Calculus and Techniques of Optimization with Microeconomic ...

Calculus and Techniques of Optimization with Microeconomic Applications John Hoag Bowling Green State University, USA

World Scientific N E W JERSEY • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • HONG KONG • TAIPEI • C H E N N A I

Contents

Acknowledgments

XI

To The Instructor

xui

To The Student Introduction

xv XVII

Section I

Sets and Functions

Section II

What is Mathematics? Theorems and Proof Styles of Proof

9 9 10

Section III

Matrix Algebra Adding and Multiplying Matrices Determinants Matrix Inverse

13 14 23 30

Section IV

Linear Algebra Linear Combinations and Dependency Rank Solutions to Linear Equations Lines and Vectors Linear Functions

37 40 50 55 61 66

viii

Calculus and Techniques of Optimization with Microeconomic Applications

Section V

Calculus of One Variable Metric Spaces Limits Limits at Infinity Continuity Derivatives Differential Maximum

69 70 80 87 90 95 104 106

Section VI

Calculus of Several Variables Derivatives Total Differential Finding the (Total) Differential Maximum Quadratic Forms Inverse Functions Implicit Function Theorem Implicit Function Theorem — Linear Equations Implicit Function Theorem — General Case Convexity

115 117 124 125 128 140 142 149 153 154 161

Section VII

Techniques of Optimization Optimization Unconstrained Maximum Problem — One Variable Case Unconstrained Maximum Problem — Several Variables Constrained Maximum Problem Employing the Implicit Function Theorem The Method of Lagrange Summary — Unconstrained and Equality Constraints Inequality Constraints Kuhn-Tucker

175 175 175 177 179 179 182 186 189 191

Section VIII

Odds and Ends Duality Homogeneous Functions Homothetic Functions Linking Homogeneous Functions and Concavity Envelope Theorem Separating Hyperplanes

201 201 203 206 206 206 209

Contents

Comparative Statics An Example of Comparative Statics

213 215

Section IX

Preferences, Utility, and Demand Elements of a Maximum Preferences and Utility Transformations of Quasiconcave Functions Utility Maximization Interpreting the First-Order Necessary Conditions Demand Applying Comparative Statics to the Consumer Problem Indirect Utility Using Duality Revealed Preference Choice Under Uncertainty Allais Paradox Risk Aversion

217 217 220 228 231 232 233 234 238 238 241 246 252 253

Section X

Supply Obtaining Total Variable Cost Profit Maximization and Supply in Terms of Output Profit Maximization, Several Inputs Profit Maximization, Several Inputs and Several Outputs Monopoly One Input and One Output

257 258 261 263 265 266 268

Section XI

Game Theory Games in Normal Form Dominant Strategies Mixed Strategies More Finding Nash Finding Nash for Three Players Interval Strategies Sets with Differentiable Payoffs Cournot with Quadratic Costs Cournot with Asymmetric Information Dynamic Games: The Stackelberg Model Extensive Form Why Nash Equilibrium?

271 274 276 282 293 305 315 317 318 320 321 324

Calculus and Techniques of Optimization with Microeconomic Applications

Section XII

Welfare and General Equilibrium Pure Trade Arrow's Theorem A Model with Production Competition and the Market System Welfare Theorems Competitive Equilibrium with Production Robinson Crusoe

327 327 332 334 337 340 341 342

Section XIII

Applications The Firm Over Time Pollution Problem Duopoly with Advertising The Averich-Johnson Effect

347 347 351 355 358

References

363

Index

365