The Baroque Instability of Efficiency: Sen's Challenge to Normative Optimisation in Law and Economics — Epoche C2
The criterion under examination Pareto efficiency — the property of a social state that no person can be made better off without making some other person worse off — is the load-bearing normative premise of the law-and-economics tradition. In Richard Posner's presentation of that tradition, legal rules are appraised by whether they maximise wealth, and large stretches of the common law are read as though they had been designed to do so, with efficiency supplying the criterion and doctrinal analysis following as application (Posner, 2011). The edifice is imposing in the manner of the colonial cathedrals of Latin America: a monumental façade raised on soft alluvial ground and on the footings of an earlier building. Amartya Sen dismantled those footings, and his case has three distinct parts, habitually run together. Only the third is about human irrationality; the first two go through even when every individual is perfectly rational. The first part is a remark about the criterion's logical form. Write $x \succeq_P y$ for "the state $x$ is at least as good as the state $y$ by the Pareto criterion", which holds exactly when $u_i(x) \ge u_i(y)$ for every individual $i$ in the society. The relation is reflexive and transitive, so it is a quasi-ordering; what it is not is complete. Two states are comparable only when every individual's welfare moves in the same direction between them, or does not move at all. Let one person gain where another loses, and the criterion says nothing whatever. How often is that? A crude but instructive count: compare two states and treat the sign of $u_i(x) - u_i(y)$ for each of the $n$ individuals as an independent coin toss. The pair is Pareto-comparable only if all $n$ signs agree, and there are two ways for that to happen out of $2^n$ sign patterns, giving a probability of $2 \cdot (1/2)^n = 2^{1-n}$. For two people that is one half; for ten it is $2^{-9}$, about one comparison in five hundred; for a polity of any realistic size it is indistinguishable from zero. Real preferences are correlated rather than independent, so this estimates nothing measured. It makes the structural point exactly: the fraction of pairs on which the criterion is defined falls by half with each additional person whose interests are at stake, and a legal system is precisely the institution that must rule on the cases where interests conflict. Where the criterion does speak, it is undemanding. In a pure exchange economy with a fixed aggregate endowment, the Pareto-efficient allocations form the contract curve, and in the two-person diagram that curve runs from one origin of the Edgeworth box to the other. The allocation that hands the entire endowment to a single agent lies on it, because taking anything away from that agent makes her worse off and there is nobody else to make worse off by giving it to her. Sen states the consequence flatly in Collective Choice and Social Welfare : a state can be Pareto optimal with some people destitute and others surrounded by luxury, provided only that relieving the destitution would require cutting into the luxury (Sen, 1970a). Silence almost everywhere and permission where it speaks — that is the criterion before anyone tries to improve it. The standard repair, and why it fails on its own terms Applied welfare economics has known this since the 1930s, and the standard repair is the compensation test, which is what "efficiency" in law and economics usually means in practice. Nicholas Kaldor's version: the move from $x$ to $y$ counts as an improvement if, starting from $y$, the goods available there could be redistributed so that everyone would be better off than they were at $x$ — whether or not that redistribution actually occurs. John Hicks's version runs the test from the other end: the move is an improvement if, starting from $x$, no redistribution could have made everyone better off than they are at $y$. Posner's wealth maximisation is a monetised form of the same idea. The attraction is plain: such tests are complete exactly where the Pareto relation is silent, because they compare not two points but two whole sets of attainable welfare distributions. Tibor Scitovsky showed in 1941 that the completeness is bought at the price of coherence (Scitovsky, 1941). Let $U_x$ be the set of pairs $(u_A, u_B)$ of welfare levels attainable by redistributing what is available in state $x$ between two people, and $U_y$ the corresponding set for state $y$. Suppose the actual outcome in $x$ is the pair $(10, 40)$ and in $y$ it is $(40, 10)$, and suppose the attainable sets are bounded by $u_A/60 + u_B/50 \le 1$ for $x$ and $u_A/50 + u_B/60 \le 1$ for $y$. Both actual outcomes are attainable, since $10/60 + 40/50 = 0.967$ and $40/50 + 10/60 = 0.967$, each below one. Now apply Kaldor's test in each direction. Starting from $y$, the pair $(12, 44)$ is attainable, because $12/50 + 44/60 = 0.973$, and it beats the actual outcome of $x$ in both coordinates; so $y$ passes the test against $x$. Starting from $x$, the pair $(44, 12)$ is attainable, because $44/60 + 12/50 = 0.973$, and it beats the actual outcome of $y$; so $x$ passes the test against $y$. Each state is "potentially superior" to the other, and the criterion has said both that we should move and that we should move back. The example is not a contrivance. The frontiers cross because the composition of what is produced differs between the states: if $x$ is rich in the good that B values and $y$ in the good that A values, then $x$ is the cheaper place in which to buy welfare for B and $y$ the cheaper place for A, so neither frontier lies wholly inside the other. Nor does anything depend on treating welfare as cardinal. The coordinates may be any increasing representation of each person's preferences, since the only facts used are which points dominate which in both coordinates, and dominance survives separate increasing transformations of the two scales. Scitovsky's own remedy was a double test — require t