lecture5full

# lecture5full - V p 1 p 2 I = α α 1-α 1-α I p α 1 p...

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Recap Multiple goods Utility was the sum of two functions Returns to scale of each function determined whether or not we had a corner solution Kinked budget constraints

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Constant Elasticity of Substitution u ( x 1 , x 2 ) = ( α x - ρ 1 + ( 1 - α ) x - ρ 2 ) - 1
Constant Elasticity of Substitution 2

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Demands and Income x 1 ( p 1 , p 2 , I ) = α 1 ρ + 1 I ( 1 - α ) 1 ρ + 1 p ρ ρ + 1 2 p 1 ρ + 1 1 + α 1 ρ + 1 p 1

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Demands and Income 2
Demands and own price x 1 ( p 1 , p 2 , I ) = α 1 ρ + 1 I ( 1 - α ) 1 ρ + 1 p ρ ρ + 1 2 p 1 ρ + 1 1 + α 1 ρ + 1 p 1 when ρ = 0, get Cobb Douglas demands: x 1 ( p 1 , p 2 , I ) = α I ( 1 - α ) p 1 + α p 1 = α I p 1

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Inferior and Giffen Goods
Demands and other prices x 1 ( p 1 , p 2 , I ) = α 1 ρ + 1 I ( 1 - α ) 1 ρ + 1 p ρ ρ + 1 2 p 1 ρ + 1 1 + α 1 ρ + 1 p 1

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Demands and prices and income Ordinary demands are HD0 in prices and income. x 1 ( tp 1 , tp 2 , tI ) = x 1 ( p 1 , p 2 , I ) x 1 ( p 1 , p 2 , I ) = α 1 ρ + 1 I ( 1 - α ) 1 ρ + 1 p ρ ρ + 1 2 p 1 ρ + 1 1 + α 1 ρ + 1 p 1
Indirect utility functions u ( x 1 , x 2 ) = x α 1 x 1 - α 2 x 1 ( p 1 , p 2 , I ) = α I p 1 x 2 ( p 1 , p 2 , I ) = ( 1 - α ) I p 2

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Properties of indirect utility functions

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Unformatted text preview: V ( p 1 , p 2 , I ) = α α ( 1-α ) 1-α I p α 1 p 1-α 2 Property 1 : HD0 in prices and income Property 2 : decreasing in prices Property 3 : increasing in income Property 4: Roy’s Identity x 1 ( p 1 , p 2 , I ) =-∂ V ( p 1 , p 2 , I ) ∂ p 1 ∂ V ( p 1 , p 2 , I ) ∂ I Why does this work? V ( p 1 , p 2 , I ) is the Lagrangian at the best choices of x 1 and x 2 V ( p 1 , p 2 , I ) = L ( x * 1 , x * 2 , λ * ) = u ( x * 1 , x * 2 ) + λ * ( I-p 1 x * 1-p 2 x * 2 ) Why does this work 2?...
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lecture5full - V p 1 p 2 I = α α 1-α 1-α I p α 1 p...

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