Social Welfare Function
What is Social Welfare Function?
A social welfare function is a rule that combines individual well-being into a single ranking of social outcomes, letting a society compare allocations.
A social welfare function takes each person's utility and returns one number for society, so outcomes can be ranked instead of only labeled efficient or not. It is the tool that fills the gap Pareto efficiency leaves, because it forces an explicit judgment about how much one person's gain is worth against another's loss. The utilitarian version simply adds utilities, W = U1 + U2 + ... + Un, and cares only about the total. The Rawlsian version takes the minimum, W = min(U1, ..., Un), so society improves only when the worst off person does. Different functions rank the same allocations differently, which is why the choice of function is a value judgment rather than an economic finding.
Social Welfare Function: a worked example
Compare two allocations for a two-person society. Allocation A gives utilities of 90 and 10; allocation B gives 55 and 40. A utilitarian function adds them: A scores 90 + 10 = 100 and B scores 55 + 40 = 95, so A wins. A Rawlsian function looks only at the worst off person: A scores 10 and B scores 40, so B wins. Neither allocation beats the other on the Pareto test, because moving from A to B helps the second person and hurts the first, which is exactly why a society needs a welfare function to break the tie.
The mistake students make with social welfare function
Students treat a social welfare function as something economists discover, like a demand curve. It is chosen, not measured; picking utilitarian over maximin is a statement of values, and economics can show the consequences of each but cannot settle which is right. A related error is assuming higher social welfare always means everyone gained. A rise in the total can hide a large loss for one group.
Social Welfare Function questions
What is the difference between a utilitarian and a Rawlsian social welfare function?
A utilitarian social welfare function adds everyone's utility and maximizes the total, while a Rawlsian one maximizes the utility of the worst off person. Utilitarianism will accept large inequality if it raises the sum; the Rawlsian rule refuses any gain that does not help the bottom. The two often rank the same pair of policies in opposite order.
Why is a social welfare function needed if we have Pareto efficiency?
A social welfare function is needed because Pareto efficiency cannot rank the many allocations that all pass its test. Almost every real policy creates winners and losers, so the Pareto criterion stays silent and something else has to choose. The welfare function supplies the missing distributional judgment.
Can a social welfare function be measured from data?
No, a social welfare function cannot be read off data, because it requires comparing how much utility one person gets against another and utility is not observable on a common scale. Economists sometimes infer implied weights from choices a government has already made, but that reverses the logic rather than measuring welfare. Any specific function is a modeling assumption you should state openly.
Formula / Example
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