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ericyu3 comments on A strange implication of critical-level utilitarianism - Less Wrong
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There are several things here I fail to understand.
Why d/dN? If you're looking for optimal income per capita, you need d/dw=0 not d/dN=0.
The result you've allegedly reached is that w = w0 exp(alpha-1) where alpha<1, which means w<w0, which means you're not actually in the regime where net utility equals N[U(w)-U(w0)], so you've been doing calculus on the wrong formulae.
Clearly utility is not only a function of income. (Even considering only money, you need to consider assets as well as income.) Of course considering only income is a handy simplification that may turn something impossibly complicated into something susceptible to analysis, but I think you should be explicit about making that simplification because the importance of things other than money is actually a pretty big deal.
This all seems like a more complicated but still minor variation on simple and familiar observations like these: (a) simple versions of utilitarianism say well-off people should give almost all they have to poorer people; (b) simple versions of average utilitarianism say we should kill all the least happy people; (c) simple versions of total utilitarianism say we should prefer an enormous population of people with just-better-than-nothing lives to a normal-sized population of very happy people. I would expect solutions to (or bullet-biting on) these problems to deal with the more complicated but similarly counterintuitive conclusions presented here (assuming for the sake of argument that either my objections above are wrong or else the conclusions remain when the errors are repaired).
Oh, I see. You're taking wage to be determined by production, which in turn is determined by population according to the Cobb-Douglas formula, and then asking "what's the optimal population?". Got it.
Yup, better now.
So, anyway, now that I understand your argument better, there's something that looks both important and wrong, but maybe I'm misunderstanding. You're assuming that A -- the constant factor in the Cobb-Douglas formula -- is the same for all countries. But surely it isn't, and surely this accounts for a large amount of the variation in productivity and wealth between countries. It seems like this would lead to big differences in w between countries even if they're all close to optimal population.
The A factor drops out of the final expression for the optimal wage. If the form of the production function is the same between two countries, their optimal wages will be the same as well. However, their optimal populations will obviously be different. For example, if country 1 has 10 times higher A than country 2, but their values of alpha are the same, then their optimum wages are the same, but country 1's optimum population is higher by a factor of 10^(1/(1-alpha)).
Here, A lumps together productivity and the amount of land a country has (so that a large poor country may have higher A than a small rich one). Obviously, increasing A will increase welfare, but it won't change the optimal wage (if the country is above that level already, increasing A will bring wages further away from the optimum) - the best thing to do (according to this model) is to increase A as much as possible, and also adjust the population level to match the optimal wage.