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