Oskar Morgenstern, “Mathematical Economics”

Oskar Morgenstern’s signed subsection of the encyclopaedia article Economics, originally published in 1931 and republished in the supplied 1937 version, combines methodological argument with a historical survey. Its central claim is that mathematics belongs in economics as a means of reasoning, not as the defining doctrine of a separate school. Mathematical economists differ substantially among themselves; their common feature is a logical instrument whose value depends on what it enables them to explain.

It is a distinction in method used rather than in subject matter and approach that differentiates mathematical economists from others, for the differences of approach among mathematical economists are almost as great as between recognized economic schools.

This distinction governs the opening defence of mathematical reasoning. Drawing on Wittgenstein’s account of mathematics as a logical method, Morgenstern treats mathematical operations as means of passing between substantive propositions rather than as economic knowledge in themselves. Objections that economic quantities are finite, discrete, or connected by nonmechanical relationships mistake the range of mathematics for the narrower capabilities of infinitesimal calculus. Whether a mathematical technique is appropriate must be decided for particular problems, not settled by an abstract prohibition against mathematics in the social sciences.

So far there have been found very few instances in which mathematics is absolutely necessary, but there are more examples in which the application of mathematics facilitates the prosecution of the argument.

The defence is thus deliberately qualified: applicability does not establish indispensability. Conclusions must illuminate concrete phenomena, and their value does not depend on whether the reasoning is expressed mathematically. Morgenstern distinguishes three uses of mathematics. Symbols and graphs can clarify exposition without generating a genuinely mathematical argument; Whewell’s algebraic rendering of classical doctrines illustrates how formal restatement may accomplish little. More fruitfully, mathematical reasoning can simplify selected problems and make their solutions precise. Marshall exemplifies this limited, illuminating application, especially in partial-equilibrium analysis, although Morgenstern notes that partial equilibrium can also be treated without mathematics.

The third use organizes economic theory as a comprehensive system of interdependent propositions. Walras and Pareto’s general-equilibrium theory represents this ambition through simultaneous equations linking economic quantities.

In contrast to the mathematics of the kind which Marshall employed, simplicity is here sacrificed for the sake of completeness.

The contrast concerns scale and explanatory purpose, not merely technical sophistication. Marshall isolates manageable parts of economic activity; the Lausanne school attempts to represent the connections among all markets. Morgenstern’s subsequent history shows how these different purposes emerged, rather than presenting mathematical economics as an uninterrupted succession of formal improvements.

The survey begins with Ceva, Beccaria, Lloyd, and Isnard, distinguishing early analytical achievements from mere algebraic translation. Isnard anticipates economic interdependence, while Canard’s work is judged unsuccessful because of mathematical errors. Von Thünen supplies influential formulae, but Cournot marks the decisive application of calculus to economic analysis. His treatment moves from monopoly revenue maximization through imperfect monopoly toward competition and gives precise expression to cost relationships. Morgenstern also records disputes over Cournot’s duopoly results, keeping analytical achievement distinct from universal acceptance.

Dupuit’s demand analysis, diminishing utility, and consumers’ rent, followed by graphical work by Mangoldt and Jenkin, extends the history beyond systematic theorists. Gossen formulates diminishing utility and marginal equalization but attracts little contemporary attention. The independent development of marginal-utility reasoning by Jevons, Menger, and Walras in 1871 then provides a foundation for connecting individual valuation with prices and exchange. Menger’s inclusion is especially significant: mathematical reasoning need not display mathematical symbols. Morgenstern thereby makes the distinction between logical method and visible notation integral to his historical account.

Walras extends exchange analysis into a system in which each market’s conditions depend on those of others. Morgenstern presents the matching of equations and unknowns as establishing theoretical solvability, then explains the extension to production. Pareto replaces fixed production coefficients with variable ones and introduces indifference curves into the equilibrium framework.

The advantage of the method of indifference curves is that it presupposes merely the comparability and not the exact measurability of value and that it makes possible a more exact study of the relations of complementarism among commodities.

Here formal refinement also changes conceptual requirements: analysis can proceed through comparisons of preferences without exact measurement of utility. The later survey differentiates Lausanne followers, English partial-equilibrium theorists, and continental contributors. Edgeworth’s contract curve brings indeterminacy into exchange analysis; Wicksell connects psychological valuation with mathematical formulation and contributes to monetary explanations of business cycles. Cassel’s simplified Walrasian system receives a sharper criticism for having more equations than unknowns.

The closing sections turn to dynamics and empirical research. Roos and Evans introduce time and variable price functions, accepting more demanding techniques to bring premises closer to reality; Amoroso explores dynamics through physical analogies. Statistical investigators, especially H. L. Moore, seek numerical counterparts to theoretical equations and inductive verification of pure theory. Morgenstern reports Moore’s claim to have converted static equilibrium into moving equilibrium without treating it as settled. The conclusion anticipates greater use of mathematics in both deduction and induction, alongside closer integration with psychological economics. The work’s enduring conceptual contribution is its measured criterion of progress: mathematics matters when it advances economic reasoning, not merely when it supplies a formal appearance.