lecture6full

# lecture6full - = p 1 h 1 p 1 p 2 u p 2 h 2 p 1 p 2 u...

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Recap Properties of demand functions income own price other prices Indirect utility functions Substitute demands into utility function HD0 in prices and income decreasing in prices increasing in income Roy’s identity

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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 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 )
Roy’s identity 2

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Expenditure minimization min x 1 , x 2 p 1 x 1 + p 2 x 2 subject to u ( x 1 , x 2 ) = u
Expenditure minimization 2 min x 1 , x 2 p 1 x 1 + p 2 x 2 subject to x α 1 x 1 - α 2 = u

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Expenditure minimization 3
Hicksian (compensated) demands

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Properties of Hicksian demands Property 1 : HD0 in prices Property 2 : Decreasing in own price
Expenditure functions In the same way that we substituted ordinary demands into the utility function to get the indirect utility function , we can substitute Hicksian demands into expenditures to get the expenditure function E ( p 1 , p 2 , u

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Unformatted text preview: ) = p 1 h 1 ( p 1 , p 2 , u ) + p 2 h 2 ( p 1 , p 2 , u ) Expenditure functions 2 Properties of expenditure functions E ( p 1 , p 2 , u ) = p α 1 p 1-α 2 u ( 1-α ) 1-α α α Property 1 : HD1 in prices Property 2 : increasing in prices Property 3 : increasing in utility Property 4: Shephard’s Lemma ∂ E ( p 1 , p 2 , u ) ∂ p 1 = h 1 ( p 1 , p 2 , u ) Expenditure functions and Indirect utility functions E ( p 1 , p 2 , u ) = p α 1 p 1-α 2 u ( 1-α ) 1-α α α V ( p 1 , p 2 , I ) = α α ( 1-α ) 1-α I p α 1 p 1-α 2 Indirect utility functions and Expenditure functions E ( p 1 , p 2 , u ) = p α 1 p 1-α 2 u ( 1-α ) 1-α α α V ( p 1 , p 2 , I ) = α α ( 1-α ) 1-α I p α 1 p 1-α 2...
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lecture6full - = p 1 h 1 p 1 p 2 u p 2 h 2 p 1 p 2 u...

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