Some mathematical methods and techniques in economics
Author:
I. N. Herstein
Journal:
Quart. Appl. Math. 11 (1953), 249-262
MSC:
Primary 90.0X
DOI:
https://doi.org/10.1090/qam/60205
MathSciNet review:
60205
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- Kenneth J. Arrow, Social Choice and Individual Values, Cowles Commission Monograph No. 12, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951. MR 0039976
---, “Alternative Proof of the Substitution Theorem for Leontief Models in the General Case” in [29], pp. 155-165.
---, “Mathematical Models in the Social Sciences” in “the Policy Sciences,” edited by Daniel Lerner and Harold Lasswell, Stanford University Press, pp. 129-155.
- Kenneth J. Arrow and Gerard Debreu, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954), 265–290. MR 77069, DOI https://doi.org/10.2307/1907353
- Martin Beckmann, A continuous model of transportation, Econometrica 20 (1952), 643–660. MR 68196, DOI https://doi.org/10.2307/1907646
- Edward G. Begle, A fixed point theorem, Ann. of Math. (2) 51 (1950), 544–550. MR 35433, DOI https://doi.org/10.2307/1969367
- G. W. Brown and J. von Neumann, Solutions of games by differential equations, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N. J., 1950, pp. 73–79. MR 0039220
- A. Charnes, Optimality and degeneracy in linear programming, Econometrica 20 (1952), 160–170. MR 56264, DOI https://doi.org/10.2307/1907845
- John S. Chipman, The multi-sector multiplier, Econometrica 18 (1950), 355–374. MR 44096, DOI https://doi.org/10.2307/1907834
Debreu, Gerard, “Real Representation of a Preference Ordering,” (Forthcoming).
---, “The Coefficient of Resource Utilization,” Econometrica 19 (1951) 273-291.
- Gerard Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 886–893. MR 50251, DOI https://doi.org/10.1073/pnas.38.10.886
- Gerard Debreu and I. N. Herstein, Nonnegative square matrices, Econometrica 21 (1953), 597–607. MR 59240, DOI https://doi.org/10.2307/1907925
Dorfman, Robert, Application of Linear Programming to the Theory of the Firm, University of California Press, 1951.
- Samuel Eilenberg, Ordered topological spaces, Amer. J. Math. 63 (1941), 39–45. MR 3201, DOI https://doi.org/10.2307/2371274
- Samuel Eilenberg and Deane Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222. MR 16676, DOI https://doi.org/10.2307/2371832
- G. C. Evans, The Dynamics of Monopoly, Amer. Math. Monthly 31 (1924), no. 2, 77–83. MR 1520361, DOI https://doi.org/10.2307/2300113
Frisch, Ragnar, “Sur un Problème d’Economie Pure,” Norsk Mathematisk Forenings Skrifter 1 (1926) 1-40.
Frobenius, G., “Über Matrizen aus Positiven Elementen,” Sitzungsberichte der Koniglich Preussischen Akadamie der Wissenschaften, (1908), pp. 471-476.
---, “Über Matrizen aus nicht-negativen Elementen,” Sitzungsberichte Preuss. Ak. Wiss. (1912) 456-477.
- Nicholas Georgescu-Roegen, Some properties of a generalized Leontief model, Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951, pp. 165–173. MR 0042674
- Murray Gerstenhaber, Theory of convex polyhedral cones, Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951, pp. 298–316. MR 0042145
Goodwin, Richard M., “The Multiplier as a Matrix,” Economic Journal 59 (1949) 537-555.
Hawkins, D. and Simon, H. A., “Some Conditions of Macroeconomic Stability,” Econometrica 17 (1949) 245-248.
- I. N. Herstein and John Milnor, An axiomatic approach to measurable utility, Econometrica 21 (1953), 291–297. MR 61356, DOI https://doi.org/10.2307/1905540
- Clifford Hildreth, Alternative conditions for social orderings, Econometrica 21 (1953), 81–94. MR 52756, DOI https://doi.org/10.2307/1906944
Hotelling, H. “The Economics of Exhaustable Research,” Journal Political Economy 39 (1931) 137-176.
- Shizuo Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 457–459. MR 4776
Koopmans, Tjalling C., Activity Analysis of Production and Allocation, Cowles Commission for Research in Economics, Monograph 13, Wiley, 1951.
---, “Analysis of Production” in [29], pp. 33-97.
---, “Alternative Proof of the Substitution Theorem for Leontief Models in the Case of Three Industries,” [29], pp. 147-155.
- Tjalling C. Koopmans and Stanley Reiter, A model of transportation, Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951, pp. 222–259. MR 0052757
Kuhn, H. W. and Tucker, A. W., Contributions to the Theory of Games, Annals of Mathematics Study, No. 24, Princeton, 1950.
---, “Nonlinear Programming” in Proceedings of the Second Berkeley Symposium of Mathematical Statistics and Probability, Berkeley, 1951.
- Oscar Lange, Theoretical derivation of elasticities of demand and supply: the direct method, Econometrica 10 (1942), 193–214. MR 7249, DOI https://doi.org/10.2307/1905464
Leontief, Wassily, The Structure of American Economy, 1919-1939; Second Edition, N. Y., Oxford University Press, 1951.
- Jacob Marschak, Rational behavior, uncertain prospects, and measurable utility, Econometrica 18 (1950), 111–141. MR 39233, DOI https://doi.org/10.2307/1907264
- J. C. C. McKinsey, Introduction to the theory of games, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1952. MR 0050248
- Lloyd A. Metzler, A multiple-region theory of income and trade, Econometrica 18 (1950), 329–354. MR 44095, DOI https://doi.org/10.2307/1907833
Morgenstern, Oskar and Von Neumann, John, Theory of Games and Economic Behavior.
Von Neumann, John, “The Personnel Assignment Problem and a Certain Two Person Game,” to appear in Contributions to the Theory of Games, Annals of Mathematics Studies.
---, “Über ein. Ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes,” Ergebnisse eines Mathematischen Kolloquiums 1 (1937) 73-83.
Pareto, V., “Anwendung der Mathematik auf Nationalökonomie,” Encyklopädie Math. Wiss. (Arithmetik und Algebra), Zweiter Teil, Leipzig, 1900-1904, pp. 1094-1120.
- C. F. Roos, A Mathematical Theory of Competition, Amer. J. Math. 47 (1925), no. 3, 163–175. MR 1506555, DOI https://doi.org/10.2307/2370550
- Paul Anthony Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, Mass., 1947. MR 0029145
---, “Abstract of a Theorem Concerning Substitutability in Open Leontief Models,” in [29], pp. 142-147.
Slutsky, E., “Sulla Teoria de Bilancio del Consumatore,” Giornale Economisti 60 (1915) 19-23.
- Robert Solow, On the structure of linear models, Econometrica 20 (1952), 29–46. MR 50255, DOI https://doi.org/10.2307/1907805
Stone, Richard, “Comptabilité Social, Agrégation, Et Invariance,” Economie Appliquee 59 (1949) 26-54.
- Abraham Wald, Statistical Decision Functions, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1950. MR 0036976
---, Über einige Gleichunzssysteme der mathematischen Ökonomie,” Zeitsch. National-ökonomie 7 (1936) 637-670. Translated in Econometrica 29 (1951) 368-403.
- H. Weyl, The elementary theory of convex polyhedra, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N.J., 1950, pp. 3–18. MR 0038088
- Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642–648 (German). MR 35265, DOI https://doi.org/10.1007/BF02230720
- Herman Wold, A synthesis of pure demand analysis, Skand. Aktuarietidskr. 1943 (1943), 85–118. MR 9296, DOI https://doi.org/10.1080/03461238.1943.10404744
Arrow, Kenneth J., Social Choice and Individual Values, Cowles Commission for Research in Economics, Monograph 12. Wiley, 1951.
---, “Alternative Proof of the Substitution Theorem for Leontief Models in the General Case” in [29], pp. 155-165.
---, “Mathematical Models in the Social Sciences” in “the Policy Sciences,” edited by Daniel Lerner and Harold Lasswell, Stanford University Press, pp. 129-155.
Arrow, Kenneth J. and Debreu, Gerard, “Existence of an Equilibrium for a Competitive Economy.” Forthcoming.
Beckmann, Martin, “A Continuous Model of Transportation,” Econometrica 20 (1952) 643-661.
Begle, E. G., “A Fixed Point Theorem,” Annals of Mathematics 51, (1950) 544–550.
Brown, G. W. and von Neumann, J., “Solutions of Games by Differential Equations,” in [33], pp. 73-79.
Charnes, A., “Optimality and Degeneracy in Linear Programming,” Econometrica 20 (1952) 160-171.
Chipman, John, “The Multi-Sector Multiplier,” Econometrica 18 (1950) 387-395.
Debreu, Gerard, “Real Representation of a Preference Ordering,” (Forthcoming).
---, “The Coefficient of Resource Utilization,” Econometrica 19 (1951) 273-291.
---, “A Social Equilibrium Existence Theorem,” Proc. Nat. Ac. 38 (1952) 886-893.
Debreu, Gerard and Herstein, I. N., “Nonnegative Square Matrices,” Cowles Commission Discussion Paper, Mathematics 417, and forthcoming in Econometrica.
Dorfman, Robert, Application of Linear Programming to the Theory of the Firm, University of California Press, 1951.
Eilenberg, Samuel, “Ordered Topological Spaces,” Amer. Journal Math. 43 (1941) 39-46.
Eilenberg, Samuel, and Montgomery, Deane, “Fixed Point Theorems for Multi-valued Transformations,” Amer. Journal Math. 68 (1946) 214-222.
Evans, G. C., “The Dynamics of Monopoly,” Amer. Math. Monthly 31 (1924), pp. 77-83.
Frisch, Ragnar, “Sur un Problème d’Economie Pure,” Norsk Mathematisk Forenings Skrifter 1 (1926) 1-40.
Frobenius, G., “Über Matrizen aus Positiven Elementen,” Sitzungsberichte der Koniglich Preussischen Akadamie der Wissenschaften, (1908), pp. 471-476.
---, “Über Matrizen aus nicht-negativen Elementen,” Sitzungsberichte Preuss. Ak. Wiss. (1912) 456-477.
Georgescu-Roegen, Nicholas, “Some Properties of a Generalized Leontief Model,” in [29], pp. 165-173.
Gerstenhaber, Murray, “Theory of Convex Polyhedral Cones,” in [29], pp. 298-317.
Goodwin, Richard M., “The Multiplier as a Matrix,” Economic Journal 59 (1949) 537-555.
Hawkins, D. and Simon, H. A., “Some Conditions of Macroeconomic Stability,” Econometrica 17 (1949) 245-248.
Herstein, I. N. and Milnor, John, “An Axiomatic Approach to Measurable Utility,” forthcoming in Econometrica.
Hildreth, Clifford, “Alternative Conditions for Social Ordering,” Econometrica 21 (1953) 81-94.
Hotelling, H. “The Economics of Exhaustable Research,” Journal Political Economy 39 (1931) 137-176.
Kakutani, S., “A. Generalization of Brouwer’s Fixed Point Theorem,” Duke Math. Journal 8 (1941) 457-459.
Koopmans, Tjalling C., Activity Analysis of Production and Allocation, Cowles Commission for Research in Economics, Monograph 13, Wiley, 1951.
---, “Analysis of Production” in [29], pp. 33-97.
---, “Alternative Proof of the Substitution Theorem for Leontief Models in the Case of Three Industries,” [29], pp. 147-155.
Koopmans, Tjalling C., and Reiter, Stanley, “A Model of Transportation,” in [29], pp. 222-260.
Kuhn, H. W. and Tucker, A. W., Contributions to the Theory of Games, Annals of Mathematics Study, No. 24, Princeton, 1950.
---, “Nonlinear Programming” in Proceedings of the Second Berkeley Symposium of Mathematical Statistics and Probability, Berkeley, 1951.
Lange, Oscar, “The Foundations of Welfare Economics” Econometrica, 10 (1942) 215-228.
Leontief, Wassily, The Structure of American Economy, 1919-1939; Second Edition, N. Y., Oxford University Press, 1951.
Marschak, Jacob, “Rational Behavior, Uncertain Prospects and Measurable Utility,” Econometrica 18 (1905) 111-141.
McKinsey, J. C. C., Introduction to the Theory of Games, RAND Series, McGraw-Hill, 1952.
Metzler, Lloyd, “A Multiple Region Theory of Income and Trade,” Econometrica 18 (1950) 329-354.
Morgenstern, Oskar and Von Neumann, John, Theory of Games and Economic Behavior.
Von Neumann, John, “The Personnel Assignment Problem and a Certain Two Person Game,” to appear in Contributions to the Theory of Games, Annals of Mathematics Studies.
---, “Über ein. Ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes,” Ergebnisse eines Mathematischen Kolloquiums 1 (1937) 73-83.
Pareto, V., “Anwendung der Mathematik auf Nationalökonomie,” Encyklopädie Math. Wiss. (Arithmetik und Algebra), Zweiter Teil, Leipzig, 1900-1904, pp. 1094-1120.
Roos, C. F., “A Mathematical Theory of Competition,” Amer. Journal Math. 67 (1925) 163-175.
Samuelson, Paul A., Foundations of Economic Analysis, Harvard, 1947.
---, “Abstract of a Theorem Concerning Substitutability in Open Leontief Models,” in [29], pp. 142-147.
Slutsky, E., “Sulla Teoria de Bilancio del Consumatore,” Giornale Economisti 60 (1915) 19-23.
Solow, Robert, “On the Structure of Linear Models, Econometrica 20 (1952) 29-47
Stone, Richard, “Comptabilité Social, Agrégation, Et Invariance,” Economie Appliquee 59 (1949) 26-54.
Wald, Abraham, Statistical Decision Function, Wiley, 1950.
---, Über einige Gleichunzssysteme der mathematischen Ökonomie,” Zeitsch. National-ökonomie 7 (1936) 637-670. Translated in Econometrica 29 (1951) 368-403.
Weyl, Herman, “Elementary Theory of Convex Polyhedra,” in [33], pp. 3-19, [translated by H. W. Kuhn from Commentarii Math. Helvet. 8 (1935) 290-306.]
Weilandt, Helmut, “Unzerlegbare, nicht-negative Matrizen,” Math. Zeitschr. 52 (1950) 642-648.
Wold, Herman, “A Synthesis of Pure Demand Analysis,” Part II, Skand. Aktuarietidskrift 26 (1943) 220-263.
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Article copyright:
© Copyright 1953
American Mathematical Society
