In this article we will discuss about the production function of a firm.
The firm produces an output by using the inputs. Generally, the more of the inputs the firm uses the more would be the quantity of output it would be able to produce. That is, the quantity of output used depends upon the quantities of the inputs used.
This functional relation (of dependence) between the quantities of inputs used by the firm and the quantity of output produced by it is known as the production function. This relation is a mathematical or an engineering relation. The definition of production function obviously tells us that such a function reflects the firm’s technology.
The construction of a firm’s production function presumes:
(i) That the technology of the firm remains unchanged, and
(ii) That the firm uses and combines the inputs with maximum (possible) operational efficiency so that it could minimise the cost of production of a given quantity of output or maximise the quantity of output at a given amount of cost.
It follows from the above two points that a particular combination of the inputs would produce one, and only one, quantity of output, and that quantity would be the maximum possible quantity, or, a particular quantity of output can be produced at a particular amount of cost, and that amount would be the minimum possible amount.
In other words, we obtain here a one-to- one correspondence between the output produced and the cost incurred. It may be noted here that if the production technology did not remain constant and/or if the firm did not operate with maximum efficiency, then we could not obtain this one-to-one correspondence.
Because, then, we would have obtained a larger output at a constant cost (prices of the inputs remaining unchanged) if the technology improved and/or the efficiency improved. If we assume that the firm uses only two inputs X and Y, and produces only one output Q, then its production function may be written as
q = f (x, y) (8.1)
where x and y are the quantities used of the two inputs and q is the quantity produced of the output.
(8.1) is the general form of the firm’s production function. This function only tells us that the quantity produced of the commodity Q depends upon the quantities used of the inputs X and Y. It does not give us any further details of this dependence.
For example, from (8.1), we cannot know what would be the value of q at x = 10 units and y = 7 units, or, it does not tell us how q would change if x changed from 10 units to 15 units and y changed from 7 units to 10 units. If we want to have these information, we have to know the specific form of the production function. For example, if we are told that the specific form of the production function is
q = f (x, y) = xy (8.2)
then it would be possible for us to obtain that the value of q would be 70 units at x = 10 and y = 7, or, q would change from 70 units to 150 units if x increased from 10 to 15 and y increased from 7 to 10. Again, two other specific forms of the production function may be
q = 2x + 7y (8.3)
and q = x + xy + y (8.4)
From the production functions (8.3) and (8.4) also, we may obtain the values of q at different combinations of the two inputs or how q would change if the input quantities did change. One important characteristic of all these production functions is that q must increase if any or both of the input quantities increase, i.e., for all these functions, we must have
∂q/∂x > 0 and ∂q/∂y > 0.