The production function is the dependence of the quantity of goods produced on the corresponding factors of production, expressed with the help of an economic and mathematical model , with which it is manufactured. Consider this concept in more detail.
A production function always has a specific look, as it is designed for a specific technology. The introduction of new technological developments entails a change or the creation of a new type of dependence.
This function is used to find the optimal (minimum) amount of costs that are necessary for the manufacture of a certain amount of goods. For all production functions, regardless of what type of production they express, these general properties are characteristic:
• the growth in the volume of goods produced due to only one factor (resource) has a finite limit (only a certain number of workers can work normally in one room, since the number of places is limited by area);
• factors of production can be interchangeable (automation of the production process) and complementary (workers and tools).
In its most general form, the production function looks like this:
Q = f (K, L, M, T, N), in this formula
Q is the volume of goods produced;
K - equipment (capital);
M - the cost of materials and raw materials;
T - used technologies;
N - entrepreneurial ability.
Types of production functions
There are many types of this relationship that take into account the influence of one or several of the most important factors. However, two main types of the production function have gained the greatest popularity: a two-factor model of the form Q = f (L; K) and the Cobb-Douglas function.
Two-factor model Q = f (L; K)
This model considers the dependence of output (Q) on labor costs (L) and capital (L). Quite often, a group of iso-quanta is used to analyze this model. An isoquant is such a curve that connects all possible points of combinations of production factors, allowing to produce a specific volume of goods. On the X axis, labor costs are usually noted, and on the Y axis, capital. Several isoquants are drawn on the same graph, each of which corresponds to a certain volume of production using a specific technology. The result is an isoquant map with different quantities of manufactured goods. It will be the production function for this enterprise.
The following general properties are characteristic of isoquants:
• the farther the curve is from the origin, the higher the volume of output;
• the concave and downward view of the isoquant is due to the fact that a decrease in the use of capital with a stable volume of manufactured goods causes an increase in labor costs;
• the concave shape of the isoquantum curve depends on the maximum permissible rate of technological substitution (the amount of capital that can be replaced by 1 additional unit of labor).
Cobb-Douglas Function
This production function, named after two American discoverers, where the total volume of output Y depends on the resources used in the production process, for example, labor L and capital K. Its formula:
Y = AKαLβ,
where α and b are constants (α> 0 and b> 0);
K and L are capital and labor, respectively.
If the sum of the constants α and b is equal to unity, then it is generally accepted that such a function has a constant effect of the scale of production. If the parameters K and L are multiplied by any coefficient, then Y also needs to be multiplied by the same coefficient.
The Cobb-Douglas model can be applied to any particular company. In this case, α is the share of total costs spent on capital, and β is the share spent on labor. Cobb-Douglas models can also contain more than two variables. For example, if N is land resources, then the production function takes the form Y = AKαLβNγ, where γ is a constant (γ> 0), and α + β + γ = 1.