Returns to scale describe an increase in the scale of production -- greater resources devoted to capital and labour -- and the corresponding increase in output. Thus, three basic situations emerge in returns to scale: increasing, decreasing and constant. In increasing returns to scale, when inputs are increased by a factor of X, the outputs increase by a factor greater than X, meaning that the increased resources devoted to production generated an even greater increase in outputs. Conversely, with decreasing returns to scale, when inputs are increased by a factor X, outputs increase by less than X and, naturally, with constant returns to scale the increase in inputs and outputs is identical. All three scenarios, however, are modelled by the same basic production function.
Establish your production function. As a simple example, assume Q= 2K + 5L, where Q stands for output, K represents capital, or input, and L represents labour, or input.
Increase the scale of production by increasing both K and L by a given factor. Returning to the example, use the variable X to represent the increase in production scale. Thus, the value X must be multiplied by both K and L, increasing the investment in production. Note that although a variable is useful to illustrate the math involved, in a real situation the factor X would be a positive number greater than 1. A value of X = 2, for instance, would indicate a doubling in the scale of capital and labour inputs, while X = 1.5 would correspond to a 50 per cent increase in inputs.
Multiply the increase by both input factors. In the example, the new function would become: Q(prime) = X(2K) + X(5L). Keep in mind that Q(prime) is not the same as the original Q value. In fact, the purpose of this calculation is to compare the two values. The first function already established that Q = 2K + 5L, but we want to know what happens to Q if we increase inputs by X. In other words, we want to define Q(prime) to determine whether the increase in Q is greater, equal or less than the increase in inputs.
Solve the equation. In the case of the example: Q(prime) = X(2K) + X(5L) =2KX + 5LX = X(2K+5L), the original equation already established that Q = 2K + 5L, so that X(2K+5L) can be rewritten as X(Q), meaning that Q(prime) = X(Q).
Determine whether the new value of outputs increased in a greater, lesser or equal proportion to inputs. In the case of the example, Q(prime) is equal to the original value of Q multiplied by a factor of X (just like the inputs were), yielding constant returns to scale because the increased proportion of inputs is exactly the same as the increase in outputs.
Tips and warnings
- Be aware that very few production functions yield constant returns to scale like the example used. Try the functions Q = KL or Q = K^0.5 * L^0.5 to see examples of increasing and decreasing returns.
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