A farmer owns 1,000 acres of cropland suitable for growing corn, soybeans, sorghum, and wheat. The expected at-harvest price for each of these crops is: $2.50 per bushel for corn, $6.35 per bushel for soybeans, $2.75 per bushel for sorghum, and $3.10 per bushel for wheat. The farmer expects the following yields for each crop: 135 bushels per acre for corn, 45 bushels per acre for soybeans, 95 bushels per acre for sorghum, and 100 bushels per acre for wheat. In order to grow each crop, the farmer has to plow the land, plant each of the crops, and harvest each of the crops. The following sum- marizes the labor requirements and variable cost for each operation:
|
Production Operation
|
Labor Requirement (hours/acre)
|
Variable Cost ($/acre)
|
|
Plow
|
0.50
|
10
|
|
Plant Corn
|
0.30
|
75
|
|
Plant Soybeans
|
0.30
|
35
|
|
Plant Sorghum
|
0.30
|
45
|
|
Plant Wheat
|
0.30
|
40
|
|
Harvest Corn
|
1.35
|
50
|
|
Harvest Soybeans
|
0.65
|
20
|
|
Harvest Sorghum
|
0.75
|
25
|
|
Harvest Wheat
|
0.60
|
44
|
Assume that the farmer is endowed with 5,000 hours of family labor and can hire up to an additional 600 hours of hired labor at a cost of $5.00 per hour. The farmer can rent up to 200 acres of land from a neighbor at a cost of $100 per acre. On the other hand, the farmer can also rent out up to 500 acres of her own 1,000 acres to another neighbor, who is willing to pay $100 per acre.
Suppose that this crop farmer wants to evaluate whether or not it would be profitable to add up to 60 dairy cows to her farm. Assume that the cost per cow is $500, and each cow can produce 15,000 pounds of milk per year which sells for $0.12 per pound. Assume that each cow requires the following annual diet, which will be produced entirely on the land owned by the farmer: (1) a concentrate made out of 35 bushels of corn mixed with 9 bushels of soybeans, and (2) 1.22 tons of hay. Therefore, in additionto corn, soybeans, sorghum, and wheat, the farmer must now grow hay. Assume that any excess hay that is grown beyond the need of the dairy herd can be sold at harvest for $75.00 per ton. The farmer expects a yield of 2.5 tons per acre for hay, a cost of $15 per acre for planting hay, and $16.00 per acre for harvesting hay. Also, it takes 0.25 hours per acre for planting hay, 0.50 hours per acre for harvesting hay, and 65 hours per cow for milking cows.
a. Formulate the LP model for this problem.
b. Solve it with Solver.
c. Derive an output supply function for milk, corn, and soybeans.
d. Derive an input demand function for land and another for family labor.