Here is a passage from Hilary Greaves's Population axiology.

In many decision situations, at least in expectation, an agent’s decision has no effect on the numbers and identities of persons born. For those situations, fixed-population ethics is adequate. But in many other decision situations, this condition does not hold. Should one have an additional child? How should life-saving resources be prioritised between the young (who might go on to have children) and the old (who are past reproductive age)? How much should one do to prevent climate change from reducing the number of persons the Earth is able to sustain in the future? Should one fund condom distribution in the developing world? In all these cases, one’s actions can affect both who is born and how many people are (ever) born. To deal with cases of this nature, we need variable-population ethics: ‘population ethics’ for short.

So more generally, I'd say that population ethics is the study of how to compare the moral value of different populations.

As an aside, even if I agreed with that definition, I don't think infinite ethics would be a subset of population ethics.

The distinctive problems of infinite ethics arise roughly when we can affect an infinite number of value-bearing locations. But this is independent of what those value-bearing locations are - in particular, they need not be people.

For example, we'd run into infinitarian paralysis if we thought that each of our actions affected the axiological value of an infinite number of paintings.

I don't think variable populations are a defining feature of population ethics - do you have a source for that?

For example, the abstract of Greaves (2017) says (emphasis mine): "Population axiology is the study of the conditions under which one state of affairs is better than another, when the states of affairs in question may differ over the numbers and the identities of the persons who ever live."

Similarly, the Wikipedia article on Population ethics starts with (emphasis mine): "Population ethics is the philosophical study of the ethical problems arising when our actions affect who is born and how many people are born in the future."

More fuzzy indications are that Part IV of Reasons and Persons is titled "Future Generations" and Gustaf Arrhenius's seminal dissertation is titled "Future Generations. A Challenge for Moral Theory". And Arrhenius says (emphasis mine): "The main problem has been to find an adequate population theory, that is, a theory about the moral value of states of affairs where the number of people, the quality of their lives, and their identities may vary."

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It also seems to me your candidate definition of "how to compare the moral value of different populations" is too broad to be useful. For example, to answer that question for a given population we also need to know what individual well-being consists in (not just how to aggregate individual welfare to get population welfare). So on your definition the whole question of well-being, including the classic debates between hedonism, desire satisfaction and objective list theories, are subsumed under population ethics! Whereas I think it's useful to view variable-population questions as their own subfield precisely because they arise no matter which view you take on the nature of well-being.

For example, one question discussed in population ethics is when a more equal population with lower total welfare is better than a less equal population with higher total welfare.

Hm, interesting - I definitely agree that this question is relevant, and discussed in the literature, for fixed-population settings. And if I thought it was part of population ethics I'd agree that "variable-population cases" wasn't a good definition of population ethics. But it wouldn't have occurred to me to subsume that question under population ethics, and I can't recall it being labeled population ethics anywhere.