Companies use production factors or inputs to produce products and offer services. A production function describes a relationship between the quantity of one or several inputs, and the quantity produced, with a given technology.
When a production function is expressed with a mathematical formula, it is generally a question of theoretical models or constructions, which allow us to analyze situations and draw general conclusions, even though it is a theoretical construction.
An example of a production function can be the following:
Q = L 0.6 K 0.4
– Q is the amount produced
– T is the number of man-hours consumed in the production
– K is the amount of capital applied to production
The Total Product is simply the number of goods produced by all workers and inputs applied to production.
Total Product = Amount of Goods Produced
Work is usually considered as a variable in the short term, while capital is only variable in the long term.
The average product is defined as the average quantity produced, for each unit of a given factor. If this factor is work, the average product is the average produced by each worker. To obtain the average product, we must divide the total product by the amount used of the factor.
Average Product = Amount of Goods Produced / Amount of Factor Used
In our example, if K = 50 and L = 10, the total product is:
Q = L 0.6 K 0.4
Q = 10 0.6 50 0.4 = 19.04
In this case, the average product of labour is 19.04 / 10 = 1,904, that is, that each worker produces on average 1,904 units of the good.
The marginal product is defined as the increase of the total product when the quantity of an input used in a unit is increased.
Mathematically it can be described in two ways:
a) When the analysis is discrete, it is described mathematically as follows:
Marginal Product = ΔQ / ΔL
b) If the analysis is infinitesimal, it is described as:
Marginal Product = dQ / dL
In our case, we derive Q with respect to L and obtain:
dQ / dL = 0.6 L -0.4 K 0.4
Try to keep other factors constant. For example, if there is an accident on the production line, production would temporarily stop. Do not use production numbers for that day as your results would be distorted.