Social welfare, compensation and Arrow's theorem

Take the contract curve of Lesson 1 and measure each person on it by her utility instead of her bundle. The result is the utility possibility frontier: every efficient split of utility between two people. It shows, in utility terms, all the efficient allocations (CUET 2026). The contract curve, the utility possibility frontier and the production possibility frontier are three different objects, and the matching question in CUET 2026 asks you to tell them apart: the contract curve shows efficient allocations of goods; the frontier shows efficient allocations in terms of utility; the PPF shows the combinations of two goods that can be made with given inputs.

Every point on the frontier is Pareto efficient, so the Pareto criterion cannot choose among them. Going from one to another helps one person and hurts the other, and the Pareto criterion is silent. Choosing needs a social welfare function (JNU 2016): it ranks every feasible allocation, which the Pareto criterion cannot.

A Bergson–Samuelson social welfare function turns everyone’s utility into one number for society: W = f(U₁, …, U_m). Abram Bergson introduced the idea in 1938; Paul Samuelson developed it in 1947. Three standard forms (JAM 2026):

Two traps in JAM 2026. Rawls’s function is the minimum of utilities, not the maximum. And a social welfare function reflects value judgements; interpersonal comparisons of utility cannot be made “on scientific grounds”. The lab draws the level curve of each function that just touches the frontier, which is the planner’s choice.

Most policies create winners and losers, so the Pareto criterion often says nothing. The compensation principle (JNU 2019) asks a different question: could the winners compensate the losers? The compensation need not actually be paid. Three tests, in order:

• Kaldor criterion (1939): the change is an improvement if the gainers could pay the losers enough to make them as well off as before and still be better off themselves.
• Hicks criterion (1939): the change is an improvement if the losers could not profitably pay the gainers to give it up.
• Scitovsky double criterion (1941): the move must pass the Kaldor test and the move back must fail it (the “reversal test”). Together this is what UGC NET calls a double Kaldor–Hicks test (UGC NET Dec 2024, JNU 2019).

Why a double test? The Kaldor test alone can approve a move and also approve the move back. That is the Scitovsky paradox, and it happens when the utility possibility frontiers cross: each position can compensate its way to the other. Use the lab below to see both.

Exams ask you to match these or put them in order, so learn the sequence. The dates below are from the original publications.

So CUET 2025’s chronology, oldest first, is Pareto, Bergson, Kaldor–Hicks, Rawls. The matching question in JNU 2019 pairs a double Kaldor–Hicks test with Scitovsky, the compensation principle with Kaldor–Hicks, a social welfare function with Bergson, and the impossibility theorem with Arrow.

A different test of a fair allocation needs no welfare function. An allocation is envy-free (a no-envy allocation) if nobody prefers someone else’s bundle to her own. Giving a cake of 1 kg half to each of two people who like cake is both efficient and envy-free (JNU 2017); giving the whole cake to one person is efficient but the other envies her; and (0.5, 0.4) is neither efficient (0.1 is wasted) nor envy-free. Efficiency and envy-freeness are different tests, and an allocation can pass one and fail the other.

Suppose three voters rank three alternatives and we use majority rule. Try the lab’s first profile: each alternative is first for exactly one voter. A beats B, B beats C and C beats A, two votes to one each time. This is Condorcet’s voting paradox: majority rule can cycle, so it gives no ranking at all.

Kenneth Arrow asked whether any rule avoids this. His answer, first published in 1950 as “A Difficulty in the Concept of Social Welfare” and in 1951 in Social Choice and Individual Values, is that none does. With at least three alternatives, no way of combining individual rankings into a social ranking satisfies all of:

• Unrestricted domain: it works for every possible set of individual rankings.
• The Pareto principle: if everyone prefers A to B, so does society.
• Independence of irrelevant alternatives: society’s ranking of A and B depends only on how individuals rank A and B.
• Non-dictatorship: no single individual’s preference decides everything.

Any rule that satisfies the first three has a dictator. The essence of the theorem is that no collective decision rule can guarantee results that are not unfair or paradoxical (Hyderabad 2010), not that efficiency is impossible, nor that collective action is impossible.