A mathematical relationship between the quantity of output of a good and the quantities of the inputs required to make it. It is written as
f( ) is the mathematical notation for ‘is a function of’, i.e. ‘is related to’ or ‘depends on’. Thus the equation would read: the quantity of output of the good depends on the quantities of the inputs x1 (e.g. labour), x2 (machinery), x3 (raw materials), and so on. This is, of course, a very general statement. The next step is to specify a precise mathematical form for the equation. For example, we might consider that the relation between output and inputs is best described by a linear equation such as
Alternatively, we might consider the relation best described by a multiplicative function of the form:
Many other functional forms exist. Choice of a particular functional form is important, since different functions have different mathematical properties and involve different assumptions about the technological characteristics of the production process being described. In choosing a particular form, the aim is to achieve a good compromise between simplicity and ease of manipulation on the one hand and accuracy in describing the technological relationships on the other. Production functions may be specified for individual firms, in which case they are useful for deriving the cost curves of firms and their demand curves for factors of production, or they may relate to the economy as a whole, in which case they are useful in growth theory, the theory of income distribution and the theory of international trade.