Chaiyuth_ConsumerTheory - Notes on Microeconomic Theory...

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Notes on Microeconomic Theory Nolan H. Miller September 5, 2003
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Contents 1 The Economic Approach 1 2 Consumer Theory Basics 5 2.1 Commodities and Budget Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Demand Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Three Restrictions on Consumer Choices . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 A First Analysis of Consumer Choices . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Requirement 1 Revisited: Walras’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.1 What’s the Funny Equals Sign All About? . . . . . . . . . . . . . . . . . . . . 12 2.5.2 Back to Walras’ Law: Choice Response to a Change in Wealth . . . . . . . . 13 2.5.3 Testable Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.4 Summary: How Did We Get Where We Are? . . . . . . . . . . . . . . . . . . 15 2.5.5 Walras’ Law: Choice Response to a Change in Price . . . . . . . . . . . . . . 15 2.5.6 Comparative Statics in Terms of Elasticities . . . . . . . . . . . . . . . . . . . 16 2.5.7 Why Bother? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.8 Walras’ Law and Changes in Wealth: Elasticity Form . . . . . . . . . . . . . 18 2.6 Requirement 2 Revisited: Demand is Homogeneous of Degree Zero . . . . . . . . . . . 18 2.6.1 Comparative Statics of Homogeneity of Degree Zero . . . . . . . . . . . . . . 19 2.7 Requirement 3 Revisited: The Weak Axiom of Revealed Preference . . . . . . . . . . 20 2.7.1 Compensated Changes and the Slutsky Equation . . . . . . . . . . . . . . . . 21 2.7.2 Other Properties of the Substitution Matrix . . . . . . . . . . . . . . . . . . . 25 3 The Traditional Approach to Consumer Theory 26 3.1 Basics of Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 i
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Nolan Miller Notes on Microeconomic Theory ver: Aug. 29, 2003 3.2 From Preferences to Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1 Utility is an Ordinal Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.2 Some Basic Properties of Utility Functions . . . . . . . . . . . . . . . . . . . 34 3.3 The Utility Maximization Problem (UMP) . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 Walrasian Demand Functions and Their Properties . . . . . . . . . . . . . . . 44 3.3.2 The Lagrange Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.3 The Indirect Utility Function and Its Properties . . . . . . . . . . . . . . . . 47 3.3.4 Roy’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.5 The Indirect Utility Function and Welfare Evaluation . . . . . . . . . . . . . 50 3.4 The Expenditure Minimization Problem (EMP) . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 A First Note on Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.2 Properties of the Hicksian Demand Functions and Expenditure Function . . . 56 3.4.3 The Relationship Between the Expenditure Function and Hicksian Demand . 59 3.4.4 The Slutsky Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.5 Graphical Relationship of the Walrasian and Hicksian Demand Functions . . 63 3.4.6 The EMP and the UMP: Summary of Relationships . . . . . . . . . . . . . . 64 3.4.7 Welfare Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.8 Bringing It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 Topics in Consumer Theory 73 4.1 Homothetic and Quasilinear Utility Functions . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.1 The Gorman Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.2 Aggregate Demand and Aggregate Wealth . . . . . . . . . . . . . . . . . . . . 78 4.2.3 Does individual WARP imply aggregate WARP? . . . . . . . . . . . . . . . . 80 4.2.4 Representative Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 The Composite Commodity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4 So Were They Just Lying to Me When I Studied Intermediate Micro? . . . . . . . . 85 4.5 Consumption With Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5.1 The Labor-Leisure Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Consumption Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.6.1 Discounting and Present Value . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ii
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Nolan Miller Notes on Microeconomic Theory ver: Aug. 29, 2003 4.6.2 The Two-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6.3 The Many-Period Model and Time Preference . . . . . . . . . . . . . . . . . . 97 4.6.4 The Fisher Separation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 Producer Theory 104 5.1 Production Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1.1 Properties of Production Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1.2 Pro fi t Maximization with Production Sets . . . . . . . . . . . . . . . . . . . . 110 5.1.3 Properties of the Net Supply and Pro fi t Functions . . . . . . . . . . . . . . . 112 5.1.4 A Note on Recoverability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 Production with a Single Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2.1 Pro fi t Maximization with a Single Output . . . . . . . . . . . . . . . . . . . . 117 5.3 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3.1 Properties of the Conditional Factor Demand and Cost Functions . . . . . . . 121 5.3.2 Return to Recoverability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 Why Do You Keep Doing This to Me?
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