Gerhard Tintner’s review of R. G. D. Allen’s Mathematical Analysis for Economists (1938) presents the textbook as a landmark in the consolidation of mathematical economics. His praise rests on a double achievement: Allen supplies economists with an accessible mathematical apparatus while using economic applications to introduce contemporary theory. The review moves from the discipline’s growing recognition through the textbook’s mathematical coverage and economic examples to its suitability for graduate teaching. Its principal reservation concerns the comparatively limited treatment of dynamic economics.
Tintner begins by placing Allen’s achievement within a change in the status of the field:
The time has evidently come to give a broad survey of the field, and this once revolutionary subject is entering the stage in which textbooks are being written about it.
Textbook production here signifies disciplinary maturity. Mathematical economics has moved beyond the struggle of its pioneers toward an established place in academic life, a development Tintner attributes especially to the Econometric Society. Allen’s book answers the resulting need for a broad, teachable survey. Its importance is therefore institutional as well as instructional: it makes an expanding specialist field accessible through a coherent course of study.
The review’s first substantial assessment concerns the selection and presentation of mathematical tools. Allen begins with numbers, variables, functions, graphs, analytical geometry, limits, and continuity, then proceeds through differential and integral calculus, differential equations, Taylor series, determinants, quadratic forms, and the calculus of variations. Tintner does not value this coverage simply for its breadth. Its governing principle is relevance to economists, in contrast to calculus textbooks whose applications come from physics and the natural sciences.
No ordinary textbook in advanced calculus presents, for instance, such a thorough treatment of the problem of constrained maxima and minima in several variables, which is so important for economic applications.
Constrained optimization exemplifies the book’s distinctive alignment of mathematical instruction with economic problems. Tintner recommends it both for self-study and as a handbook for economists with some college mathematics, including those whose knowledge needs refreshing. The practical standard of success is participation in current research: mastering Allen’s exposition should prepare readers to follow most articles in Econometrica.
The second purpose emerges from the applications themselves. Although the book’s organization follows mathematical topics, Tintner judges its examples to offer an exceptionally strong introduction to mathematical economics. Allen’s own contributions to economic theory enter the exposition, and the discussions of consumer choice, monopoly, and production receive particular praise. Tintner nevertheless specifies the limits of this achievement:
The economic problems are only introduced as applications and examples in connection with certain mathematical theorems.
The distinction qualifies his enthusiastic description of the book as the best available textbook on mathematical economics. It provides neither a systematic nor a continuous development of economic theory; its theoretical instruction is distributed across mathematical demonstrations. For Tintner, this organization does not diminish the examples’ value. Working through them offers a particularly effective route into contemporary economic analysis.
The closing assessment recommends the book for graduate work while identifying its predominantly static orientation. The chapter on the calculus of variations is somewhat short, and dynamic problems appear less frequently than their growing importance warrants. Tintner proposes G. C. Evans’s Mathematical Introduction to Economics (1930) as a complement, with Allen supplying enough introductory preparation to make Evans’s exposition accessible. His final judgment remains emphatically favorable: the textbook equips economists for modern analytical work, though fuller preparation in dynamics requires supplementary study.
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