Sen's Reassessment of Efficiency and Distribution in Welfare Economics — Epoche C2
Two things "efficient" can mean, and the doctrine that keeps distribution out When a legal rule or a public project is called efficient, one of two precise things is normally meant: that the allocation it produces is Pareto optimal, so that no one can be made better off without someone else being made worse off; or that it passes the Kaldor–Hicks test, so that those who gain could in principle compensate those who lose and still be ahead. Amartya Sen's Collective Choice and Social Welfare (1970) is usually filed as a contribution to the mathematics of preference aggregation, but its effect is to show that neither criterion can carry the normative weight law and economics places on it, and that the reason is not a taste for equality over efficiency but a restriction on the information the criteria may use. The doctrine under attack deserves its strongest statement, because it rests on theorems and not merely on habit. The first fundamental theorem of welfare economics says that where agents take prices as given, markets are complete and preferences are locally non-satiated, every competitive equilibrium allocation is Pareto optimal; the second says that if preferences and production sets are also convex, every Pareto optimal allocation can be sustained as a competitive equilibrium after a suitable reassignment of endowments. Together they license the division of labour that dominates applied welfare analysis: fix the allocation through prices and property rules, and repair the distribution afterwards by transfers. That division depends on a hypothesis buried in the second theorem. The reassignment it invokes must be lump-sum: conditioned on characteristics no agent can alter in response to the transfer itself. Condition transfers instead on income, consumption or transactions — the only quantities a revenue authority can observe — and they change the marginal terms on which people work and trade, the equilibrium shifts, and the conclusion no longer follows. The separability doctrine thus rests on premises that fail in every actual fiscal system. Sen's argument is more radical, because it bites even where the hypotheses hold: it concerns what the criteria being separated are made of. The compensation test, and where it fails on its own terms Pareto optimality ranks almost nothing, since any change harming one person however slightly falls outside its jurisdiction, and essentially every legal reform harms someone. The compensation tests were designed to extend its reach. Kaldor (1939) proposed that a move from state $x$ to state $y$ counts as an improvement if those who gain in $y$ could, out of their gains, compensate the losers and remain better off than at $x$. Hicks (1939) proposed the mirror test: the move counts as an improvement if the losers could not profitably bribe the gainers to stay at $x$. In both, the compensation is hypothetical. The first objection is therefore immediate and is not the interesting one: an unpaid compensation leaves a real loser, so the test certifies as an improvement a change that is not a Pareto improvement at all. The interesting objection is internal, and it was made by Scitovsky (1941). Consider the utility possibility frontier of a state: the set of utility distributions attainable by redistributing the aggregate bundle that state makes available. Kaldor's test asks whether the point realised at $x$ lies inside the frontier through $y$. Nothing prevents two such frontiers from crossing, since a change in production or in prices alters the terms on which one person's utility trades against another's; and if they cross, the point realised at $x$ may lie inside $y$'s frontier while the point realised at $y$ lies inside $x$'s. The test then pronounces $y$ superior to $x$ and $x$ superior to $y$. A criterion that ranks a pair in both directions is not a criterion. Scitovsky's repair — require that the Kaldor test pass and the reverse test fail — restores asymmetry only by adding a condition whose satisfaction is an empirical accident. There is a further point, and it is the one that connects the compensation tests to Sen. Take a redevelopment scheme of the kind routinely appraised in the fast-growing cities of Southeast Asia. Suppose it displaces 200 stallholders, each losing net trading revenue with a present value of 2,000 currency units, so that the aggregate loss is $200 \times 2{,}000 = 400{,}000$, and suppose the land-value uplift plus the consumer surplus of the new users comes to 1,000,000. The Kaldor test passes, with a headline net gain of $1{,}000{,}000 - 400{,}000 = 600{,}000$. But that arithmetic sums money, and money can be summed only if a unit of income counts the same in every hand. Suppose instead that the social value of a marginal unit of income is inversely proportional to the recipient's consumption — the standard iso-elastic weighting with elasticity one — and that the gainers consume ten times what the stallholders consume. The weight on the stallholders is then ten times that on the gainers, and the weighted comparison is $1{,}000{,}000 \times 1 - 400{,}000 \times 10 = -3{,}000{,}000$. The verdict's sign is reversed by a parameter the unweighted test did not announce it was setting. The point is not that the elasticity should be one rather than zero; it is that zero is a choice too, and the appearance of neutrality comes from concealing it. Arrow's theorem: the statement, and what each hypothesis is doing Sen's case that this concealment is structural rather than accidental runs through Arrow's theorem, which must therefore be stated with the hypotheses it actually requires. Let $X$ be a set of social alternatives with $|X| \ge 3$, and let the individuals number $n$, finite and at least two. Each individual has a preference ordering: a complete and transitive ranking of $X$. A social welfare function $f$ assigns to every profile of individual orderings a social ordering, itself complete and transitive. Arrow's conditions, in the fo