Skills Compound

Humans are pretty bad at intuiting about exponential growth. Whether it’s pollution, population growth, or the way a virus spreads, our brains are bad at understanding the impact of such growth.

Exponential growth tends to feel slow at first, and then at some point you realize it’s all happening quite quickly.

Another popular example of this is the growth of wealth versus labour. Why is there so much wealth inequality and concentration today? It’s quite simple, the returns on wealth compound at a higher rate than the returns on labour (which, depending on how you look at it, have been diminishing or stagnant in most places).

How do we use this information to our advantage? It starts with figure out how to recognize exponential growth. Anything that compounds will grow exponentially. Anything that is 1) growing and 2) has its future value dependent on its past values is probably growing exponentially.

If you’re unsure, sometimes it can be as simple as creating a plot and seeing if the growth is accelerating all the time, which can sometimes be ascertained visually.

But anyway, this post isn’t so much about exponential growth, but rather it’s about skills. Skills compound, and therefore they grow exponentially (provided you continue to learn).

To understand how skills compound, we can think about learning skills as a loop with steps that are something like this:

You need to solve problem Z, which requires skill X and Y
You know how to do X, but you don’t know Y yet
You invest time to learn Y, and this gets added to your set of skills
You apply X and Y to solve Z
Not only have you mastered X and Y, but combined they also create a new skill, Z (which is a superset, but could also be considered distinct on its own)

Additionally, skills can often compound in weird ways. If X and Y are worth 5 skill points on their own, but Z is worth 12 skill points, we can see there’s some magic happening where the combination of these skills is worth more than the sum of their parts.

Why does this happen? It’s due to what I’ll call this the learned premium. In other words, there’s value in the fact that you have experience with this combination of skills, and that experience is worth a premium because there’s value in knowing how X and Y combine to make Z, the difference of which is the learned premium.

The total value $v$ of your skills can be modeled as:

$$ v = \displaystyle\sum_{i=1}^ns = sᵢp $$

Where $sᵢ$ is a skill, and $p$ is the learned premium.

Which is equivalent to:

$$ v = \displaystyle\sum_{}s = s_{1}p + s_{2}p + s_{3}p + s_{4}p + s_{5}p + \cdots + s_{n}p $$

You can assume that $n$ is just a function of time, where the longer you spend working to develop your skills the greater the value of $n$ becomes. On the day you’re born a fresh new baby, $n = 0$. Every time you learn a new skill, you get $n = n + 1$.

The individual value for each instance of $s_{n}$ is kind of irrelevant, as over time the compounding of $p$ and $n$ makes $s_{n}$ less significant. In fact, so long as $s_{n} >= 1$ and $p > 1$, your total skill value $v$ will always grow exponentially as $n$ grows (in other words, if you continuously learn new skills).

In dayjob parlance we talk about these things as skills ($s_{n}$) and experience ($n$). If you are more skilled at learning (which is–on its own–a valuable skill) then your learned premium $p$ increases. That’s right, $p$ is actually a function of $n$ too. We can model it this way:

$$ p = n^\lambda $$

Where $\lambda$ is a made-up constant that represents the risk-free learning premium (i.e., the minimum amount of premium you can attain from simply learning any skill at any level).

Okay neat, but what do I as an individual do with this information? In short, optimize for $p$ and $n$. Going deep (depth) into a subject or domain will increase your $p$ quite a bit, by increasing $\lambda$ through acquisition of specialized knowledge, however it’s probably easier to increment $n$ by going wide (breadth) and learning lots of different (and easier to acquire) skills from various domains.

Or, at least I think so.