"2 years of 10% and 0% growth is not the same as 2 years of 5% growth."

In comments here the other day, Jim said:

Its not very meaningful to average percentages. 2 years of 10% and 0% growth is not the same as 2 years of 5% growth.

Jim said pretty much the same thing a few years back. I can't find that older remark, but it definitely got stuck between my ears. Since it came up again now, I thought this time I'd actually take a look at some numbers. Because without looking at the numbers I can't make progress, and it'll be a loose end in my head forever.

So I opened up Excel and typed 5% Every Year. And then I typed 10%, 0%, 10%, ...

And I looked at what I had, and those were the two conditions Jim specified: one of constant and one of alternating growth. I thought about it. And then I added another one: Random growth one year, and then enough growth the next year to bring the two-year total to 10%. (The third year again random, and the next bringing the two-year total to 10%; etc.)

I generated numbers enough to fill the cells below the three labels, but not more than I could see on the screen. But this seemed unsatisfactory because it only amounted to about 30 years of growth. I should figure more than 30 years, I thought. So I added some more, and the screen scrolled, and when I got 40 years of growth I figured that was plenty.

(These decisions I make, based on impressions and impulses and a desire to keep most of my data visible on the screen... is this how science is done?)

I was going to number the first row of data "Year 0". My starting point. That way the second row, the first year of growth, would get a "1" in the Year column. But then, no. Because I knew I'd be looking at exponential trend line equations to see the growth rates for my different columns of data. And exponential trend lines want my "Year" numbers to start with 1. Not zero.

So I made my first year Year 1 even though it's just a starting point and not a measure of change. That made the first change appear in Year 2. And that made the 40th change appear in Year 41. I hate this one-year discrepancy, because if I don't explain it it looks like a mistake. And if I do explain it, it looks like I'm making excuses. But it's just the way Excel works. Either that, or Excel and I have a serious misunderstanding.

Here is the graph I came up with:

Graph #1: A Comparison of Constant and Averaged Growth Rates

I put trend lines and trend line equations on the graph, then hid the trend lines because I just need the equations. The trend line equations on the graph are color-coded to match the lines and the legend. Blue shows 5% growth every year. The trend line equation shows a 4.88% growth rate; this differs from 5% because of compounding, I suppose. Nothing is easy.

Red shows alternating growth: 10% one year, 0% the next, and repeating. This gives an average growth rate of 5% but of course the red line is not a smooth curve. It is up-and-down, up-and-down, like a saw-tooth. The trend line equation shows an exponential growth rate of 4.77%.

That's close to 4.88%, I told myself.

Green uses a random number between 0 and 10 as the growth rate for one year, and then 10 minus that growth rate as the rate for the next year. Then for the following year a new random number; and for the year after that, again the difference from 10. This gives a series of two-year averages each equal to 5%; the trend line equation shows 4.88%, just the same as the blue line.

Now I'm going to ignore the green and concentrate on the red.


The red exponential rate is a little low. And at the right side of the graph, you can see that my red line ends at a low point of the sawtooth. I thought these two lows might be related. So I added one more year to the graph, to make that red line end at a high point of the sawtooth. You can see the red line scoot up at the end:

Graph #2: Like Graph #1 but with One Additional Year

However, there is no change in the 4.77% exponential trend rate. I should have known.

Looking at that red line now, it appears that for the first half of the graph, the high points of the sawtooth are noticeably above the blue line. But in the latter half, the lows of the sawtooth being below the blue is what stands out. This seems to be telling me that the red line is not growing as fast as the blue line.

4.77% versus 4.88% growth: I should have known.

Okay. So I figured, let's look at the difference between red and blue. The one minus the other. Get rid of the exponential upward sweep, and consider the red line relative to the blue.

Also, the difference between green and blue.

Graph #3: Differences from the Constant Rate of the Blue Line

Okay: The red line is definitely not keeping up with the blue.

The green line is easily explained with one word: "random". But the red one, the red one is interesting. At the start, red is almost entirely above the zero line: Red is greater than blue. By the end, red is almost entirely below the zero line: Red is less than blue.

If the graph kept going for another 40 years, I expect we'd see the red continue downward.

Notice also that the zigs and zags of the red grow longer as we go from left to right. The difference between red and blue is getting bigger. But don't forget, the red and blue on Graph 1 and 2 also get bigger as we go from left to right. So the question now is: Is the difference increasing faster than the blue line, or slower, or is it keeping pace? And that's our next graph:

Graph #4: Differences as a Percent of the Blue Line

Now the red zigs and zags are all of equal length, or nearly equal anyway. So I want to say the red difference is keeping pace with blue. And yet, for the years shown, the red difference starts out on average at around 2% of the value of blue on the high side; and 35 or 40 years later we find it on average again around 2% the value of blue, but on the low side.

I have no doubt that if we continued the graph out for another 40 years or so we would see the red zig-zag continue on its downward path. Actually, what's happening here reminds me of Bruce Boghosian's The Growth of Oligarchy in a Yard-Sale Model of Asset Exchange (PDF, 15 pages). See The Mathematics of Inequality at TuftsNow:

This is essentially a coin toss, and you might think that in the end both sides would end up with the same amount of wealth. But it turns out those who have more keep getting more.

Yeah, I thought "both sides would end up with the same amount of wealth". Or in this case, with the same amount of growth. Now I'm seeing this is not the case.

In Fair coin, foul outcome I looked at some Excel graphs of the process Boghosian describes. My last graph there shows the same zig-zag pattern of decline that turned up in Graph #4 above. And I figured out how the process works:

Consider a two-transaction sequence. On average in a two-transaction sequence, because the coin is fair, the rich guy and the poor guy should each win one of the wealth transfers. There are two ways this can happen:

1. The rich guy wins the first transfer, and the poor guy wins the second. Or
2. The poor guy wins the first transfer, and the rich guy wins the second.

If the rich guy wins first, the first transfer reduces the poor guy's wealth. So the second wealth transfer is smaller. The poor guy loses the larger transfer and wins the smaller one. Therefore, his wealth is reduced.

If the poor guy wins first, the first wealth transfer increases the poor guy's wealth. So the second wealth transfer is larger than the first. The poor guy loses this larger transfer. Therefore, his wealth is reduced.

Either way, the poor guy loses.

I had in mind to say this of the red pattern on Graph #4: Jim is right. Two years of 10% and 0% growth is not the same as two years of 5% growth. And, all else equal, it looks like low volatility of economic growth will over the long term produce greater growth than high volatility. However, the economy never grows in a persistent alternating pattern like the "10%, 0%, 10%..." example. Not over the long term.

And I was going to say: "So Jim is right, but the effect on the economy must be of little consequence." I was going to say that, but now, no. And now also I must omit the "However, the economy never grows in a persistent alternating pattern" and the rest of that paragraph.

Beyond that, I have to think about this some more.