Suppose you have a population of  identical people, each with the same income . Critical-level utilitarianism (CLU) says that you should maximize utility above a certain "critical level" - in this case, if each person's utility function is , we want maximize  for some income level  below which you don't think life is worth living. (Critical-level utilitarianism doesn't help us pick which value of  to use, though.) To choose the optimal level of , we need to know which combinations are feasible. Cobb-Douglas production functions (which are essentially power functions) are frequently used to model how much output can be produced with a given amount of input, so I will use them here. If total income is given by the Cobb-Douglas production function , then both the average and the marginal productivity of labor are proportional to , so the average income should scale the same way. If we assume that each person's utility scales as the logarithm of their income, we get that . Optimizing, we set , so  and . Usually,  is greater than 0 but less than 1, so the optimal per-capita income ranges from the critical income level to -fold above it. A pessimistic production function with  would have an optimal per-capita income that's a factor of  above the critical level.

This calculation uncovered an interesting (to me, at least) paradox. The current variation in per-capita incomes across countries is far larger than -fold or even -fold, so according to this model, regardless of the critical income level you choose, either a lot of countries are far below the critical level (and the lives of the people in them are not worth living by a large margin), or a lot of countries are far above the optimal income level (and their populations should increase until their per-capita incomes decrease by a large amount). This implies some strange priorities: if the critical level is high, then it'd be good to reduce the population of a low-income country even if it decreases their per-capita income greatly, and if the critical level is low, poverty reduction in high-income is unimportant, since it'd be much better to increase the birth rate of poor (in high-income countries) people.

Although this model is very simple, it's fairly robust to changes in the production function or the utility function, since the differences in income between countries are so large. The major assumption I made is that CLU is reasonable. This post applies to total utilitarianism as well, since it is a special case of CLU.

What do you think? Do you agree with CLU or a similar theory? If you do, what are the problems with my model? If you agree with CLU and think this model is basically correct, what does it imply about what an effective altruist's priorities should be?