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Answer Key - Problem Set 3

# Answer Key - Problem Set 3 - Prof Akbulut Answer Key...

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Prof. Akbulut Answer Key - Problem Set 3 (10 points) 1. (2 pts.) Let U x 1/3 y 2/3 be the utility function where x and y are two goods. p x and p y are respectively the prices of the two goods x and y, and M is the income of the consumer. (a) Derive the indirect utility function V p x , p y , M max U x , y x 1/3 y 2/3 s.t. p x x p y y M Lagrangian: £ x 1/3 y 2/3 M p x x p y y £/ x 1 3 y x 2 3 p x 0 £/ y 2 3 x y 1 3 p y 0 £/ M p x x p y y 0 x p x , p y , M 1 3 M p x y p x , p y , M 2 3 M p y V p x , p y , M 1 3 p x 1 3 2 3 p y 2 3 M (b) Derive the expenditure function E p x , p y , U E p x , p y , U 0 U 0 p y 2/3 2 3 p x 1/3 1 3 just by rearranging the idirect utility function Notice that we can get the expenditure function from the following as well: The dual problem to the above problem is min M p x x p y y s.t. x 1/3 y 2/3 U 0 Lagrangian: £ p x x p y y x 1/3 y 2/3 U 0 £/ x p x 1 3 y x 2 3 0 £/ y p y 2 3 x y 1 3 0 £/ U 0 x 1/3 y 2/3 0 x p x , p y , U 0 p y 2 p x 2 3 U 0 y p x , p y , U 0 2 p x py 1 3 U 0 E p x , p y , U 0 p x x p y y U 0 p y 2/3 2 3 p x 1/3 1 3

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