*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.

Because:

Given the imprecise nature of the question, the moment mathematical precision was introduced, I became extremely skeptical this would be productive. I was not disappointed, though I understand the math well enough. My issue is not with your formulae but with their relevance.

The two biggest problems in analyzing the value of self-improvement are that we don't know what it's worth and, worse, it's endogenous - improving ourselves yields direct utility (if we value our "character," "virtue," or what-h... (read more)

Great observation! But...

I dislike "It is left as an exercise for the reader". I don't know why there's a factor of g-1 in the equation. It's likely I will never have the time to check this post again; so you may have lost your one chance to convince me that it's correct.

I sometimes avoid revealing an answer in a post, but only when it's either a teaser for a future post, or when I want the reader to guess first because I want their guess to be used as evidence.

No, we shouldn't. Please explain. It's your idea; try to make it accessible.

I'd like to see this applied to common self-improvement strategies. 2 I've mentioned several times here are spaced repetition software (see also my own little article and dual n-back.

We could probably get somewhere fi... (read more)

The maths is fine of course. The same analysis was used by many people to decide it would be a good idea to borrow money to build lots of houses - in places like the US, Ireland, Spain and so forth. The outcome, as we all know now, wasn't terribly rational. The maths isn't limited to self-improvements - any kind of improvement or construction activity has the same economics.

There isn't anything wrong with the mathematics. The difficulty is that it requires us to speculate on what the future looks like. Low interest rates dramatically increase the importanc... (read more)

Actually, I'm not sure it does. You seem to have gotten through to a couple of people on the strength of your math, but one way of wording a critique I see repeated in the comments is that there's

no such thingas the "instantaneous annual value" of self-improvement in the real world. I tend to agree.What was your intention when you decided to compute the instantaneous annual value of different strategies? Sometimes it makes sense to let a model deviate from reality in order to make it simpler, clearer, or... (read more)

Thanks for the careful explanations. Even I was able to follow your math, which is pretty rare. The evaluation of the present value of an asset part was very interesting. I join other people in being skeptical of the immediate usefulness of the 'when to self-improve' part, but please do make the post on the practical side.

I don't understand some of the variables.

fandpappear to be the same thing: the annual income they would get from coding full-time.Also, if you can grow your wealth-creation skills faster than your discount rate, then obviously you should put all your effort into growing those skills and none into earning money now.

[ETA: so anyone wanting a justification for staying in their parents' basement hacking, there you are!]

Obvious within the terms of the model, at least. In practice, you have to use the skills as you develop them -- that's part of learning the... (read more)

Thank you for daring to use math! (How did you make the equations?)

You might be interested in John Holland's theorem showing that the genetic algorithm optimizes (on average) the tradeoff between exploration (trying out new things) and exploitation (doing things you already know work pretty well). I can't find a good link on it; you'd probably need to read his 1975 book "Adaptation in natural systems". Or try googling /Holland exploitation exploration "multi-armed bandit"/.

Quick caveat: this analysis assumes

ris constant. It is possible for this assumption to be violated without self-contradiction.I will remark further after I have derived the equation.

John:

Do you suggest any practical way to calculate how steep is my discounting curve, in real life?

The first equation can't be right. If r=1, you have no discount rate; you will ask for $2000 for your website. But the equation a/lnr gives the answer infinity (ln1 = 0).

Oh, I see it! That's pretty clever (although you assume constant

gas well). I would like to see "When to Self-Improve in Practice" posted here.(Actually, I would have probably posted both in one essay, were it not too long.)

The 'made-for-Adsense pop culture site that is bringing in $2000 a year' link goes to a Reddit 404 page.