The Math of When to Self-Improve

by John_Maxwell3 min read15th May 201072 comments

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Productivity
Personal Blog

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.

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Because:

  • It's rational to be motivated entirely by the net present value of one's future earnings.
  • Programmers are immortal (and never switch career.)
  • A programmer's skill increases in a smooth, deterministic, predictable and exponential manner over time.
  • A programmer's skill is a scalar-valued quantity proportional to their (potential) income.
  • Cows are spherical.
4John_Maxwell12yThere might be other relevant sources of utility for the developer, but did you really want me to complicate the post by discussing them? I think your discount rate is supposed to take the probability of your dying each year into account. Ditto the above regarding career switching/retirement. Not necessary. For my model to be useful, you just need to know the expected rate at which the programmer's wealth-creation ability will improve. The formula gives the programmer's expected rate of instantaneous value creation. Taboo skill. Anyway, the real question is whether using this model or some approximation of it produces better results than relying on one's native intuition. Relevant link [http://www.technologyreview.com/computing/24556/].
0gwern11yYour link is paywalled; try http://www.technologyreview.in/computing/24967/ [http://www.technologyreview.in/computing/24967/]

Requires understanding of integrals.

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)

0orthonormal12yThe context was optimizing job earnings, not transhumanist brain modifications. I think the model is reasonable in that context, if a bit hard to apply.
0Psychohistorian12yWhen I read "self-improvement" I don't immediately think "investment to enhance one's future earnings," though I admit it does make some sense. The endogeneity problem largely disappears if you define your utility in monetary terms, but uncertainty still abounds, and actual problems may remain (most noticeably, self-investment may lead to lower utility if it doesn't pay off, since you feel like you're worth more than you get; while important, this is not reflected in a purely monetary model). Since your actual concern is probably utility and not money, that issue is significant. Also, "transhumanist brain modifications" are hardly necessary to generate utility function changes. Most forms of self-improvement in the personal (as opposed to professional) sense are likely to either require or result in changes in one's utility function.
4orthonormal12yI don't think we disagree on anything substantive. You might find the post's title misleading for a limited model like this, but I prefer it to something more disclaimer-heavy. For instance: "A Toy Model Of Optimizing A Scalar-Valued Function Given Some Predictable Ability To Spend Time On Increasing The Rate of Change, But With A Discount Rate Included; Which Model May Be Of Some Analogous Application To Simple Work-Related Self-Optimization (Not Counting Self-Optimization Of Types That May Substantively Change One's Goals And Valuations)".
1Psychohistorian12yI agree on the first part. The rephrasing is perhaps a straw man. "The Math of When to Invest in Oneself," would get the exact point across without the ambiguity of "self-improvement."
1orthonormal12yFair enough; it was just too fun not to post. (Of course, they actually did titles like that in the 17th century.)
0John_Maxwell12yI think I have a simpler utility function than you do :)

Great observation! But...

You should try to figure out why the equation makes sense for yourself.

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.

2gwern11yI think it comes from the units in the definition of g: At that point, you've accounted for current present value of your current skills, but now you want to add in the value of future growth. The '1' in '1.5' is what you have now, and is always what you have now by definition; the '.5' is additional growth. To remove what you already have, you do '-1'. Hence you multiply by 'g-1'
1PhilGoetz11yOh, right. I was forgetting he defined g that way. Thanks!

You should try to figure out why the equation makes sense for yourself.

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

3timtyler11yDeja Vu: http://lesswrong.com/lw/28e/the_math_of_when_to_selfimprove/2100?c=1 [http://lesswrong.com/lw/28e/the_math_of_when_to_selfimprove/2100?c=1]
0thomblake11yweird
1PhilGoetz11yRedundant, yes. Not surprising that I had the same reaction twice to the same post, though. Delete newer one (Y/N)?
4FAWS11yPlease don't. It's sort of funny, this sub-thread would become incomprehensible, and it might be a nice example to point out for people who have odd ideas about what predicting human actions would have to entail.

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.

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)

2John_Maxwell11yI'm pretty sure you'd be crazy to value the time and energy you spend on these activities at only $8/hour. Another way of thinking the cards would be "if it takes me 30-40 seconds to memorize this card for 3 years, how many seconds do I expect to spend looking up the information on this card if I don't bother to memorize it?" Would you really spend 30-40 seconds looking up information on the median card you'd consider memorizing? As for dual-n-back, first there's the critique you mention in your FAQ, which seems very damning. Second it's quite unpleasant in my experience at least, and therefore uses up a lot of energy. And third, I'm inherently skeptical that simple mental activities like dual-n-back and nintendo brain age could increase your IQ faster than complex activities like writing software that someone might be willing to pay for or becoming proficient in economics [http://introecon.com/]--and these last activities have significant side benefits above and beyond possible IQ increase. If you just want to increase your income I suspect there are much more efficient and direct ways of doing it. I have a few arbitrage-type ideas I can share via private message with anyone who expresses a desire to donate a lot of their income to preventing human existential risk. For what it's worth, when I first wrote this post many months ago, I thought time spent on pure self-improvement activities was generally time well spent. Now I think it's generally not time well spent unless you're at the point where you spend eight hours a day browsing reddit or something like that. My view is that if you're able to spend a decent chunk of the day working on worthwhile projects, you should do that, and your self-improvement efforts should be limited to running day-long or week-long self-experiments on yourself that don't cost you many productive hours and recording the results of those. (Example of such a self-experiment: if you're a programmer, write down every bug you have, esti
2gwern11yWhy? I'm not a high-powered lawyer who earns thousands per hour and can easily take on new clients if he discovers he has 10 extra hours a week. Nor am I homo economicus - I am quite biased and crazy. (I think this goes for us all.) I should mention the implicit model was simplified for programming. What you get from memorizing a card is not just knowing it, but knowing you know it. Even if I don't remember exactly one of the many quotes in my Mnemosyne database, I know that I know it and it's in Mnemosyne. Useful for discussions. (Google is not helpful. If you don't remember an exact long substring, its results are pretty worthless. As I have discovered to my ruth many times.) Yes, there are others. As my enthusiasm for n-backing wanes, I've been falling behind on the other null-results. Here's a recent one: "Improvement in working memory is not related to increased intelligence scores" [http://groups.google.com/group/brain-training/browse_frm/thread/dfefba647545fb57] . As for n-backing versus writing software or learning economics, well, the latter are paradigmatic 'crystallized intelligence' as opposed to the 'fluid intelligence' that n-backing is supposed to help. I don't know any good way to calculate their relative values, although it's obvious to me that n-backing would be most valuable for children. (The Google Group's uploaded files includes 1 or 2 studies showing that working-memory exercises helped childrens' grades and behavior.)
1John_Maxwell11yI suspect that a profit-maximizing human as smart as you appear to be who only had the credentials for a minimum-wage job would be working the minimum number of hours each week they could get away with in order to feed, clothe, and house themselves while they used as much of their spare time as possible to cook up something better. At this point converting time to money confuses things because their utility for money drops off so quickly once they have their basic needs covered. Upon reflection, I revise my opinion to "if your lack of credentials and chutzpah means the best jobs available to you pay minimum wage only, improving your credentials/chutzpah is a more profit-maximizing use of time than intelligence enhancement." If you're a student then you're already working on your credentials and your earning power is going to go up soon, so if you're going to work for money it makes sense to put it off until then, and you should value your time at something rather close to the amount you'll make after graduating (assuming you've got a typical discount rate). Try http://www.diigo.com/ [http://www.diigo.com/] -- it lets you do full-text searches on the pages you've bookmarked (along with a bunch of other cool features). At least, if they haven't removed that in the latest version. I know that knowledge of economics is considered crystallized intelligence. I don't see what this has to do with the possibility that the process of learning something new and wrapping your head around it builds fluid intelligence. If fluid intelligence doesn't help me learn stuff faster, is it really worth having? Doesn't it seem likely that learning things makes you better at learning things? If this is true, could an increase in fluid intelligence be the mechanism for it?
2gwern11yWell... I suspect we may be having vocabulary issues here. Gf [http://en.wikipedia.org/wiki/Fluid_intelligence] is defined as "the capacity to think logically and solve problems in novel situations, independent of acquired knowledge." If your existing Gc already applies to a situation - say, your algebra applies to the economics you're learning - then to some extent the problems of economics are not 'novel'. It's only a pure-Gf problem when the problems are highly novel. In that case I find it intuitively plausible that a lot of irrelevant Gc wouldn't help much. Example: if I memorize a couple thousand English words (pronunciation & definition) for the GRE [http://en.wikipedia.org/wiki/Graduate_Record_Examination] for a large increase in my Gc, why should I expect any increased ability to write proofs in mathematical set theory [http://en.wikipedia.org/wiki/Set%20theory] which will initially draw on Gf as a strange and alien subject?
0John_Maxwell11yI agree that memorizing words wouldn't help your fluid intelligence. If doing your first few set-theory proofs draws on Gf heavily, then strictly intuitively speaking it seems to me that this ought to improve Gf just about as fast as anything. Of course, solid experimental results rank above my intuition--but the dual-n-back result isn't solid.
0wedrifid11yYes, you have to understand stuff before your can learn it. And be able to tell the difference between nonsense and things actually worth learning. Yes, it does make you better at learning things. There has been considerable research done on the subject. Basically, no. It's not that it couldn't be, just that it isn't. People's fluid intelligence is extremely hard to change. Very few things improve fluid intelligence and (unfortunately) learning stuff isn't one of them. Dual-n-back training does give a modest effect, as does exercise (and particularly cerebellar targetted exercise). Fortunately, learning stuff will improve your performance at all sorts of activities, even if your fluid intelligence isn't much altered. Fluid intelligence is overrated.
2gwern11yCould you expand on that? I had not heard that exercise actually affected Gf [http://en.wikipedia.org/wiki/Fluid_and_crystallized_intelligence] or that there was such a thing on cerebellar-targeted exercise. I know of occasional results like the prefrontal cortex's cells enlarging after aerobic exercise, but that's not an increase in Gf.
0gwern11yI've added 2 studies to the criticism section: http://www.gwern.net/DNB%20FAQ#criticism [http://www.gwern.net/DNB%20FAQ#criticism]
0John_Maxwell11yThis gives a 403 error: http://community.haskell.org/~gwern/static/docs/chein2010.pdf [http://community.haskell.org/~gwern/static/docs/chein2010.pdf]
0gwern11yI believe all the PDF links should be working at the FAQ's new home.
0John_Maxwell11yOK. You might want to make this page do a 301 redirect to the new FAQ location. http://www.google.com/search?client=ubuntu&channel=fs&q=dual+n+back+faq&ie=utf-8&oe=utf-8 [http://www.google.com/search?client=ubuntu&channel=fs&q=dual+n+back+faq&ie=utf-8&oe=utf-8]
0gwern11yYeah, for the PDFs you have to go to the actual wiki repo: https://patch-tag.com/r/gwern/Gwern/home [https://patch-tag.com/r/gwern/Gwern/home] . It is not a very good static copy of my wiki, I'll admit.

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)

2gwern11yCorporations also have many issues that a truly unitary immortal entity might not; the principle-agent conflict [http://en.wikipedia.org/wiki/Principal-agent_problem] is everywhere in corporations, especially the financial industry types you excoriate.

Requires understanding of integrals

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 thing as 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)

1John_Maxwell12yThat's a danger I hadn't thought of. Thanks. Maybe it's best to discuss new methods of using math only with those who think they're super-competent at it, so they won't be impressed if you do something tricky and will shoot down anything that doesn't make sense to them (instead of assuming you're just more clever). Could you be a little more precise here? I'm talking about someone's rate of value production in terms of dollars per year. I used the word instantaneous to emphasize the fact that this rate isn't necessarily going to hold steady over even one year. The idea is there are two ways of creating value: directly, or by self-improving. To know how much value you are creating by self-improving, you could start by estimating the percentage increase in your effectiveness as a result of your self-improvement efforts. This seems like something that one could possibly estimate. But that still wouldn't be enough to compare the two sorts of value creation if you weren't able to value yourself as an asset. If you value yourself as an asset using net present value, then you can treat the value produced through self-improvement just like the value you create directly. The formula is for the instantaneous rate of value creation because you only need to know what you can do to maximize the rate at which you're creating value right now. Perhaps, although there is a probability distribution over the time at which a person stops producing. So there might be a better way to correct for this. Well the ending formula would have been more complicated: (g%20-%201)) Fooling around with a calculator, you can see that if a person's discount rate is 1.05 and z = 20, then the term you suggest is almost 40% of the size of the term it's being subtracted from, which is significant. However, if you change the person's discount rate to 1.15, the term you suggest is 6% of the first term's size. My high discount rate might have been what caused me think the term you suggested was unim
0Mass_Driver12ySure! I'll try to distinguish between 3 concepts: (1) is average value production over a meaningfully long period of time, e.g. twelve months. Even if we don't know how productive you are on any given day, we can get a decent estimate of your productivity over twelve months by extrapolating from past performance and from your honest statements about what you plan to do next. If you say you plan to write code for immediate profit, and, in the past, that activity has earned you between $2,000 and $9,000 a month, then we might crunch the numbers and estimate that you'll make something like $57,500 a year, with wide error bars. (2) is the net present value of (1). If you figure that after coding for twelve months, you'll have earned $57,500, and your discount rate is 1.15, then your net present value of coding for twelve months is $50,000. Unless you get paid on a biweekly basis, in which case your net present value might be more like $54,000. (3) is the slope of the curve used to estimate (1). The units are expressed in $ per year, but the quantity itself is fundamentally connected with a very short period of time. If you assume, as a trivial and obviously inaccurate example, that the formula for a pure code-writing strategy is Income(t) = ($2750 t t) + $52,000, then the "instantaneous value" of Income(t) is $0 when you start out, $2750 six months into the year, and $5500 at the end of the year. My point is that (3) is not a very useful metric, because we are very unlikely to have anywhere near enough information about the typical person's production curve to start calculating derivatives. Extrapolating future income based on past income already taxes the predictive powers of our data set to the limit. If you want to put yourself in a reference class of similarly situated programmers, fine, but that raises a host of other theoretical issues, e.g. which reference classes are most relevant. Obviously I agree that (2) is an interesting metric -- that's why I want to
1John_Maxwell12yI appreciate you said that, because I realized that despite my claim of a high discount rate, I haven't actually borrowed any money. Probably if I had a steady stream of income I would. I really did mean (3), and I'm not ashamed of it. My thinking is that if you're an individual who's trying to be as effective as possible, you're going to want to guess what you can do to be maximally effective right now, and I might as well fit my formulas for you. Edit: It's true that there is higher variance in a person's output over a short time period. But I'm not sure we should avoid a question just because it's hard to answer.
0Mass_Driver12yMakes sense. Just because you have a high discount rate doesn't mean you have a high tolerance for risk; there's a fine line between wanting to redirect your future income toward the present and wanting to spend now at the cost of going bankrupt later .>I really did mean (3), and I'm not ashamed of it. Well, at least that clears up your motives -- they're pure. Sorry I doubted you. I thought maybe for a moment that you just liked showing off calculus, but I guess you were just trying to attack a really hard problem. This comment is sincere, not sarcastic. Avoid, no. Save for slightly later, yes. In my opinion, the much easier and nearly as useful problem of calculating medium-term net present value should have been solved first, and then once we all understand that and have begun to apply it to our everyday lives, then it's time to try to solve the much harder and only marginally more useful problem of calculating instantaneous net present value. But, you know, you're the one doing the work, and (not knowing you) I have no reason to distrust your estimate of your ability to solve a really hard problem. So, good luck!
1RobinZ12yI would add that this is another reason to simplify the math - doing so eliminates the need for exercises by making the answers less confusing.

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. f and p appear 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)

0John_Maxwell12yIf they're doing some self-improvement activity that requires that they stop wealth-creation altogether, p = 0. If they're coding half-time, p = 0.5f. Etc. You're right about f. OK, but what if you have one project that's a little more educational and one that's a little more profitable? The equation should help you decide between them. BTW, you've done a good job of explaining why I think college for programmers is stupid. (But I'm going anyway, for the friends/prestige.) Hm. I tend to think that someone can often improve their skill faster per minute by reading advice than doing. The things I learned from my last project (~1 month of work) would easily fit in a blog post. Plus what about reading about a subject like probability, game theory, or history that could potentially transform the way you look at things? One hole in my model is it has no way of taking in to account self-improvement effects that are temporary, such as spending the time necessary to think of a really good idea for a project (significantly helps your wealth creation skill, but only until you finish that project).

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"/.

6John_Maxwell12yIf you put LaTeX [http://www.latex-project.org/intro.html] code after " http://www.codecogs.com/png.latex? [http://www.codecogs.com/png.latex?]", you should get a png of the equation the code represents that you can insert in the post editor using the image insertion tool. Codecogs' own equation editor [http://www.codecogs.com/latex/eqneditor.php] is good if you don't know LaTeX. Use this [http://lwlatex.appspot.com/] thing I coded just today if you want to insert LaTeX in a comment, as there's a lot of nasty escaping that needs to go on. Sounds interesting, but wouldn't one's definition of "optimized" depend on one's discount rate? I guess in Holland's model exploration requires resources? That's not a factor in my model, but maybe it should be. Even if my independent software developer had all their living expenses covered, they might still be able to "explore" faster with more resources by hiring software developers in third-world countries to read blogs for them :)
3komponisto12yTesting: %20dx%20=%20\sum_{n=1}%5E{\infty}a_n%20+%20\sqrt[17]{84}) Thank you for creating this!
3John_Maxwell12yYou're welcome!
1Jordan12yFantastic. Skimming through these comments and seeing so much nicely formatted LaTeX makes me smile. Thanks for the little app. If we could get this supported natively in the comments that would be doubly good.

Quick caveat: this analysis assumes r is constant. It is possible for this assumption to be violated without self-contradiction.

I will remark further after I have derived the equation.

3rhollerith_dot_com12yThere is a good reason that the discount rate (r) is assumed to be constant: if your discount rate is not constant, you are a money pump. For more, look for web pages that contain "discount rate" and "preference reversal".
2RobinZ12yI'm not suggesting "treat r as a constant, but let it vary" - I'm suggesting "treat r as a fixed function of time t". For example, someone might use =\frac{1}{(T-t)^2}) to value resources proportionately to how long before time T they are acquired. This kind of valuation scheme is not vulnerable to money-pumping. Edit: My math looks wrong on the equation, but I hope the example illustrates what I mean when I say "r is not constant".
1rhollerith_dot_com12yWhen you say, "Someone might use r(t) = . . . to value resources . . ." and then refer to the equation "r(t) = . . ." as a "valuation scheme," you lead me to believe that you believe that r represents utility. But that is not how John is using r. In John's section on net present value, John has postulated that the utility experienced by our plucky entrepreneur at time t == u(t) == a / r ^ t where a and r do not vary over time. Although John did not explicitly mention the function u, I do so now because you seem to have confused u with r. You wrote that r can very over time "without self-contradiction", to which I replied, "not without our plucky entrepreneur becoming a money pump," which I still believe to be the case. Of course, John's model does not capture the full complexity of the choices and constraints facing an entrepreneurial software developer, but there is a good reason why most treatments of net present value assume that the discount rate does not vary over time.
2RobinZ12yI did not misunderstand. The discount over a time period dt with a constant r is 1/r^dt. If we want a time-varying discount rate q(s), we can use the transform }{log(r)}) and produce the same problem, so long as log(q) is uniformly the same sign as log(r).
1rhollerith_dot_com12yI am sad because my attempt to teach you about preference reversals has almost certainly failed. ADDED. For reference, here is the whole dialog on preference reversals. [http://lesswrong.com/lw/28e/the_math_of_when_to_selfimprove/20i2?context=1#comments] ADDED. On most subjects, I would have let my esteemed interlocutor have the last word so as to keep the peace and so as not to appear as a self-aggrandizing jerk who cannot stop trying to get one up on the person I am disagreeing with. I humbly suggest however that in subjects like math where there often is an objectively-correct fact of the matter, everyone benefits a lot from writers not being too afraid to be confrontational. One of those benefits is "clarity" (something concrete for the reader's mind to latch onto), something easily lost in the abstractions in conversations about math. In other words, I humbly suggest that a competent writer involved in a dialog about math will appear to observers who are not used to good dialogs about math to be unnecessarily domineering, rude, dogmatic or otherwise socially inept even if he is not. ADDED. In other words, in internet discussions on math (or programming languages), if you care too much about not insulting or embarrassing your interlocutor, my experience has been that the whole discussion tends to become a hazy fog. ADDED. I am open to learning from others here how to improve the social side of my communications in dialogs like this.
7Nick_Tarleton12yI suspect there's confusion over what it means to have different discount rates / utility functions at different times. This could mean either that utility depends on the time (call it τ) at which it's computed, or on the time (call it t) at which utility-bearing events occur. The latter alone is always OK, whether or not the relationship is exponential. The former alone might create dynamic inconsistency, and if so, probably (always?) a money pump. Dependence on t-τ (i.e., 'discounting' as usually conceived of) is dynamically consistent if and only if the relationship is exponential.
3RobinZ12yThat agrees with my suspicions - thank you.
3orthonormal12yGiven that the confusion between you and RobinZ was dispelled below [http://lesswrong.com/lw/28e/the_math_of_when_to_selfimprove/20oa], a good piece of advice might be to be careful when you think you have an interlocutor trapped between a simple theorem and a hard place; it's often turned out (in my experience) that some condition of the theorem doesn't apply to the particular case the other person is suggesting, and that the divergence of opinions can be traced elsewhere. Most of the regulars here are smart enough to get the point on preference reversals when pointed out— the fact that RobinZ said he understood but was talking about something different should have counted as evidence to you.
2RobinZ12yI am almost certain that I am simply not conveying what I mean - I don't think you're self-aggrandizing, I think you're as frustrated as I am with this obstinate (apparent?) disagreement. I'm going to describe a concrete example. If you're right, you should be able to either (a) explain how to perform a money-pump on the agent described, or (b) explain why the agent described constitutes a special case. If I'm right, you should be able to describe the difference between the agent that would suffer preference reversals and the agent described. Let t represent the number of years since 2000 C.E. Let E(t) represent an earnings stream - between time t and time t+dt, the agent gains revenue E(t)*dt. Let r(t) represent the instantaneous discount rate at time t. And let P(E) represent the value of earnings stream E to the agent at the year 2000. (The agent is indifferent between earnings stream E and immediate revenue P.) When r(t) = r is a constant, we can easily calculate the present value of any instantaneous future earnings dE at time t: which corresponds to the simple formula I maintain that this last formula, still holds when r is no longer a constant, and therefore (as dE = E(t)dt): \exp{\int_0^z-\log{r(y)}dy}dz) Note that for the special case of F_t - future earnings at time t - we have }dz})
0RobinZ12ySorry, the concrete example. Take %20=%201%20+%200.001%20t) and point future income functions ;%20F_2%20=%20\$100%20\times%20\delta(t-20)) which (using the Dirac delta function [http://en.wikipedia.org/wiki/Dirac_delta_function]) correspond to instantaneous incomes at times t = 10 and 20. That is, 2010 and 2020. Using these functions, =\$100\times\exp\left(%20\int_0%5E{10}-\log(1+0.001t)dt\right)\approx\$95.14) and =\$100\times\exp\left(%20\int_0%5E{20}-\log(1+0.001t)dt\right)\approx\$81.98) Note that to find (say) the value of F_2 in 2010, you would write =\$100\times\exp\left(%20\int_{10}%5E{20}-\log(1+0.001t)dt\right)\approx\$86.17) which is not equal to P(F_1).
0Dan_Moore12yThe OP gives two examples of market pricing - the market price for a website, and a perhaps more subjective price of acquiring a marketable skill set. The question of how to value cashflows to determine a market price has been pretty well studied. The fundamental theorem of arbitrage-free pricing [http://en.wikipedia.org/wiki/Fundamental_theorem_of_arbitrage-free_pricing] basically boils down to saying that to avoid arbitrage possibilities in pricing, risk-adjusted cashflows must be discounted at a risk-free rate. The scope of this theorem is continuously traded securities; it seems reasonable to apply inductive logic to extend this result to any commodity well modeled by a Walrasian auction [http://en.wikipedia.org/wiki/Walrasian_auction]. This would include, I think, a marketable skill set. When the OP talks about 'my discount rate', he must be referring to his personal preferences - i.e., his utility function.
0RobinZ12yI don't know much economics, but I think the point I was making was that other utility functions were possible. I don't have any comment on pricing risk.
0[anonymous]12yThe problem with your math is that your r(t) isn't the same as my r. If ) gives the factor by which you discount a future resource acquisition that will occur in the future, then r is for people who have a ) of the form = \frac{1}{r}^{\Delta%20t}) Your equation is for people who have a ) of the form = \frac{k}{(\Delta t)^{2}}) where k is some constant.
2John_Maxwell12yHm, interesting. Also, I forgot to mention that it approximates by making the software developer's product release continuous, when in reality they might be releasing a product, say, every few months.

John:

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

0gwern11yI think you could just compare sums of money. Would you trade 10$ now for 20$ in a year? I would, so your discount is <2. Would you trade 10$ for 15$ in a year? Then your discount is <1.5. And so on. If you just have trouble comparing dollars, then maybe you could compare coffee, or books, or something.
[-][anonymous]12y 0

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).

[-][anonymous]12y 0

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

2John_Maxwell12yI don't think constant g is important. If your g changes you can just recompute the formula. Remember, the formula gives your instantaneous rate of value creation. Glad you liked it. I decided on two posts in case some people didn't know integrals or weren't concerned with the derivation of the equation and were willing to take my word that it made sense.
0RobinZ12yI was looking at the "I don't think constant *g* is" in the Recent Comments sidebar when that occurred to me. Even cases where it's training are dealt with - just treat the payoff as distributed over the period of training, I suppose. That's a good thought - sort of an inverse of what EY did with technical appendicies [http://lesswrong.com/lw/1rv/demands_for_particular_proof_appendices/].
2John_Maxwell12yIt doesn't for me. Can you read this [http://diigo.com/0axtq] version?
2cupholder12yYeah, that one works. Edit - the problem must be at Reddit's end, even going to reddit.com gives me a 404 here.