This article is from the Anarchy FAQ, by Bryan Caplan with numerous contributions by others.
The most widely-used concept in theoretical welfare economics is "Pareto optimality" (also known as "Pareto efficiency"). An allocation is Pareto-optimal iff it is impossible to make at least one person better off without making anyone else worse off; a Pareto improvement is a change in an allocation which makes someone better off without making anyone else worse off. As Hal Varian's Microeconomic Analysis explains, "[A] Pareto efficient allocation is one for which each agent is as well off as possible, given the utilities of the other agents." "Better" and "worse" are based purely upon subjective preferences which can be summarized in a "utility function," or ordinal numerical index of preference satisfaction.
While initially it might seem that every situation is necessarily Pareto optimal, this is not the case. True, if the only good is food, and each agent wants as much food as possible, then every distribution is Pareto optimal. But if half of the agents own food and the other half own clothes, the distribution will not necessarily be Pareto optimal, since each agent might prefer either more food and fewer clothes or vice versa.
Normally, economists would expect agents to voluntarily trade in any situation which is not Pareto optimal; but neoclassical theorists have considered a number of situations in which trade would be a difficult route to Pareto optimality. For example, suppose that each agent is so afraid of the other that they avoid each other, even though they could both benefit from interaction. What they need is an independent and powerful organization to e.g. protect both agents from each other so that they can reach a Pareto-optimal allocation. What they need, in short, is the state. While economists' examples are usually more elaborate, the basic intuition is that government is necessary to satisfy the seemingly uncontroversial principle of Pareto optimality.
Anarchists of all sorts would immediately object that the very existence of deontological anarchists shows that Pareto optimality can never justify state action. If even the slightest increase in the level of state activity incompensably harms the deontological anarchist, then obviously it is never true that state action can make some people better off without making any others worse off. Moreover, virtually all government action makes some people better off and other people worse off, so plainly the pursuit of Pareto improvements has little to do with what real governments do.
Due to these difficulties, in practice economists must base their judgments upon the far more controversial judgments of cost-benefit analysis. (In the works of Richard Posner, this economistic cost-benefit approach to policy decisions is called "wealth-maximization"; a common synonym is "Kaldor-Hicks efficiency.") With cost-benefit analysis, there is no pretense made that government policy enjoys unanimous approval. Thus, it is open to the many objections frequently made to e.g. utilitarianism; moreover, since cost-benefit analysis is based upon agents' willingness to pay, rather than on agents' utility, it runs into even more moral paradoxes than utilitarianism typically does.
In the final analysis, welfare economists' attempt to provide a value-free or at least value-minimal justification of the state fails quite badly. Nevertheless, economic analysis may still inform more substantive moral theories: Pareto optimality, for example, is a necessary but not sufficient condition for a utilitarian justification of the state.