reuters (4 January and 19 June 2018); Debenhams company history.
Total revenue (TR) is the money a firm receives from sales: price × quantity sold. It is the same as consumers’ total spending on the good. On a demand diagram it is the rectangle from the origin to the point on the demand curve.
A price change moves TR in two opposite ways: the firm gets less (or more) for each unit it was already selling, and sells more (or fewer) units. Which effect wins depends on PED. As always, only the price changes (ceteris paribus).
A shop cuts all its prices by 20%.
It sells more, as you would expect.
What happens to its total revenue?
Revenue is price × quantity. A price cut lowers one and raises the other. If quantity rises by more than 20%, revenue rises; if by less, it falls. That is exactly what PED tells you.
High up the demand curve: £16 and 4 units. Total revenue is the rectangle, £16 × 4 = £64.
Cut the price to £14 and sales rise to 6. New TR is £14 × 6 = £84. Revenue has gone up.
Check: The price falls to £14 and sales rise to 6. What is the new total revenue?
£14 × 6 = £84. Revenue has risen by £20.
Why? Two rectangles. The red one is lost: £2 less on each of the 4 units already sold, £8. The green one is gained: 2 extra units at £14, £28. The gain is bigger, so TR rises. This is the elastic part of the curve: quantity rose by 50% for a 12.5% price cut.
Now low down, where demand is inelastic. Cut the price from £6 to £4. Lost: £2 on each of 14 units, £28. Gained: 2 extra units at £4, £8. The loss is bigger: TR falls from £84 to £64.
So with inelastic demand it works the other way round: raise the price and TR rises, because sales fall proportionately less. That is why firms selling goods with few substitutes, and governments taxing them, can raise prices without losing much revenue.
Check: Demand for a good is price inelastic. To raise its total revenue, a firm should:
With inelastic demand, quantity falls proportionately less than price rises, so TR rises.
At the midpoint PED = −1. A small price change changes quantity by the same percentage, so TR does not change: here it is at its highest, £10 × 10 = £100. Move away from the midpoint either way and TR falls.
Suppose a price cut does raise a firm’s total revenue.
Check: A price cut raises a firm’s total revenue. Must its profit rise too?
Profit is total revenue minus total cost. The extra units cost money to make, and that extra cost can be bigger than the extra revenue.
Back to Debenhams. Its price cuts did not bring in enough extra shoppers, while demand for department stores was also falling, so revenue fell. And revenue is not profit: extra sales also raise costs, so even a price cut that raises revenue can lower profit.
They shaded the triangle above the price, which is consumer surplus. Total revenue is the rectangle price × quantity: from the origin up to £12 and across to 8 units.
Explain why a price cut may lower a firm’s total revenue.
Total revenue is price multiplied by quantity sold. A price cut has two opposite effects: the firm receives less for each unit it was already selling, but it sells more units.
[Diagram: a demand curve with two TR rectangles, the area lost (old quantity × fall in price) and the area gained (extra quantity × new price).]
If demand is price inelastic, for example because the good has few close substitutes, quantity demanded rises proportionately less than the price falls. The revenue lost on existing sales is then bigger than the revenue gained on the extra sales, so total revenue falls.
Only when demand is price elastic does a price cut raise total revenue, and at unitary elasticity total revenue does not change.
Debenhams: Reuters (January and June 2018); company history to 2021. Diagrams drawn with Rough.js and perfect-freehand, both MIT licensed. Your answers are saved only on this device.