In many cases, there is a close relationship between top-level business indicators.For example, to hold Super strong impact About the evaluation of the subscription business. Or, the percentage of vacant seats is very important to the airline.A little fun Toy model What I can do creates a strange relationship between conversion rates and revenue.

## Intuition

Take a look at e-commerce companies. For example, I work for a company that has an online mortgage loan.

My intuition is roughly:

- The higher the conversion rate, the higher the volume. It’s an obvious primary effect.
- The higher the conversion rate, the better the unit economics. This means you’ll be able to win a large number of customers that previously didn’t make sense. You can continue to increase volume until the marginal acquisition cost catches up with the new break-even point.

Let’s formalize it and make some assumptions. First, you need to make some assumptions about how acquisition costs increase with volume. The marginal CAC (customer acquisition cost) is not constant. This is because you have to spend a little more effort on every lead you get. Therefore, I would like to choose a function that grows fairly slowly.

Please note that it is very important not to confuse *limit* Acquisition cost (cost to acquire user $$ n $$) *average* Acquisition cost or *total* Acquisition cost.The total acquisition cost is an integral of the marginal acquisition cost, so it always increases. *More* Than linear (because the margin acquisition cost does not decrease).

## model

I will go to the limbs here. This is the model. The acquisition cost of the read $$ n $$ is proportional to $$ n ^ {0.4} $$. Let’s examine the math and then return to the (slightly arbitrary) function selection.

This function has the property of being reasonably slow growing. Using $$ n ^ {0.4} $$, the cost of acquiring read # 2000 is about 32% higher than the cost of acquiring read # 1000.

Note what i’m saying *lead* $$ n $$, not a user. The difference is that not all leads are converted to users. The difference is the conversion rate.Therefore, the total cost of acquisition *user* Is $$ n ^ {0.4} / r $$. Where $$ r $$ is the conversion rate.

As long as you make a profit from the lead, you are acquiring the lead, so until the marginal acquisition cost becomes constant (basically, the profit minus the subsidy cost). Therefore, $$ n ^ {0.4} / r = C $$, and $$ n = (Cr) ^ {2.5} = mathcal {O} (r ^ {2.5}) $$. The strange $$ mathcal {O} $$ symbol is just a fancy notation from computer science, meaning that all constants can be ignored.

So this is pretty interesting. Basically, a 20% increase in conversion rate is said to increase the total amount by 58%. This is a very non-linear relationship between conversion rate and volume.

## How is your total profit?

Ignoring all the fixed costs of running a company, the profit for each unit is constant minus acquisition costs.

$$ int_0 ^ n left (C_1-C_2 m ^ {0.4} / r right) dm = left[ C_1 m – C_2/1.4 m^{1.4}/r right]_0 ^ n = C_1 n-C_2 / 1.4 n ^ {1.4} / r $$

If you plug in the previous expression for $$ n $$, you get:

$$ = C_1 (r ^ {2.5})-C_2 / 1.4 ((Cr) ^ {2.5}) ^ {1.4} / r = C_3 r ^ {2.5} = mathcal {O} (r ^ {2.5}) $$

Magically, this is the same here. For example, if you increase your conversion rate from 5% to 6% (20% increase), your total gross profit will increase by 58%.If you *double* For conversion rates (which isn’t quite unreasonable for early-stage startups using non-optimized conversion funnels), the gross profit is 5.7x. A kind of sweet, and again the same non-linear relationship between two variables.

Interpretation of the above graphic. The “flater” the acquisition curve, the greater the dollar’s profit. This is because a flat acquisition cost curve means that the amount of break-even points needs to move to the right very quickly.

## More rumination

Do you think $$ n ^ {0.4} $$ is not rational? Okay, set it to $$ sqrt {n} $$. You can do the same, but instead of $$ mathcal {O} (r ^ {2.5}) $$, you get $$ mathcal {O} (r ^ 2) $$.

Of course, it doesn’t have to be a polynomial. Logarithm is fine. Lambert W function.. In fact, in all these cases, the results are even more dramatic. The important thing here is that with slow-growing functions, *Inversion* That makes us a fast-growing feature.As long as you choose a function to grow *Less than linear*, *Super linear* Relationship between conversion rate and gross profit.

So what is the correct choice of features? I think this depends on the industry. If you are selling pet rabbit tiaras, the size of the entire market is quite small, and running out can significantly increase acquisition costs. Revenue as a function of conversion rate is almost flat because the inverse of the soaring function is a stagnant function.

But let’s say you sell a mortgage (I do!) Or groceries or petrol. After that, the market size is huge, and the acquisition cost slowly increases from 1000 users to 10,000 users and 100,000 users. Wherever you see large companies, it shows that marginal acquisition costs must grow very slowly (otherwise competing with small businesses is exorbitantly expensive). .. Non-linear method.

Feel free to drill holes in this theory. If so, please let me know!

Tagging with: Startup, Math