Gerhard Tintner: Review of Martin J. Beckmann’s Lineare Planungsrechnung (Linear Programming) (1959)

Gerhard Tintner’s review presents Beckmann’s book as an exceptionally successful introduction to linear programming because it explains the economic significance of optimization rather than concentrating on computational procedures. His assessment proceeds through the introduction and five chapters, then weighs their strengths and limitations before making a broader appeal for German-speaking economists to engage with the subject. The review’s central criterion is how effectively mathematical methods illuminate economic problems, particularly the short-run production and cost decisions of firms under free competition.

Tintner emphasizes the book’s progression from concrete examples to general principles. The introduction explains cost minimization through ice-cream production. The first chapter, on production techniques with constant coefficients, represents von Thuenen’s location theory and its generalizations graphically as linear-programming problems. A fictitious furniture factory then provides a detailed model of short-run industrial decisions. These examples establish the practical economic setting before the reader encounters the general theory.

The second chapter develops a general model of the firm with elementary mathematical tools. Tintner singles out Dantzig’s corner principle and, especially, Koopmans’s price theorem, which supports much of the subsequent discussion. Duality connects maximization and minimization problems, while marginal productivity and comparative statics make the method particularly relevant to economists. He locates Beckmann’s distinctive approach through a comparison with another major contemporary treatment:

Im Gegensatz zu dem bekannten Buche von R. Dorfman, P. A. Samuelson und R. M. Solow (Linear Programming and Economic Analysis, New York 1958), das die Beziehungen dieser Methode zu Problemen des allgemeinen wirtschaftlichen Gleichgewichtes betont, hebt Beckmann die Beziehungen zur neoklassischen Methode (partielles Gleichgewicht) hervor.

English translation: In contrast to the well-known book by R. Dorfman, P. A. Samuelson and R. M. Solow (Linear Programming and Economic Analysis, New York 1958), which emphasizes this method’s connections with problems of general economic equilibrium, Beckmann stresses its connections with the neoclassical method (partial equilibrium).

This distinction gives the review its conceptual focus. Beckmann integrates linear programming with familiar economic analysis rather than presenting it simply as a new calculating technique. Frequent references to Marshall, together with discussions of Hayek’s Ricardo effect, non-increasing increments of output, and the Giffen paradox, place optimization in conversation with established theoretical problems.

The remaining chapters broaden applications and supply extensions and mathematical foundations. Chapter three covers transport, distribution and location, minimum diets, production adjustment over time, international or interfirm division of labour, and personnel assignment, with a brief connection to input-output analysis. Chapter four introduces Pareto maxima, maximizing a minimum, game theory, convex programming, gradient methods, stochastic programs, and integer problems. Tintner praises its twelve exercises as carefully chosen aids to learning the methods through use. Chapter five supplies matrices and vectors, proofs of the price theorem and corner principle, and an introduction to the simplex method, followed by historical remarks and an outlook.

The review’s praise nevertheless includes specific reservations. Tintner finds the mathematical apparatus perhaps too compressed, as well as the treatments of consumption and Leontief models. These qualifications identify uneven coverage without undermining his judgment of the book’s principal achievement:

Im Gegensatz zu vielen anderen Einführungen in die Methode der linearen Programme, die rechnerische Probleme (Simplexmethode) vielleicht zu ausführlich behandeln, betont das vorliegende Buch die ökonomische Bedeutung der Methoden.

English translation: In contrast to many other introductions to the method of linear programming, which perhaps treat computational problems (the simplex method) too extensively, the present book emphasizes the economic significance of the methods.

Tintner considers this emphasis especially effective for competitive firms’ short-run production and cost problems. He consequently ranks the book as by far the best introduction then available and recommends it to economists, business scholars, industrial engineers, and mathematical economists. Its accessibility matters to that recommendation: most of the book requires no knowledge of higher mathematics.

The closing paragraph extends the assessment from pedagogy to disciplinary participation. Tintner approves Beckmann’s recognition of German-language antecedents, naming von Thuenen, von Stackelberg, Schneider, von Neumann, and Wald. This historical acknowledgment supports a forward-looking hope:

Dürfen wir hoffen, dass dieses schöne Buch dazu beitragen wird, dass sich auch die deutschsprachige Wissenschaft mehr mit diesen bisher meist in den angelsächsischen Ländern behandelten Problemen befassen wird?

English translation: May we hope that this fine book will help German-speaking scholarship, too, engage more with these problems, which have hitherto been addressed mostly in the Anglo-Saxon countries?

The review thus values Beckmann’s work both as an economically grounded teaching text and as a means of widening participation in linear-programming research. Its strong endorsement rests on concrete explanatory successes, while its criticisms remain directed at particular areas needing fuller treatment.