Scalability hack: think concavity instead of convexity

Scalability hack: think concavity instead of convexity

Scalability may well represent one of the most overused words in startup financing discussions. VCs use the word (and I’m guilty of this) as shorthand to refer to a startup’s ability to grow dramatically enough to deliver outsize returns to the investor.

I like to simplify it as a joke that there are three kinds of businesses: a) work once, get paid once; b) get paid once, work forever; and c) work once, get paid forever. You pay me five bucks to babysit your kid is an example of (a). A consultant who agrees to a fixed-price bid and then experiences project scope creep that becomes a mission that never ends is an example of (b). Software more closely resembles the third category, i.e. work once to write the code, then get paid forever with each incremental software license sold. This third category represents the businesses VCs like to fund.

A more elegant way to think about this same concept are the notions of concavity and convexity. Let’s start with some simple math. As you may remember from high school geometry, in two dimensional space the direction of a line is described by its slope. The following diagram illustrates a straight line with a positive slope.


Non-linear shapes, like those in the next diagrams, are reflected by equations with power or logarithmic functions.


The slope of all three curves is positive (i.e. the curves are all going up); this is often referred to as the ‘delta’. However the ‘slope of the slope’ — the rate at which the slope changes (derivatives traders like to call it the ‘gamma’) — varies dramatically for each of these three curves. In Figure 1, the gamma is zero, since a straight line has a constant, un-changing slope. In Figure 2, the gamma is negative, since the curve keeps going up but at decreasing steepness In Figure 3, the gamma is positive: not only the does the curve keep going up, but the steepness increases.

Figure 2 depicts a curve that is ‘convex upward’, whereas Figure 3 shows a curve that is ‘concave upward’.

I intentionally did not label the axes, as they could pertain to a variety of business attributes. The horizontal axis represents inputs into a business, such as time, labor, or capital. The vertical axis represents the output of the effort: think revenues, downloads, clicks, game plays, user registrations, etc.

A business whose outputs follow a linear path, like in Figure 1, is not really ‘scalable’ in the VC sense of the term. Like in most service businesses, growth is possible, but it requires a consistent level of effort year in, year out.

Convex upward or ‘negative-gamma’ businesses, i.e. those whose performance resembles Figure 2, can be the most dangerous from a shareholder perspective. A sharp rise in results comes early, almost too easily, with little effort. But then the effect of diminishing returns kicks in, and entrepreneurs persevere by working harder, moving faster, perhaps even convincing investors to pour more money into the business. However, these well-intentioned heroic efforts deliver decreasing results, leading eventually to burnout.

Concave upward businesses, in contrast, are characterized by slow visible progress in the early phases followed by torrential performance as everything starts to click. Entrepreneurs practicing a lean startup methodology often witness a growth curve that is concave upward, as relentless prototyping and hypothesis-testing delivers limited visible results but valuable learning. As insight into the product-market fit is gained, performance soars as the startup ratchets up investment.

So when trying to hack growth for your business, think concavity.