The upcoming discussion will update you about the difference between short-run and long-run production functions.
The firm may change only the quantities of the variable inputs in the short run when the quantities of the fixed inputs remain unchanged.
That is, in the short run, the output quantity can be increased (or decreased) by increasing (or decreasing) the quantities used of only the variable inputs. This functional relationship (of dependence) between the variable input quantities and the output quantity is called the short run production function.
We have to remember here, of course, that in the short-run, the firm uses a particular combination of fixed inputs, and its short-run production function is obtained in respect of that combination.
In the long run, however, all the inputs used by the firm, the variable inputs and the so called fixed inputs, all are variable quantities and the firm’s production is a function of all these inputs. This functional relation of dependence between all the inputs used by the firm and the quantity of its output is called the long run production function of the firm.
We may illustrate the difference between the short-run and the long run production functions in the following way. Let us suppose that the firm uses only two inputs X and Y to produce its output of one commodity, Q, and of these two inputs X is a variable input and Y is a fixed input.
Therefore, in this case, the firm’s short-run production function may be written as:
q = f(x, y̅) (8.5)
where y̅ is the fixed quantity of the fixed input y. The firm’s long run production function in this example would be:
q = f(x, y) (8.6)
where x and y are the variable quantities of the inputs X and Y.
We may write the firm’s short-run production function (8.5) in the following form also:
q = h(x) (8.7)
For, in our example, in the short-run, the change in the firm’s output depends on the change in the quantity used of the input X only.