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PS3+solution

# PS3+solution - Rice University Professor Diamond Economics...

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Rice University Economics 370 Professor Diamond Microeconomic Theory SOLUTION TO PROBLEM SET 3: UTILITY AND DEMAND AND INCOME AND SUBSTITUTION EFFECTS I. Suppose that an individual’s indifference curves are described by straight lines with a slope equal to –b , and that, as usual, prices and income, p 1 , p 2 , m are determined exogenously. 1. What is the demand function for x 1 ? The budget line is x 2 m/p 2 slope=-p 1 /p 2 So, x 1 = m/p 1 if p p 1 < b p 2 m/p 1 x 1 0 if 1 > b p 2 p p 1 0 x 1 m/p 1 if = b 2 2. What is the demand function for x 2 ? x 2 = m/p 2 if p p 1 > b p 2 0 if 1 < b p 2 p p 1 0 x 2 m/p 2 if = b 2

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2 2 II. Suppose an individual’s utility function (defined over two goods) is expressed as u ( x 1 , x 2 ) = x 1 x 2 . 1. Derive the demand function for x 1 . 2 2 2 1 2 1 2 2 1 1 2 2 2 2 1 2 1 1 2 1 2 1 1 2 2 1 1 1 2 1 2 1 1 , 2 2 2 2 2 3 u u mu x mu xx x x mu p x x mu p xx x p x x p px p x m p px p x m p m x p = = = = = = = = + = + = = 1 2. Derive the demand function for 1 1 2 1 2 2 1 2 2 2 3 3 p p m x x p p p = = = x 2 . 2 m p III. Suppose now that the utility function is u ( x 1 , x 2 ) = ln( x 1 ) + 2 ln( x 2 ) . x 1 and x 2 . 1. Again, derive the demand functions for both
4 1 2 1 1 2 1 1 2 1 2 1 2 2 1 2 1 1 1 2 1 2 1 1 1 1 2 1 2 2 1 2 1 2 , 2 2 2 3 2 2 2 3 3 u u mu mu 2 x x x mu p x p x x x mu p x p p px p x m p m x p p p m m x x p p p p = = = = = = = + = = = = = 2. Are your answers for II and III the same? Why or why not?

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