AEB 6184 – Simulation and
Estimation of the Primal
Elluminate - 6
Cobb-Douglas Parameters
 Y w1
w1 x1
 
 
pY
 x1 p
 Y w2
wx
  
  Ax1 x2 x3  w1 x1  w2 x2  w3 x3    
  2 2
p
pY
 x2
 Y w3
w3 x3

 

p
pY
 x3
Field Data for Corn
Fuel Use (Field Operations)
Fertilizer
Dry Spread
0.15
Nitrogen
91.80
Disk-Chisel
1.70
Phosphorous
36.58
Field Cultivate
0.70
Potash
23.50
Planting
0.40
Spraying
0.10
Combine
1.45
Total Diesel
4.50
Diesel Price
2.64
Total Fuel
11.88
Total Fertilizer
151.88
Revenues
Total Costs and Revenues
Total Fuel
Parameters
11.88
α
0.0195
Total Fertilizer
151.88
β
0.2500
Total Labor
33.06
γ
0.0544
Total Variable Cost
196.82
Diesel (gal.)
Corn Yield
135.0
Fertilizer (tons)
Corn Price
4.50
Labor (hours)
Total Revenue
607.50
Profit per Acre
410.68
4.50
0.4116
4.15
Prices
2007
2008
Corn Price (FL)
4.00
4.50
Corn Price (GA)
4.50
4.60
Fertilizer Price
367
614
Fuel Price (Diesel)
2.639
3.393
Labor Price
7.97
8.91
Cobb-Douglas Function
Y  Ax1 x2 x3  A 
Y
135

 151.484
  
x1 x2 x3 0.8911
Y  151.484  4.500.0195  0.41160.2500  4.150.0544   135
Drawing Prices
 We need to think about three four prices




Corn Price
Diesel Price
Fertilizer Price
Labor Price
 We could assume normality
 How to choose Ω?
 Possible negative prices?
 p
 
 w1 
 w2 
 
 w3 
  4.50  

 
2.639
,
N 
  367  
 
 
  7.97  
 We could choose a uniform distribution.
 For our purpose here, let’s assume that the standard
deviation is 1/3 of the value of each price.
 In addition, let’s assume that the input prices have a
correlation coefficient of 0.35 and the output price is
uncorrelated.
 The variance matrix then becomes
 2.2500
 0.0000

 0.0000

 0.0000
0.0000
0.0000
0.0000 
0.7738
5.3806
0.3505 
5.3806 14,965.4444 12.2499 

0.3505
12.2499
7.0579 
Drawing Random Samples
 In the univariate form, given a mean of  and a
standard deviation of  we would create the random
sample by drawing a z from a standard normal
distribution
xi     zi
 In the multivariate world, we use the Cholesky
decomposition of the variance matrix and use the
vector of the means
xi    Pzi : xi , zi ,   M 41 , P,  M 44 , PP  
Price Draws
Corn
Fuel
Fertilizer
Labor
4.0317966
-7.1139459
207.26474
13.085498
6.2839929
3.7932016
376.34403
15.001602
7.4239697
3.1434042
371.20099
8.1599310
4.5525423
11.179552
505.75759
7.6772579
4.3730528
3.9382136
370.57437
7.5128182
5.6384301
-9.1074158
153.62440
6.2727747
1.6477721
5.5279925
445.25053
7.2384782
1.8284618
10.856964
522.64669
7.9042016
2.5232567
2.0854507
351.84155
12.358072
Production Levels
p
Yp 
*
x1  
 x1  

w1
w1 
1
p
Yp 


*
  
x2  
 x2     Y  A  x1Y   x2Y   x3Y    Y   Ax1 x2 x3  1   
w2
w2 
p
Yp 
*
x3  
 x3  

w3
w3 
Input Demands and Output Levels
Fuel
Fertilizer
Labor
Output
4.7781260
0.61742454
3.3704676
147.90853
7.8305355
0.85013512
8.4152978
170.02817
0.9229998
0.26157066
3.7495936
116.23514
2.8592925
0.38957212
4.1813763
132.05003
0.4462693
0.07103387
0.9507833
76.77729
0.2432353
0.06477872
0.9320548
74.06527
2.3879263
0.18145925
1.1241763
101.21032
5.2533449
0.31587940
13.261536
135.01835
Estimation
 First stage estimation – Ordinary Least Squares
ˆ1   x ' x 
1
 x ' y   x  1| ln  x   , y  ln  y 
 Second stage – System Ordinary Least Squares using
the first-order conditions.
 From the first-order conditions
x1  
pY
pY
 x1  k20  k21
w1
w1
x2  
pY
pY
 x2  k30  k31
w2
w2
Estimation (Continued)
 Each of the observations in the system can then be
expressed as
ln  y   k10  k11 ln  x1   k12 ln  x2   k13 ln  x3   1
x1  k20  k21
pY
 2
w1
x2  k30  k31
pY
 3
w2
Estimation (Continued)
 Imposing the cross equation restrictions
ln  y   k10  k11 ln  x1   k12 ln  x2   k13 ln  x3   1
x1  k20  k11
pY
 2
w1
x2  k30  k12
pY
 3
w2
Estimation (Continued)
 First estimation (without heteroscedasticity)
ln  y1  


x
 11 
yy   x21  ,




 x 
 2n 
ˆ2   xx ' xx 
1
1

0


xx  0



0

 xx ' yy 
ln  x11 
p1Y1
w11
ln  x21 
0
0
p1Y2
0
pnYn
w21
w2 n
ln  x31  0 0 

0
1 0


0
0 1



0
0 1

Estimation (Continued)
 With heteroscedasticity
V   

n
ˆ3  xx  I nn  Vˆ  xx
1
 
1
xx  I nn  Vˆ 1  yy

Results
Parameter
1
2
3
True
Alpha 0
2.1926
2.1837
2.1834
2.1804
Alpha 1
0.0069
0.0194
0.0193
0.0195
Alpha 2
0.2587
0.2503
0.2499
0.2500
Alpha 3
0.0519
0.0511
0.0541
0.0544
Kappa 1
0.3827
0.4233
Kappa 2
-0.0003
0.0003