Thread by @Gergyl

f' = k.f, k > 0

But you knew that.

Hence f(t) = f(0).eᵏᵗ, k > 0

A fascinating function with important characteristics that few seem to grasp.

Because it explodes? No, because it is *always exploding*. The shape is perfectly fractal. It does not depend on the position on the t-axis, nor on the scale of the t-axis.

Is just the growth equation -- growth at constant rate k, the instantaneous growth rate per unit t. Think of k as a percentage, so say 10% growth per year (continuous, not discrete) would be:

f(at year t) = f(at year 0).e⁰˙¹ᵗ

Which goes: 1.0, 1.1, 1.22, 1.35, 1.49...

That is years zero to four ...

year five is 1.65
year ten is 2.72 (e!)
year 20 is 7.4
year 50 is 148
year 100 is 22,026
year 200 is 485,165,195
year 500 is 5,184,705,522,859,000,000,000, approximately.

It is an interesting function.

Ten percent constant* annual growth produces a five-billion-trillion-fold increase in just 500 years. Contemplate that for a moment. Then contemplate the mind of a classical economist who argues that some such process must be sustainable, because... (there is never a because).

*Distributed, not discrete, though the discrete result (for a lump addition each year instead of a distributed, instantaneously evaluated addition) is in fact very similar.

There are no sustainable f' = k.f, k > 0 processes in the observable universe. None at all. They do not exist. Except that humans routinely pretend that they do.

There are many f' = k.f, k > 0 processes, and not just in biological systems; it's just that none of them are sustainable. When you add yeast to malted barley, it immediately grows exponentially (f' = k.f). Then, later, it all dies. We call the result beer.

When you ignite a nuclear bomb (or a type Ia supernova), the reaction rate goes f' = kf ... for a while, until the thing blows itself apart. These are not sustainable processes. They cannot be. The equation says so.

What then is 3% economic growth? Obviously f' = k.f, k > 0 ... where f is some real-dollar (inflation corrected) measure of GDP, being all that we 'produce', or more commonly all that we consume, narrowly defined (but broad defined, economically...).

...Produce in one year, as estimated quarterly. Your classical economists will tell you this growth process is sustainable, because the meaning of 'dollar value' is constantly being adjusted (despite that inflation is removed), which voids the explosion problem.

...Which is to say that the definition of 'dollar' is non-physical, which is true, partly. Except look at the function ... a five-billion-trillion-fold increase in 500 years? (Less; that was 10%.) Is this a mere definitional change of what 'inflation-corrected dollar' means?

At 3% growth, the 500-year factor is times 3,269,017 just by the way. So what of 3% population growth? Sustainable? That's what, three million times as many people 500 years from now? That would be about seventy-five thousand billion people in Australia. Yes, *billion*.

"But that isn't what I meant." Ok, so what did you mean? It's what the word means -- 'sustainable', able to be sustained. It says nothing about for how long. It means indefinitely.

There are sustainable growth processes, it's just that constant percentage growth, f' = k.f, isn't. It is not even close. Yet that is nearly always what politicians and 'development' promoters mean by 'sustainable growth'. Simply liars.

Maybe we'll write something about sustainable growth, but first a little more on f' = k.f. That's a differential equation (in cute Lagrangian notation). It says that the rate of change (f', the 'derivative', in units of f) is a constant (k) times the current total (f).

It's nearly always the case that the differential equation for a physical process (about *change*) is simpler than the explicit equation (about aggregates). Why? Dunno, is. But it's often not the case that the differential equation is explicitly solvable. Except this one is.

We already wrote the solution: f(t) = f(0).eᵏᵗ. You don't need the math, but it is rather pretty -- 'e' is 2.7182818... Why? Because that is the only number the powers of which have the derivative the same as the whole (f' = k.f !!). How's that? I dunno; ask your deity. Is.

(The k carries. It's one of those basic rules of differentiation; don't worry about it.)

(Deities seem to have much complex to say about philosophy and human behaviour, but nothing at all complex about math. Perhaps it's 'inordinately fond of secret mathematics' ...rather than those beetles.)

(If Abraham had only bothered to write down e to 20 decimals, convinced.)

Humans seem to assimilate the behaviour of f' = k.f better if they notice that the result has a characteristic doubling time, which is constant for a given k. (...And a characteristic *halving* time for negative k, but that's a more familiar concept, called 'half life'.)

...Such that

f(2T) = 2.f(T), where big-T is the doubling time

So...

f(0).e²ᵏᵀ = 2.f(0).eᵏᵀ
e²ᵏᵀ / eᵏᵀ = 2

And take logs...

2kT - kT = ln(2)
T = (1/k) ln(2)

And if k is in percent...

T = 69.3 ÷ k ...Voila!

...Which is the 'rule of 70' of compound interest. (Because compound interest is just f' = k.f, k > 0 ...except usually with discrete instead of distributed additions.)

If you divide the percentage rate of increase into ~70 (69.3147 for precision), that gives the (fixed) doubling time of the total. So 10% constant-rate growth doubles the total about every 7 years (6.93147 years); 3% doubles it about every 23 years.

Which means that, for f' = k.f with a k of 10%pa, if we start with 1 at year zero, it goes:
2 at year 7
4 at year 14
8 at year 21
16 at year 28

...and so on to ~4,722,000,000,000,000,000,000 at year 504.

(which is approximate because we used 70 instead of 69.314...)

But, point is, it doubles and then doubles again and then doubles and doubles and doubles some more, until it reaches ridiculous amount. It explodes. It is the opposite of 'sustainable'.

"In each new doubling interval, as much is added as has ever existed before", the late Al Bartlett.

(Which is dense tautology, but it's surprising how many, upon hearing it, go, oh shit, yeah, you mean that...)

Said we would write something about *sustainable* growth. There's an infinite number of sustainable growth pathways, it's just that constant proportional ('exponential') growth isn't one of them; in fact it's the exact opposite. Best known is a thing called 'logistic growth':

Sorry, not my figures ... stolen from below (which is an excellent summary if you can't follow mine, but we aren't going to use its notation).

khanacademy.org/science/biolog…

Logistic growth is observed in many systems in many many fields. It is *very* common. Here it is in beer brewing (the 'biomass' curve ... at least for a while...):

It starts out exponential (growth proportional to total), then feels the influence of an upper limit (a 'carrying capacity'), and curves around towards it. So the math has some of f' = k.f and some of something else, an upper limit ... we'll call L.

f' = k.f (1 - f/L)

Ain't super hard, is it. More later.

That's the differential equation, which as usual is the simplest view. Again it is explicitly solvable, but the solution is a bit more of a handfull:

f(t) = L.f(0).eᵏᵗ / (L + f(0).(eᵏᵗ - 1))

...Which of course people like to play maths on to simplify. Turns out the thing is symmetrical, and the easiest fix is to make the exponent the distance from the midpoint (the inflection point):

f(t) = L / (1 + e⁻ᵏ⁽ᵗ⁻ᵗ⁰⁾)

where t₀ is the midpoint of the curve.

Whatevs. The point is that there is a simple sustainable growth process, one that is physical, common in our universe, and well understood ... except by politicians and endless growth promoters.

Next time some clown is talking sustainable growth, ask them what their L is, their upper limit. Because if they don't have one, they're unlikely to be describing a sustainable process. More likely they're applying the mathematics of a nuclear warhead.