Economic Foundations of Symmetric Programming

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Foreword......Page 17
Preface......Page 19
1 Introduction......Page 21
Duality, Symmetry, and the Euler-Legendre Transformation......Page 24
Duality without Constraints......Page 26
Asymmetric Duality with Constraints......Page 29
Symmetric Dual Nonlinear Programs......Page 30
Appendix 1.1 – The Euler-Legendre Transformation......Page 32
References......Page 34
2 Lagrangean Theory......Page 35
Unconstrained Maximization......Page 36
Concave and Convex Functions......Page 37
Constrained Maximization......Page 38
Saddle Point Problem......Page 40
Homogeneous Functions......Page 41
A Symmetric Lagrangean Function......Page 42
Exercises......Page 45
Reference......Page 47
3 Karush-Kuhn-Tucker Theory......Page 48
Concave Nonlinear Programming......Page 52
Alternative Specifications of Nonlinear Problems......Page 58
Interpretation of Karush-Kuhn-Tucker Conditions......Page 60
Equilibrium Problem......Page 62
How to Solve Nonlinear Programming Problems......Page 64
Exercises......Page 65
Appendix 3.1: Constraint Qualification......Page 66
References......Page 68
4 Solving Systems of Linear Equations......Page 69
Product Form of the Inverse......Page 72
A Numerical Example of the Pivot Method......Page 74
The Geometric Meaning of a Solution......Page 79
Exercises......Page 81
Appendix 4.1: Determinants and Minors......Page 83
Appendix 4.2: Solution of a Linear System of Equations......Page 85
Preliminaries......Page 86
Asymmetric Quadratic Programming......Page 88
The Dual of the Least-Squares Problem......Page 90
Symmetric Quadratic Programming......Page 92
A Special Case of Self-Duality......Page 94
Numerical Example: The Dual of the Least-Squares Problem......Page 95
GAMS Command File: Least-Squares Example......Page 96
Exercises......Page 98
Appendix 5.1: Differentiation of Linear and Quadratic Forms......Page 100
Appendix 5.2: Eigenvalues and Eigenvectors......Page 101
GAMS Command File for Computing the Eigenvalues of an (n × n) Symmetric Matrix......Page 103
Appendix 5.3: Integrability Conditions......Page 105
References......Page 107
6 Linear Complementarity Problem......Page 108
The Complementary Pivot Algorithm......Page 110
Example of Symmetric Quadratic Programming as an LC Problem......Page 117
Input Data File for the Lemke Computer Program......Page 120
Output file from the Lemke Computer Program......Page 121
Solving the LCP by Quadratic Programming......Page 123
Example of Solution of LCP by Quadratic Programming......Page 125
Exercises......Page 127
References......Page 129
7 The Price Taker......Page 130
Derivation of the Dual LP Problem......Page 134
General Linear Model of Joint Production......Page 138
Numerical Example 1: Linear Joint Production......Page 143
Numerical Example 2: Two Plants, One Market......Page 145
GAMS Command File: Numerical Example 2......Page 146
The Primal Simplex Algorithm......Page 149
Numerical Example 3: The Primal Simplex Algorithm......Page 152
The Dual Simplex Algorithm......Page 153
Numerical Example 4: The Dual Simplex Algorithm......Page 155
Guidelines to Set Up LP Problems......Page 156
Exercises......Page 158
References......Page 160
Pure Monopolist......Page 161
Perfectly Discriminating Monopolist......Page 165
Discriminating Monopolist......Page 168
Perfectly Discriminating Monopolist with Multiple Plants......Page 171
Pure Monopolist with Asymmetric D Matrix......Page 172
Numerical Example 1: Pure Monopolist with Asymmetric D Matrix......Page 174
GAMS Command File: Numerical Example 1......Page 175
Numerical Example 2: Perfectly Discriminating Monopolist with Symmetric D Matrix......Page 177
Numerical Example 3: Perfectly Discriminating Monopolist with Asymmetric D Matrix: An Equilibrium Problem......Page 178
GAMS Command File: Numerical Example 3......Page 179
Numerical Example 4: Discriminating Monopolist with One Physical Plant and Two Markets......Page 181
Numerical Example 5: Discriminating Monopolist with Two Physical Plants and Two Markets......Page 183
GAMS Command File: Numerical Example 5......Page 185
Exercises......Page 188
Pure Monopsonist......Page 192
Perfectly Discriminating Monopsonist......Page 195
Perfectly Discriminating Monopsonist Respecified......Page 197
Perfectly Discriminating Monopolist and Monopsonist by SQP......Page 199
Pure Monopolist and Pure Monopsonist by SQP......Page 201
Pure Monopolist and Pure Monopsonist with Asymmetric D and G Matrices......Page 203
Perfectly Discriminating Monopolist and Perfectly Discriminating Monopsonist with Asymmetric D and G Matrices......Page 204
Numerical Example 1: Price Taker and Pure Monopsonist......Page 206
GAMS Command File: Numerical Example 1......Page 207
Numerical Example 2: Pure Monopolist and Pure Monopsonist by SQP with Asymmetric D and E Matrices......Page 211
Numerical Example 3: Price Taker and Perfectly Discriminating Monopsonist......Page 213
GAMS Command File: Numerical Example 3......Page 214
Exercises......Page 218
10 Risk Programming......Page 222
Risky Output Prices......Page 223
Risky Output Prices and Input Supplies......Page 227
Chance-Constrained Interpretation of Risk Programming......Page 229
Risky Output Prices and Input Supplies with Covariance......Page 231
Risky Technology......Page 233
Generalization......Page 238
Extension of the Primal-Dual Algorithm to Concave Programs......Page 240
Freund's Numerical Example of Risk Programming......Page 242
GAMS Command File: Freund's Example of Risk Programming......Page 246
Exercises......Page 251
References......Page 254
11 Comparative Statics and Parametric Programming......Page 255
Parametric Programming in LP Models......Page 256
Comparative Statics in QP Models......Page 260
LP Parametric Programming: Variation in Input Quantity b1......Page 261
First Increment of b1......Page 263
Second Increment of b1......Page 264
Second Decrement of b1......Page 265
LP Parametric Programming: Variation in Output Price c2......Page 266
First Increment of c2......Page 268
Second Increment of c2......Page 269
Second Decrement of c2......Page 270
Parametric Quadratic Programming by LCP......Page 271
Exercises......Page 278
References......Page 279
Model 1: Final Commodities......Page 280
Model 2: Intermediate and Final Commodities......Page 286
Model 3: Endogenous Income......Page 289
Model 4: Spatial Equilibrium – One Commodity......Page 291
Model 5: Spatial Equilibrium – Many Commodities......Page 296
Numerical Example 1: General Market Equilibrium Final Commodities......Page 297
GAMS Command File: Numerical Example 1......Page 298
Numerical Example 2: General Market Equilibrium Intermediate and Final Commodities......Page 302
GAMS Command File: Numerical Example 2......Page 304
Numerical Example 3: Spatial Equilibrium – One Commodity......Page 308
GAMS Command File: Numerical Example 3......Page 311
Numerical Example 4: Spatial Equilibrium – Many Commodities......Page 315
GAMS Command File: Numerical Example 4......Page 319
Appendix 12.1: Alternative Specification of GME......Page 324
Appendix 12.2: A Detailed Discussion of Spatial Equilibrium......Page 328
Appendix 12.3: Spatial Equilibrium, Many Commodities......Page 332
Exercises......Page 335
References......Page 337
13 Two-Person Zero- and Non-Zero-Sum Games......Page 338
Two-Person Zero-Sum Games......Page 339
Two-Person Non-Zero-Sum Games......Page 345
Algorithm for Solving a Bimatrix Game......Page 348
A Numerical Example of a Bimatrix Game......Page 349
Maximizing Expected Gain......Page 355
Exercises......Page 358
References......Page 359
14 Positive Mathematical Programming......Page 360
Phase I – Estimation of Output Marginal Cost......Page 368
Phase II – Estimation of the Cost Function......Page 370
Phase III – Calibrating Model and Policy Analysis......Page 372
Empirical Implementation of PMP......Page 373
Recovering Revenue and Cost Functions......Page 374
Phase II – Estimation of Marginal Costs......Page 375
Phase IV – Calibrating Model......Page 376
Symmetric Positive Equilibrium Problem – SPEP......Page 377
Phase II of SPEP – the Total Cost Function......Page 379
Phase III of SPEP – Calibrating Model for Policy Analysis......Page 380
Dynamic Positive Equilibrium Problem – DPEP......Page 381
The Dynamic Framework......Page 382
Phase I of DPEP – Estimation of Marginal Costs......Page 383
Phase II of DPEP – Estimation of the Cost Function......Page 387
Phase III of DPEP – Calibration and Policy Analysis......Page 388
Numerical Example 1: Dynamic Positive Equilibrium Problem......Page 389
GAMS Command File for Numerical Example 1......Page 396
Revisiting the Three Phases of the Traditional PMP......Page 417
Numerical Example 2: Arfini-Donati PMP Specification......Page 420
GAMS Command File for Numerical Example 2......Page 423
References......Page 430
MOS in Linear Programming......Page 432
Example of Primal MOS in LP......Page 433
Dealing with MOS......Page 435
MOS in QP Models......Page 438
Determining the Number of Solutions......Page 440
Example 1: Kaneko's Necessity but not Sufficiency......Page 442
Example 2: Kaneko's Necessity and Sufficiency......Page 444
Computing All Basic Feasible and Complementary Solutions......Page 446
Example 3: LP Problem Revisited......Page 448
Example 4: QP Problem (15.11) Revisited......Page 453
Exercises......Page 457
References......Page 458
(i) Symmetric Quadratic Programming (SQP)......Page 459
(ii) Asymmetric Quadratic Programming (AQP)......Page 460
(iv) Symmetric and Asymmetric Equilibrium Problems (SEP, AEP)......Page 461
II. Method......Page 462
V. Input......Page 465
Input Format......Page 466
VI. Output......Page 468
VII. How to Use the Lemke Program......Page 469
INPUT DATA in Packed Form......Page 470
OUTPUT (obtained in a (datafile.lis) file)......Page 471
INPUT DATA in Free Format......Page 473
OUTPUT (obtained in a (datafile.lis) file)......Page 474
OUTPUT (obtained in a (datafile.lis) file)......Page 476
INPUT DATA in Free Format......Page 478
OUTPUT (obtained in a (datafile.lis) file)......Page 479
INPUT DATA in Free Format......Page 482
OUTPUT (obtained in a (datafile.lis) file)......Page 483
References......Page 486
README File for Lemke7......Page 487
COMMON.FOR Subroutine......Page 489
initia.f Subroutine......Page 493
initia7.f Subroutine......Page 494
lemke.f Main......Page 498
matrd.f Subroutine......Page 503
matrix.f Subroutine......Page 504
newbas.f Subroutine......Page 520
newbas7.f Subroutine......Page 521
opnfil.f Subroutine......Page 522
pinput.f Subroutine......Page 523
pinput6.f Subroutine......Page 525
pivot.f Subroutine......Page 527
pprint.f Subroutine......Page 528
pprint5.f Subroutine......Page 534
pprint6.f Subroutine......Page 539
pprint7.f Subroutine......Page 542
sort.f Subroutine......Page 547
symval.f Subroutine......Page 549
vecprnt.f Subroutine......Page 557
Index......Page 559