review_ch5 - Chapter 5 Review: Choice 5.1 Optimal choice...

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Unformatted text preview: Chapter 5 Review: Choice 5.1 Optimal choice The choice ( x * 1 ,x * 2 ) is an optimal choice for the consumer, i.e. it is a point on the highest indifference curve that just touches the budget line. 1 2 CHAPTER 5. REVIEW: CHOICE 5.2 Finding the optimal choice The utility maximization problem can be framed as a constrained maxi- mization problem as follows: max x 1 ,x 2 u ( x 1 ,x 2 ) such that p 1 x 1 + p 2 x 2 = m There are two ways to solve this (at least). The first way is simply to solve the constraint for one of the variables in terms of the other and then substitute it into the objective function and apply calculus. That is, x 2 ( x 1 ) = m p 2- p 1 p 2 x 1 (5.1) 5.2. FINDING THE OPTIMAL CHOICE 3 Now substitute this back into the utility function to get the unconstrained maximization problem max x 1 u ( x 1 ,m/p 2- ( p 1 /p 2 ) x 1 ) This kind of problem can be solved just by differentiating with respect to x 1 and setting the result to zero in the usual way. In other words, the F.O.C is ∂u ( x 1 ,x 2 ( x 1 )) ∂x 1 + ∂u ( x 1 ,x 2 ( x 1 )) ∂x 2 dx 2 dx 1 = 0 (5.2) Differentiating (5.1) gives dx 2 dx 1 =- p 1 P 2 and substituting this into (5.2) givesand substituting this into (5....
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This note was uploaded on 03/14/2010 for the course ECON microecono taught by Professor Yy during the Spring '10 term at Seoul National.

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review_ch5 - Chapter 5 Review: Choice 5.1 Optimal choice...

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