ch11-constrained-optimization

# ch11-constrained-optimization - Envelope theorem The change...

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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.

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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

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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 = +

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( 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?

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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 1 1 1 * 1 2 1 2 2 2 2 ( , , ) ( , ) ( , , ) , , ) 0 ( , , ) , , ) 0 ( , , ) , , ) 0 p w w py w x w x g x x y

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