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indi_choice - 7 Individual choice Comparative statics Engel...

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7 Individual choice Comparative statics: Engel curves: Response of x to changes in m . O ff er curves: Response of x i to change in p i ( p j ). Example: Excise and Income taxes.
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7.1 Slutsky equation Theorem: ∂x j ( p , m ) ∂p i = ∂h j ( p , v ( p , m )) ∂p i ∂x j ( p , m ) ∂m x i ( p , m ) . Proof: Let x ( p , m ) and u = u ( x ) be a solution to Umax, then h j ( p , u ) x j ( p , e ( p , u )) . Di ff erentiate w.p. to p i and evaluate at p , obtain ∂h j ( p , u ) ∂p i = ∂x j ( p , m ) ∂p i + ∂x j ( p , m ) ∂m ∂e ( p , u ) ∂p i | {z } . = x i ( p ,m ) 7.2 Hicks and Slutsky compensations Setup: Prices change = Consumption changes = Indirect utility changes. Idea of compensations: Quantify ($) the changes. Hicks: m , such that original utility is a ff ordable. Corresponds to income e ff ect change. Slutsky: m , such that original consumption bundle is a ff ordable. Corresponds to the total e ff ect. Example: Excise and income taxes.
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7.3 Duality in consumption Prices normalized to make m = 1. Indirect utility: v ( p ) = max x u ( x ) , s.t. px = 1 . Direct utility (dual problem): u ( x ) = min p v ( p ) , s.t. px = 1 . Example: Cobb-Douglas. 7.4 Revealed Preferences Observe: ( p t , x t ) for some t . Suppose p t x t p t x , then u ( x t
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