Assignment:
If fibers had a higher marginal profit per unit input than gasoline, gallons of crude should be switched from gasoline production to fiber production. Here is concrete example. Suppose the functions are:
Gasoline: G = 72MG - 1.5MG^2
Fiber: F = 80MG - 2MG^2
Here gasoline output is measured in thousands of gallons, fiber output in thousands of square feet, and crude oil in thousands of barrels. The products profits per unit output are $0.50 per gallon for gasoline and $0.75 per square foot for fiber. Then the respective marginal profits are:
MpG = ($0.50)(72 - 3MG) = 36 - 1.5MG
MpF = ($0.75)(80 - 4MF) = 60 - 3MF
Setting these equal to each other and rearranging gives:
MF = 0.5MG + 8
Solving this equation and the constraint MG + MF = 20 implies MG = 8 thousand barrels and MF = 12 thousand barrels. This allocation generates 480 thousand gallons of gasoline and 672 thousand square feet of fiber. The firm's total profit is $744 (less the cost of the crude).
Question: Find the optimal crude oil allocation in the preceding example if the profit associated with fiber were cut in half, that is, fell to $0.375 per square foot.