Please check

Here gasoline output is mesured in thousands of gallons, fiber output in the thousands of square feet, and crude oil in thousands of barrels. The product's profits per unit output are $.50 per gallon for gasoline and $.75 per square foot for fiber. The the respective marginal profits are:

Mg = ($.50)MPg = ($.50)(72 - 3Mg) = 36 - 1.5Mg

Mf = ($.75)MPf = ($.75)(80 - 4Mf) = 60 - 3Mf
Setting these equal to each other and rearranging gives:
Mf = .5Mg + 8.

Solving this equation and 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 sq ft of fiber. The firm's total profit is $744 thou.

I have to find the optimal crude oil from the above equation I did. The profit associtated with the fiber if it were cut in half, that is, fell to $.375 per sq ft.

Mf = ($.375)(80 - 4Mf) = 30 - 1.13Mf
Mg = 4 thou
Mf - 16 thou
480 thousand gallons of gasoline
336 square feet of fiber

Would the profit still remain the same?

Please check my math? Thanks

To check if the profit would remain the same when the profit associated with fiber is cut in half, we need to calculate the new profit.

First, let's calculate the new marginal profit for fiber, Mf_new:

Mf_new = ($.375)(80 - 4Mf_new) = 30 - 1.5Mf_new

Next, let's solve the equations Mf_new = .5Mg + 8 and Mg + Mf_new = 20 to find the new allocations of gasoline and fiber.

Mg + Mf_new = 20
Mg + (.5Mg + 8) = 20
1.5Mg = 12
Mg = 8 thousand barrels

Mf_new = .5Mg + 8
Mf_new = .5(8) + 8
Mf_new = 12 thousand barrels

Now we can calculate the new outputs of gasoline and fiber using the new allocations:

Gasoline output = 480 thousand gallons
Fiber output = 672 thousand sq ft

To calculate the new profit, we need to multiply the new outputs by the new profit per unit:

New profit = ($0.50 per gallon) * (480 thousand gallons)
+ ($0.375 per sq ft) * (672 thousand sq ft)

New profit = $240 thousand + $252 thousand = $492 thousand

Comparing the new profit ($492 thousand) to the previous profit ($744 thousand), we can see that the profit is not the same when the profit associated with fiber is cut in half.

Therefore, the profit would not remain the same if the profit associated with fiber fell to $0.375 per square foot.