# Slope as a marginal quantity > What one more unit adds: marginal vs average CAC and cost, operating leverage as 1 ÷ margin of safety, and price elasticity as a ratio of relative changes. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Lines and models - Lesson 18/36, ~28 min - Source: https://productovi.com/courses/math-basics/slope-as-a-marginal-quantity ## The next step, not the whole road **Alex Tutor:** Back on Klasna's product team: marketing tested its social channel at four budgets. 50,000 UAH a month brought 25 new families, 100,000 brought 40, 150,000 brought 50, and 200,000 brought 55. **Sofia:** At 200,000 that's 200,000 ÷ 55 ≈ 3,636 UAH per family. Cheaper than it looked. **Alex Tutor:** That's the **average** CAC: everything spent ÷ everyone acquired. Now ask what the **last** 50,000 bought: 55 − 50 = 5 families. That's 50,000 ÷ 5 = 10,000 per family. $$ \text{marginal} = \frac{\Delta y}{\Delta x} \qquad \text{CAC}_{marginal} = \frac{\Delta \text{spend}}{\Delta \text{customers}} $$ *Δ (delta) means "the change in". A marginal quantity compares two neighbouring points; an average divides the total by the total.* **Alex Tutor:** That's the **marginal** CAC: the slope between two neighbouring points, flipped. On a straight line through zero the two are equal. On a curve that flattens, the marginal one runs away from the average. ![Families acquired flatten as spend grows; at 200,000 UAH the average cost is 3,636 per family, but the last 50,000 bought only 5 families, 10,000 each.](https://productovi.com/courses/math-basics/diagrams/m5-l2-average-vs-marginal.png) *The average is the slope from zero; the marginal is the slope of the last step.* **Alex Tutor:** First, a quick check on what the average hides. > **Exercise:** Klasna's social channel brought 50 new families at 150,000 UAH a month and 55 at 200,000, an average CAC of about 3,636 UAH at 200,000. So the families bought by the last 50,000 also cost about 3,636 each. > **Exercise:** Klasna's social channel brought 40 new families at 100,000 UAH a month and 50 at 150,000. What is the marginal CAC of going from 100,000 to 150,000, in UAH per family? **Alex Tutor:** Now the question a budget owner actually asks. Suppose a new family leaves Klasna 6,000 UAH of contribution over its life with the school. **Sofia:** Then I compare 6,000 with what the family costs. But which cost: the average or the marginal one? **Alex Tutor:** Ask what the decision changes. Going from 150,000 to 200,000 doesn't touch the first 150,000; it only adds the last step. **Sofia:** So only the families that step brings, and only the money it costs. > **Exercise:** A new family is worth 6,000 UAH of contribution to Klasna. The channel spends 150,000 a month and brings 50 families; at 200,000 it would bring 55. Should it go to 200,000? ## Marginal and average cost **Alex Tutor:** The same split works for costs. Remember the own video classroom from last time: 120,000 UAH a month plus 8 per lesson. **Sofia:** Each extra lesson costs 8: that's the slope, the marginal cost. The average is total ÷ lessons, so 120,000 ÷ n + 8. **Alex Tutor:** Right. The average falls with volume because the fixed part is spread thinner; the marginal stays at 8. And a step cost flips it: the 1,201st learner brings a whole new coordinator, 38,000 UAH, at the margin. **Sofia:** So for "should we take one more", the marginal number decides. The average is for "how did we do overall". > **Exercise:** The own video classroom costs 120,000 UAH a month plus 8 UAH per lesson. What is the average cost per lesson at 6,000 lessons a month? **Alex Tutor:** Now the step cost from a moment ago, at the margin. > **Exercise:** Support coordinators cost ⌈learners ÷ 400⌉ × 38,000 UAH a month. At 1,200 learners that is 3 × 38,000 = 114,000 UAH, 95 UAH per learner on average. How much coordinator cost does learner number 1,201 add? ## Operating leverage: a slope of percentages **Alex Tutor:** Sometimes we compare relative changes instead. **Operating leverage** asks: if lessons grow by 1%, by how many percent does profit grow? In March it was 4.15: +10% lessons gave about +41% profit. $$ DOL = \frac{\%\Delta \text{profit}}{\%\Delta \text{volume}} = \frac{\text{contribution}}{\text{profit}} \approx \frac{1}{MoS} $$ *DOL: degree of operating leverage; MoS: margin of safety, (actual − break-even) ÷ actual.* **Sofia:** And 1 ÷ 0.2411 ≈ 4.15, the margin of safety again. Closer to break-even, bigger swings. **Alex Tutor:** Yes. At 10% above break-even, DOL ≈ 10: a 5% dip in lessons wipes out half the profit. > **Exercise:** A month runs 20% above break-even (margin of safety = 0.20). By about how many percent does profit grow if lessons grow by 10%? ## Price elasticity **Alex Tutor:** The same ratio of relative changes measures how buyers react to price. **Price elasticity** = %ΔQ ÷ %ΔP. In the finance course, 10% off brought 30% more packages: +30% ÷ −10% = −3. **Sofia:** Negative because price and quantity move in opposite directions. And |−3| > 1 means buyers react strongly? **Alex Tutor:** Yes, demand is **elastic**, so revenue rose: 0.9 × 1.3 = 1.17, +17%. But contribution fell 12.6%, because the cut came out of a thin margin. Elasticity tells you about revenue; profit needs the margin too. ![A 10% price cut brought 30% more packages, an elasticity of −3; revenue rose 17% while contribution fell 12.6%.](https://productovi.com/courses/math-basics/diagrams/m5-l2-elasticity.png) *Elastic demand lifts revenue; it can still lower profit.* > **Exercise:** Klasna raises a package price by 5%, and sales of that package fall by 4%. What is the price elasticity? **Alex Tutor:** And what does elasticity mean for revenue? Try a price rise. > **Exercise:** Klasna raises a package price by 10%, and sales of that package fall by 15%. By how many percent does the package's revenue change? Give one decimal, with a minus sign for a fall. ## Your turn: the next 50,000 **Alex Tutor:** Leo saw the channel table and wrote: "At 200,000 a family costs 3,636 and brings 6,000. Let's go straight to 200,000." **Sofia:** He's reading the average, and the last step is the expensive one. **Alex Tutor:** Then give him the right budget and the numbers behind it, step by step. > **Exercise:** Reply to Leo, Klasna's CEO, in 3–5 sentences. The channel: 50,000 → 25 families, 100,000 → 40, 150,000 → 50, 200,000 → 55; a family is worth 6,000 UAH. Which budget would you choose and why? Use marginal CAC. ## Where you will meet this > **In short** > > - A **marginal** quantity is Δy ÷ Δx between two neighbouring points; an **average** is total ÷ total. They match only on a line through zero. > - **Marginal CAC** = Δspend ÷ Δcustomers. Buy the next step while its marginal CAC is below a customer's value. > - With fixed costs, the **average cost** falls with volume while the **marginal cost** stays at the slope; a step cost makes one unit very expensive. > - **Operating leverage** = %Δprofit ÷ %Δvolume ≈ 1 ÷ margin of safety. **Price elasticity** = %ΔQ ÷ %ΔP; |e| > 1 is elastic. > **These lessons use it** > > - [Break-even and margin of safety](/courses/finance-for-product/break-even): operating leverage 4.15 and 1 ÷ MoS. > - [Price vs volume: how many customers can a discount lose?](/courses/finance-for-product/price-vs-volume): +30% packages for −10% price. > - [Case: double marketing before exam season?](/courses/finance-for-product/marketing-budget-case): the marginal CAC of the second 300,000. > - [Channels and marketing attribution](/courses/product-analytics/channels): marginal CAC per channel against its users' value. > - [Referrals and community](/courses/bootstrapped-product/referrals-and-community): the cost per extra customer, not per customer. **Alex Tutor:** Next time: formulas with several inputs at once, and how to find which input matters most. - Previous lesson: [The linear function](https://productovi.com/courses/math-basics/the-linear-function.md) - Next lesson: [Several variables](https://productovi.com/courses/math-basics/several-variables.md) --- Productovi Learning. 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