An explanation of constrained optimization for economists:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Toronto [u.a.]
Univ. of Toronto Press
2015
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | XVI, 480 S. graph. Darst. |
| ISBN: | 9781442614468 9781442646537 |
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Record in the Search Index
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| adam_text | Contents^
List of Figures xii
List of Tables xvii
1 Introduction 1
2 Basics 6
2d Space.............................................................. 6
2.2 Real Vector Space................................................ 7
2.3 Some Linear Algebra.............................................. 9
2 4 Direct Products of Sets......................................... 20
2 5 Addition of Sets................................................ 22
2-6 Convex Sets...................................................... 26
2 7 Partial Derivatives............................................. 33
2-8 Total Derivatives................................................ 38
2.9 Continuous Differentiability.................................... 44
2*10 Contour Sets ................................................... 46
2.11 Marginal Rates of Substitution ................................. 47
2-12 Gradient Vectors................................................ 50
2*13 Inner Products.................................................. 51
2.14 Gradients, Rates of Substitution, and Inner Products............ 55
2*15 Quadratic Forms................................................. 61
2.16 Critical Points................................................. 70
2-17 Hessian Matrices................................................ 70
2*18 Quadratic Approximations of Functions........................... 71
2*19 Saddle-Points................................................... 76
2*20 Metrics......................................................... 79
2.21 Norms........................................................... 82
vii
CONTENTS
viii
2.22 How to Proceed from Here...................................... 83
2.23 Problems...................................................... 84
2.24 Answers....................................................... 89
3 Basics of Topology 104
3.1 Topology....................................................... 104
3.2 Topological Bases.............................................. 107
3.3 Open and Closed Sets .......................................... 109
3.4 Metric Space Topologies....................................... Ill
3.5 Bounded Sets.................................................. 114
3.6 Compact Sets.................................................. 115
3.7 How Do Our Topological Ideas Get Used? ....................... 116
3.8 Problems...................................................... 118
3.9 Answers....................................................... 119
4 Sequences and Convergence 123
4.1 Sequences..................................................... 123
4.2 Bounded Sequences............................................. 124
4.3 Convergence................................................... 125
4.4 Subsequences.................................................. 129
4.5 Cauchy Sequences.............................................. 132
4.6 Remarks....................................................... 137
4.7 Problems...................................................... 139
4.8 Answers....................................................... 141
5 Continuity 145
5.1 Basic Ideas about Mappings.................................... 146
5.2 Continuity.................................................... 151
5.3 Semi-Continuity and Continuity................................ 155
5.4 Hemi-Continuity and Continuity................................ 158
5.5 Upper-Hemi-Continuity by Itself............................... 169
5.6 Joint Continuity of a Function................................ 175
5.7 Using Continuity.............................................. 176
5.8 Problems...................................................... 177
5.9 Answers....................................................... 181
CONTENTS ix
6 Hyperplanes and Separating Sets 191
6.1 Hyperplanes.................................................... 192
6.2 Separations of Sets............................................ 193
6.3 Separating a Point from a Set.................................. 197
6.4 Separating a Set from a Set ................................... 203
6.5 How Do We Use What We Have Learned?............................ 203
6.6 Problems....................................................... 205
6.7 Answers........................................................ 206
7 Cones 216
7.1 Cones ......................................................... 216
7.2 Convex Cones................................................... 218
7.3 Far leas’s Lemma............................................... 220
7.4 Gordan’s Lemma................................................. 224
7.5 Remarks ....................................................... 231
7.6 Problems....................................................... 232
7.7 Answers....................................................... 233
8 Constrained Optimization, Part I 238
8.1 The Constrained Optimization Problem........................... 238
8.2 The Typical Problem............................................ 241
8.3 First-Order Necessary Conditions............................... 242
8.4 What Is a Constraint Qualification?............................ 247
8.5 Fritz John’s Theorem........................................... 249
8.6 Constraint Qualifications and Farkas’s Lemma................... 255
8.7 Particular Constraint Qualifications........................... 257
8.8 Remarks........................................................ 265
8.9 Problems....................................................... 267
8.10 Answers....................................................... 269
9 Lagrange Functions 283
9.1 What Is a Lagrange Function?................................... 283
9.2 Revisiting the Karush-Kuhn-Tucker Condition.................... 286
9.3 Saddle-Points for Lagrange Functions........................... 287
9.4 Saddle-Points and Optimality................................... 291
9.5 Concave Programming Problems .................................. 293
9.6 Remarks........................................................ 297
X CONTENTS
9.7 Problems...................................................... 298
9.8 Answers....................................................... 299
10 Constrained Optimization, Part II 302
10.1 Necessary vs. Sufficient Conditions........................... 302
10.2 Second-Order Necessary Conditions............................ 304
10.3 Geometry of the Second-Order Necessary Condition............... 311
10.4 Second-Order Necessary Condition Testing....................... 320
10.5 Sufficient Conditions.......................................... 326
10.6 Second-Order Sufficient Condition Testing...................... 329
10.7 Remarks ....................................................... 331
10.8 Problems..................................................... 332
10.9 Answers........................................................ 334
11 Optimal Solutions and Maximum Values 341
11.1 Questions to Be Answered....................................... 341
11.2 Solution Correspondences, Maximum Values....................... 342
11.3 Existence of an Optimal Solution............................... 347
11.4 Theorem of the Maximum......................................... 351
11.5 Comparative Statics Analysis................................... 358
11.6 Problems...................................................... 360
11.7 Answers........................................................ 364
12 Comparative Statics Analysis, Part I 373
12.1 What Is Comparative Statics Analysis?........................ 373
12.2 Quantitative Comparative Statics Analysis...................... 374
12.3 Implicit Function Theorems..................................... 377
12.4 Starting Points................................................ 381
12.5 Assumptions................................................... 381
12.6 Using the First-Order Conditions............................... 382
12.7 Using the Maximum-Value Function............................... 391
12.8 Qualitative vs. Quantitative Analyses......................... 405
12.9 Problems....................................................... 407
12.10 Answers....................................................... 413
13 Comparative Statics Analysis, Part II 431
13.1 Qualitative Comparative Statics Analysis....................... 431
13.2 Primal and Primal-Dual Problems................................ 432
CONTENTS xi
1.5.3 Primal-Dual First-Order Conditions.......................... 434
13.4 The Primal-Dual Second-Order Condition ..................... 437
13.5 Qualitative Comparative Statics Results . ................... 440
13.0 An Example................................................... 442
13.7 The Second-Order Envelope Theorem........................... 446
13.8 Le Chatelier’s Principle..................................... 456
13.9 More Comparisons of Comparative Statics...................... 459
13.10 Problems.................................................... 460
13.11 Answers..................................................... 461
Bibliography 474
Index
475
AN EXPLANATION OF CONSTRAINED OPTIMIZATION
FOR ECONOMISTS
In a constrained optimization problem, the decisionmaker wants to
select the “optimal” choice ֊ the one most valuable to him or her ֊ that
meets all of the constraints imposed by the problem. Such problems
are at the heart of modern economics, where the typical behavioral
postulate is that a decisionmaker behaves “rationally”; that, is, chooses
optimally from a set of constrained choices.
Most books on constrained optimization are technical and full of
jargon that makes it hard for the inexperienced reader to gain an holis-
tic understanding of the topic. Peter B. Morgan’s Explanation of Con-
strained Optimization for Economists solves this problem by emphasiz-
ing explanations, both written and visual, of the manner in which many
constrained optimization problems can be solved. Suitable as a text-
book or a reference for advanced undergraduate and graduate students
familiar with the basics of one-variable calculus and linear algebra, this
book is an accessible, user-friendly guide to this key concept.
PETER B. MORGAN is an associate professor in the Department of
Economics at the University at Buffalo.
|
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| spelling | Morgan, Peter B. 1949- Verfasser (DE-588)170104281 aut An explanation of constrained optimization for economists Peter B. Morgan Toronto [u.a.] Univ. of Toronto Press 2015 XVI, 480 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Optimierung (DE-588)4043664-0 gnd rswk-swf Nebenbedingung (DE-588)4140066-5 gnd rswk-swf Wirtschaftsmathematik (DE-588)4066472-7 gnd rswk-swf Optimierung (DE-588)4043664-0 s Nebenbedingung (DE-588)4140066-5 s Wirtschaftsmathematik (DE-588)4066472-7 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027863728&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027863728&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Morgan, Peter B. 1949- An explanation of constrained optimization for economists Optimierung (DE-588)4043664-0 gnd Nebenbedingung (DE-588)4140066-5 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd |
| subject_GND | (DE-588)4043664-0 (DE-588)4140066-5 (DE-588)4066472-7 |
| title | An explanation of constrained optimization for economists |
| title_auth | An explanation of constrained optimization for economists |
| title_exact_search | An explanation of constrained optimization for economists |
| title_full | An explanation of constrained optimization for economists Peter B. Morgan |
| title_fullStr | An explanation of constrained optimization for economists Peter B. Morgan |
| title_full_unstemmed | An explanation of constrained optimization for economists Peter B. Morgan |
| title_short | An explanation of constrained optimization for economists |
| title_sort | an explanation of constrained optimization for economists |
| topic | Optimierung (DE-588)4043664-0 gnd Nebenbedingung (DE-588)4140066-5 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd |
| topic_facet | Optimierung Nebenbedingung Wirtschaftsmathematik |
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