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.

## Murphy on Growth Limits

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.

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.)