9781118041581

Why must this be the case? If marginal products differed (say, MPAMPB),

barrels should be shifted from the low-MP plant (refinery A) to the high-MP

plant (refinery B).

Let’s apply this rule in a specific example. Based on extensive studies, sup-

pose that management has estimated the following production functions for

the refineries:

where gasoline outputs are measured in thousands of gallons and quantities of

crude oil are measured in thousands of barrels. Marginal products are

Figure 5.4 shows the marginal product curve for each refinery and two possible

allocations. One is a naiveallocation calling for an equal split between the two

facilities: MAMB5 thousand barrels. Using the production functions, we find

total output to be 182.5 thousand gallons. However, the figure immediately points

out the inefficiency of such a split. At this division, the marginal product of the last

barrel of crude at refinery A greatly exceeds the marginal product of the last bar-

rel at refinery B (19 10). Barrels should be reallocated toward refinery A.

We can readily identify the optimal solution from Figure 5.4: MA8 thou-

sand barrels and MB2 thousand barrels.^10 At these allocations, each refin-

ery’s marginal product is 16. (To check this, refer to the marginal product

expressions just given.) The total output of gasoline from this allocation is 196

thousand gallons—a considerable improvement on the naive allocation.

Furthermore, no other allocation can deliver a greater total output.

Multiple Products

Firms often face the problem of allocating an input in fixed supply among dif-

ferent products. The input may be a raw material—for instance, DRAM com-

puter chips allocated to the various models of personal computers

manufactured by the firm—or it may be capital. Frequently, the input in short-

est supply is managerial labor itself. Which products of the firm are in greatest

need of managerial attention? Which top-level managers are best suited to

improve performance in a given product line?

Refinery B: MPB 20 2MB.

Refinery A: MPA 24 MA

Refinery B: QB20MBMB^2 ,

Refinery A: QA24MA.5MA^2

212 Chapter 5 Production

(^10) We can find these allocations directly using the facts that MPAMPBand MAMB10. Equating
marginal products implies 24 MA 20 2MB. Solving this equation and the quantity constraint
simultaneously gives the solution, MA8 thousand barrels and MB2 thousand barrels.
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