Slope as a marginal quantity

Two numbers answer two different questions.

• The average divides a total by a total: everything spent ÷ everyone acquired. It answers "how did we do overall?"
• The marginal quantity compares two neighbouring points: Δy ÷ Δx, the change in one over the change in the other (Δ as in Notation: indices, bars, hats and Greek letters). It answers "what does the next step buy?"

Klasna's social channel brought 25 new families at 50,000 UAH a month, 40 at 100,000, 50 at 150,000 and 55 at 200,000. At 200,000 the average CAC is 200,000 ÷ 55 ≈ 3,636 UAH. But the last 50,000 bought only 55 − 50 = 5 families: 10,000 each.

On a straight line through zero the two numbers are equal. On a curve that flattens, as most channels do once the easy audience is reached, the marginal number runs away from the average. The average is the slope of the line from zero to the point; the marginal is the slope of the last step.

Marginal CAC = Δspend ÷ Δcustomers, taken between two neighbouring budgets. Order the budgets, then divide each step's extra spend by the extra customers that step brought.

Example: a search-ads test at Klasna brought 20 new families at 30,000 UAH a month, 32 at 60,000 and 38 at 90,000.

• 0 → 30,000: 30,000 ÷ 20 = 1,500 per family;
• 30,000 → 60,000: 30,000 ÷ 12 = 2,500;
• 60,000 → 90,000: 30,000 ÷ 6 = 5,000.

The averages look calm: 1,500, then 60,000 ÷ 32 = 1,875, then 90,000 ÷ 38 ≈ 2,368. The marginal numbers show the real trend: each step costs about twice the one before.

Two slips mix a total with a step: dividing the whole budget by the extra customers (90,000 ÷ 6) and dividing the extra spend by all customers (30,000 ÷ 38). Both Δs must belong to the same step.

When a decision only adds a step, judge it by what that step adds. Buy the next step of spend while its marginal CAC is below what a new customer is worth; stop where it rises above.

Say a new family leaves Klasna 4,000 UAH of contribution over its time with the school, and the search-ads steps cost 1,500, 2,500 and then 5,000 per family (20 families at 30,000 a month, 32 at 60,000, 38 at 90,000). The first two steps pay; the third doesn't. Stop at 60,000. The last 30,000 would bring 6 × 4,000 = 24,000 of contribution: a loss of 6,000 a month.

Meanwhile the average at 90,000, 90,000 ÷ 38 ≈ 2,368, still sits well below 4,000. It looks fine only because the cheap early families are mixed in. A decision about the extra money needs the extra numbers.

The average answers a different question, whether the channel as a whole paid back, and is the right number for that one.

Costs split the same way. Klasna's own video classroom would cost 120,000 UAH a month plus 8 UAH per lesson, a line as in The linear function.

• Marginal cost: what one more lesson adds, the slope: 8 UAH at any volume.
• Average cost: total ÷ lessons = 120,000 ÷ n + 8. It falls as volume grows, because the fixed part is spread thinner: 5,000 lessons give 24 + 8 = 32 UAH a lesson, 12,000 lessons give 10 + 8 = 18 UAH.

The average comes ever closer to the marginal cost but never reaches it.

Which one to use? "Should we take one more lesson, or one big extra order?" needs the marginal cost. "What did a lesson cost us this month?" and any price that must cover the fixed part need the average. Pricing one special deal at the marginal 8 UAH can be fine; pricing everything there leaves the 120,000 uncovered.

A step cost turns the marginal picture upside down. Support coordinators cost ⌈learners ÷ 400⌉ × 38,000 UAH a month: one coordinator per 400 learners, rounded up.

At 1,200 learners Klasna pays 3 coordinators, 114,000 UAH, an average of 95 UAH per learner. Learners 801 to 1,200 cost nothing extra at the margin: the third coordinator has room for them. But learner 1,201 needs a fourth coordinator, so at the margin that one learner costs 38,000 UAH.

Near a step, the marginal cost is either zero or a whole salary, and the average hides both. In practice: right before a step, growth is nearly free; right after it, the new capacity is paid for before it fills up.

So plan growth in blocks, here of 400 learners, and judge the block rather than the single learner: once the new coordinator is full, 38,000 ÷ 400 = 95 UAH per learner again.

Sometimes the slope is measured in percentages. Operating leverage (DOL) asks: if volume grows by 1%, by how many percent does profit grow? DOL = %Δprofit ÷ %Δvolume = contribution ÷ profit.

A quick estimate: DOL ≈ 1 ÷ margin of safety, where the margin of safety, (actual − break-even) ÷ actual, says how far a month sits above break-even. In March Klasna ran 24.11% above break-even, so DOL ≈ 1 ÷ 0.2411 ≈ 4.15: +10% lessons gave about +41% profit.

Why so strong? Fixed costs don't grow with volume, so the extra contribution goes straight to profit, and profit is a small base. The closer to break-even, the smaller that base and the larger the swings: at 10% above break-even DOL ≈ 10, and a 5% dip in lessons wipes out half the profit.

It works in both directions, which is what people forget: a business with high leverage loses as fast in a bad month as it gains in a good one.

Price elasticity compares two relative changes: e = %ΔQ ÷ %ΔP, the percentage change in quantity sold over the percentage change in price.

In the finance course, a 10% discount brought 30% more packages: e = +30% ÷ −10% = −3. The sign is negative because price and quantity usually move in opposite directions, so people compare the size, |e|:

• |e| > 1: elastic demand, buyers react more strongly than the price moved;
• |e| < 1: inelastic demand, they react less.

Watch the order: quantity on top, price below. Swapping them gives −0.33 instead of −3 and the opposite reading. Use percentage changes for both, never one in UAH and the other in percent.

An elasticity measured for one change, such as a 10% discount in exam season, is a local number: it says little about a 40% cut or another season.

Revenue is price × quantity, so after a change it is multiplied by (1 + %ΔP) × (1 + %ΔQ), a product of factors as in Products of shares: funnels and metric trees.

• Elastic demand and a cut: −10% price, +30% packages → 0.9 × 1.3 = 1.17, revenue +17%.
• Inelastic demand and a rise: +5% price, −4% sales → 1.05 × 0.96 = 1.008, revenue +0.8%.

With elastic demand a cut lifts revenue and a rise lowers it; with inelastic demand it's the other way round. Don't just add the percentages: −10% + 30% = +20% overstates the 17%.

Revenue is not profit, though. The 10% cut above lowered contribution by 12.6%, because the discount came out of a thin margin while every extra package still cost tutors' time. Check contribution before calling a price change a win.

How it fits together: a marginal quantity is the slope of one step, Δy ÷ Δx; marginal CAC and marginal cost decide "one more step?", while operating leverage and elasticity are the same idea in percentages. The next lesson handles formulas with several inputs and finds which one matters most.