Production Function

The production function is a mathematical equation determining the relationship between the factors and quantity of input for production and the number of goods it produces most efficiently. It answers the queries related to marginal productivity, level of production, and cheapest mode of production of goods.

Four major factors of production are – entrepreneurship, labor, land, and capital. They form an integral part of inputs in this function. The production function helps the producers determine the maximum output that firms and businesses can achieve using the above four factors. In addition, it aids in selecting the minimum input combination for maximum output production at a certain price point.

• The production function is a mathematical function stating the relationship between the inputs and the outputs of the goods in production by a firm.
• Entrepreneurship, labor, land, and capital are major factors of input that can determine the maximum output for a certain price.
• Analysts or producers can represent it by a graph and use the formula Q = f(K, L) or Q = K+L to find it. 
• There are two types of productivity function, namely long run, and short run, depending on the nature of the input variable.

Production function means a mathematical equation/representation of the relationship between tangible inputs and the tangible output of a firm during the production of goods. A single factor in the absence of the other three cannot help production. In simple words, it describes the method that will enable the maximum production of goods by technically combining the four major factors of production- land, enterprise, labor and capital at a certain timeframe using a specific technology most efficiently. It changes with development in technology. J H Von was the first person to develop the proportions of the first variable of this function in the 1840s.

This function depends on the price factor and output levels that producers can easily observe. Moreover, every manufacturing plant converts inputs into outputs. Hence the factors necessarily determine the production level of goods to maximize profits and minimize cost. Therefore, the production function is essential to know the quantity of output the firms require to produce at the said price of goods. It determines the output and the combination inputs at a certain capital and labor cost.  

It is a common phenomenon that a firm's marginal cost starts to increase at higher production levels, which is known as diminishing returns to scale. The diminishing returns to scale lead to a lesser proportional increase in output quantity by increasing the input quantities. Moreover, the increase in marginal cost is identifiable by using this function. 

The Leontief production function is a type of function that determines the ratio of input required for producing in a unit of the output quantity. Also, producers and analysts use the Cobb-Douglas function to calculate the aggregate production function.

Here is the production function graph to explain this concept of production:

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 futher, the curve slopes downwards due to diminishing marginal product of labor. 

The general production function formula is:

Q= f (K, L), 

Here Q is the output quantity, 

L is the labor used, and 

K is the capital invested for the production of the goods. 

The f is a mathematical function depending upon the input used for the desired output of the production. For example, it means if the equation is re-written as: 

Q= K+ L for a firm if the company uses two units of investment, K, and five units of labor. As a result, the producer can produce 5+2 = 7 units of goods. Hence, increasing production factors – labor and capital- will increase the quantity produced. 

Another formula that this function uses is the Cobb-Douglas function denoted by:

Y= AK α L β, 

Where A is the technology improvement factor, 

K is the capital, 

L is the labor, 

Alpha (α) is the capital-output elasticity, and Beta (β) is the labor elasticity output.  

One must always note that α + β is: 

 One under constant returns to scale 

>1 under decreasing returns to scale and

<1 under increasing returns to scale

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. 

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.

This has been a guide to Production Function & its definition. We explain types, formula, graph of production function along with an example. You can learn more about accounting from the following articles: –

• Supply Function
• Mass Production
• Consumption Function