PS4_Sol

PS4_Sol - Problem Set 4 Microeconomics Spring 2008 1. An...

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Unformatted text preview: Problem Set 4 Microeconomics Spring 2008 1. An economy consists of three types of consumers: 1, 2 and 3. A consumer of type i has the utility function U i ( x i ;y i ) = A i x & i i y 1 & & i i and has income I i ; where i = 1 ; 2 ; 3 . All consumers face the same prices p x and p y for goods X and Y; respectively. (Note: x i and y i are the amounts of X and Y consumed by the type i consumer.) (a) Set up the type i consumer&s contrained utility maximization problem. Choose ( x i ;y i ) to max A i x & i i y 1 & & i i subject to p x x i + p y y i = I i (b) Construct the Lagrangian function and set up the corresponding unconstrained problem. L i = A i x & i i y 1 & & i i + & i [ I i & p x x i & p y y i ] Choose ( x i ;y i ;& i ) to max A i x & i i y 1 & & i i + & i [ I i & p x x i & p y y i ] (c) What are the necessary FOC s for maximization? L i x i = 0 ) A i i x & i & 1 i y 1 & & i i = & i p x L i y i = 0 ) A i (1 & i ) x & i i y & & i i = & i p y L i i = 0 ) p x x i + p y y i = I i (d) Interpret and express the FOC s as the relevant economic conditions. ( L i x i = 0) & ( L i y i = 0) ) A i & i x & i & 1 i y 1 & & i i A i (1 & & i ) x & i i y & & i i = i p x i p y ) & i y i (1 & & i ) x i = p x p y i.e.i....
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This note was uploaded on 02/15/2011 for the course ECON 101 taught by Professor Womer during the Spring '08 term at NYU.

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PS4_Sol - Problem Set 4 Microeconomics Spring 2008 1. An...

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