301l4s - Microeconomics 301 Sergei Severinov Lecture 4 Part...

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Unformatted text preview: Microeconomics 301 Sergei Severinov Lecture 4, Part 2 • Consumer Choice Problem and con- sumer Demand. How to identify corner solutions? Ad- vanced. We can formulate the consumer’s choice problem as utility maximization problem: max u ( x 1 ,...,x n ) s.t. p 1 x 1 + ... + p n x n ≤ I 1 Solving the consumer’s choice problem using the Lagrangian method: Step 1. Form the Lagrangian function. L = u ( x 1 ,...,x n )+ λ ( I- p 1 x 1- ...- p n x n ) λ is a so-called Lagrange multi- plier on the budget constraint. 2 Solution is characterized by the following first-order conditions: x i × ∂ L ∂x i = x i u i- λp i = 0 That is: either ∂ L ∂x i = u i- λp i = 0 or x i = 0 and λ ∂ L ∂λ = λ ( I- p 1 x 1- ...- p n x n ) = 0 . (Second-order conditions: con- cavity of u ( .,.,. ).) If x i > 0, then rearranging, we get: u i p i = u j p j = λ or u i u j = p i p j 3 Consider the following example: max u ( x 1 ,x 2 ) = x 1 / 3 1 x 1 / 3 2 when the budget constraint is: p 1 x 1 + p 2 x 2 ≤ I . The Lagrangian is: L = x 1 / 3 1 x 1 / 3 2 + λ ( I- p 1 x 1- p 2 x 2 ) The first -order conditions are: x 1 ∂ L ∂x 1 = x 1 1 3 x 1 / 3 2 x 2 / 3 1- λp 1 = 0 x 2 ∂ L ∂x 2 = x 2 1 3 x 1 / 3 1 x 2 / 3 2- λp 2 = 0 λ ∂ L ∂λ = λ ( I- p 1 x 1- p...
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301l4s - Microeconomics 301 Sergei Severinov Lecture 4 Part...

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