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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).
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