Sunday, December 05, 2021

The professor of impossibility

There is scarcely any field of economic theory that has not felt the impact of Kenneth Arrow’s seemingly complex models.

Written by Arunava Sen |
February 27, 2017 12:00:29 am

Kenneth Arrow, who died on February 21, has been described by fellow Nobel Laureate Alvin Roth as “the Albert Einstein of economics”. It would be no exaggeration to say that virtually all modern economic theory rests in some way on his work.

Arrow’s most celebrated contribution is his so-called Impossibility Theorem that appeared in his 1951 book, Social Choice and Individual Values. Arrow was interested in aggregating diverse opinions. For instance, suppose the government wanted to rank all the states in India in terms of the well-being of the country’s citizens. This ranking would draw on several other rankings — per-capita income, the performances of the states in life expectancy and literacy rate. There are an astronomically large number of procedures to arrive at the overall ranking even when the numbers of states and criteria are small. However, Arrow’s Impossibility Theorem proves that there is no procedure that orders these individual rankings into a satisfactory evaluation of well-being in the country. In other words, the theorem proves that meaningful aggregation is impossible from individual choices. Arrow showed that any procedure that satisfies two plausible requirements has to be dictatorial.

The Arrow impossibility result led to the birth of an entirely new field of research for economists and political scientists, called the social choice theory. The theory, whose development also owes a great deal to the work of Amartya Sen, explores the structure of voting procedures from several perspectives. Virtually all models to analyse political competition originate in some way or the other from Arrow’s impossibility result.

Arrow’s overall body of work is staggering in its breadth. One measure of its range is the fact that his Collected Papers published in 1983 run into six volumes. Each volume has a different title — Social Choice and Justice, General Equilibrium, Individual Choice and Uncertainty, The Economics of Information, Production and Capital, and Applied Economics. He wrote fundamental papers in these areas, resolving classical questions and providing a stream of new ideas that are now the bedrock of economic theory.

Arrow won the Nobel Prize in 1972 and is the youngest to have won the prize in economics. Interestingly, his prize citation only refers to his work in general equilibrium, prompting many in the profession to remark that he deserved two or three more Nobel Prizes. In his 1954 paper co-authored with Gerard Debreu, he settled a question in general equilibrium theory that had vexed economists for over a century. An important feature of markets is that there are “spillover” effects. A change in the wage rate will affect consumer incomes and the demand for commodities. That will, in turn, affect the labour market and the wage rate. Using sophisticated mathematical arguments, Arrow and Debreu showed the existence of commodity and factor prices that simultaneously ensure the equality of demand and supply in all markets (Lionel McKenzie arrived a similar result independently). This made the classical competitive model (price-taking agents and the absence of market imperfections) coherent; the model is now referred to as the Arrow-Debreu model.

The competitive model is clearly an idealisation and contemporary economic theory is largely concerned with investigating departures from its assumptions. Two such departures are the introduction of unobservable actions (moral hazard) and unobservable information (adverse selection) in models of interactions between economic agents. For instance, owners of firms are unable to monitor the risk-taking behaviour of managers, even though these actions directly affect the firm’s profitability. Similarly, while selling spectrum, the government is unable to observe the true valuations of the buyers of various units. Theory has sought to understand the efficiency costs of such imperfections and to design ways to ameliorate these costs. Arrow was a pioneer in developing research in these areas. His work guided policy in different parts of the world. His 1963 paper on the moral hazard in health insurance markets, for example, is fundamental to the understanding of the issues involved in devising universal healthcare schemes such as Obamacare in the US.

It is worth mentioning three other disparate areas (among many) where Arrow made foundational contributions. He
developed the standard measure of  risk-aversion (called the Arrow-Pratt measure) in finance. He showed that discrimination against groups could persist even without an explicit “taste” for discrimination due to presence of incomplete information in labour markets. Such discrimination is referred to as statistical discrimination. Finally, together with the Indian economist B.S. Minhas (and two others), he developed a flexible form of a production function that Minhas later used extensively in the context of Indian agriculture.

Arrow continued to research till his last days. In fact, he wrote more than a hundred articles after his Collected Papers were published, branching into areas such as sustainable development and climate change. His intellectual powers were formidable and his speed of thought was legendary. In seminars, he was invariably ahead of the speaker, anticipating arguments and often finding better ones during the seminar itself. He was, however, unfailingly courteous and always generous to colleagues and students. He was a lifelong supporter of liberal and progressive social causes.

It is hard to imagine that there will ever be another Kenneth Arrow.

The writer is professor, Indian Statistical Institute, Delhi Centre, New Delhi

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