Case 082Operating cases and estimationWarm up

A paint maker with a 40% gross margin is considering a 5% price rise. How much volume can it lose before gross profit falls, and what would you need to believe about customers to go ahead?

The situation

Rangvik Paints sells decorative paint through dealers. On every Rs 100 of sales it spends Rs 60 on materials and other direct costs, a 40% gross margin. Raw material prices have been stable, but management thinks the brand can carry a 5% price rise.

The sales head worries that dealers will push cheaper brands and volumes will drop. The CFO asks you for the arithmetic before the board meeting.

Your task

Find the volume loss at which gross profit is unchanged, explain the formula, and list what you would need to believe about customers and competitors.

Quick check

Before the arithmetic: how much volume can Rangvik lose after a 5% price rise before gross profit falls?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Rangvik can lose up to 11.1% of volume before gross profit falls. The 5% rise lifts the margin per unit from Rs 40 to Rs 45, so volume can drop to 40/45 of today. The rule is price change over the new margin, 5/45. Going ahead means believing fewer than about one customer in nine walks away, which turns on brand loyalty, dealer behaviour and whether rivals follow.

Step 1Why does a small price rise buy so much room?

Look at where the extra Rs 5 goes. Costs per tin do not change, so the whole price rise drops into margin: Rs 40 of gross profit per Rs 100 of old sales becomes Rs 45, a 12.5% increase in profit per unit. A tea stall that adds Rs 1 to a Rs 10 cup costing Rs 6 to make lifts its profit per cup from Rs 4 to Rs 5, a quarter more, so it can lose a fifth of its customers and still take home the same. The same logic gives Rangvik its cushion.

The relationship

40 \times V_0 = 45 \times V_1 \quad\Rightarrow\quad \frac{V_1}{V_0} = \frac{40}{45} = 0.889 \quad\Rightarrow\quad \text{break-even loss} = \frac{\Delta p}{m + \Delta p} = \frac{5}{45} = 11.1\%

40	gross profit per Rs 100 of sales today
45	gross profit per unit after a 5% price rise, costs unchanged
V0, V1	volume before and after
m, Delta p	the margin and the price change, both as a share of today's price

What it says in wordsGross profit is unchanged when the wider margin times the smaller volume equals the old margin times the old volume.

Step 2Does the same rule work for a price cut?

Yes, and it shows why cuts are dangerous. A 5% price cut shrinks the margin from Rs 40 to Rs 35, so volume has to rise by 5/35, 14.3%, just to stand still. The asymmetry comes from dividing by the new margin: a rise divides by a bigger number and a cut by a smaller one. The thinner the starting margin, the wider the gap. At a 20% margin the same 5% rise could afford to lose 20% of volume, and a cut would need 33% more.

After a 5% price rise Rangvik's gross profit stays above today's level unless volume falls more than 11.1%, while a 5% price cut needs 14.3% more volume just to break even.

Step 3What would you need to believe to go ahead?

Turn the break-even into a statement about customers. The rise pays if fewer than about one buyer in nine switches away, which is a price elasticityThe percentage change in volume for a one per cent change in price. Here volume can fall 2.2 per cent for each 1 per cent of price before profit drops. below about 2.2. Three things decide it. Brand: does the customer ask for Rangvik by name, or does the painter choose? Dealers: will they push a rival that leaves them a fatter cut? Competitors: will they follow within a quarter, as they often do when costs are common to all?

Then state the limit of the arithmetic. Gross profit is not the whole picture: lost volume also loses absorption of fixed factory and distribution costs, so the real break-even sits a little below 11.1%. The test is cheap. Raise prices in one region or one product range first, watch dealer offtake for a quarter, and compare it with a region where prices held. A case answer that ends with how to test the belief is stronger than one that ends with a yes.

Where candidates lose it

The usual mistake is dividing the price change by the old margin, 5/40, and saying 12.5%. That overstates the cushion because after the rise each unit earns Rs 45, not Rs 40.

The second is answering with intuition, that customers hate price rises, without the number. Interviewers want the break-even first and the customer judgement second, attached to it.

What the interviewer asks next

Raw material costs rise 8% at the same time. What price rise keeps gross profit per unit unchanged?
Rangvik could instead offer dealers an extra 2% margin to push volume. How much volume must that add?
How would you estimate Rangvik's real price elasticity from its own data?

Company names and figures are illustrative.