Nolan Miller

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Notes on Microeconomic Theory
Nolan H. Miller September 5, 2003

Contents
1 The Economic Approach 2 Consumer Theory Basics 2.1 2.2 2.3 2.4 Commodities and Budget Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Demand Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three Restrictions on Consumer Choices . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 5 8 9

A First Analysis of Consumer Choices . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5

Requirement 1 Revisited: Walras’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.5.8 What’s the Funny Equals Sign All About? . . . . . . . . . . . . . . . . . . . . 12 Back to Walras’ Law: Choice Response to a Change in Wealth . . . . . . . . 13 Testable Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Summary: How Did We Get Where We Are? . . . . . . . . . . . . . . . . . . 15 Walras’ Law: Choice Response to a Change in Price . . . . . . . . . . . . . . 15 Comparative Statics in Terms of Elasticities . . . . . . . . . . . . . . . . . . . 16 Why Bother? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 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 2.7.2 Compensated Changes and the Slutsky Equation . . . . . . . . . . . . . . . . 21 Other Properties of the Substitution Matrix . . . . . . . . . . . . . . . . . . . 25 26

3 The Traditional Approach to Consumer Theory 3.1

Basics of Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

i

Nolan Miller 3.2

Notes on Microeconomic Theory

ver: Aug. 29, 2003

From Preferences to Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1 3.2.2 Utility is an Ordinal Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Some Basic Properties of Utility Functions . . . . . . . . . . . . . . . . . . . 34

3.3

The Utility Maximization Problem (UMP) . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 Walrasian Demand Functions and Their Properties . . . . . . . . . . . . . . . 44 The Lagrange Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 The Indirect Utility Function and Its Properties . . . . . . . . . . . . . . . . 47 Roy’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 The Indirect Utility Function and Welfare Evaluation . . . . . . . . . . . . . 50

3.4

The Expenditure Minimization Problem (EMP) . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.4.7 3.4.8 A First Note on Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Properties of the Hicksian Demand Functions and Expenditure Function . . . 56 The Relationship Between the Expenditure Function and Hicksian Demand . 59 The Slutsky Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Graphical Relationship of the Walrasian and Hicksian Demand Functions . . 63 The EMP and the UMP: Summary of Relationships . . . . . . . . . . . . . . 64 Welfare Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Bringing It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 73

4 Topics in Consumer Theory 4.1 4.2

Homothetic and Quasilinear Utility Functions . . . . . . . . . . . . . . . . . . . . . . 73 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.1...
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