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.

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.