Scitovsky Paradox

Scitovsky Paradox in Cost-Benefit Analysis

The Scitovsky paradox is a logical contradiction inside the compensation test of welfare economics, and it appears when the switch between two social states and the switch back again can each be shown to let the gainers compensate the losers. Tibor Scitovsky identified it in 1941, soon after Nicholas Kaldor and John Hicks proposed the test. The test still supplies the welfare justification for cost-benefit analysis of health programmes, so the paradox warns health economists that a positive net benefit does not on its own prove that society has more to share. It needs a change in the quantities of goods available, which makes it a concern for large reallocations of resources, not for small projects valued at fixed prices. QALY-based cost-effectiveness analysis in UK health technology assessment avoids it by a different route, one that sits at the centre of the welfarist and extra-welfarist debate.

How the compensation test can reverse itself

Most policy changes create losers, so a strict Pareto improvement is rare. Feldman's dictionary entry on Kaldor-Hicks compensation explains that Kaldor and Hicks proposed in 1939 that a change is desirable if those who gain could in theory compensate those who lose and remain better off, whether or not the compensation is paid. Fleurbaey's encyclopedia entry on normative economics gives the two versions. Under Kaldor's criterion the gainers could compensate the losers after the change, and under Hicks's the losers could not compensate the gainers.

$$x \to y \text{ passes the Kaldor test if } \exists\, y' \in A(y) \text{ with } u_i(y') \ge u_i(x) \text{ for every } i \text{ and } u_j(y') > u_j(x) \text{ for some } j$$

where $x$ is the initial state, $y$ is the new state, $A(y)$ is the set of allocations that can be reached by redistributing the totals of goods available in $y$, and $u_i$ is a utility function for person $i$. Any ordinal representation of each person's preferences will do, because the test never adds one person's utility to another's.

Feldman describes Scitovsky's model as two people and two goods, with two alternative allocations that are each Pareto efficient given their own totals. Between them the total of one good rises and the total of the other falls. Scitovsky showed that the switch to $y$ and the switch back to $x$ can both be Kaldor-Hicks moves, so the test says $y$ is better than $x$ and $x$ is better than $y$ at the same time. Feldman reports that Scitovsky called this an "absurd result".

Seeing the reversal with utility possibility frontiers

Feldman shows how Samuelson, writing in 1950, analysed the problem with utility frontiers. Each state has a frontier, the outer boundary of the utility combinations that redistribution of its goods could produce. The switch to $y$ passes the Kaldor test if the point reached in $x$ lies inside the frontier of $y$, and the reverse switch passes if the point reached in $y$ lies inside the frontier of $x$. A reversal therefore needs the two frontiers to cross.

Hausman's encyclopedia entry draws the consequence for cost-benefit analysis. Because each of two states can be a potential Pareto improvement over the other, such an improvement does not imply a larger economic pie to share.

Scitovsky's double criterion and later repairs

Scitovsky's remedy, quoted by Feldman, was a two-edged test. A new situation counts as better only if redistribution in it could make everybody better off than in the initial situation, and redistribution in the initial situation could not reach a position that everybody prefers to the new one. The switch must pass the Kaldor test and its reversal must fail.

The double criterion removes the direct contradiction but leaves other problems. Feldman points out that it can rank a redistributed version of $y$ above $x$, although that version has the same possibility frontier as $y$, which is no more attractive than that of $x$. Samuelson argued that the comparison should depend on all the positions possible in each situation, so that $y$ is declared better only when its utility frontier lies entirely outside that of $x$. Fleurbaey adds that combining the Kaldor and Hicks criteria does not prevent intransitive social judgements.

Money measures fare no better. Feldman describes Boadway's 1974 example in which two efficient allocations share the same totals of goods, yet the sum of compensating variations is positive in both directions, and he concludes that this money test is inconsistent whenever the Kaldor-Hicks test is, and sometimes when it is not. Culyer and Evans, answering Pauly's 1994 case for judging health services by adding up consumers' surpluses, quoted Blackorby and Donaldson's conclusion that there is an overwhelming case against that practice, a warning for any health study that sums willingness to pay across people.

Worked example with health care and other consumption

All numbers in this example are illustrative. A region can use its resources in one of two ways. State X produces 10 units of health care and 2 of other consumption, and state Y produces 4 units of health care and 8 of other consumption. Person A, who has a long-term condition, values a unit of health care at twice a unit of other consumption, and person B values it at half. Their utilities are written $u_A = 2H + C$ and $u_B = H + 2C$, where $H$ and $C$ are the units of health care and other consumption each person holds.

1. Starting allocations. In state X, A holds 2 units of health care and B holds 8 units of health care plus 2 of consumption, so $u_A = 2 \times 2 = 4$ and $u_B = 8 + 2 \times 2 = 12$. In state Y, A holds 4 units of health care and 6 of consumption and B holds 2 of consumption, so $u_A = 2 \times 4 + 6 = 14$ and $u_B = 2 \times 2 = 4$.

2. Both allocations are efficient. The only trade both would welcome sends health care to A and consumption to B, but A has no consumption to offer in state X and B has no health care to give up in state Y.

3. The switch to Y passes the Kaldor test. A gains (4 rises to 14) and B loses (12 falls to 4). Redistributing the goods of state Y so that A holds 3 units of health care and B holds 1 unit of health care and all 8 units of consumption gives $u_A = 2 \times 3 = 6$ and $u_B = 1 + 2 \times 8 = 17$. Both exceed their levels in state X, so the gainer could compensate the loser.

4. The switch back to X also passes. Now B gains and A loses. Redistributing the goods of state X so that A holds 8 units of health care and B holds 2 units of health care and 2 of consumption gives $u_A = 2 \times 8 = 16$ and $u_B = 2 + 2 \times 2 = 6$. Both exceed their levels in state Y, so this switch passes the same test.

5. The frontiers cross. The highest utility A can reach is 2 × 10 + 2 = 22 in state X and 2 × 4 + 8 = 16 in state Y, while for B the figures are 10 + 2 × 2 = 14 and 4 + 2 × 8 = 20. Both frontiers pass through the point where A has 12 and B has 8. The frontier of Y lies outside when A has less than 12 and the frontier of X when A has more, so each actual allocation sits inside the other state's frontier.

6. Repairs. Because each switch and its reversal both pass, the Scitovsky double criterion ranks neither state, and with crossing frontiers the Samuelson criterion is silent too. If the goods of state Y were shared as in step 3, giving utilities of 6 and 17, the double criterion would rank that version of Y above X, since no redistribution in state X gives B more than 14.

The verdict on a health-care-heavy against a consumption-heavy use of resources thus depends on how the goods happen to be shared. Here the compensation test endorses whichever switch is proposed, and only an explicit distributional judgement, such as a social welfare function, can settle the choice.

What the paradox means for health appraisal

Feldman notes that money gains and losses can be aggregated without logical problems when there is only one good or when relative prices do not change, and that invariant relative prices are the standard underlying assumption of cost-benefit studies. The risk grows with changes large enough to shift relative prices and the mix of goods, such as a major reallocation of national resources between health care and other spending.

Claxton, Walker, Palmer and Sculpher treat transfers between the health care system and the wider economy as a form of compensation test. They argue that a sequence of marginal decisions can add up to non-marginal effects, and that unless compensating transfers are made the implied reallocation between sectors may not be socially desirable, particularly if an explicit welfare function cannot be completely specified.

Just, Schmitz and Zerbe show that when two states lie on one fixed production possibilities frontier and only normal goods are present, a reversal requires the luxury good for one person to be the necessity for the other, conditions they regard as highly unlikely in reality. The example above reaches a reversal with linear preferences only because its allocations sit at corners, where the relevant marginal rate of substitution is B's in state X and A's in state Y. They suggest that requirement explains why the paradox is rarely if ever found in empirical benefit-cost studies, so its lesson for health economists is mainly about what a positive net benefit can and cannot be taken to mean.

Fleurbaey notes that weighted sums of willingness to pay, or consistent social welfare functions, avoid the problems of the compensation tests. In HM Treasury's Green Book (2026) distributional weights give a higher social value to benefits for lower-income households, with weighted estimates presented alongside unweighted ones. Equity-weighted analysis and distributional cost-effectiveness apply the same idea to health gains.

Welfarist and extra-welfarist readings

The paradox belongs to the welfarist tradition, in which social welfare is built from individual utility. Brouwer, Culyer, van Exel and Rutten argue that the potential Pareto criterion compromises the Pareto principle, because applied cost-benefit analysis assigns an equal unit weight to each monetary gain or loss whoever receives it. In extra-welfarism, they explain, uncompensated changes can be judged improvements or deteriorations by appeal to additional ethical criteria, such as the distribution of need.

The NICE technology appraisal manual (PMG36) justifies cost-effectiveness analysis by NICE's focus on maximising health gains from a fixed NHS and personal social services budget (section 4.2.15), and its reference case gives an additional QALY the same weight regardless of who receives it, except in specific circumstances (table 4.1). Summing QALYs across people works like Feldman's one-good case, since gains and losses share a single unit and the ranking cannot reverse, at the price of an open comparison between people through the decision to count each QALY equally. Culyer and Evans add that it has never been clear why compensation that does not in fact take place should influence the ranking of states.

How the paradox relates to neighbouring ideas

The Scitovsky paradox is a flaw in the Kaldor-Hicks test, sometimes called Kaldor-Hicks efficiency. The Boadway paradox is the parallel flaw in summed compensating variations, possible even with fixed totals. The paradox of voting concerns cycles produced by majority voting over individual orderings, and Arrow's impossibility theorem shows that no aggregation rule meeting a few mild conditions avoids such failures in general, while the Scitovsky paradox concerns one test that approves a change and its reversal.

Limitations and common misreadings

The paradox does not show that cost-benefit analysis is always inconsistent. It needs changes in the totals of goods and preference patterns that Just, Schmitz and Zerbe consider unusual. Fleurbaey also notes that if compensation is actually paid the Pareto criterion alone suffices, so the tests matter only when the losers remain losers.

The logical problem is also separate from the ethical objections. Hausman notes that willingness to pay depends on wealth, so cost-benefit analysis weights the preferences of the rich more than those of the poor, whether or not a reversal is present. A health study valued by equivalent variation or compensating variation should state which measure it used and whether relative prices are assumed fixed.

Sources

• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• Claxton K, Walker S, Palmer S, Sculpher M. Appropriate perspectives for health care decisions. CHE Research Paper 54. York: Centre for Health Economics, University of York; 2010. https://www.york.ac.uk/media/che/documents/papers/researchpapers/rp54_appropriate_perspectives_for_health_care_decisions.pdf
• Culyer AJ, Evans RG. Mark Pauly on welfare economics: normative rabbits from positive hats. Journal of Health Economics. 1996;15:243-251. Abridged reprint in Cookson R, Claxton K, editors. The Humble Economist. York: University of York and Office of Health Economics; 2012. Chapter 4. https://www.york.ac.uk/media/che/documents/books/The%20Humble%20Economist%20corrected%20PDF%20version.pdf
• Feldman AM. Kaldor-Hicks compensation. In: Newman P, editor. The New Palgrave Dictionary of Economics and the Law, volume 2. London: Macmillan; 1998. p. 417-421. https://benjaminmoll.com/wp-content/uploads/2024/10/Feldman_Kaldor_Hicks.pdf
• Fleurbaey M. Normative economics and economic justice. Stanford Encyclopedia of Philosophy. First published 28 May 2004; substantive revision 6 May 2026. https://plato.stanford.edu/entries/economic-justice/
• Hausman DM. Philosophy of economics. Stanford Encyclopedia of Philosophy. First published 12 September 2003; substantive revision 4 September 2018. https://plato.stanford.edu/entries/economics/
• HM Treasury. The Green Book (2026). Updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Just RE, Schmitz A, Zerbe RO. Scitovsky reversals in benefit-cost analysis with normal goods. Journal of Benefit-Cost Analysis. 2013;4(3):411-413 (erratum to their 2012 paper). https://doi.org/10.1515/jbca-2013-0034
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Section 4.2.15 and table 4.1. https://www.nice.org.uk/process/pmg36