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).

I discuss something similar to that a bit on page 41, but mainly focusing on whether depletion could make it harder for civilisation to re-emerge. Ultimately, it still looks to me like it would be easier and faster the second time around.