Kenneth Arrow, 1921-2017
|Arrow in 2009 at the San Francisco AEA. For him, the environment was the world's biggest problem. For me it was conflict. We called it quits after a while, and I took this photograph.
Professor Dipak Banerjee, my teacher, introduced me to Kenneth Arrow in 1974, who appeared (much in the manner of Hindu god[desse]s for whom my mother has special reverence) in the form of a small yellow paperback. I acquired Social Choice and Individual Values from Dasgupta and Co. of College Street, and still have it. I was a first year undergraduate. That little book was a repository of the most profound logical thought. I had never seen anyone distill what appeared to be an abstract question in political economy into a theoretical device that cut sharply, and cut deep.
What was the question? Briefly, it was well known from the so-called Condorcet paradox that majority voting could produce nasty cycles in choices, even when the individual preferences involved in that voting process were perfectly reasonable. That led to the question: was there any political system that could “reasonably” aggregate individual preferences? Now think about the question for a second: we know what majority voting is, but there is in principle an infinity of other systems. How could one ever formulate such a problem, let alone attempt to answer it? The very formulation — as axioms placed on an abstract mapping that connected individual preferences to their social counterpart — was sheer genius. But the apparatus was not only beautiful: it could also speak. It argued that under the minimal desiderata placed on the aggregator, there was no way of putting together individual preferences into a satisfactory social ordering; one that was cycle-free.
Arrow had received the Nobel Prize just two years before this encounter with him. He was just 51 years old, by far the youngest Laureate then (or since) in Economics. I ran off to the National Library in Calcutta to dig out the Nobel citation, excitedly anticipating a homage to my beloved Impossibility Theorem. Yet oddly, the The Nobel citation mentions Arrow’s monumental theorem only at the very end, and almost in passing. It focused instead on Arrow’s (and Hicks’s) contributions to general equilibrium theory:
“[Arrow] provided the basis for a radical reformulation of the traditional equilibrium theory. Through this reformulation, which was based on the mathematical theory of convex sets, the general equilibrium theory gained both in generality and in simplicity… The model presented in this paper became the starting point for the major part of further research in this field. Among Arrow's many important contributions should also be mentioned his development of the theory of uncertainty and its incorporation within the frame of general equilibrium theory and, furthermore, his analysis of the possibilities for decentralized decisions in a society where the price system is fixed by the central authority… As perhaps the most important of Arrow's many contributions to welfare theory appears his ‘possibility theorem,’ according to which it is impossible to construct a social welfare function out of individual preference functions.”
This was disappointing as far as my current passion was concerned. But it was also exhilarating: because there was more! (Later, I realized just how much more.) Back I went to Professor Banerjee. I wanted to know why general equilibrium was not just a question of several equations in the same number of unknowns, and what all this was about “the mathematical theory of convex sets.” In response, Dipak-babu helpfully produced another small tome by Gerard Debreu. This was based on the work with Arrow. Though certainly more mainstream this time in its questions, the techniques went way over my 17-year old head. Briefly again, the theory of general equilibrium in its purest form would need to deal with highly interactive systems of demand and supply, to which the simplistic logic of counting equations and unknowns did not apply well. Moreover, one would need to allow for not just single-valued functions describing supply and demand, but for choices that would sometimes be multi-valued, both for consumer and for producers. The resulting search for equilibrium would have to come from a deeper mathematical base. As I slowly began to follow the argument, I realized that this was no mere technicality. The idea was to take the philosophy of Adam Smith to its logical end, to establish the most fundamental conditions under which a general equilibrium could be said to exist, and to display its welfare properties. It was another tour de force in philosophy, of the kind that philosophers rarely would — or more aptly, rarely could — engage in.
It is noteworthy that Professors Arrow and Debreu had very different goals that drove their joint research. For Debreu, the theory of general equilibrium was the philosophical culmination of his work. He personally told me that he considered the two welfare theorems celebrating Smith’s invisible hand to be the crowning glory of economic theory. In contrast, for Arrow, the very same results delineated an idealized frontier beyond which markets ceased to function with full efficiency. The theory of general equilibrium, with its attendant theorems that spoke to the magic of markets, were stakes driven into the ground to mark his explorations beyond. In April 1978, Arrow summarized this view in a lecture delivered at Columbia University:
“[I interpreted] neoclassical economic theory and particularly the then new and rapidly developing discipline of welfare economics as pointing to an ideal efficient economy rather than the actual one, marked both by massive unemployment and by monopolistic distortion… In true Hegelian fashion, capitalist instability and the socialist counterattack seemed to be synthesized: it seemed possible to have an economy that retained much of capitalist drive and initiative and yet gave room for the government to intervene to avoid at least the worst inefficiencies of unemployment and the idling of other resources. I accepted provisionally what seemed to be a widespread consensus in the euphoria of postwar economic growth. The state had an active role to play in maintaining effective demand and in dealing with the many imperfections of the market system revealed by theoretical welfare economics — the overcoming of market failures and monopoly and the realization of economies of scale…
I have spoken of a provisional acceptance. I still felt it important to explore more deeply the possibility that socialism was a superior possibility. I was more aware of the complexities of operation of a socialist system and sought to develop more deeply the theory of such a system. I also sought to explore more fully the criteria for a democratic social organization… [Today,] the apparent pause in economic growth, the crisis in stabilization policy occasioned by the current inflationary threats and realities, and the loss of purpose in redistributional measures all combine to raise anew the question of alternatives to capitalism.”
Yes, Arrow did make a cautious case for socialism. To me, it was particularly interesting that in the end, the case was made not on the positive grounds of inevitable destruction of the capitalist system, but rather on the normative grounds that such a system could be rife with inefficiencies and unequal treatment.
But the similarities with wishy-washy proponents of one “system” over another end there. Arrow was already deeply concerned with the problems posed by the asymmetry of information, and the efficiency with which the market could deal with such asymmetries. There were others who followed similar paths: among them Leonid Hurwicz, Roy Radner, Jacob Marschak, George Akerlof and Michael Spence. Asymmetries of information lay at the heart of theories of organization, contractual relationships, peculiarities in markets such the market for health, and even theories of racial discrimination based on statistical considerations. It is fair to say that Arrow made deep contributions to all these areas of research. In Arrow’s own words:
“My research, even before 1972, moved in directions beyond those cited for the Nobel Memorial Prize. Most of it, in one way or another, deals with information as an economic variable, both as to its production and as to its use. Two 1962 papers studied the efficiency with which the market encourages innovation and the implications of learning by doing for economic growth. In 1963 and later papers, I pointed out that the special market characteristics of medical care and medical insurance could be explained by reference to differences in information among the parties involved. Later themes included a specification of the demand for information and the implications of information as an economic input for returns to scale. Another area of study was the economics of racial discrimination.”
Arrow’s contributions to the economics of information are fundamental. In similar vein, I believe that his research into learning by doing and economic growth is also an attempt to break free of the first-best world so lovingly described in Debreu's A Theory of Value. The short paper on learning by doing is a modern classic, foreshadowing a modern literature on endogenous growth:
“Though doubtless no economist would ever have denied the role of technological change in economic growth, its overwhelming importance relative to capital formation has perhaps only been fully realized with the important empirical studies of Abramovitz and Solow. These results do not contradict the neoclassical view of the production function as an expression of technical knowledge. All that has to be added is the obvious fact that knowledge is growing in time. Nevertheless a view of economic growth that depends so heavily on an exogenous variable, let alone one so difficult to measure as the quantity of knowledge, is hardly intellectually satisfactory…
"The theorems about the economic world presented here differ from those in most standard economic theories; profits are the result of technical change; in a free-enterprise system, the rate of investment will be less than the optimum; net investment and the stock of capital become subordinate concepts, with gross investment taking a leading role.”
As a graduate student, reading all this in the late 70s and early 80s, I viewed Arrow as a thinker who could both ask the deepest questions in economic and political philosophy, and at the same time use mathematical arguments with ease and utility to answer them to a substantive degree. This was someone who could put past accomplishments into perspective, leave them behind, and seek to look beyond to newer and more difficult concerns. Such were the nature of his forays into questions of incomplete information, pervasive externalities, increasing returns and interpersonal equity. There was just one word for it: inspirational. The inevitable reaction was not long in coming: I wanted to be like Ken Arrow. Surprise surprise, that was not destined to happen. But something else did: I had the immense good fortune to become his colleague at Stanford.
I was at Stanford on a job flyout, at the very beginning of 1982. I had been warned about the meeting with Arrow. Apparently, all I had to do was tell him the assumptions of my model and then, before I could get any further, he would proceed to tell me all the results that could conceivably be proved from those assumptions. This was unnerving news. But nothing of the sort happened.
I walked into his office. It was small and cluttered, full of books that went up high. It had a vertical rather than horizontal feel. Arrow himself gave the opposite impression. He was shorter than I had expected, strong, rooted to the ground. The man seemed to be in constant motion. He was both brisk and welcoming. He was wearing brightly colored suspenders, and there was an enormous bicycle helmet on his desk. (I did not then realize that these, along with the perennially flipped pencil at seminars, would be an intrinsic part of my later memory of him.) I began talking about my work. It was a bit of an out-of-body experience. I could see myself talking to him. Arrow listened very closely. There was an intensity of gaze that never wavered, except when he would start speculating, during which he would look up at the ceiling and back to me. He asked questions non-stop. He talked very fast, the words tripping over one another, the tone uneven, the sentences clearly struggling to keep up with the flow of thoughts. He didn’t exactly anticipate my results. But after 15 minutes, I had a second eerie sensation: that I was talking to someone who had thought about my problem for a very long time. This was a weird feeling that I came to associate with Arrow over the next few years.
When we finished our meeting, I summoned up the courage to quote Joan Robinson on Bagicha Minhas and the CES production function: “It is a sad comment on the state of our education that a talented young man be brought from India to be bamboozled like this.” He roared with laughter and said, you can put that up on your office door if you come here. I did.
The interaction with Arrow was pretty much a constant thing. He was teaching History of Economic Thought at the time. I was ploughing through Schumpeter’s book and sat in on a few lectures. I would have thought his favorite economist was Walras. But it wasn’t, it was Cournot, and the choice now makes complete sense. Arrow had long moved away from general equilibrium and he was now firmly in the world of imperfect competition. Cournot’s notion of equilibrium — now Nash’s — spoke to him more forcefully than a price-taking system. (He had, of course, long worried about the existence of the Walrasian auctioneer, who was supposedly setting these prices in the first place.) Arrow invariably spoke of Cournot with great enthusiasm.
Less enthusiasm was shown towards the neo-Ricardians. Once I had mustered up enough courage, I would talk with Arrow with complete freedom, and before long I had told him that I had once spent an entire day with Piero Sraffa at Cambridge, and that he had given me a signed copy of his book (another famous little tome that did the rounds in Calcutta, albeit in somewhat different circles than Professor Banerjee’s.) I told him that I admired the book for its apparent demonstration that the distribution of income across labor income and profit could not be fully pinned down by economics — that some reference to the political system was needed. Arrow looked at me with a mix of irritation and pity: “I’ll get you out of that soon enough.” And of course he did convince me that Production of Commodities By Means of Commodities had an extra degree of freedom in it that generated a fake indeterminacy, though I still harbor a sneaking suspicion that Sraffa was on to something.
Arrow was obviously deeply concerned about the role of information and constantly sought to bring that central idea up in all the sensible contexts he could find. As an example, I was working then on non-convexities in labor markets via nutrition and how this could generate inequality. Arrow impressed upon me the parallel with informational nonconvexities and how this could generate high rates of return to individuals (or holders of hedge funds), thereby resulting in persistent and growing inequality. Of course, interested as he was in formal structure, he delighted in the possibility that the same theoretical setup could potentially be applied to matters as disparate as undernutrition and information. That same delight in formal structure helped me on many other occasions; ranging from game theory to general equilibrium, where he took the time to read my work and was always both critical and encouraging. These interactions have never left me, and many decades later, when I am almost as old now as Ken was when I met him, they continue to inform my own thinking.
When I left for India in 1986, he was initially skeptical, but then immensely encouraging. I think he did feel that the tenure-track system had its constraints; that except in the most creative and courageous of researchers it could foster inhibition and an unquestioned adherence to the status quo. In the end, he understood my reluctance to be in that system and to go back to India instead. He kept in touch with my research and always responded to my occasional questions, and remained a guide.
There are many stories about Ken Arrow. Some are semi-apocryphal. Some we can vouch for. For instance, I have seen him nod off during talks (including one that I gave) and then wake up to ask a remarkable question. And he did flip pencils in seminars, and I have seen him on at least one occasion attend a talk with his bicycle helmet on. Once Doug Bernheim and I, convinced that a speaker was wrong, paid no further attention to the seminar and tried to construct a counterexample together. Arrow somehow knew that that was what we were up to, because after the seminar he walked into Doug’s office (where we were still at it), wrote the required example on the board, did a little jig and walked away — complete with bicycle helmet. But one story possibly is apocryphal, and yet fully sums up the Ken Arrow that I was so fortunate to know. Arrow was in class, teaching. He was speaking fast, running as he always did with his thoughts. Students were frantically taking notes as the disembodied sentences emerged. And then, suddenly: “Stop, stop! That’s all wrong!” As the students frantically began to erase their jottings, he continued: “No, no, not what I said, what I was going to say.”
That was Kenneth Arrow: self-effacing, razor-sharp, a genius; ever ahead of himself, ever ahead of his time.
PS: Some asked me about another Arrow story elsewhere in this blog, which they couldn't find. Here's the link.