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:
MπG = ($0.50)MPG = ($0.50)(72-3MG) = 36 - 1.5MG
MπF = ($0.75)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 thousand (less the cost of the crude).
Find the optimal crude oil allocation in the preceding example if the profit associated with fiber were cut in half, that is, fell to $.375 per square foot.