I’ve never heard of the Bekenstein bound; thanks for this sharing this additional way to estimate the limits of economic efficiency.
An upper bound for information density is given by https://en.wikipedia.org/wiki/Bekenstein_bound and it is exceedingly large, so large that there isn’t a fundamental limit on the time frames considered here.
This doesn’t seem right. Specifically it seems like the Bekenstein bound might be larger than the limit Holden discusses in his post, but not so large as to not be reachable with Business As Usual exponential growth in a short timeframe.
Can we quantify what the Bekenstein bound actually is here to check? Here’s my attempt:
Using the Earth as the system and the equation from Wikipedia, it looks like the Bekenstein bound is 9.82*10^74 bits. (Let me know if this is incorrect.) There are 10^50 atoms in Earth, so that’s 10^25 bits/atom.
For our galaxy, it looks like the number is 8*10^36, assuming again that I set up the math right. Holden’s claim is that the limit to economic efficiency is likely less than the efficiency of an economy that is a factor of 10^70 times larger than today’s world economy and that uses less than 10^70 atoms.
Note however that (a) there’s not enough time to colonize the galaxy in 8,200 years even at the speed of light and (b) economic growth during colonization of the galaxy is quadratic, not exponential, since it is limited by the speed of light expansion of civilization. So given these two considerations, I think it makes more sense to look at the Earth-system rather than the Galaxy-system.
Using the Earth as the system instead, the similar claim would be that the maximum size economy that could be sustained by the Earth would be an economy less than 10^50 times as large as today’s world economy.
At 2% annual economic growth, it would take 5,813 years for the economy to grow by a factor of 10^50.
If we make the very conservative claim that today’s economy only uses 1 bit of information [1], then to grow the economy by a factor of 10^75 would presumably mean that the resulting economy would have at least 10^75 bits (I think this is a reasonable assumption; let me know if it’s not), i.e. the Bekenstein bound for the Earth.
At 2% annual growth, it would take 8,720 years to grow the economy by this factor, i.e. definitely still in the range of time frames considered in this post.
In reality, our current world economy uses more than 1 bit, and the Bekenstein bound would presumably thus be reached before the economy is able to grow by a factor of 10^75. E.g. Maybe the actual maximum factor according to the Bekenstein bound might be closer (on a log scale) to 10^50 than 10^75 (?). I have no idea and would be interested in hearing from anyone who thinks they do have a way to estimate this.
The calculations for total bits in the systems is correct: 9.82*10^74 bits (earth) and 4×10^106 bits (galaxy). The bit limit grows quadratically as you expand out the radius and energy-mass content.
12,395 years of 2% growth to achieve the the upper bound of the galaxy’s information content given perfect mass-energy efficiency.
Stepping back, this exercise in extrapolating 2% growth to extremes can be reduced to just the mathematical statement that any exponential growth will exceed a (constant or sub-exponentially growing) limit in finite time. Yes… QED.
Solutions like expanding this future-humanity’s Boltzmann-economy at a 2% radius per year get you to faster-expansion than the speed of light quickly. Quadratic (or similar orders) is probably the best one can do in the long run.
TLDR: in bits, exponential growth at a fixed growth rate forever of an economy is impossible (ad absurdum), but something like quadratic growth forever is not impossible.
An important point that I don’t think we’ve said yet is that information density is of course not the same as economic productivity.
What would the Gross Galactic Product be of a maximally-efficient galaxy economy that had reached the 4×10^106 bit information density limit? It would necessarily be close to $10^106 or close to 10^106 times greater than the size of today’s GWP, right?
Similarly, if annual GWP increases at 2%/year, that does not necessarily mean that the economy’s information density (or perhaps more accurately, the information density of the system the economy is enclosed in) is increasing at close to 2%/year, does it?
I avoided the ‘productivity’ or ‘economic value’ to focus on something physically tangible. Markets put an objective value on those, but there’s no physical laws to help here.
Generally you’d expect the marginal value of information to fall as more information is created. Aristotle’s works vs another youtube video or terrabytes of system logs. The information-density-value-efficiency gets lower as you get bigger. Our own hard drive’s content is a good additional example: probably <5% of the contents are high value, contrast to when we all had much smaller storage (e.g. 2.5 inch floppy drives).
That said, this is analogous to diminishing marginal value (or returns) to scale in economic activity.
Efficiency of any economic system with respect to fundamental resource usage (information, energy) probably is almost certainly declining in scale. Friction adds up.
I’ve never heard of the Bekenstein bound; thanks for this sharing this additional way to estimate the limits of economic efficiency.
This doesn’t seem right. Specifically it seems like the Bekenstein bound might be larger than the limit Holden discusses in his post, but not so large as to not be reachable with Business As Usual exponential growth in a short timeframe.
Can we quantify what the Bekenstein bound actually is here to check? Here’s my attempt:
Using the Earth as the system and the equation from Wikipedia, it looks like the Bekenstein bound is 9.82*10^74 bits. (Let me know if this is incorrect.) There are 10^50 atoms in Earth, so that’s 10^25 bits/atom.
For our galaxy, it looks like the number is 8*10^36, assuming again that I set up the math right. Holden’s claim is that the limit to economic efficiency is likely less than the efficiency of an economy that is a factor of 10^70 times larger than today’s world economy and that uses less than 10^70 atoms.
Note however that (a) there’s not enough time to colonize the galaxy in 8,200 years even at the speed of light and (b) economic growth during colonization of the galaxy is quadratic, not exponential, since it is limited by the speed of light expansion of civilization. So given these two considerations, I think it makes more sense to look at the Earth-system rather than the Galaxy-system.
Using the Earth as the system instead, the similar claim would be that the maximum size economy that could be sustained by the Earth would be an economy less than 10^50 times as large as today’s world economy.
At 2% annual economic growth, it would take 5,813 years for the economy to grow by a factor of 10^50.
If we make the very conservative claim that today’s economy only uses 1 bit of information [1], then to grow the economy by a factor of 10^75 would presumably mean that the resulting economy would have at least 10^75 bits (I think this is a reasonable assumption; let me know if it’s not), i.e. the Bekenstein bound for the Earth.
At 2% annual growth, it would take 8,720 years to grow the economy by this factor, i.e. definitely still in the range of time frames considered in this post.
In reality, our current world economy uses more than 1 bit, and the Bekenstein bound would presumably thus be reached before the economy is able to grow by a factor of 10^75. E.g. Maybe the actual maximum factor according to the Bekenstein bound might be closer (on a log scale) to 10^50 than 10^75 (?). I have no idea and would be interested in hearing from anyone who thinks they do have a way to estimate this.
The calculations for total bits in the systems is correct: 9.82*10^74 bits (earth) and 4×10^106 bits (galaxy). The bit limit grows quadratically as you expand out the radius and energy-mass content.
12,395 years of 2% growth to achieve the the upper bound of the galaxy’s information content given perfect mass-energy efficiency.
Stepping back, this exercise in extrapolating 2% growth to extremes can be reduced to just the mathematical statement that any exponential growth will exceed a (constant or sub-exponentially growing) limit in finite time. Yes… QED.
Solutions like expanding this future-humanity’s Boltzmann-economy at a 2% radius per year get you to faster-expansion than the speed of light quickly. Quadratic (or similar orders) is probably the best one can do in the long run.
TLDR: in bits, exponential growth at a fixed growth rate forever of an economy is impossible (ad absurdum), but something like quadratic growth forever is not impossible.
An important point that I don’t think we’ve said yet is that information density is of course not the same as economic productivity.
What would the Gross Galactic Product be of a maximally-efficient galaxy economy that had reached the 4×10^106 bit information density limit? It would necessarily be close to $10^106 or close to 10^106 times greater than the size of today’s GWP, right?
Similarly, if annual GWP increases at 2%/year, that does not necessarily mean that the economy’s information density (or perhaps more accurately, the information density of the system the economy is enclosed in) is increasing at close to 2%/year, does it?
I avoided the ‘productivity’ or ‘economic value’ to focus on something physically tangible. Markets put an objective value on those, but there’s no physical laws to help here.
Generally you’d expect the marginal value of information to fall as more information is created. Aristotle’s works vs another youtube video or terrabytes of system logs. The information-density-value-efficiency gets lower as you get bigger. Our own hard drive’s content is a good additional example: probably <5% of the contents are high value, contrast to when we all had much smaller storage (e.g. 2.5 inch floppy drives).
That said, this is analogous to diminishing marginal value (or returns) to scale in economic activity.
Efficiency of any economic system with respect to fundamental resource usage (information, energy) probably is almost certainly declining in scale. Friction adds up.