Extrema of Smooth Functions: With Examples from Economic Theory

ISBN-13: 9783642767951 / Angielski / Miękka / 2011 / 197 str.

Mohamed A. El-Hodiri

Opis

It is not an exaggeration to state that most problems dealt with in economic theory can be formulated as problems in optimization theory. This holds true for the paradigm of "behavioral" optimization in the pursuit of individual self interests and societally efficient resource allocation, as well as for equilibrium paradigms where existence and stability problems in dynamics can often be stated as "potential" problems in optimization. For this reason, books in mathematical economics and in mathematics for economists devote considerable attention to optimization theory. However, with very few exceptions, the reader who is interested in further study is left with the impression that there is no further place to go to and that what is in these second hand sources is all these is available as far as the subject of optimization theory is concerned. On the other hand the main results from mathematics are often carelessly stated or, more often than not, they do not get to be formally stated at all. Furthermore, it should be well understood that economic theory in general and, mathematical economics in particular, must be classified as special types of applied mathematics or, more precisely, of motivated mathematics since tools of mathematical analysis are used to prove theorems in an economics context in the manner in which probability theory may be classified. Hence, rigor and correct scholarship are of utmost importance and can not be subject to compromise.

Spis treści

I: Finite Dimensional Problems.- Chapter: No Major Constraints.- 1. Existence.- 2. First Order Necessary Conditions.- 3. Profit Maximizing Firms 1.- 4. First Order Sufficient Conditions.- 5. Second Order Necessary Conditions.- 6. Second Order Sufficient Conditions.- 7. Concavity, Second Order Conditions and Quadratic Forms.- 8. Profit Maximizing Firms 2.- 9. Non-negativity Constraints.- 10. Utility Maximizing Consumers 1.- 2: Equality Constraints.- 1. First Order Necessary Conditions.- 2. Utility Maximizing Consumers 2.- 3. First Order Sufficient Conditions.- 4. Second Order Necessary Conditions.- 5. Second Order Sufficient Conditions.- 6. Utility Maximizing Consumers 3.- 7. Cost Functions.- 8. Further Notes On Second Order Conditions.- 9. Notes On The Literature.- 3: Inequalities as Added Constraints.- 1. First Order Necessary Conditions.- 2. First Order Sufficient Conditions.- 3. Second Order Necessary Conditions.- 4. Second Order Sufficient Conditions.- 5. Notes On The Literature.- 4: Extensions and Applications.- 1. Relation To Saddle Value Problems.- 2. Extension To Vector Maxima.- 3. A Discrete Time Optimal Control Problem.- 4. Consumer’s Optimum, Once More.- 5. A Problem In Welfare Economics.- 6. Odds and Ends.- 7. More About Concavity And Its Generalizations.- 8. Notes On The Literature.- II: Variational Problems.- 5: The Problem of Bolza with Equality Constraints.- 1. Unconstrained Problem.- 2. Statement of The Problem of Bolza.- 3. First Order Necessary Conditions.- 4. Weierstrass And Second Order Necessary Conditions.- 5. Sufficient Conditions.- 6. Characterization For Problem A’.- 6: The Problem of Bolza with Equality-Inequality Constraints.- 1. Necessary Conditions.- 2. Sufficient Conditions.- 7: Extensions and Applications.- 1. An Optimal Control Problem.- 2. Extensions.- 3. Two Static Economic Examples.- 4. Investment Demand.- 5. Skilled Worker’s Pay.- Constrained Extrema of Differentiable Functionals, Asurvey.- 1. Introduction.- 2. Finite Dimensional Problems.- 3. A Variational Problem.- 4. A Problem In Topological Linear Spaces.- References.- Epilogue.