A production function shows how much can be produced with a certain set of resources. Generally, when looking at production, we assume there are two factors involved in production: capital (K) and labour (L), as this allows us graphical representations of isoquants. However, any analysis made with 2 factors can mathematically be extended to n factors.
Therefore, a production function can be expressed as q = f(K,L), which simply means that q (quantity) is a function of the amount of capital and labour invested. In the adjacent figure, qx is function of only one factor, labour, and it can be graphically represented as shown (green).
It is well worth introducing here another concept: marginal productivity, which is how much more quantity we could produce by adding one unit more of a factor. As is logical, this will depend on how we are employing the factors we already have. The marginal product is the partial derivative of the production function with respect to the factor we are examining:
Marginal productivity decreases with each additional unit, as it can be seen in the above figure (cyan). At a certain point, the more workers we have, for example, the more each additional worker will be redundant if we do not invest in other necessary factors. This is the same as saying that the second derivative is negative. At that point, A, production is as efficient as possible.
Video – Production function: