How to Calculate a Social Welfare Function
A utilitarian social welfare function adds every person's utility, W = U1 + U2 + ... + Un, while a Rawlsian one scores an allocation by the lowest utility, W = min(U1, ..., Un).
The Social Welfare Function formula
Calculator
Enter each person's utility under two allocations to get utilitarian and Rawlsian welfare, and which one each rule picks.
How well off the first person is before any change.
Same measure applied to the second person.
Same measure applied to the third person.
The first person's utility if the change goes ahead.
The second person's utility under the same change.
The third person's utility under the same change.
Adding every person's utility gives 90, with no weight on how it is split.
- Utilitarian welfare, proposed allocation
- 105
- Rawlsian welfare, current allocation
- 20
- Rawlsian welfare, proposed allocation
- 10
- A utilitarian picks
- The proposed allocation
- A Rawlsian picks
- The current allocation
The same sum under the proposal comes to 105.
The worst-off person holds 20 utils, and that alone is the Rawlsian score.
Under the proposal the lowest utility is 10.
The utilitarian rule ranks by the total, so a large gain to one person can outweigh a smaller loss to another.
The two rules disagree, and that gap is exactly why the choice of social welfare function has to be stated.
How to calculate Social Welfare Function, step by step
- 1List every person's utility under each allocation. You need one utility number per person, measured the same way, for every allocation you plan to rank.
- 2Add them up for the utilitarian score. W = U1 + U2 + ... + Un counts a util the same whoever holds it, so only the total matters and the split is ignored.
- 3Take the smallest for the Rawlsian score. W = min(U1, ..., Un) scores an allocation by whoever is worst off, so gains above the bottom add nothing to it.
- 4Rank the allocations under each rule. The higher W wins. The two rules can crown different winners whenever the gain is concentrated at the top.
- 5Say which rule you used. A welfare ranking only means something once the social welfare function is named, because the ranking follows from the rule and not from the numbers alone.
Worked example: Social Welfare Function
Three people hold utilities of 40, 30 and 20 under the current allocation, so utilitarian welfare is 40 + 30 + 20 = 90 and Rawlsian welfare is min(40, 30, 20) = 20. A proposed reform would give them 70, 25 and 10, so utilitarian welfare rises to 70 + 25 + 10 = 105 while Rawlsian welfare falls to min(70, 25, 10) = 10. A utilitarian backs the reform because 105 beats 90, and a Rawlsian rejects it because the worst-off person drops from 20 to 10. Same numbers, opposite verdicts, and the social welfare function is what settles the argument.
Social Welfare Function questions
Why can a utilitarian and a Rawlsian rank the same two allocations differently?
A utilitarian counts only the total, so a change that hands 30 extra utils to the best-off person while taking 10 from the worst-off raises the sum and lowers the minimum at the same time. One rule approves it, the other rejects it.
Does a higher social welfare score mean everyone is better off?
No. A change that raises W can still leave some people worse off. Only a Pareto improvement makes at least one person better off with nobody worse off, and most policy changes are not Pareto improvements.
What does a social welfare function assume about utility?
That utility can be compared across people. Adding one person's utils to another's, or asking who holds the fewest, both need interpersonal comparison, which is the standard objection to the whole approach.
Is there anything between the utilitarian and Rawlsian rules?
Yes. Weighting people differently, or adding utilities only after passing them through a concave function, gives a family of rules between counting the total and counting only the bottom. The more concave the function, the more weight the worst-off person carries.
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