Growing by Multiplying

Growth that adds a fixed amount and growth that multiplies by a fixed factor behave nothing alike. Telling them apart, and computing a doubling time, is the whole of this page.

A quantity can get bigger in two ways, and almost everything in this course

follows from the difference between them.

In the first, each step adds a fixed amount. A salary that rises by two

thousand a year, a tank filling at ten litres a minute, a counter incremented

once per request. In the second, each step multiplies by a fixed factor. A

population where each individual has the same chance of reproducing, money

earning interest on the interest, a model whose training budget rises by half

again each year.

Over the first few steps these are hard to tell apart and may genuinely be

close. Over twenty steps they are not comparable quantities any more.

FIG 1Adding three a step, and multiplying by 1.3 a step

starts at 10, adds 3 each stepstarts at 10, multiplies by 1.3 each step

Both start at ten, and for the first four steps the straight line is ahead. A reader shown only that stretch would reasonably conclude that adding three a step is the faster process, and would be wrong in a way that no further data could make worse: by step fourteen the multiplying process is more than seven times the adding one, and it has only just begun. Nothing about the early part of this chart indicates what the later part does, which is exactly why the two kinds have to be told apart by arithmetic rather than by eye.

The test that separates them

The numbers themselves will not say which process produced them; the gaps

between the numbers will.

Take consecutive values and subtract. If the differences are steady, the growth

is additive. Take the same values and divide each by the one before it. If the

ratios are steady, the growth is multiplicative. One of the two tests comes out

flat, and whichever it is has settled the question.

FIG 2The same two processes, with both tests applied

step	step	adding three	difference	multiplying by 1.3	ratio
1	0	10		10.0
2	1	13	3	13.0	1.3
3	2	16	3	16.9	1.3
4	3	19	3	22.0	1.3
5	4	22	3	28.6	1.3
6	5	25	3	37.1	1.3
7	10	40	3 throughout	137.9	1.3 throughout

7 steps

The third column is flat for the additive process and the fifth is flat for the multiplicative one. Notice that the second column does have ratios and the fourth does have differences; they are simply not steady, which is the whole content of the test. In practice the data is noisy and neither column is perfectly flat, so the question becomes which one is closer to flat, and that is usually not a close call.

There is a useful corollary. For an additive quantity, the average increase per

step is a meaningful summary, and it is what people reach for. For a

multiplicative one it is not, because the increase is proportional to how much

there already is: quoting an average increase of fifty units a step for a

process that gained two units early and four hundred late describes neither

part of it. The steady number for a multiplicative process is the ratio, and

that is what should be quoted.

Writing it down once

Every multiplicative process can be written in one shape, and getting it into

that shape is worth doing immediately because it separates the parts that

matter from the parts that do not.

FIG 3A multiplicative process, in the only form it needs

: how much there is after t steps. The brackets are the function notation of the previous course: a machine that takes a number of steps and returns an amount
: how much there was at the start, when t was nought. The subscript here is a label rather than a counter: there is no N_1 in this notation
: the factor, the number each step multiplies by. Above one and the quantity grows, below one and it shrinks, exactly one and nothing happens
: how many steps have been taken, and the whole reason this is a different kind of thing from a power: the variable is in the exponent

Two of these four do all of the work and the other two are bookkeeping. The starting amount scales everything uniformly, which means it never changes the shape and rarely changes a conclusion. The factor decides everything about the shape. A reader given an unfamiliar growth claim should go looking for r first, and should be suspicious of any claim that does not make it recoverable.

A note on the exponent, because it is the distinction the previous course ended

on. In $t^2$ the variable is the thing being multiplied and the exponent is

fixed; in $r^t$ the variable is the number of times, and the thing being

multiplied is fixed. These are not variations on a theme. For large $t$ , every

expression of the first kind is eventually overtaken by every expression of the

second kind, whatever the constants: $1.01^t$ eventually exceeds $t^{100}$ ,

and it does so permanently.

Doubling time

The factor is the honest summary of a multiplicative process, and it is also

nearly useless in conversation, because it depends on the step size. A factor

of 1.0002 per hour and a factor of 1.3 per year may describe exactly the same

process, and nobody can see that by looking.

The fix is to quote a duration instead: how long until there is twice as much.

That number divides out the starting amount, divides out the units, and, in a

genuinely multiplicative process, does not depend on when you start measuring.

Twice as much from here takes the same time as twice as much from anywhere

else, which is the defining property of this kind of growth and the thing that

makes the summary meaningful.

Computing the doubling time exactly requires the logarithm of the next lesson.

Computing it approximately requires nothing.

FIG 4Doubling time by counting, and by the rule of seventy

python

def doubling_steps(r):
    amount, steps = 1.0, 0
    while amount < 2.0:        # multiply until there is twice as much
        amount *= r
        steps += 1
    return steps

for pct in (1, 3, 5, 7, 10, 15):
    r = 1 + pct / 100
    print(pct, doubling_steps(r), 70 / pct)

1 70 70.0                      # counted 70, rule says 70.0
3 24 23.3
5 15 14.0
7 11 10.0
10 8 7.0                       # the drift begins to show here
15 5 4.7

The loop is the definition: multiply until there is twice as much, and count. The approximation is what to use when there is no computer, and the table shows it is accurate enough to argue with. It fails gently, drifting high for large rates, and the drift only matters above about fifteen per cent a step, where the whole idea of a smooth rate is usually shaky anyway.

Seventy divided by the percentage growth rate is the doubling time, and it is

worth committing to memory because it converts a rate nobody can feel into a

duration everybody can. Three per cent a year is a doubling in a working

lifetime's quarter. Seven per cent is a doubling in a decade, so a tenfold

increase in about thirty three years. Twenty per cent a month, which is a

plausible figure for something early and successful, is a doubling every three

and a half months and a thousandfold in under three years.

Why this is consistently underestimated

The reason multiplicative growth surprises people is not that the arithmetic is

difficult. It is that the arithmetic contradicts the only model of change most

of us carry, which is built on adding.

Here is the sentence that does the damage. In a doubling process, each step

adds more than the sum of every step before it. Not more than the previous

step, which would be unremarkable: more than all of them put together. If a

pond's lily cover doubles daily and the pond is full on the thirtieth day, it

was half empty on the twenty ninth and a nearly invisible three per cent on the

twenty fifth. Anybody watching the pond on day twenty five and extrapolating by

eye would predict a full pond some time next year.

Three practical consequences follow, and they are the reason this page exists

at the front of the course.

A multiplicative process that looks flat is not slow, it is early. The flat

part is the same process as the steep part, drawn at a scale where the steep

part has not happened yet. This is why the next lessons spend so much effort on

a chart that makes the two parts look alike.

Extrapolating by eye always undershoots, and the size of the undershoot grows

with the distance extrapolated. Any forecast made by looking at a curve and

continuing it by hand is an additive forecast, whatever the curve was doing.

The gap between visible and overwhelming is short. Because the last step adds

more than everything before it, the interval between a quantity being large

enough to notice and large enough to dominate is one or two doubling times.

For anything where the doubling time is months, that is the difference between

a quarter's planning and none at all.

None of this requires a logarithm. But each of these statements is clumsy to

state and clumsier to compute with, because they are all about exponents and we

have been handling exponents by counting. The next lesson introduces the one

question that turns all of it into arithmetic: given an amount, what exponent

produced it?

Recap

There are two ways a quantity can change at each step: it can gain a fixed amount, or it can be multiplied by a fixed factor. The first draws a straight line and the second does not, and no amount of staring at the first few values will tell them apart. The test is to divide each value by the one before it, and see whether the ratio holds steady.
Multiplicative growth is written as a starting amount times a factor raised to the number of steps. Every quantity in this course that spans many orders of magnitude has that shape, and the number that matters most in it is the factor, not the starting amount.
A factor per step corresponds to a doubling time, which is how long until there is twice as much. Doubling time divides out the starting amount and the units, which is why it is the number people can actually reason with, and the reason seventy divided by a percentage growth rate is worth remembering.

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

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