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.
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.