Cost Function: Cobb-Douglas and
CES Technology with Money Metric
Utility Function
Dr. Hemlata Manglani
Reduced Form of Cobb-Douglas
Technology
C ( w1 , w2 , y )  w1 x1 ( w1 , w2 , y )  w2 x2
b


 
b
1


b


b
 
     b  b  b

A
w1 w2 y
   
 b 
b 


  b 1
1
 b
 A 1
Contd.
   b       b
      w1 w2 y
b 

 b 

  b b b    b  w1 w2b y

 


1


b  1    b  w1 w2b y
  
  b
b    1 w1 w2 y
 b 


Contd.
  b   b
 b 
w1 w2 y

 b 
   1   b
  b   w1 w2 y
b 


 1   b
  (1   ) 
w1 w2 y

1   
   (1   ) 1 w1 w2b y


 Kw1 w2b y
 K    (1   ) 1
CES Technology



1
 
f ( x1 , x2 )  x1  x2
min w1 x1  w2 x2


such  that  x1  x2  y
p
First Order Conditions
w1  px1 1  0
w2  px2 1  0
x1  x2  y p

 1

 1

 1

 1
x1  w1 (p )
x2  w 2 (p )
Contd.
1
 1
x1 ( w1 , w2 , y )  w1
1
 1
2
x2 ( w1 , w2 , y )  w

 1

 1
 w1  w 2 




 1

 1
 w1  w 2 


1

y

1

y
Contd.
c( w1 , w2 , y )  w1 x1 ( w1 , w2 , y )  w2 x2 ( w1 , w2 , y )



 1



 1
 1
 1
 y  w1  w 2   w1  w 2 

 


 1

 1
 y  w1  w 2 


 1
p

1

Contd.
r   /  1


1
r r
2
c( w1 , w2 , y )  y w1r  w
Money Metric Utility Functions
Direct Money Metric Utility Function
There is a nice construction involving the expenditure
function that comes up in a variety of places in welfare
economics. Consider some prices p and some given
bundle of goods x.
We can ask the following question:
How much money would a given consumer need at the
prices p to be as well off as he could be by consuming
the bundle of good X?
Contd.
Good 2
The money metric utility function gives
the minimum expenditure at prices p
necessary to purchase a bundle at least as
good as x.
min pz
z
x
such  that  u ( z )  u ( x)
x
m( p, x)  e( p, u ( x))
Good 1
Indirect Money Metric Utility Function
Good 2
This function gives the minimum expenditure at
prices p for the consumer to be as well off as he
would be facing prices q And having income m
Optimal bundle at prices p and income u(p,q,m)
(q,m)
Optimal bundle at prices q and income
Good 1
Contd.
u ( p, q, m)  e( p, v(q, m))
This measures how much money one would need
at prices p to be as well off as one would be
facing prices q and having income m.
Example: Direct Money Metric Utility
Function

e( p1 , p2 , u )  Kp1 p
1
2
u
m
v( p1 , p2 , m) 
 1
Kp1 p2
Money Metric Utility function

m( p, x)  Kp1 p
1
2
 Kp1 p12 x 1 x12
u ( x1 , x2 )
Ex: Indirect Money Metric Utility
Function

1
1
2
u ( p, q, m)  Kp p v(q1 , q2 , m)

1
 Kp p
1
2
m
Kq1 q12
 p1 p12 q1 q2 1m
Example: For The CES Utility Function
u ( x1 , x2 )  ( x 1 , x 2 )1 / 
e( p, u )  ( p1r  p2r )1 / r u
r 1 / r
2
v ( p, m)  ( p  p )
r
1
m
m( p, x )  ( p1r  p 2r )1 / r ( x 1 , x 2 )1 / 
u ( p, q, m)  ( p1r  p 2r )1 / r ( q1r  q 2r ) 1 / r m
Roy’s Identity
 v( p, m) / p1 1 / r ( p1r  p2r ) (11 / r ) mrp1r 1
x1 ( p, m) 

v( p, m) / m
( p1r  p2r ) 1 / r
r 1
1
p m
 r
( p1  p2r )