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AEB 6184 – Simulation and Estimation of the Primal
Elluminate - 6
Cobb-Douglas Parameters
1 1 1
1
2 2 21 2 3 1 1 2 2 3 3
2
3 3 3
3
w w xY
x p pY
w w xYAx x x w x w x w x
x p pY
w w xY
x p pY
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
Field Data for Corn
Total Costs and Revenues Parameters
Total Fuel 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.) 4.50
Corn Yield 135.0 Fertilizer (tons)
0.4116
Corn Price 4.50 Labor (hours)
4.15
Total Revenue 607.50
Profit per Acre 410.68
Revenues
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
1 2 3
1 2 3
0.0195 0.2500 0.0544
135151.484
0.8911
151.484 4.50 0.4116 4.15 135
YY Ax x x A
x x x
Y
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?
Drawing Prices
1
2
3
4.50
2.639,
367
7.97
p
wN
w
w
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 becomes2.2500 0.0000 0.0000 0.0000
0.0000 0.7738 5.3806 0.3505
0.0000 5.3806 14,965.4444 12.2499
0.0000 0.3505 12.2499 7.0579
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
In the multivariate world, we use the Cholesky decomposition of the variance matrix and use the vector of the means
Drawing Random Samples
i ix z
4 1 4 4: , , , , ,i i i ix Pz x z M P M PP
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
Price Draws
Production Levels
*1 1
1 1
1* 1
2 2 1 2 3 1 2 32 2
*3 3
3 3
p Ypx x
w w
p Ypx x Y A x Y x Y x Y Y Ax x x
w w
p Ypx x
w w
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
Input Demands and Output Levels
First stage estimation – Ordinary Least Squares
Second stage – System Ordinary Least Squares using the first-order conditions. From the first-order conditions
Estimation
1
1̂ ' ' 1| ln , lnx x x y x x y y
1 1 20 211 1
2 2 30 312 2
pY pYx x k k
w w
pY pYx x k k
w w
Each of the observations in the system can then be expressed as
Estimation (Continued)
10 11 1 12 2 13 3 1
1 20 21 21
2 30 31 32
ln ln ln lny k k x k x k x
pYx k k
w
pYx k k
w
Imposing the cross equation restrictions
Estimation (Continued)
10 11 1 12 2 13 3 1
1 20 11 21
2 30 12 32
ln ln ln lny k k x k x k x
pYx k k
w
pYx k k
w
First estimation (without heteroscedasticity)
Estimation (Continued)
11 21 31
1 11
1111
1 221
21
2
2
1
2
1 ln ln ln 0 0
ln 0 0 0 1 0
, 0 0 0 0 1
0 0 0 0 1
ˆ ' '
nn n
n
x x x
p Yywx
p Yyy xxx w
x p Yw
xx xx xx yy
With heteroscedasticity
Estimation (Continued)
11 1
3ˆ ˆ ˆ
n n n n
V n
xx I V xx xx I V yy
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
Results