Multiple Plants

employing different degrees of automation), what is the effect on output of expanding the labor force (for instance, adding extra shifts)? Does the industry exhibit economies of scale and, if so, over what range of outputs? (That is, will a 40 percent increase in plant scale deliver more than a 40 percent increase in output?)

Production data—though subject to measurement errors—are very useful to managers. Based on these data, the manager (often with the help of an operations research specialist) can estimate the mathematical relationship between levels of inputs and quantity of output. The principal statistical method for carrying out this task is regression analysis (the most important elements of which were discussed in Chapter 4). The end product of this analysis is a tangible representation of the firm’s production function.

OTHER PRODUCTION DECISIONS

Within the limits of its production technology, the firm’s managers face a number of important decisions. We have already discussed finding the optimal use of single input in the short run and choosing the best mix of inputs in the long run. We now consider two other decisions: (1) the allocation of a single input among multiple production facilities and (2) the use of an input across multiple products.

Multiple Plants

Consider an oil company that buys crude oil and transforms it into gasoline at two of its refineries. Currently it has 10 thousand barrels of oil under long-term contract and must decide how to allocate it between its two refineries. The company’s goal is to allocate supplies to maximize total output from the refineries. Let MA and MB represent the crude input at each refinery and QA and QB the gasoline outputs. The firm’s problem is:

The key to maximizing total output is to compare marginal products at the two refineries. Barrels of crude first should be allocated to the refinery at which the marginal product is greater. Let’s say this is refinery A. As additional barrels are allocated to this refinery, its marginal product diminishes, and it becomes worthwhile to allocate oil to refinery B as well.

In the final allocation of all 10 thousand barrels, output is maximized if and only if the marginal products of both refineries are equal, that is, when

Maximize Q QA QB, subject to MA MB 10 thousand.

MPA MPB.