by Edwin Ivanauskas | Tags: Personal Finance, Investing, Banking

A lot of confusion (or is it conscious data manipulation?) in quite a few subjects stems from the misuse of exponential data. To see what I'm talking about check out how inflation is calculated in debates about the gold standard and how people analyze deficits.

The main culprit is using real numbers rather than looking at them more objectively. As an example, for inflation people will graph the prices in different time periods and show how quickly they have accelerated in the recent past and then point to the cause for that acceleration (whatever demon they want to pick on at the time tends to be the cause).

So this article is meant to be a primer on exponential growth so you can understand the concept and apply it to relevant subjects. This will also help you catch someone who is manipulating data to make a point.

The Allowance (or Salary) I want

I'll start off with the basic example as I remember it being taught to me in grade school. I was told to go to my parents and ask for an allowance of 1 penny and ask them to double it every week. Initially the teacher was met with shock, "What can I do with 1 penny?!?!" The teacher then proceeds to go through the steps week by week.

At this point, the students realize how quickly the money grows and how many things they could buy with it. All they have to do is convince their parents to fall for the trap they themselves fell for. The teacher continues the lesson by graphing these numbers.

*yes it was done roughly on a chalkboard, but this is easier for me

This is the point at which the students lost interest and is likely why most of us don't remember today. But now that you are financially savvy adults, we will continue.

There is a very noticeable acceleration of the allowance between weeks 12-14. Is there anything that triggered this acceleration? No.

This type of model is called exponential growth and will always happen when an equation takes a percentage of the previous number. In this case each week is 200% of the previous week.

To further emphasize that nothing special happens between weeks 12 and 14, I graphed this example out to 100 weeks where the allowance would be $6,338,253,001,141,150,000,000,000,000.00.

Here we might be led to believe that something significant happened at around week 91, because that's when the acceleration begins. But as we now know, this is just a property of an exponential equation.

How to Ignore Exponential Growth

How do we deal with these exponential growth models obscuring valuable information? Well there are a variety of ways; the general concept is to graph the data objectively.

Natural Log
The proper way to eliminate the effects of exponential growth mathematically is to use the natural log (ln(x) in our math classes). When I graph the first 15 weeks of my example above and take the natural log I get this:

As we can clearly see, the line is absolutely straight and there is no acceleration. You can see an example of this in the real world in my article about the gold standard and inflation.

Controls
Other controls can be used to get rid of the effects of exponential growth. A common one is using an objective measurement to map the number you are measuring against.

Exponential Growth in the Real World

So out in the real world how do we know whether the numbers we are looking at are a case of exponential growth? The simplest way to answer this is if every number is a certain proportion of the previous number. Here are some examples:

Inflation
Inflation is measured as a percentage, 3% for example. This means the level of all prices is 3% higher than it was the previous year. This can be controlled for by graphing the natural log.

GDP
As a nation grows and expands its productivity, the amount of output increases accordingly (measured by GDP). This type of increase is exponential because the amount of growth possible in one year is based on the productivity and capital resources of a previous year.

Government Budgets
As GDP grows, so does a government's revenues or deficits. Just as a nation's output grows exponentially, a government's revenue or deficits will do the same. This can be controlled for by graphing budgets against GDP.

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