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Midterm 2014 Solutions - Econ 310 Microeconomic Theory II...

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Econ 310 Microeconomic Theory II Solutions to Midterm Exam, Winter 2014 Solution 1 The utility-max problem is max x 1 ; x 2 min h x 1 = 4 1 ; 16 x 1 = 4 2 i s:t: p 1 x 1 + p 2 x 2 ° y: Following the usual arguments regarding the min-utility function, it must be that x 1 = 4 1 = 16 x 1 = 4 2 at the optimum. That is, x 1 = 16 4 x 2 . The budget constraint must hold with equality by non-satiation. Then use it to solve for the Marshallians: x 2 = y 16 4 p 1 + p 2 and x 1 = 16 4 y 16 4 p 1 + p 2 : The indirect utility function is v ( p 1 ; p 2 ; y ) = x 1 = 4 1 = 4 s 16 4 y 16 4 p 1 + p 2 : (b) To get the indirect utility function, apply the duality of v ( p ;e ( p ;u )) = u ) 4 s 16 4 e ( p ;u ) 16 4 p 1 + p 2 = u ) e ( p ;u ) = (16 4 p 1 + p 2 ) u 4 16 4 : To get the Marshallians, there are two ways : (i) Use duality of x h 1 ( p ;u ) = x 1 ( p ;e ( p ;u )) = 16 4 16 4 p 1 + p 2 ± (16 4 p 1 + p 2 ) u 4 16 4 = u 4 x h 2 ( p ;u ) = x 2 ( p ;e ( p ;u )) = 1 16 4 p 1 + p 2 ± (16 4 p 1 + p 2 ) u 4 16 4 = u 4 16 4 : (ii) Apply Shepard°s lemma. A student gets full credit for correctly using either method.
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