# Can the stock market show indefinite exponential growth?

In a comments on a question on money.SE, the following dialog took place:

Eventually there will not be enough matter to represent all the money, so we know for certain that the answer is "No" for a long enough term.

--yters

What do you mean by "represent?" Do you mean that we won't be able to build computer chips that can store a digital representation of someone's stock income? Do you mean that we will no longer be able to cash out our stock portfolio into a fixed commodity like gold, at a fixed dollar price? The former seems wrong to me, and the latter irrelevant.

-- BenCrowell

@BenCrowell: the stock market is not abstract, it depends on investors expectations on the real economy. Its obvious that the economy cannot grow forever, as the planet (and its resources) are finite.

-- Martin Argerami

I'm not convinced by the argument that Martin Argerami claims is obvious. It seems to me to rest on a false assumption that value is measured by resources, e.g., that a dollar's "true" value is measured by how many milligrams of gold it can buy. But I'm not an economist. Would anyone like to take a shot at explaining who is right?

• We have inflation which seems to be indefinite and exponentialyl growing. Do you maybe mean inflation adjusted capitalizations on the stock market? Then I would say only if there are equally large debts somewhere else. – Trilarion Nov 10 '17 at 13:16

May I rephrase your question into the broader question "Can economic growth continue indefinitely?"

(In response to objections that eventually the sun will burn out or the universe will suffer heat death, I take indefinite to mean "lasting for an unknown or unstated length of time" (OED). So I am thinking of 100s, 1000s, or even 10000s of years ahead. But I am not thinking of billions of years ahead or the "infinite future".)

Non-economists commonly believe the answer to be "no", giving some reason along the lines of "resources are finite!"

But the economist's answer to this is "Yes, of course economic growth can continue indefinitely." And so to answer also your narrower question, "Yes, of course the stock market can show indefinite exponential growth." (By "can", I mean that this is at least conceivable. Whether it will is a different matter altogether. After all the world might end tomorrow in a nuclear apocalypse.)

I think we can distinguish between two common fallacies at work here.

Fallacy #1. "Economic growth is about making ever more "stuff", digging ever more gold and other natural resources out of the ground, burning ever more energy, etc." (This caricature is perhaps why many non-economists and especially environmentalists are averse to economists and the idea of economic growth.) The fallacy typically continues, "Resources/the universe is finite. Therefore economic growth must also be finite."

But this is wrong. Economic growth is about improving human well-being, broadly conceived.

It is true that for a long time (the past few centuries), improvements in human well-being were largely through improvements in material well-being and highly-correlated with making ever more stuff and burning ever more energy. After all, it was not two centuries ago that the vast majority of mankind lived at bare subsistence level. (Indeed, even today, many people still do.)

But going forward, it is perfectly conceivable that we make ever less "stuff", dig ever less "stuff" out of the ground, and burn ever less energy, and yet still become ever more well-off. This is actually already happening today in the rich countries (see e.g. falling energy use, briefly analyzed below).

Since the 1930s and 1940s, we've measured economic growth mostly by GDP growth. But economists have always recognized that GDP is a very flawed measure of well-being. Economists are working on alternatives that better capture the notion of improvements in human well-being or equivalently, economic welfare. I do not expect that in 100 years, the current concept of GDP without fundamental modifications will still be used to as the primary measure of economic well-being.

(Footnote: Perhaps in the future, we will also include non-human well-being in our conception of economic growth. But for now, we still restrict attention mostly to human well-being.)

Fallacy #2. "Bad things (like the consumption of food or resources) will grow rapidly or even exponentially. In contrast, offsetting good things (like technology) can at best grow arithmetically. Therefore, there are necessarily limits to growth."

This fallacy is not new. Here's an example of doom-and-gloom predictions from each of the past three centuries, all of which proved to be wrong.

Malthus began with two “fixed laws of our nature.” First, men and women cannot exist without food. Second, the “passion between the sexes” drives them to reproduce.

He explained that, if unchecked, people breed “geometrically” (1, 2, 4, 8, 16, etc.). But, he continued, the production of food can only increase “arithmetically” (1, 2, 3, 4, 5, etc.). “The natural inequality of the two powers of population and of [food] production in the earth,” he declared, “form the great difficulty that to me appears insurmountable [impossible to overcome].”

Malthus concluded: “I see no way by which man can escape from the weight of this law.” In other words, if people keep reproducing in an uncontrolled geometric manner, they will eventually be unable to produce enough food for themselves. The future, Malthus argued, pointed not to endless improvement for humanity, but to famine and starvation.

Writing in the Times of London in 1894, one writer estimated that in 50 years every street in London would be buried under nine feet of manure. Moreover, all these horses had to be stabled, which used up ever-larger areas of increasingly valuable land. And as the number of horses grew, ever-more land had to be devoted to producing hay to feed them (rather than producing food for people), and this had to be brought into cities and distributed—by horse-drawn vehicles. It seemed that urban civilization was doomed.

Our attempts to use even the most optimistic estimates of the benefits of technology in the model did not prevent the ultimate decline of population and industry, and in fact did not in any case postpone the collapse beyond the year 2100 (p. 145.).

This was a highly-influential best-seller that has sold over 16M copies in over 30 languages.

Take their example of gold. On p. 56, they calculate that if gold use continued growing exponentially AND there was 5 times as much gold available as there were known gold reserves (they thought this was a very optimistic assumption), gold would be depleted in 29 years, or in 2001.

Surprisingly, 2001 came and went and gold continued to be mined. Indeed, more than ever. Gold mining graph (source):

Roughly every 5 years since 1972, The Limits to Growth folks (AKA the Club of Rome) have released a new update to their 1972 book, each time explaining why they had been correct all along (of course) and sometimes pushing back their predictions about when the eventual collapse will set in. In their 30-Year Update, they make no mention whatsoever of gold.

The following is the response by two critics to The Limits to Growth, also quoted by Robert Solow in a Newsweek article:

The authors load their case by letting some things grow exponentially and others not. Population, capital and pollution grow exponentially in all models, but technologies for expanding resources and controlling pollution are permitted to grow, if at all, only in discrete increments.

(Footnote: Doomsday-mongering was especially fashionable in the West around the 1970s. See also the famous Simon-Ehrlich wager at around the same time.

Predictions at the polar extremes capture the public's attention. Ray Kurzweil comes to mind as someone who makes similar predictions, but at the polar opposite.

In contrast, the median economist is cautiously optimistic, believing merely that slow but steady, sustained growth is possible. No doomsday, no stagnation, but no impending Singularity either. Not exactly a position that sells many books.)

In 2012, a physics professor wrote a somewhat-influential blogpost: Exponential Economist Meets Finite Physicist, exhibiting both of the above fallacies. That someone as intelligent as a physics professor could commit both fallacies shows that economists must do a far better job at educating the public.

There is plenty that is wrong in that blogpost and perhaps I will do a sentence-by-sentence dissection elsewhere, but this is probably not the proper avenue. Here I'll merely point out one obvious factual error that's of particular relevance. He claims as fact that

energy growth has far outstripped population growth, so that per-capita energy use has surged dramatically over time—our energy lives today are far richer than those of our great-great-grandparents a century ago [economist nods]. So even if population stabilizes, we are accustomed to per-capita energy growth: total energy would have to continue growing to maintain such a trend [another nod].

As Tim Harford points out, this is FALSE. In recent decades, energy growth per person in many countries has actually been falling, even as GDP per capita has risen. Graph (data from World Bank, June 1st 2017 update):

In every rich country, per-capita energy use peaked years ago and has been falling ever since. In fact, in some countries, it peaked DECADES ago (peaked in 1978 in the US, in 1979 in Germany, and in 1973 in the UK).

(One would've hoped that a physics professor backed up his factual claims with something more than a fictitious and bumbling economist who repeatedly nods.)

See also falling energy intensity (energy use per unit of GDP) (source):

The highest per-capita energy use ever attained was the US in 1978. My prediction is that global average human well-being will keep improving, but global per-capita energy use will never hit the US 1978 peak (at least not until we start populating other planets and stars).

• Which trap are you referring to exactly, when you talk about the "Exponential Economist Meets Finite Physicist" article? You should be very specific in pointing what the supposed logical flaws are. You think that this non-economist (Tom Murphy) doesn't understand economy? Well, he thinks most economists don't understand physics. And he does try to prove that the economy cannot be completely decoupled from the physical world. – Eric Duminil Jun 27 '17 at 10:50
• Important part from the article : " If the flow of energy is fixed, but we posit continued economic growth, then GDP continues to grow while energy remains at a fixed scale. This means that energy—a physically-constrained resource, mind—must become arbitrarily cheap as GDP continues to grow and leave energy in the dust." – Eric Duminil Jun 27 '17 at 10:57
• Another one : "But if energy became arbitrarily cheap, someone could buy all of it, and suddenly the activities that comprise the economy would grind to a halt. Food would stop arriving at the plate without energy for purchase, so people would pay attention to this. Someone would be willing to pay more for it. Everyone would. There will be a floor to how low energy prices can go as a fraction of GDP." – Eric Duminil Jun 27 '17 at 10:57
• While it is true that material wealth is not a great indicator of well being, that is what the stock market measures. The fact that it isn't the greatest thing to measure doesn't change the fact that is what it actually measures. In what way is saying that maybe one day "the market will completely transform and start measuring something completely different than what it does now" not the same thing as saying No, it can't continue on forever?? – Shane Jun 27 '17 at 20:47
• @EricDuminil: The trap is to believe that economic growth must always somehow be tied to an increase in energy use. More broadly, as stated a couple times in my answer above, the trap is the common layperson's misconception that economic growth is about making ever more "stuff", digging ever more "stuff" out of the ground, and burning ever more energy and "stuff". – Kenny LJ Jun 28 '17 at 2:15

To answer your question, one needs to know a little economics and a little physics. I only really know anything about the former. That caveat aside, stock market values and GDP are metrics that reflect productive activity. Some vector of inputs $\mathbf{x}$ (n.b. these need not be material resources) are subjected to a production process yielding outputs $f(\mathbf{x})$ (n.b. this output also does not need to be in material consumables).

There are two ways to increase economic output (i.e., to achieve economic growth).

• The first is to increase the supply of inputs to $\mathbf{y}>\mathbf{x}$. This is the pre-industrial mode of econmic development: work harder, extract more resources, cultivate more land, and thereby produce more stuff.

• The second way to increase growth is to develop a new production technology, $g$, such that $g(\mathbf{x})>f(\mathbf{x})$ (more is produced with the same inputs). For example, computers allow us to produce more output from fewer inputs than do typewriters. Indeed, if technology improves at a sufficient rate then it is (theoretically) possible to sustain exponential economic growth even as the the consumption of inputs decreases faster than exponentially.

In the extreme long-run, this observation interacts with the laws of physics (brace yourself for hand-waving). Provided there is some region of the universe that is not in a state of maximum entropy then there is scope for humans to effect a process of production and, with sufficiently advanced technology, for this production to grow exponentially (note: the claim is not that it will happen or that it is likely, only that there does not seem to be anything in either economics or physics that makes it fundamentally impossible).

On the other hand, if the entire universe converges to a uniform state of maximum entropy (commonly known as heat death) then, roughly, continuation of economic growth would require $f(\mathbf{0})>\mathbf{0}$—i.e. that valuable outputs spontaneously emerge from a fully entropic system. This seems implausible and probably contradicts the first law of thermodynamics.

So my assessment of the situation is as follows:

Q: must the economy's long-run rate of economic growth eventually be less than exponential as a matter of theoretical necessity?

A: If the universe converges to a uniform state of maximum entropy then yes, otherwise no.

This leaves the question of the long-run fate of the universe. To end on a note of optimism, here's a quote from Wikipedia:

"The role of entropy in cosmology remains a controversial subject since the time of Ludwig Boltzmann. Recent work has cast some doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general. Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly, moving the universe further from the heat death with time, not closer. This results in an "entropy gap" pushing the system further away from the posited heat death equilibrium. Other complicating factors, such as the energy density of the vacuum and macroscopic quantum effects, are difficult to reconcile with thermodynamical models, making any predictions of large-scale thermodynamics extremely difficult."

• You could stop your argument right after "Indeed, if technology improves at a sufficient rate then it is (theoretically) possible to sustain exponential economic growth even as the the consumption of inputs decreases faster than exponentially.". Done! The answer to the question is therefore : No. – Eric Duminil Jun 28 '17 at 10:04
• This might not be a popular opinion on this website, but here goes : the economy is based on a physical world. It means that it's fine to talk about economics with only physics knowledge, but it's not okay to apply economics principle to the physical world. – Eric Duminil Jun 28 '17 at 10:09
• @EricDuminil Also, the claim that it is fine to talk about economics with only physics knowledge is ridiculous. Physics might say something about the infinite-horizon outcome, but tells us virtually nothing about macroeconomic phenomena at meaningful time scales, and absolutely nothing about phenomena in other branches of economics. – Ubiquitous Jun 28 '17 at 10:30
• Indefinite exponential growth in efficiency leading to Indefinite exponential economic growth has more problems than simply saying that "Indefinite exponential economic growth is possible". There is far, far less room to improve efficiency than there is room to increase inputs. ie: we'd have to invent entirely new branches of physics to increase efficiency that much, not invent new technologies. – Shane Jun 28 '17 at 14:57
• In reality, there are only likelihoods. It isn't impossible for you to walk off the top of Trump tower and to fly away. It isn't impossible for the universe to spontaneously replace itself with a custard pie. It isn't impossible for us to find new physics that will allow for the efficiency gains you describe. But to actually say that it is possible for these things to happen renders the word meaningless. – Shane Jun 28 '17 at 18:22

It depends on what time scales you are talking about, but no.

"The greatest shortcoming of the human race is our inability to understand the exponential function."

-Albert Allen Bartlett, emeritus professor of physics at the University of Colorado at Boulder, USA.

You can watch the talk that goes over the math, but the conclusion inescapable: Exponential growth -- consistent growth over time -- cannot proceed indefinitely within a finite system. Ie: planet Earth or the universe.

Even if you (correctly) believe that value doesn't come from gold, you have to agree that it comes from somewhere. Eventually, that somewhere/something will run out. Even if you believe that the economy will be digital, and we go green, and come up with sustainable growth of computers and the digital economy, eventually just their waste heat will cook the planet. Forget co2!

EDIT:

Do you mean that we won't be able to build computer chips that can store a digital representation of someone's stock income? This seems wrong to me.

Yes! That is exactly what it means. It means that if you somehow managed to store the value of a digit in one planck length^2 (note this is MANY, MANY, MANY, orders of magnitude beyond what is theoretically possible, even in theory) eventually, the representation of the value will fill up the ENTIRE universe. Then on the next doubling, you would need TWO universes to just to contain the representation of the value of the stock market. Then four universes...

The reason this seems wrong to you is because one of the greatest shortcomings of the human race is our inability to understand the exponential function.

One day the sun will burn out. One day, the last star will burn out. One day, the universe be totally cold and dark. On that day, even electrons will be too cold to spin around their atom. On that day, all physical processes stop. Yes, that does include the stock market.

On that depressing note, I leave you with the parable of the Indian king and his grain.

There's a famous legend about the origin of chess that goes like this. When the inventor of the game showed it to the emperor of India, the emperor was so impressed by the new game, that he said to the man