Kenneth Arrow’s immense contributions

One can feasibly claim that modern economics is but a series of footnotes to his works


Kenneth Arrow’s work on general equilibrium was an exact mirror image of his work on social choice. Photo: AP
Kenneth Arrow’s work on general equilibrium was an exact mirror image of his work on social choice. Photo: AP

Kenneth Arrow died on 21 February. Philosopher and mathematician Alfred Whitehead once quipped that Western philosophy is but a series of footnotes to Plato. One can claim that modern economics is but a series of footnotes to Arrow’s works.

His first major breakthrough was the impossibility theorem (he proved it as part of his PhD work). This eponymous theorem has been described as one of the most profound results in social sciences. There are two ways of understanding it. One is, you open a social choice textbook and try to follow mathematical gymnastics. Not everyone’s cup of tea. The other way is to forget the technicalities and look at the broad picture. The technicalities will follow. Arrow’s impossibility theorem essentially answers the following question. Let us say there is a group of individuals: a club, board of directors, government committee, electorate of an entire country, whatever it may be. Differences among the group are reconciled by a voting procedure.

Is there a way of arriving at group decisions that are as “rational” and “coherent” as a five-year-old child’s would be? Rather surprisingly, the answer is no, not even in principle. For every conceivable voting scheme, there are voting patterns where the choice of a five-year-old will demonstrably be superior to the group decision. Let us see a few examples.

Have you seen a child getting deadlocked between various flavours of ice cream? The social choice term for no deadlock condition is “completeness”. Individual choices are usually complete. But imagine a parliament that has two houses. Law-making requires consensus. One of them is dominated by the ruling party, the other by the opposition. And you have a deadlock. This is more or less the actual story of the current Indian parliament.

Let us say a child, when confronted with a red and a blue balloon, chooses the red one. If we include a green balloon in the box, will her choice flip to the blue balloon? Extremely unlikely. Why should introduction of an irrelevant alternative change her choice? But think of the 2000 US presidential election. The presence of candidate Ralph Nader, who did not have the slightest chance of making it to the White House, did affect the national choice between George Bush and Al Gore.

Arrow’s philosophical point is that group decisions do not inherit the rationality properties of individual choice, no matter how clever the voting scheme is. His actual technical conditions are chosen to make this point precise.

Arrow’s work on general equilibrium was an exact mirror image of his work on social choice. His impossibility result identifies a set of innocuous axioms and shows them to be inconsistent. His general equilibrium model lays down a set of conditions so general and with so many moving parts that one may think it should be trivial to spot inconsistency. In a grand climax, Arrow (together with Gérard Debreu) then shows that the model is in fact consistent.

Imagine a musical group sans a music director. Members of the group never meet in person. Each member practises in isolation, improvises on his tune separately. But their performance on stage is perfectly synchronized. Remarkable but unlikely, you might say. Now substitute economy for the musical group. There are billions of individuals making intricate consumption, production, saving and investment decisions. Somehow their plans and decisions hang together: markets clear. This is so remarkable that except for perceptive observers such as Adam Smith, few even noticed it. Before Arrow, consistency of the market mechanism was a vague conjecture. After Arrow, it became a precisely defined and proven theorem.

Speaking of this work, Nobel laureate Robert Aumann has remarked: “Arrow’s mathematical work falls into that lasting category of things that are very far from obvious but are still basically simple—simple, not in the sense of easy, but in the sense of clean, like the best modern architecture.”

Much of Arrow’s work deals with fundamental issues. One area where his work has immense policy significance is information economics. Is there a case for macro-prudential regulation in financial markets? Should participation in a collective insurance scheme be made compulsory? Has securitization made the world more risky? The answer to each of these questions boils down to whether a set of broad conditions identified by Arrow is met in practice. His work is so fundamental that in any debate on information economics, all the parties end up implicitly invoking his arguments.

Arrow remains the youngest recipient of the Nobel Memorial Prize in economic sciences. At least four of his students were awarded the Nobel prize as well. He made fundamental contribution in half a dozen sub-fields of economics. Most of his papers launched new research projects. Arrow was surely an intellectual par excellence, arguably the greatest economist of the 20th century.

Economist Yew-Kwang Ng has recounted an interesting anecdote about young Arrow. In his boyhood, Arrow was not only precocious but also mischievous. His mother used to ground him in a room to constrain his activities. The punishment failed to have the desired deterrent effect. On investigation, it was found that the room also contained a copy of an encyclopaedia and Arrow would immerse himself in reading it. So the “punishment” was in fact a reward in disguise. His mother (she was Arrow’s mother after all!) then devised an unusual punishment. Whenever he did something cheeky, he was sent to play outside. The trick worked.

In a sense this anecdote captures all the key themes of Arrow’s intellectual journey. A preference for learning, the importance of incentives and a subtle departure from cookie-cutter rationality. His was an extraordinary life devoted to learning and expanding the horizon of human knowledge.

Avinash M. Tripathi is an independent contributor on economic issues.

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