9781118041581

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 a concrete example. Suppose the production functions are

Here gasoline output is measured in thousands of gallons, fiber output in thou-

sands of square feet, and crude oil in thousands of barrels. The products’ prof-

its per unit output are $.50 per gallon for gasoline and $.75 per square foot for

fiber. Then the respective marginal profits are

Setting these equal to each other and rearranging gives

Solving this equation and the constraint MGMF20 implies MG8 thou-

sand barrels and MF12 thousand barrels. This allocation generates 480 thou-

sand gallons of gasoline and 672 thousand square feet of fiber. The firm’s total

profit is $744 thousand (less the cost of the crude).

MF.5MG8.

MF($.75)MPF($.75)(804MF) 60 3MF.

MG($.50)MPG($.50)(723MG) 36 1.5MG

Fiber: F80MF2MF^2

Gasoline: G72MG1.5MG^2

214 Chapter 5 Production

CHECK

STATION 5

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.

FINAL REMARKS With respect to both the plant and product decisions, two

comments are in order. First, the appropriate marginal conditions are

extended easily to the case of multiple (more than two) plants and decisions.

(For instance, if there are three plants, the marginal product condition

becomes MPAMPBMPC.) Second, the decision framework changes sig-

nificantly if management is able to vary the amount of the input. If manage-

ment has access to as much crude oil as it wants (at a price), the problem can

be dealt with plant by plant or product by product. Indeed, we have already

considered the solution to this decision earlier: For each plant or each prod-

uct, use of the input should be expanded to the point where its marginal rev-

enue product equals its marginal cost per unit input (i.e., the input’s price).

Aluminum vs.

Steel in Cars

and Trucks

Pushed by stricter fuel-efficiency standards, steel and aluminum companies are

fighting to build the next generation of lightweight cars and trucks. The more

manufacturers can reduce the weight of vehicles, the more they can raise the

mileage per gallon (MPG) rating. By replacing steel with aluminum (which is

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