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

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