discount break-even volume

Also called: d / (m − d), volume a discount needs

The extra volume a discount needs to keep total contribution unchanged: ΔV = d ÷ (m − d), where m is the contribution margin ratio and d the share of price the discount takes out of margin. New volume = V × m ÷ (m − d).

A discount comes almost entirely out of the margin, so it needs far more volume than its own size. If m is the contribution margin ratio (CM ÷ price) and d the share of price the discount removes from margin, total contribution stays the same only if volume grows by

$\Delta V = \frac{d}{m - d} \qquad V_{new} = V \cdot \frac{m}{m - d}$

With a percentage fee f on the price, the margin loses only d × (1 − f) of the price, so use that as d. The denominator m − d shrinks fast: as the discount approaches the margin, the volume needed explodes, and at d ≥ m no volume is enough.

The same formula read the other way tells how many customers a price rise can lose: −d ÷ (m + d).

Example

A programming lesson: price 800, fee 2.5%, tutor 520, video 30, so CM = 230 and m = 0.2875.

10% off: d = 0.10 × 0.975 = 0.0975; ΔV = 0.0975 ÷ 0.19 ≈ +51%.
Check: CM falls to 720 × 0.975 − 550 = 152, and 230 ÷ 152 ≈ 1.51.
3,500 lessons must become 3,500 × 0.2875 ÷ 0.19 ≈ 5,296.

Common mistakes

"10% off needs 10% more." It needs d ÷ (m − d), often five times more.
Forgetting the fee. A percentage fee shrinks with the price, so d is discount × (1 − f).
Gifting loyal buyers. The volume target counts everyone who gets the discount, not only new customers.

Learn it in the course

Rearranging a formula: solve for what you need · Do the same to both sides and undo in reverse order: break-even Q = F/c, runway, the regression intercept, sample size, a fraction inside a denominator, and why a gross-up divides by (1 − f).
Contribution margin and break-even · The unit-economics formulas as pure math: CM = P(1 − f) − T − V, CM%, break-even ⌈F ÷ c⌉, margin of safety, step costs, the volume a discount needs, gross-up and ROI.