Exponential economist meets Physicist

Here is an imaginary dialogue between a physicist and an economist who is not Georgescu-Roegen.  The physicist is skeptical about the prospect for continued exponential growth, excerpt:

Physicist: Well, we could (and do, somewhat) beam non-thermal radiation into space, like light, lasers, radio waves, etc. But the problem is that these “sources” are forms of high-grade, low-entropy energy. Instead, we’re talking about getting rid of the waste heat from all the processes by which we use energy. This energy is thermal in nature. We might be able to scoop up some of this to do useful “work,” but at very low thermodynamic efficiency. If you want to use high-grade energy in the first place, having high-entropy waste heat is pretty inescapable.

…we’re too close to an astounding point for me to leave it unspoken. At that 2.3% growth rate, we would be using energy at a rate corresponding to the total solar input striking Earth in a little over 400 years. We would consume something comparable to the entire sun in 1400 years from now. By 2500 years, we would use energy at the rate of the entire Milky Way galaxy—100 billion stars! I think you can see the absurdity of continued energy growth. 2500 years is not that long, from a historical perspective. We know what we were doing 2500 years ago. I think I know what we’re not going to be doing 2500 years hence.

For the pointer I thank Sam Penrose, Jim Nichols, Jason Ketola, and Mark Weaver.  It is interesting throughout, though I expect war to intervene at some point to break the exponential growth.

Addendum: Here is a related paper by Robin Hanson.


Actually, I've been reading the "Do the Math" blog, where your post came from, for a few months now. Later posts have great explanations of the theoretical limits on all the the major alternative energy sources.

I think that first link is broken. It just leads to a 404 on the MR site.

http:// is missing. Copy & paste the link location, delete everything before "physics", press enter.

Yup, figured it out. Just a heads up so it can get fixed

Why is growth synonymous with energy usage? To the extent energy is used for travel and logistics we may be close to maximum capabilities there. What if we just transported much better things at the same energy usage for the transportation. If you include the amoebas in your growth rate calculations Mr. Physicist Amoeba would be amazed.

It's not synonymous, but at least so far the correlation is very strong and efficiency gains don't seem to be powerful enough to effectively reverse that trend. That's not to say anyone can predict the future, but it's hard to read the blog (which I very much recommend even though I don't agree with quite a big part of it) and not come away with the impression that continued growth will be at least a good deal harder to achieve than most people are willing to admit to themselves.

Here's a mechanism for the observation. Production + allocation are synonymous with energy use, since you can't produce or move something without doing work.

That appears to be an equivocation on "work".


I guess Facebook is producing and moving electrons. But I wouldn't argue that their product is contributing to exponential energy usage.

Compared to what? Before Facebook? Between all their servers and all the computers that connect to them, it must be a pretty big number, and that they've only been around for 6-7 years, that's a lot of power.

I expect war to intervene at some point to break the exponential growth
Isn't that just great and dandy?! Exponential growth for some time, then a war to destroy accumulted capital — a sort of civilizational house burning — then back to square onethree. Rinse, repeat.

I expect war to intervene at some point to break the exponential growth

At which point a biologists points out that usually the bacteria on the Petri dish consume all the nutrients and choke on their own waste.

Where is the evidence that such growth follows an exponential function?

What do you mean?

The formula for exponential growth is below. The growth rate is r, time t is discrete intervals, x_0 is the value of x at time 0.

x_t = x_0(1+r)^t

Is energy use really following this function?

If you tweak the value of r enough, you can make this approximate a linear function.

This is not physics; it is simple math. It's just a rehash of the old math problem "if I give you a penny today, two pennies tomorrow, four pennies the next day...". In a short time you become a millionaire; the problem is that no one will agree to pay you on this formula, and it is an equally unrealistic formula for energy usage.

It makes it sound like we are going to run out of resources in a few hundred years, but actually the "limit" this hits is thousands of times the current per-capita energy uses. Is everyone going to need their own multi-megawatt power plant someday? I doubt it, no matter how much the economy grows, but that is the basis of this analysis. It is far more realistic to assume that long term growth in energy usage will approach a linear, rather than an exponential, trend. And it could grow linearly pretty much indefinitely without hitting the ceiling.

See my post below. Steve beat me to it.

Oops, this was meant as a reply to Neal above.

Sorry about these errant comments. It seems to be an Ipad problem.

Most people, including economists, tend to assume that trends in measures of economic activity can be described as percentage change per year. The point the physicist makes is that over time scales of civilisations, compound interest leads to absurd figures. An increase of 1% per year amounts to a factor of more than 60,000,000,000 after 2500 years. In the last 2500 years (which implies starting the clock in the early Roman Republic), the 1% per year model broke down. It will break again. I don't know how it will break down and neither does anyone else here.

This isn't complicated. Human population growth is slowing, and will stop (and maybe even begin declining) around 2050-75. Energy usage will plateau accordingly. No wars or famines or plagues required (although those may still occur)

Why is this hard?

Hang on, isn't this the basic fallacy that continual economic growth implies increasing use of energy or other physical resources? Economic growth is about increasing satisfaction of *preferences*. We don't know what that might look like.

Gosh guys, do follow the link and read the article. It's not that long and it answers these objections, which is half the point of the article.

Saturos and Anonymous,

Yes this is an example of the fallacy that increasing satisfaction requires increasing resources. If you read the comments section you can see that we get Tom to acknowledge this. He is now shifting his argument to economic growth will have to be different in the future than it has in the past. I agree with this rephrased argument, but the initial argument is clearly incorrect.

I think the premise is false, or at least misleading: that economic *growth* must continue. Of course the physicist chooses this as his starting point, nothing physical can grow forever.

But so what? From an economic perspective, if we reach a stage where every human has access to everything he or she needs that society can give them, then who cares if we cant provide *more* tomorrow? Thats the misleading part, the assumption that economic prosperity and human happiness *requires* continual growth, that we wont reach some point where we are happy the way we are.

The physicist sort of skips past this when his strawman economist brings up the fact that human population is unlikely to continue to grow. The psyicist's response is more or less "well yes, but we will continue to use more energy. Really? Each individual human will continue to use energy without bounds? We wont reach some saturation point where everyone has the energy they need to do what they want?

I think of it like if everyone lived in a desert, then there would seem to always be demand for more and more water. However, eventually, that demand does get sated.

On the other hand, at the current rate of energy consumption our civilization needs something like 10 million years to spend the equivalent of 1 second of the Sun's total energy output. There is still quite a bit of room to grow!

This is the real bubble. It is posible to think and wish whatever you want about posible eficiency, future techs, and futuramas. the data say this:
It is no turn back. Some few maybe can construct some kind of futurama with zero emisions, but do you really think that it is posible for 10000,000,000 of human beings? (people eat, shit, go to the beach, buy sunglasses and envy their neighbour's tv.

The entire solar output is about a trillion times our power consumption. 2^14 is nowhere near a trillion. I'm getting 4000 years just for the sun at his "doubling every 100 years" rate.

Dan, the term "three-peat" was trademarked by Pat Riley.

If you three-peat your "oops," you may have to pay him a royalty.

Hmmmm, I'd better be careful then.

These numbers may seem absurd, but they are not. As long as we can improve energy efficiencies to the point where it becomes feasible to start mining stuff for energy outside earth, the game is up. We don't need to be manually "mine" the sun or the solar system or the milky way, we just need to build self-replicating robots just smart enough to do it themselves. Increased automation is one of the strongest trends in the production of everything, and one of the most important factors of productivity growth.

So, why not?

Dyson spheres are hard to build.

And whatever you want to call a galaxy-covering sphere, I do not think we could make one in the next 10,000 years or so.

Its still called a Dyson spheres, its just /lots/ of Dyson spheres around lots of stars.

Well, we don't have to build them. We just have to build the robots that will build the robots that will build them!

OT but the dialogue author Tom Murphy has a great lecture on the UCTV website about finding a lost Soviet reflector on the moon and other cool stuff.

Whew! He had me worried for a while there. Now I can rest much easier.

on the one hand, it's still only 4,000 years, on the other, I don't have any idea how one of us gets 2^14 and the other gets 2^40

Something's gone wrong with the reply system. This is the second time I've attempted to reply to a specific posting (to careless above, in this case) but the software has accepted it as a non-reply comment. Something broke in the last few days.

What is your browser? This happened to me using Safari on the IPad.

Space-based robots and automated, on-site space production are a wild card in this situation. A much richer civilization with that type of technology could pay for the launch of these robots, which would then mine resources to send back to earth, set up space-based solar panels in geosynchronous orbit, and so forth. As long as your civilization's population is relatively stable, that allows you to continue to enhance your energy consumption to whatever feels comfortable.

Did you miss the whole thing about the entire solar output not being enough for that growth in the next few thousand years?

Did you miss the part where I wrote "to whatever feels comfortable" and "as long as your civilization's population remains stable"? We're not going to get that type of exponential growth of energy use under those conditions, particularly not once technological advancement plateaus.

So if you assume no exponential growth, there will be no exponential growth. And it works out especially well if you stop at a level you assume will be easy to provide nearly indefinitely.

Beaming power to earth does not solve the problem of waste heat.

I think this economist has it wrong when he says the price of energy will go down compared to incomes. The key concept here is “economic rents” which are payments in excess of marginal cost that accrue to the owners of property that is valuable, scarce, and in demand. Rents can accrue to patent holders, land owners, members of size-restricted professional associations like doctors, etc. (Talk to an economist about rents and David Recardo to learn more).

What I think will happen once we reach the limit of how much energy we can make use of is this: the owners of the means of this energy’s production will start extracting economic rents from the rest of us. The energy lords will be the people who own the rights to the last of the oil wells and uranium mines, permits to emit carbon or heat (or whatever limits we have in place), landowners in solar-panel-friendly and windy locations, and the like. The price they can charge for the use of their property (the ability or rights to extract, the ability or right erect a wind farm, etc.) will go up and up and up as society gets richer since no one can compete with them. As we run out of efficiency options and substitutes for energy (as you described), the rents will go higher and higher.

We may reach a steady-state economy size if we come to the point where any increase in production requires an increase in rents paid to the energy-production property of equal amount. At that point we’re done making the economy bigger. We are back to the economy that Thomas Malthus described with respect to land and food production (before we solved that problem for a few centuries with technology).

This can only end in serfdom (where the energy production property owners are the lords) or socialism (where governments seize or buy this property from the lords and distribute the rents to the people). I predict socialist revolutions.

You are thinking about current energy production. Thorium based fission is very likely in the near term, and no one is getting a cartel on thorium. Hydrogen based fusion is likely within 200 years, and I'm pretty sure you would agree no one will get rents from hydrogen.

The article is about when we run into the physical limit for how much energy we can produce and how efficient all of our equipment is. Presumably this would include having run out of thorium or something. We're talking about hundreds of years in the future when we're using 1000x more energy as a world economy than we are now. So I think in THAT world "unlimited thorium" is not a good assumption.


In the mid-1800s, U.S. citizens who had the money to travel did so by using trains, but most people simply stayed home. Electricity was not available. Lighting was by tallow candles or whale-oil lamps. Homes were heated by wood, primarily.
Let us ignore the energy expended by horses, mules, and oxen, and ignore the wind energy that drove the ships at sea (often whaling vessels), and ignore the energy from farm windmills and small hydro-powered grist mills and factories. Then the annual per-capita energy used by U.S. citizens of the era amounted to 110 gigajoules. On a year-round average, this amounts to 3400 joules per second, or 3.4 kilowatts.

[Note: the watt is a measure of power --- energy divided by time --- whether that power be electrical or not.]

In the late 1990s, the picture is much different. We scurry around in automobiles, reheat our coffee in microwave ovens, live in centrally heated homes, and have air-conditioned homes, offices, and cars. We fly around the country in jet planes, sitting next to passengers using lap-top computers. We send merchandise all around the country in trucks. Our present per-capita consumption of energy is 360 gigajoules per year, amounting to an around-the-year average consumption of 11,400 joules per second, of 11.4 kilowatts.

What's that? We only use 3.3 times as much energy per capita per year as our forbears of the 1800s?

That is correct.

[Citations desperately needed]

I'm not saying indefinite exponential growth will happen, but the energy angle is bunk. An advanced civilizations access to energy is not in anyway confined to the light emissions from local stars. Conveniently stars have this lovely way of storing this energy, by creating heavy elements. Even today we can fission heavy elements like Uranium and Plutonium, as well as breed more from non-fissile Thorium and Uranium, to generate energy. Surely in the span of 400 years we'll have mastered the efficient fusion of light elements to generate energy. Who knows what other sources of energy could be discovered in the meantime?

It's not very healthy hanging around impending supernovas, waiting for fissionables to fly at you.

But as I pointed out below, even with complete annihilation of matter for energy production, we're talking about turning entire solar systems into waste heat and neutrinos every year 15000 years down the line with the growth rate he's talking about.

Well, a few points:

One, the Sun is actually a very poor source of energy -- it has roughly the mass-equivalent output of a compost heap (that's because it does p-p fusion; it's very hot mainly because it has excellent confinement). So "using energy at a rate corresponding to the total solar input striking Earth" or even the entire output of the Sun is not as absurd as it sounds on its face.

Two, the energy usage trend is probably going to level off because efficiency is increasing, population increase is leveling off, and there are limits to how much energy people really want to use.

And three, it's useful to ask what we thought was possible 100 or 2500 years ago and how that compares to today. There is a lot of physics we still don't have a firm grasp on.

As to one, 5 billion pounds of matter to energy per second ain't bad. And we're talking about an Earth mass into energy every 15 minutes in 2500 years by his math (looks more like 5 times that by my estimate)

"And we’re talking about an Earth mass into energy every 15 minutes in 2500 years by his math"

Which is a very good sign that his math is bad. Or more accurately his assumptions are bad. This is similar to the classic Malthusian argument and almost certainly suffers from the same type of flaw.

Ah here we go:

Since fusing hydrogen into helium releases around 0.7% of the fused mass as energy,[48] the Sun releases energy at the mass-energy conversion rate of 4.26 million metric tons per second

So that is actually about right.

But again, this is roughly the same energy release as a Sun-sized compost heap. Almost every terrestrial source of power has a much greater power density than the Sun.

I think you're confusing the mass of the reactants with the mass of of the matter converted to energy -- he Sun fuses 620 million metric tons of hydrogen each second, but the vast majority of that mass is still matter at the end of the p-p cycle.

Ah here we go:

Since fusing hydrogen into helium releases around 0.7% of the fused mass as energy,[48] the Sun releases energy at the mass-energy conversion rate of 4.26 million metric tons per second

So that is actually about right.

But again, this is roughly the same energy release as a Sun-sized compost heap. Almost every terrestrial source of power has a much greater power density than the Sun.

This is silly. There is no reason we can't be using that much energy on those timescales!
I point at the work of another physicist, Freeman Dyson.

We don't even need Dyson spheres, we just need to master fusion. It seems hard right now but I can't imagine us not having fusion power in 1000 years.

Assuming we haven't moved onto to something better than fusion power by then. 1,000 years is a very long time at our current rate of technological progress.

It's one thing to assume that we'll move on from chemical and fission power to more efficient sources. It's another to assume that we'll find power sources vastly more efficient than the most efficient vaguely theoretical sources we have at the moment.

If you're going to assume we can come up with energy from the vacuum, you might as well assume we can come up with habitable land, water, fertilizer, air, etc at will then. That's just defining scarcity away.

"you might as well assume we can come up with habitable land, water, fertilizer, air, etc at will then."

Thats even easier than coming up with energy from the vacuum. Space is big, and has a lot of stuff.

Good until act 3. After that there seems to be some confusion on real and nominal variables, and the importance of relative prices.

This is why we don't let people judge their own half-remembered debates.

I am sceptical of a claim that arguments fall if they rely on transformational economic growth. How much economic growth has actually been marginal as opposed to transformational? Books were big. So were steam engines.

The Administration called and reminded me that this is exactly why we need CAFE standards. Consider yourselves chastised.

"The economists who devise a functioning steady-state economic system stand to be remembered for a longer eternity than the growth dudes."

Too bad that the evenly rotating economy is apparently a confused concept:

This is the bloody Laffer Curve of "eco-physics" if you want to call it that. Of course economic growth rates can't be maintained forever, any more than Moore's Law can hold forever, or revenue will always rise with rising tax rates. The useful question (as opposed to just wise physics guy dazzling mere mortals with Science! and its wonders) is roughly how long we can expect this rise to continue. Indeed, if you look at his site, he's obsessed with reducing energy use to small fractions of today's consumption. THAT's his agenda here, and all this stuff about thousands of years from now is tinsel and bullshit.

If history is any indicator, the rate of increase will increase. (What Robin Hanson, another physicist describes as a singularity)

Yeah, I agree. This whole argument is bullshit. Who cares what happens 100 years from now? Unless it involves destroying the ecosystem, it doesn't matter (which is why global warming is the one thing that does matter). If all you care about is the result of exponential growth over indefinitely long time frames, no fixed reduction today is going to do a damn thing. But projecting growth over a centuries long time frame is useless. How can that possibly inform policy or decision making today?

Because of our ability to use language, human beings have recursion at the heart of how they function. This allows us to have a literally infinite potential and ability to expand our knowledge. This has been the main source of economic growth and why we can predict it increasing exponentially as we learn new things and this unlimited growth of knowledge fuels other improvements in our lot.

We're the only animals that we know of that have this ability. The only question then is if there is an inherent barrier in nature due to entropy that can cancel off this recursion. If there is a way around entropy its likely we'll find it as long as we survive long enough. So the only questions would then really be whether we'll survive long enough, after we reach the initial barrier of energy production, and if there is indeed a solution to be found.

"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"

This sentence is completely full of fail. By "trying to buy all of it" he raises the price so its no longer arbitrarily cheap. Energy has a /very/ inelastic demand curve (the reason it does is because so much depends on it), which means that its extremely sensitive to changes in demand, and attempts to corner the market in energy fail comically and quickly.

One neat thing about the price system is the items that are most important for other things, also are the hardest to corner, even if their price is currently very low.

I'm surprised no one caught his mistake. I guess its because this is a macro blog, not a micro blog.

These arguments always strike me as pointless. It seems pretty obvious that there are limits to economic growth. If the goal of economics is to rearrange things to improve people's well-being; well there much be an optimum arrangement for everyone. And once we reach that (or approach it) then we have no more economic growth. I think it's unlikely we are close to that, but could we be close in 400 years? Sure.

Andy - Can you imagine your own "optimum arrangement"? And are you sure you'd feel the same once you got there? Also, assuming new people continue to get born (we don't discover immortality and stop having children), we have a continual inflow of unique new individuals with unique new preferences, and associations of preferences.

The interesting question is whether increasing satisfaction of wants continues to come in the form of marketable commodities or through some other facet of social experience.

To dan1111, yes, I'm also using Safari on the Mac. And this comment is being entered after hitting the reply button for your comment, but will likely end up at the bottom of this thread. Until about a week or so ago, the reply mechanism was working fine.

World energy growth per capita annually is 0.79%, 1971-2009 So that 2.3% energy use growth rate he assumes requires population growth of 1.5% annually for 400 years -- he is projecting a world population of 2.3 trillion.

Thus he is really saying that level of population growth is unsustainable (though he doesn't seem to know it). But does anyone think differently? So what is the point supposed to be?

At 0.79% growth after 100 years energy use merely doubles, and after 400 years still is, well,*considerably* short of all the solar energy hitting the earth.

Moreover, in advanced economies energy use *declines* per capita. E.g. in the USA, in million btus: 1979: 359; 2009: 308; change -14%. Energy use per dollar of GPD has been falling since 1949, is down 50% since the 1970s. EIA.gov. So the 0.79% will be declining in the future as economies advance. . And of course population growth is declining -- with population declining outright in many of the most advanced, highest energy-use economies (Japan, etc, even China soon) further dampening energy consumption.

The UN's middle-case population projection for 100 years from now is 9 billion (up 28% from today's 7 billion). Thus we can make a reality based projection of world energy use being less than 2.6 times today's and probably *declining*.

You do the math. It seems the exponential physicist is using exponents to refute the possibility of *very finite* economic growth.

"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."

Hurrwhuh? I can't let this escape unmentioned, even though I'm a week and a half late finding this. It makes it difficult to read the rest of it with an open mind.

Apparently somebody is going to barge in and take over the market, which even if cheap and operating on thin margins would have to be large and involve a big distribution system. And the rest of us are going to sit there passively while nefarious forces buy it up, which almost necessarily involves acquisition of a huge number of different companies at different parts of the chain (resource, production, distribution) then sit there passively and not present any competition or substitute. Plus, what are the chances that you'd be able to maintain the necessary funding without other investors (who may eventually defect the intended cartel) or with only trusted confederate investors?

Why would anybody pile up the enormous resources necessary to undertake this task, with or without outside investors, only to force the market to grind to a halt? If you wanted to destroy the world economy, an evil genius could be much more cost-efficient by releasing a terrible virus, or forcing nuclear detonations, or any other of a dozen Bond-villain plots. I don't think you're going to get a lot of banks to loan you the capital necessary to buy a thin-margin, high-cost energy production and distribution market (even in a world with no antitrust bureaucrats and no utility regulations) simply so they can destroy the world market and lose the value of their entire investment portfolios (except maybe for stores of potable water and shotgun shells).

Seems like anti-market superstition. It's pretty silly to think that it's so easy to simply take over a very large market without any notable bidding competition and then operate the business without any competitors or substitutes. It would take government assistance to attempt that sort of thing, and then government assistance to stamp out any black market (if monopolistic prices for energy rose too high, you would see a black market in generators or batteries, or in straightforward siphoning off the grid).


Please see http://markbahner.typepad.com/random_thoughts/2012/05/exponential-engineer-meets-finite-physicist.html .

Comments welcome. (Non-snarky preferred, but snarky never censored on "Random Thoughts.")

"Gosh guys, do follow the link and read the article. It’s not that long and it answers these objections, which is half the point of the article."

The article does not address the fact that GDP is the total value of all goods and services traded in an economy in a year, and that "value" is not physical.

In fact, the way Tom Murphy addresses this problem with his assessment--i.e., that "value" is not physical--is: 1) to essentially decide that *he* should be the one who assigns values to things (e.g., that a Ho-Ho is just as valuable as an expensive dessert), and 2) to say that even if welfare is increasing, GDP isn't increasing...it's just "development."

Why is growth synonymous with energy usage?

"It’s not synonymous, but at least so far the correlation is very strong..."

What is the correlation coefficient between per-capita energy use and per-capita GDP in the U.S. from 1949 to 2010?

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