# Arrow Kenneth J Essays In The Theory Of Risk-Bearing Definition

## Kenneth J. Arrow, 1921-2017

One of the most prominent economic theorists of the twentieth century, Kenneth J. Arrow has made fundamental contributions to numerous fields, most of then concentrated around Neo-Walrasiangeneral equlibrium theory and welfare economics, of which he can be considered one of the primary architects.

Kenneth Arrow was a thorough New York City product: born and raised in the city, educated at City College (CCNY) and subsequently a graduate student in mathematical statistics at Columbia under Harold Hotelling and Abraham Wald. It was at Columbia that he was nudged towards economics. Finishing his Ph.D coursework in 1942, Arrow's subsequent dissertation was to become a decade-long affair. After a stint during World War II in the Weather Division of the USAF (which resulted in his 1949 article), Arrow returned to Columbia in 1946 and, with dissertation topic still missing, he began taking steps to move into the private sector - passing a series of actuarial examinations and searching for jobs in the insurance industry. During this time period, Arrow attended Jacob Marschak's mathematical seminar at the New School. Hotelling and Wald, exasperated at Arrow's drifting, convinced him to join Marschak at the Cowles Commission in Chicago as a research associate in 1947.

At the Cowles Commission, Arrow absorbed much of the of the Walrasian research programme being set up under Marschak and Koopmans. Arrow wrote an article with Marschak and Harris on inventory policy and contributed a piece to the famous 1951 Koopmans-edited Cowles monograph on activity analysis.

After two highly inspiring years at Cowles, Arrow moved to the RAND Corporation, a research institute associated with the United States Air Force. He would subsequently move on to teach at Stanford - where he has remained until today, save for an interlude at Harvard from 1968 to 1979.

One of the research items at RAND was the then-novel use of game theory to analyze international conflicts and strategy. However, game theory itself presupposed the the parties possessed some sort of utility functions. For parlor games with individual people, this is acceptable, but when dealing with nations as a whole involved in strategic affairs, the question of "America's" utility and "Soviet Union's" utility was an altogether different issue. Thus, Arrow wondered, under what conditions might it be reasonable to assume that collectivities such as nations possessed nicely-behaved utility functions. At long last, a dissertation topic was found, written and subsequently published as the momentous classic, *Social Choice and Individual Values* (1951).

At the heart of Arrow's 1951 dissertation was the presumption that social choice orderings could be derived from individual choice orderings in a simple axiomatic manner. Arrow's astounding conclusion, since then referred to as the "Arrow Impossibility Theorem", was that a certain set of quite acceptable axioms on social choice orderings necessarily implied that there would be a "dictator" (i.e. that a single agent's own preferences over outcomes would dominate everybody else's). This celebrated result has since created quite an research industry in social welfare theory to which Arrow himself has contributed numerous follow-ups (e.g. 1952, 1967, 1973, 1977) which has sometimes led him far afield into the philosophy of ethics and justice (e.g. 1967, 1973, 1977, 1978).

Arrow's dissertation was concurrently published with another formidable piece of Cowles-inspired work: his proof of the First and Second Welfare theorems in general equilibrium theory (1951). Gerard Debreu, then at Cowles, had independently proved the same theorem. Arrow and Debreu began their famous collaboration which culminated in the celebrated "Arrow-Debreu" proof of existence of a competitive equilibrium (1954).

All the while, Arrow had been groping for incorporating uncertainty into a general equilibrium context. In a path-breaking article, Arrow (1953) suggested that a simple procedure would be to consider markets for "state-contingent" commodities. Consequently, he demonstrated that in equilibrium with a full set of state-contingent markets, there would be an optimal allocation of risk However, Arrow, however, noted that a full set of state-contingent commodities might seem too unrealistic. In the same article, Arrow proposed the famous contrivance of "Arrow securities", i.e. securities which paid a unit of account in a particular state and nothing otherwise. Arrow demonstrated that a full set of state-contingent commodity markets could be replaced by a considerably smaller set of Arrow securities which spanned the various possible states - and that, consequently, the optimal allocation of risk would be identical as in an Arrow-Debreu model with a full set of state-contingent markets.

Arrow subsequently turned his attention to a new topic - the issue of "stability" of competitive equilibrium with multiple markets. His interest in this topic was initiated by his work on "D- stability" with McManus (1958) which was followed up by a famous article with Hurwicz (1958) on local stability. Perhaps his most famous achievement in this regard was to provide sufficient conditions (i.e. WARP) for global stability of equilibrium in an article with Block and Hurwicz (1959). His subsequent extensions and clarifications with Hurwicz (1960, 1961) outlined both the achievements and problems of stability theory.

Arrow had, in the meantime, maintained his interest in mathematical programming. With Hurwicz and Uzawa (1961), he produced a famous rather weak constraint qualification (which replaced Kuhn-Tucker's) for obtaining the saddle-point characterization of a local solution to a non-linear programming problem. With Enthoven (1961) he produced several famous results in characterizing optimization problems when both the maximand and constraint functions are quasi-concave.

He moved concurrently into the issues of production and growth. In 1961, with Hollis Chenery, B.S. Minhas and Robert Solow, Arrow introduced the famous "Constant Elasticity of Substitution" (CES) production function. In 1962, he produced two famous pieces on "learning-by-doing", the predecessor of modern endogenous growth theory.

His landmark contributions continued: putting his early experience with actuarial work to good use, Arrow produced his famous 1963 paper on the medical insurance that introduced the concept of moral hazard into economics and announced the dawn of information theory. His 1965 lectures *Aspects of the Theory of Risk Bearing* introduced the famous "Arrow-Pratt" measure of risk-aversion as well as popularized asymmetric information and the terms "moral hazard" and "adverse selection". Much of Arrow's work on moral hazard, optimal insurance and optimal risk-bearing allocations was collected in his famous 1971 book, *Essays in the Theory of Risk- Bearing*. Arrow's work on education and racial discrimination (1972, 1973) have since become famous classroom applications of signalling and screening mechanisms under conditions of asymmetric information.

Arrow had linked the theory of public investment with uncertainty in a famous article with Lind (1970) arguing for the risk-bearing role of government. His concern with mathematical programming and public policy led him naturally to the topic of optimal policy - in particular, to the use of optimal control theory as a guide to resource allocation, inventory policy, public investment, etc. In a series of articles with Mordecai Kurz (1969, 1970), culminating in their famous 1970 book, Arrow presented numerous applications and extensions of the then-rare use of Hamiltonians. Arrow's famous "sufficiency" conditions for an optimum generalized the Mangasarian conditions.

In 1971, Kenneth Arrow and Frank H. Hahn produced their famous treatise/textbook *General Competitive Analysis* (1971) which remained, until recently, a definitive treatment of Walrasian general equilibrium theory. Although research in general equilibrium have since moved in very different directions than those considered by Arrow and Hahn, it was at least partly due to their critical assessments of the treatment of money, uncertainty and stability in a G.E. context which led economists to recast these issues in different light. In 1981-3, Arrow lent his hand as co-editor of the *Handbook of Mathematical Economics*, which once again summarized the state of the art in G.E.

Although Arrow has produced more than a lion's share of groundbreaking contributions to general equilibrium theory, social welfare theory, growth, production, uncertainty, information and optimal public policy, he has not rested on his laurels and continues to produce apace. For instance, his 1979 work with Radner on the theory of "teams" and with Chang on the theory of natural resources (1980) has opened new avenues in the theory of organization and resource allocation. Even more recently, Arrow has teamed up with Hahn again to began tackling the issue of "endogenous uncertainty" (1999).

Kenneth Arrow is perhaps one of the most respected and admired living economists. In many ways, his life is exemplary in that the almost incredible success that have accompanied him have not, in any way, hardened into that arrogance and pettiness so common among professional scholars. By all accounts, Arrow ranks highly among economists and non-economists, orthodox and heterodox, for his scholarly depth, his wide-ranging interests, his personal and intellectual generosity and openness, and his consistent refusal to engage in ideological quibbling. If nothing else, Arrow is positive proof of Pushkin's conjecture that "villainy and genius are two things that can never go together" (Pushkin, 1832).

However, we must also remind ourselves that his achievements were neither the result of wild luck nor were they quickly or cheaply bought by hack work, but rather the outcome of an often painful but nonetheless continuous dedication to the task of the scholar. It is evident throughout his work that Arrow has maintained the highest standards of rigor, avoiding oversimplification and ideological rhetoric, clearly aware of and indeed actively demarcating the limits of applicability of economic theory. In so doing, has achieved and granted to us a far deeper understanding of both economics and the economic process than might otherwise have been possible.

Kenneth J. Arrow shared the Nobel memorial prize in 1972 with another exemplary scholar, John Hicks, for his "pioneering contributions to general economic equilibrium theory and welfare theory". It might be informative to recall that it was Hicks's own *Value and Capital* (1939) that Arrow counts as the most influential work in his early intellectual career.

**Kenneth J. Arrow**, in full **Kenneth Joseph Arrow**, (born August 23, 1921, New York, New York, U.S.—died February 21, 2017, Palo Alto, California), American economist known for his contributions to welfare economics and to general economic equilibrium theory. He was cowinner (with Sir John R. Hicks) of the Nobel Prize for Economics in 1972. Perhaps his most startling thesis (built on elementary mathematics) was the “impossibility theorem” (or “Arrow’s theorem”), which holds that, under certain conditions of rationality and equality, it is impossible to guarantee that a ranking of societal preferences will correspond to rankings of individual preferences when more than two individuals and alternative choices are involved.

In one of his earliest articles, published in 1951, Arrow showed that a competitive economy in equilibrium is efficient. Furthermore, he demonstrated that an efficient allocation could be reached if a government uses lump-sum taxes to transfer wealth and then lets the market work toward equilibrium. One implication of his findings is that, if a government chooses to redistribute income, it should do so directly rather than through price regulations that could hamper the free market. Arrow’s early work on equilibrium still stands as one of the reasons many economists oppose price controls.

After receiving his Ph.D. from Columbia University in 1951, Arrow taught at the University of Chicago (1948–49), at Stanford University (1949–68), and at Harvard University (1968–79). In 1979 he returned to Stanford University as Joan Kenney Professor of Economics and Professor of Operations Research. Arrow became professor emeritus at Stanford in 1991.

Arrow received numerous honours and awards, including the John von Neumann Theory Prize (1986), for notable contributions to operations research and management science, and the National Medal of Science (2004), the highest scientific honour in the United States. He has also been a fellow of several academic societies, including the Econometric Society, the American Economic Association (AEA), the Institute of Mathematical Statistics, and the American Association for the Advancement of Science.

Among Arrow’s major publications are *Social Choice and Individual Values* (1951), *Essays in the Theory of Risk Bearing* (1971), and *The Limits of Organization* (1974).

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