Essay about Mathematical Economics

33067 Words Feb 16th, 2012 133 Pages
Lecture Notes on Mathematics for Economists

Chien-Fu CHOU

September 2006

Contents
Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Static Economic Models and The Concept of Equilibrium Matrix Algebra Vector Space and Linear Transformation Determinant, Inverse Matrix, and Cramer’s rule Differential Calculus and Comparative Statics Comparative Statics – Economic applications Optimization Optimization–multivariate case Optimization with equality constraints and Nonlinear Programming General Equilibrium and Game Theory 1 5 10 16 25 36 44 61 74 89

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1

Static Economic Models and The Concept of Equilibrium

Here we use three elementary examples to illustrate the general
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In microeconomics, we even attempt to include every commodity (including money) in a general equilibrium model. Qd1 = D1 (P1 , . . . , Pn ; a) Qd2 = D2 (P1 , . . . , Pn ; a) . . . Qdn = Dn (P1 , . . . , Pn ; a) Qsn = Sn (P1 , . . . , Pn ; a) Qs2 = S2 (P1 , . . . , Pn ; a) Qs1 = S1 (P1 , . . . , Pn ; a) Qdn = Qsn Qd2 = Qs2 Qd1 = Qs1 Qdi : quantity demanded of commodity i Qsi : quantity supplied of commodity i Pi : market price of commodity i a: a factor that affects the economy Di (P1 , . . . , Pn ; a): demand function of commodity i Si (P1 , . . . , Pn ; a): supply function of commodity i We have three variables and three equations for each commodity/market.

2 Substituting the demand and supply functions, we have D1 (P1 , . . . , Pn ; a) − S1 (P1 , . . . , Pn ; a) ≡ E1 (P1 , . . . , Pn ; a) = 0 D2 (P1 , . . . , Pn ; a) − S2 (P1 , . . . , Pn ; a) ≡ E2 (P1 , . . . , Pn ; a) = 0 . . . . . . Dn (P1 , . . . , Pn ; a) − Sn (P1 , . . . , Pn ; a) ≡ En (P1 , . . . , Pn ; a) = 0. For a given a, it is a simultaneous equation in (P1 , . . . , Pn ). There are n equations and n unknown. In principle, we can solve the simultaneous equation to find the ∗ ∗ equilibrium prices (P1 , . . . , Pn ). A 2-market linear model: D1 = a0 + a1 P1 + a2 P2 , S1 = b0 + b1 P1 + b2 P2 , D2 = α0 + α1 P1 + α2 P2 , S2 = β0

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