solutionproblemset4 - For the expenditure function we just...

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UCLA Economics 11 – Fall 2006 Professor Mazzocco Problem Set 4 Answer Key 1). A consumer has the following utility function: U = β Ln(X) + (1- β )Ln(Y) a) Using the expenditure minimization approach, find the Hicksian (compensated) demands for X and Y and the expenditure function. b) Starting from the expenditure function find the indirect utility. c) Starting from the Hicksian demands find the Marshallian Demands. Answer Answer : We will solve the Lagrangian: L = p x X + p Y Y + λ ( U - β Ln(X) - (1- β )Ln(Y)) F.O.C. L x1 = P x λβ (1/X) = 0 (1) L x2 = P Y λ (1- β )(1/Y) = 0 (2) L = U - β Ln(X) + (1- β )Ln(Y)= 0 (3) Using equations (1) and (2), we get P X /P Y = β (1/X) /(1- β )(1/Y), and using this and equation (3), we get: X = exp(U-(1- β )ln((1- β )P X / β P Y )) [ ] β β β β β β - - = - - = 1 / ) 1 ( ) exp( )) / ) 1 ln(( ) 1 exp(( ) exp( Py Px U X Py Px U X and Y = exp(U- β ln( β P Y /(1- β )P X ))
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[ ] β β β β β β Px Py U Y Px Py U Y ) 1 /( ) exp( )) ) 1 /( ln( exp( ) exp( - = - = The above are compensated demands for good X and good Y.
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Unformatted text preview: For the expenditure function we just replace the compensated demands we have found into E = p x X + p Y Y . E = P X exp(U-(1-β )ln((1-β )P X / β P Y )) + P Y exp(U-β ln( β P Y /(1-β )P X )) E = [ ] [ ]---+-Px Py U Py Py Px U Px ) 1 /( ) exp( / ) 1 ( ) exp( 1---= 1 1 ) 1 ( ) exp( Py Px U E b) From E* we can solve directly for the utility level U ---= 1 1 ) 1 ( ) exp( Py Px I U -=--1 1 ) 1 ( Py Px I Ln U c) Solution for X: From the compensated demand: [ ]--=--= 1 / ) 1 ( ) exp( )) / ) 1 ln(( ) 1 exp(( ) exp( Py Px U X Py Px U X Plugging back the Indirect Utility function we get: X= β I/Px The same process for Y yields: Y=(1-β )I/Py...
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