# Lecture 6 - Economics 101A (Lecture 6) Stefano DellaVigna...

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Economics 101A (Lecture 6) Stefano DellaVigna September 15, 2009

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Outline 1. Utility maximization 2. Utility maximization — Tricky Cases 3. Indirect Utility Function
1 Utility Maximization Nicholson, Ch. 4, pp. 114—124 (94—105, 9th) X = R 2 + (2 goods) Consumers: choose bundle x =( x 1 ,x 2 ) in X which yields highest utility. Constraint: income = M Price of good 1 = p 1 , price of good 2 = p 2 Bundle x is feasible if p 1 x 1 + p 2 x 2 M Consumer maximizes max x 1 ,x 2 u ( x 1 2 ) s.t. p 1 x 1 + p 2 x 2 M x 1 0 2 0

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Maximization subject to inequality. How do we solve that? Trick: u strictly increasing in at least one dimension. ( º strictly monotonic) Budget constraint always satis f ed with equality Ignore temporarily x 1 0 ,x 2 0 and check after- wards that they are satis f ed for x 1 and x 2 .
Problem becomes max x 1 ,x 2 u ( x 1 ,x 2 ) s.t. p 1 x 1 + p 2 x 2 M =0 L ( x 1 2 )= u ( x 1 2 ) λ ( p 1 x 1 + p 2 x 2 = M ) F.o.c.s: u 0 x i λp i for i =1 , 2 p 1 x 1 + p 2 x 2 M

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Moving the two terms across and dividing, we get: MRS = u 0 x 1 u 0 x 2 = p 1 p 2 Graphical interpretation.
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## Lecture 6 - Economics 101A (Lecture 6) Stefano DellaVigna...

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