Kenneth Joseph Arrow (b. 1921)

Kenneth J. Arrow’s name is associated with two famous components of microeconomic theory: Arrow’s (im)possibility theorem of social choice and the so-called Arrow- Debreu model. The publication of Arrow’s impossibility theorem in 1950 marks the birth of a whole scientific field (social choice theory) that goes well beyond economics and overlaps with several other subjects, including political science, political philosophy and social ethics.

Arrow’s scientific life is well documented. There are, among others, two papers authored by Arrow himself (1992, 2009), a number of interviews - for example, Kelly (1987 [2010]) and Feiwel (1987a, 1987b) - and the prefaces to the six volumes of his col­lected papers as well as the short historical introductions to many individual papers in this collection (Arrow 1984a, 1984b, 1984c, 1984d, 1985a, 1985b).

In the book edited by Szenberg (1992), eminent economists were asked to write a 20-page essay on their life philosophy. Arrow responded with a beautifully written piece. We learn there about his interest in literature (Proust, Joyce, Kafka, Shelley and Keats) and in music (Wagner). His interest or even passion for abstraction had implications for his taste regarding the fine arts, which is also reflected in his admiration of Mondrian.

Life

Arrow was born in New York on 23 August 1921, into a wealthy family. He told Feiwel (1987b): “my first ten years were spent in considerable affluence, my next ten years in very considerable poverty”. After high school, he attended the City College of New York because it was then an institution that did not charge tuition fees. He gradu­ated in 1940 with a degree of Bachelor of Science in Social Science, but with a major in mathematics. During his high school and college years he was particularly interested in mathematics and more specifically in mathematical logic.

Back in Columbia, he had difficulties finding a dissertation topic. He contemplated the idea of redoing Hicks’s Value and Capital from a more rigorous standpoint, using second-order properties and so on. Although he heard about the problem of the exist­ence of a general competitive equilibrium, it was not from Wald who had made major advances in this area while he was still in Vienna before the Second World War (see Weintraub 1985; Duppe and Weintraub 2014). However, Wald discouraged him from working on this question because of its difficulty. In the spring of 1947, Arrow got a posi­tion at the Cowles Commission then located in Chicago. There he met Jacob Marschak, Tj ailing Koopmans, Leonid Hurwicz (with whom he collaborated later on several major projects), Lawrence Klein and others. He spent the summer 1948 at the RAND Corporation. Summers at RAND are now famous among laymen since the publica­tion of Nash’s biography (Nasar 1998). It is at RAND that Arrow had the definite idea leading to his impossibility theorem. It is also at RAND that he met a number of game theorists (Shapley, Nash and von Neumann), applied mathematicians (Bellman), and statisticians (Blackwell).

In Breit and Hirsch (2009), Arrow identifies his three major contributions: social choice, general equilibrium, and asymmetric information. A fourth is added, even if it is based on a short paper: the abstract approach to revealed preference theory.

Social Choice

Surprisingly, Arrow’s interest in collective decision-making was prompted by his desire to write a PhD dissertation whose objective (“grandiose” as he wrote in Arrow 1984a) was to redo Hick’s Value and Capital. One of the points he wished to improve upon concerned the theory of the firm. In Hicks, there is a single owner, which does not apply to most actual firms, in particular large ones. In a firm with several owners, even if we can accept that they have a common objective, maximization of profits, they may have different expectations of the future, and, consequently, different preferences regard­ing, say, investments. Decisions should then be taken using majority rule, possibly weighted according to shares. Arrow quickly discovered the Condorcet paradox and, even though he was fairly sure that this was known, he did not know that it was attrib­uted to Condorcet. He considered at that time that this type of intransitivity/cycle was a nuisance and he did not pursue his research until, in the winter of 1947-48, he had the idea that, if we assume some homogeneity in individual preferences - for instance, on the basis of a political spectrum left-right - we could obtain a transitive collective ranking.

To simplify the presentation, assume that everything is finite, both the set of options (social states, candidates, allocations, and so on, depending on the context) and the set of agents - this is a crucial assumption that is needed not only to make the proofs simpler: with an infinite set of agents, the theorem is no longer true. Each agent is rational in the sense that he or she can rank the options with possible ties (indifferences). Given that each agent has a ranking of the options, the Arrovian social choice problem is to deter­mine a (unique) social ranking. Rules or procedures that associate a social ranking with a list of individual rankings (one ranking per individual) are called Arrovian social welfare functions, or social welfare functions for short. The list of individual rankings will be called a “profile”. Of course, you can imagine that the social ranking is systematically the ranking of a specific individual. This would be a dictatorial social welfare function. Arrow (1963) imposes four conditions on the social welfare function.

The first condition states that individuals can have any ranking, so that there is no super-rationality condition as in the case of Black’s single-peakedness. Admitting ties, this means that, for three options, one has 13 rankings (six rankings without ties, three with two top options, three with two bottom options and one where the three options are tied).

The second condition concerns the level of information that should be used in the aggregation process. This information is purely ordinal, or, more precisely, binary within individuals and non-comparable across individuals. Given two profiles and two options, if we have the same binary information regarding these two options in the individual rankings, then the binary collective relation regarding the two options must be identi­cal. For instance, consider options a and b, if individual 1 prefers a to b in the first and the second profile, individual 2 is indifferent between a and b in the first and the second profile, individual 3 prefers b to a in the first and the second profile, and so on, then the social/collective preference between a and b must be identical, say, a is collectively pre­ferred to b for both profiles. This is obtained in case, say, individual 1 ranks a first and b last (supposing we have 20 options) in the first profile and a third and b fourth in the second profile. What is taken into consideration is only the fact that individual 1 prefers a to b in both profiles. This is independent of the fact that there are 18 options ranked between a and b in the first profile and none in the second profile. This is the reason why Arrow called this condition “independence of irrelevant alternatives”.

The third condition is very simple. It says that the collective preference must respect unanimity. If in a profile every individual agrees that some option is better than some other option, the social preference must be identical to the unanimous preference. In spite of appearing innocuous, this condition has an important conse­quence. Given a very small amount of diversity among the individual rankings, this condition excludes constant social welfare functions. Such constant functions would give the same collective ranking for every profile, that is, the social ranking would be independent of the agents’ preferences. In the 1951 version of the theorem, this unanimity (sometimes called Pareto) condition was the consequence of a condition of non-imposition associated with independence of irrelevant alternatives and some monotonicity assumption.

Loosely speaking, the fourth condition excludes that the social preference be system­atically the preference of a specific individual; it excludes dictatorship. More precisely it says that there is no individual (a dictator) whose “strict” preference becomes the social “strict” preference. There is an indeterminacy in case the dictator is indifferent.

The theorem states that if there are at least two agents and three options, there is no social welfare function satisfying the four conditions. With three options and three indi­viduals there are 13²¹⁹7 possible social welfare functions, a really huge number. The four conditions are sufficient to annihilate all these functions: no function among this huge number satisfies the four conditions.

In Arrow (1951a), there is also a discrete version of single-peakedness and a possibility theorem for majority rule (majority rule is then a social welfare function satisfying the last three conditions).

General Equilibrium

The Walrasian description of an economy as a system of simultaneous equations rep­resenting the consumers’ demands and producers’ supplies with the equilibrium condi­tion that for each market demand is equal to supply largely left aside the fundamental problem of existence. A prevalent idea was that it was sufficient to have a number of equations equal to the number of unknowns. At the end of note XXI in the mathemati­cal appendix to Marshall’s Principles of Economics (1920: 856) we read: “Thus, however complex the problem may become, we can see that it is theoretically determinate, because the number of unknowns is always exactly equal to the number of the equations which we obtain.” According to Schumpeter (1954: 1006), Walras did not believe this. Schumpeter writes: “Of all the unjust or even meaningless objections that have been leveled at Walras, perhaps the most unjust is that he believed that this existence question is answered as soon as we have counted ‘equations’ and ‘unknowns’ and have found that they are equal in number” (ibid.). The best we had at the beginning of the 1940s were Wald’s papers in German, in particular Wald (1936 [1951]). Wald was one of Arrow’s teachers at Columbia, but his interests had switched to mathematical statistics. From Wald, Arrow learnt that the existence problem was very difficult and, given Wald’s mathematical expertise, this was rather discouraging. Arrow’s visit to RAND in 1948 was crucial because it is there that he learnt a great deal about convex analysis, in par­ticular results about hyperplanes separating convex sets that would reveal fundamental tools for the second theorem of welfare economics. What was also crucial was Nash’s very short paper (Nash 1951), which uses Kakutani’s fixed point theorem to prove the existence of an equilibrium point in n-person games. While he was working on this exist­ence problem, Arrow learnt that Debreu had obtained essentially similar results. Arrow and Debreu then decided to write a joint paper (Arrow and Debreu 1954). At about the same time, McKenzie also had very similar intuitions (see McKenzie 2002; Duppe and Weintraub 2014).

Although recent advances in mathematics were important for this result, Arrow and Debreu (1954) cannot be evaluated only on the basis of the use of the relevant technical tools. In fact Arrow and Debreu - in this paper but also in two previous papers about the so-called welfare economics theorems (Arrow 1951b; Debreu 1951) - created the basic framework that would pervade modern microeconomic theory.

To simplify consider an exchange economy. The supply side is given by the initial endowments of agents. Agents have a budget set, defined in terms of a given vector of prices and their initial endowments, and maximize their preference relation over this budget set. Formal conditions of convexity and continuity guarantee that there is a vector of quantity of goods (not necessarily unique) that maximizes this preference relation. This defines the individual demand (a correspondence or with more restrictive conditions a function). A simple operation of vector addition (one can also define the addition of sets of vectors) gives the (global) demand. Subtracting the global supply (the sum of the individual initial endowments), one gets the excess demand correspondence. The equilibrium existence question is then to find a vector of prices such that the nul- vector belongs to the excess demand correspondence.

The relations between equilibrium allocations and Pareto optimality are known as the classical theorems of welfare economics. An allocation is a vector composed of vectors of quantities of goods, one vector for each agent, such that their sum is equal to the sum of the agents’ initial endowments. If the allocation corresponds to a zero of the excess demand function, given an equilibrium vector of prices, it is said that it is an equilibrium allocation. An allocation is Pareto optimal if there is no other vector such that each individual finds his vector component at least as good and one individual finds his vector component better. The first welfare economics theorem states that an equilibrium allocation is Pareto optimal. The second welfare economics theorem states that, given a Pareto optimal allocation, one can find a vector of prices and a distribution of total resources as initial endowments that will guarantee that the given allocation is an equilibrium allocation. Although the proof of the first theorem is elementary, to prove the second one, one has to use the mathematical results about hyperplanes sepa­rating convex sets.

Arrow and Hahn (1971) offers a detailed view of Arrow’s work in general equilibrium theory with further developments on stability.

The Economics of Information

The Arrow-Debreu model of a competitive economy of 1954 does not consider uncer­tainty. Of course, both Arrow and Debreu knew that this was a limitation of their analy­sis. As mentioned previously, Arrow’s interest in social choice was indirectly prompted by problems related to uncertainty in the decision-making process in the theory of the firm. In a paper published in French (Arrow 1953 [1963-64]), Arrow proposed to deal with uncertainty via contingent contracts. Arrow (1953 [1963-64]) is often considered as the foundational paper for the theory of incomplete markets and the theory of finance. The following description is in Debreu (1959: 98): “A contract for the transfer of a com­modity now specifies, in addition to its physical properties, its location and its date, an event on the occurrence of which the transfer is conditional.” The purpose was to have a treatment of uncertainty that was not reducible to probabilities and would allow a straightforward generalization of the results of Arrow-Debreu (1954).

A number of economists were not satisfied by this treatment of uncertainty, in spite of its elegance. Arrow was one of them. At the beginning of the 1960s, he was asked by the Ford Foundation to survey medical economics from a theoretical point of view. In his historical introduction to Arrow (1963) he explains that he “started the survey in a conscientious catalogue fashion, but felt that the whole study lacked focus” (Arrow 1985b: 15). He knew that a “key component” was asymmetric information, between the physician and the patient, as well as between both of them and the insurer. Arrow noted that health insurance creates an incentive to spend more freely than neces­sary, a phenomenon known as “moral hazard”. In 1963 the question of asymmetric information was hardly mentioned in microeconomic theory. Nowadays, it is one of the major topics of the microeconomic textbooks as exemplified by Mas-Colell et al. (1995) and Jehle and Reny (2001). Arrow played an instrumental role in this development.

Revealed Preference Theory

Feiwel (1987a) rightly mentions that Arrow’s work on choice theory is underappreci­ated. Arrow himself is rather modest saying, “It has certainly been surpassed by Sen and Richter” (Feiwel 1987a: 223). However, one may wonder whether without his short paper (1959) the set-theoretic treatment of revealed preference theory would have seen this major development that culminates (at this time) in Bossert and Suzumura (2010).

In standard microeconomic theory of the consumer, the basic concept is the agent’s (ordinal) utility function (or her preference in the Arrow-Debreu model). Given her initial endowment and the price system, the agent maximizes this utility function over her budget set. Given appropriate assumptions, we know that there is a point (a vector of quantities of each good) in the budget set that does maximize the function. Making further assumptions (for instance, strict quasi-concavity) this point can be unique. We have then a demand function that associates this maximizer to a price-vector and income (the income is the value of the initial endowment given this price-vector). In a fundamen­tal paper, Samuelson (1938: 71) proposed to inverse the analysis “to develop the theory of consumer’s behavior freed from any vestigial traces of the utility concept.” In his 1950 paper, Samuelson stated the weak axiom of revealed preference in the following way: “If at a price and income of situation A you could have bought the goods actually bought at a different point B and if you actually chose not to, then A is defined to be ‘revealed better than’ B. The basic postulate is that B is never to reveal itself to be also ‘better than’ A” (Samuelson 1950: 370). The idea was then to go from a demand function (that would satisfy this axiom and other conditions) to a utility function having all the properties necessary for its maximization. This is the integrability problem. This problem is quite demanding from a mathematical point of view (see Chipman et al. 1971).

Arrow’s purpose was to develop revealed preference theory in a more general frame­work than the framework of consumer theory. He defines a binary relation of revealed preference in terms of choice (a choice function is defined over subsets of a general set and selects, for each subset in its domain, some options belonging to the subset). Then an option a is revealed preferred to an option b if there is a subset in which a is selected and b is not. The weak axiom of revealed preference asserts that if a is revealed preferred to b, then there is no subset to which a belongs in which b is chosen. Arrow introduced other consistency conditions on choice functions and on the binary relation of revealed prefer­ence. In particular, one of these consistency conditions on choice functions was shown to be equivalent to the weak axiom of revealed preference. It states that if A and B are two subsets in the domain of the choice function and if A is included in B the options selected in the larger subset that also belongs to the smaller one are the options that are selected in the smaller subset. To give a clarifying illustration, consider a department store that includes a food department. You are doing your weekly shopping. What you are buying includes food and, say, household products, clothes and toilet products. Let us suppose that you have to leave all these products and are asked to do your shopping again but only within the food department. Then you will buy exactly the same food products as when it was possible to visit all the departments. Arrow suggested that on the basis of a choice function you can derive a binary relation of (normal) preference by saying that a is chosen in a subset that is limited to a and b if and only if a is at least as good as b. Arrow then demonstrated that if the choice function satisfies the weak axiom of revealed preference, the binary relation derived from this choice function is a complete pre-order (a complete ranking with possible ties). The complete pre-order is said to rationalize the choice function, and one can, loosely speaking, consider that this rationalizability cor­responds in the set-theoretic framework to the integrability a la Samuelson.

Other Works

There are many other important contributions by Arrow. These include his works on growth theory, capital and production theory, technical progress theory (learning by doing), stability of equilibrium, risk analysis (Arrow 1970), multicriterion decision­making (Arrow and Raynaud 1986), public investments (Arrow and Kurz 1970), applied mathematics (quasi-concave programming), health economics, and so on.

Maurice Salles

See also:

Jeremy Bentham (I); Abram Bergson [Abram Burk] (I); James M. Buchanan (I); Marie-Jean-Antoine-Nicolas Caritat de Condorcet (I); Gerard Debreu (I); Formalization and mathematical modelling (III); General equilib­rium theory (III); John Richard Hicks (I); Lausanne School (II); John Stuart Mill (I); John Forbes Nash Jr (I); Arthur Cecil Pigou (I); Public choice (II); Paul Anthony Samuelson (I); Amartya Kumar Sen (I); Social choice (III); Uncertainty and information (III); Utilitarianism and anti-utilitarianism (III); Welfare economics (III).

References and further reading

Arrow, K.J. (1950), ‘A difficulty in the concept of social welfare’, Journal of Political Economy, 58 (4), 328-46. Arrow, K.J. (1951a), Social Choice and Individual Values, 2nd edn 1963, New York: Wiley.

Arrow, K.J. (1951b), ‘An extension of the basic theorems of classical welfare economics’, in J. Neyman (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA: University of California Press, pp. 507-32.

Arrow, K.J. (1953), ‘Le role des valeurs boursieres pour la repartition la meilleure des risques’, in Econometrie, Colloques Internationaux du Centre National de la Recherche Scientifique, vol. 11, pp. 41-7, Paris: Editions du CNRS, English trans. (1963-64), ‘The role of securities in the optimal allocation of risk-bearing’, Review of Economic Studies, 31 (2), 91-6.

Arrow, K.J. (1959), ‘Rational choice functions and orderings’, Economica, 26 (102), 121-7.

Arrow, K.J. (1963), ‘Uncertainty and the welfare economics of medical care’, American Economic Review, 53 (5), 941-73.

Arrow, K.J. (1970), Essays in the Theory of Risk-Bearing, Amsterdam: North-Holland.

Arrow, K.J. (1984a), Collected Papers 1: Social Choice and Justice, Oxford: Blackwell.

Arrow, K.J. (1984b), Collected Papers 2: General Equilibrium, Oxford: Blackwell.

Arrow, K.J. (1984c), Collected Papers 3: Individual Choice under Certainty and Uncertainty, Oxford: Blackwell. Arrow, K.J. (1984d), Collected Papers 4: The Economics of Information, Oxford: Blackwell.

Arrow, K.J. (1985a), Collected Papers 5: Production and Capital, Oxford: Blackwell.

Arrow, K.J. (1985b), Collected Papers 6: Applied Economics, Oxford: Blackwell.

Arrow, K.J. (1992), ‘“I know a hawk from a handsaw”’, in M. Szenberg (ed.), Eminent Economists: Their Life Philosophies, Cambridge: Cambridge University Press, pp. 42-50.

Arrow, K.J. (2009), ‘Kenneth J. Arrow’, in W. Breit and B.T. Hirsch (eds), Lives of the Laureates: Twenty Six Nobel Economists, Cambridge, MA: MIT Press, pp. 35-47.

Arrow, K.J. and G. Debreu (1954), ‘Existence of an equilibrium for a competitive economy’, Econometrica, 22 (3), 265-90.

Arrow, K.J. and F.H. Hahn (1971), General Competitive Analysis, San Francisco, CA: Holden-Day.

Arrow, K.J. and M. Kurz (1970), Public Investment, the Rate of Return, and Optimal Fiscal Policy, Baltimore, MD: Johns Hopkins Press.

Arrow, K.J. and H. Raynaud (1986), Social Choice and Multicriterion Decision-Making, Cambridge, MA: MIT Press.

Black, D. (1948), ‘On the rationale of group decision making’, Journal of Political Economy, 56 (1), 23-34.

Breit, W. and B.T. Hirsch (2009), Lives of the Laureates: Twenty Six Nobel Economists, Cambridge, MA: MIT Press.

Bossert, W. and K. Suzumura (2010), Consistency, Choice and Rationality, Cambridge, MA: Harvard University Press.

Chipman, J.S., L. Hurwicz, M.K. Richter and H.F. Sonnenschein (eds) (1971), Preference, Utility, and Demand, New York: Harcourt, Brace, Jovanovich.

Debreu, G. (1951), ‘The coefficient of resource utilization’, Econometrica, 19 (3), 273-92.

Debreu, G. (1959), Theory of Value. An Axiomatic Analysis of Economic Equilibrium, New York: Wiley.

Duppe, T. and E.R. Weintraub (2014), Finding Equilibrium. Arrow, Debreu, McKenzie and the Problem of Scientific Credit, Princeton, NJ: Princeton University Press.

Feiwel, G.R. (1987a), ‘Oral history I: an interview with Kenneth J. Arrow’, in G.R. Feiwel (ed.), Arrow and the Ascent of Modern Economic Theory, Basingstoke: Macmillan, pp. 191-242.

Feiwel, G.R. (1987b), ‘Arrow on Arrow: an interview with Kenneth J. Arrow’, in G.R. Feiwel (ed.), Arrow and the Foundations of the Theory of Economic Policy, Basingstoke: Macmillan, pp. 637-57.

Hicks, J.R. (1939), Value and Capital, 2nd edn 1946, Oxford: Oxford University Press.

Jehle, G.A. and P.J. Reny (2001), Advanced Microeconomic Theory, 2nd edn, Boston, MA: Addison-Wesley.

Kelly, J.S. (1987), ‘An interview with Kenneth J. Arrow’, Social Choice and Welfare, 4, 43-62, reprinted in K.J.Arrow, A.K. Sen and K. Suzumura (eds) (2010), Handbook of Social Choice and Welfare, vol. 2, Amsterdam: Elsevier, pp. 4-24.

Marshall, A. (1920), Principles of Economics, 8th edn, London: Macmillan.

Mas-Colell, A., M.D. Whinston and J. Green (1995), Microeconomic Theory, New York: Oxford University Press.

McKenzie, L.W. (2002), Classical General Equilibrium Theory, Cambridge MA: MIT Press.

Nasar, S. (1998), A Beautiful Mind, New York: Simon & Schuster.

Nash, J.F. (1951), ‘Equilibrium points in n-person games’, Proceedings of the National Academy of Sciences, 36 (1), 48-9.

Samuelson, P.A. (1938), ‘A note on the pure theory of consumer’s behaviour’, Economica, 5 (17), 61-71.

Samuelson, P.A. (1950), ‘The problem of integrability in utility theory’, Economica, 17 (68), 355-85. Schumpeter, J.A. (1954), History of Economic Analysis, London: Allen and Unwin.

Szenberg, M. (ed.) (1992), Eminent Economists: Their Life Philosophies, Cambridge: Cambridge University Press.

Wald, A. (1936), ‘Uber einige Gleichungssysteme der mathematischen Okonomie’, Zeitschrift fur Nationalokonomie, 7, 637-70, English trans. 1951, ‘On some systems of equations of mathematical econom­ics’, Econometrica, 19 (4), 368-403.

Weintraub, E.R. (1985), General Equilibrium Analysis: Studies in Appraisal, Cambridge: Cambridge University Press.