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## Production Function

The production function of an organization is a relationship between inputs used and output produced by the organization. For various quantities of inputs used, it gives the maximum quantity of output that can be produced. Or, in other words, the production function gives a quantitative perception of the connection between the sources of inputs and outputs. The inputs are the various factor of production- land, labor, capital, and venture while the outputs are the final products and services.

To put it differently, the production function can provide us with the greatest goods and services that we can produce utilizing a given amount of inputs. Further, it can also help us in deciding the inputs we need to accomplish a minimum level of production. Note that a production function is characterized by a given state of innovation.

If there are two-factor inputs-  labor let it be L and capital let it be K, then production function can be written as-

Qx = f (L, K) where Qx is the quantity of output of commodity x, f is the function, and L and k are the units of labor and capital respectively.

The above equations meant that the quantity of output depends on units of labor on capital used in production.

According to Richard H.Lefthich: The term production function is applicable to the physical relationship between a firm’s input of resources and its output of goods or services per unit of time leaving prices aside.

By Richard H.Lefthich

Moving away from complex definitions, we can say that the production function is a catalog of output possibilities. Quantitatively, we can express the production function in the form of an equation in which the output is a dependent variable. Further, this output is a function of the inputs which are the independent variables. The equation is as follows:

q= f (a, b, c……, n)

Here ‘q’ stands for the rate of output of the given commodity and a, b, c, d…..n are the different inputs used per unit of time.

### Cobb-Douglas Production Function

The Cobb-Douglas production function, named after Paul H. Douglas and C.W. Cobb, is a famous statistical production function. It was derived to study the whole of American manufacturing industries. The Cobb-Douglas production function is as follows:

Q= KLª[C^(l-a)]

Here Q is output, L is the quantity of labor, C is the quantity of capital, l and a are positive constants. This study led to the conclusion that labor contributes about 3/4th and capital about 1/4th of the increase in manufacturing production.

The production function is determined by the state of technology-

### Short-run production function

A production function that shows the changes in output when only one factor is changed while the other factor remains constant is termed a short-run production function.

In the above example of the production function, Labor (L) is considered as the variable factor which can be changed to influence the level of output.

The other factor capital (K) is a fixed factor that cannot be changed.

The underlying theory to the short-run production function is the “Law of variable proportion” or “Returns to a factor”.

### Long-run production function

A long-run production function studies the impact on output when all the factors of production can be changed simultaneously and in the same proportion. So in the long-run size of operation of the firm can be expanded or contracted depending on the fact that the factors of production are increased or decreased.

Returns to a factor and returns to scale are two important laws of production. Both laws explain the relation between inputs and output. Both laws have three stages of increasing, decreasing, and constant returns.

## Returns to a factor

Return to factor means a change in physical output of a good when one quantity of one factor is varied while that of other factors remains constant. It is a short-run concept. There are three parts of return to a factor:

(I) Increasing Returns to a Factor
(ii) Constant Returns to a Factor
(iii) Diminishing Returns to a Factor

### Increasing Returns to a Factor

This stage starts at the beginning point 0 and continues until the Total productivity(TP) curve’s point of inflection. Total productivity(TP) increases at a faster rate during this interaction, which is accompanied by a rising Marginal productivity (MP) curve. Most of the point of the MP curve compares to the mark of inflection. Average productivity (AP) continues to grow during this stage.
Reasons behind Increasing Factor Returns:

#### The Fixed factors are underutilized

There aren’t sufficient labor units in the primary progressive phase to totally utilize the fixed factors. Thus, the firm’s output can be increased basically by combining more work inputs with the fixed factor, bringing about an increment in the result of the output of the additional unit of labor.

#### Division of work

As the work input develops, the division of work becomes possible, increasing labor production and efficiency considerably more.

#### Specialization of work

Due to the division of work, individual work units become more specialized, expanding in overall production and efficiency. Subsequently, the MP curve keeps on rising, while the TP curve continues to rise.

### Constant Returns to a Factor

Constant Returns to a Factor or Returns to a Factor plays a vital role in the study of the Theory of Production. This factor shows the short-run production functions in which one element shifts while the others are fixed. Also, when you get an additional output on applying an additional unit of the input, then, at this point, this output is either equivalent to or not exactly the output that you acquire from the previous unit.

Constant Returns to a Factor concerns themselves with the way the output changes when you increase the number of units of a variable factor. Subsequently, it alludes to the impact of the evolving factor-ratio on the output.

All in all, the Constant Returns to a Factor between the units of a variable factor and the amount of output in the short-term. This is expected that all other factors are constant. This relationship is also called the Constant Returns to a Factor.

The constant returns to a factor that keeps different factors constant, when you increase the variable factor, then, at that point, the total product initially increases at an increments rate, then, at that point, increments at a diminishing rate, and at last, its begins declining.

#### Reasons behind constant returns to a factor

In this, the marginal product increases with an expansion in the variable factors.

Hence, the producer can utilize more units of the variable to proficiently utilize the fixed factors. Hence, the producer would like to not stop in this situation however will attempt to expand further.

producers generally prefer not to work in this situation. In this stage, there is a decrease in total product and the marginal product becomes negative.

To order to increase the output, producers reduce the number of variable factors. However, in this situation, he causes higher costs and also gets lesser revenue thereby getting reduced profits.

### Diminishing Returns to a Factor

Diminishing returns to a factor refers to a situation where the absolute result will tend to increase at the reducing rate when additional units of the variable factor are joined with the fixed factors of production. Diminishing returns to a factor might happen because of the following reasons:

#### Fixed nature of the factor-

The fixed nature of the factor is the principal reason that clarifies the event of diminishing returns to a factor. As an increasing number of units of the variable factor keep on being mixed with the fixed factors, the latter gets overutilized.

#### Imperfect substitution-

Factors of production are imperfect substitutes for one another. Increasingly more workers, for instance, can’t be persistently utilized instead of additional capital. Appropriately, diminishing return to the variable factor becomes inescapable.

## Laws of Return

In the long run, all factors of production are variable. The factor is not fixed. Accordingly, the scale of production can be changed by changing the quantity of all factors of production.

Laws of returns to scale refer to an increase in output due to an increase in all factors in the same proportion. Such an increase is called returns to scale. The concept of returns to scale is associated with the tendency of production that is observed when the ratio between the factors is kept constant but the scale is expanded, that is the use of all the factors is changed in the same proportion.

According to Koutsoyiannis “The term returns to scale refers to the changes in output as all factors change by the same proportion.”

By Koutsoyiannis

According to Leibhafsky “Returns to scale relates to the behavior of total output as all inputs are varied and is a long-run concept.”

By Leibhafsky

Returns to scale are of the following types:-

• Increasing Returns to scale.
• Constant Returns to Scale
• Diminishing Returns to Scale

### Increasing Returns to Scale

When the ratio between the factors of production is kept fixed and the scale is expanded, initially output increases in a greater proportion than the increase in the factors of production.

For instance, to deliver a specific item, assuming if the quantity of inputs is multiplied and the increment in output is more than double, it is supposed to be an increasing return to scale. At this point when there is an increase in the scale of production, the average cost per unit produced is lower. This is because at this stage an organization appreciates high economies of scale.

A movement from a to b indicates that the amount of input is doubled. Now, the combination of inputs has reached 2K+2L from 1K+1L. However, the output has increased from 10 to 25 (150% increase), which is more than double. Similarly, when input changes from 2K-H2L to 3K + 3L, then output changes from 25 to 50(100% increase), which is greater than a change in input. This shows increasing returns to scale. There are a number of factors responsible for increasing returns to scale.

### Constant Returns to Scale

Increasing returns to scale can be obtained only up to a point. After this point is reached, expansion of scale only leads to an equal proportionate change in output. This production is said to produce constant returns to scale when the proportionate change in input is equivalent to the proportionate change in output.

For instance, when inputs sources are doubled, the output should also be doubled, then, at that point, it is a case of constant returns to scale.

When there is a movement from a to b, it indicates that input is doubled. Now, when the combination of inputs has reached 2K+2L from IK+IL, then the output has increased from 10 to 20.

Similarly, when input changes from 2Kt2L to 3K + 3L, then output changes from 20 to 30, which is equal to the change in input. This shows constant returns to scale. In constant returns to scale, inputs are divisible and the production function is homogeneous.

### Diminishing Returns to Scale

Diminishing returns to scale ensure that the size of the productive firms cannot be infinitely large. Generally, after a limit when the quantity of the factors of production is increased in such a way that the proportion of the factors remains unchanged, output increases in a smaller proportion as compared to increases in the amounts of the factors of production. Diminishing returns to scale refers to a circumstance when the proportionate change in output is not less than the proportionate change in input.

For instance, when capital and labor are doubled however the output generated is less than doubled, the return to scale would be named as a diminishing return to scale.

When the combination of labor and capital moves from point a to point b, it indicates that input is doubled. At point a, the combination of input is 1k+1L and at point b, the combination becomes 2K+2L.

However, the output has increased from 10 to 18, which is less than a change in the amount of input. Similarly, when input changes from 2K+2L to 3K + 3L, then output changes from 18 to 24, which is less than the change in input. This shows the diminishing returns to scale.

Diminishing returns to scale is due to diseconomies of scale, which arise because of managerial inefficiency. Generally, managerial inefficiency takes place in large-scale organizations. Another cause of diminishing returns to scale is limited natural resources. For example, a coal mining organization can increase the number of mining plants, but cannot increase output due to limited coal reserves.

When all the factors of production (labor, capital, etc.) are increased in the conditions of constant techniques, three possibilities arise-

1. Output increases in a greater proportion as compared to the increase in the factors of production. This is the case of increasing returns to scale.
2. Output increases in the same proportion as the increase in the amount of the factors of production. This is the case of constant returns to scale.
3. Output increases in a smaller proportion as compared to the increase in the amounts of the factors of production. This is the case of diminishing returns to scale.

The differences between Returns to Factor and Returns to Scale are as follows-

### Short vs Long-Run

Returns to Factor is a short-run production function. While Returns to Scale is a long-run production function.

### Number of variable factors

Returns to Factor relate to the production function where only one factor is varied keeping the other factor fixed in order to have more output, Whereas Returns to Scale relate to the production function when a firm changes its scale of production by changing one or more of its factors.

### Respective Law

The law of Variable Proportions is the law explaining Returns to Factor. On the other hand, The Law of Returns to Scale explains the returns to scale concept.

### Variability Proportion vis-a-vis units of factor

In the case of the returns to factor concept, the factor- proportion varies as more and more of the units of the variable factor are employed to increase output. Factor proportion called scale does not vary. Factors are increased in the same proportion to increase output.

### Causes

Returns to variable proportions(factors) are caused by indivisibility of certain fixed factors, specialization of certain variable factors, imperfect substitutability of factors, or sub-optimal factor proportions. Returns to scale can be attributed to economies and diseconomies of scale caused by technical and/or managerial indivisibilities, exhaustibility of natural and managerial resources, or depreciation of certain factors.

### Decreasing vs Negative Returns

Returns to a factor end up in negative returns. Returns to scale end up in decreasing returns but not in negative.

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## General FAQ

### What production function?

The production function of an organization is a relationship between inputs used and output produced by the organization. For various quantities of inputs used, it gives the maximum quantity of output that can be produced.

### What is the Cobb-Douglas production function?

The Cobb-Douglas production function, named after Paul H. Douglas and C.W. Cobb, is a famous statistical production function. It was derived to study the whole of American manufacturing industries.

### What is a short-run production function?

A production function that shows the changes in output when only one factor is changed while the other factor remains constant is termed a short-run production function.

### What is a long-run production function?

A long-run production function studies the impact on output when all the factors of production can be changed simultaneously and in the same proportion. So in the long-run size of operation of the firm can be expanded or contracted depending on the fact that the factors of production are increased or decreased.

### What are returns to a factor?

Return to factor means a change in physical output of a good when one quantity of one factor is varied while that of other factors remains constant. It is a short-run concept.

(I) Increasing Returns to a Factor
(ii) Constant Returns to a Factor
(iii) Diminishing Returns to a Factor

### What are increasing returns to a factor?

This stage starts at the beginning point 0 and continues until the Total productivity(TP) curve’s point of inflection. Total productivity(TP) increases at a faster rate during this interaction, which is accompanied by a rising Marginal productivity (MP) curve. Most of the point of the MP curve compares to the mark of inflection. Average productivity (AP) continues to grow during this stage.

### What are constant returns to a factor?

Constant Returns to a Factor or Returns to a Factor plays a vital role in the study of the Theory of Production. This factor shows the short-run production functions in which one element shifts while the others are fixed. Also, when you get an additional output on applying an additional unit of the input, then, at this point, this output is either equivalent to or not exactly the output that you acquire from the previous unit.

### What are diminishing returns to a factor?

Diminishing returns to a factor refers to a situation where the absolute result will tend to increase at the reducing rate when additional units of the variable factor are joined with the fixed factors of production.

### What are the laws of return?

In the long run, all factors of production are variable. The factor is not fixed. Accordingly, the scale of production can be changed by changing the quantity of all factors of production.

Laws of returns to scale refer to an increase in output due to an increase in all factors in the same proportion. Such an increase is called returns to scale. The concept of returns to scale is associated with the tendency of production that is observed when the ratio between the factors is kept constant but the scale is expanded, that is the use of all the factors is changed in the same proportion.

### How many types of returns to scale?

Returns to scale are of the following types:-
a. Increasing Returns to scale.
b. Constant Returns to Scale
c. Diminishing Returns to Scale

### What is Increasing Returns to Scale?

When the ratio between the factors of production is kept fixed and the scale is expanded, initially output increases in a greater proportion than the increase in the factors of production.

### What are constant returns to scale?

Increasing returns to scale can be obtained only up to a point. After this point is reached, expansion of scale only leads to an equal proportionate change in output. This production is said to produce constant returns to scale when the proportionate change in input is equivalent to the proportionate change in output.

### What are diminishing returns to scale?

Diminishing returns to scale ensure that the size of the productive firms cannot be infinitely large. Generally, after a limit when the quantity of the factors of production is increased in such a way that the proportion of the factors remains unchanged, output increases in a smaller proportion as compared to increases in the amounts of the factors of production. Diminishing returns to scale refers to a circumstance when the proportionate change in output is not less than the proportionate change in input.

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