First written: Jul. 2017; Last update: Nov 2022.
Exponential growth is a common pattern found throughout nature. Yet it is also a pattern that tends not to last, as growth rates tend to decline sooner or later.
In biology, this pattern of exponential growth that wanes off is found in everything from the development of individual bodies — for instance, in the growth of humans, which levels off in the late teenage years — to population sizes.
One may of course be skeptical that this general trend will also apply to the growth of our technology and economy at large, as innovation seems to continually postpone our clash with the ceiling, yet it seems inescapable that it must. For in light of what we know about physics, we can conclude that exponential growth of the kinds we see today, in technology in particular and in our economy more generally, must come to an end, and do so relatively soon.
Limits to growth
Physical limits to computation and Moore’s law
One reason we can make this assertion is that there are theoretical limits to computation. As physicist Seth Lloyd’s calculations show, a continuation of Moore’s law — in its most general formulation: “the amount of information that computers are capable of processing and the rate at which they process it doubles every two years” — would imply that we hit the theoretical limits of computation within 250 years:
If, as seems highly unlikely, it is possible to extrapolate the exponential progress of Moore’s law into the future, then it will only take two hundred and fifty years to make up the forty orders of magnitude in performance between current computers that perform 1010 operations per second on 1010 bits and our one kilogram ultimate laptop that performs 1051 operations per second on 1031 bits.
Similarly, physicists Lawrence Krauss and Glenn Starkman have calculated that, even if we factor in colonization of space at the speed of light, this doubling of processing power cannot continue for more than 600 years in any civilization:
Our estimate for the total information processing capability of any system in our Universe implies an ultimate limit on the processing capability of any system in the future, independent of its physical manifestation and implies that Moore’s Law cannot continue unabated for more than 600 years for any technological civilization.
In a more recent lecture and a subsequent interview, Krauss said that the absolute limit for the continuation of Moore’s law, in our case, would be reached in less than 400 years (the discrepancy — between the numbers 400 and 600 — is at least in part because Moore’s law, in its most general formulation, has played out for more than a century in our civilization at this point). And, as both Krauss and Lloyd have stressed, these are ultimate theoretical limits, resting on assumptions that are unlikely to be met in practice, such as expansion at the speed of light. What is possible, in terms of how long Moore’s law can continue for, given both engineering and economic constraints is likely significantly less. Indeed, we are already close to approaching the physical limits of the paradigm that Moore’s law has been riding on for more than 50 years — silicon transistors, the only paradigm that Gordon Moore was talking about originally — and it is not clear whether other paradigms will be able to take over and keep the trend going.
Limits to the growth of energy use
Physicist Tom Murphy has calculated a similar limit for the growth of the energy consumption of our civilization. Based on the observation that the energy consumption of the United States has increased fairly consistently with an average annual growth rate of 2.9 percent over the last 350 odd years (although the growth rate appears to have slowed down in recent times and been stably below 2.9 since c. 1980), Murphy proceeds to derive the limits for the continuation of similar energy growth. He does this, however, by assuming an annual growth rate of “only” 2.3 percent, which conveniently results in an increase of the total energy consumption by a factor of ten every 100 years. If we assume that we will continue expanding our energy use at this rate by covering Earth with solar panels, this would, on Murphy’s calculations, imply that we will have to cover all of Earth’s land with solar panels in less than 350 years, and all of Earth, including the oceans, in 400 years.
Beyond that, assuming that we could capture all of the energy from the sun by surrounding it in solar panels, the 2.3 percent growth rate would come to an end within 1,350 years from now. And if we go further out still, to capture the energy emitted from all the stars in our galaxy, we get that this growth rate must hit the ceiling and become near-zero within 2,500 years (of course, the limit of the physically possible must be hit earlier, indeed more than 500 years earlier, as we cannot traverse our 100,000 light year-wide Milky Way in only 2,500 years).
One may suggest that alternative sources of energy might change this analysis significantly, yet, as Murphy notes, this does not seem to be the case:
Some readers may be bothered by the foregoing focus on solar/stellar energy. If we’re dreaming big, let’s forget the wimpy solar energy constraints and adopt fusion. The abundance of deuterium in ordinary water would allow us to have a seemingly inexhaustible source of energy right here on Earth. We won’t go into a detailed analysis of this path, because we don’t have to. The merciless growth illustrated above means that in 1400 years from now, any source of energy we harness would have to outshine the sun.
Essentially, keeping up the annual growth rate of 2.3 percent by harnessing energy from matter not found in stars would force us to make such matter hotter than stars themselves. We would have to create new stars of sorts, and, even if we assume that the energy required to create such stars is less than the energy gained, such an endeavor would quickly run into limits as well. For according to one estimate, the total mass of the Milky Way, including dark matter, is only 20 times greater than the mass of its stars. Assuming a 5:1 ratio of dark matter to ordinary matter, this implies that that there is only about 3.3 times as much ordinary non-stellar matter as there is stellar matter in our galaxy. Thus, even if we could convert all this matter into stars without spending any energy and harvest the resulting energy, this would only give us about 50 years more of keeping up with the annual growth rate of 2.3 percent.1
Limits derived from economic considerations
Similar conclusions to the ones drawn above for computation and energy also seem to follow from calculations of a more economic nature. For, as economist Robin Hanson has argued, projecting present economic growth rates into the future also leads to a clash against fundamental limits:
Today we have about ten billion people with an average income about twenty times subsistence level, and the world economy doubles roughly every fifteen years. If that growth rate continued for ten thousand years[,] the total growth factor would be 10200.
There are roughly 1057 atoms in our solar system, and about 1070 atoms in our galaxy, which holds most of the mass within a million light years. So even if we had access to all the matter within a million light years, to grow by a factor of 10200, each atom would on average have to support an economy equivalent to 10140 people at today’s standard of living, or one person with a standard of living 10140 times higher, or some mix of these.
Indeed, current growth rates would “only” have to continue for three thousand years before each atom in our galaxy would have to support an economy equivalent to a single person living at today’s living standard, which already seems rather implausible (not least because we can only access a tiny fraction of “all the matter within a million light years” in three thousand years). Hanson does not, however, expect the current growth rate to remain constant, but instead, based on the history of growth rates, expects a new growth mode where the world economy doubles within 15 days rather than 15 years:
If a new growth transition were to be similar to the last few, in terms of the number of doublings and the increase in the growth rate, then the remarkable consistency in the previous transitions allows a remarkably precise prediction. A new growth mode should arise sometime within about the next seven industry mode doublings (i.e., the next seventy years) and give a new wealth doubling time of between seven and sixteen days.
And given this more than a hundred times greater growth rate, the net growth that would take 10,000 years to accomplish given our current growth rate (cf. Hanson’s calculation above) would now take less than a century to reach, while growth otherwise requiring 3,000 years would require less than 30 years. So if Hanson is right, and we will see such a shift within the next seventy years, what seems to follow is that we will reach the limits of economic growth, or at least reach near-zero growth rates, within a century or two. Such a projection is also consistent with the physically derived limits of the continuation of Moore’s law; not that economic growth and Moore’s law are remotely the same, yet they are no doubt closely connected: economic growth is largely powered by technological progress, of which Moore’s law has been a considerable subset in recent times.
The conclusion we reach by projecting past growth trends in computing power, energy, and the economy is the same: our current growth rates cannot go on forever. In fact, they will have to decline to near-zero levels very soon on a cosmic timescale. Given the physical limits to computation, and hence, ultimately, to economic growth, we can conclude that we must be close to the point where peak relative growth in our economy and our ability to process information occurs — that is, the point where this growth rate is the highest in the entire history of our civilization, past and future.
Peak growth might lie in the past
This is not, however, to say that this point of maximum relative growth necessarily lies in the future. Indeed, in light of the declining economic growth rates we have seen over the last few decades, it cannot be ruled out that we are now already past the point of “peak economic growth” in the history of our civilization, with the highest growth rates having occurred around 1960-1980, cf. these declining growth rates and this essay by physicist Theodore Modis. This is not to say that we most likely are, yet it seems that the probability that we are is non-trivial.
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 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. This tentatively suggests that we should not expect to see growth rates significantly higher than those of today sustained in the future.
Could we be past peak growth in science and technology?
That peak growth lies in the past may also be true of technological progress in particular, or at least many forms of technological progress, including the progress in computing power tracked by Moore’s law, where the growth rate appears to have been highest around 1990-2005, and to since have been in decline, cf. this article and the first graphs found here and here. Similarly, various sources of data and proxies tracking the number of scientific articles published and references cited over time also suggest that we could be past peak growth in science as well, at least in many fields when evaluated based on such metrics, with peak growth seeming to have been reached around 2000-2010.
Yet again, these numbers — those tracking economic, technological, and scientific progress — are of course closely connected, as growth in each of these respects contributes to, and is even part of, growth in the others. Indeed, one study found the doubling time of the total number of scientific articles in recent decades to be 15 years, corresponding to an annual growth rate of 4.7 percent, strikingly similar to the growth rate of the global economy in recent decades. Thus, declining growth rates both in our economy, technology, and science cannot be considered wholly independent sources of evidence that growth rates are now declining for good. We can by no means rule out that growth rates might increase in all these areas in the future — although, as we saw above with respect to the limits of Moore’s law and economic progress, such an increase, if it is going to happen, must be imminent if current growth rates remain relatively stable.
Might recent trends make us bias-prone?
How might it be relevant that we may be past peak economic growth at this point? Could it mean that our expectations for the future are likely to be biased? Looking back toward the 1960s might be instructive in this regard. For when we look at our economic history up until the 1960s, it is not so strange that people made many unrealistic predictions about the future around this period. Because not only might it have appeared natural to project the high growth rate at the time to remain constant into the future, which would have led to today’s global GDP being more than twice of what it is; it might also have seemed reasonable to predict the growth rates to keep on rising even further. After all, that was what they had been doing consistently up until that point, so why should it not continue in the following decades, resulting in flying cars and conversing robots by the year 2000? Such expectations were not that unreasonable given the preceding economic trends.
The question is whether we might be similarly overoptimistic about future economic progress today given recent, possibly unique, growth trends, specifically the unprecedented increase in absolute annual growth that we have seen over the past two decades. The same may apply to the trends in scientific and technological progress cited above, where peak growth in many areas appears to have happened in the period 1990-2010, meaning that we could now be at a point where we are disposed to being overoptimistic about further progress.
Yet, again, it is highly uncertain at this point whether growth rates, of the economy in general and of progress in technology and science in particular, will increase again in the future. Future economic growth may not conform well to the model with roughly symmetric growth rates around the 1960s, although the model certainly deserves some weight. All we can say for sure is that growth rates must become near-zero relatively soon. What the path toward that point will look like remains an open question. We could well be in the midst of a temporary decline in growth rates that will be followed by growth rates significantly greater than those of the 1960s, cf. the new growth mode envisioned by Robin Hanson.2
Implications: This is an extremely special time
Applying the mediocrity principle, we should not expect to live in an extremely unique time. Yet, in light of the facts about the ultimate limits to growth seen above, it is clear that we do: we are living during the childhood of civilization where there is still rapid growth, at the pace of doublings within a couple of decades. If civilization persists with similar growth rates, it will soon become a grown-up with near-zero relative growth. And it will then look back at our time — today plus minus a couple of centuries, most likely — as the one where growth rates were by far the highest in its entire history, which may be more than a trillion years.
It seems that a few things follow from this. First, more than just being the time where growth rates are the highest, this may also, for that very reason, be the time where individuals can influence the future of civilization more than any other time. In other words, this may be the time where the outcome of the future is most sensitive to small changes, as it seems plausible, although far from clear, that small changes in the trajectory of civilization are most significant when growth rates are highest. An apt analogy might be a psychedelic balloon with fluctuating patterns on its surface, where the fluctuations that happen to occur when we blow up the balloon will then also be blown up and leave their mark in a way that fluctuations occurring before and after this critical growth period will not (just like quantum fluctuations in the early universe got blown up during cosmic expansion, and thereby in large part determined the grosser structure of the universe today). Similarly, it seems much more difficult to cause changes across all of civilization when it spans countless star systems compared to today.
That being said, it is not obvious that small changes — in our actions, say — are more significant in this period where growth rates are many orders of magnitude higher than in any other time. It could also be that such changes are more consequential when the absolute growth is the highest. Or perhaps when it is smallest, at least as we go backwards in time, as there were far fewer people back when growth rates were orders of magnitude lower than today, and hence any given individual comprised a much greater fraction of all individuals than an individual does today.
Still, we may well find ourselves in a period where we are uniquely positioned to make irreversible changes that will echo down throughout the entire future of civilization.3 To the extent that we are, this should arguably lead us to update toward trying to influence the far future rather than the near future. More than that, if it does hold true that the time where the greatest growth rates occur is indeed the time where small changes are most consequential, this suggests that we should increase our credence in the simulation hypothesis. For if realistic sentient simulations of the past become feasible at some point, the period where the future trajectory of civilization seems the most up for grabs would seem an especially relevant one to simulate and learn more about. However, one can also argue that the sheer historical uniqueness of our current growth rates alone, regardless of whether this is a time where the fate of our civilization is especially volatile, should lead us to increase this credence, as such uniqueness may make it a more interesting time to simulate, and because being in a special time in general should lead us to increase our credence in the simulation hypothesis (see for instance this talk for a case for why being in a special time makes the simulation hypothesis more likely).4
On the other hand, one could also argue that imminent near-zero growth rates, along with the weak indications that we may now be past peak growth in many respects, provide a reason to lower our credence in the simulation hypothesis, as these observations suggest that the ceiling for what will be feasible in the future may be lower than we naively expect in light of today’s high growth rates. And thus, one could argue, it should make us more skeptical of the central premise of the simulation hypothesis: that there will be (many) ancestor simulations in the future. To me, the consideration in favor of increased credence seems stronger, although it does not significantly move my overall credence in the hypothesis, as there are countless other factors to consider.5
Appendix: Questioning our assumptions
Caspar Oesterheld pointed out to me that it might be worth meditating on how confident we can be in these conclusions given that apparently solid predictions concerning the ultimate limits to growth have been made before, yet quite a few of these turned out to be wrong. Should we not be open to the possibility that the same might be true of (at least some of) the limits we reviewed in the beginning of this essay?
Could our understanding of physics be wrong?
One crucial difference to note is that these failed predictions were based on a set of assumptions — e.g. about the amount of natural resources and food that would be available — that seem far more questionable than the assumptions that go into the physics-based predictions we have reviewed here: that our apparently well-established physical laws and measurements indeed are valid, or at least roughly so. The epistemic status of this assumption seems a lot more solid, to put it mildly. So there does seem to be a crucial difference here. This is not to say, however, that we should not maintain some degree of doubt as to whether this assumption is correct (I would argue that we always should). It just seems that this degree of doubt should be quite low.
Yet, to continue the analogy above, what went wrong with the aforementioned predictions was not so much that limits did not exist, but rather that humans found ways of circumventing them through innovation. Could the same perhaps be the case here? Could we perhaps some day find ways of deriving energy from dark energy or some other yet unknown source, even though physicists seem skeptical? Or could we, as Ray Kurzweil speculates, access more matter and energy by finding ways of travelling faster than light, or by finding ways of accessing other parts of our notional multiverse? Might we even become able to create entirely new ones? Or to eventually rewrite the laws of nature as we please? (Perhaps by manipulating our notional simulators?) Again, I do not think any of these possibilities can be ruled out completely. Indeed, some physicists argue that the creation of new pocket universes might be possible, not in spite of “known” physical principles (or rather theories that most physicists seem to believe, such as inflationary theory), but as a consequence of them. However, it is not clear that anything from our world would be able to expand into, or derive anything from, the newly created worlds on any of these models (which of course does not mean that we should not worry about the emergence of such worlds, or the fate of other “worlds” that we perhaps could access).
All in all, the speculative possibilities raised above seem unlikely, yet they cannot be ruled out for sure. The limits we have reviewed here thus represent a best estimate given our current, admittedly incomplete, understanding of the universe in which we find ourselves, not an absolute guarantee. However, it should be noted that this uncertainty cuts both ways, in that the estimates we have reviewed could also overestimate the limits to various forms of growth by countless orders of magnitude.
Might our economic reasoning be wrong?
Less speculatively, I think, one can also question the validity of our considerations about the limits of economic progress. I argued that it seems implausible that we in three thousand years could have an economy so big that each atom in our galaxy would have to support an economy equivalent to a single person living at today’s living standard. Yet could one not argue that the size of the economy need not depend on matter in this direct way, and that it might instead depend on the possible representations that can be instantiated in matter? If economic value could be mediated by the possible permutations of matter, our argument about a single atom’s need to support entire economies might not have the force it appears to have. For instance, there are far more legal positions on a Go board than there are atoms in the visible universe, and that’s just legal positions on a Go board. Perhaps we need to be more careful when thinking about how atoms might be able to create and represent economic value?
It seems like there is a decent point here. Still, I think economic growth at current rates is doomed. First, it seems reasonable to be highly skeptical of the notion that mere potential states could have any real economic value. Today at least, what we value and pay for is not such “permutation potential”, but the actual state of things, which is as true of the digital realm as of the physical. We buy and stream digital files such as songs and movies because of the actual states of these files, while their potential states mean nothing to us. And even when we invest in something we think has great potential, like a start-up, the value we expect to be realized is still ultimately one that derives from its actual state, namely the actual state we hope it will assume, not its number of theoretically possible permutations.
It is not clear why this would change, or how it could. After all, the number of ways one can put all the atoms in the galaxy together is the same today as it will be ten thousand years from now. Organizing all these atoms into a single galactic supercomputer would only seem to increase the value of their actual state.
Second, economic growth still seems tightly constrained by the shackles of physical limitations. For it seems inescapable that economies, of any kind, are ultimately dependent on the transfer of resources, whether these take the form of information or concrete atoms. And such transfers require access to energy, the growth of which we know to be constrained, as is true of the growth of our ability to process information. As these underlying resources that constitute the lifeblood of any economy stop growing, it seems unlikely that the economy can avoid this fate as well. (Tom Murphy touches on similar questions in his analysis of the limits to economic growth.)
Again, we of course cannot exclude that something crucial might be missing from these considerations. Yet the conclusion that economic growth rates will decline to near-zero levels relatively soon, on a cosmic timescale at least, still seems a safe bet in my view.
I would like to thank Brian Tomasik, Caspar Oesterheld, Duncan Wilson, Kaj Sotala, Lukas Gloor, Magnus Dam, Max Daniel, and Tobias Baumann for valuable comments and inputs. This essay was originally published at the website of the Foundational Research Institute, now the Center on Long-Term Risk.
1. One may wonder whether there might not be more efficient ways to derive energy from the non-stellar matter in our galaxy than to convert it into stars as we know them. I don’t know, yet a friend of mine who does research in plasma physics and fusion says that he does not think one could, especially if we, as we have done here, disregard the energy required to clump the dispersed matter together so as to “build” the star, a process that may well take more energy than the star can eventually deliver.
The aforementioned paper by Lawrence Krauss and Glenn Starkman also contains much information about the limits of energy use, and in fact uses accessible energy as the limiting factor that bounds the amount of information processing any (local) civilization could do (they assume that the energy that is harvested is beamed back to a “central observer”).
2. It should be noted, though, that Hanson by no means rules out that such a growth mode may never occur, and that we might already be past, or in the midst of, peak economic growth: “[…] it is certainly possible that the economy is approaching fundamental limits to economic growth rates or levels, so that no faster modes are possible […]”
3. The degree to which there is sensitivity to changes of course varies between different endeavors. For instance, natural science seems more convergent than moral philosophy, and thus its development is arguably less sensitive to the particular ideas of individuals working on it than the development of moral philosophy is.
4. One may then argue that this should lead us to update toward focusing more on the near future. This may be true. Yet should we update more toward focusing on the far future given our ostensibly unique position to influence it? Or should we update more toward focusing on the near future given increased credence in the simulation hypothesis? (Provided that we indeed do increase this credence, cf. the counter-consideration above.) In short, it mostly depends on the specific probabilities we assign to these possibilities. I myself happen to think the far future should dominate, as I assign the simulation hypothesis (as commonly conceived) a very small probability.
5. For instance, fundamental epistemological issues concerning how much one can infer based on impressions from a simulated world (which may only be your single mind) about a simulating one (e.g. do notions such as “time” and “memory” correspond to anything, or even make sense, in such a “world”?); the fact that the past cannot be simulated realistically, since we can only have incomplete information about a given physical state in the past (not only because we have no way to uncover all the relevant information, but also because we cannot possibly represent it all, even if we somehow could access it — for instance, we cannot faithfully represent the state of every atom in our solar system in any point in the past, as this would require too much information), and a simulation of the past that contains incomplete information would depart radically from how the actual past unfolded, as all of it has a non-negligible causal impact (even single photons, which, it appears, are detectable by the human eye), and this is especially true given that the vast majority of information would have to be excluded (both due to practical constraints to what can be recovered and what can be represented); whether conscious minds can exist on different levels of abstraction; etc.