Production Function and Returns to Scale
Suppose the Production Function is given by:
Q = f (K, L)
Now, let us assume a Cobb-Douglas production function which is homogeneous of degree one, which in turn means that when all inputs are increased in the same proportion, and this proportion can be factored out.
For instance, if all inputs are increased in proportion by ‘h’ and if the output will also increase by ‘h’, then the production function is said to be of homogeneous of degree one. This is also known as a linear homogeneous production function, and it implies constant returns to scale.
Such a function can be expressed as follows:
hQ = f (hK, hL)
hQ = hf (K, L)
However, if all inputs are increased in the same proportion by ‘h’, but the output does not increase by that proportion, then in such a case, a production function may be written as:
kQ = f (hK, hL)
Where ‘k’ denotes the k-times increase in output as a result of the h-times increase in both inputs.
Hence, ‘k’ may be greater than ‘h’, or equal to ‘h’ or less than ‘h’. If:
- k>h, then it represents an increasing return to scale
- k=h, then it represents constant returns to scale
- k<h, then it represents decreasing returns to scale
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