Section 8A

Growth: Linear versus Exponential



A quantity changes with time.

That may be: The height of a tree, the population of earth, the amount of money invested into a bank account, and anything else.

We want to describe/model that change.

Assume that this is growth.

Two basic growth patterns

Linear growth occurs when a quantity grows by the same absolute amount in each unit of time

Exponential growth occurs when a quantity grows by the same relative amount in each unit of time

Same definitions of linear and exponential apply to decay instead of growth.

Examples of linear growth

A car is driving at a constant speed along a highway. Its distance from the starting point grows linearly. Indeed, if the speed is 60 mph, this distance grows by 1 mile every minute.

Mattress saving plan. If I put a spare dollar every day under my mattress, the amount of money saved this way grows linearly. Namely, this amount grows by a dollar per day. Same happens if I make use of a saving plan with zero interest rate.

Example of exponential growth

The first one to think about is an investment with some interest rate. We studied that a lot already.

We will have more examples today, but firstly let us concentrate on this one

Linear versus exponential

Assume that we want to deposit \(\$ 100 \), and have to choose a bank with better offer.

Linear bank offers just \(\$100 \) interest for every month.

Exponential bank offers 240% APR compounded monthly, which is 20% per month.

We calculate. After the first month, we will have
\(100 + 100 = \$ 200\) with linear bank, and only
\(100 + 0.2 \times 100 = \$ 120 \) with exponential bank.
After the second month,
\(200 + 100 = \$ 300\) with linear bank, and only
\(120 + 0.2 \times 120 = \$ 144 \) with exponential bank.

Linear versus exponential ... two banks

It looks that the linear offer is much better so far.

Let us do long-range calculations.

After 1 year, with linear bank we will have

\( 100 + 12 \times 100 = \$1,300 \)

After 1 year with exponential bank

\(A=P \times \left(1+\frac{APR}{n}\right)^{\left(n \times Y\right)} = 100 \times (1+0.2)^{(12 \times 1)} \approx \$ 892\)

Although linear bank is still ahead, exponential seemingly is doing better than expected.

Indeed, after 2 years, we will have \(\$2,500 \) in linear bank, and \(\$ 7,949.7 \) in exponential.

Linear versus exponential ... two banks ... continued

In fact, we have observed a general phenomenon:

Whatever sluggish may seem to be exponential growth in the beginning, with time it beats any linear growth.

The amount of money in the linear bank grows linearly : by \(\$ 100 \) every month.

The amount of money in the exponential bank grows exponentially : by 20% of what it was every month.

Further examples of exponential growth

Chessboard parable.

One puts one grain of wheat on the first square of chessboard, two grains of wheats on the second, four on the third, and so further doubling the amount of wheats every time, and passing to the next square.

The procedure looks not too bad: the grains are small, and one starts only with one grain.

A simple calculation shows that there should be \(2^{63} \) grains on the last (64-th) square.

Indeed, there are \(2=2^1\) on the second square, \(4=2^2\) on the third square, \(8=2^3\) on the fourth, and so on...

Chessboard parable ... continued

However, the number \(2^{63} \approx 9.2 \times 10^{18} \) is already too big. Not only that amount of grains does not fit the square of a chessboard, it is larger than all amount of wheat on the planet

Further examples of exponential growth

The magic penny

This is a combination of the chessboard and exponential bank from previous examples

Magic penny doubles into two magic pennies every day

This way, on day \(n\) one has \(2^n\) magic pennies.

As a calculation in the textbook suggests, on day 51, the amount of money suffices to pay off the national debt of this country

Further examples of exponential growth

Bacteria in a bottle

This is a bottle which initially contains one bacteria at 11:00am

The bacteria belongs to a fictional breed which is capable to double every minute no matter what

By 11:01am there are two of them, by 11:02am, there are four, there are 8 by 11:03am, 16 by 11:04am, and so forth ...

Assume that the bottle is full exactly at noon.

A surprising observation: at 11:59am, just a minute before they exhaust the space, only half of the bottle was filled with the bacteria. The end did not seem to be that close!

Bacteria in a bottle ... continued ...

One can expand our observation a bit

Clearly, the amount of bacterias in the bottle is \(2^n\) at
\(n\) minutes past eleven.

Full bottle means \(2^{60}\) bacterias.

11:50am, ten minutes before the bottle is full, there are \(2^{50} \)

and they occupy \(\frac{2^{50}}{2^{60}} =2^{-10} \approx 10^{-3} \), a thousands part of the bottle.

One may already see kind of mold in the bottle but one can hardly predict that the bottle is going to become full just in 10 minutes

Bacteria in a bottle ... continued ...

Yet another interesting observation

If we have another bottle to put half of the bacteria in it while our bottle still contains not too much,

we will not make significant difference: two bottles will be full in 12:01pm.

More generally, just 1 hour after the beginning, it is the amount of full bottles of bacteria, which is going to double every minute.

These bacteria will fill whole universe quite quickly

Bacteria in a bottle ... continued ...

To sum up

Exponential growth actually takes place in real world quite often

However, it never lasts long enough, so that we do not frequently witness its catastrophic consequences