My Favorite Chart on Earth

Is there a simple explanation for non-economics types like me of why the linear graph is misleading and the log one is not?

I love how economists make all sorts or ridiculous claims by looking at the historical stock market. Two or three data points is not a trend, if you have 3 users and they all register, that doesn't mean your conversion rate is 100%.

Sometimes we forget that 100 years of data is nothing compared to human history and does not signify anything about the future.

Many things that affect the economy depend upon how much "stuff" is already there. A broader technology base means more innovation (= higher real GDP) in the future, because innovations build off one another. A larger money supply (= inflation) requires a larger growth in absolute dollar terms to achieve the same effect, because the important measure is the percentage growth in the money supply, not the absolute growth. More people (= population) have more babies, which leads to more population growth in absolute terms. Whenever you have to multiply out changes instead of adding them, you get exponential growth instead of linear.

There're some notable exceptions, eg. growth in the unemployment rate is usually linear, because it's already computed as a percentage. Growth in the absolute number of people unemployed is exponential.

Yes, here are a few explanations:

Because of compound interest money grows exponentially - the more money you have, the faster it grows.

Suppose I start with earning 10,000 a year, but my neighbor earns 20,000. Now suppose we both each 5% raises each year. After 10 years I earn 16288, and he earns 32577.

Now graph my income over the years and look at the gap between our incomes. In a linear graph the separation keeps getting larger and larger (we started with a 10,000 separation, and now have a 16289 separation). But that's wrong - we are both gaining equally as well. In a logarithmic graph the separation will stay exactly the same (and notice how when we started he earned twice as much as me, and he still earns twice as much as me after 10 years).

Suppose I'm in the business of selling stuff.

If I earn $1 per widget, and I can afford to keep 1000 widgets in stock, then I earned $1000. My neighbor however can stock 2000, and earns $2000. Now suppose both our profits go up to $1.50 per widget - now I earn $1500, and he earns $3000.

Now compare: In a linear graph he had $1000 more than me, and now has $1500 more than me. If you graph this over time he will look like he's doing far better than me - the income gap will keep growing - but really, he's doing exactly as well as me, it's just he has a higher base.

A logarithmic graph will show the relative difference between us - he has twice as much as me, and that doesn't change at all.

These explanations would be so much better with some graphs - is there a web service that will plot these things for me as live data?

One final example:

Suppose we are both businessmen, and we want to see who is better, I start with $100 and manage to grow it to $200, my friend started with $300 and grew it to $450. Who did better? I earned $100, but he earned $150 - clearly he did better right? But if you notice while I doubled my money, he only 1.5'd it. I'm clearly the better businessman, yet I earned less money.

On a linear graph it will look like he did better, since he now has so much more. But on a logarithmic graph the truth comes out - I did better. (And over time I will far exceed his income, and a logarithmic graph will show this correctly, but a linear one won't.)