PRODUCTION

FUNCTION
 INTRODUCTION
• Production function can be defined as a technological relationship
between the physical inputs and physical output of the
organisation. It is a statement of the relationship between a firm’s
scarce resources (i.e. its inputs) and the output that results from the
use of these resources.
• Inputs include the factors of production, such as land, labour,
capital, whereas physical output includes quantities of finished
products produced. The long-run production function (Q) is usually
expressed as follows:
Q= f (LB, L, K, M, T, t)
Where, LB= land and building L = labour K = capital
M = raw material T = technology t = time
In other words, it is a mathematical equation that shows how much output can be produced from
a given set of inputs.

•
 Production Function Definition
• Production function is the name given to the relationship between
the rates of input of productive services and the rate of output.
- Stigler
• Production Function is the technological relationship, which
explains the quantity of production that can be produced by a
certain group of inputs. It is related with a given state of
technological change.
- Samuelson
• The relation between a firm’s physical production (output) and the
material factors of production (input) is referred to as production
function.
- Watson
In a standard equation, the
Production function is
represented by Q, Labour
(Variable element) is
represented by L, and Capital
(Fixed element) is represented
by K.

For example, When there are

4 units of labour and 5 units
of capital, the equation for
the production function is
Q = f(4,5).
 Features of Production Function
• 1. Complementary: A producer will have to combine the inputs to
produce outputs. Outputs can not get generated without the use of
inputs.
• 2. Specificity: For any given output, the combination of inputs that may be
used is clearly defined. What type of factors are needed for the
production of a particular product is clearly mentioned before the actual
production gets started.
• 3. Production Period: The period of the production process is clearly
explained to the production unit. Each stage of production is given some
specific time. Production generally gets completed over a long period of
time.
 Types of Production Function

Short Run Long Run

Production Production
Function Function
‘Law of Variable ‘Law of Returns
Proportions.’ to Scale’
1. Short Run Production Function:
• Short Run is a period of time where output can only be changed by changing the level
of variable inputs. In the short run, some factors are variable and some are fixed. Fixed
factors remain constant in the short run like land, capital, plant, machinery, etc.
Production can be raised by only increasing the level of variable inputs like labour.
Therefore, the situation where the output is increased by only increasing the variable
factors of input and keeping the fixed factors constant is termed as Short Run
Production Function. This relationship is explained by the ‘Law of Variable
Proportions.’
• The short-term production function is written as:
Q = f (L) where, ‘Q’= Quantity of output and ‘L’=
Labour
For example, an agricultural firm has 10 units of labor and 6 acres of land. Here, land is
the constant factor and labor is the variable factor. The firm initially uses only one unit of
labor (variable factor) on its land (constant factor). Hence the land-labor ratio is 6: 1. If
the firm uses 2 units of labor, the land-labor ratio becomes 6: 2 or 3: 1.
 2. Long Run Production Function:
• Long Run is a span of time where the output can be increased by
increasing all the factors of production whether it is fixed (land, capital,
plant, machinery, etc.) or variable (labour). Long run is enough time to
alter all the factors of production. All factors are said to be variable in
the long run. Therefore, the situation where the output is increased by
increasing all the inputs simultaneously and in the same proportion is
termed Long Run Production Function. This relationship is explained by
the ‘Law of Returns to Scale.’

• For example, the builder can build a new building or expand an existing
building, buy more new machines and equipment, open more branches
of a production unit, buy more land to expand the firm’s size, can hire
more and skilled workers, change production and managerial
technology, etc.
 Uses of Production Function
• Helps in making short-term decisions, such as optimum level of output.
• Helps in calculating the least cost combination of various factor inputs at a
given level of output.
• It gives logical reasons for making decisions. For example, if price of one input
falls, one can easily shift to other inputs.
• It may be used to compute the least-cost combination of inputs for a given
output.
• It may be used by the manager to obtain the most appropriate combination of
input which yields the maximum level of output with a given level of cost.
• Helps the managers in deciding the additional value of variable input employed
in the production process.
• Production functions help the managers in taking long-run decisions.
 Concept of Product
Product or output refers to the volume of the goods that the company produces using inputs during a
specified period of time. The concept of product can be looked at from three different angles
1. Total Product (TP):
Total Product (TP) refers to the total quantity of goods that the firm produced during a given course of
time with the given number of inputs. For example, if 6 labours produce 10 kg of wheat, then the total
product is 60 kg.

2. Average Product (AP):

Average Product refers to output per unit of a variable input. AP is calculated by dividing TP by units of
the variable factor. For example, if the total product is 60 kg of wheat produced by 6 labours (variable
inputs), then the average product will be 60/6, i.e., 10 kg.
Average Product (AP)=Total Product (TP) / Units of Variable Factor

3. Marginal Product (MP):

Marginal Product refers to the addition to the total product when one more unit of a variable factor is
employed. It calculates the extra output per additional unit of input while keeping all other inputs
constant.
MPn = TPn – TPn-1
Here, MPn = Marginal product of nth unit of the variable factor,
TPn = Total product of n units of the variable factor, and
TPn-1 = Total product of (n-1) units of the variable factor
Law of Variable Proportion
 Law of Variable Proportion
• The Law of Variable Proportions states that as we increase the
quantity of only one input while keeping other inputs fixed, the
total product increases initially at an increasing rate, then at a
decreasing rate, and finally at a negative rate.

• As per the law of variable proportions, the changes in TP and MP

can be categorised into three phases:
• Phase 1: TP rises at an increasing rate, and MP increases.
• Phase 2: TP rises at decreasing rate, MP decreases and is
positive.
• Phase 3: TP falls, and MP becomes negative.
• For example, Let’s say a farmer has 1 acre of land (i.e., fixed factor) and wants to
use labour (i.e., variable factor) to improve the production of rice there. The
output increased initially at an increasing rate, then at a decreasing rate, and
finally at a negative rate as he employed more and more units of labour. The
below table displays the output behaviour in this case.

Fixed Factor Variable Factor TP MP

(Land) (Labour) (units) (units) Phase

1 1 5 5 Phase I:
Increasing
Returns to a
1 2 20 15
Factor

1 3 32 12 Phase II:
Decreasing
1 4 40 8
Returns to a
Factor
1 5 40 0

Phase III:
1 6 35 -5 Negative Returns
to a Factor
Phase I: Increasing Returns to a Factor whereTP increases
at an increasing rate
• In the initial stage, each additional variable component raises the total production by an
increasing amount. This indicates that each variable’s MP rises and that TP rises at an
increasing rate.
• It occurs as a result of the initial variable input quantity being too small in comparison to
the fixed input. Due to the division of labour, efficient use of the fixed input during
manufacturing increases the productivity of the variable input.
• One labour generates 5 units, as shown in the schedule and diagram, whereas two labours
produce 20 units. It means that MP rises until it reaches its maximum point at point P,
which signifies the end of the first phase, while TP rises at an increasing rate (up to point
Q).
• Point of Inflexion: A point from where the slope of TP curve changes is known as
point of inflexion. Till the point of inflexion, TP increases at an increasing rate, and from
this point downwards, it increases at a diminishing rate.
 Phase II: Decreasing Returns to a Factor where TP
increases at a decreasing rate
• Every extra variable in the second phase increases the output by a
less and smaller amount. This indicates that when the variable
factor increases, MP decreases, and TP rises at a decreasing rate.
This stage is known as the diminishing returns to a factor.
• This occurs as a result of pressure on fixed inputs that results in a
decline in variable input productivity after a certain level of
output.
• When MP is zero (point S), and TP is at its maximum (point M) at
40 units, the second phase comes to an end.
• The second phase is highly important because a rational producer
will always try to produce during this time because MP and TP
are both positive for each variable factor.
 Phase III: Negative Returns to a Factor
where TP falls
• The third phase shows a decline in TP due to the use of more variable
factors. MP has now become negative. As a result, this stage is
referred to as negative returns to a factor.
• It occurs when the amount of variable input exceeds the fixed input by
a great difference, which causes TP to decrease.
• The third phase in the above graph begins after points S on the MP
curve and M on the TP curve.
• In the third phase, MP for each variable factor is negative. Therefore,
no company would deliberately decide to operate at this phase.
 Phase of Operation
• A logical or rational producer will always attempt to operate in
Phase II of the Law of Variable Proportion at all times.
• Every additional unit of a variable factor used in Phase I results in
an increase in production or marginal product. Therefore, if
production is increased with more units of the variable factor,
there is scope for greater profits.
• In Phase III, each variable’s marginal product is negative.
Therefore, this phase is eliminated due to technical inefficiency,
and a rational manufacturer would never engage in the third
phase of production.
• This leads us to the conclusion that a producer will seek to
operate in Phase II since the MP of each variable factor is positive
and TP is at its highest level.
 ISOQUANTS
• An isoquant in economics is a curve that, when plotted on a graph, shows
all the combinations of two factors that produce a given output. Often
used in manufacturing, with capital and labor as the two factors,
isoquants can show the optimal combination of inputs that will produce
the maximum output at minimum cost.
• An isoquant or an iso-product curve is the line which joins together different
combinations of the factors of production (L, K) that are physically able to
produce a given amount of output.
• Assumptions of Isoquant Curve:
1. There are only two factor inputs, labour and capital, to produce a
particular product.
2. Capital, labour and goods are divisible in nature.
3. Capital and labour are able to substitute each other up to a certain
limit.
4. Technology of production is given over a period of time.
5. Factors of production are used with full efficiency.
• Fig. 1 shows two typical isoquants —
capital use is measured on the
vertical axis and labour use on the
horizontal. Isoquant Q1 shows the
locus of combinations of capital and
labour yielding 100 units of output.
The producer can produce 100 units
of output by using 10 units of capital
and 75 of labour or 50 units of
capital and 15 of labour, or by using
any other combination of inputs on
Q1 = 100. Similarly, isoquant
Q2 shows the various combinations
of capital and labour that can
produce 200 units of output.
 Types of isoquants
Depending on the degree of substitutability of factors, the production
isoquant can take on a variety of shapes. They are:

1. Linear isoquant

2. Input-output isoquant

3. Kinked isoquant

4. Smooth, convex isoquant

Linear isoquant assumes that all
factors of production are
perfectly substitutable.
Isoquant will assume the shape
of a straight line sloping
downwards from left to right.
Given commodity may be
produced using only capital,
only labor, or an infinite mix of K
and L.

However, linear isoquant does

not have existence in the real
world.
P1, P2, P3, P4 are different processes
Law of Returns to Scale
 Law of Returns to Scale
• When a firm changes the quantity of all inputs in the long run, it changes the
scale of production for the goods.
According to Watson, “Returns to Scale is related to the behaviour of total
output as all inputs are varied in same proportion and it is a long run concept.”

• According to the Law of Returns to Scale, when all the factor inputs are varied
in the same proportions, then the scale of production may take three forms:
1. Increasing Returns to Scale (IRS): If output increases more than
proportionate to the increase in all inputs.
2. Constant Returns to Scale (CRS): If all inputs are increased by
some proportion, output will also increase by the same proportion.
3. Decreasing Returns to Scale (DRS): If increase in output is less
than proportionate to the increase in all inputs.
• For example, if all inputs are doubled, we have constant, increasing or decreasing returns to
scale, respectively, if output doubles, more than doubles or less than doubles. The firm
increases its inputs from 3 to 6 units (K, L) producing either double (point B), more than
double (point C) or less than double (point D) output (Q) as shown in Figure 7.2.
 I. Increasing Returns to Scale (IRS)
• In the first stage of Returns to Scale, the proportionate increase in total output is more than the
proportionate increase in inputs. In simple terms, if all the inputs increase by 100%, then the increase
in output will be more than 100%.
• If all the factors of production are increased in a particular proportion and the output increases by
more than that proportion then the production function is said to exhibits IRS

Inputs (Units) Percentage Percentage

(K = Capital, L = Output Increase Increase
Labour) (Units) in Inputs in Outputs

2K + 4L 200 – –

4K + 8L 450 100% 160%

6K + 12L 600 100% 120%

• For example, in many industrial processes
if all inputs are doubled, factories can be
run in more efficient and effective ways,
there by actually more than doubling
output. This is shown in Figure 7.4. To
produce X units of output, L units of
Labour and K units of output are needed.
If labour is doubled to 2L units and
capital to 2K units, an output greater
than 2X is produced (point c lies on a
higher isoquant than point b).
 Causes of IRS
Indivisibility of factors: Machines, management, labour , finance etc cannot be
available in very small sizes, so when business expands the returns to scale
increases because these indivisible factors are used to their maximum capacity

Specialisation & division of Labour: Efficiency increases when work is divided into
small tasks and workers are divided

Internal economies: As firm expands, it may be able to install better machines,

sell its product more easily, borrow money cheaply thereby increasing returns to
scale proportionatey

External economies: when a large no. of firms are concentrated at one place,
skilled labour, credit & transport facilities are easily available thereby increasing
returns to scale.
 II. Constant Returns to Scale (CRS):
• If all the factors of production are increased in a particular proportion and
the output increases in exactly that proportion then the production
function is said to exhibit CRS. Thus if labour and capital are increased by
10% and the output also increases by 10% then the production function is
CRS. Once the firm has achieved the point of optimum capacity, it
operates on Constant Returns to Scale. After the point of
optimum capacity, the economies of production are
counterbalanced by the diseconomies of production.

Inputs (Units) Percentage Percentage

(K = Capital, L = Output Increase Increase
Labour) (Units) in Inputs in Outputs

6K + 12L 600 – –

8K + 16L 1,000 100% 100%

10K + 20L 2,300 100% 100%

• If you look at Figure 7.3, to
produce X units of output, L
units of labour and K units
of capital are needed (point
a). If labour and capital are
now doubled (as is possible
in the long run), so that
there are 2L units of labour
and 2K units of capital, the
output is exactly doubled
i.e., equals 2X (point b).
Similarly, trebling input
achieves treble the output
and so on.
 III. Decreasing Returns to Scale (DRS)
• If the factors of production are increased in a particular proportion and the output
increases by less than that proportion than the production function is said to exhibit
DRS. For example, if capital and labour are increased by 10% and output rises by
less than 10% the production function is said to exhibit decreasing returns to scale.

Inputs (Units) Percentage Percentage

(K = Capital, L = Output Increase Increase
Labour) (Units) in Inputs in Outputs

10K + 20L 2,300 – –

12K + 24L 4,600 100% 80%

14K + 28L 6,000 100% 75%

If you look at Figure 7.5, to
produce X units of output L
units of labour and K units of
capital are required. By
doubling the input, the output
increases by less than twice its
original level. For example, if
inputs are 2L and 2K, output
level ‘a’ is reached, which lies
below the one showing 2X.
Causes of Diminishing Returns to Scale
• The main reason behind Diminishing Returns to Scale
is Diseconomies of Large Scale. Diseconomies of Scale mean that
the firm has now become so large that it has become difficult to
manage its operations. Diseconomies of Scale are of two types;
viz., Internal Diseconomies and External Diseconomies.
• Internal Diseconomies: Internal Diseconomies means the
disadvantages of the large-scale production that a firm has to
suffer because of its own operations. For example, Technological
Diseconomies because of the heavy cost of wear and tear.
• External Diseconomies: External Diseconomies mean the
disadvantages of large-scale production that all the firms of the
industry have to suffer when the industry as a whole
expands. For example, stiff competition, etc.
 COBB-DOUGLAS PRODUCTION FUNCTION
• The Cobb-Douglas production function is based on the empirical
study of the American manufacturing industry made by Paul H.
Douglas and C.W. Cobb. It is a production function which takes into
account two inputs, labour and capital, for the entire output of the
manufacturing industry.
• In its original form, this production function applies not to an
individual firm but to the whole of manufacturing in the United
States. In this case, the output is manufacturing production and
inputs used are labour and capital. Cobb-Douglas production function
is stated as:
Q = ALa Cβ
• where Q is output and L and С are inputs of labour and capital
respectively. A, a and β are positive parameters where = a
(alpha)> O, β (beta)> O.

•
• The equation tells that output depends directly on L and C,
and that part of output which cannot be explained by L and С
is explained by A which is the ‘residual’, often called technical
change.
• The production function solved by Cobb-Douglas had 1/4
contribution of capital to the increase in manufacturing
industry and 3/4 of labour so that the C-D production function
is
Q = AL3/4 C1/4
which shows constant returns to scale because the total of the
values of L and С is equal to one: (3/4 + 1/4), i.e.,(a + β = 1)
• The C-D production function showing
constant returns to scale is depicted
here. Labour input is taken on the
horizontal axis and capital on the vertical
axis.
• To produce 100 units of output, ОС1
units of capital and OL1 units of labour
are used. If the output were to be
doubled to 200, the inputs of labour and
capital would have to be doubled. ОС2 is
exactly double of ОС1 and of OL2 is
double of OL2.
• Similarly, if the output is to be raised
three-fold to 300, the units of labour
and capital will have to be increased
three-fold. OC3 and OL3 are three times
larger than ОС1, and OL1, respectively.
• Another method is to take the scale line
or expansion path connecting the
equilibrium points Q, P and R. OS is the
scale line or expansion path joining
these points.
• It shows that the isoquants 100, 200 and
300 are equidistant. Thus, on the OS
scale line OQ = QP = PR which shows
that when capital and labour are
increased in equal proportions, the
output also increases in the same
proportion.
 Importance of Cobb-Douglas Production
Function
• Despite these criticisms, the C-D function is of much importance.
• It has been used widely in empirical studies of manufacturing
industries and in inter-industry comparisons.
• It is used to determine the relative shares of labour and capital in
total output.
• Its parameters a and b represent elasticity coefficients that are used
for inter-sectoral comparisons.
• This production function is linear homogeneous of degree one
which shows constant returns to scale, If α + β = 1, there are
increasing returns to scale and if α + β < 1, there are diminishing
returns to scale.
• Economists have extended this production function to more than
two variables.
 Criticisms of Cobb-Douglas Production
Function
• The C-D production function considers only two inputs, labour
and capital, and neglects some important inputs, like raw
materials, which are used in production. It is, therefore, not
possible to generalize this function to more than two inputs.
• In the C-D production function, the problem of measurement of
capital arises because it takes only the quantity of capital available
for production. But the full use of the available capital can be
made only in periods of full employment. This is unrealistic
because no economy is always fully employed.
• The C-D production function is criticised because it shows
constant returns to scale. But constant returns to scale are not an
actuality, for either increasing or decreasing returns to scale are
applicable to production.
 Criticisms of Cobb-Douglas Production
Function
• It is not possible to change all inputs to bring a proportionate change in
the outputs of all the industries. Some inputs are scarce and cannot be
increased in the same proportion as abundant inputs. On the other
hand, inputs like machines, entrepreneurship, etc. are indivisible. As
output increases due to the use of indivisible factors to their maximum
capacity, per unit cost falls.
• Thus when the supply of inputs is scarce and indivisibilities are present,
constant returns to scale are not possible. Whenever the units of
different inputs are increased in the production process, economies of
scale and specialization lead to increasing returns to scale.
• In practice, however, no entrepreneur will like to increase the various
units of inputs in order to have a proportionate increase in output. His
endeavour is to have more than proportionate increase in output,
though diminishing returns to scale are also not ruled out.
 Criticisms of Cobb-Douglas Production
Function
• The C-D production function is based on the assumption of
substitutability of factors and neglects the complementarity of
factors.
• This function is based on the assumption of perfect competition
in the factor market which is unrealistic. If, however, this
assumption is dropped, the coefficients α and β do not represent
factor shares
• One of the weaknesses of C-D function is the aggregation
problem. This problem arises when this function is applied to
every firm in an industry and to the entire industry. In this
situation, there will be many production functions of low or high
aggregation. Thus the C-D function does not measure what it
aims at measuring.