The S-curve Killed Exponential Growth

The End of Days Fund: A Thought Experiment

Here's a fun hypothetical.  What if while she was pregnant Mary got together with her husband Joseph and they decided to invest in the future for their son Jesus.  Now, they know he's the promised messiah, but they probably didn't realize he would be crucified and rise again in a couple thousand years.  But let's say for the sake of the hypothetical they knew about this.  Suppose Joseph said to Mary, "What if our son gets crucified and then after a couple thousand years comes back to Earth.  Let's start a savings fund for him to use on his ultimate Return.  Instead of sending him to college, we should start an End of Days Fund for Jesus when he comes again."

Now, the problem with starting a fund at this point is that Joseph and Mary are poor.  We know this because they were required to make a sacrifice for their firstborn son, but the Law of Moses allows poor people to substitute two turtle doves if they can't afford a lamb.  Joseph and Mary sacrificed doves for Jesus' birth, which they would only do if they were too poor to afford a proper lamb.

A little while later, when Jesus was a "young child", some wise men came along and gave them nice gifts.  Continuing our hypothetical, Joseph suggests maybe they could use the money from those gifts to pad the EoD Fund.  Meanwhile, Herod is looking for Jesus and starts a 'slaughter of the innocents'.  God tells Joseph to go on an emergency 'vacation' to Egypt, and they end up having to pawn the gifts to fund their escape.  The money to start the fund is once again out of reach.

Fortunately, Mary figures out basic equations of compound interest while staying with her cousin Elizabeth.  She says, "Don't worry Joseph.  I've figured out a solution for the EoD Fund.  I've done the math and, as I figure it, we don't need to start off with lots of money.  You see, we have lots of time to work with, and that's as good as money.  It'll be a while before our Son is resurrected, leaves, and comes again.  Let's just invest in a safe, low-interest fund.  Over the course of a couple millennia compound interest will allow it to grow to an unfathomable size.  By the time our boy comes back He'll be loaded!"  So they invest the equivalent of $1 and set up a perpetual fund-management firm to watch over the investment.  They play it safe and target a paltry annual return of 2% per year.  Then they let it run, undisturbed, for a couple millennia.  It's just sitting there, growing at 2% per year, waiting for the Return.

Let's say Jesus comes back in the year 2020.  He'd have well over $92 quadrillion in the bank upon his return!  It has been said that compound interest rules the world, and if our hypothetical worked as planned it would easily allow Christ to rule the world on its own.

Except it won't work.  It's impossible.

As I related that story, something in the back of your mind should have been itching with the idea that this could never happen.  The question is why is it impossible?  The problem isn't with the math.  If you use a simple time value of money calculator, you find that a $1 investment over 2020 years comes out to $92 quadrillion.  I once had an engineer as a roommate, and in situations like this one he would always insist on doing what he called a "reasonableness test".  The idea is that you ignore all the detailed analysis about how difficult it would be for Joseph and Mary to find a good investment firm two millennia ago, or getting continuous investment vehicles, and focus on the simple implications of what your calculations tell you.

Let's apply that to the EoD Fund.  With over $92 quadrillion, Jesus could purchase pretty much everything on Earth a couple times over.  He would, in theory, have more money than actually exists.  But by definition he can't own more than everything that exists.  The hypothetical fails the reasonableness test, because you can't own investments worth more than everything.
People invest money, but from the perspective of thousands of years they can only invest for short periods.  Perhaps the problem with our hypothetical is that nobody can really live that long.  However, there are corporations and other legal entities that are perpetual.  They don’t die, so they could participate in our hypothetical as ever much as Joseph and Mary.  Is it not feasible for an institution that expects to be in existence for hundreds of years to pursue this strategy?   They could set aside a paltry sum of money for a long period of time and then come back to it later and basically rule the world.  Is that possible?

Again no.  But why?

This whole hypothetical relies on the idea of perpetual compound interest.  Compound interest is a great idea in theory.  In practice over the course of a lifetime it is what fuels most retirement accounts, and successfully so.  It has made countless people wealth and it can make you rich, with a little patience.  But it can’t make you so rich you rule the world.  This is because it exists within a finite system, and our calculations extending out into the quadrillions of dollars incorrectly assumed we are living in an infinite system.  The difference: no finite system can grow exponentially forever.  That’s what it means to be finite, you can’t grow infinite sums.

Here’s how that usually plays out:

  • There is an initial phase where you see what looks like exponential growth.  If you match it against a graph of exponential growth it will line up perfectly.  Some people run around claiming it is exponential growth.  They make wild proclamations based on that prediction.  "Doomsday is approaching!"The singularity is near!"Eternal prosperity is upon us!"
  • But there's always an inflection point where practical considerations start to take over.  Investments are driven by real-world exchanges of one person trading with another person.  On a large scale we can ignore those considerations.  But at some point you’ve invested so much money into the system that you’re borrowing from the last person on Earth willing to do so.  They aren’t going to give you the same rate of return, or perhaps they’re a really poor investment risk.  Long before that point, you’ll start having problems finding new investors who are of as good a quality as your first choice.  Dozens of other factors will come into play as you reach into the billions of dollars, let alone the trillions or quadrillions.  But importantly, they don’t matter at the beginning - before you’ve exhausted your initial pool of good resources.  It isn't until later on that things you once took for granted become problematic.  We call these rate-limiting factors.  Every finite system has them. 
  • Eventually, the practical considerations become more important than the thing that's causing exponential growth.  You get down to that last man on Earth and he’s only willing to deal with you by paying you a 0.001% interest rate compounded every decade.  Instead of a 2% return per year, eventually you get down very close to a 0% return.
This  sequence of events produces what is called an S-curve.  We call it that because of the shape, and it's a hallmark of finite systems.  It looks remarkably similar to an exponential curve, and a common mistake you find among fans of science is they will mistake a developing S-curve for an infinite exponential curve.

This seems rather obvious when we look at the case of Jesus' apocalypse fund.  For example, after he dies, 33 years in, he's still got less than $2 in the fund.  His mother is still alive, and John is charged with taking care of her, so early on the fund survives.  Fast forward 250 years, Constantine is emperor, and he sees there might be some value in getting into this Christian action.  When he looks at the Apocalypse Fund he's not impressed.  It's only got about $141 to its name.  Constantine has bigger fish to fry, and he leaves it alone.  So far so good, so much exponential growth.

By the time Islam is taking over the Levant, the fund has grown to almost $145,000.  It's starting to get pretty sizeable, but there are people who are richer than that hanging about (in inflation-adjusted dollars).  It's not hard to find people willing to invest in the small but growing fund.

By 1000 years, the fund has almost $400 million.  Someone is going to embezzle from the fund.  And in the unlikely chance it survives intact, always yielding a 2% rate, you can bet that by 1250 when the fund is worth $56 billion there will be a lot of people interested in stealing from it.  Around this time or before, you'll see returns dipping well below 2% as you struggle to keep up with the impossible task of making the kind of returns you started out with.  You're still making lots of money - because you have a lot to start with, but the rate at which you're making money is starting to slow down.

With the incentives for fraud and outright theft, plus the difficulty of finding enough people who want to borrow at 2% when there's all this money circulating from the EoD Fund, you're going to struggle to not lose money before 1500.  You certainly won't see the hypothesized continuation of exponential growth up near $8 trillion.

Other practical considerations include the impossibility of cashing out any kind of savings vehicle that large.  Who is going to give you the money?  By definition, they have to have money or something valued equivalent to the owed sum in order for you to realize gains from your investments.  Sure, on paper they may be committing to pay you 2% next year on what you gave them this year, but that doesn't mean they will be able to pay up when the time comes.  So if you're looking to do some time travel investment a la the above thought experiment, don't expect you'll be able to own the world by investing a dollar.  Compound interest is powerful, but over hundreds of years this power runs into the rate-limiting factors of finite systems.

This scenario is only going to break down if you let it run for a couple millennia.  Within normal human lifespans it never comes into play.  So why is it a useful thought experiment?  Because it forces you to think about how expanding finite systems can look like infinitely expanding exponential systems - until they don't.  We live in a finite universe, but most people are trained to think in terms of infinite mathematical concepts.  Once you catch the idea of how finite systems have to differ from infinite ones you can start to see this concept everywhere.  And you can adjust your expectations accordingly.

The important idea to take away from this thought experiment is the distinction between infinite and finite trends.  There are a lot of systems like this, where we should expect S-curves, but instead we see journalists and futurists erroneously projecting indefinite exponential growth.  The cause of the error is simple: they’re projecting an infinite trend line onto a finite system.  Learn to see the finite limitations on the system and you'll beat the popular predictors.  Let's look at one well-known example.

Life expectancy versus lifespan

There have been significant advances in life expectancy over the past century or so.  Some people look at this trend and conclude that of course it will continue to increase.

But simply living longer isn't the only thing this medical advance gives us.  An interesting thing happens if you continue to extend the line of improving life expectancy farther out into the future.  Eventually you start seeing annual improvements of life expectancy of more than one year.  If you extend average human life expectancy each year by at least one year, people stop dying of old age.  If we get to the point of medical advancement where we extend life expectancy by five years every three years that's the same as living forever!

There's a big problem with this story, though.  Life expectancy is not a measure of how long you're likely to live.  It's a measure of how long the average person actually lives.  This includes babies.  If you have 100 people and 50 of them die in their mid-eighties, but the other 50 die under the age of five you end up with a life expectancy in the mid fourties.  This is called a bimodal distribution, and it's the bane of all people who want to use an average to understand a subject.

In this scenario, the average age of death (life expectancy) is in the middle of the graph, exactly where life expectancy is at its lowest.  You could easily look at the single statistical output and come to the wrong conclusion.  "Life expectancy of 45?  That means few people are going to live to fifty!"  So far, we've been dealing with a perfect hypothetical, where everyone either dies young or lives to old age.  And although the real world is more complex than that, this hypothetical is not too far from reality.

One of the biggest advances in medicine over the past century or so has been the reduction of infant and early childhood mortality from very high to very low.  Looking at that bimodal graph again, most people aren’t dying during middle age.  That’s because the young and elderly are generally more susceptible to things that kill.  You rarely hear of healthy middle-aged people dying of the flu, but a baby with small lungs or an elderly person with a weakened immune system is much more likely to struggle during flu season.

Thus, people who survive their first decade or two of life are highly likely to survive the next three.  The distribution on our bimodal graph starts to make more sense.  We can see why there’s such a trough in the middle.  We can also see how modern medicine can all but erase that huge bump on the left side of the graph (the children who die young).  It can still pull up the average by leaps and bounds without any movement on the right side of the graph - the elderly.  Fewer babies dying pushes the average age of death (life expectancy) out close to the age most of us expect to kick the bucket.  This is a situation we can only mine once.  There are only so many dead children bringing down the average that we can save.

More importantly, this sheds a different perspective on the idea that we're already making the kind of advances that will eventually have us pushing death back, eventually extending our lives by five years for every three years a person lives.  People back in the 1800's didn't die at age 35.  They died in their fifties, sixties, and seventies.  They were still a bit younger on average than you and I, because modern medicine does help extend life in the twilight years as well.  But we're not living twice as long.

The solitary number "life expectancy" leads us in the wrong direction when we don't ask deeper questions of the data.  When we look at the change in distribution more closely, this statistic doesn't tell us we're going to live longer.  It tells us we're much less likely to die as babies.  But once you've reduced infant mortality (less than 1 year old) from 10% down to less than 1%, you can expect the gains next century aren't going to be exponentially higher than those of last century.  Instead we should expect to see diminishing returns from reductions in early life mortality.

But people who get excited about medicine finding a way for us to live forever aren't really thinking about improvements in childhood mortality as a way to get there.  That means the statistic they're really interested in is not life expectancy.  We're interested in lifespan: how long a person is able to live.  When it comes to lifespan there's no evidence we're making any progress.  In fact, we may be moving in the wrong direction here.  For a while now, the three biggest killers in developed nations have been heart disease, diabetes, and cancer.  Given that obesity is a strong predictor for all three of these conditions, and given that obesity is seeing a dramatic rise - especially in the USA - it's only a matter of time before we start seeing significant declines in life expectancy.  Hopefully we can reverse that with future discoveries that will help people maintain healthy body weight and/or advances against cancer, diabetes, and heart disease.  But that's about making up lost ground, not extending the metric we really care about: lifespan.

Extending lifespan is a separate project

Lifespan isn't getting longer due to amazing new medical treatments.  Helping kids not die is noble and wonderful, and I’m not breaking any philosophical ground by saying we should keep at it.  But we're not going to all start three hundred years by making sure kids don't die young.  Those are two separate projects.  That observation is obvious when we stop staring at the graph of improving life expectancy over time and start considering the reasons we’re making finite gains in the finite system we’re operating under.

In order to extend lifespan we have to turn our attention to keeping old people live longer.  That has proven to be a tough nut to crack.  Usually cause of death isn't declared as "old age" even when that's basically what happened.  Every once in a while you'll get a physician who puts down something like, "age-related illness", but more often it's something like heart failure or pneumonia.  Seeing this, there's a temptation to say, "but if we found better treatments for heart failure and geriatric pneumonia we could extend lifespans!"  Probably not by much.

Old people don't get pneumonia from the common cold by accident.  They get sick from common illnesses (think shingles) because their immune system weakens over time.  This gradual decline is part of an aging process scientists still only vaguely understand.  We should absolutely work to reduce suffering by working on problems such as geriatric pneumonia and heart failure.  But we should do that because it alleviates current illness, not because we expect it to significantly prolong life.

There's a temptation to think we can just go in and fix each problem and the patient will live longer.  Obviously, if someone was going to die of pneumonia and they don’t, you’ve extended that person’s lifespan.  So what if they get heart disease, you fix their heart and move on to the next problem.  Over the course of dozens of advancements, eventually you'll see solid gains to lifespan! 

Extend by how much?  Are we talking about weeks, months, or years?  We've grown accustomed to thinking of and expecting big gains in life expectancy from earlier work on infant mortality.  When you save an infant from dying right after childbirth, they'll soon start to thrive and enter that period of long low-probability of dying.  There's no reason to expect the same sort of gains when treating the elderly.  If you cure a very elderly person of one disease, the likelihood they'll develop a new medical problem goes up, not down.

The older a person gets, the higher the likelihood they'll run into a new medical problem.  If we figured out a way to get most people to live past 100, we should expect that would happen by playing an increasingly frenzied game of clinical whack-a-mole with disease.  Of course we'll continue to find new ways to take care of our aging population, and we'll try to do so by helping to prolong their lives.  But the realities of medical treatment shouldn't cause us to project medical advancements against disease to produce genuine improvements to lifespan.  Indeed, on this side of the graph we see more reason to expect medical advances to produce diminishing returns.

There is a way to improve lifespan.  Someday we will understand aging better, and someday after that we'll figure out how to tweak the process.  Perhaps then we can realize the dream of living forever.  It could just be a matter of research.  How much we can extend lives with some future invention is as unknown as the future.  Biology is complex, so it probably won't be an easy or simple process.  I'll save that for a post another day.  Let's look again at the promise of living forever by extending lifespan through new discoveries in medical science.  The idea was that we're seeing lots of new discoveries in medicine that save lives.  Eventually this has to extend lifespan, right?  But if the only way we're going to see meaningful extensions of lifespan is for an underdeveloped field that hasn't produced anything like clinical treatments in the past to one day find a 'cure' to aging, we're not extrapolating from the curve anymore.  Now we're wishing for future advances in science and technology that may or may not happen.  It's like projecting that everyone will be riding hoverboards and flying cars in twenty years.  It could happen!  But that's what they said twenty years ago, and we're no closer today than we were back then.  (Unless you count air taxis as a kind of proto-flying car.  Still, where are the hoverboards?)

From all this, you might assume I think extending lifespan is a futile project.  Actually, I don't think that at all.  I think it's a great thing to pursue, and I hope researchers will have success in tackling this difficult challenge.  But realistically, extending lifespan is a different project than improving life expectancy.  For life expectancy, we're nearing the end of what we can do in the developed world.  We're nearing the top of the S-curve.  We can expect diminishing returns from here on out.  That means as we develop more and more treatments, people will increasingly have to make the decision to die a couple weeks or months earlier, or endure endless visits to the hospital to fight for each additional day.  Alternately, in the project of truly extending lifespan, it's not clear we're even at the beginning.

What to do with projections of exponential growth
It's easy to see when something is an S-curve once the growth starts to slow down.  It's much harder to identify an S-curve while it's still undergoing exponential growth.  The phenomenon looks like it will continue growing faster and faster forever.  Since the world is finite, it's tempting to claim that all phenomena will resolve from exponential growth down to an S-curve.  For example, we've experienced decades - even centuries - of exponential improvements in science and technology.  Will those improvements continue exponentially forever, or will they cease at some point?  Will economic prosperity continue improving exponentially, or will it slow down to the point of diminishing returns?

Perhaps the most difficult problem of all is defining the inflection point of an S-curve.  Remember that the curve is defined by the fact that it's part of a finite system.  All finite systems have limiting factors that impede exponential growth.  So finding the inflection point should be as simple as finding the limiting factor.  If for science that limiting factor is the volume of knowledge that remains unknown, we'll never know where the inflection point is.  Because we'll never know how much is still out there waiting to be discovered.

There doesn't have to be one single limiting factor, though.  For example, with economic prosperity one limiting factor is the amount of energy coming to us from the sun.  We might calculate this and project how much energy could be harnessed by human ingenuity, then calculate an inflection point from there.  But maybe before we hit that point we discover a different limiting factor, perhaps a political one, that places limits on economic prosperity.  If we were aware of the solar emissions limiting factor, we might erroneously calculate the inflection point to be hundreds of years off.  We'd be surprised after a couple decades to learn of the new inflection point much closer than we thought possible.

Past generations have experienced systems that acted like exponential growth long past the lives of those experiencing them.  There are likely systems we live under today that will continue exponential growth far into the future.  They will keep going long enough that for all practical purposes we can count on them being infinite exponential growth curves.  The point isn't to say, "you're projecting exponential growth, and that's all just a myth!"  The heuristic is to recognize exponential extrapolations for what they are: an assumption that because something has been expanding rapidly that it will naturally continue to do so forever.  Can infinite growth occur?  Maybe.  But if you're in the middle of exponential growth, remember that finite systems produce S-curves, and at some point you'll reach your limit.


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