The revenue version
Raise prices by x, and revenue survives losing up to x ÷ (1 + x) of clients. A 10% rise breaks even at 9.1% client loss; 15% at 13%; 25% at a full 20%. The asymmetry surprises people in the right direction: the rise "pays for" almost its own percentage in churn, because every remaining client pays more on the whole book.
The margin version — the honest one
Revenue is vanity here too. A departing client takes their revenue but returns their cost-to-serve; a price rise on stayers is pure margin. On contribution margin m, the break-even loss rises to x ÷ (m + x): at 50% margins, a 10% rise breaks even at 16.7% churn — nearly one client in six, for revenue neutrality, with a lighter workload thrown in. The higher your margin, the more courage the arithmetic funds.
And the comparison that decides the meeting: put expected churn (honestly guessed — most firms overestimate it) against break-even churn. Every point of gap between them is free money currently being donated to politeness.
What the formula omits
Selection: the clients most likely to leave over price are disproportionately the low-margin, high-maintenance ones, which biases the true break-even further in your favour. Momentum: a rise resets the anchor for every future negotiation. And the one real risk — concentration: the formula treats clients as fungible percentages, and a book where one client is 30% of revenue does not churn in percentages. Check concentration before courage.
The house position
Price Rise or Client Loss runs both versions on your numbers and prints the gap between break-even and expected churn — the figure the meeting was always about. For the one-big-client problem the formula can't see, the concentration instrument's logic applies to revenue books as well as portfolios.