Exponential Growth


8 minute read

Exponents are counterintuitive. When we measure change, in common life we are more often than not, looking at absolutes. We rarely in daily life, except some professional use cases track the rate of change. The difference in looking at linear and exponent is the difference between looking at absolutes and relatives.

During the current Coronavirus spread, this term has been going around. Yet the world seems to have not been able to comprehend the speed or scale of the spread of the virus. Most countries have underestimated the danger and the extent of the virus. In the initial days, when the numbers were low the intuitive sense had not imagined how the numbers will grow. A graph that now most people are familiar with

People who were able to plot these numbers and had done the research, including credible people like Bill Gates, were ignored. The reason for this still remains that exponential growth is counterintuitive.

This is not to say, that we don’t understand exponents. We look at exponents all the time. Everything big happening at the world scale is generally happening at an exponential rate.

Real-world applications where understanding exponential growth helps

  • Population expansion
  • A startup in the growth phase
  • GDP of a any place
  • Amount of waste that humans produce
  • The virality of a post
  • Total Energy consumption of the world
Energy consumption of the world
GDP of the world

All of these change in exponential levels, yet in our everyday life, until you sit down with a pen and paper and plot a graph, it is hard to comprehend what exponential change looks like.

Absolute vs Relative

Exponents only apply to the rate of change. These come into the picture for things where your base or current levels matter. Let’s imagine you go shopping and want to buy T-shirts. When you go out to buy a T-shirt, you are not looking to buy 20% of your wardrobe, you are looking to buy a fixed number of T-shirts. Though if you did, your house will be full of T-shirts within a short amount of time.

Let’s say you love T-shirts just like me, and decide for the foreseeable future you are going to buy T-shirts every month. You currently have 10 T-shirts in your cupboard. So in one case, you decide you are going to buy 2 new T-shirts every month. In the other case, you decide you are going to go exponential and you will buy 20% of your cupboard every month.

In the first month, you will go out and buy the same number of T-shirts, so you got 2 cool new T-shirts. You make a nice graph to see how many T-shirts you will have over the next few months to have enough space to keep all the T-shirts.

You see that if you buy linearly, you get 22 T-shirts by the end of 6 months and 28 T-shirts if you buy exponentially. It doesn’t really seem that much, but you want to make sure that you have enough space for a year, so you expand the graph to a year.

You see the graphs and thin you don’t need so many T-shirts in the first year. You say let’s lower the amount from 20% to 1%, and forget about the graph excited about the T-shirts. You put all of this on an online website, attach your credit card and instruct them to send over T-shirts every month. You like to automate and just want the T-shirts to arrive every month. You decide to keep an eye on it and to check it every year

  • Year 1: You seem to be doing worse than what you wanted. You only reached 24 T-shirts after all the new calculations. You knew you should have taken the linear path. But you let it continue.
Year 1 comparison
  • Year 2: Now you seem satisfied, you just passed what you would have got if you took the linear path. You give yourself a pat on the back and keep the automation going.
Year 2 comparison
  • Year 3: Now you are excited, you see the exponential path was helpful. Now you have more than double the T-shirts than your linear counter-part. You like the winning streak and decide to let it continue for a couple of years.
Year 3 comparison
  • Year 4: You have been wondering why it seems that your entire room is filled with T-shirts over the last year. You now have 605 T-shirts. Your linear counter-part has 108 T-shirts. You quickly decide to remove your credit card from the website. Thank god you had set up a yearly review. But you continue to wonder how it would have looked if you had not removed your card.
Year 4 comparison
  • Year 5: Exponential purchase would have led you to 1890 while a linear purchase model would have got you 130. You would have maxed out your credit card by now.
Year 5 comparison
  • Year 6: You are glad you did not go bankrupt. You would have had to probably sell your house to pay for the T-shirts now. The exponential purchase would have led to own 5922 T-shirts in pale comparison to the linear purchase model by which you would own 154.
Year 6 comparison

You decide never to look at purchases in exponents every again. You just averted a financial crisis because of T-shirts. Damn you T-shirts.

One of the reasons why exponents are counter-intuitive is because in most of your daily life you are looking at absolutes. In most of your decisions that govern personal life, you look at numbers and not growth rates. So while we understand exponents, applying it to real-life problems is still hard.

Time Scale

Another reason why exponential growth looks linear is the time scale. Population growth is a classic example. If you are not studying the population growth professionally, it is very hard to imagine how the population changes over time.

Population growth of the world

While the actual population growth has been affected by a variety of factors and the growth rates have changed over different times, let’s doa thought experiment with our own conditions.

Let’s start with a population of 10 people and add a growth rate of 1%(the current growth rate of the world). Let’s run this simulation for 300 years.

Population growth simulation of 300 years

Well, to start off, this graph becomes useless. The problem with the graph is that the Y-axis, which represents the population is linear. As the population growth is dependent on the base or the current population, as the population grows the numbers are of a different order. But what is interesting is that the population seems to explode in the last step. All we need to do is change the projection to 250 years and then we will be able to see clearly.

Population growth simulation of 250 years

While the numbers on the axis seem to have changed, but the graph still looks similar. You can’t make sense of the graph and the population seems to have exploded only in the last few years again. The curve at 230 years was missing in the last graph it seems.

So exponential growth is counterintuitive even in graphs. At least in graphs that were meant to plot linear growth. To plot exponential growth we need to use logarithmic graphs. In a log graph, the Y-axis changes by the order of the value. We would not go in the details of Logarithm, but to brush up you can watch the video. For our current understanding, you can remember that our Y-axis will change somewhat in the following way.

Log 10
Population growth simulation of 300 years on the log-scale y-axis

Let’s replot our 300-year simulation of the population on a Y-axis that is log scaled.

Now the graph is legible, and we can see what is happening.

010
70102
1111,042
14010,644
162102,942
1811,136,999
19610,728,756
209100,327,728
2211,038,009,758
23211,457,518,030
241100,341,176,867
2501,076,851,378,398
25810,693,024,376,314
266128,424,802,438,041
2731,337,769,415,982,320
27911,401,408,762,677,100
285110,862,590,483,662,000
2911,239,888,876,792,430,000
29610,383,102,114,216,300,000

The simulation took 70 years to reach the first hundred people, and even after another 70 years the population had just reached 10,000. This is almost half the time of our total simulation. If you jump another 70 years the population has now hit a 100 million people. That is a steep jump, but if you jump another 70 years you are now at it has crossed 10 quadrillion(1000 trillion).

The pattern here is that every 70 years or so the population multiplies itself with itself. And this is why even after understanding exponential growth, it remains counterintuitive until you actually plot it out.

The number 1

Small changes in the growth rate have a huge impact on the outcome in the long run. Let’s plot comparison between 1% growth, 5% growth, 1% decline and 5% decline.

Graph showing % change on a log scale

A number lower than 1, will drag you down to 0 in the long run and a number higher than 1 will take you up to infinity. Till the rate of change is constant and not the amount of change, you are looking at exponential growth or decay.

Just as exponential growth, there is also Exponential decay. The good part about exponential decay is that it will only tend to 0 and never reach 0, without any external factors.

Half-life, for example, is an important Scientific method that Scientists use to find out how old something in nature is. If an element is going to reduce to half its size in a fixed amount of time, always, we can calculate how far back was it put there if we know the starting quantity and the current quantity.

Exponents are counterintuitive because, in our daily lives, we look at absolutes in short time frames.

PS: If you feel like playing around with exponents graphs, you can use copy this template and fiddle around.

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