production Function

Meaning of Production Function:

In simple words, production function refers to the functional relationship between the quantity of a good produced (output) and factors of production (inputs).

“The production function is purely a technical relation which connects factor inputs and output.” Prof. Koutsoyiannis

In this way, production function reflects how much output we can expect if we have so much of labour and so much of capital as well as of labour etc. In other words, we can say that production function is an indicator of the physical relationship between the inputs and output of a firm.

The reason behind physical relationship is that money prices do not appear in it. However, here one thing that becomes most important to quote is that like demand function a production function is for a definite period.

It shows the flow of inputs resulting into a flow of output during some time. The production function of a firm depends on the state of technology. With every development in technology the production function of the firm undergoes a change.

The new production function brought about by developing technology displays same inputs and more output or the same output with lesser inputs. Sometimes a new production function of the firm may be adverse as it takes more inputs to produce the same output.

Mathematically, such a basic relationship between inputs and outputs may be expressed as:

Q = f( L, C, N )

Where Q = Quantity of output

L = Labour

C = Capital

N = Land.

Hence, the level of output (Q), depends on the quantities of different inputs (L, C, N) available to the firm. In the simplest case, where there are only two inputs, labour (L) and capital (C) and one output (Q), the production function becomes.

Q =f (L, C)

Features of Production Function:

Following are the main features of production function:

1. Substitutability:

The factors of production or inputs are substitutes of one another which make it possible to vary the total output by changing the quantity of one or a few inputs, while the quantities of all other inputs are held constant. It is the substitutability of the factors of production that gives rise to the laws of variable proportions.

2. Complementarity:

The factors of production are also complementary to one another, that is, the two or more inputs are to be used together as nothing will be produced if the quantity of either of the inputs used in the production process is zero.

The principles of returns to scale is another manifestation of complementarity of inputs as it reveals that the quantity of all inputs are to be increased simultaneously in order to attain a higher scale of total output.

3. Specificity:

It reveals that the inputs are specific to the production of a particular product. Machines and equipment’s, specialized workers and raw materials are a few examples of the specificity of factors of production. The specificity may not be complete as factors may be used for production of other commodities too. This reveals that in the production process none of the factors can be ignored and in some cases ignorance to even slightest extent is not possible if the factors are perfectly specific.

Production involves time; hence, the way the inputs are combined is determined to a large extent by the time period under consideration. The greater the time period, the greater the freedom the producer has to vary the quantities of various inputs used in the production process.

In the production function, variation in total output by varying the quantities of all inputs is possible only in the long run whereas the variation in total output by varying the quantity of single input may be possible even in the short run.

Production Function Graph

Here is the production function graph :

This graph shows the short-run functional relationship between the output and only one input, i.e., labor, by keeping other inputs constant. The X-axis represents the labor (independent variable), and the Y-axis represents the quantity of output (dependent variable). 

The curve starts from the origin 0, indicating zero labor. It gets flattered with the increase in labor. One can notice that with increasing labor, the level of output increases to a level. Further, it curves downwards. It is because the increase in capital stock leads to lower output as per the capital’s decreasing marginal product. In short, the short-run curve slopes upwards till the product reaches the optimum condition; if the producers add more labor further, the curve slopes downwards due to diminishing marginal product of labor. 

Example

Here is a production function example to understand the concept better. 

Let us consider a famous garments company that produces the latest designer wear for American customers. It requires three types of inputs for producing the designer garments: cloth, industrial sewing machine, and tailor as an employee. 

The variables- cloth, tailor, and industrial sewing machine is the variable that combines to constitute the function. The Production function will then determine the quantity of output of garments as per the number of inputs used. The industrial sewing machine can sew ten pieces of garments every hour. The tailor can use these sewing machines to produce upto five pieces of garment every 15 minutes. The length of clothing that the tailor will use per piece of garment will be 2 meters. After including the data into the above formula, which is 

Quantity of output, Q = min (input-1, input-2, input-3) where input1= cloth, input 2= industrial sewing machine and input 3 = tailor

Production function Q, in one hour = min (input 1, input 2, input 3) = min (cloth+ tailor + industrial sewing machine) = min (2mtrs per piece, 20 pieces by tailor, 20 pieces by machine) = min (40 meters, 20 pieces, 20 pieces)

From the above, it is clear that if there are:

• Only 100 mtrs cloth are there then only 50 pieces of the garment can be made in 1 hour
• With only one machine, 20 pieces of production will take place in 1 hour. 
• Only one tailor can help in the production of 20 pieces.

Therefore, the best product combination of the above three inputs – cloth, tailor, and industrial sewing machine- is required to maximize the output of garments. 

Types Of Production Function 

There are two main types of productivity functions based on the input variables, as discussed below.

1) Long Run :- In the long-run production function, all the inputs are variable such as labor or raw materials during a certain period. Therefore, the operation is flexible as all the input variables can be changed per the firm’s requirements. Furthermore, in the production function in economics, the producers can use the law of equi-marginal returns to scale. It leads to a smaller rise in output if the producer increases the input even after the optimal production capacity. It means the manufacturer can secure the best combination of factors and change the production scale at any time. Therefore, the factor ratio remains the same here. Moreover, the firms are free to enter and exit in the long run due to low barriers.

2) Short Run:- The firm cannot vary its input quantities in the short-run production function. The law of variable proportion gets applicable here. There is no change in the level of activity in the short-run function. The ratio of factors keeps changing because only one input changes concerning all the other variables, which remain fixed. The manufacturing firms face exit barriers. As a result, they can be shut down permanently but cannot exit from production.

For any production company, only the nature of the input variable determines the type of productivity function one uses. If one uses variable input, it is a short-run productivity function; otherwise, it is a long-run function.