Will you still need me, will you still feed me, when I'm 640

Imagine a world where humans lived much longer than their current 70-80 year range (for Westerners).  Imagine they lived 640 years.  Earning just a single 1% above the prevailing inflation rate would transform one unit of capital into 601.  That's surely going to be enough to retire on.  One presumption here is nothing about the economy changes, but in a sense this could change everything - for a start there'd be a lot more capital seeking a return.  Also, at what point in those 640 years would we decide to stop working?  Nevertheless, that's an assumption of this post.

Would it be enough to retire on?  To answer that, I'd first like to know how a typical salaried person's salary growth would slow down over the centuries.  We work from 20 to 60, approximately, and we see over that period a growth in salary.  This (again in the Western world) represents a career trajectory which, I think, we can't extend onward for centuries. The pattern of real  lifetime wage growth, I strongly suspect, would flatten out after a while and we'd have a more or less inflation-stable income.  Of course, so much is uncertain here.  Would we be as productive or less so, aged 200?  We're in the realm of science fiction, for sure but that's useful to imagine a flat-lining, since one could then conceive some parameter, ω, ranging typically between 0 and 1, which, when achieved might lead us to retire.  The parameter represents the fraction of our mature stable salary S such that we'd be happy to retire on ωS for our remaining time alive. By 'retire' of course, I don't mean become inactive, I mean having in essence the ability to self-fund a liveable income.

To translate this into capital terms, how much capital would one need to accumulate so that it earned us a real return of ωS indefinitely.  Let's further assume for simplicity that we immediately start earning S at the beginning of our working career.    In other words, how much capital would you need to accumulate in order to be able to pay for a perpetuity worth, in real terms, ωS paid to you yearly forever. Given the length of time here, it is fine to approximate the fixed term annuity with a perpetuity, since they'll both amount to a similar value, and the maths for a perpetuity is simpler.

This capital amount R would be our retirement trigger such that  $R=\omega S/r$.  With $r$ the real rate of return in the above example set at 1%, $R=100 \omega S$.  A general guideline of 67% is often given for the expected final pension of retiring Westerners.  This means on the ultra conservative estimate, you'd better have 67 times your salary before you can retire.  That's a lot.

How long would you have to work when you could put some savings fraction $\delta$ of your salary away every year and until you reached  $R=67 S$? I.e. how many annual payments of $\delta S$, growing each year in a retirement pot for you at a real rate of return again of 1% would result in a pot of size  $R$?  This second problem isn't a simple annuity problem, since even though you're paying a fixed amount each year for $n$ years, the real point is that each year your pool of retirement capital grows, and it is this larger pool which is subject to the following year's growth of 1% real return.  This compounding element will mean many fewer years to wait for freedom from wage slavery.  But how many years?  This structure isn't a plain annuity but more like a sinking fund, whose formula is $\frac{Kr}{(1+r)^n-1}$ where $K$ is the target amount you're planning to need in $n$ years.  Assuming annual compounding and real growth of $r$ which you can consistently receive on your growing fund.

For my current needs, I'm saying that $\delta S = \frac{r\omega S/r}{(1+r)^n-1}$.  I now want to rearrange this to solve for $n$.  First of all I notice that on the top line the rates cancel, so I can write
$(1+r)^n-1= \frac{\omega S}{\delta S}$ and rather conveniently the capital amounts cancel,  $(1+r)^n-1= \frac{\omega}{\delta}$,  The capital amounts cancelling merely reminds me that this simplistic analysis would hold, given the same simplifying assumptions, for any wage slave, regardless of their actual income level.  Moving on, $(1+r)^n= \frac{\omega}{\delta}+1$ and if I take logs on both sides $n \ln(1+r)= \ln(\frac{\omega}{\delta}+1)$ before finally arriving at $n = \frac{\ln(1+\frac{\omega}{\delta})}{\ln(1+r)}$.

Let's plug some sample values in.  Stick with $\omega=\frac{2}{3}$.  Now, we all try to save 5% of our salary at least into the pension pot each year with our current life timeline.  Let's assume this doesn't change.  $\delta = \frac{1}{20}$.  Again let us make the real return 1%.  That's $\frac{1.1583}{0.0043}$ or 268 years (or 41% of your extended life of 640.  For reference purposes, 41% of 60 working life years is about 25 years.  So if you start at 20 and die at 80, saving 5% a year, on the expectation of two thirds final salary means you can retire at 80-25 or 55.

What if you earned a real 2% on your annual saving, all else staying the same?  You get 134 years of saving.  And if you were prepared to forgo 10% of your salary each year for pension saving, all other things the same?  You'd work for 205 years.  Next, if you got a 2% real rate and saved 10% of your salary, you'd take 103 years (16% of your potential working life) before you could retire.

According to the Fed, 5.89% is the Western world's long term current real rate of return.  So, unless you were unlucky enough to hit a world war, this rate of return on 10% pension contributions would have you working for only 35 years, out of your 640 years of living.

By the way, 67% salary as an annuity, discounted at 5.89% real, costs you about 11.4 times your salary.  The major element I leave out of the above is the fact that the annuity your retirement pot buys you is not going to grow with inflation.

UK working age income is currently (2018) 18k p.a.  So you'd need at least 204k in your pot.  For richer folks, say on 100k, you'd need more than 1.1 million in your pension pot to get you a 67k lifestyle (less, assuming the power of inflation).  I am, of course, ignoring the UK government state pension.