We have the notion of total utilitarianism, in which the government tries to maximize the sum of the utility values of each of its constituents. This leads to "repugnant conclusion" issues in which the government generates new constituents at a high rate until all of them are miserable.

We also have the notion of average utilitarianism, in which the government tries to maximize the average of the utility values of each of its constituents. This leads to issues -- I'm not sure if there's a snappy name -- where the government tries to kill off the least happy constituents so as to bring the average up.

Not quite. If our societal utility function S(n) = n x U(n), where n is the number of people in the society, and U(n) is the average utility gain per year per person (which decreases as n increases, for high n, because of over crowding and resource scarcity), then you don't maximise S(n) by just increasing n until U(n) reaches 0. There will be an optimum n, for which 1 x U(n+1) - the utility from yet one more citizen, is less than n x ( U(n) - U(n+1) ) - the loss of utility by the other n citizens from adding that person.