O(n) vs. O(n^2) Startups

Summary

I recently saw a tweet[1] about how people should go about starting startups/businesses, and it caused me to formalize my intuitions around two distinct types of tech businesses I am familiar with. I’ll call them $O(n)$ startups and $O(n^{2})$ startups. Throughout this essay, let n represent the time elapsed since launch. An $O(n)$ startup grows its key metric (revenue, users, etc.) roughly linearly with time—double the time, double the metric. An $O(n^{2})$ startup accelerates, with growth compounding super-linearly over time. Here are a few popular examples: $O(n)$: Mailchimp, 37signals -> steady ARR, high margins, no outside capital. $O(n^{2})$: Slack (pre-acq), Uber -> heavy spend, but every user fuels ecosystem/network effects. what do i mean? Businesses generally grow following a few patterns. They generally have some TAM to saturate and saturate the TAM at some rate. This rate can be vaguely linear, what I call $O(n)$, or vaguely superlinear, what I call $O(n^{2})$. The reason I borrow the asymptotic notation is because it implies the growth rate is an upper bound (best case scenario) and generalizes away specific constant factors and sums. The analogy breaks down when you force $n$ or $n^{2}$ imply something numerically specific about your growth rate, or introduce functions with different growth rates like logs or exponentials. For now we will (somewhat unprincipledly) stick with two sole classes: $O(n)$ and $O(n^{2})$.[2] I think businesses aim to improve their growth rate via constant factors (eg $O(n^{2})$ -> $O(2*n^{2})$)[3] but fundamentally can’t change them. Some businesses are doomed to grow linearly (lifestyle businesses) and some are able to grow superlinearly. implications VCs generally only invest in $O(n^{2})$ companies. While $O(n)$ businesses are far more common. $O(n)$ businesses can’t be fund returners. Notice how this growth rate is an upper bound and ignores intercepts? This essentially means a company could underperform their growth rate...