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Unformatted text preview: Envelope theorem The change in the value function when a parameter changes is equal to the derivative of the Lagrangian function with respect to the parameter, evaluated at the optimum choices. Example 1 Lagrange multiplier as shadow price Recall Lagrange Method – bivariate case The Lagrangian function for constrained maximization (or minimization) problem with the objective function subject to the equality constraints that 1 2 ( , ) f x x 1 2 ( , ) is g x x c = ( 29 1 2 1 2 1 2 ( , , ) ( , ) ( , ) x x f x x g x x c λ λ =   L ( 29 1 2 1 2 1 2 * ( , , ) ( , ) ( , ) x x f x x g x x c f c c λ λ λ =   ∂ ∂ = = ∂ ∂ L L • Recall the Soup & Salad example Budget (c) = 6 Price of Soup (P S )= 0.25 Price of Salad (P V )= 0.50 What is the marginal effect of budget changes in the utility? 1 1 ( , ) ln( ) ln( ) 2 2 U S V S V = + ? U c ∂ = ∂ • Utility from the consumption of soup and salad is 1 1 ( , ) ln( ) ln( ) 2 2 U S V S V = + ( 29 * * * * 1 1 ( , , ) ln( ) ln( ) 2 2 1 6 12 6 1 6 S V S V S V P S P V c S V U c c λ λ λ λ = + + = = = ∂ ∂ = = = ∂ ∂ L L Example 2 Hotelling’s lemma • A firm uses two inputs x 1 and x 2 , that cost w 1 and w 2 , to produce good y, which sells for price p • Production function is g(x 1 , x 2 ) • How does profit change as prices change? • Firm chooses x 1 and x 2 to maximize its profit 1 2 1 1 2 2 1 2 ( , , ) subject to ( , ) p w w py w x w x g x x y π =   = ( 29 1 2 1 1 2 2 1 2 * 1 2 1 2 * 1 2 1 2...
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This note was uploaded on 12/04/2011 for the course ECON 300 taught by Professor Cramton during the Fall '08 term at Maryland.
 Fall '08
 cramton

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