Wednesday, November 23, 2011

Thoughts On Exponential Growth

Matthew Iglesias talks about automation using a framework from the book Race Against the Machine, which we covered earlier:
I offered my complaint about Erik Brynjolfsson and Andrew McAffee Race Against The Machine yesterday, but I also want to praise one extremely important insight in the book that really changed my way of thinking about something. This is what they call “the back half of the chessboard” and they derive it from an old story about a Persian king who makes a deal in which he promises to pay someone as follows. On the first day, one grain of rice is placed on one square of a chessboard. On the second day, two grains go in the second square. On the third day, four grains go in the third square. On the forth day, it’s eight grains in the forth square. The king agrees, and of course it turns out that 2^64 grains of rice bankrupts the kingdom. But the point about the back of the chessboard is that even though the mathematical pattern is evident throughout the process, the actual impact is amazingly backloaded.

The point of this, in terms of technological progress, is that we’ve gotten so accustomed to Moore’s Law that we sometimes overlook the implication that the deeper we get into the chessboard, the bigger the changes. We all know that computers advanced a lot between 1991 and 2011, but we should expect the scale of change over the next 20 years to dwarf those changes. This is a straightforward application of a well-known principle and some pretty basic math, but it’s usually not discussed in quite the right way. We think we’re used to the idea of rapid improvements in information technology, but we’re actually standing on the precipice of changes that are much larger in scale than what we’ve seen thus far.
I've never heard the term "back half of the chessboard" before, but it's a great term, and a good framework for understanding so I'll use it from now on. As I've noted earlier, I'm a skeptic when this is extrapolated to something like progress, which is nonquantifiable, even though I do believe we are in for a lot more automation in the future. But I should note that the "back half of the chessboard" concept has to do with not just automation, but ANY sort of exponential growth, including economic growth.

Another analogy is the water lily problem. It goes like this: If a single lily pad began doubling on a pond on the first day of June and doubled each day thereafter until the entire pond was covered by the end of the month, what percentage of the pond would be covered with lily pads after twenty days?

The answer is one-tenth of one percent. That’s right, 0.1%! What happens over the next 10 days is a little short of amazing—the entire pond gets covered. And, since it doubles every day, even on day 29, only half the pond would be uncovered! From no problem to catastrophe would occur in a single day! Like the chessboard, the problem is heavily "back-loaded." It is not noticed until it is too late.

Folding a piece of paper gives an even more striking analogy. Since each time you fold a piece of paper, it is a doubling of the previous paper's total thickness, it is analagous to exponential growth. So, although twenty pieces of paper is insignificantly small, folding a piece of paper only twenty times gives you a thickness taller than a skyscraper! Of course, folding it two or three times is no problem; we do it all the time. Again, such is the nature of exponential growth - it's normal until it isn't.

This is why exponential functions always lead to absurdity. Extract it out far enough, and it grows infinitely in no time. That's why debt is destroying the global economy - debt grows exponentially. Eventually, it grows rapidly enough to no longer be serviceable. This is one reason why a debt-based economic system will always be subject to boom and bust. The real, underlying productive economy cannot grow exponentially forever, as it subject to the laws of thermodynamics, demographics, etc. Since the supply of money and debt (essentially the same thing) always increases faster than the productive economy can, there is always inflation, followed by a crash as debt cannot be paid back. Then there is a deleveraging making you even poorer than you were before the crash! Right now, debt is being used as a weapon to eliminate the safety net and sell of the nation's productive assets to wealthy private investors. As you can see, such a debt crisis is not only possible, it is inevitable!

Exponential growth also applies to population. The most notable essay on exponential growth was written by Thomas Malthus. He noted that populations grow exponentially, while resources do not. Food production is limited by the supply of land; you can add some new arable land over time, but it cannot grow exponentially, as population can (when you bring new fields into production, they are not double the size of the previous ones). Thus, he famously predicted there would be inevitable population crashes and dieoffs.

He was supposedly proven wrong - population kept doubling in spite of his predictions. But we all know why - the use of fossil fuels allowed more fields to be farmed, nitrogen to be extracted from the air and dumped onto the fields (eliminating the need for crop rotation), and food to be shipped all over the place. New high-yielding strains of wheat and rice were also developed.

I would argue, however, that Malthus was not entirely wrong - all those things came at a terrible cost. Expanded agriculture has wreaked havok on the environment. And the quality of our diet has declined - those people are being kept alive by cheap carbohydrates - alive and malnourished. Sure they're alive, but that's about it - they're not healthy or thriving. Even in the developed world, cheap food has come at a cost; there's evidence that our highly processed carbohydrate-rich diet may be behind many of the diseases we suffer from. Some also contend that the genetically-modified dwarf varieties of wheat that have taken over world agriculture are contributing to the obesity epidemic (although this is controversial). Much of our economic growth has already come at the cost of quality - just look at the crappy particle board furniture we use today.

In his now famous newsletter to investors, Jeremy Grantham pointed out the absurdity of expecting exponential growth to continue forever:
I briefly referred to our lack of numeracy as a species, and I would like to look at one aspect of this in greater detail: our inability to understand and internalize the effects of compound growth. This incapacity has played a large role in our willingness to ignore the effects of our compounding growth in demand on limited resources. Four years ago I was talking to a group of super quants, mostly PhDs in mathematics, about finance and the environment. I used the growth rate of the global economy back then – 4.5% for two years, back to back – and I argued that it was the growth rate to which we now aspired.

To point to the ludicrous unsustainability of this compound growth I suggested that we imagine the Ancient Egyptians (an example I had offered in my July 2008 Letter) whose gods, pharaohs, language, and general culture lasted for well over 3,000 years. Starting with only a cubic meter of physical possessions (to make calculations easy), I asked how much physical wealth they would have had 3,000 years later at 4.5% compounded growth. Now, these were trained mathematicians, so I teased them: “Come on, make a guess. Internalize the general idea. You know it’s a very big number.” And the answers came back: “Miles deep around the planet,” “No, it’s much bigger than that, from here to the moon.”

Big quantities to be sure, but no one came close. In fact, not one of these potential experts came within one billionth of 1% of the actual number, which is approximately 10^57, a number so vast that it could not be squeezed into a billion of our Solar Systems. Go on, check it. If trained mathematicians get it so wrong, how can an ordinary specimen of Homo Sapiens have a clue? Well, he doesn’t. So, I then went on. “Let’s try 1% compound growth in either their wealth or their population,” (for comparison, 1% since Malthus’ time is less than the population growth in England). In 3,000 years the original population of Egypt – let’s say 3 million – would have been multiplied 9 trillion times!
That's the problem with exponential growth - by the time the problem reveals itself, it is too big to tackle. Population, economics, resource depletion, all are problems with an exponential component. It amazes me that economists, who are supposedly so wise in the ways of mathematics, cannot see that we're on the back half of the chessboard, and eternal economic growth is not possible, or even desirable. It just seems like common sense.



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