The returns-to-scale concept can be clarified by reexamining the production data in Table 7.1. Assume that the production system represented by those data is currently operating with one unit of input X and three units of input Y. Production from such an input combination would be 35 units. Doubling X and Y results in an input combination of X = 2 and Y = 6. Output from this input combination would be 72 units. A 100 percent increase in both X and Y increases output by 37 units (= 72 - 35), a 106 percent increase (= 37/35 = 1.06). Over this range, output increases more than proportionately to the increase in the productive factors. The production system exhibits increasing returns to scale over this range of input use.
The returns to scale of a production system can vary over different levels of input usage. Consider, for example, the effect of a 50 percent increase in X and Y from the input combination X = 2, Y = 6. increasing X by 50 percent results in employment of three units of that factor (= 2 X 1.5), whereas a 50 percent increase in Y leads to nine units (= 6 X 1.5) of that input being used. The new input combination results in 89 units of production. Therefore, a 50 percent increase in input employment generates only a 24 percent [= (89 - 72)/72] increase in output. Because the increase in output is less than proportionate to the underlying increase in input, the production system exhibits decreasing returns to scale over this range.
Isoquant analysis can be used to examine returns to scale for a two-input, single-output production system. Consider the production of Qj units of output by using the input combination of (X1,Y1). if doubling both inputs shifts production to Q2, and if Q2 is precisely twice as large as Qv the system is said to exhibit constant returns to scale over the range (X1,Y1) to (2X1,2Y1). if Q2 is greater than twice Q1, returns to scale are increasing; if Q2 is less than double Q1, the system exhibits decreasing returns to scale.
Returns to scale can also be examined graphically, as in Figure 7.10. in this graph, the slope of a curve drawn from the origin up the production surface indicates whether returns to scale are constant, increasing, or decreasing.7 A curve drawn from the origin with a constant slope indicates that returns to scale are constant. if a curve from the origin exhibits a constantly increasing slope, increasing returns to scale are indicated. if a production function increases at a decreasing rate, decreasing returns to scale are indicated.
A more general condition is a production function with first increasing, then decreasing, returns to scale. The region of increasing returns is attributable to specialization. As output increases, specialized labor can be used and efficient, large-scale machinery can be used in the production process. Beyond some scale of operation, however, further gains from specialization are limited, and coordination problems may begin to increase costs substantially. When coordination expenses more than offset additional benefits of specialization, decreasing returns to scale set in.
7 Both inputs X and Y can be plotted on the horizontal axis of Figure 7.10 because they bear constant proportions to one another. What is actually being plotted on the horizontal axis is the number of units of some fixed input combination.
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