What about a cubic growth model? The volume of space or amount of resources used in each time period could grow cubically with the time period as we colonize space in every direction with an expanding sphere (roughly), and we continue to occupy and generate value from the interior of the sphere in each time period. Or do we need to non-renewably exhaust a good portion of the resources of the interior, so we only approximately generate value within an area near the frontier of expansion?
Thanks! Cubic growth is an important possibility to consider.
If anyone is up for crunching some numbers, I only took a very rough stab at the numbers at this comment and I think it would be really valuable to do better. I’d be happy to add an acknowledgment in the final paper, or provide your libation of choice.
I guess I would want to say two things about cubic growth (ordered roughly in the importance they’d have in an academic discussion, although maybe in reverse order of importance for present purposes):
1. Cubic growth is a strong assumption: Most academic modelers are deeply uncomfortable using cubic growth models. The standard response in many economics circles to a model of sustained cubic value growth would be a desk rejection.
The reason why academics are uncomfortable using cubic growth models is that cubics grow fast. [Edit: Numbers here were initially too high] For example, on a cubic growth model, 20 centuries from now we’d be at 8,000v; 100 centuries from now we’d be at 1 million v; and a thousand centuries from now we’d be at a billion v. We need to make sure we have adequate scientific grounding for growth assumptions of this strength.
Even the few academics who do use cubic growth models often qualify them. So for example, Christian’s model (as well as the argument you gave about an expanding sphere) doesn’t need to, and probably shouldn’t assume cubic growth in the next 10-20 centuries. Rather, it assumes the limiting behavior of value growth will be cubic. That’s not such a big deal for someone like Christian, but it’s an awfully big deal when we’re looking at time of perils models, since these models just give far less weight to the limiting behavior than many other models would.
2. Cubic growth isn’t enough: I hope that the table in the text goes some way towards suggesting that building more growth into the model can only do so much to help the pessimist. One way to make the point more rigorously would be to draw on point that the mathematician David Holmes made to me.
His point was that across all models (including the Time of Perils model) we have V[WX]−V[W]=V[W]fr/(1−r). I didn’t go back and prove that (maybe try to scratch it out and I’ll give it a go if you’re having trouble?). But if that’s right, the lesson is that we can’t really get astronomical value for existential risk reduction without pumping up V[W], the value of the world itself. So for example, with a pessimistic r = 0.2, and a fractional reduction f = 0.1 in risk, the value of existential risk reduction would be V[W]/​40.
And it that is right, then we can see that value growth alone isn’t enough by looking at V[W] across growth modes. For example (and here I’m relying on Wolfram Alpha rather than crunching numbers explicitly, but I think that’s reasonably accurate) on a cubic growth model we’d get V[W] = 2420v, valuing risk reduction at 60.5v, and on a quartic (!) growth model we’d get V[W] = 43,380v, valuing risk reduction at about 1,085v, and on a quintic (!!!) growth model we’d get V[W] = 972,020v, valuing risk reduction at about 24,000v.
That’s not to say that value growth is totally impotent. But it takes a surprising amount of value growth to do the same work that we could do instead by being more optimistic about starting levels of risk.
I’ll have to go back and check the math again here. Actually I’d really appreciate some help if anyone else wanted to run some numbers here!
What about a cubic growth model? The volume of space or amount of resources used in each time period could grow cubically with the time period as we colonize space in every direction with an expanding sphere (roughly), and we continue to occupy and generate value from the interior of the sphere in each time period. Or do we need to non-renewably exhaust a good portion of the resources of the interior, so we only approximately generate value within an area near the frontier of expansion?
Christian Tarsney uses a cubic growth model in The epistemic challenge to longtermism.
Thanks! Cubic growth is an important possibility to consider.
If anyone is up for crunching some numbers, I only took a very rough stab at the numbers at this comment and I think it would be really valuable to do better. I’d be happy to add an acknowledgment in the final paper, or provide your libation of choice.
I guess I would want to say two things about cubic growth (ordered roughly in the importance they’d have in an academic discussion, although maybe in reverse order of importance for present purposes):
1. Cubic growth is a strong assumption: Most academic modelers are deeply uncomfortable using cubic growth models. The standard response in many economics circles to a model of sustained cubic value growth would be a desk rejection.
The reason why academics are uncomfortable using cubic growth models is that cubics grow fast. [Edit: Numbers here were initially too high] For example, on a cubic growth model, 20 centuries from now we’d be at 8,000v; 100 centuries from now we’d be at 1 million v; and a thousand centuries from now we’d be at a billion v. We need to make sure we have adequate scientific grounding for growth assumptions of this strength.
Even the few academics who do use cubic growth models often qualify them. So for example, Christian’s model (as well as the argument you gave about an expanding sphere) doesn’t need to, and probably shouldn’t assume cubic growth in the next 10-20 centuries. Rather, it assumes the limiting behavior of value growth will be cubic. That’s not such a big deal for someone like Christian, but it’s an awfully big deal when we’re looking at time of perils models, since these models just give far less weight to the limiting behavior than many other models would.
2. Cubic growth isn’t enough: I hope that the table in the text goes some way towards suggesting that building more growth into the model can only do so much to help the pessimist. One way to make the point more rigorously would be to draw on point that the mathematician David Holmes made to me.
His point was that across all models (including the Time of Perils model) we have V[WX]−V[W]=V[W]fr/(1−r). I didn’t go back and prove that (maybe try to scratch it out and I’ll give it a go if you’re having trouble?). But if that’s right, the lesson is that we can’t really get astronomical value for existential risk reduction without pumping up V[W], the value of the world itself. So for example, with a pessimistic r = 0.2, and a fractional reduction f = 0.1 in risk, the value of existential risk reduction would be V[W]/​40.
And it that is right, then we can see that value growth alone isn’t enough by looking at V[W] across growth modes. For example (and here I’m relying on Wolfram Alpha rather than crunching numbers explicitly, but I think that’s reasonably accurate) on a cubic growth model we’d get V[W] = 2420v, valuing risk reduction at 60.5v, and on a quartic (!) growth model we’d get V[W] = 43,380v, valuing risk reduction at about 1,085v, and on a quintic (!!!) growth model we’d get V[W] = 972,020v, valuing risk reduction at about 24,000v.
That’s not to say that value growth is totally impotent. But it takes a surprising amount of value growth to do the same work that we could do instead by being more optimistic about starting levels of risk.
I’ll have to go back and check the math again here. Actually I’d really appreciate some help if anyone else wanted to run some numbers here!
I agree that cubic growth will likely be outweighed by a high exponential risk rate r.