If fibers had a higher marginal profit per unit input than gasoline, gallons of

Question

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 $.50 per gallon for gasoline and $.75 per square foot for fiber. Then the respective marginal profits are
MpG=($.50) MPG=($.50)(72-3MG )=36-1.5MG
MpG=($.75) MPG=($.75)(80-4MG )=60-3MG
Setting these equal to each other and rearranging gives
MF= .5MG+8.
Solving this equation and the constraint MG+ MF=20 implies MG=8 thousand barrels and M_F=12 thousand barrels. This allocation generates 480 thousand gallons of gasoline and 672 thousand square feet of fiber. The firms 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 $3.75 per square foot.

Explanation / Answer

If fiber is cut in half, then the second marginal profit equation becomes .375 (80-4MF)= 30-1.5 MF So 30-.1.5 MF= 36-1.5MG 1.5MG-1.5MF=6 MG+MF= 4 MG +MF=20 2MG= 24 (add equations together) MG=12 MF=8 New optimal allocation is 12 to gas and 8 to fiber.

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