Welfare Economics: Theorems, Social Welfare Functions, and Their Limits

The Hardest Question

Every economic argument, if you push it far enough, comes down to a question that economics alone cannot answer: is this outcome better than that one? Is a world with lower unemployment but higher inflation better than one with higher unemployment but lower inflation? Is a policy that makes most people richer but some people poorer an improvement? Is economic growth worth the environmental damage it causes?

These are not technical questions. They are moral and political questions. But economists have spent more than a century trying to develop rigorous frameworks for addressing them — frameworks that would allow us to evaluate economic outcomes without simply appealing to personal preferences or political ideology. The project is called welfare economics, and its history is a story of extraordinary ambition, genuine achievement, and unavoidable failure.

Understanding welfare economics — what it can do, what it cannot do, and where the line between analysis and value judgment lies — is essential for anyone who wants to think clearly about economic policy. Because every policy recommendation, whether it comes from the left or the right, from a government agency or an academic journal, rests on a welfare judgment, whether or not that judgment is made explicit.

The Two Fundamental Welfare Theorems

The foundation of modern welfare economics is a pair of results known as the Fundamental Welfare Theorems. We have discussed them briefly in the context of Pareto efficiency, but they deserve a fuller treatment here, because they define the framework within which welfare economics operates.

The First Welfare Theorem states that, under certain conditions (perfect competition, no externalities, complete markets, perfect information), a competitive equilibrium is Pareto efficient. In plain language: if markets work perfectly, the outcome they produce has the property that no one can be made better off without making someone else worse off.

The Second Welfare Theorem states that, under additional conditions (convex preferences, divisible goods), any Pareto-efficient allocation can be achieved as a competitive equilibrium, given the right initial distribution of resources. In plain language: if you want a particular efficient outcome — one with more equality, or one that favors a particular group — you can achieve it by redistributing initial endowments and then letting markets work.

Together, these theorems establish the intellectual framework for thinking about the relationship between markets and welfare. The First Theorem tells you what perfectly functioning markets can achieve: efficiency. The Second tells you that markets can also achieve any efficient allocation, provided you get the initial distribution right. The policy implication is that you should separate efficiency from equity: use markets to produce efficient outcomes, and use redistribution (taxes and transfers) to achieve whatever distributional goals you want.

This framework is clean, powerful, and deeply influential. It is also, in important respects, misleading. The conditions required for the theorems are never met in practice. Real markets have externalities, market power, incomplete information, and missing markets. Real redistribution is costly and distortionary — there are no lump-sum taxes in the real world. And the definition of “efficiency” that the theorems use — Pareto efficiency — is, as we have seen, far weaker than it appears. The theorems establish a benchmark, not a description of reality. The question is how useful the benchmark is.

The welfare theorems tell us about efficiency, but they say nothing about how to choose among the many possible efficient outcomes. An economy in which one person has everything is Pareto efficient, and so is one in which resources are distributed equally. The theorems are silent on which is better. To make such judgments, economists developed the concept of a social welfare function — a mathematical function that aggregates individual well-being into a single measure of social welfare.

The idea is seductively simple. If we could measure each individual’s well-being (utility), and if we had a function that combined these individual utilities into a single number representing “social welfare,” then we could rank any two economic outcomes by comparing their social welfare values. The outcome with higher social welfare is better.

The details, however, are fiendishly difficult. The most common social welfare functions include:

Utilitarian: Social welfare is the sum of individual utilities. This treats all individuals equally in the sense that a unit of utility counts the same regardless of who receives it. The utilitarian function tends to favor policies that maximize aggregate well-being, but it can justify significant inequality: if transferring income from the poor to the rich somehow increases total utility (perhaps because the rich are more “efficient” at producing utility), utilitarianism endorses the transfer.

Rawlsian (maximin): Social welfare equals the utility of the worst-off individual. The best outcome is the one that maximizes the well-being of the least advantaged member of society. This function reflects John Rawls’s philosophical argument that rational individuals, choosing behind a “veil of ignorance” (not knowing what position they would occupy in society), would choose institutions that protect the worst off. The Rawlsian function prioritizes equality and insurance against bad outcomes, but it can be extreme: it would reject a policy that makes 999 people vastly better off if it makes one person slightly worse off.

Bergson-Samuelson: A general social welfare function that allows any weighting of individual utilities. This is the economist’s all-purpose tool: by choosing different weights, you can derive the utilitarian, Rawlsian, or any other social welfare function as a special case. The flexibility is both a strength and a weakness. It allows the economist to analyze the implications of different value judgments without endorsing any particular one. But it also means that the framework has no content unless you specify the weights — and specifying the weights is precisely the value judgment that the framework was supposed to help with.

Arrow’s Impossibility Theorem

In 1951, Kenneth Arrow published Social Choice and Individual Values, which contained one of the most important results in social science: Arrow’s impossibility theorem. The theorem proves that there is no method of aggregating individual preferences into a social ranking that satisfies a small set of seemingly reasonable conditions.

The conditions are:

Unrestricted domain: The method must work for any possible set of individual preferences.
Pareto principle: If every individual prefers outcome A to outcome B, the social ranking must rank A above B.
Independence of irrelevant alternatives: The social ranking of A vs B should depend only on individuals’ preferences between A and B, not on their preferences regarding other alternatives.
Non-dictatorship: No single individual’s preferences automatically determine the social ranking.

Arrow proved that no social welfare function (or voting rule) can satisfy all four conditions simultaneously. Any method that satisfies the first three must be a dictatorship; any non-dictatorial method must violate at least one of the other conditions.

The implications are profound. Arrow’s theorem does not just say that we have not yet found a perfect method of aggregating preferences; it says that no such method can exist. Any democratic procedure for making collective decisions must involve some compromise among these conditions. Majority rule violates transitivity (Condorcet’s paradox). Point systems (like Borda counts) violate independence of irrelevant alternatives. Unanimity rules give every individual a veto (which makes them effectively dictatorial).

For welfare economics, Arrow’s theorem means that the project of constructing a social welfare function from individual preferences alone — without imposing some external value judgment — is logically impossible. You cannot get from individual well-being to social well-being through a purely mechanical aggregation procedure. Somewhere in the process, a value judgment has to be made about how to weigh competing claims, and that judgment cannot be derived from the preferences themselves.

The Compensation Principle and Its Failures

Because welfare comparisons are so difficult, economists have often tried to avoid them by using the compensation principle (also known as the Kaldor-Hicks criterion). The idea, as discussed in our article on Pareto efficiency, is that a policy change is an improvement if the winners could, in principle, compensate the losers and still be better off.

The compensation principle is attractive because it seems to avoid interpersonal comparisons of utility: you do not need to weigh one person’s gain against another’s loss; you just need to check whether the total gains exceed the total losses. This is the basis of cost-benefit analysis, the most widely used tool for evaluating public policies. A new highway is justified if its total benefits (time saved, increased economic activity, reduced accidents) exceed its total costs (construction, environmental damage, displacement of residents). A regulation is justified if the value of the harms it prevents exceeds the costs of compliance.

The problems with the compensation principle are well known but often underappreciated. First, the compensation is hypothetical. The winners typically do not actually compensate the losers, which means that the policy creates real losers whose losses are justified by gains they never receive. Second, the Scitovsky paradox shows that the criterion can produce contradictory recommendations. Third, cost-benefit analysis requires putting monetary values on things — a human life, clean air, the preservation of a wetland — that do not have market prices, and the methods for doing so (willingness-to-pay surveys, hedonic pricing, the value of a statistical life) are controversial and often produce results that seem morally arbitrary.

Fourth, and perhaps most fundamentally, cost-benefit analysis assumes that a dollar is worth the same to everyone. But it is not. A dollar is worth much more to a poor person than to a rich person — the poor person might use it for food, while the rich person might not notice it. Unweighted cost-benefit analysis systematically favors policies that benefit the rich (who have a higher willingness to pay) over policies that benefit the poor (who have lower willingness to pay, simply because they have less money). This is not a technical error; it is a consequence of using willingness to pay as the measure of value in a world of unequal income.

Sen’s Capability Approach

The most influential alternative to traditional welfare economics is Amartya Sen’s capability approach, developed over a series of works beginning in the 1980s. Sen argues that well-being should be measured not by utility (subjective satisfaction) or income (command over commodities) but by capabilities — the real freedoms that people have to live lives they have reason to value.

The distinction is crucial. A person who is well fed because they choose to diet is in a very different situation from a person who is well fed because they have access to adequate nutrition. Both may have the same utility level (if the dieter is happy with their choice) and the same income level. But the person with access to adequate nutrition has a capability — the freedom to be well nourished — that the person who cannot afford food lacks. Sen’s approach says that what matters is not the outcome (how much you consume, how happy you are) but the set of real opportunities available to you.

The capability approach has been enormously influential in development economics and in the construction of the United Nations Human Development Index (HDI), which measures development not just by income but also by health and education. It has also inspired a rethinking of poverty measurement, gender inequality, and disability.

From the perspective of welfare economics, the capability approach offers a way around some of the limitations of the traditional framework. It does not require interpersonal comparisons of utility (it compares capabilities, which are in principle observable). It does not treat all Pareto-efficient outcomes as equally good (it distinguishes between those that expand capabilities and those that do not). And it provides a vocabulary for discussing the relationship between efficiency and justice that is richer than the Pareto criterion or the compensation principle.

But the capability approach has its own limitations. Which capabilities matter? How should they be measured? How should we weigh competing capabilities when they conflict? Sen has deliberately left these questions open, arguing that they should be resolved through democratic deliberation rather than philosophical fiat. This openness is intellectually honest but practically challenging: policymakers want concrete guidance, and the capability approach does not always provide it.

Cost-Benefit Analysis in Practice

Despite its theoretical limitations, cost-benefit analysis (CBA) remains the most widely used tool for evaluating public policy. Every major regulation in the United States is subject to a cost-benefit analysis by the Office of Management and Budget. The World Bank uses CBA to evaluate development projects. Governments around the world use it to assess infrastructure investments, environmental regulations, and health policies.

The practice of CBA involves several steps that illustrate both the power and the limitations of welfare economics. First, identify all the effects of the policy — who gains, who loses, and by how much. Second, assign monetary values to these effects. Third, discount future effects to present value (a dollar today is worth more than a dollar in ten years, but how much more?). Fourth, compare total benefits to total costs. If benefits exceed costs, the policy passes the cost-benefit test.

Each step involves judgment calls that are partly technical and partly ethical. What discount rate should be used for long-term environmental policies? (A high discount rate means future environmental damage counts for little; a low discount rate means it counts for a lot. The choice between a 3% and a 7% discount rate can flip the conclusion of a climate policy analysis.) How should we value a statistical life? (The U.S. government uses a figure around $10-12 million, derived from studies of how much people are willing to pay to reduce small risks of death. But willingness to pay varies with income, which means a rich person’s life is implicitly valued more than a poor person’s.) Should we count benefits to non-citizens? Should we adjust for distributional effects?

These are not technical questions with technical answers. They are ethical questions that are resolved by convention, regulation, and political pressure. The appearance of objectivity in cost-benefit analysis — the precision of the numbers, the formality of the methodology — can obscure the fact that every CBA embeds contestable value judgments in its assumptions.

The Unavoidable Value Judgment

The deepest lesson of welfare economics is that value judgments cannot be eliminated from economic analysis. They can be hidden, deferred, or disguised in technical language, but they cannot be avoided.

The choice to use the Pareto criterion rather than the Rawlsian criterion is a value judgment. The choice to discount the future at 3% rather than 7% is a value judgment. The choice to measure well-being by income rather than by capabilities is a value judgment. The choice to count only market-priced goods rather than unpaid care work, ecosystem services, or social cohesion is a value judgment. These choices are inevitable, because evaluating economic outcomes is not a purely technical exercise; it requires deciding what matters, how much it matters, and whose interests count.

The danger is not that welfare economics involves value judgments — every form of policy analysis does. The danger is pretending that it doesn’t. When an economist says that a policy is “efficient” without specifying what is included in the cost-benefit calculation, who is affected, and how the gains and losses are distributed, they are smuggling in value judgments under the cover of technical analysis. When a government report concludes that the “benefits exceed the costs” without disclosing the discount rate, the value of a statistical life, or the treatment of distributional effects, it is presenting a value-laden conclusion as if it were a scientific finding.

The remedy is not to abandon welfare economics but to practice it honestly. State the value judgments explicitly. Present the results under different assumptions. Show how the conclusions change when you use a different social welfare function, a different discount rate, or a different measure of well-being. Acknowledge that the line between analysis and advocacy is thinner than economists often pretend.

Welfare economics, at its best, is a discipline of structured moral reasoning — a way of thinking carefully and rigorously about trade-offs, competing claims, and the consequences of collective choices. At its worst, it is a way of dressing up political preferences in the language of science. The difference between the two lies not in the techniques but in the honesty of the practitioner.