The Sigmoid Function in Business aka the S-Curve

Finance Mar 23, 2021

There's much talk in the FIRE world about exponential growth. Yet that is only half the story, because in the real world the sigmoid reigns supreme.

It goes by several names which, unlike fly-by-night strangers in the real world, is actually a good sign in mathematics. Multiple names means lots of people found it useful, and it's not some seedy back-alley piece of maths. You can call it the s-curve, the logistic function, or the sigmoid function:

The standard s-curve: 1 / (1 + e^-x)

You'll notice that my graph for the sigmoid is focused on the area near the zero-input, since that's the interesting bit. At the extremes of positive and negative, it's just about one or zero. We all know that unchanging things are largely uninteresting, so we won't dwell on the extremities much.

What's important to understand is that the sigmoid starts at near-zero height in negative inputs, goes to one half at the inflection point at zero, and then goes to one for positive values. That is to say, it ranges from zero to one. Like a binary bit, on or off, but with some fuzzy bits in the middle where it transitions.

See that bit at the start where it looks exponential? That's the early stage of growth, where things seem limitless. Then you reach an inflection point at the zero input, and that so-called explosive growth tapers off into a carrying capacity instead.

Natural Systems

Carrying capacity is a consequence of natural systems. An environment has a maximum load that it can support, and usually we talk in practical terms of populations. Too many animals and they start starving, too few and they breed their numbers up until they reach the equilibrium of the carrying capacity.

Growth when under the carrying capacity of a system tends to take the form of a sigmoid, where it starts fast with exponential growth, then tapers off to the equilibrium when other factors start dominating the equation (like attrition).

Khan Academy has a video based on differential equations that introduces carrying capacity:

Logistic models & differential equations (Part 2) (video) | Khan Academy
The logistic differential equation dN/dt=rN(1-N/K) describes the situation where a population grows proportionally to its size, but stops growing when it reaches the size of K.

Another place you'll see sigmoids is in the number of customers a business has. They start with none, hustle their first few customers, and then if it's a sound business it will start growing exponentially. The actual growth rate depends on the industry and firm, but in my line of work I've seen typical numbers like 7% - 10% growth year over year. Then it'll taper off as the business matures.

The thing is, there's only so many customers in the market at any given time. For example let's say the market interested in neon orange faux-leather studded paddles is a total of 1,000 people in the world. At the start nobody owns such a paddle already. Then a business comes along and starts selling them. It takes a while to make their first customers, but after that they start selling to more customers each period – say, 30 customers a period. That can go on for a while and feel great.

Then suddenly they lose that growth. They start trickling down to 25, 20, 15 customers per period. The business is "over the hill," and has passed the inflection point of a sigmoid so they're now slowing down. This happens because after a while, the market is saturated with kitschy kink apparatus. Nearly every one of those original 1000 people already owns the paddle, and they're not buying it again.

That slowdown in growth isn't necessarily the fault of the firm – it's a natural byproduct of the environment and carrying capacity. That product line has run its course, its life cycle is completed.

In fact, there's a business life cycle that's used in portfolio management that is helpful to understand this.

Portfolio Management

One of the things we did when I learned about portfolio management was classify businesses in four groups: the dogs, the cash cows, the stars, and the unknowns. Although those four groups are based on market share and profits, it also can be applied to business life cycle maturity.

Dogs are mature firms with limited growth potential and limited profits – it's best to let sleeping dogs lie. Or to drop them from your portfolio, if you're profit-driven and a "dog" firm doesn't provide any other non-monetary benefits.

Cash cows are famous and beloved mature businesses because they are a license to print money. They have limited growth but great profits. They've likely reached their carrying capacity in market share, but have a way of generating income without acquiring new customers. Methods like subscriptions, or replacing non-durable goods for existing clients.

Stars are winning businesses that have captured a dominant market share with lots of room to quickly grow into. In our current era, a lot of these are big tech successes. These are often businesses at the start of their growth curve, who seem to be exponentially taking off.

Finally, there's the unknowns with their large potential for growth but lack of market share. They tend to be new businesses, just started and raring to go. Only time will tell when they'll hit their carrying capacity and transition into a different class of business.

In terms of a business' life cycle, you have two main paths. It starts as an unknown, becomes a star, then becomes a cash cow – or it starts as an unknown and eventually reaches its fill and turns into a dog.

Non-linear Thinking and the Business World

Well wait, why don't all businesses vanish over the centuries? How do businesses cope with this carrying capacity limit and stick around? The short answer is that they innovate, adapt, release new products/services, and make new customers therefore.

To get the long answer, we should talk about non-linear thinking and the adaptation businesses undergo as they struggle with multiple s-curve swerves in momentum.

First of all, non-linear behaviour is hard for most people to predict and intuit. There's all sorts of posts by other bloggers that try to develop an intimate understanding of exponential growth. Yet that is only exponential – what about weirder curves like the sigmoid?

Well for one example of an intuition, sigmoids give rise to plateau shaped growth trends – instead of linear or exponential growth:

Plateau shaped growth featuring two sigmoid curves

This model is characterised by plenty of growth, then a tail off, followed by a business adaptation that introduces a new breakthrough growth. As long as you can chain together new adaptations, you get a new sigmoid and a further burst of growth.

Businesses face different problems depending on where they are on the sigmoid. At the very start they face an acquisition problem: finding their first customers. Then as growth begins to take off it's a marketing problem: trying to reach as many customers as fast as possible. Then as it tails off it's a retention problem: trying to keep existing customers happy. Then as it flattens out entirely it's an innovation problem: trying to find something new that'll start the cycle anew.

That flat point after things slow down is called an inflection point too, albeit this time it's a business term business rather than pure mathematics. Leaders need to recognise these inflection points to respond to them appropriately.

In the business world if you reach a plateau you have three options. You can adapt and resume growing, you can maintain your position, or you can lose it. Adapting means finding new customers, maintaining your position requires retaining existing clients, and loss is self-evident. Loss is also the frequent result of inaction in the status quo, as well as too much panicked fruitless action.

Mindset of Limitless Growth

We live in the era of big tech companies, and their growth is so abundant that tech  begins to feel like a base business necessity – like electricity became. If you're caught up in it, tech can make you feel like there's limitless growth to be had.

Tech companies seemingly changed the game for growth companies, because tech has a relatively low up front cost to develop, but almost limitless potential to stick around and keep selling. Yet they still have some costs to operate, and most tech companies are rent-seeking so they adopt behaviours that create churn. Stuff like constant feature re-shuffling that requires constant development, or finding new ways to monetise user attention into a stream of cash.

I like to think that it's worthwhile to take a step back and review the big picture. Nobody is currently building computers designed for the next century. It's all short-term focus, riding primarily on hype and profit highs and the fact that we've not really seen the limit yet for tech companies' growth.

Tech is definitely a game-changing paradigm shift, don't mistake that. Between things like the open source community making it easier to leverage ideas into reality, and the massive scale at which you can interact with the world, tech is a powerful tool that's here to stay. With all the double-edged danger that powerful tools carry...

Yet I see things like $TSLA's valuation and can't help but worry that it's driven to such highs without substance to support it. That investors are essentially gambling on its stock price rather than caring for its business.

I'm also worried about FAANG companies making the headlines for their greed and misuse of employees.  Too many tech companies strip away any hope of dignity in labour, in a perverse reversal of the responsibility of leadership; leaders should nurture, not exploit.

The mindset of limitless growth sees its worst reality check with gig-economy companies like Uber and Lyft. At first they seemed like rare unicorns, with all this potential for growth. Yet within several years, they discovered that they burnt out a huge portion of their workforce. There just aren't limitless pools of people out there willing to be used like that.

The internet is another realm where it all seems to be limitless. Reaching millions, even billions of people are not unheard of. It can feel like there's always more people out there. Yet common sense should tell us that there's a limit to the number of people you can reach by internet. You're not going to get more than the world's population, for instance.

Which is really what the belief in limitless growth is about. When you're on the upward curve of a sigmoid, it feels like you're perpetually growing. That it will just keep climbing, or that you don't know when it will slow down. It's hard to predict where you'll end up.

Yet all things come to a close. At some point that crucial inflection point of the sigmoid function is passed, and growth slows to its carrying capacity.



Mistress of the Home, responsible for all matters financial. A loving Domme tempered with ambition and attention to detail.