The production function describes the relationship between physical inputs and outputs, indicating how much can be produced with a given amount of inputs based on technological knowledge. It is categorized into short-run and long-run production functions, each reflecting different variable factors, leading to distinct theories such as the law of variable proportions and returns to scale. The law of variable proportions explains how output responds to changes in one variable factor while other factors remain constant, and returns to scale describes changes in output when all input factors change proportionally.

1. Production Function
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2. Meaning
Production function expresses the technological relationship between physical
inputs and physical quantities of output . In other words it shows that with a
given state of technological knowledge and during a particular period of time
how much can be produced with given amount of inputs .
Q = f ( L , C , N )

3. Definitions
L. R. Klein, " The production function is a technical or engineering relation
between input and output . As long as the natural laws of technology remain
unchanged , the production function remains unchanged . "
Prof . George J. Stigler, " Production function is the relationship between
inputs of productive services per unit of time and outputs of product per
unit of time .
Watson, " The relationship between a firm's physical production ( output )
and the material factors of production ( input ) is referred to as production
function . "

4. Types of Production Functions
• Short-run Production Function
It is the type of production function that shows changes in output with the change in one
variable. Here, the other factors remain constant.
There are two theories based on short-run production function:
1. Law of Variable Proportion
2. Return to Factor
• Long-run Production Function
All the variables are changed in this production function but in the same proportion. Here, all
the factors are variable, and the size operation can be increased or decreased as required.
The theory based on the long-run production function is ‘Return to Scale’.

5. 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.

6. 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

7. Law of Production
The law of production describe the ways which are
technically possible to in crease the level of
production . We have already discussed that the output
can be increased in following two ways :
Law of
Production
Law of Variable
Proportions
Return to a Factor
Short Run
Production
Function
Law of Returns to
Scale
Long Run
Production
Function
The response of output to changes in
the amount of a variable factor , when
other factors of production remains
constant, is referred to as returns to
factors or law of variable proportions .
In long run the output can be increased by increasing all the factors , i.e. ,
the scale of production itself . The response of output to changes in the
size of scale of all the factors ( in the same proportion ) is termed as
Returns to Scale .

8. Law of Variable Proportions Or
Law of Diminishing Returns
• The law which studies the relationship
between one variable factor of production
(say labour) and output, keeping the
quantities of other factors fixed, is called the
law of variable proportion.
• According to this law as the proportion of
factors is changed, the total production at
first increases more than proportionately,
then equi-Proportionally and finally less than
proportionately. The classical economists
called it as The Law of Diminishing Returns .

9. Assumptions of Law of Variable
Proportion
Law of variable proportion holds good under certain
circumstances, which will be discussed in the following lines.
Constant state of Technology: It is assumed that
the state of technology will be constant and with
improvements in the technology, the production
will improve.
Variable Factor Proportions: This assumes that
factors of production are variable. The law is not
valid, if factors of production are fixed.
Homogeneous factor units: This assumes that all
the units produced are identical in quality,
quantity and price. In other words, the units are
homogeneous in nature.
Short Run: This assumes that this law is applicable
for those systems that are operating for a short

10. Illustration of Law

12. Causes of increasing Returns
Indivisibilities: Indivisibility means that certain factors are available only in some
minimum sizes. Such inputs cannot be divided into small sizes to suit the small
scale of production. For e.g. there cannot be half a machine, half a computer or
half a manager. Such inputs have to be employed even if the scale of production is
small. Therefore, as the scale of production increases, these indivisible factors are
utilized better and more efficiently. This leads to increasing returns to scale.
Internal economics: Internal economics are those economics which are opened
to an individual firm when its size expands. They emerge within the firm after its
expansion. There can arise several types of internal economics when a firm
expands its output. Like Efficiency, cheap source of finance, Reasonable rates etc.

13. Causes of Diminishing Returns
Dis-economics causes for diminishing returns: Be beyond a
stage - beyond the optimum stage - increase in the quantity of the
variable factor only disrupts the existing organisation . It spoils
division of labour and breeds inefficiency. In short when the fixed
factors reach the point of optimum utilization, every additional unit
of variable factor will bring in less than proportionate returns i.e. ,
marginal product will fall .
Imperfect substitutability of factors of production: Diminishing
returns occur because the factors of production are imperfect
substitutes for one another . For example , we take two factors -
land as a fixed factor and labour as a variable factor . In the
beginning with additional doses of labour, production goes on
increasing at an increasing rate. In other words, up to certain point,
substitution is beneficial. Once that optimum point is reached the

14. Causes of Negative Returns
This situation arises when excessive
variable factors adversely affect the
efficiency of fixed factor. The proverb
"too many cooks spoil the broth",
aptly applies to this situation.

15. Stage of Rational Decision
The law of variable proportions is based on the fundamental technical fact
that if we increase the proportion of one factor of production , keeping other
inputs fixed , then the variable factor must eventually get diminishing
returns . Hence in order to achieve maximum profits , the rational decision of
the firm will be to operate in stage II . Why ?
In Stage I : The fixed factor is abundant and
remains under utilized , it will pay the producer to
continue employing additional units so that the output
increases . Thus , the firm will have an incentive to
expand through stage I.
No rational producer would like to operate in
Stage III , because in ' this stage , the marginal product
of the variable factor becomes negative and there is
actual decline in the total production.
Thus the rational decision of the firm will be neither to operate in stage I nor in
stage III . It will operate in stage II where ( a ) TP is increasing at a decreasing
rate , ( b ) MP and AP are falling but are positive .

16. Law of Returns to Scale
Returns to scale in economics refers to a term that states that the
degree of change in input factors changes the output proportionally and
concurrently during the production process. It reflects the quantitative
change that applies in the long-term using similar technology. It forms the
basis of measuring a firm’s or industry’s efficiency of production capacity.
Definitions:
“The term returns to scale refers to the changes in output as all factors
change by the same proportion.” Koutsoyiannis
“Returns to scale relates to the behaviour of total output as all inputs are
varied and is a long run concept”. Leibhafsky

17. Assumption:
1. All factors are variable but enterprise is fixed.
2. Input ( labour & capital ) are used in fixed proportion.
3. No change in state of technology.
4. There is perfect competition.
5. The product is measured in quantities.

18. Returns to scale are of the following three types
1. Increasing Returns to scale: An increasing
returns to scale occurs when the output increases
by a larger proportion than the increase in inputs
during the production process.
2. Constant Returns to Scale: A constant returns
to scale means that the proportionate increase in
input is exactly equal to the increase in output.
3. Diminishing Returns to Scale: A decreasing
returns to scale occurs when the proportion of
output is less than the desired increased input
during the production process.

19. Explanation
In the long run, output can be increased by increasing all factors in the same
proportion. Generally, laws of returns to scale refer to an increase in output due
to increase in all factors in the same proportion. Such an increase is called
returns to scale.

20. Increasing Returns to Scale:
Increasing returns to scale or diminishing cost refers to a
situation when all factors of production are increased, output
increases at a higher rate. It means if all inputs are doubled,
output will also increase at the faster rate than double. Hence, it is
said to be increasing returns to scale. This increase is due to many
reasons like division external economies of scale. Increasing
returns to scale can be illustrated with the help of a diagram 8.
In figure 8, OX axis represents increase in labour and capital while
OY axis shows increase in output. When labour and capital
increases from Q to Q1, output also increases from P to P1 which is
higher than the factors of production i.e. labour and capital.

21. Diminishing Returns to Scale:
Diminishing returns or increasing costs refer to that production situation,
where if all the factors of production are increased in a given proportion,
output increases in a smaller proportion. It means, if inputs are doubled,
output will be less than doubled. If 20 percent increase in labour and
capital is followed by 10 percent increase in output, then it is an instance
of diminishing returns to scale.
The main cause of the operation of diminishing returns to scale is that
internal and external economies are less than internal and external
diseconomies. It is clear from diagram 9.
In this diagram 9, diminishing returns to scale has been shown. On OX
axis, labour and capital are given while on OY axis, output. When factors
of production increase from Q to Q1 (more quantity) but as a result
increase in output, i.e. P to P1 is less. We see that increase in factors of
production is more and increase in production is comparatively less, thus
diminishing returns to scale apply.

22. Constant Returns to Scale:
• Constant returns to scale or constant cost refers to the
production situation in which output increases exactly in the
same proportion in which factors of production are increased.
In simple terms, if factors of production are doubled output
will also be doubled.
• In this case internal and external economies are exactly equal
to internal and external diseconomies. This situation arises
when after reaching a certain level of production, economies of
scale are balanced by diseconomies of scale. This is known as
homogeneous production function. Cobb-Douglas linear
homogenous production function is a good example of this
kind. This is shown in diagram 10. In figure 10, we see that
increase in factors of production i.e. labour and capital are
equal to the proportion of output increase. Therefore, the
result is constant returns to scale.

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