Social Welfare Function
A social welfare function is a function that ranks different allocations of resources according to their desirability from the standpoint of society as a whole. It aggregates the well-being of individuals into a single measure of social welfare, and it provides a criterion for evaluating policies and for analysing the trade-off between efficiency and equity. The concept is central to welfare economics and to normative public economics, and its properties determine what can be said about optimal policy.
Definition
A social welfare function maps the utilities of individuals into a measure of social welfare, and the form of the function embodies judgments about how the well-being of different individuals should be weighed. A utilitarian function adds the utilities of all individuals, treating each equally and implying that social welfare depends on the total, not on the distribution. A function with diminishing marginal social welfare gives more weight to the well-being of the worse off, and it implies a concern for equality.
The choice of the function involves value judgments. Different functions imply different rankings of allocations and different prescriptions for policy, and there is no single correct function. The analysis of policy therefore involves an explicit or implicit choice of a welfare function, and the transparency of that choice is important for the evaluation of the conclusions.
Properties
Social welfare functions are often assumed to satisfy certain properties. The Pareto principle requires that an allocation be ranked higher if it makes someone better off and no one worse off. Anonymity or symmetry requires that the identity of individuals not matter, only their well-being. The independence of irrelevant alternatives, in social choice theory, requires that the ranking of two alternatives depend only on the ranking of those alternatives by individuals, not on other options.
These properties can conflict. The theorem of Arrow shows that no social welfare function can satisfy a set of seemingly reasonable conditions simultaneously, which implies that any aggregation of individual preferences involves a trade-off among desiderata. This result, the impossibility theorem, is fundamental to social choice theory and it clarifies the limits of aggregation and the role of value judgments in social decisions.
Efficiency and equity
The social welfare function provides a framework for analysing the trade-off between efficiency and equity. Efficiency concerns the size of the total to be distributed, while equity concerns its distribution. A policy that raises total output but worsens the distribution may increase or decrease social welfare, depending on the weights given to different individuals and on the magnitude of the changes.
The framework shows that the goal of policy is not to maximise output alone but to maximise social welfare, which may require accepting some loss of efficiency to achieve a more equal distribution. The optimum balances the marginal gain from redistribution, in terms of social welfare, against the marginal cost in terms of reduced output and distorted incentives. The analysis of optimal taxation applies this framework to determine the best structure of taxes and transfers.
Applications
Social welfare functions are used in the evaluation of policies, in the design of tax and transfer systems, and in cost-benefit analysis. When comparing policies that affect different groups differently, the welfare function provides a way to aggregate the effects and to rank the alternatives. Distributional weights, which reflect the social value of income to different groups, are used in cost-benefit analysis to incorporate equity considerations alongside efficiency.
The framework also informs the analysis of inequality and poverty. Measures of social welfare that are sensitive to distribution, such as the Atkinson index, are derived from social welfare functions, and they provide a way to summarise the distribution in a single number that reflects the degree of inequality aversion. These measures are used to compare countries and periods and to evaluate the effects of policy on distribution.
Utilitarian, Rawlsian, and other forms
Different forms of the social welfare function embody different ethical positions. A utilitarian function sums the utilities of individuals, implying that social welfare depends on the total and that a gain to one can offset a loss to another as long as the sum rises. A Rawlsian function, based on the maximin principle, evaluates social welfare by the position of the worst-off individual, implying a strong concern for the least advantaged.
Intermediate forms give weight to all individuals but with diminishing weight as their well-being rises, reflecting a concern for equality that is stronger than in the utilitarian case but weaker than in the Rawlsian extreme. The choice among these forms reflects judgments about fairness and about the trade-off between the total and its distribution, and it determines the prescriptions of the analysis. Different forms can imply different policies, and the choice is therefore consequential.
Inequality aversion
The degree of inequality aversion is captured in the curvature of the social welfare function, and it can be parameterised in measures such as the Atkinson index. A higher degree of inequality aversion gives more weight to the well-being of the worse off and implies a greater willingness to sacrifice total income for a more equal distribution. The parameter is a normative choice, though it can be informed by evidence about preferences and by democratic decisions.
The measurement of inequality aversion affects the evaluation of policies. A policy that raises total income but increases inequality is beneficial under a low degree of aversion and harmful under a high degree. The choice of the parameter therefore determines the conclusions, and it must be made explicit in the analysis. The use of a range of parameters, and the reporting of results for different values, allows the sensitivity of the conclusions to be assessed.
Social choice and aggregation
The aggregation of individual preferences into a social welfare function raises problems that are studied in social choice theory. Arrow's impossibility theorem shows that no aggregation rule can satisfy a set of reasonable conditions simultaneously, including the Pareto principle, the independence of irrelevant alternatives, and non-dictatorship. The result implies that any aggregation involves trade-offs among desiderata and that the choice among rules is normative.
The aggregation problem has practical implications. It means that the construction of a social welfare function involves judgments that cannot be derived purely from individual preferences, and that the analysis of policy must be transparent about the judgments made. It also motivates the use of procedures, such as deliberation and democratic decision, to resolve the conflicts that the theory identifies, and it informs the design of institutions that make social choices.
Applications in policy analysis
Social welfare functions are used in the analysis of taxation, of public spending, and of regulation. The analysis of optimal taxation uses a social welfare function to balance the efficiency costs of taxes against their distributional benefits, and the resulting prescriptions depend on the degree of inequality aversion and on the structure of the economy. The analysis of public spending compares the benefits to different groups and evaluates them using the welfare function.
In cost-benefit analysis, distributional weights derived from the social welfare function are used to adjust the value of benefits and costs that accrue to different groups, so that the analysis reflects the social valuation of gains to rich and poor. The weights depend on the degree of inequality aversion and on the incomes of the groups affected, and their use allows the analysis to incorporate equity alongside efficiency. The approach is used in the appraisal of public projects and in the evaluation of policies with distributional effects.