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201Lecture102006

# 201Lecture102006 - Economics 201BSecond Half Lecture 10 The...

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Economics 201B–Second Half Lecture 10 The Index Theorem Definition : If p is a regular equilibrium price, define index ( p ) = ( 1) L 1 sign det D ˆ z | ˆ p For L = 2, index ( p ) = ( 1) 1 sign det D ˆ z | ˆ p = sign ˆ z p ) 1. index ( p ) = +1 means that ˆ z p ) < 0; that means de- mand is downward sloping, so we are in the “normal” case in which an increase in ˆ z , the excess demand for good 1, results in an increase in the equilibrium price of good 1. 2. index ( p ) = 1 means that ˆ z p ) > 0; that means de- mand is upward sloping, so we are in the “abnormal” case in which an increase in ˆ z , the excess demand for good 1, results in an decrease in the equilibrium price of good 1. For L = 3, index ( p ) = ( 1) 2 sign det D ˆ z | ˆ p = sign det D ˆ z | ˆ p The sign of the determinant is +1 if orientation is preserved, -1 if orientation is reversed: 0 1 is obtained from 1 0 1

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by a counterclockwise rotation. D ˆ z | ˆ p 0 1 is obtained from D ˆ z | ˆ p 1 0 by a rotation; counterclockwise (orientation preserved) if det D ˆ z | ˆ p > 0 , index (ˆ p ) = +1 and clockwise (orientation reversed) if det D ˆ z | ˆ p < 0 , index (ˆ p ) = 1 For L = 3, index ( p ) = ( 1) 3 sign det D ˆ z | ˆ p = sign det D ˆ z | ˆ p The sign of the determinant is +1 if orientation is preserved, -1 if orientation is reversed: 1 0 0 , 0 1 0 , 0 0 1 is a right-handed system.

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201Lecture102006 - Economics 201BSecond Half Lecture 10 The...

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