Why are certain trends so precisely exponential?

I don't have an answer to your question, but I do have a concern: beware of positive bias. Don't just look for positive examples--how many trends are not precisely exponential? I'm pretty sure the answer is "a lot." (I know it sounds basic, but it bears repeating.)

Any time the current growth rate depends linearly on the size of the current population, you get exponential growth.

For instance, given effectively boundless food, the number of bacteria will double every regular time period because, in that time period, each bacterium can split into more.

Similarly, human populations grow exponentially (up to environmental bounds like food supply, war, etc.) because after roughly twenty years, humans beget more humans. With a population as gargantuan as Earth's, that growth rate is practically continuous.

Kurzweil's argument is that this applies to many technological industries, too. Faster, more powerful computers make it easier and faster to design the next technological generation. Thus when you measure computing power with any one well-defined scale like transistor count, you'll find that the growth rate depends on the current number. Ergo, exponential.

I'm not sure what's going on with the GDP, but if it's really keeping so nicely to an exponential curve like that, then I'd expect it means that the growth in GDP depends primarily on the current GDP and that it has been this way for a very long time.

I don't know whether this is what's really going on. As others here have pointed out, there might be some biases playing a role here. And as I suspect has been beaten to death here before, Kurzweil hides a number of dubious assumptions about market forces and human creativity in his projections. But I do know that if you really, honestly have a strongly exponential growth pattern, it's most likely that you'll find some reason why the growth rate depends on the current population.

Abusing linear regression makes the baby Gauss cry... It's true that fitting lines on log-log graphs is what Pareto did back in the day when he started this whole power-law business, but "the day" was the 1890s. There's a time and a place for being old school; this isn't it. - Cosma Shalizi

For comparison, here's a relevant example of a log-log graph (also from FiveThirtyEight) of an exponentially growing quantity which doesn't appear to show such a steady rate of growth:

The difference between this graph and the GDP graph is pretty striking.

Fascinating question. I share your curiosity, and I'm not at all convinced by any attempted explanations so far. Further, I note that the trend makes a prediction: an economic crunch will be followed by a swell of corresponding magnitude. So who wants to go invest in the US stock market now?

Would they look so precisely exponential, with exactly the same exponent, if we fit 5-10 log-linear lines to the graph rather than one? Eyeballing it, looks like the answer is no...

Moore's Law is one big idea (Feynman's "There is plenty of room at the bottom") filtered through the conservative engineering practice of not doing too much in one go. Much of engineering proceeds in 20% or 50% increments. Steam trains get more powerful, bridges longer, mines deeper, ships larger, all in percentage increments over the previous record holder.

Technological advances affect GDP as they become mainstream. This requires knowledge propagation in which teachers teach a students, some students become teachers and teach more students,... They say that Faraday discovered electromagnetic induction on 29th August 1831, but by the time this hit the GDP figures with the electrification of American industry in the 1920's and 1930's it had been smeared out an exponential take up on a decades time-scale determined by human factors.

Hmm, my observations don't actually lead to a prediction of a steady rate of growth.

One the other hand, the attention to human factors hints that there might be a fairly low maximum rate of growth of around 3.5% a year, set by the proportion of good students in a population. One notices how steam railway trains stagnated from err, 1910 (not too sure) but they got to 4% thermal efficiency and then all the bright young engineers went into automobiles and aviation, and steam trains remained crap until diesel -electrics came along.

http://www.greatplay.net/offsite/us-econ-growth.png

When you look at a graph of the growth rate itself, the data doesn't look like a perfect fit for a line anymore.

Instead, I think the 3.5% is an average that looks good when you have a graph that is rather zoomed out, but overall isn't that much of some sort of pattern that needs explaining.

Instead, the economy seems to fluctuate and cycle a lot, and we already have some theories of business cycles to explain why that is so.

You can also see points in history where certain geopolitical events caused lower or higher than average growth, so the idea that the GDP is seemingly immune to geopolitical events is hardly there in actuality.

~

I can't explain Moore's Law (that's an area I haven't studied at all), but I'd be willing to wager that if you looked at the graph of growth like this, you would see some fluctuations as well. (No negative growth though, obviously.)

It would be interesting to see. If the Moore's Law graph really didn't fluctuate much, then I think you have something in need of explaining.

~

Also, you might be wondering something like "if technology has improved drastically, why hasn't GDP growth shot up to something wild like 7% per year?". But growth itself requires more and more change every year just to keep at a constant growth percentage, and as Moore's Law shows, technology itself improves at a mostly exponential rate, so we would only expect the same exponential increase in GDP, not something more-exponential-than-exponential.

If there's no convincing natural explanation, there's likely a human one. The human sense of scale or progress is vaguely exponential, so it's possible that some trends are exponential because humans work to make it that way so that things seem like steady progress.

Intel works to maintain this exponential trend.

In the last 5 years serial speed gains in computers fell off, while genomics took off. Both changes were quite abrupt.