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Collection of utility functions and correspondingindirect utility, expenditure and demand functions corrected versionThomas HerzfeldAdvanced Microeconomics ECH-32306September 23, 2011 Introduction This list serves as a tool to check your own work. Starting from the variousutility functions try to ﬁnd the corresponding demand, indirect utility andexpenditure functions. Prove their respective properties. CES utility function  u ( x ) = ( x ρ 1  +  x ρ 2 ) 1 /ρ where 0   =  ρ <  1Marshallian demand functions: x 1 ( p ,y ) =  p r − 11  y p r 1  +  p r 2 and  x 2 ( p ,y ) =  p r − 12  y p r 1  +  p r 2 with  r  =  ρ/ ( ρ − 1)Indirect utility function: v ( p ,y ) =  y (  p r 1  +  p r 2 ) − 1 /r Expenditure function: e ( p ,u ) =  u (  p r 1  +  p r 2 ) 1 /r Hicksian demand functions: x h 1 ( p ,u ) =  p r − 11  u (  p r 1  +  p r 2 ) 1 − 1 /r  and  x h 2 ( p ,u ) =  p r − 12  u (  p r 1  +  p r 2 ) 1 − 1 /r Cobb-Douglas utility function  u ( x ) =  x α 1 x 1 − α 2  where 0  < α <  1Marshallian demand functions: x 1 ( p ,y ) =  αy p 1 and  x 2 ( p ,y ) = (1 − α ) y p 2 Indirect utility function: v ( p ,y ) =  y  α p 1  α  1 − α p 2  1 − α 1  Expenditure function: e ( p ,u ) =  u   p 1 α  α   p 2 1 − α  1 − α Hicksian demand functions: x h 1 ( p ,u ) =  u αp 2 (1 − α )  p 11 − α and  x h 2 ( p ,u ) =  u (1 − α )  p 1 αp 2 α Quasi-linear utility function  u ( x ) =  v ( x 1 ) +  x 2 Let’s look at a concrete example of this function  u ( x ) =  x 1 / 21  +  x 2 .Marshallian demand functions: x 1 ( p ,y ) =   y p 1 if   y  ≤  p 2  p 22 4  p 21 if   y > p 2 x 2 ( p ,y ) =   0 if   y  ≤  p 2 y −  p 1 x 1  p 2 =  y p 2 −  p 2 4  p 1 if   y > p 2 Indirect utility function: v ( p ,y ) =  y p 1  1 / 2 if   y  ≤  p 2 y p 2 +  p 2 4  p 1 if   y > p 2 Expenditure function: e ( p ,u ) =   p 1 u 2 if   y  ≤  p 2 up 2  −  p 22 2  p 1 +  p 22 4  p 1 =  up 2  −  p 22 4  p 1 if   y > p 2 Hicksian demand functions: x h 1 ( p ,u ) =   u 2 if   y  ≤  p 2  p 22 4  p 21 if   y > p 2 x h 2 ( p ,u ) =   0 if   y  ≤  p 2 u −  p 2 2  p 1 if   y > p 2 Linear utility function  u ( x ) =  x 1  +  x 2 Marshallian demand functions: x 1 ( p ,y ) =  y/p 1  if   p 1  < p 2 any number between 0 and  y/p 1  if   p 1  =  p 2 0 if   p 1  > p 2 x 2 ( p ,y ) =  0 if   p 1  < p 2 any number between 0 and  y/p 1  if   p 1  =  p 2 y/p 2  if   p 1  > p 2 2  Indirect utility function: v ( p ,y ) =   y/p 1  if   p 1  < p 2 y/p 2  if   p 1  > p 2 Expenditure function: e ( p ,u ) =   up 1  if   p 1  < p 2 up 2  if   p 1  > p 2 Hicksian demand functions: Apply Shephard’s lemma to the expendi-ture function yields straight vertical Hicksian demand functions. x h 1 ( p ,u ) =  u if   p 1  < p 2 x h 2 ( p ,u ) =  u if   p 2  > p 1 Stone-Geary utility function  u  = ( x 1 − a 1 ) b 1 ( x 2 − a 2 ) b 2 where  b 1 ,b 2  ≥  0and  b 1  +  b 2  = 1This is the utility function underlying the Linear Expenditure System.Marshallian demand functions: x 1 ( p ,y ) =  a 1  +  b 1  p 1 ( y  −  p 2 a 2 ) and  x 2 ( p ,y ) =  a 2  +  b 2  p 2 ( y  −  p 1 a 1 )Indirect utility function: v ( p ,y ) =  b 1  p 1 ( y  −  p 2 a 2 )  b 1  b 2  p 2 ( y  −  p 1 a 1 )  b 2 Expenditure function: e ( p ,u ) =  p 1 a 1  +  p 2 a 2  +  e u   p 1 b 1  b 1   p 2 b 2  b 2 Hicksian demand functions: x h 1 ( p ,u ) =  a 1 + b 1  p 1 e u   p 1 b 1  b 1   p 2 b 2  b 2 and  x h 2 ( p ,u ) =  a 2 + b 2  p 2 e u   p 1 b 1  b 1   p 2 b 2  b 2 Leontief utility function  u  = min { x 1 ,x 2 } Marshallian demand functions:  x 1  =  x 2  =  x ( p ,y ) =  y p 1 +  p 2 Indirect utility function:  v ( p ,y ) =  y p 1 +  p 2 Expenditure function:  e ( p ,u ) =  u (  p 1  +  p 2 )Substitution between commodities in the angle would not change util-ity. Therefore, the Hicksian demand functions are constant  x h 1  =  u and  x h 2  =  u 3

Jul 23, 2017

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Jul 23, 2017
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