Murphy on Growth Limits

Physicist Tom Murphy says he argued with “an established economics professor from a prestigious institution,” on whether economic growth can continue forever. They both agreed to assume Earth-bound economies, and quickly also agreed that total energy usage must reach an upper bound within centuries, because of Earth’s limited ability to discard waste heat via radiation.

Murphy then argued that economic growth cannot grow exponentially if any small but necessary part of the economy is fails to grow, or if any small fraction of people fail to give value to the things that do grow:

Not everyone will want to live this virtual existence. … Many would prefer the smell of real flowers. … You might be able to simulate all these things, but not everyone will want to live an artificial life. And as long as there are any holdouts, the plan of squeezing energy requirements to some arbitrarily low level fails. …

Energy today is roughly 10% of GDP. Let’s say we cap the physical amount available each year at some level, but allow GDP to keep growing. … Then in order to have real GDP growth on top of flat energy, the fractional cost of energy goes down relative to the GDP as a whole. … But if energy became arbitrarily cheap, someone could buy all of it. … There will be a floor to how low energy prices can go as a fraction of GDP. … So once our fixed annual energy costs 1% of GDP, the 99% remaining will find itself stuck. If it tries to grow, energy prices must grow in proportion and we have monetary inflation, but no real growth. …

Chefs will continue to innovate. Imagine a preparation/presentation 400 years from now that would blow your mind. … No more energy, no more ingredients, yet of increased value to society. … [But] Keith plopped the tuna onto the bread in an inverted container-shaped lump, then put the other piece of bread on top without first spreading the tuna. … I asked if he intended to spread the tuna before eating it. He looked at me quizzically, and said—memorably, “It all goes in the same place.” My point is that the stunning presentation of desserts will not have universal value to society. It all goes in the same place, after all. (more; HT Amara Graps)

While I agree with Murphy’s conclusion that the utility an average human-like mind gains from their life cannot increase exponentially forever, Murphy’s arguments for that conclusion are wrong. In particular, if only a fixed non-zero fraction of such minds could increase their utility exponentially, the average utility would also increase exponentially.

Also, the standard power law (Cobb-Douglas) functional form for how utility depends on several inputs says that utility can grow without bound when one sector of the economy grows without bound, even when another needed sector does not grow at all and takes a fixed fraction of income. For example, if utility U is given by U = Ea N1-a, where E is energy and N is non-energy, then at competitive prices the fraction of income going to the energy sector is fixed at a, no matter how big N gets. So N can grow without bound, making U grow without bound, while E is fixed.

My skepticism on exponential growth is instead based on an expectation of strongly diminishing returns to everything, including improved designs:

Imagine that … over the last million years they’ve also been searching the space of enjoyable virtual reality designs. From the very beginning they had designs offering people vast galaxies of fascinating exotic places to visit, and vast numbers of subjects to command. (Of course most of that wasn’t computed in much detail until the person interacted with related things.) For a million years they have searched for possible story lines to create engaging and satisfying experiences in such vast places, without requiring more computational resources behind the scenes to manage.

Now in this context, imagine what it means for “imagination” to improve by 4% per year. That is a factor of a billion every 529 years. If we are talking about utility gains, this means that you’d be indifferent between keeping a current virtual reality design, or taking a one in a two billion chance to get a virtual reality design from 529 years later. If you lose this gamble, you have to take a half-utility design, which gives you only half of the utility of the design you started with. …

It may be possible to create creatures who have such strong preferences for subtle differences, differences that can only be found after a million or trillion years of a vast galactic or larger civilization searching the space of possible designs. But humans do not seem remotely like such creatures. (more)

Neither mass, nor energy usage, nor population, nor utility per person for fixed mass and energy can grow exponentially forever.

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  • I read that article a couple of days ago and thought it would be interesting to see a Blogging Heads episode between you two discussing this idea. What about it?

  • Vaniver

    Robin: one of Murphy’s points was that, when energy is constrained, he expects the share of GDP that energy represents to fall. The economist agreed with this, and went so far as to claim that it would asymptotically approach zero (at least, until Murphy asked him directly, he didn’t see a reason for there to be a lower bound).

    I thought about this and came to the conclusion that a static quantity of energy in no way implies a static price of energy- and that we should expect the price of energy to grow with the rest of GDP, so the share of GDP that energy takes up is irrelevant. (Indeed, the natural currency for a Dyson Sphere civilization is energy, and things become much clearer when you think about the cost of goods or services in watts.)

    Is my reasoning right? (I agree that exponential growth is impossible in the long term for reasons you describe but I think Murphy gets it wrong.)

    • V

      I think he uses that as a reductio ad absurdum: continued GDP growth would require energy prices to fall to an arbitrarily small fraction, but since energy prices can’t fall to an arbitrarily small fraction of GDP, GDP growth can’t continue indefinitely.

  • Robert Koslover

    It seems to me that this problem, albeit in a somewhat different era and context, has been thoroughly addressed previously: See, for example, , among others.

    So growth (economic or otherwise) can’t go on forever. Yes indeed, I believe you. And not to be too argumentative, but so what? The universe itself very possibly won’t last forever either. But we seem far, far removed (to put it mildly) from having to deal with any of the implications of that, right?

  • Poelmo

    Energy is 100% of GDP, or rather, it will be in the future. It’s not difficult to understand this: right now we’re harvesting natural resources faster than they are being replenished and assigning value to that practice. Naturally this can’t go on forever, copper for example will be depleted before the year 2100 and many other resources will follow. The solution to this problem is recycling, which costs energy. In fact, given enough energy everything can be recycled and even petroleum and diamonds can be manufactured, instead of mined.

    Once we reach the age of recycling (and hopefully replicators), energy will limit the size of the economy. Of course the amount of matter in use at any given time (if everyone hoards stuff instead of bringing it in for recycling once they’re done with it then that limits how much stuff can be traded) and the state of technology will determine how efficiently we will use that energy. Energy will be the gold of the future, gold itself would be worthless (when you “buy” it you’ll only pay for the energy cost of manufacturing, recycling and transport of the gold, not for the gold itself, because given enough energy you could make/recycle the gold yourself).

    An economy’s growth would be limited by technological progress, so a capitalist economy would have to try to find a mechanism that prevents investment from outpacing technological progress (otherwise you get inflation and increased inequality, to those of you who don’t think that’s a bad thing: go live like a beggar in Calcutta for a week and then we’ll see if you still think only the AVERAGE income per capita matters).

    Attempts at finding such a mechanism, making it work in practice and sticking to it will probably fail (or if it succeeds progress would someday become very slow, at least for a period long enough to tank the capitalist economy) and either lead to some kind of 70s sci fi movie dystopia or to a new economic system that has no problems with slow or zero growth, this will either be a community based barter system (money may still exist but will be of limited use) where everyone just lives off their work and nobody ever gets rich, like in a medieval village, or a moneyless system of energy accounting.

    P.S. On the radiation aspect. Can’t we just use some energy to funnel waste heat into space more quickly than the Earth does naturally (maybe build a huge greebled black tower that extends far into space, make the topside of clouds black, etc…) and/or actively cool the Earth? I understand there are limits to this too but it could postpone trouble for a long time.

    • KPres

      “An economy’s growth would be limited by technological progress, so a capitalist economy would have to try to find a mechanism that prevents investment from outpacing technological progress”

      Explain to me how the profit mechanism fails in this regard, because on the surface, it seems absurd. If all growth comes from technological progress, and investment outpaces that growth, then you get a lot of negative returns, and consequently investment falls. Where have I gone wrong?

      • Poelmo

        There are ways in modern capitalism to keep making more money and stashing it away, giving you more and more power if other people don’t stash their money as much as you do.

        If I were to limit my investment to things like mining, construction, education, consumer goods and other stuff that represents more than just paper value then I would suffer negative returns. But that’s a naive image of capitalism: these days a big bank can have a central bank print money for them and then the big bank can use that money to bet against other big banks on Greece’s bankruptcy . If they loose they pay off the winner with the printed money (suffering a tiny loss because of inflation), if they win they get to keep the printed money (minus that tiny inflation loss), so the big banks control an ever increasing share of the money in the world and if there is no (or very slow) real economic growth that means they’ll control an ever increasing share of the goods and services in the world (either by owning them or just having the power to flood the market and buy them at any moment).

        You don’t even have to use such a scheme: you can get much the same results by speculating with real estate (drive up prices by keeping buildings empty and such).

  • The thing that struck me the most about the conversation, is an implicit switch in the expectations of the future that takes place part way through. Early on in the conversation, Murphy gets his unnamed economist to accept that they’re not going to talk about leaving the planet, and they never revisit this assumption. But after 20 or so doublings, they’re well past the point that every individual on the planet has control of enough resources and intelligence to design, build and launch a settlement of another body in the solar system. If Murphy is serious about the growth curve, and the rest of the argument he’s making, limiting things to Earth is a huge mistake.

    • Poelmo

      Colonizing planets would lead to a linear increase in resources, while Murphy and Hanson were talking about exponential growth of resource consumption. They talked about energy consumption doubling every 23 years, this means that colonizing 10 Earth-like planets would delay the problem by only 80 years, and that’s assuming humanity’s population growth won’t accelerate at all, even though there are 10 new planets for the taking.

      • Hanson is talking about exponential growth, but Murphy is very clearly talking about any growth at all. His epilogue makes it clear, he is talking about “a model in which GDP is fixed—under conditions of stable energy, stable population, steady-state economy.” Even the linear growth of space colonization is more than Murphy wants to accept.

      • V

        Even the linear growth of space colonization is more than Murphy wants to accept.

        If I understand correctly his arguments, he accepts that linear growth, while ultimately unsustainable, could continue for a very long time.

        He makes and argument that space colonisation is probably either impossible or so impractical and slow that it will not make a large difference, at least in the foreseeable future.

    • V

      We don’t get 20 resource doublings. Even if we managed to leave the Solar System, exponential energy growth at the assumed rate would require outpacing the speed of light in a few more centuries.

  • Roger

    As Robin responded to me in the comments on his original post…. ” Growth could continue for a while and then stop at a high level and then capacities could stay high forever after. Not such a bad outcome at all. But still, growth does end.”

    Tom the physicist is arguing that energy limits utility improvement. Robin is arguing that the world can become so awesome that we can no longer fathom why we should make it any better.

    I agree with Robin.

  • Steve

    Still need to read more Freeman Dyson.

  • The end state of a civilization is a Dyson sphere collecting solar energy on the inside and radiating heat at the 3 degree K black body temperature on the outside. That allows the civilization to use superconducting circuits to minimize losses, allows the total energy of their star to be utilized and maximizes the thermodynamic efficiency of energy conversion.

    Because the lifetime of a star on the main sequence is limited, and its rate of energy production increases as it ages, a civilization would want to remove mass from their star to prolong its life.

    The answer to Fermi’s Paradox might be that they are in Dyson spheres. They only radiate at the background black body temperature so they are invisible. It turns out that the oldest and smallest globular galaxies also have the highest concentrations of “dark matter”. Maybe “dark matter” is the mass enclosed by Dyson spheres that is invisible because it radiates at the background black body temperature.

    • V

      Good luck building a Dyson sphere, especially one that is so efficient that radiates at background temperature.
      I would guess that, even if you could find the materials, the energy cost of building it would probably exceed its energy output within its expected lifetime.

      The most likely answer to Fermi’s paradox is that alien civilisations, if they exist, are so sparse that detection, not to mention contact, is impossible.

  • V

    Murphy refers specifically to GDP growth, which is what policy makers and economists giving advice on public policy refer to when they are talking about economic growth.

    IIUC, his argument is that physical limits impose a maximum size on some vital sectors of the economy (food production, manufacturing, etc.) and this in turn imposes a cap on GDP (because the fraction of income spent on these vital products can’t become arbitrarily small). It seems to me that his argument is correct.

    In the end of the article, he appears to remark that this argument doesn’t imply a cap on subjective utility, even if he doesn’t use the term.

  • We will find solutions and substitutes around our major problems. Just take a look at this bet:

    Put me down as being in the camp that says “I welcome the day the sun burns out, because man kind will have created its own sun long before it.”

  • Michael Wengler

    I had such a strong initial reaction to this post and was so sure I must be right.  So I did something reasonable: I read the source post of the discussion.  

    It supported in spades my initial reaction.  I HAVE learned something over time, and I suspect that Robin and any serious reader of this blog falls on the same side of this.  

    The physicist is right.  The problem isn’t in the details, either.  It is in the difference between a fuzzy concept of infinity (or forever as they name it here) and an actual concept of forever.  

    Before reading Robin’s blog I did think that humanity’s future was rosy, that Malthus was horribly misinformed by living before modern times.  Since reading this blog, it seems more likely that we have had a really good few centuries that actually pushed productivity out in front of population pressure.  But absent some kind of biological innovation we have never yet seen the likes of, the slow but steady exponential of biological growth will eventually regain on the episodic-but-ultimately-non-exponential growth of a finite universe.  

    Infinity is non-physical, non-real.  If there is ANY example of an argument where, in a mathematical sense, inifinity is the right answer, I have not yet seen it.  You can, I think, be assured a good living if you can find money bets against infinity and without further thought always take the side against infinity.  


    My favorite finite fact in the discussion was that if energy use on earth continues growing at 3%/year, the average surface temperature of the earth is boiling point in 2500 years.  So maybe we devote certain parts of the earth to glowing at 10,000 C so the rest of earth can be kept at 23C or so, but what does that gain us, a factor of 3?  10?  1000?  It certainly doesn’t gain us a factor of infinity.  Whether 100 C average surface temperature is the limit or we are clever and get past that, the actual limit is some finite multiple of that.  

    There are only 10^70 particles in the universe.  10^71 is 10X the actual universe.  10^100 is insanely larger than the actual universe.  10^1000, 10^100^100? Or as less wrong likes to fantasize 3^^^3?  No matter how you wind up counting it, finity is GIGANTICALLY LESS than infinity no matter how big you make finity.  And Infinity is Bullshit.  (Robin, that looks like a good post title for you.)

    • the physicist is right when talking about numbers, as usual. Later in his article he concedes much ground and only allows himself to say “Will [growth] be at the 2% per year level (factor of ten better every 100 years)? I doubt that.” Which is really much more of a concession then the economist ever made during the conversation. Of course, due to his typical ivy league physicist ego, he was still absolutely stunned at how much more right he was than the economist.