Transcript PowerPoint

AEB 6184 – Direct Estimation of
Cost Functions
Elluminate 11
Formulation of the Cobb-Douglas Cost
Function

Starting with the basic Cobb-Douglas production function
formulation
min w1 x1  w2 x2
x1 , x2
s.t. Y  A0 x1 x2

Forming the Lagrangian
 L
 x  w1  
 1
L  w1 x1  w2 x2   Y  A0 x1 x2   
 L  w  
2
 x2
Y
0
x1
Y
0
x2

Solving for x1 in terms of x2
 Y x
w1

w2  Y

1
x2
w2 
 x2

 x1 
x2
 x1
w1 
Substituting this result into the production function and solving


for x2
w  
w  
A0  2 x2  x2  y  0  Y  A0  2 x2  x2
 w1  
 w1  
Y

A0

 w1  
 

  x2
 w2  
Y 
 x2 Y , w1 , w2    
 A0 
1
 

 w1     


w

 2 

Input demands and cost function
Y 
x1 Y , w1 , w2    
 A0 
1
 
1
 

 w2     


w

 1 

Y 
 w1     
x2 Y , w1 , w2    


A
w

 2 
 0
1


 Y      w2     

C Y , w1 , w2   w1  


 A
w

 1 
 0 

1


 
 w  Y 

2 

A

 0 


 w1  


w

 2 

 





Estimation

Defining a two equation system
1
 
   Yt    
1t  A0 ,  ,    Ct   w1t  

   A0 
 
1

  Yt    
 2t  A0 ,  ,    x1t    
  A0 


 w2t  


w

 1t 

 
1


 
  w   Yt 

2t  

A

 0 


 w1t  


w

 2t 

 







 w2t      

 
w

 1t  

Next, we construct an error matrix with 45 rows and 2
columns
   A , ,   A , , 
11
0
21
0



A
,

,

A
,

,




12
0
22
0

  A0 ,  ,    





A
,

,

A
,

,




 1N 0

2N
0

Maximum likelihood estimation is then
T   A0 ,  ,     A0 ,  ,  
min L  A0 ,  ,    ln
A0 , , 
2
T

The variance matrix for the estimates is given by the Hessian
matrix at the optimal value by the Cramer-Rao lower bound.
Estimation using the KLEM dataset
Parameter
A0
Estimate
0.1485
Std. Dev.
0.1756
T-Ratio
0.8456


0.7386
0.8456
0.2868
0.4288
2.5751
1.9722
Exercise


Extend the Cobb-Douglas function two four inputs and
refit the KLEM data.
Implement the same estimation for the Constant
Elasticity of Substitution function.