The Math of When to Self-Improve

by John_Maxwell 2 min read15th May 201072 comments


An economic analysis of how much time an individual or group should spend improving the way they do things as opposed to just doing them.  Requires understanding of integrals.

An Explanation of Discount Rates

Your annual discount rate for money is 1.05 if you're indifferent between receiving $1.00 now and $1.05 in a year.  Question to confirm understanding (requires insight and a calculator): If a person is indifferent between receiving $5.00 at the beginning of any 5-day period and $5.01 at the end of it, what is their annual discount rate?  Answer in rot13: Gurve naahny qvfpbhag engr vf nobhg bar cbvag bar svir frira.

If your discount rate is significantly different than prevailing interest rates, you can easily acquire value for yourself by investing or borrowing money.

An Explanation of Net Present Value

Discount rates are really cool because they let you assign an instantaneous value to any income-generating asset.  For example, let's say I have a made-for-Adsense pop culture site that is bringing in $2000 a year, and someone has offered to buy it.  Normally figuring out the minimum price I'm willing to sell for would require some deliberation, but if I've already deliberated to discover my discount rate, I can compute an integral instead.

To make this calculation reusable, I'm going to let a be the annual income generated by the site (in this case $2000) and r be my discount rate.  For the sake of calculation, we'll assume that the $2000 is distributed perfectly evenly throughout the year.

Question to confirm understanding: If a person has a discount rate of 1.05, at what price would they be indifferent to selling the aforementioned splog?  Answer in rot13: Nobhg sbegl gubhfnaq avar uhaqerq avargl-gjb qbyynef.

When to Self-Improve

This question of when to self-improve is complicated by the fact that self-improvement is not an either-or proposition.  It's possible to generate value as you're self-improving.  For example, you can imagine an independent software developer who's trying to choose between improving their tools and working on creating software that will turn a profit.  Although the developer's skills will not improve as quickly through the process of software creation as they would through tool upgrades, they still will improve.

My proposed solution to this problem is for the developer to analyze themself as an income-generating asset.

The first question is what the software developer's discount rate is.  We'll call that r.

The second question is how much income they could produce annually if they started working on software creation full-time right now.  We'll call that amount f.  (If each software product they produce is itself an income-generating asset, then the developer will need to estimate the average net present value of each of those assets, along with the average time to completion of each, to estimate their own income.)

Then, for each of the tool-upgrade and code-now approaches, the developer needs to estimate

  • What their instantaneous annual income from software development is from pursuing that strategy.  (For the tool-upgrade approach, that annual income will obviously be 0.)  We'll call that p for present.
  • What the instantaneous annual growth factor in their full-time development income is from pursuing that strategy.  (For example, if working on improving their tools currently offers the software developer the opportunity to improve their wealth creation skills at a rate of a 50% increase in their ability per year, their growth factor would be 1.5.)  We'll call that g for growth.

Given all these parameters, the developer's instantaneous annual value production in a given scenario will be

You should try to figure out why the equation makes sense for yourself.  If you're having trouble post in the comments.

If this post goes over well, I'm thinking of writing a sequel called When to Self-Improve in Practice where I discuss practical application of the value-creation formula.  Feel free to comment or PM with ideas, questions, or a description of your situation in life so I can think of a new angle on how this sort of thinking might be applied.  (Exercise for the reader: Modify this thinking for a college student who's trying to decide between two summer internships and has one year left until graduation.)

Edit: Making LaTeX work in comments manually is a royal pain.  Use this instead.