How Large Can Growth Rates Really Get?
30 Dec 2023Among the general public there is demand that the interest rate/investment return should be positive and be somewhere around 5%, perhaps more or perhaps less. The expectation is that compound interest/reinvested returns will never stop. On a finite planet, endless exponential growth will lead to the overexhaustion of the biosphere followed by collapse, but we can still entertain the idea of interstellar travel, right? Space is so large. Surely we can fit a 5% growth rate in there somehow or at least 3%, right? Unfortunately it isn’t that easy. Exponential growth not only hits the limits of our planet startingly quickly, any extra terrestial growth is bounded by the speed of light. Maintaining a specific exponential growth rate may be possible over a handful of centuries, but over thousands of years, even the speed of light will pose a significant barrier to high growth rates. To disprove the idea of a large growth rate r, which is necessary to justify an interest/investment return rate of the same value, it is already sufficient to prove that r must be below some finite limit for a finite duration. For that reason I have chosen 2000 years or two millenia. The reason is quite boring. It is the year 2023 so I have simply chosen the round number 2000. The last two thousand years clearly have proven that we can have exponential growth, so why shouldn’t we have another two thousand years of exponential growth? The history of human civilization is much longer than that. The probability of humanity going extinct should be low. At least much lower than if I picked a time frame of a million years. So yes, I am making an assumption here. I am assuming that humanity will not go extinct within the next two thousand years.
How much growth would there be, at 5% exponential growth, over the next 2000 years?
Surprisingly (or maybe unsurprisingly?), a lot. So much that you are unable to comprehend it in absolute terms. I could tell you the numbers and you wouldn’t even be able to process them. You might read them and think “yeah, sounds about right” and basically show off your ignorance about the magnitude of the numbers involved.
The current world GDP as of 2021 is 96.51 trillion USD [3] or $96,510,000,000,000.
| Year | GDP | Multiplier |
|---|---|---|
| 2021 | 96,510,000,000,000 | 1 |
| 2071 | 1,106,004,600,000,000 | 11.46 |
| 2121 | 12,691,065,000,000,000 | 131.50 |
| 2221 | 1,668,906,895,800,000,000 | 17,292.58 |
| 2421 | 28,859,708,728,172,400,000,000 | 299,033,351.24 |
| 2821 | 8,630,015,417,305,864,776,916,500,000,000 | 89,420,945,159,111,644.15 |
| 3621 | 771,704,135,353,195,724,523,069,472,891,550,658,945,600,000,000 | 7,996,105,433,148,852,186,540,974,747,606,990.56 |
| 4021 | 230,765,273,767,253,930,151,559,627,085,198,497,545,700,268,256,000,000,000 | 2,391,102,204,613,552,275,946,115,709,099,559,605,695,785.60 |
I would honestly quit at this point. The numbers simply don’t add up. There is no way this is going to happen in 2000 years. 1 billion years? Maybe, but certainly not 2000! As a bonus: How large would the universal GDP be in the year 10000? (7979 years of 5% growth).
The multiplier is 11,733,221,761,155,213,243,003,859,229,331,861,858,641,289,448,176,397,180,092,538,404,309,116,320,403,053,010,109,065,462,059,741,617,924,210,521,327,054,266,990,170,156,615,523,454,599,379,162,663,369,664,600,749,936,410,771.54.
The GDP would be 1,132,373,232,169,089,630,082,302,454,222,817,987,977,470,844,643,504,091,850,730,881,399,872,816,082,098,646,005,625,907,743,385,663,545,865,557,413,274,007,307,221,321,814,964,168,603,386,082,988,641,806,330,618,376,363,003,561,488,815,416,111.8666258315.
Don’t worry. These numbers don’t make any sense to me either, yet they are what people subconsciously want, when they say that exponential growth should go on forever.
How much growth do we have left on earth?
If we assume a $1 trillion GDP per capita in the year 4021 (yes, that means the average person will earn more in a year than the top 5 wealthiest people own as of 2023), then this means earth would need a population of 230,765,273,767,253,930,151,559,627,085,198,497,545,700,268 humans. The volume of earth’s atmosphere is 4,200,000,000 cubic kilometers [1]. This means every human would get a measily 0.000000000000000000000000000000000018 cubic kilometer of space. Ok, but what about people who don’t want to count zeroes? It is around 18 cubic nanometers. Enjoy fitting inside of that space!
But surely we can just build up, right?
What exactly prevents us from building space elevator style skyscrapers? The sky wouldn’t be the limit anymore. The biggest limitation would be the lack of a breathable atmosphere and a shortage of building materials. If we give every human 50m^2 of living space and pave the entire planet with housing, including the oceans, we will have 510,072,000 km^2 [0] at our disposal. 10,200,000,000,000 people (10.2 trillion people) can live on the first floor! Sounds good right? Except that there wouldn’t even be enough space for the elevator to leave the apartment, let alone produce food, but lets not sweat the details!
How many floors of this wonderful structure would we need? You’re in luck! The answer is 22,624,046,447,769,993,152,113,688,929,921. Assuming a ceiling that is 4m tall, the structure would be 90,496,185,791,079,972,608,454,755,719,684m tall. The sun is 150 million kilometers away [4]. This means our structure would frequently collide with the sun! or pluto…. Actually. We would be shooting way past anything you could imagine. 90 * 10^27km or better yet 105.7 trillion light years. Of course, you may now object, that building such a structure on earth is stupid to begin with. After all, it is basically an extremely long rope or tendrill originating from earth. This would result in some space elevator spaghetti and the project would have to be canceled during construction. Clearly, the answer is to have a solid sphere with no natural sunlight, it’s not like being 105.7 trillion light years into the boonies there was any sunlight to begin with (the universe is only a few billion years old after all). Given the previous numbers we assume a volume of 200m^2 per human. Our sphere must have a volume of at least 46,153,054,753,450,786,030,311,925,417,039,699,509,140,053,600 m^3 ignoring walls. The sphere would have a radius of 2,225,000,000,000km (that is only 0.2352 light years!). It would take light only 85 days to travel from the center to the outer layer! We can build this structure in 2000 years, right? How about no, you idiot!
It’s okay, we can go to space! I’ve always wanted humanity to become an interstellar species anyway!
Is what you might use as an objection, but the problem is that space just isn’t dense enough. Yes, space is big, but we don’t want empty space. We want some nice crusty planets to colonize. 230,765,273,767,253,930,151,559,627,085,198,497,545,700,268 people and if we assume 8 trillion people per planet (I don’t want people to accuse me of limiting the population of a planet to earth style planets). Then we would need 28,845,659,220,906,741,268,944,953,385,649 planets!
Assuming 400 billion stars per galaxy [5] and two planets per star (this is optimistic), then all we have to do is colonize just 36,057,074,026,133,426,586 galaxies! I don’t know about you, but for me that is a really, really big number. Our goal is not only to develop interstellar travel within two thousand years, but also to develop inter galatic travel and the timeline does not get better, if we extend it to 4000 or more years. Given 2 trillion galaxies in the observable universe, our problem will be, uh, that we have run out of galaxies, in fact, we would have to colonize 18,028,537 observable universes within the next 2000 years. Millions. Let me repeat. MILLIONS of observable universes. We only have ONE!
The speed of light limits the reachable subset of the universe to a sphere
How many stars are within 50, 100, 500, 1000 and 2000 light years? [6]
The volume formula for a sphere is V = 4/3 π r³.
Assuming we launch all our space ships in the year 2021, they will colonize a volume according to this table:
| Year / Light years | Stars | Volume |
|---|---|---|
| 2071 (50) | - | 523,599(ly)^3 |
| 2121 (100) | 59722 | 4,188,790(ly)^3 |
| 2221 (200) | 118443 | 33,510,322(ly)^3 |
| 2421 (400) | 185868 | 268,082,573(ly)^3 |
| 2821 (800) | 253114 | 2,144,660,585(ly)^3 |
| 3621 (1600) | 303418 | 17,157,284,679(ly)^3 |
| 4021 (2000) | 316637 | 33,510,321,638(ly)^3 |
Looking at these numbers, you might say that the speed of light is not a big deal. After all, it gives you a massive headstart compared to exponential growth in the first years. The GDP multiplier is 11.46 for 2071 and 131.50 for 2121. Exponential growth ain’t got nothing on cubic growth, right? Beyond the year 400, exponential growth will rapidly catch up. In fact, comparisons like these kind of make you question the concept of exponential growth as being efficient. Let’s set aside the fact that space is mostly empty. If cubic growth indeed is faster, why aren’t we on the cubic growth bandwagon then? Why bother with slow exponential growth? If exponential growth is unsustainable beyond cubic growth, then we are basically wasting our time for nothing.
The speed of light is irrelevant, we can use time dilation to travel long distances in two thousand years!
The argument is that from the observer’s frame of reference, time passes much more slowly and therefore actual time spent traveling is much shorter than the distance itself would imply. However, this ignores that every person represents a unique frame of reference. Humanity collectively cannot use time dilation. People stuck on earth do not experience time dilation. In fact, the only way to make this work is by cloning a single person, sending them to every planet in the observable universe by the trillions and then pretending that every clone is the same person. No communication between the clones is necessary and they all count towards the utility function of a single person. “We have successfully beat the speed of light!” is what an insane person would say.
Millions of observable universes? Really?
And this is the part where you are going to call me stupid, because the premise is obviously ridiculous and oh and actually, the interest rate/investment return also contains a risk premium, oh and there is also an inflation adjustment component and don’t forget about the fact that the entire money system could be phased out and replaced by a different currency, the economy could collapse due to political instability, war or hyperinflation and then your money would be worthless. There is no way, anyone could earn 2000 years of uninterrupted compound interest/investment returns with a 100% guarantee, that just isn’t possible! Your concerns about not finding a second planet to colonize will turn to dust and fade away, because we won’t have a currency that lasts more than 200 years or even 100 years. And you would be right, but that doesn’t mean I would be wrong. The purpose of this blog post is to show you what would happen if the economy existed in a vacuum and we blindly wanted to satisfy this endless exponential growth fantasy. The answer is that we would shockingly quickly reach the end. There would be nothing to do whatsoever, within less than 2000 years. Growth would no longer be exponential and 2000 years is just an arbitrary number that just happens to coincide with the birth date of jesus. That’s how the calender is set up. It is the 21th century after all. Two millenia is a nice round number. Do you want me to pick uglier numbers? Like 859?
The end of exponential growth or the beginning of sub exponential growth?
The problem with endless exponential growth is quite simple. It is so simple it only takes a single sentence to explain. The derivative of the exponential function is the exponential function… Yes, that is all I have to say to show you how ridiculous the idea of endless exponential growth is. What this means is that if you were to look at absolute growth only, the rate of this absolute growth also rises exponentially. The rate of change itself is growing exponentially! So in the beginning, it will look like everything is fine, but in the end, everything will be not so fine because the rate of change itself will exceed the capacity of our planet. We will have to keep adding earth sized planets every year in the near future (next 200 years). No, this isn’t colonizing Mars in 200 years, it is colonizing a planet - every - single - year.
If we bound the growth rate to colonizing the entire observable universe within 2000 years, what growth rate would we get? Let’s recap. One observable universe, two trillion galaxies, 400 billion stars per galaxy, two habitable planets per star, eight trillion people per habitable planet and one trillion dollars GDP per capita giving us a universal GDP of $12,800,000,000,000,000,000,000,000,000,000,000,000,000,000,000, which is a growth multiplier of 132,628,743,135,426,380,685,939,280,903,533.312. That’s a lot. The exponential growth rate is going to remain very large even if growth itself stops after two thousand years!
| Years | Universe | Galaxy | Speed of light Sphere |
|---|---|---|---|
| 50 | 338% | 149.1% | - |
| 100 | 109.5% | 57.84% | 44.5% |
| 200 | 44.7% | 25.63% | 20.6% |
| 400 | 20.3% | 12.08% | 9.9% |
| 800 | 9.68% | 5.87% | 4.9% |
| 1600 | 4.73% | 2.8% | 2.4% |
| 2000 | 3.77% | 2.3% | 1.94% |
| 3200 | 2.33% | 1.41% | - |
| 6400 | 1.16% | 0.71% | - |
| 12800 | 0.57% | 0.35% | - |
Ouch. If we want to colonize the entire universe with a GDP per capita of a trillion dollars, our upper bound for the growth rates is 3.77%. If we can only colonize the local galaxy, then the upper bound falls down to 2.3%. Meanwhile the maximum growth rate bounded by the reachable sphere limited by the speed of light is 1.94% and all of the growth rates have a tendency to go to 0% over 12800 years. Please remember that these numbers represent an upper bound that assumes some breakthrough technology that grows the GDP per capita by a factor of 81739414 and the population is a thousand times higher than on earth as of now. 1.94% year over year growth over the next two thousand years will require faster than light travel. These are not predictions of how fast the economy will grow, they are predictions on the fastest possible growth rate over a long time period. It is possible to grow faster in the early years and then not grow at all for the rest. Just as a reminder: The population has to grow to 8 trillion humans on earth. We are not on such a growth path. In fact, we are struggling to keep the populations stable as of now. The expectation of 1.94% growth will therefore be disappointed. There will be less growth.
Ok, but less than exponential growth is good right? So what, let’s just have linear growth then, after we exhausted our exponential growth potential. Linear growth means that if the economy grows by 1 trillion this year, it will grow by 1 trillion next year. Surely this can be sustainable? It can be! The problem is that linear growth prohibits compound interest/returns. In other words, all interest/returns payments must be consumed eventually. However, this also means borrowers can always refuse additional debt and force lenders/investors to consume their interest payments and investment returns. Does that sound remotely like the real world to you? Imagine you lend money to a borrower or an investor to a company and he can just go and say “I won’t pay interest/dividends unless I know you spend them on consumption and you have no choice but to obey!”. Ridiculos, right?
Conclusion
Exponential growth over short periods of time such as two millenia, rapidly result in illogical numbers. Any planetary limit is guaranteed to be broken an unimaginable times over in two thousand years if we don’t leave the planet and if we do leave the planet, then our problem is that the speed of light is too slow and the universe too small, even if speed of light travel was invented today. Assuming we have enough spaceships on earth to colonize the entire universe, the resulting growth rates over the next two thousand years are bounded to single digit percentages below 2%. Growth after ten thousand years is predicted to be non-existent. Once the lack of population growth in the real world is taken into account, the exponential growth rate is expected to be closer to 0% than to 5% or even 3%. Numbers of this magnitude are often demanded by neoclassical or Austrian economists, who insist that growth should simply continue forever to validate their economic theory.
References
[0] https://en.wikipedia.org/wiki/Earth
[3] https://de.wikipedia.org/wiki/Liste_der_L%C3%A4nder_nach_Bruttoinlandsprodukt (I know the number isn’t perfectly accurate but you know what? Calculating these large numbers is painful)
[4] https://solarsystem.nasa.gov/news/1164/how-big-is-the-solar-system/
[5] https://en.wikipedia.org/wiki/Milky_Way
[6] https://lovethenightsky.com/stars-within-100-light-years/