# The Math of When to Self-Improve

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. $\lim_{z\to\infty}\int_0^z\frac{a}{r^t}\,\mathrm{d}t=\lim_{z\to\infty}-\frac{a}{r^{t}\ln{r}}\Big|_{t=0}^{t=z}=\frac{a}{\ln{r}}$ 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 $p+\frac{f}{\ln{r}}(g - 1)$ 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: • 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. • It’s ra­tio­nal to be mo­ti­vated en­tirely by the net pre­sent value of one’s fu­ture earn­ings. There might be other rele­vant sources of util­ity for the de­vel­oper, but did you re­ally want me to com­pli­cate the post by dis­cussing them? Pro­gram­mers are im­mor­tal (and never switch ca­reer.) 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. A pro­gram­mer’s skill in­creases in a smooth, de­ter­minis­tic, pre­dictable and ex­po­nen­tial man­ner over time. 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. A pro­gram­mer’s skill is a scalar-val­ued quan­tity pro­por­tional to their (po­ten­tial) in­come. 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. • 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-have-you), in­di­rect util­ity (im­prov­ing our abil­ity to ob­tain ad­di­tional goals), and may it­self change our util­ity func­tion (e.g. self-mod­ify­ing to be a per­son who cares more about phys­i­cal fit­ness has al­tered the co­effi­cient of many junk foods in my util­ity func­tion). It’s not wholly ir­rele­vant, but the in­puts are so ill-defined as to ren­der for­mal­iz­ing it of no prac­ti­cal value. I’m think­ing of writ­ing a se­quel called When to Self-Im­prove in Practice If this is an ac­cu­rate de­scrip­tion, I’d be very much in­ter­ested in read­ing it. • The con­text was op­ti­miz­ing job earn­ings, not tran­shu­man­ist brain mod­ifi­ca­tions. I think the model is rea­son­able in that con­text, if a bit hard to ap­ply. • When 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. • I 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)”. • I 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.” • Fair enough; it was just too fun not to post. (Of course, they ac­tu­ally did ti­tles like that in the 17th cen­tury.) • I 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. • I think it comes from the units in the defi­ni­tion of g: (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. 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’ • Oh, 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. • weird • Re­dun­dant, yes. Not sur­pris­ing that I had the same re­ac­tion twice to the same post, though. Delete newer one (Y/​N)? • Please 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 figur­ing out their value. For ex­am­ple, the Su­per­memo docs es­ti­mate that the av­er­age flash­card will re­quire “30–40 sec­onds” per 3 years; I be­lieve I’ve seen Anki-re­lated ma­te­rial es­ti­mat­ing the to­tal time­cost of re­view at around 5 min­utes for 20-30 years. (The ex­po­nen­tial back­off for re­view in­her­ent to spaced rep­e­ti­tion means that any fur­ther out is hard to guess at. At that point, you’re go­ing years be­tween re­views.) Both Anki and Su­per­memo keep de­tailed statis­tics, nor have I heard any­thing from Mnemosyne devs con­tra­dict­ing it, so this seems likely to be quite ac­cu­rate to me. At a min­i­mum wage of$8/​hr, that im­plies each card needs to be worth 66 cents to know. Or does it?

Dual n-back is harder to an­a­lyze. In my FAQ I have col­lected a num­ber of re­ports of in­creased IQ and gen­eral anec­do­tal benefits.

My own per­sonal regime is a daily 5 rounds at 270 sec­onds each, or roughly 22 min­utes, or roughly $3 at min­i­mum wage. I have done any nback­ing on 221 days ($663?). If that trans­lates into an effec­tive IQ boost of 10 points (seems to be the mid-range of those who did see boosts, and con­sis­tent with the Jaeggi 2008 re­sults), how much is that boost worth and was it worth­while? If we jump from mere cor­re­la­tion to cau­sa­tion and take a high es­ti­mate of an in­come in­crease of 25%, then it would cer­tainly seem to have been worth­while. But this isn’t tak­ing into ac­count dis­count­ing, and is quite sloppy.

• I’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—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, es­ti­mate how long it takes to solve, and once it’s solved record how long it ac­tu­ally took to solve and the de­tails of your solu­tion. Then pe­ri­od­i­cally re­view for trends. I’ve done a lit­tle bit of this; it was highly un­pleas­ant/​bor­ing but it seems in­tu­itively like it could be quite benefi­cial so I’m plan­ning to do it more.) • be crazy to value … at only$8/​hour.

Why? 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.)

Another way of think­ing the cards would be

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.)

first there’s the cri­tique you men­tion in your FAQ, which seems very damning

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”.

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.)

• Why? 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 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).

(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.)

Try 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.

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 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?

• 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?

Well… I sus­pect we may be hav­ing vo­cab­u­lary is­sues here. Gf 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 for a large in­crease in my Gc, why should I ex­pect any in­creased abil­ity to write proofs in math­e­mat­i­cal set the­ory which will ini­tially draw on Gf as a strange and alien sub­ject?

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.

• If fluid in­tel­li­gence doesn’t help me learn stuff faster, is it re­ally worth hav­ing?

Yes, you have to un­der­stand stuff be­fore your can learn it. And be able to tell the differ­ence be­tween non­sense and things ac­tu­ally worth learn­ing.

Doesn’t it seem likely that learn­ing things makes you bet­ter at learn­ing things?

Yes, it does make you bet­ter at learn­ing things. There has been con­sid­er­able re­search done on the sub­ject.

If this is true, could an in­crease in fluid in­tel­li­gence be the mechanism for it?

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.

• Dual-n-back train­ing does give a mod­est effect, as does ex­er­cise (and par­tic­u­larly cere­bel­lar tar­get­ted ex­er­cise).

Could you ex­pand on that? I had not heard that ex­er­cise ac­tu­ally af­fected Gf 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.

• I’ve added 2 stud­ies to the crit­i­cism sec­tion: http://​​www.gw­ern.net/​​DNB%20FAQ#criticism

• 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 im­por­tance of the fur­ther-away fu­ture—if you halve the in­ter­est rate, you (more or less) dou­ble the time into the fu­ture that is rele­vant to your calcu­la­tion of whether your ac­tivity will be prof­itable. And the fu­ture is un­cer­tain, as all post-crunch prop­erty de­vel­op­ers (and their bankers) know. The pre­vailing low in­ter­est rates en­couraged the spec­u­la­tion into the far fu­ture that we now know is differ­ent from that en­visaged.

Cor­po­ra­tions are (con­trary to AlephNeil) po­ten­tially im­mor­tal(ish), mo­ti­vated by the net pre­sent value of their fu­ture earn­ings, and don’t like switch­ing ca­reers. They are no­to­ri­ously short-term plan­ners in many cases, and un­cer­tainty is the rea­son why. Most think that im­prove­ments should pay off in 3-5 years, or they aren’t worth do­ing. It’s not the pre­vailing in­ter­est rate that counts here.

Self-im­prove­ment is the same—it’s pretty un­cer­tain—you don’t know what you’re go­ing to get for it. Well, in the less in­ter­est­ing cases like gain­ing a qual­ifi­ca­tion, you might have a rea­son­able es­ti­mate. But when do­ing self-im­prove­ment at the fore­front of hu­man knowl­edge, you se­ri­ously don’t know what you’re go­ing to get. But that’s by far the most in­ter­est­ing place to self-im­prove, I’d say.

I guess ev­ery­one on this blog is a self-im­prover al­most by defi­ni­tion—those who don’t im­prove just be­cause they like to won’t find this place in­ter­est­ing, and prob­a­bly won’t un­der­stand it ei­ther. I think most of us here will there­fore already have an­swered the ques­tion, and had some ex­pe­rience of what that an­swer leads to.

• They are no­to­ri­ously short-term plan­ners in many cases, and un­cer­tainty is the rea­son why. Most think that im­prove­ments should pay off in 3-5 years, or they aren’t worth do­ing.

Cor­po­ra­tions also have many is­sues that a truly uni­tary im­mor­tal en­tity might not; the prin­ci­ple-agent con­flict 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 more tractable. I don’t see how your model ac­com­plishes this. On the con­trary, it seems to me like com­put­ing the value of differ­ent strate­gies on an in­stan­ta­neous ba­sis com­pli­cates your model by re­quiring the use of in­te­grals.

A model that con­fined it­self to calcu­lat­ing the net pre­sent value of a finite # of months or years of var­i­ous strate­gies would have been both (a) more ac­cu­rate, in the sense of bet­ter re­flect­ing real-world con­cerns, and (b) eas­ier to un­der­stand, in that it would have re­quired less math.

I do hope you pub­lish the ‘prac­ti­cal’ half of your ar­ti­cle, but I urge you to be care­ful not to let your abil­ity to do math get in the way of your abil­ity to de­velop and teach use­ful mod­els.

You also may wish to avoid mimick­ing the for­mal style of text­books, e.g., “ex­er­cise for the reader,” “ques­tion to con­firm un­der­stand­ing.” This tone of voice can be easy to use, but it’s odd and un­pleas­ant for me to read it, given that (in the­ory) we’re all peers here. You may have some­thing to teach us, but you’re not ex­actly my pro­fes­sor.

• You seem to have got­ten through to a cou­ple of peo­ple on the strength of your math

That’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).

there’s no such thing as the “in­stan­ta­neous an­nual value” of self-im­prove­ment in the real world.

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.

What was your in­ten­tion when you de­cided to com­pute the in­stan­ta­neous an­nual value of differ­ent strate­gies?

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.

A model that con­fined it­self to calcu­lat­ing the net pre­sent value of a finite # of months or years of var­i­ous strate­gies would have been both (a) more ac­cu­rate, in the sense of bet­ter re­flect­ing real-world concerns

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.

(b) eas­ier to un­der­stand, in that it would have re­quired less math.

Well the end­ing for­mula would have been more com­pli­cated:

$\\int\_\{0\}^\{z\}\\frac\{a\}\{r^\{t\}\}dt = \-\\frac\{a\}\{r^\{t\}\\ln\{r\}\}\\Big |\_\{t=0\}^\{t=z\} = \\frac\{a\}\{\\ln\{r\}\} \- \\frac\{a\}\{r^\{z\}\\ln\{r\}\}$

$p \+ \(\\frac\{f\}\{\\ln\{r\}\} \- \\frac\{f\}\{r^\{z\}\\ln\{r\}\}\$(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 unim­por­tant.

You also may wish to avoid mimick­ing the for­mal style of text­books, e.g., “ex­er­cise for the reader,” “ques­tion to con­firm un­der­stand­ing.” This tone of voice can be easy to use, but it’s odd and un­pleas­ant for me to read it, given that (in the­ory) we’re all peers here.

OK, so maybe the ques­tion/​an­swer should just go in ital­ics be­tween sec­tions, like

Who is John Galt? An­swer in rot13: fhcrezna.

so you wouldn’t think I was mak­ing a ploy for high sta­tus. (Heh, I can’t see how us­ing rot13 mimicks the for­mal style of text­books :)

• 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.

Sure! 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 read your next ar­ti­cle. I’m just con­fused about what good you think (3) is ac­com­plish­ing.

Per­haps you didn’t mean to re­fer to (3) at all. Per­haps you just used the phrase “in­stan­ta­neous value” to mean “net pre­sent value.” That would be some­what con­fus­ing. Espe­cially in a post mak­ing heavy use of sim­ple in­te­grals, I as­so­ci­ate the word “in­stan­ta­neous” with the idea of deriva­tives and slopes.

My high dis­count rate might have been what caused me think the term you sug­gested was unim­por­tant.

I’m cu­ri­ous as to how ac­cu­rate your self-es­ti­mate of your dis­count rate is. Are you heav­ily in debt or oth­er­wise deeply lev­er­aged? You should be able to find all kinds of op­por­tu­ni­ties to bor­row at less than 15% in­ter­est.

OK, so maybe the ques­tion/​an­swer should just go in ital­ics be­tween sec­tions.

That would cer­tainly be eas­ier for me to read. Know­ing how the LW com­mu­nity works, I sus­pected you weren’t ac­tu­ally mak­ing a ploy for higher sta­tus. It’s a men­tal en­ergy drain, though, to have to sit there re­mind­ing my­self that you’re just us­ing a funny reg­ister, and not ac­tu­ally try­ing to be an au­thor­ity figure. The en­ergy drain takes away from my abil­ity to read and en­joy and learn from your post, and I sus­pect at least some other peo­ple would feel the same way. And, yes, rot13 is a clue that you’re not ac­tu­ally full of your­self. :-)

• I’m cu­ri­ous as to how ac­cu­rate your self-es­ti­mate of your dis­count rate is. Are you heav­ily in debt or oth­er­wise deeply lev­er­aged? You should be able to find all kinds of op­por­tu­ni­ties to bor­row at less than 15% in­ter­est.

I 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.

• Prob­a­bly if I had a steady stream of in­come I would.

Makes 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.

But I’m not sure we should avoid a ques­tion just be­cause it’s hard to an­swer.

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!

• You also may wish to avoid mimick­ing the for­mal style of text­books, e.g., “ex­er­cise for the reader,” “ques­tion to con­firm un­der­stand­ing.” This tone of voice can be easy to use, but it’s odd and un­pleas­ant for me to read it, given that (in the­ory) we’re all peers here. You may have some­thing to teach us, but you’re not ex­actly my pro­fes­sor.

I would add that this is an­other rea­son to sim­plify the math—do­ing so elimi­nates the need for ex­er­cises by mak­ing the an­swers less con­fus­ing.

• 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 skills. Choose a pro­ject that you need some new skill to com­plete. A real pro­ject, cre­at­ing a product that that you are go­ing to sell.

I be­lieve this ap­plies gen­er­ally to any form of self-im­prove­ment: do­ing stuff, and learn­ing to do stuff bet­ter, can­not be sep­a­rated.

• f and p ap­pear to be the same thing: the an­nual in­come they would get from cod­ing full-time.

If 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.

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. 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 skills. Choose a pro­ject that you need some new skill to com­plete. A real pro­ject, cre­at­ing a product that that you are go­ing to sell.

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.)

I be­lieve this ap­plies gen­er­ally to any form of self-im­prove­ment: do­ing stuff, and learn­ing to do stuff bet­ter, can­not be sep­a­rated.

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).

• 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.

• 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”/​.

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

If you put LaTeX code af­ter “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 is good if you don’t know LaTeX. Use this 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.

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).

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 :)

• Use this 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.

Test­ing:

$\\int\_a^b f\(x\$%20dx%20=%20\sum_{n=1}%5E{\in­fty}a_n%20+%20\sqrt[17]{84})

Thank you for cre­at­ing this!

• You’re wel­come!

• Fan­tas­tic. 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.

• There 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”.

• I’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

$r(t\$=\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”.

• When 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.

• I 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

$\frac{dt}{ds}=\frac{log(q\$}{log(r)})

and pro­duce the same prob­lem, so long as log(q) is uniformly the same sign as log(r).

• I am sad be­cause my at­tempt to teach you about prefer­ence re­ver­sals has al­most cer­tainly failed.

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.

• I 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.

• That agrees with my sus­pi­cions—thank you.

• I am open to learn­ing from oth­ers here how to im­prove the so­cial side of my com­mu­ni­ca­tions in di­alogs like this.

Given that the con­fu­sion be­tween you and RobinZ was dis­pel­led be­low, 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.

• I 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:

$dP=\frac{dE}{r^t}$

which cor­re­sponds to the sim­ple formula

$\log{\frac{dP}{dE}}=-t\log{r}=\int_0^t-\log{r}{dt}$

I main­tain that this last for­mula,

$\log{\frac{dP}{dE}}=\int_0^t-\log{r}{dt}$

still holds when r is no longer a con­stant, and there­fore (as dE = E(t)dt):

$P=\int_0^{t}E(z\$\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

$P=F_{t}\exp{\int_0^t-\log{r(z\$}dz})

• Sorry, the con­crete ex­am­ple. Take

$r\(t\$%20=%201%20+%200.001%20t)

and point fu­ture in­come functions

$F\_1 = \\100 \\times \\delta\(t\-10\$;%20F_2%20=%20\$100%20\times%20\delta(t-20)) which (us­ing the Dirac delta func­tion) cor­re­spond to in­stan­ta­neous in­comes at times t = 10 and 20. That is, 2010 and 2020. Us­ing these func­tions, $\\mathbf\{P\}\(F\_1\$=\$100\times\exp\left(%20\int_0%5E{10}-\log(1+0.001t)dt\right)\ap­prox\$95.14) and $\\mathbf\{P\}\(F\_2\$=\$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 $\\mathbf\{F\_\{2010\}\}\(F\_2\$=\$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). • The 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 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. 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. • I 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. • The prob­lem with your math is that your r(t) isn’t the same as my r. If $v(\\Delta t\$) gives the fac­tor by which you dis­count a fu­ture re­source ac­qui­si­tion that will oc­cur $\\Delta t$ in the fu­ture, then r is for peo­ple who have a $v(\\Delta t\$) of the form $v(\\Delta t\$ = \frac{1}{r}^{\Delta%20t}) Your equa­tion is for peo­ple who have a $v\(\Delta t\$) of the form $v\(\Delta t\$ = \frac{k}{(\Delta t)^{2}}) where k is some con­stant. $v\\Delta$ • Hm, 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? • I 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. • 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).

• $\Delta t$

• 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.)

• I 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.

• I 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.

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.

That’s a good thought—sort of an in­verse of what EY did with tech­ni­cal ap­pen­di­cies.

• It doesn’t for me. Can you read this ver­sion?

• Yeah, that one works.

Edit—the prob­lem must be at Red­dit’s end, even go­ing to red­dit.com gives me a 404 here.