Finding equilibrium. Arrow, Debreu, McKenzie and the problem of scientific credit (Q2874643)

The book of well-known methodologists, scientific historians and researchers in economic theory (in mathematical mode), is a scrutinizing of one of dramatic events in the creation of the modern economic mainstream which are the neoclassical economic theory and its core -- the general equilibrium theory (GET). The base of the GET was founded by Lion Walras in his treatise of the seventieth years of the nineteenth century where he has set a purpose to give a mathematical representation of the main conjecture of Adam Smith on the functioning of a competitive market. The Walrasian representation was an `equilibrium system' of equations which defined the market prices providing equality of a demand and supply in a competitive market. Walras, being an engineer and self-educated mathematician, understood that it is necessary to prove existence of a solution of the equilibrium system. However, he couldn't do it, and he left this difficult problem for ``future generations of mathematicians''.NEWLINENEWLINEIt was the Austrian mathematician Abraham Wald, who has firstly found (in early thirties) a proof of equilibrium existence, as well as its uniqueness, for Walrasian system in Gustav Cassel setting where individual consumers were been excluded and the aggregate market demand was been taken as initially given. The Wald's achievement was based on some condition that he put on the market demand. A bit later (in 1938) Paul Samuelson suggested this condition as a rationality principle that was an alternative with respect to utility maximization and this principle was named as weak axiom of revealed preference. However, the demand theory has been created for the individual consumer, and it has been established at fortieth years that the weak axiom (Wald's condition) has fulfilled for an individual demand rationalized by a utility function of the consumer but in the common case it has not fulfilled for a sum of individual demands. So, the Wald's achievement was estimated by the leading mathematical economists (particularly, by Samuelson) as `economically illegitimate', and the battle for the proving the equilibrium existence was continued.NEWLINENEWLINEThe book under reviewing is devoted to presentation of the story of finding an `economically legitimate' proof of equilibrium existence, as it is understood in the economic mainstream till nowadays. This story about three protagonists of the `equilibrium finding', Kenneth Arrow, Gerard Debreu, and Lionel McKenzie, their similarities and diversities in origin, fate, character, career, the process of proving the equilibrium existence, and `scientific crediting'.NEWLINENEWLINEThe book consists of eight chapters collected in three parts. The first part ``People'' consists of three chapters each of them is devoted to one of the protagonists, their lives from childhood to the late of forties when their ways crossed in the University of Chicago where the Cowless commission for research in economics was situated. What is more important for understanding following diversity in their scientific crediting, Arrow and Debreu, which would obtain the highest form of credit (Nobel Prizes in 1972 and 1983 correspondingly), were graduates of prestigious the Columbia University (Arrow) and the École Normale Supérieure (Debreu), and they have had successful start of their careers. Especially Arrow, who at that time had obtained the truly classical result -- the `theorem of impossibility' (impossibility of democracy in the ideal perception). McKenzie was a provincial fellow from the Georgia State who has graduated the Duke University, also prestigious one; however, his first steps were not very successful. At 1948 he attempted to obtain the PhD on economics at Oxford (with J. Hicks' supervising) but was failed.NEWLINENEWLINEThe second part is ``Context''. Chapter 4, describes sites where after war time the mathematicians and economists were located, that were concerned about developing economic problems, practical as well as theoretical, on base of new mathematical tools. There were the RAND corporation and the Cowless commission. Chapter 5 represents their scientists community and corresponding intellectual atmosphere where new mathematical tools, such as linear and nonlinear programming, and `activity analysis', were been formed.NEWLINENEWLINEThe third part is ``Credit''. Chapter 6 represents details of presentations at the Chicago econometric society meeting (in 1952) of equilibrium existence results by McKenzie and, jointly, Arrow and Debreu, and submission corresponding papers at `Econometrica'. These papers were: L. W. McKenzie, ``On equilibrium in Graham's model of world trade and other competitive systems''; K.J. Arrow and G. Debreu, ``Existence of an equilibrium for a competitive economy.'' The results were different with respect to economic content but similar mathematically, and McKenzie submitted his paper by several months earlier. His paper was published in 1954, no 3, and the paper by Arrow and Debreu was published in the next no 4.NEWLINENEWLINEChapter 7 describes the protagonists' scientific ways after issuance their papers on equilibrium existence in the context of growing estimation of their contribution in the new economic theory in mathematical mode. The described period extends to 1972 when Kenneth Arrow received the Nobel Prize (NP) in memory of Alfred Nobel (shared with John Hicks).NEWLINENEWLINEIn the last, eighth chapter the authors `describe questions of [scientific] credit generically, as emerging of our protagonists' work.' (p. 205). They scrutinize the discussion in scientific community about NP rewarding, firstly Arrow, without Debreu -- coauthor of their collaborative work of 1954, accepted by professional community as a seminal work, then Debreu in 1983, and non-rewarding McKenzie. It is interesting that Debreu felt dissatisfaction with respect to awarding him the prize as to an economist. He wanted to accept it as a mathematician.NEWLINENEWLINEIt could be said, the book by Düppe and Weintraub shows that McKenzie was unfairly excluded from the Nobel Prizes which both Arrow and Debreu, but separately, won for the similar scientific results. In the 'Conclusion' the authors explained this unfairness as the ``Matthew effect'': ``For unto every one that hath shall be given, he shall have abundance: but from him that hath not shall be taken away even that which he hath.'' (p. 241) [hath=has, V.G.]NEWLINENEWLINEAlso, for deeper comprehension of this example of the failure of `scientific crediting', it is necessary take into account that further researches for the economy model of Arrow and Debreu, with respect to uniqueness of the equilibrium, revealed essential inconsistency of the model with reality. The inconsistency was a consequence of the non-fulfilling of the weak axiom for the excess market demand functions. It was established that a very arbitrary set of prices could be the set of equilibrium prices under very good individual preferences (`anything goes', or Sonnenschein-Mantel-Debreu theorem, 1972-74). It is impotent that Debreu was one of contributors of this shattering result.NEWLINENEWLINEThe resolution of the market demand problem and the problem of non-uniqueness of the equilibrium has been presented in (Russian) works by \textit{V. K. Gorbunov}, particularly, in ``To the problem of aggregations of consumers' demand'' [Zh. Èkon. Teor. 2009, No. 1, 85--94 (2009; Zbl 1183.91087)], ``Economic equilibrium and buyers aggregation: rehabilitation of the Wald's theorem'' [ibid. 2011, No. 3, 130--144 (2011)], and ``To the theory of market demand: regularity and economic equilibrium'' [Ekonomicheskaya Nauka Sovremennoy Rossii. 2013, No. 4(63), 19--36 (2013)]. The market demand problem has been resolved on the basis of notion the \textit{statistical ensemble of consumers'} which has been taken as the origin of theory and modeling. In frame of this approach to the demand theory it has been shown that a modification of the holistic type equilibrium model of Cassel-Wald is more adequate to reality with respect to the individualistic model of Arrow-Debreu. Correspondingly, the contribution of A. Wald in the GET also has been unfairly under-estimated.