A farmer owns 500 acres of land, which is suitable for growing corn, soybeans, and sunflowers. His expectations are that the net profit from producing each crop is: $55 per acre for corn, $60 per acre for soybeans, and $50 per acre for sunflowers. He and his family can supply 3,000 hours per year in performing all the farm operations nec- essary to grow these crops. In addition, he is endowed with the equivalent of 4,500 hours of tractor time necessary to grow these crops. Assume that the only resources necessary in crop production are land, labor, and tractor time; the technological rela- tionships are summarized below.
|
Resource (unit)
|
Corn
|
Crop
Soybeans
|
Sunflowers
|
Resource Endowment
|
|
Land (acres)
|
1.0
|
1.0
|
1.0
|
500
|
|
Labor (hours)
|
0.4
|
0.2
|
0.3
|
3,000
|
|
Tractor (hours)
|
0.5
|
0.2
|
0.4
|
4,500
|
Assume that the farmer's objective is to maximize total profits.
a. Solve this problem using the simplex method.
b. Derive and graph the output supply functions for each crop. Output supply in this context is the relationship between acres produced and net profitability of the crop.
c. Derive and graph the input demand functions for land and labor.