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The Rise of Mathematical Economics

In the Preface they quote Arrow and Hahn “the notion that a social system moved by independent actions in pursuit of different values is consistent with a final coherent state of balance,…is surely the most important intellectual contribution that economic thought has made to the general understanding of social processes”. They proceed: “Speaking of Leon Walras – the first economist to construct a theoretical system based on the idea of economic equilibrium – Paul Samuelson, with equal enthusiasm, places him on the same level as Newton, and Schumpeter calls him ‘the greatest of all economists’ whose system represents the only work in economics comparable with the achievements of theoretical physics”. (ix)

“Our thesis is that the problem of mathematization is no secondary feature of general economic equilibrium but rather one of the basic reasons for its creation and development”. (x)

Chapter 1 Introductory Remarks and Basic Concepts

“From the outset it has been the aim of the theory whose historical development, main achievements, and state of art are chronicled in this book to use mathematics as its basic tool in describing and analyzing economic reality. This was not a secondary feature but a central programmatic aim: to transform economics into a rigorously quantitative discipline, into a mathematical science on a par with astronomy and physics. In more modern terms – and with a certain shift in meaning – this aim would be defined as the transformation of economics into a ‘formalized science’”. (1)

“Furthermore, the history of the theory of economic equilibrium has the specific characteristic of allowing previous statements to be given a concrete value that is certainly much greater than a mere statement of principle. As will be shown throughout this book, the highly different and even divergent programs (or paradigms) that succeed one another in the history of the theory retain an almost intact core that can be identified with the aim to demonstrate the existence, the uniqueness, and the global stability of the equilibrium [my italics]. This core will be called the invariant paradigmatic nucleus [author’s italics]. Indeed, we shall see that the theory axiomatised by Gerard Debreu, although completely different in its form and content from Walras’s theory, not only retains this nucleus intact but also represents an early attempt to demonstrate at least the property of the existence of equilibrium and to analyze the conditions in which uniqueness occurs.” (3)

Chapter 2 The Origins charts the spread of Newtonianism and the social sciences

There is a chapter on Walras’s forerunners, Achylle-Nicolas Isnard, engineer, administrator and economist, 1749-1803, Nicolas-Francois Canard, Jules Dupuit (1804-1866) and Augustin Cournot (1801-1877) the sole acknowledged master in the eyes of the Lusanne school. Mathematician and student of demographic statistics.

Then a chapter on Leon Walras (1834-1910), mining engineer, turned to philosophy, lit crit and the social sciences.

Then a chapter on Vilfredo Pareto (1848-1923) another engineer!

“The period around 1910 appears to have been a moment of real crisis for general equilibrium theory…[at least for] that version of the theory whose central paradigm was the model of classical mathematical physics and especially mechanics. The crisis of a great ‘reductionist’ project was thus coming to a head.” (170)

Reductionism and determinism were under threat from relativity and quantum theory. “The central position that mechanical analogy had played in classical reductionism was now assigned to mathematical analogy. It was thus no longer a matter of reducing the laws of phenomena to the form of mechanical laws [as per Walras] but rather the formal unification of different laws by means of mathematical frameworks bringing out their basic analogy (‘empty schemata of possible contents’ as the exponents of axiomatic mathematics were to call it)”.

“Not only is mathematics no longer the language of nature but all claims to derive mathematical laws from nature are dropped, along with any excessive demands for the verifiability of such laws”.

“The foundations were thus laid for promising new developments in our discipline. The evolution of mathematical modeling (based precisely upon the idea of mathematical analogy) made it possible to again attempt to found a mathematical economics worthy of the name of science without losing ones way in the maze of problems concerned with the relationship between theory and empirical reality…Without a word having been said on the matter, it was clear that the paradigmatic climate was wholly different”. (171)

The plot thickens in the 1930s as the action moves to Vienna. Here we encounter J von Neuman, Oskar Morgenstern [joint authors of the book on game theory that Popper took along on the boat from New Zealand to England], Wald, Karl Menger and the three stooges of the Vienna Circle. Hicks, on a different continent, also made a major contribution to spread the word in the economics profession. Samuelson, on yet another continent, also made a major contribution to the acceptance of the new regime.

The authors revert for a few pages to the crisis in classical physics with the usual talk about undecidability, indeterminism, relativity, etc.

“A consequence of the new scientific approach was the crumbling of the unified fabric of classical science and, at the same time, a fragmentation of scientific work, an increasingly accentuated process of specialization”. (182)

They move on to consider the role of David Hilbert (1862-1943) at Gottingen, especially his axiomatic program. They note that he rejected some of the developments that followed from his work. He remained attached to a “unitary” concept of science (like Newton). “He fought against abstraction for its own sake and stressed the crucial role of problems in research.” (184)

Von Neuman emerges as the man who went all the way with abstraction [in maths, physics, and economics, it seems].

“It was, in fact, John von Neuman who interpreted the Hilbertian viewpoint with the most clear sighted consistency and, at the same time, the most scrupulous awareness of the deeper currents of contemporary research” (184). [For an alternative view, see comments on von Neuman by Popper in Schism.]

The authors quote a commentator on von Neuman’s ‘naïve and optimistic faith in mathematical machinery’. Under his leadership “The old reductionism was replaced by a sort of neoreductionism, whose key idea was the centrality of mathematics, understood as a purely logicodeductive schema” (185)

The authors suggest that “Some years earlier non Neuman had carried out a similar operation in the field of physics in order to exorcise the problems raised by quantum mechanics” quoting M Cini that von Neuman’s axiomatization of quantum mechanics codifies ‘in the form of real and proper vetoes, formulated in scientific language, the two ideological strong points of the Gottingen-Copenhagen school: a) the ultimate and definitive nature of QM; b) the impossibility of an objective description of reality because of the indispensable role of the observer. Both these assertions of a metaphysical nature were transferred by von Neuman into propositions that belong to the theory itself…it is exactly here that we find the proof of the axiomatization of QM is an operation of definition of the boundaries of the discipline, boundaries of whose integrity the community had to become guarantor’. [sounds like a labour union picket line!] (186)

So the scene is set for foundational work along the lines of a new sort of reductionism, a neoreductionism not inspired by mechanics but by the axiomatic proofs of mathematics. The Newtonian flux of time has been ousted by “a static and atemporal mathematics”. The new mathematics included von Neuman’s “decisive use of Brouwer’s theorem” to assign a key role to the mathematics of fixed point theorems which led to the work of Arrow and Debreu in the 1950s.

The authors then backtrack to sketch in the role of Abraham Wald and the Vienna Circle who provided a hospitable niche for von Neuman and Morgenstern to develop their ideas, as did the symposium convened by Karl Menger (where Karl Popper was so impressed by a presentation by Morgenstern that he thought mathematical economics had found its Galileo).

Actually it seems that Morgenstern remained in touch with the world and was scathing about some mathematical economists who merely counted equations and unknowns. He was also concerned about the unrealistic assumption of perfect foresight on the part of economic actors. His reaction to this problem was to explore the mathematics of game theory to deal more adequately with real market transactions. It appears that the two-person zero-sum games which they explored did not capture enough of the reality of real-world events to lead anywhere significant, apart from some innovations in mathematics.

Chapter 8 Equilibrium in Time and Hicks’s Contribution

Hicks was concerned with the interrelation of markets and the need to develop analytical techniques to handle these interrelations.

According to the authors he wanted to combine the static theory of the determination of relative prices, the theory of capital and interest [following Keynes?] and the theory of the trade cycle.

Note his two innovations regarding consumer preferences (represented by indifference curves) and his elaboration of the concept of temporary equilibrium.

Morgenstern was highly critical and rubbished the crudeness of the mathematical analysis.

Chapter 9 New Trends in the United States

Irving Fisher (1867-1947) spent some years working on general equilibrium mathematics. He was a pupil of Willard Gibbs (1839-1903) a major mathematical

physicist, who took an interest in Fisher’s work.

The mathematician Hotelling worked with Wald at Columbia Uni.

The debate on general equilibrium theory merged with the debate on central economic planning when Oskar Lange formalized the Walrasian notion of “tatonnement”, the “central auctioneer” in debate with Hayek. Lange worked in Chicago with the Cowles Commission for Research in Economics which became the major sponsor of mathematical economic theory in the US during the 1930s.

[As an aside, the authors note the formation of various professional associations of economists during the 1930s. This development would provide material for exploration of the positive and negative effects of professionalisation, as in the case of the philosophy of science]

Another student of Gibbs (E. B. Wilson) exerted influence on Paul Samuelson who remembered Wilson as his revered teacher of mathematical economics and statistics.

The authors pause in their narrative to provide an overview of four developing lines of research that can be discerned.

First, von Neuman’s decisive step towards the strongest possible mathematical approach. Developments in this line mostly employed convex analysis and fixed point theorems.

Second, the application of game theory to economic behaviour.

Third, adherence to the “ideal” Walrasian model, “the boldest and most consistent program of axiomatization, which was carried out by G Debreu and achieved the greatest successes of GET.” (259)

Fourth, Paul Samuelson’s work, which assimilated the Hicks approach but provided a more sophisticated mathematical apparatus and “a deeper understanding of physicomathematical culture.” (259)

Samuelson was a “child prodigy” in economic theory. His move from Chicago to Harvard, in his own words ‘put me right in the forefront of the three great waves of modern economics: the Keynesian revolution…the monopolistic or imperfect-competition revolution, and finally, the fruitful clarification of the analysis of economic reality resulting from the mathematical and econometric handling of the subject – including an elucidation for the first time of the welfare economic issues that had concerned economists from the days of Adam Smith and Karl Marx to the present’. (quoted on page 260)

“Samuelson’s work [during the 1930s and 1940s] marks a decided step forward in the mathematization of the discipline. It is the first treatise of economic theory in which the formal apparatus is not confined to the appendices but is one with the main argument”. (262)

Samuelson insisted on the importance of empirical verification and the need for “operationally meaningful theorems” [meaning theorems that might be falsified] but he never went very far in that direction [Stanley Wong shredded the work on revealed preferences which won Samuelson the Nobel Prize] and the authors concluded “It is doubtful whether GET – the peak of the neoclassical synthesis – could come up to the criteria laid down so firmly by Samuelson with reference to the natural sciences. However, it does not appear that he himself ever seriously attempted to get to grips with the problem”. (269).

The theory of games and activity analysis. Morgenstern and von Neumann started this movement rolling when they met at Princeton in 1939 [where they sometimes dined with Bohr, Einstein and Weyl!!!]. Their book appeared in 1944 [Popper took a copy to read on the boat from NZ to Britain in 1946].

This line of work reached its highest development in the 1950s in the hands of Kenneth J Arrow and Gerard Debreu. Their work was made possible by J Nash [A Beautiful Mind] at Princeton who generalized some results from von Neumann and Morgenstern and made a link between game theory and the theory of the existence of equilibrium.

Parallel work proceeded on activity analysis and linear programming. In 1949 the Cowles Commission hosted a conference on linear programming.

Kenneth Arrow: welfare, equilibrium, and the rationality of collective choice. Arrow (1921 - ) became interested in welfare economics as a result of course in mathematical economics by Hotelling. [It appears that mathematical welfare economics has rather little to do with the welfare of real people]. He worked at the Cowles Commission from 1947 to 1949.

He was further inspired by Hick’s pioneering work in the field - “Value and Capital” (noted previously). Arrow was concerned with the logic of collective choice (the topic appears to manifest a holistic error from the start, and it is not surprising that negative results emerged).

The “invisible hand’ in recent interpretations of the axiomatic model of general equilibrium.

It seems that there has been a bifurcation of thinking in this area, between one strand represented by Arrow and Hahn, and another represented by Debreu. At the moment I cannot make out the difference but if it matters (which for the purpose of economics proper is unlikely) it may be necessary to get clear on this.

Gerard Debreu

“Debreu’s approach marks a new and decisive paradigmatic shift in the theory. In the first place, it involves – in accordance with the canons of the axiomatic approach – emptying the theory radically and uncompromisingly of all empirical reference” (286). So much for Gerard Debreu as a scientist.

Chapter 10 The Question of the Existence of Equilibrium

The last three chapters of the book indicate the progress that has been made in the last three decades to solve the three basic problems of general equilibrium theory (GET) – namely the questions of existence, uniqueness and stability.

They note that theory and empirical interpretation have lived “parallel lives” since the 1930s. In other words the field has no empirical reference.

Chapter 11 The Question of the Uniqueness of Equilibrium

The authors report that after Arrow and Debreu’s joint work early in the 1950s, Arrow spent some years on the problem of uniqueness. Debreu did not see scope for development in that direction.

“We shall now give a rapid description of the results obtained in this field. While obviously not closely connected with uniqueness, these themes are mentioned in this chapter since they are …prior to the problems of both uniqueness and stability. Moreover, as we shall see, the results achieved are so clear as to leave little doubt about the limits within which research on uniqueness and stability must work.” (315)

Debreu was concerned that the microeconomic foundations of the theory might not imply a sufficient structure to describe an aggregate description. This concern has been aggravated by work by Sonnenschein. Not only is the way barred towards solving problems of uniqueness and stability, there is doubt about the whole of Samuelson’s program to obtain significant results by starting from very general hypotheses about the behaviour of economic agents.