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# Production Function

## Introduction

In economics, it is sometimes necessary to relate input with output. In order to understand the input invested and how much of it is required to gain a certain amount of output, economists establish production functions that show the relationship between inputs and output.

## What is the Production Function?

**The production function of an organization or enterprise is an association between inputs used and the utmost amount of outputs that can be manufactured.** In other words, the production function is the relationship that shows the relationship between inputs and outputs for any given amount of inputs invested and the maximum amount of outputs obtained in the production process.

### Example

Let us assume that a car manufacturer is mentioned in the introduction of the production and cost.

Let us assume that he utilizes only 2 inputs to manufacture a car: the spares made of metal and labor. A production function would then show the maximum number of cars he can manufacture using the resources and number of hours he invests (*labor*) in the process. Suppose the manufacturer uses 20 units of spares and 2 hours of time a day to produce 1 car in a week. A function that shows this association is called a production function.

Suppose we represent the function as -

**q = K × L**

Where q is the number of cars produced,

K is the spares required, and

L is the number of hours worked in a day.

Production functions are dependent upon technology. As new technology is used, the production of units usually goes up. This means that introduction of technology works as an enhancer of the production function. So, when technology is upgraded, we will have a new production function for the given example above.

The inputs used in the process of production functions are known as **factors of production**. Usually, companies need different amounts of different inputs in order to get the desired amount of output. If we contemplate that a company needs only two factors of production to get desired outputs. These two factors of production, let’s say, are capital and labor.

Therefore, the production function will be

**q = f (C, L)**

where q is the output,

C is the capital required and

L is the labor invested.

## Types of Production Function

There are two main types of Production Function:

Short-Run Total Production Function and

Long-Run Total Production Function

Production functions can be of two types depending on the number of variable factors. We have seen that there can be two or more factors associated with the production of a product. In the case, that there are more than two variables, or even if there are only two variables, we can keep one variable changeable while other variables are fixed.

On the other hand, we can also keep all variables changeable. Depending on the changing nature of variables, production functions can be categorized into two groups.

### Short-Run Production Function

**The short run production function considers the change in the level of output when only one variable is changeable and other variables are kept constant**. The law of returns to a factor is an example of such a production function. In the case of the short-run production function mentioned above, let's consider that capital is kept as a variable factor while labor is the fixed factor.

Then we will get a function where the outcome of the function will only vary depending on only changing capital. When the amount of capital is increased, production will go up as labor is constant. Consequently, production levels will come down when capital is decreased.

** For example**, let's say that a firm uses 20 units of capital and 6 units of labor. Let’s assume that the firm initially uses only one unit of labor. So, its labor-to-capital ratio is 6:1. Now if the capital is increased to two units, the labor-to-capital ratio will be 6:2 or 3:1. So, the labor-to-capital ratio or the short-run production function will decrease when capital is increased.

### Long-Run Production Function

**In the case of the long-run production function, the variables are changed proportionally**. Instead of keeping all variables fixed except one like short-run production functions, in the case of long-run functions, all variables change proportionately. The law of returns to scale is used to express such functions.

The change in output in the case of long-run production functions is more complex to derive than short-run production functions. As all the variables change, the ultimate outcome of the production function may have an incremental or detrimental effect.

For the above example of capital and labor, in the case of a long-run production function when the capital is increased to two units, the labor will increase to 12 units, so the ultimate outcome will be the same for the labor to capital ratio. However, for functions with more than two variables, the changes may not be the same.

## Total Product, Marginal Product, Average Product

### Total Product

Given the inputs and the total period of time, the total product refers to the final amount of products produced by a firm. It is the actual quantity of goods produced by a firm and can be expressed in units. The total products produced by a firm show its productivity and can be used to determine the strength of production or the production function.

### Marginal Product

The increase in output received due to the addition of a variable input is known as the marginal product. It is the number of goods obtained extra from the total product due to an increase of input by one unit.

$$\mathrm{Marginal\: Product\:=\:\frac{Change\: in\: Output}{Change\: in\: Input}}$$

*It can also be noted that Total Product is the summation value of marginal products.*

$$\mathrm{\mathit{Total\:product}\:=\:\sum Marginal\:product}$$

### Average Product

It is calculated as output per unit factor of inputs. In other words, the average product is the average of the total product per unit of inputs. It can be obtained by dividing the total product by the total number of variable inputs.

$$\mathrm{\mathit{Average\:product}\:=\:\frac{Total\:product}{Units\:of\:Variable\:Factor\:Input}}$$

## Conclusion

The production function is an important tool to consider the production levels of an enterprise. It helps to identify the strength of production for a given set of variables. Production functions can also be used to determine the utility of functions that are related to the production of a good or service. Enterprises can evaluate their efficiency by using the production function and they can also monitor whether the production is at the optimum efficiency level while the variables of production change.

In simpler words, the production function helps in the determination of the true value of outputs against the inputs invested for production. It is a tool to get a holistic view of the overall production process.

## FAQs

**Qns 1. What are Returns to a Factor?**

**Ans**. Returns to a factor is a concept where the behavior of physical output is calculated when only one variable is allowed to change and all other factors are kept fixed. It is a short-run philosophy.

**Qns 2. What is the relation between total product and marginal product?**

**Ans**. The relation is that the total product is the sum total of marginal products.

$$\mathrm{TP\:=\:\sum MP}$$

**Qns 3. What is the relation between the total product and the average product?**

**Ans**. The relation is that the average product is the per unit total product of the variable factor.

Or,

$$\mathrm{AP\:=\:\frac{Total\:product}{Units\:of\:Variable\:Factor\:Input}\:\frac{TP}{L}}$$

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