The Mathematical Economists

Economics faces six fundamental questions: (i) will the society organized on the principles of exchange (the Market Economy) stay composed or will it fall apart (the question of existence of equilibrium)?, (ii) will such an equilibrium be unique (a multiplicity of equilibria poses difficult and embarrassing questions)?, (iii) will such an equilibrium be robust (the question of stability of equilibrium)?, (iv) will such an economy (society) be efficient?, (v) will it grow or expand forever?, and (vi) will it be just? The classical economists, Adam Smith in particular, answered all these questions affirmatively using a characteristic methodology. However, Karl Marx challenged the entire structure of faith in the merits of the exchange economy and shattered all optimism regarding the said order. The neoclassicists, mostly using their own new (mathematical, marginalist, rationalistic, atomistic, hedonistic, etc) methodology set out to prove that answers to all those six questions were in affirmative. They restructured the faith in the said order. In so doing, they had to distance themselves from the reality and they did not mind doing so. This endeavour made neoclassical economics dogmatic and religious in nature. Leijonhufvud (1973) characterized neoclassical economics in the most sarcastic manner.

In his A History of Western Philosophy Bertrand Russsell wrote: "The origin ... is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. ... It gradually became clear that a great part of philosophy can be reduced to something that may be called "syntax". ... when errors in syntax are avoided, a philosophical problem is thereby either solved or shown to be insoluble. ... there can be no doubt that the utility of philosophical syntax in relation to traditional problems is very great." (pp. 783-785).

The economists who were guided by the philosophy hinted at by Russell made serious attempts to mathematically formulate (often the Neo-Classical) economic theories in the mathematical syntex and purge the economic philosophy (the system of theories) of the prevailing fallacies and slipshod reasoning. Those economists are the Mathematical Economists. One of the most impoertant findings of mathematical economics is, e.g. Sonnenschein–Mantel–Debreu theorem, that shows, among other points that in general equilibrium model  the excess demand function for an economy is not restricted by the usual rationality restrictions on individual demands in the economy. Thus, microeconomic rationality assumptions have no equivalent macroeconomic implications. The theorem's main implications are that, with many interdependent markets within the economy, there might not exist a unique equilibrium point. Frank Hahn regarded the theorem as the most dangerous critique against the micro-founded mainstream economics.  It amounts to say that the Walrasian aggregate excess-demand function inherits only certain properties of individual excess-demand functions, namely continuity, homogeneity of degree zero, Walras' law, and boundary condition assuring that, as prices approach zero, demand becomes large. These inherited properties are not sufficient to guarantee that the aggregate excess-demand function obeys the weak axiom of revealed preference. The consequence of this is that the uniqueness of the equilibrium is not guaranteed : the excess-demand function may have more than one root – more than one price vector at which it is zero (the standard definition of equilibrium in this context). The range of implications is however not limited to just the absence of uniqueness. The findings entail that there are problems with establishing general results on uniqueness (Ingrao and Israel 1990; Kehoe 1985, 1991; Mas-Colell 1991), stability (Sonnenschein 1973; Ingrao and Israel 1990;, Rizvi 1990), comparative statics (Kehoe 1985; Nachbar 2002, 2004), econometric identification (Stoker 1984), microfoundations of macroeconomics (Kirman 1992; Rizvi 1994), and the foundations of imperfectly competitive general equilibrium (Roberts and Sonnenschein 1977; Grodal 1996). Subfields of economics that relied on well-behaved aggregate excess-demand for much of their theoretical development, such as international economics, were also left in the lurch (Kemp and Shimomura 2002; Rizvi 2006). All these do not go well with the Neo-Classical economics.

Mathematical economics is, unfortunately, often thought to be as any and every application of mathematics to represent economic theories and analyze problems posed in economics. If one argues in the same vein, any and every use of mathematics in physics will amount to doing mathematical physics. Now since no branch of physics is such where mathematics is not used; ergo, the entire body of physics is mathematical physics. But that will not do. Mathematical physics is a term used in a very special sense. It denotes research aimed at (or the body of knowledge at hand) studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics. In the same manner, mathematical economics denotes research aimed at (or the body of knowledge at hand) studying and solving problems inspired by economics within a mathematically rigorous framework (such as the one we have in Arrow's impossibility theorem). It refers to the axiomatic development of mutually consistent theories in economics making a coherent system. It is not the representation of an 'indifference curve' by, say, a rectangular hyperbola and its tangency with a downwards sloping price line. It is concerned with the implications of the said mathematical specification on the theory of consumer's behaviour immediately and the theory of general (competitive) equilibrium finally. It is also concerned with representing a cardinally conceived price line tangent to an ordinally conceived indifference curve (innocently specified as a rectangular hyperbola) in the same plane.

There is one more important point to note as indicated by Nicholas Georgescu-Roegen: "There are endeavors that now pass for the most desirable kind of economic contributions although they are just plain mathematical exercises, not only without any economic substance but also without any mathematical value". Georgescu-Roegen cites as an example a paper that assumes more traders in existence than there are points in the set of real numbers.

The Mathematical Economists
Léon Walras Vilfredo Pareto
Francis Y. Edgeworth Kenneth J. Arrow
John von Neumann Tjalling Charles Koopmans
Gérard Debreu Lionel W. McKenzie
Hugo F. Sonnenschein Rolf Mantel
Franklin M. Fisher Herbert Scarf
Lloyd S. Shapley Martin Shubik
John F. Nash, Jr. Frank H. Hahn
Michael Mandler Takashi Negishi
Graciela Chichilnisky Ariel Rubinstein

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