### Hyperbolic growth

(Reposted from Facebook,  I would prefer put it here than have it lost to the sands of time.)

My view of the future is strongly influenced by the history of economic output. I think it’s instructive to look at the sequence of doubling times—how long did it take economic output to double, and then how long did it take it to double again, and so on? Here are the lengths of the first 8 doublings since 0 AD (followed by the year that the doubling ends):

1000 (ending in 1000)
600 (ending in 1570)
200 (ending in 1765)
100 (ending in 1860)
40 (ending in 1900)
40 (ending in 1940)
20 (ending in 1960)
15 (ending in 1975)
20 (ending in 1995)
>20 (to present)

(Source: Brad DeLong stitched together some estimates in 2006 which I believe are roughly right. These are the “ex-Nordhaus” here. The last two doublings are what Tyler calls the “great stagnation“, also accompanied by flat median wages.)

Here’s the same data as a graph, for visual thinkers:

And with a log transform:

Growth has accelerated tremendously over human history, but the change was gradual. If we measure time by economic doublings rather than calendar years, then about half of history was part of the rapid acceleration surrounding the industrial and agricultural “revolutions.” So it seems more appropriate to think in terms of periods of rapid acceleration and slow acceleration, with both conditions being relatively common.

We have been in a period of slowing growth for the last forty years. That’s a long time, but looking over the broad sweep of history I still think the smart money is on acceleration eventually continuing, and seeing something like this (though I have little idea whether it will start in 2025 or 2045 or 2075):

20 (ending in 2025)
10 (ending in 2035)
5 (ending in 2040)
2 (ending in 2042)
1 (ending in 2043)
0.5 (ending in June 2043)…

This is consistent with my view of particular technologies that are currently progressing rapidly (especially computing hardware, AI, robotics, and energy).

If you are alive in 10,000 BC and you are trying to predict when humans will achieve some ambitious technological milestone, it basically doesn’t matter what you are trying to predict—when we’ll go to space, or cure any particular disease, or automate fishing, or double our life expectancies, or whatever else. You can get to within a few percent by just estimating when the industrial revolution will occur. And you can estimate the timing of the industrial revolution pretty well, by noticing that the world economy is doubling every 4000 years and just predicting “a few doublings from now.”

Similarly, if you are alive today and want to forecast the time at which any particular ambitious technological development will occur, I think you should just be trying to predict when the world economy will be doubling every few years—which I think will be happening within “a few doublings,” something like 20-80 years.

This kind of curve extrapolation is speculative. That said, I think that it’s less speculative than simply extrapolating the last 40 years of slowing growth into the indefinite future—the fact that we’ve lived through one trend doesn’t make extrapolating it any more responsible, it just makes it feel more like common sense.

## 18 thoughts on “Hyperbolic growth”

1. “We have been in a period of slowing growth for the last forty years. That’s a long time, but looking over the broad sweep of history I still think the smart money is on acceleration eventually continuing, and seeing something like this (though I have little idea whether it will start in 2025 or 2045 or 2075)

20 (ending in 2025)
10 (ending in 2035)
5 (ending in 2040)
2 (ending in 2042)
1 (ending in 2043)
0.5 (ending in June 2043)…”

Perhaps it would be worth noting that various economic reports (e.g. https://www.pwc.com/gx/en/world-2050/assets/pwc-the-world-in-2050-full-report-feb-2017.pdf; https://www.oecd.org/eco/growth/Long-term-projections-of-the-world-economy-a-review.pdf — the latter is “a review of publicly available
projections of GDP per capita over long time horizons” ) do not predict that growth will be accelerating by 2050. Indeed, they in fact predict a continual decline in growth. These projections may be wrong, of course, yet they do at least cast doubt upon the notion that the smart money is on a growth acceleration occurring within the next few decades. (I’m aware your main claim is not that it will happen over the next few decades in particular, but that was the example you gave.)

“This kind of curve extrapolation is speculative. That said, I think that it’s less speculative than simply extrapolating the last 40 years of slowing growth into the indefinite future—the fact that we’ve lived through one trend doesn’t make extrapolating it any more responsible, it just makes it feel more like common sense.”

I’m curious why you consider higher future growth less speculative, also considering the fact that Moore’s law seems likely to end within a decade, and the fact that progress has already slowed down (cf. >>”Our cadence today is closer to two and a half years than two.” Intel is expected to reach the 10 nm node in 2018, a three-year cadence.<< https://en.wikipedia.org/wiki/Moore%27s_law#Near-term_limits ). This is not a very strong reason against believing in higher future growth rates, of course, but the question is what (stronger) reasons we have that would favor a belief in such growth rates. Referring to the larger pattern across history does not provide a very strong reason, IMO, and I think one can even argue that a careful examination of this broader trend actually weakly suggests that peak growth lies in the past, cf:

“A relevant data point here is that the global economy has seen three doublings since 1965, where the annual growth rate was around six percent, and yet the annual growth rate today is only a little over half — around 3.5 percent — of, and lies stably below, what it was those three doublings ago. In the entire history of economic growth, this seems unprecedented, suggesting that we may already be on the other side of the highest growth rates we will ever see. For up until this point, a three-time doubling of the economy has, rare fluctuations aside, led to an increase in the annual growth rate.

And this “past peak growth” hypothesis looks even stronger if we look at 1955, with a growth rate of a little less than six percent and a world product at 5,430 billion 1990 U.S dollars, which doubled four times gives just under 87,000 billion — about where we should expect today’s world product to be. Yet throughout the history of our economic development, four doublings has meant a clear increase in the annual growth rate, at least in terms of the underlying trend; not a stable decrease of almost 50 percent. ”
https://foundational-research.org/the-future-of-growth-near-zero-growth-rates/

2. Elliot says:

I think you have an error in your list of doublings.

You have one ending in 1995, and list that the next doubling is still being awaited in 2017. But GWP was 33 trillion in 1995 and got to 78 trillion in 2014 (https://en.wikipedia.org/wiki/Gross_world_product), so the doubling time was less than 20 years.

1. Good catch! I took these numbers from Brad De Long’s estimates here (https://delong.typepad.com/print/20061012_LRWGDP.pdf). His GDP estimates are split into an “ex-nordhaus” figure, measuring wealth in terms closer to “how much food could you make?”, and then a separate bonus for the introduction of new types of goods (which become an important part of the typical consumption bundle and are rapidly falling in price).

I used his ex-nordhaus estimates, which gives lower growth estimates than reported GDP, because I had more confidence in its comparability to historical figures and it may be more directly relevant to the possibility of explosive growth. Brad deLong prefers to use the full figure.

But I haven’t sanity-check this very much and don’t know how he produces those numbers.

Hopefully it doesn’t change the basic picture too much / doesn’t undermine the basic perspective in this post.

3. Chantal says:

Hi Paul, I am posting on your blog seeking your input on a few questions I had about hyperbolic growth describing world GDP over time. I noticed in this post https://slatestarcodex.com/2019/04/22/1960-the-year-the-singularity-was-cancelled/ you were cited as mentioning a hyperbola fitting the GDP vs. years before 2020 plot. This is in reference to this plot https://slatestarcodex.com/blog_images/demographics_capitadouble2.png.

I noticed that you had this current post on hyperbolic growth. I was wondering if you knew more about the mathematics of why hyperbolic growth describes the data in these posts.

How do we show that a hyperbolic function fitting GDP vs. years before 2020 corresponds to the linearly decreasing log-log plot of doubling time of GDP vs. years before 2020?

Context:
I am letting D(x) be a function describing the doubling time of GDP at time x. I understand that if we have a hyperbolic function f(x) = -1/(x – 10000) where x ranges from 0 to 9999, then its doubling time is D(x) = -(x – 10000)/2 = 5000 – x/2 (obtained via solving f(x + D(x))/f(x) = 2). Converting this to log-log, we have log(D(x)) = log(5000 – x/2), so log(D(x)) = log(5000 – e^(log(x))/2). If we let Y=log(D(x)) and X=log(x) then Y = log(5000 – e^X/2). Plotting this we get a curve that does not look linear: https://www.wolframalpha.com/input/?i=Plot%5Blog%285000+-+e%5Ex%2F2%29%2C+%7Bx%2C+0%2C+log%2810000%29%7D%5D

It makes sense to me that the doubling rate of GDP over time is linearly decreasing if the function describing GDP over time is hyperbolic (this was shown above), but I am not clear on how to show that the log of the doubling rate vs log of time is linearly decreasing.

I am wondering how to find a function such that the log of its doubling rate is linearly decreasing with respect to log of time. In other words, if we started with this doubling time plot https://slatestarcodex.com/blog_images/demographics_capitadouble2.png how would we recover the original function?

Here are my thoughts on how to approach this problem so far:
We can express as log (D(x)) = m*log(x) + b. For simplicity let’s say m = -1 and b = 0

This simplifies to D(x) = 2^(-log_2(x)) = 1/x

Let f(x0) = y0 where x0 and y0 are arbitrary positive values. The time it takes to double at x0 is defined by D(x0) = 1/x0 so f(x0 + 1/x0) = 2y0. The time it takes to double at x0 + 1/x0 is 1/(x0 + 1/x0) so f(x0 + 1/x0 + 1/(x0 + 1/x0)) = 4y0. This process can be extended for the next points that keep incrementing with the reciprocal of the previous input to f.

I am wondering how to find a closed form for f that fits this continued-fraction like input in its definition.

How do we find the class of functions whose doubling time decreases linearly with respect to time? I understand that the hyperbolic function (as described in question 1) is one of the functions that matches this criteria, but how do we prove that it is either the only function that matches the criteria of linearly decreasing doubling time, or find the class of functions which has linearly decreasing doubling time?

1. Chantal says:

Hello Paul I am curious about your thoughts on this post whenever you get the chance. Thanks

1. This form of decreasing doubling times occur whenever you have x’ = x^{alpha} for alpha>1. Or whenever you have x’ = x * A (think A = technology) with A’ = x^{alpha}. These have solutions like GDP(year t) = 1 / (2040 – t)^{alpha-1}, which you can just check by differentiating. You get similar forms of solution if you have similar differential equations.

You can also see David Roodman’s writeup for some thoughts on the stochastic differential equation version of this which makes more sense: https://www.openphilanthropy.org/research/modeling-the-human-trajectory/

1. Chantal says:

Hello, I am retyping this comment since the previous version seems to have gotten lost. Using the equation you gave, $\displaystyle \mathrm{GDP}(t) = \frac{1}{(2040 - t)^{\alpha - 1}}$, I find the doubling time by solving for $D(t)$ in $\mathrm{GDP}(t + D(t))/\mathrm{GDP}(t) = 2$ which gives $D(t) = (2^\frac{1}{1 - \alpha} - 1)t + 2040(1 - 2^{\frac{1}{1 - \alpha}})$.

You can see that this equation is linear in $t$, which would make the log of the doubling time not linear in $\log t$. How do you reconcile this with this image showing a linearly decreasing log-log plot of doubling time? https://slatestarcodex.com/blog_images/demographics_capitadouble2.png

1. In your equation it seems like D(t) is a constant times (2040-t). So log D(t) is a constant plus log(2040-t). Isn’t that linear? (Sorry if I’m missing something.)

In the SSC graph the coefficient on the log-log plot seems to be 1.3 (ln(20)/ln(10)) instead of 1, but I don’t think the estimated coefficient is that precise, and you shouldn’t expect it to be given the amount of noise.

4. Samantha says:

Hi Paul, I posted a comment on this post a long time ago but perhaps it got caught in the spam filter.

1. Sorry about that. I don’t see it in the pending or spam queue, but it might have been deleted.

1. Chantal says:

I was posting under the name Samantha as well. The comment in question is the longer one above posted on October 8 2020. Thanks

5. Chantal says:

Hello-I left a reply about 2 months ago and it is not displaying so it seems to be yet again caught in spam filter.

1. I don’t see it in “pending” or “spam,” not sure what happened? Sorry about unreliable blog comments.

1. Chantal says:

Ok np I reposted it

2. Chantal says:

Hello, I am retyping this comment since the previous version seems to have gotten lost. Using the equation you gave, $\displaystyle \mathrm{GDP}(t) = \frac{1}{(2040 - t)^{\alpha - 1}}$, I find the doubling time by solving for $D(t)$ in $\mathrm{GDP}(t + D(t))/\mathrm{GDP}(t) = 2$ which gives $D(t) = (2^\frac{1}{1 - \alpha} - 1)t + 2040(1 - 2^{\frac{1}{1 - \alpha}})$.

You can see that this equation is linear in $t$, which would make the log of the doubling time not linear in $\log t$. How do you reconcile this with this image showing a linearly decreasing log-log plot of doubling time? https://slatestarcodex.com/blog_images/demographics_capitadouble2.png