You could generalize a bit further by looking at the behavior of
The integral of the ratio of the world under two interventions, ∫∞0u(t)v(t) or ∫∞0u(t)−v(t). This integral could have a value even if the integral of each intervention is indefinite.
The ratio of the limit of integrals under two interventions, limt→∞∫t0u(t)∫t0v(t) or limt→∞∫t0u(t)−∫t0v(t). This could likewise have a value even if ∫∞0u(t) isn’t defined
I didn’t want to get too distracted with these complication in the piece, but I’m sympathetic to these and other approaches to avoid the technical issue of divergent integrals of value when studying longterm effects.
In the case in question (where u(t) always equals k v(t)) we get an even stronger constraint the ratio of progressively longer integrals doesn’t just limit to a constant, but equals a constant.
There are some issues that come up with these approaches though. One is that they are all tacitly assuming that comparing things at a time is the right comparison. But suppose (contra my assumptions in the post) that population was always half as high in one outcome as the other. Then it may be doing worse at any time, but still have all the same people eventually come into existence and be equally good for all of them. Issues like this where the ratio depends on what variable is being integrated over don’t come up in the convergent integral cases.
All that said, the integrating to infinity in economic modelling is presumably not to be taken literally, and for any finite time horizon — no matter how mindbendingly large — my result that the discounting function doesn’t matter holds (even if the infinite integral were to diverge).
You could generalize a bit further by looking at the behavior of
The integral of the ratio of the world under two interventions, ∫∞0u(t)v(t) or ∫∞0u(t)−v(t). This integral could have a value even if the integral of each intervention is indefinite.
The ratio of the limit of integrals under two interventions, limt→∞∫t0u(t)∫t0v(t) or limt→∞∫t0u(t)−∫t0v(t). This could likewise have a value even if ∫∞0u(t) isn’t defined
I agree!
I didn’t want to get too distracted with these complication in the piece, but I’m sympathetic to these and other approaches to avoid the technical issue of divergent integrals of value when studying longterm effects.
In the case in question (where u(t) always equals k v(t)) we get an even stronger constraint the ratio of progressively longer integrals doesn’t just limit to a constant, but equals a constant.
There are some issues that come up with these approaches though. One is that they are all tacitly assuming that comparing things at a time is the right comparison. But suppose (contra my assumptions in the post) that population was always half as high in one outcome as the other. Then it may be doing worse at any time, but still have all the same people eventually come into existence and be equally good for all of them. Issues like this where the ratio depends on what variable is being integrated over don’t come up in the convergent integral cases.
All that said, the integrating to infinity in economic modelling is presumably not to be taken literally, and for any finite time horizon — no matter how mindbendingly large — my result that the discounting function doesn’t matter holds (even if the infinite integral were to diverge).