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Returns to Scale
How many hours of labor should the firm hire, and how much output should it produce?
To answer these questions, we apply the fundamental rule
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First, observe that labor’s marginal product is MPL dQ/dL 60 2L. In turn, labor’s marginal revenue product is MRPL ($2)(60 2L) 120 4L. Setting this equal to $16, we obtain 120 4L 16. The optimal amount of labor is L 26 hours. From the production function, the resulting output is 884 units. Finally, the firm’s operating profit (net of its labor cost) is ($2)(884) ($16)(26) $1,352.
MRPL MCL.
PRODUCTION IN THE LONG RUN
In the long run, a firm has the freedom to vary all of its inputs. Two aspects of this flexibility are important. First, a firm must choose the proportion of inputs to use. For instance, a law firm may vary the proportion of its inputs to economize on the size of its clerical staff by investing in computers and software specifically designed for the legal profession. In effect, it is substituting capital for labor. Steeply rising fuel prices have caused many of the major airlines to modify their fleets, shifting from larger aircraft to smaller, fuel-efficient aircraft.
Second, a firm must determine the scale of its operations. Would building and operating a new facility twice the size of the firm’s existing plants achieve a doubling (or more than doubling) of output? Are there limits to the size of the firm beyond which efficiency drastically declines? These are all important questions that can be addressed using the concept of returns to scale.
Returns to Scale
The scale of a firm’s operations denotes the levels of all the firm’s inputs. A change in scale refers to a given percentage change in all inputs. At a 15 percent scale increase, the firm would use 15 percent more of each of its inputs. A key question for the manager is how the change in scale affects the firm’s output. Returns to scale measure the percentage change in output resulting from a given percentage change in inputs. There are three important cases.
Constant returns to scale occur if a given percentage change in all inputs results in an equal percentage change in output. For instance, if all inputs are doubled, output also doubles; a 10 percent increase in inputs results in a 10 percent increase in output; and so on. A common example of constant returns to scale occurs when a firm can easily replicate its production process. For instance, a manufacturer of electrical components might find that it can double its
outputby replicating its current plant and labor force, that is, by building an identical plant beside the old one.
Increasing returns to scale occur if a given percentage increase in all inputs results in a greater percentage change in output. For example, a 10 percent increase in all inputs causes a 20 percent increase in output. How can the firm do better than constant returns to scale? By increasing its scale, the firm may be able to use new production methods that were infeasible at the smaller scale. For instance, the firm may utilize sophisticated, highly efficient, large-scale factories. It also may find it advantageous to exploit specialization of labor at the larger scale. As an example, there is considerable evidence of increasing returns to scale in automobile manufacturing: An assembly plant with a capacity of 200,000 cars per year uses significantly less than twice the input quantities of a plant having a 100,000-car capacity. Frequently, returns to scale result from fundamental engineering relationships. Consider the economics of an oil pipeline from well sites in Alaska to refineries in the contiguous United States. Doubling the circumference of the pipe increases the pipe’s crosssectionalarea fourfold—allowing a like increase in the flow capacity of the pipeline. To sum up, as long as there are increasing returns, it is better to use larger production facilities to supply output instead of many smaller facilities.
Decreasing returns to scale occur if a given percentage increase in all inputs results in a smaller percentage increase in output. The most common explanations for decreasing returns involve organizational factors in very large firms. As the scale of the firm increases, so do the difficulties in coordinating and monitoring the many management functions. As a result, proportional increases in output require more than proportional increases in inputs.
Outputelasticity is the percentage change in output resulting from a 1 percent increase in all inputs. For constant returns to scale, the output elasticity is 1; for increasing returns, it is greater than 1; and for decreasing returns, it is less than 1. For instance, an output elasticity of 1.5 means that a 1 percent scale increase generates a 1.5 percent output increase, a 10 percent scale increase generates a 15 percent output increase, and so on.
Reexamine the production function in Table 5.1. Check that production exhibits increasing returns for low levels of input usage and decreasing returns for high levels of usage. Can you find instances of constant returns in the medium-input range?
A study of surface (i.e., strip) coal mining estimated production functions for deposits of different sizes.5 The study was based on a survey of Illinois mines that included information (for each mine) on the production of coal (in tons), the amount of labor employed (in hours), the quantity of earth-moving capital (in dollars), and the quantity of other capital (also in dollars). Significant
5G. A. Boyd, “Factor Intensity and Site Geology as Determinants of Returns to Scale in Coal Mining,” Review of Economics and Statistics (1987): 18–23.
