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Contents
1 Preliminaries on Modern Economics and Mathematics 1
1.1 Nature of Modern Economics . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Modern Economics and Economic Theory . . . . . . . . . . . . . . 1
1.1.2 Key Assumptions and Desired Properties Commonly Used Economics 3
1.1.3 The Basic Analytical Framework of Modern Economics . . . . . . . 4
1.1.4 Methodologies for Studying Modern Economics . . . . . . . . . . . 6
1.1.5 Roles, Generality, and Limitation of Economic Theory . . . . . . . 8
1.1.6 Roles of Mathematics in Modern Economics . . . . . . . . . . . . . 10
1.1.7 Conversion between Economic and Mathematical Languages . . . . 11
1.1.8 Distinguish between Necessary and Su±cient Conditions as well
Normative and Positive Statements . . . . . . . . . . . . . . . . . . 11
1.2 Language and Methods of Mathematics . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Separating Hyperplane Theorem . . . . . . . . . . . . . . . . . . . . 14
1.2.3 Concave and Convex Functions . . . . . . . . . . . . . . . . . . . . 14
1.2.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.5 The Envelope Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.6 Point-to-Set Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.7 Continuity of a Maximum . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.8 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 26
I Individual Decision Making 30
2 Consumer Theory 32
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