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gjm comments on A strange implication of critical-level utilitarianism - Less Wrong
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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.