"A valuable and useful resource for researchers on this topic, especially those in operations research, input-output analysis, computable general equilibrium models, and applied linear algebra, as well as for economists and others. The book collects and unifies a large body of research."—Chris Shannon, University of California, Berkeley

• PREFACE ix 1. Nonparametric Analysis 3 1.1 Introduction 3 1.2 Quantitative Analysis 4 1.3 Nonparametric Analysis 6 1.4 An Example 8 1.5 Organization of the Book 11 2. Qualitative Comparative Statics and Stability 13 2.1 Introduction 13 2.2 Sign Solvability-Background 13 2.3 The Algorithmic Approach to Strong Sign Solvability 15 2.4 The Algebraic Approach to Sign Solvability 20 2.5 Sign Stability 27 2.6 Potential Stability 36 2.7 Conclusions 38 Appendix: Special Topics in Matrix Analysis 40 3. Information and Invertibility 49 3.1 Introduction 49 3.2 An Algebraic Analysis of Qualitative Invertibility 50 3.3 The Elimination Principle 56 3.4 Categorizing Information 60 3.5 The Invertibility of Matrices with Ranked Entries 63 3.6 The Invertibility of Matrices with Entries Specified within Given Intervals 81 3.7 Summary of the Algebraic Method 85 3.8 Stability 86 4. Applications in Qualitative Comparative Statics 92 4.1 Introduction 92 4.2 Algorithmic Principles 93 4.3 A Qualitative Analysis of the Oil Market Simulation Model 98 4.4 A Qualitative Analysis of the Oil and Gas Supply Model 104 4.5 Klein's Model 1 117 4.6 Summary 121 5. The Maximization Hypothesis 122 5.1 Introduction 122 5.2 Unconstrained Maximization 122 5.3 Application: Profit-Maximizing Choices of Inputs by a Competitive Firm 134 5.4 Constrained Maximization 136 5.5 Application: Minimizing the Cost of Production 141 Appendix: Advanced Topics in Matrix Analysis 143 6. The Correspondence Principle 150 6.1 Introduction 150 6.2 An Example 152 6.3 Stability and Sign Solvability 157 6.4 GM-Matrices 161 6.5 Scope of the Correspondence Principle 165 6.6 Interpretation 168 7. The Competitive Equilibrium: Comparative Statics 170 7.1 Introduction 170 7.2 A Competitive Economy 171 7.3 Some Historical Comments 174 7.4 Restatement of Sign Solvability 176 7.5 Sign Solvability under Walras's Law 180 7.6 Sign Solvability under Walras's Law and Homogeneity 196 8. The Competitive Equilibrium: Stability206 8.1 Introduction 206 8.2 Stability under Walras's Law and Homogeneity 208 NOTES 223 BIBLIOGRAPHY 227 NAME INDEX 239 SUBJECT INDEX 241

"A valuable and useful resource for researchers on this topic, especially those in operations research, input-output analysis, computable general equilibrium models, and applied linear algebra, as well as for economists and others. The book collects and unifies a large body of research."—Chris Shannon, University of California, Berkeley

• PREFACE ix 1. Nonparametric Analysis 3 1.1 Introduction 3 1.2 Quantitative Analysis 4 1.3 Nonparametric Analysis 6 1.4 An Example 8 1.5 Organization of the Book 11 2. Qualitative Comparative Statics and Stability 13 2.1 Introduction 13 2.2 Sign Solvability-Background 13 2.3 The Algorithmic Approach to Strong Sign Solvability 15 2.4 The Algebraic Approach to Sign Solvability 20 2.5 Sign Stability 27 2.6 Potential Stability 36 2.7 Conclusions 38 Appendix: Special Topics in Matrix Analysis 40 3. Information and Invertibility 49 3.1 Introduction 49 3.2 An Algebraic Analysis of Qualitative Invertibility 50 3.3 The Elimination Principle 56 3.4 Categorizing Information 60 3.5 The Invertibility of Matrices with Ranked Entries 63 3.6 The Invertibility of Matrices with Entries Specified within Given Intervals 81 3.7 Summary of the Algebraic Method 85 3.8 Stability 86 4. Applications in Qualitative Comparative Statics 92 4.1 Introduction 92 4.2 Algorithmic Principles 93 4.3 A Qualitative Analysis of the Oil Market Simulation Model 98 4.4 A Qualitative Analysis of the Oil and Gas Supply Model 104 4.5 Klein's Model 1 117 4.6 Summary 121 5. The Maximization Hypothesis 122 5.1 Introduction 122 5.2 Unconstrained Maximization 122 5.3 Application: Profit-Maximizing Choices of Inputs by a Competitive Firm 134 5.4 Constrained Maximization 136 5.5 Application: Minimizing the Cost of Production 141 Appendix: Advanced Topics in Matrix Analysis 143 6. The Correspondence Principle 150 6.1 Introduction 150 6.2 An Example 152 6.3 Stability and Sign Solvability 157 6.4 GM-Matrices 161 6.5 Scope of the Correspondence Principle 165 6.6 Interpretation 168 7. The Competitive Equilibrium: Comparative Statics 170 7.1 Introduction 170 7.2 A Competitive Economy 171 7.3 Some Historical Comments 174 7.4 Restatement of Sign Solvability 176 7.5 Sign Solvability under Walras's Law 180 7.6 Sign Solvability under Walras's Law and Homogeneity 196 8. The Competitive Equilibrium: Stability206 8.1 Introduction 206 8.2 Stability under Walras's Law and Homogeneity 208 NOTES 223 BIBLIOGRAPHY 227 NAME INDEX 239 SUBJECT INDEX 241

"A valuable and useful resource for researchers on this topic, especially those in operations research, input-output analysis, computable general equilibrium models, and applied linear algebra, as well as for economists and others. The book collects and unifies a large body of research."—Chris Shannon, University of California, Berkeley

• PREFACE ix 1. Nonparametric Analysis 3 1.1 Introduction 3 1.2 Quantitative Analysis 4 1.3 Nonparametric Analysis 6 1.4 An Example 8 1.5 Organization of the Book 11 2. Qualitative Comparative Statics and Stability 13 2.1 Introduction 13 2.2 Sign Solvability-Background 13 2.3 The Algorithmic Approach to Strong Sign Solvability 15 2.4 The Algebraic Approach to Sign Solvability 20 2.5 Sign Stability 27 2.6 Potential Stability 36 2.7 Conclusions 38 Appendix: Special Topics in Matrix Analysis 40 3. Information and Invertibility 49 3.1 Introduction 49 3.2 An Algebraic Analysis of Qualitative Invertibility 50 3.3 The Elimination Principle 56 3.4 Categorizing Information 60 3.5 The Invertibility of Matrices with Ranked Entries 63 3.6 The Invertibility of Matrices with Entries Specified within Given Intervals 81 3.7 Summary of the Algebraic Method 85 3.8 Stability 86 4. Applications in Qualitative Comparative Statics 92 4.1 Introduction 92 4.2 Algorithmic Principles 93 4.3 A Qualitative Analysis of the Oil Market Simulation Model 98 4.4 A Qualitative Analysis of the Oil and Gas Supply Model 104 4.5 Klein's Model 1 117 4.6 Summary 121 5. The Maximization Hypothesis 122 5.1 Introduction 122 5.2 Unconstrained Maximization 122 5.3 Application: Profit-Maximizing Choices of Inputs by a Competitive Firm 134 5.4 Constrained Maximization 136 5.5 Application: Minimizing the Cost of Production 141 Appendix: Advanced Topics in Matrix Analysis 143 6. The Correspondence Principle 150 6.1 Introduction 150 6.2 An Example 152 6.3 Stability and Sign Solvability 157 6.4 GM-Matrices 161 6.5 Scope of the Correspondence Principle 165 6.6 Interpretation 168 7. The Competitive Equilibrium: Comparative Statics 170 7.1 Introduction 170 7.2 A Competitive Economy 171 7.3 Some Historical Comments 174 7.4 Restatement of Sign Solvability 176 7.5 Sign Solvability under Walras's Law 180 7.6 Sign Solvability under Walras's Law and Homogeneity 196 8. The Competitive Equilibrium: Stability206 8.1 Introduction 206 8.2 Stability under Walras's Law and Homogeneity 208 NOTES 223 BIBLIOGRAPHY 227 NAME INDEX 239 SUBJECT INDEX 241