Meaning of the Production Function

The technical process that links inputs to the output of a good is known as the production function of that good. It is defined as the technological relationship between inputs and output giving the maximum output that can be produced from various input combinations. The production function analyses the relationship between factors inputs and output within a certain period. It is a catalog of different output possibilities. Here the article explains the meaning and types of the production function.

Features of Production Function

• It is the expression of the input-output relationship in an equation or the mathematical form. A production function expresses a physical relation as both inputs and outputs are expressed in physical terms rather than monetary units.
• The production function is a mathematical association between inputs and output of a good or service
• It can be expressed in the form of a table, graph, or mathematical equation. The production function is generally expressed in a mathematical equation.
• Production function tells that production of a commodity depends on certain specific inputs.
• It can also represent the technology of a firm, of an industry, or an economy.
• A production function is expressed regarding a particular period, say a day or a month, since both inputs and output involve flow.
• Production function describes a purely technological relation because what can be produced with a given amount of inputs depends upon the state of technology.

Therefore, a production function shows the maximum quantity of a commodity that can be produced per unit of time with the given amount of inputs, when the best production technique available is applied. Alternatively, it shows the minimum inputs that are required to produce a certain level of output with the use of the best alternative technique of production. A production function is written as.

Q= f (L, K, L_a, E, R, T…)

Where Q stands for the amount of output as a dependent variable, L stands for labour, K stands for capital, L_a stands for land, E stands for energy or fuel, R stands for raw materials, T stands for technology and these all are independent variables.

Here all the factors of production are categorized into fixed and variable factors based on whether their quantity can be changed within a short run time or not. The above production function is the multivariable production function.

If we assume that there are only two inputs, namely labour (L) and capital (K), then the simple production function can be stated as

Q_X= f (L, K)

In the given simple production function, the quantity of commodity X produced is the function that is dependent on the quantity of labour (L) and capital (K).

‘Production function is the relationship between the quantity of inputs used to make a good and the quantity of output of that good’– N.G. Mankiw

‘Production function is a purely technical relation which connects factor inputs and output.’– Anna Koutsoyiannis   

Categories of Production Function

In economic theory, we refer to two types of important production functions namely the short run and the long-run production function. these two types of production functions are briefly explained below. 

Short-Run Production Function

The short-run is the time interval that is too short to change all the factors of production. The entrepreneur cannot change the amount of capital such as machines, tools, equipment, factory, houses, etc. He can change the amounts of variable factors of production such as labour. The factors of production which are fixed in quantity in the short run are known as a fixed factor of production. The factors of production whose quantity can be changed are called variable factors of production.   

So, in the short run, there are both the fixed as well as variable factors of production and a short-run production function refers to the technical or functional relationship between inputs and output where quantities of some inputs are kept constant and quantities of some are varied. It means a production function in which there is only one input is variable, and all other inputs are assumed to be constant is known as a short-run production function. Here, we study the effect of change in the quantity of one input in total output, while keeping other inputs constant. At any point in time, there is a certain capital stock that is a set of machines and if more workers are used on the given factors to produce more, the relationship between variable inputs (labour) and output is called the short-run production function.

Some major features of the short-run production function are listed below.

• In the short-run production function, there is only one variable factor input and others are assumed fixed
• There are both variable and fixed inputs in the short-run production function
• The change in total output in the short-run production function is only caused by a single variable factor-like labour
• The behavior of the short-run prediction function or the input-output relationship in the short-run is studied under the law of variable proportions
• It is also called the production function with one variable input
• The behavior of output when only one factor of production is changed, and other factors are fixed is called ‘returns to a factor’.

The short-run production function can be expressed as

Q= f (L, K̅)

Where Q is total output, f is functional relation, L is labor units, and it is variable and K̅ is capital which is a fixed factor.

The short-run production function is alternatively expressed below.

Q= f (N_vt, K̅)

Where N_vt stands for units of variable factors of production.

Long-Run Production Function

Considering the existing firms, the long run refers to a period that is long enough to allow these firms to change the quantities of all resources employed, including plant capacity. It means the long run is the time in which a firm can change all its factors of production on inputs. In the long run, the firm changes its output by changing all inputs or factors of production like building, machinery, etc. as all inputs are variable. Therefore, a long-run production function refers to a situation in which all the inputs are variable, and the long-run production function studies change in total output when all inputs used in the production of a commodity are changed simultaneously and in the same proportion. The long-run production function is the subject matter of returns to scale.

Major features of the long-run production function are expressed below.

• It is the production function having all the inputs are variable
• The long-run production function is the technical or functional relationship between inputs and output when quantities of all inputs are variable
• The input-output relationship in the long-run production function or the behavior of the long-run production function is explained by the law of returns to scale
• It is also known as the production function with two variable inputs

A long-run production function can be expressed as.

Q= f (L, K)

Where Q is total output, L is labour, K is capital, and L and K are variable inputs.

Hence, all inputs are variable in the long run and the long-run production function shows how the total output is changed when the producer builds new building or expand the existing building, purchase new and more machines and equipment, open up more branches of the production unit, buy more land to expand the size of the firm, hire more and skilled workers, change production and managerial technology, and so on.  

This is how you can elaborate on the meaning and types of the production function.