Choice Under Uncertainty

and Intertemporal Choice

Block 2
Production and Cost

101
Consumer Theory
 Production Function with
UNIT 5 PRODUCTION FUNCTION WITH ONE One and More Variable
Inputs
AND MORE VARIABLE INPUTS
Structure
5.0 Objectives
5.1 Introduction
5.2 Production Function
5.2.1 Short-run Production Function
5.2.2 Law of Variable Proportions
5.2.3 Long-run Production Function
5.2.4 Isoquants
5.2.5 Marginal Rate of Technical Substitution
5.2.6 Producer’s Equilibrium
5.2.7 Elasticity of Technical Substitution
5.2.8 Economic Region of Production
5.3 Homogenous and Homothetic Functions
5.3.1 Homogeneous Function
5.3.2 Homothetic Function
5.4 Types of Production Functions
5.4.1 Linear Production Function
5.4.2 Leontief Production Function
5.4.3 Cobb-Douglas Production Function
5.4.4 The CES Production Function
5.5 Technological Progress and the Production Function
5.5.1 Hick’s Classification of Technological Progress
5.6 Let Us Sum Up
5.7 References
5.8 Answers or Hints to Check Your Progress Exercises

5.0 OBJECTIVES
After going through this unit, you should be able to:
• understand the concept of production function and its types;
• mathematically comprehend various concepts of production theory
introduced in Introductory Microeconomics of Semester 1;
• explain the concepts of homogeneous and homothetic functions along
with their properties;
• analyse different types of production functions, viz. Linear, Leontief,
Cobb-Douglas and CES production function; and
• discuss the impact of technical progress on the production function or
an isoquant. 103
Production and Cost
5.1 INTRODUCTION
Production in Economics means creation or addition of value. In production
process, economic resources or inputs in the form of raw materials, labour,
capital, land, entrepreneur, etc. are combined and transformed into output.
In other words, firm uses various inputs/factors, combines them with
available technology and transforms them into commodities suitable for
satisfying human wants. For example, for making a wooden chair or table,
raw materials like wood, iron, rubber, labour time, machine time, etc. are
combined in the production process. Similarly, cotton growing in nature
needs to be separated from seeds, carded, woven, finished, printed and
tailored to give us a dress. All the activities involved in transforming raw
cotton into a dress involve existence of some technical relationship between
inputs and output.
The present unit is an attempt to build up on the foundation of the Theory
of Production you learnt in your Introductory Microeconomics course of
Semester 1. Units 6 and 7 of the Introductory Microeconomics course
comprehensively discussed Production function with one variable input and
with two or more variable inputs, respectively. This theoretical base shall be
combined with the mathematical tools you have already learnt in your
Mathematical Economics course of Semester 1. Section 5.2 will give a brief
review along with the Mathematical comprehension of what we already
know about the production theory. Section 5.3 shall explain the concepts of
Homogeneous and Homothetic functions along with their properties.
Further, in Section 5.4 we will elaborate upon the types of production
functions, viz. Linear, Leontief, Conn-Douglas and CES production functions.
This Unit ends with representation of the impact of technological progress
on the production function, along with the Hick’s classification of technical
progress.

5.2 PRODUCTION FUNCTION

A firm produces output with the help of various combinations of inputs by
harnessing available technology. The production function is a technological
relationship between physical inputs or factors and physical output of a
firm. It is a mathematical relationship between maximum possible amounts
of output that can be obtained from given amount of inputs or factors of
production, given the state of technology. It expresses flow of inputs
resulting in flow of output in a specific period of time. It is also determined
by the state of technology. Algebraically, production function can be written
as:
Q = f (A, B, C, D,….)
where Q stands for the maximum quantity of output, which can be
produced by the inputs represented by A, B, C, D,…, etc. where f (.)
represents the technological constraint of the firm.

104
5.2.1 Short-run Production Function Production Function with
One and More Variable
Inputs
A Short run production function is a technical relationship between the
maximum amount of output produced and the factors of production, with at
least one factor of production kept constant among all the variable factors.
A two factor short run production function can be written as:

Q f (L, K)
where, Q stands for output, L for Labour which is a variable factor here, K for
Capital, and f (.) represents functional relationship. A bar over letter K
indicates that use of capital is kept constant, that is, it is a fixed factor of
production. Supply of capital is usually assumed to be inelastic in the short
run, but elastic in the long run. This inelasticity of the factor is one of the
reasons for it to be considered fixed in the short run. Hence, in the short
run, all changes in output come from altering the use of variable factor of
production, which is labour here.
Total Product (TP)
Total Product (TP) of a factor is the maximum amount of output (Q)
produced at different levels of employment of that factor keeping constant
all the other factors of production. Total product of Labour (TPL) is given by:
TPL = Q = f (L)
Average Product (AP)
Average product is the output produced per unit of factor of production,
given by:
Q
Average Product of Labour, APL = and Average Product of Capital,
�
Q
APK = .
�
Marginal Product (MP)
Marginal Product (MP) of a factor of production is the change in the total
output from a unit change in that factor of production keeping constant all
the other factors of production. It is given by: Marginal Product of Labour,
∆� �� ∆� ��
MPL = or and Marginal Product of Capital, MPK = or , where ∆
∆� �� ∆� ��
stands for “change in” and � denotes partial derivation in case of a function
with more than one variable [here we are considering a production function
with two factors of production, Q = f (L,K)].

Law of Diminishing Marginal Product

The law of diminishing marginal product says that in the production process
as the quantity employed of a variable input increases, keeping constant all
the other factors of production, the marginal product of that variable factor
may at first rise, but eventually a point will be reached after which the
marginal product of that variable input will start falling.
105
Production and Cost
5.2.2 Law of Variable Proportions
Also called the law of non-proportional returns, law of variable proportions
is associated with the short-run production function where some factors of
production are fixed and some are variable. According to this law, when a
variable factor is added more and more to a given quantity of fixed factors in
the production process, the total product may initially increase at an
increasing rate to reach a maximum point after which the resulting increase
in output become smaller and smaller.
G
MPL= 0

F
TPL TPL
Stage I Stage II Stage III

0 Labour (L)
APL/ MPL

H
J

APL
K
0 Labour (L)
MPL

Fig. 5.1: Law of Variable Proportion

Stage 1: This stage begins from origin and ends at point F (in part (a) of the
Fig. 5.1). Corresponding to the point F, you may see the APL reaches
maximum and APL = MPL represented by point J in part (b) of Fig. 5.1. Point E
where the total product stops increasing at an increasing rate and starts
increasing at diminishing rate is called point of inflexion. At point E, TPL
changes its curvature from being convex to concave.
Stage 2: This stage begins from point F and ends at point G (in part (a) of the
Fig. 5.1).
Corresponding to the point F, you may see the AP curve reaches its
maximum (point J) and both AP and MP curves are having falling segments
along with MP reaching 0 i.e., MP curve touches the horizontal axis (at point
K). From point F to point G, the total product increases at a diminishing rate,
marginal product falls but remains positive. At point K marginal product of
the variable factor reduces to zero. Since both the average and marginal
products of the variable factor fall continuously, this stage is known as stage
of diminishing returns.

106
 Stage 3: Beginning from point G, the total product declines and slopes Production Function with
One and More Variable
downward. Marginal product of variable factor is negative. Given the fixed Inputs
factor, the variable factor is too much in proportion and hence this stage is
called stage of negative returns.
Remember: *
����
At point H, slope of MPL = 0, i.e., ��
=0
�� ���
Up to point E (the inflection point) ���
> 0 (denoted by the convexity of
�� ���
the TPL), and from point E onwards till point G, ���
< 0 (denoted by the
concavity of the TPL).
�� �
At point F and J, MPL = APL, i.e., =
�� �
����
At point J, slope of APL = 0, i.e., ��
=0
��
At point G, slope of TPL, i.e. MPL or �� = 0
Point K onwards, MPL < 0
Relationship between Average Product and Marginal Product
1) So long as MP curve lies above AP curve, the AP curve is sloping
upwards. That is, when MP > AP, AP is rising.
2) When MP curve intersects AP curve, this is the maximum point on the
AP curve. That is, when MP = AP, AP reaches its maximum.
3) When MP curve lies below the AP curve, the AP curve slopes
downwards. That is, when MP < AP, AP is falling.
�(�)
Proof: Consider total product of Labour TPL = f (L), then APL = .
�
For maximisation of APL differentiating it w.r.t. L and putting it equal to 0,
� � �(�)
First order condition (FOC): (APL ) = 0 ⇒ � �=0
�� �� �
��(�) � �(�)
. − =0
�� � ��
��(�) �(�)
= …(1)
�� �
⇒ MPL = APL when APL reaches its maximum.
�� (��� ) � � � ��(�)
Second order condition (SOC): ���
≤ 0 ⇒ �� ��� AP� � ≤ 0 ⇒ �� � ��
�≤0

� ��(�) � �(�) �’ (�)�� �(�).�

Using Equation 1, we get, �� � ��
� = �� � �
�= ��

�� (��� ) � �(�) �(�)

Thus, ���
= � �f’(L) − �
� ≤ 0 ⇒ f’(L) ≤ �
⇒ MP� ≤ AP�

* Refer Unit 10 of your course on Mathematical Methods in Economics during first

semester (BECC-102) for the derivative criterion for finding maxima, minima,
point of inflexion. 107
Production and Cost
5.2.3 Long-run Production Function
In long run, all factors can be varied, thus, for a long-run production function
all inputs vary proportionally. Consider a long-run two factor production
function:
Q = f (L, K)
where, Q stands for output, L for Labour and K for Capital (here, K is without
bar, that is, it represents a variable factor like L).
The basic assumption we make about the production function is
monotonicity, which means as the factor labour (L) increases, given the
factor capital (K), the production Q also increases. Similarly as the factor
capital (K) increases, given the factor labour (L), the production Q increases.
Thus, the first derivative of the production function is positive w.r.t. L and K
i.e., f′(L) > 0, f′(K) > 0. In other words the marginal product of labour and
capital are positive. The second assumption we usually make about the
production function is with the curvature. The assumption is the concavity
i.e., f′′(L) < 0, f′′(K) < 0; diminishing returns to the marginal product of L and
K. But the second order cross partial derivative i.e.,
� ��(�,�) � ��(�,�) �� �(�,�)
��
� �� � = �� � �� � = ���� > 0. The two second order cross partial
derivatives are equal by Young’s theorem.
�2 f(K,L) �� �(�,�)
Note that, = f′′(L) and ���
= f′′(K) are the second order own
�L2
partial derivative w.r.t L and K respectively.

Output Elasticity of a Factor

Given the production function, X = f (L) the elasticity of output with respect
to factor (L) is given by the ratio of proportionate change in output (X) to
proportionate change in use of factor (L). Output Elasticity of Factor L (eL) is
given by:
� (��� �) %∆� �� � ��
eL = � (��� �) = %∆� = �� × � = �� �
�

Example 1
Consider a production function as follows: Q = 6K2L2 − 0.10K3L3, where Q is
the total output produced, and K and L the two factors of production. With
factor K fixed at 10 units, determine
a) The total product function for factor L (TPL)
b) The marginal product function for factor L (MPL)
c) The average product function for factor L (APL)
d) Number of units of input L that maximises TPL
e) Number of units of input L that maximises MPL
f) Number of units of input L that maximises APL
g) The boundaries for the three stages of production
108
Solution: Production Function with
One and More Variable
a) TPL = 6 (10)2 L2 − 0.10 (10)3 L3 = 600L2 − 100L3 Inputs

�(��� )
b) MPL = = 1200 L – 300 L2
��
���
c) APL =
�
��� �� ���� ��
= = 600 L – 100 L2
�
�(��� )
d) For maximisation of TPL put =0
��
⇒ 1200 L – 300 L2 = 0
⇒ L (1200 – 300 L) = 0 ⇒ L = 0 or L = 4
�� (��� )
Checking for second order condition for maximisation, i.e., < 0,
���
we get L = 4 that maximises TPL.
�(��� )
e) Condition for maximisation of MPL is given by, =0
��
1200 – 600 L = 0 ⇒ L = 2 maximises MPL
�(��� )
f) Condition for maximisation of APL is given by, =0
��
600 – 200 L = 0 ⇒ L = 3 maximises APL
g) Stage I: Labour units 0 - 3, Begins from origin till the point where APL
reaches its maximum.
Stage II: Labour units 3 - 4, begins at point where APL reaches its
maximum till the point when MPL reduces to 0 with TPL reaching its
maximum.
Stage III: Labour units 4 - ∞, begins at point where MPL = 0 till the point
where MPL < 0.
Example 2
The production function of firm is given by X = 8L + 0.5L2 – 0.2L3 where X is
the output produced and L denotes 100 workers.

a) Determine the point at which MPL = APL

b) Find the range over which production function exhibits the property of
diminishing marginal productivity of labour?

c) How many workers should be employed so that MPL becomes zero?

d) Find TPL, MPL and APL when the firm employs 150 workers.

109
Production and Cost Solution:
a) Total product of labour TPL = X = 8L + 0.5L2 – 0.2L3
��
MPL = �� = 8 + L − 0.6L�
�
APL = � = 8 + 0.5L − 0.2L�

The point at which MPL = APL is given by

8 + L – 0.6L2 = 8 + 0.5L – 0.2L2
0.5L – 0.4L2 = 0
L (0.5 – 0.4L) = 0
�.�
Either L = 0 or L = �.� = 1.25 or 125 workers

Hence the point at which MPL = APL is L = 125 workers

b) The production function exhibits diminishing marginal productivity of
����
labour over range where <0
��
�
That is, 1 – 1.2L < 0 or L > �.� ⇒ L > 0.83 or 83 workers.
Hence the range over which production function exhibits the property
of diminishing marginal product of labour is L > 0.83 or more than 83
workers.
c) The number of workers for MPL to become zero is given by solution
8 + L – 0.6L2 = 0
or 0.6L2 – L– 8 = 0
�� √����.�
Therefore, L = = 4.58 (approx). The other root being negative is
� �.�
neglected. Thus when MPL = 0 about 458 workers are being employed.
d) When firm employs 150 workers, L = 1.5
Substituting this value in TPL, MPL and APL we get
TPL = 8L + 0.5L2 – 0.2L3
= 8 × 1.5 + 0.5 × (1.5)2 – 0.2 (1.5)3 = 12.45 units
MPL = 8 + L – 0.6L2
= 8 + 1.5 – 0.6(1.5)2 = 8.15 units
APL = 8 + 0.5L – 0.2L2
= 8 + 0.5 × 1.5 – 0.2(1.5)2 = 8.3 units

110
5.2.4 Isoquants Production Function with
One and More Variable
Inputs
Isoquants is the locus of all possible input combinations which are capable
of producing the same level of output (Q). In Fig. 5.2, all the possible
combinations of Labour (L) and Capital (K), for instance (L1,K1), (L2,K2) and
(L3,K3), produce a constant level of Output (Q).

Fig. 5.2: Isoquant

Properties of Isoquants
1) Isoquants are negatively sloped.
2) A higher isoquant represents a higher output.
3) No two isoquants intersect each other.
4) Isoquants are convex to the origin. The convexity of isoquant curves
implies diminishing returns to a variable factor.
Isoquant Map
An Isoquant map is a family of isoquant curves, where each curve represents
a specified output level. Three such curves with different output levels (Q1,
Q2 and Q3) forming an Isoquant map is given in Fig. 5.3.

Fig. 5.3: Isoquant Map

111
Production and Cost Isocost Line
An Isocost line represents various combinations of two inputs that may be
employed by a firm in the production process for a given amount of Budget
�
and prices of the factors. Slope of an Isocost line is given by , that is, the
�
factor price ratio, where, w and r represent the prices paid to Labour and
Capital factors, respectively. Refer Fig. 5.4, where we have three different
isocost lines with different budget outlays represented by C1, C2 and C3, such
that C3 > C2 > C1.

Capital (K)

C1 C2 C3
0
Labour (L)

Fig. 5.4: Isocost Lines

5.2.5 Marginal Rate of Technical Substitution

Marginal rate of technical substitution (MRTS) is the rate at which one factor
can be substituted for another along an Isoquant. Along an Isoquant output
remains constant, i.e. (dQ = 0)
K.MPK L.MPL 0

K.MPK L.MPL
K MPL
L MPK

MPL
MRTS LK
MPK

As quantity of labour is increased and quantity of capital employed is

reduced, the amount of capital that is required to be replaced by an
additional unit of labour so as to keep the output constant will diminish.

5.2.6 Producer’s Equilibrium

A rational producer attempts to maximise his profits either by maximising
the production of output for a given level of cost of production or by
minimising the cost of production of a given level of output. Either way the
producer chooses, it will result in employment of an optimum combination
112
of resources in the production process so that MRTSLK equals the price ratio Production Function with
One and More Variable
of the factors. That is, producer’s equilibrium is given by the condition: Inputs
� ��� �
MRTS LK = ⟹ =
� ��� �

The equilibrium condition represents the tangency between the isoquant

�
and the isocost line. In Fig. 5.5, at point E, MRTSLK = .
�
Capital (K)

0
Labour (L)

Fig. 5.5: Producer’s Equilibrium

5.2.7 Elasticity of Technical Substitution

Elasticity of Technical substitution in production is a measure of how easy it
is to shift between factors in the production process. It is given by :

Proportionate change in ratio of inputs (K & L) used

σ=
Proportionate change in marginal rate of technical substitution of L for K

Proportionate change in K/L

=
Proportionate change in MRTSLK

K/L
K/L
MRTSLK / MRTSLK

∆�/�
� �/�
At equilibrium, MRTS LK = , therefore we get, � = ∆�/�
�
�/�

5.2.8 Economic Region of Production

The economic theory focuses on only those combinations of factors which
are technically efficient; i.e., where the marginal products of factors are
diminishing but positive. These combinations, forming the efficient region of
113
Production and Cost production, are represented by the downward sloping and convex to the
origin isoquants. Refer to the following Fig. 5.6 where factors L and K are
assumed to be substitutable but not perfectly. When firm goes on
��
substituting L for K, a point like P is reached where MRTSLK given by �� �
�
diminishes to 0 (as MPL = 0 at point P). This implies, at point P, no more K
can be given up for having more of L. Beyond point P, as L rises, MPL
becomes negative. In order to produce the fixed output (Q1), the
mismanagement caused by the excessive L units needs to be corrected. This
is done by increasing the employment of factor K (since MPK > 0) as L
increase beyond point P. This gives us the positively sloped isoquant below
ridge line OB. Similarly, at point R, MPK = 0. So, as K increases beyond point
R, to make up for the negative MPK, L would also have to be increased.

Ridge Lines
The ridge line OA is the locus of those points of isoquants where marginal
product of capital is zero and ridge line OB is the locus of those points of
isoquants where marginal product of labour is zero. See the following Fig.
5.6. A rational producer will operate in the region bound by the two ridge
lines called the economic region of production. The regions outside the
ridge lines are called regions of economic nonsense (technically inefficient
region).
Capital (K)

A
(MPK) < 0
Economic Region of
B Production

Q2
R

Q1
P
(MPL) < 0

O Labour (L)

Fig. 5.6: Economic Region of Production

5.3 HOMOGENOUS AND HOMOTHETIC FUNCTIONS

5.3.1 Homogenous Function
A function f(X1, X2,…, Xn) is said to be homogenous of degree k if
f (mX1, mX2,…, mXn) = mk f (X1, X2,…, Xn)
where m is any positive number and k is constant.
A zero-degree homogeneous function is one for which
114
 f (mX1, mX2,…, mXn) = m0 f (X1, X2,…,Xn) Production Function with
One and More Variable
In a similar way, Homogeneous production function of first degree can be Inputs
expressed as
f (mX, mY) = m1 f (X, Y)
Here X and Y are the two factors of production. It simply says if factors X and
Y are increased m times, total production also increases m times.
In case of a Linear Homogeneous production function or Homogeneous
production function of first degree with k = 1, if all factors of production are
increased in a given proportion, output also increases in the same
proportion. This represents the case of constant returns to scale (CRS).
When k > 1, production function yields increasing returns to scale (IRS),
whereas when k < 1, it yields decreasing returns to scale (DRS).
Euler’s Theorem
For a homogenous of degree k function f(X1,X2,…,Xn), Euler’s theorem gives
the following relationship between a homogeneous function and its partial
derivatives:
�� �� ��
X1 �� + X2 �� + … + Xn �� = k f(X1,X2,…,Xn)
� � �

Properties of Homogenous Functions

1) If f(X1, X2,…, Xn) is homogenous of degree k then it’s first order partial
derivatives will be homogenous of degree (k – 1).
2) For a homogenous of degree k function f (.), if f(X) = f(Y), then f(tX) =
f(tY).
Proof: We have f(tX) = tkf(X) and f(tY) = tkf(Y) (1)
Given, f(X) = f(Y) (2)
From (1) and (2), we get
f(tX) = f(tY)
3) Level curves of a homogenous function f(X, Y) have constant slopes
along each ray from the origin. That is, if f(X, Y) is a homogeneous
production function of degree k, then the MRTS is constant along rays
extending from the origin.
Returns to Scale
Returns to scale are a measure of technical property of a production
function that examines how output changes subsequent to a proportional
change in all the factors of production. If proportional change in output is
equal to the proportional change in factors, then there are constant returns
to scale (CRS). If proportional change in output is less than the proportional
change in factors, there are decreasing returns to scale (DRS), whereas if
proportional change in output exceeds the proportional change in factors,
there are increasing returns to scale (IRS).
115
Production and Cost A homogenous function of degree k, exhibits
i) Constant Returns to Scale if k = 1
ii) Increasing Returns to Scale if k > 1
iii) Decreasing Returns to Scale if k < 1

5.3.2 Homothetic Function

A Homothetic function is a monotonic transformation* of a homogeneous
function. It is given by the form:

H (X1, X2,…, Xn) = F [f (X1, X2,…, Xn)]

where, f (X1, X2,…, Xn) represents a homogeneous function of degree k, and

F (.) is a monotonically increasing function. In case of such a function, H(X1) ≥
H(X2) ⟺ H(tX1) ≥ H(tX2) for all t > 0, where symbol ⟺ represents “if and only
if.”

Given a homogeneous function of degree 2, f(X, Y) = XY, a homothetic

function H(X, Y), which is the monotonic transformation of f(X, Y) could be
in the following forms:
H1(X, Y) = XY + 2;
H2(X, Y) = (XY)2;
H3(X, Y) = X3Y3 + XY;
H4(X, Y) = ln X + ln Y (where ln stands for “natural log”)
H5(X, Y) = eXY
Important:
A Homogeneous production function implies that it is homothetic as well,
but converse is not true, for instance, f(X,Y) = XY + 1 is homothetic, but not
homogeneous [proof: f(tX, tY) = t2XY + 1 and t2 f(X,Y) = t2XY + t2, here f(tX,
tY) ≠ t2 f(X,Y), hence f(X,Y) = XY + 1 is not homogeneous].

Properties of Homothetic Functions

1) Level curves of a Homothetic function are radial expansion of one
another, i.e., H(X1, Y1) = H(X2, Y2) ⟺ H(mX1, mY1) = H(mX2,mY2) for all
t > 0.
2) Level curves of a Homothetic function H(X, Y) have constant slopes
along each ray from the origin.

* Monotonic transformation— A transformation of a set that preserves the order

of that set. For instance, if the original function is f (X, Y), a monotonic
transformation is represented by F [f(X, Y)] so that, if f1 > f2, then F(f1) > F(f2), where
116 F(.) is the strictly increasing function on f(.).
When we have a homothetic production function, the above property Production Function with
One and More Variable
implies that Isoquants have a constant slope (MRTSXY) along any ray from Inputs
�
origin. MRTSXY only depends on factor proportion ���, that is, it is a
homogenous function of degree 0.
Check Your Progress 1
1) As the quantity of a variable input increases, explain why the point
where marginal product begins to decline is reached before the point
where average product begins to decline. Also explain why the point
where average product begins to decline is reached before the point
where total output begins to decline?
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
2) Given a firm’s production function, X = 50 + 30L – L2, where X is the
output produced and L the amount of Labour employed, also the
Average Revenue function is AR = 1200 – 3X, answer the following:
a) Find MPL and the value of L at which MPL = 0
b) Does the production function show diminishing marginal
productivity of labour?
c) Write down an expression of MRPL as a function of L and find its
value where L = 10.
d) Is it profitable to employ more or less labourers? Explain.
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
3) Which of the following functions is homogenous? Write their degrees of
homogeneity.
�
a) �

b) X + Y�
c) X �/� Y�/� + 2X
d) 4X � Y + 5X � Y � − 2X � Y �
e) X � Y + 3X � Y � − 2X � Y �
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
……………………………………………………………………………………………………… 117
Production and Cost
5.4 TYPES OF PRODUCTION FUNCTIONS
In this sub-section we are introducing some functions of more than one
independent variable which have certain unique mathematical properties.
These properties facilitate derivation of certain interesting results which are
attractive for economic analysis.

5.4.1 Linear Production Function

A Linear production function is given by the following form:
Q = αK + βL
Where, Q stands for output, K and L, the two inputs in production, α and β,
the two constant terms. Production function of this form represents inputs
which behave as perfect substitutes to each other in the production process.
For this reason MRTS remains constant along an isoquant resulting in a
straight line downward sloping Isoquant curves. Refer Fig. 5.7 for Isoquants
of a linear production function.
Capital (K)

Q1 Q2 Q3

0 Labour (L)

Fig. 5.7: Isoquants for a Linear Production function

��� �
MRTSLK = = , which is a constant. , Now, Elasticity of Technical
��� �
�
∆
��
�
�
Substitution is given by, � = ∆ ����� . Here along the Isoquant, MRTS
����
remains constant, so that ∆ MRTS = 0. This implies that Elasticity of
Technical Substitution(�) = ∞ for a linear production function. That is,
inputs are perfectly substitutable for each other in the production process.

5.4.2 Leontief Production Function

Production technology sometimes could be such that factors of production
must be employed in a fixed proportion. For instance, to produce a unit of
output, capital and labour must be employed in proportion of 2:1, so that no
118
output increase could be possible by increasing the units of capital alone or Production Function with
One and More Variable
of labour alone, or of both in a different ratio than 2:1. Leontief production Inputs
technology represents the case where inputs must be combined in fixed
proportions, for this reason it is also called a Fixed-proportions production
function. The Leontief production function is given as the following form:
� �
Q = min �� , � �
� �

Where Q is the output produced, K and L represent the factors of

production, θ� and θ� are the unit input requirements. That is, to produce a
single unit of output, θ� unit of factor K and θ� units of factor L are needed.
Consequently for Q units of output, θ� Q units of factor K and θ� Q units of
factor L will be needed. Thus, the fixed proportion of factors to produce
� �
output is given by � = �� . Factors of such production function behave as
�
Perfect compliments to each other in the production process. Refer Fig. 5.8
for L-shaped Isoquants of a Leontief production function.
Capital (K)

��
K= L
��

K3 Q3
K2 Q2
K1 Q1

0 L1 L2 L3 Labour (L)

Fig. 5.8: Isoquants for a Leontief Production function

� �
In Fig. 5.8 you may notice, if K = K1 and L = L2, then we have � � < �� , thus ,
� �

�
Q = � � . In this case, the technically efficient level of L factor would be given
�
�� � �
by, � = � ⟹ L = � � K� , which is, as you may notice from the figure is
� � �
given by L1. The equation for the line from the origin, at which factor
� �
proportion equals �� , is given by K = �� L.
� �

Since there exist no possibility for altering the factor proportion, any change
in MRTS, does not result in change in factor proportion which remains fixed.
�
That is ∆ � � � = 0. This, implies that Elasticity of Technical Substitution(σ) = 0
for a Leontief production function.

5.4.3 Cobb-Douglas Production Function

A widely used form of production function is the Cobb-Douglas production
function which takes the following form: 119
Production and Cost
Q = ALαKβ
where Q is the output, L and K the factors of production, and A, α, β are all
positive constants. The Isoquants of this production function are hyperbolic,
asymptotic to both the axis (i.e. it never touches any axis). Refer Fig. 5.9.

Capital (K)
Q3
Q2
Q1
0
Labour (L)

Fig. 5.9: Isoquants for a Cobb-Douglas production function

Some properties of a Cobb-Douglas production function are as follows:

Returns to Scale
For the Cobb-Douglas production function Q = ALαKβ, when
α + β = 1 there are constants returns to scale (CRS)
α + β > 1 there are increasing returns to scale (IRS)
α + β < 1 there are decreasing returns to scale (DRS)

Average and Marginal Product of Factors

For a Cobb-Douglas Production Function Q = AL� K �
��� ��
Average Product of Labour, APL =
�

= AL��� K �
��� ��
Average Product of Capital, APK =
�

= AL� K ���
��
Marginal Product of Capital, MPk =
��

= βAL� K ���

120
 �� Production Function with
Marginal Product of Labour, MPL = One and More Variable
��
Inputs
= αAL��� K �
Marginal Rate of Technical Substitution (MRTS)
Now, we have MP� = αAL��� K � and MP� = βAL� K ���
��� ������ �� ��
MRTS = = =
��� ���� ���� ��

Product Exhaustion Theorem

In case of a homogeneous production function of degree one, if each unit of
every factor of production is given a reward equal to the marginal product
of that factor, the total output will be exactly divided among those factors.
This is what is referred to as the Product Exhaustion theorem.

Consider a CRS Cobb Douglas production function, Q A.L K , where

.

K L
Now, MPL A. and MPK A.
L K

According to Euler Theorem, if production function is homogeneous of first

degree, then
Total Output (Q) = L. MPL + K.MPK

K L
Q L.A K.A.
L K

Q A. L1 K A. L K1

A.(1 )L1 K A. L K1

A.L1 K
A.L K
=Q
Thus in Cobb Douglas production with 1 if wage rate = MPL and rate
of return on capital (K) = MPK, then total output will be exhausted.
Elasticity of Substitution
�
������������ ������ �� �����
�
es or � =
������������ ������ �� ������

� �
�� � �� �
=
������� ⁄������
121
Production and Cost

� �
�� � �� �
= �� ��
��� � ���� � �

� �
�� � �� �
= � � � � =1
�
.�� � ���� � �

Output Elasticity of Factors

��
�� � ��
Elasticity of Output of Labour = e� = . = �
�� �
�

��� ������ ��
= =
��� ����� ��

e� = α

Q K MPK
Elasticity of Output of Capital = e� = .
K Q APK

���� ����
e� =
��� ����

e� = β
Example 3
Consider the Cobb-Douglas production function below:
Q = 10L0.45K0.30
Where Q is the output produced using factors L (Labour) and K (Capital).
Calculate
a) Output Elasticities for Labour and Capital.
b) Change in Output, when Labour increases by 15%
c) Change in output, when both Labour and Capital increase by 15%
Solution:
�� � ��
a) Elasticity of output with respect to Labour, eL = �� . � = �� �
�

���.�����.�� ��.��
=
�����.�� ��.��
= 0.45
122
 �� � �� Production Function with
Elasticity of output with respect to Capital, eK = �� . � = �� � One and More Variable
�
Inputs
���.����.�� ���.��
=
����.�� ���.��
= 0.30
%∆�
b) eL =
%∆�

We have eL = 0.45 and %∆L = 15, therefore %∆Q = eL× %∆L = 6.75.
Hence, output will increase by 6.75%.
c) Change in Output when Labour increase by 15% = 6.75% (calculated in
part b)
Similarly, change in Output when Capital increases by 15% = eK× %∆K =
4.5%

Therefore, Change in Output when both Capital and Labour increase by

15% = (6.75 + 4.5)% = 11.25%
Example 4
Suppose a commodity (Q) is produced with two inputs, labour (L) and capital
(K) and production function is given by Q 10 LK . What type of returns to
scale does it exhibit?
Solution:
1/2 1/2
The above production function can be rewritten as Q 10.L K

Now, increase both the factors by a positive constant λ.

Q 10( L)1/2 ( K)1/2

1/2 1/2
10L1/2 K1/2
Q Q

Increasing L and K by λ results in increase in output (Q) by λ. Hence this

shows constant returns to scale.

5.4.4 The CES Production Function

Linear, Leontief and Cobb-Douglas production functions are a special case of
the Constant Elasticity of Substitution (CES) production function, which has
been jointly developed by Arrow, Chenery, Minhas and Solow. CES
production function is a general production function wherein elasticity of
factor substitution can take any positive constant value. The function is
given by the following equation:
Q = C [αKρ + (1 − α)Lρ ] 1/ρ
Where, Q stands for output.

123
Production and Cost ‘C ’ is an efficiency parameter, a measure of technical progress. The value of
C > 0 and any change in it resulting from technological or organisational
change causes shift in the production function.
‘α ’ is a distribution parameter, determining factor shares and 0 ≤ α ≤1. It
indicates relative importance of capital (K) and labour (L) in various
production processes.
ρ is a substitution parameter, used to derive elasticity of substitution (σ)
�
between factors K and L, given by σ = ��� . The value of ρ is less than or
equal to 1 and can be −∞. The two extreme cases are when ρ → 1 or ρ
→ −∞.
i) When ρ → 1, the elasticity of substitution tends towards ∞, the case
representing Linear Production function where factors are perfect
substitutes to each other in the production process giving straight line
Isoquants.
ii) When ρ → −∞, the elasticity of substitution tends towards 0, the case
representing Leontief Production function where factors are perfect
compliments to each other in the production process giving L-shaped
Isoquants.
iii) When ρ = 0, the elasticity of substitution = 1, then CES production
function becomes a Cobb-Douglas production function giving convex
Isoquants.
CES Production function are extensively used by economists in the empirical
studies of production processes because it permits the determination of the
value of elasticity of factor-substitution from the data itself rather than prior
fixing of the value of substitution elasticity (σ).
Check Your Progress 2
1) Consider the following production function:

Q L0.75K0.25
a) Find the marginal product of labour, and marginal product of
capital.
b) Show that the law of diminishing returns to the variable factor
holds.
c) Show that if labour and capital are paid rewards equal to their
marginal products, total product would be exhausted.
d) Calculate the marginal rate of technical substitution of capital for
labour.
e) Find out the elasticity of substitution.
f) Show that the function observes constant returns to scale.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
124
2) Consider the following CES production function: Production Function with
One and More Variable
Q = [αLρ + (1 − α)Kρ ] 1/ρ Inputs

Where 0 ≤ α ≤ 1, ρ ≤ 1, Q is output, L and K, the two factors of

production.
a) Find marginal productivities of both the factors L and K.
b) Also give the expression for MRTSLK.
c) Is this function Homothetic?
…………………………………………………………………………………………………………

5.5 TECHNOLOGICAL PROGRESS AND THE

PRODUCTION FUNCTION
Technological progress has been one of the major forces behind Economic
growth overtime. It enables output to rise even when the factors of
production remain at a constant level. Technical progress could be shown
with an upward shift of the production function (Refer Fig. 5.10 A, where
with same level of Labour L1, more of output could be produced, X1 > X) or a
downward movement of the production isoquant (Refer Fig. 5.10 B, where
the same level of output X could be produced by fewer quantities of factors
of production K1 and L1, with K1 < K0 and L1 < L0).

Fig. 5.10A: Technological change and production Fig. 5.10B: Technological change and Isoquant
function

5.5.1 Hicks Classification of Technological Progress

Hicks had distinguished three types of technical progress depending on its
effect on rate of substitution of factors of production. They are as follows:

Capital Deepening Technical Progress

Technical progress is said to be capital-deepening (or Labour saving) when
shifted Isoquant due to technical progress has lower MRTSLK at the
125
Production and Cost equilibrium points. This results as MPK increases more than MPL. It simply
means that the technical progress has resulted in increasing capital per
worker or capital intensity in the economy. Refer Fig. 5.11, where as we
move closer to the origin, MRTSLK falls along the equilibrium points.

Capital (K)

Q
Q

0 Labour (L)
Fig. 5.11: Capital Deepening Technical Progress

Labour Deepening Technical Progress

Technical progress is said to be Labour-deepening (or Capital saving) when
shifted Isoquant due to technical progress has higher MRTSLK at the
equilibrium points. This results as MPL increases more than MPK. This results
when technical progress decreases capital per worker or capital intensity in
the economy. Refer Fig. 5.12, where as we move closer to the origin, MRTSLK
rises along the equilibrium points.
Capital (K)

0 Labour (L)

Fig. 5.12: Labour Deepening Technical Progress

Neutral Technical Progress

Technical progress is said to be neutral when for a shifted Isoquant due to
technical progress MRTSLK does not change at the equilibrium points. Here,
MPK and MPL both increase at same proportion. This is represented in
126
Fig. 5.13, where as we move closer to the origin, MRTSLK remains constant Production Function with
One and More Variable
along the equilibrium points. Inputs
Capital (K)

0 Labour (L)
Fig. 5.13: Neutral Technical Progress

5.6 LET US SUM UP

A production function is a technological relationship between inputs and the
output in a production process. The unit began with defining a production
function, along with its two types— the short run and the long run
production function. A short run production function is a technical
relationship between output and inputs with at least one fixed input,
whereas a long run function is a relationship between output and inputs
with all inputs being variable. Unit proceeded with giving a brief review of
the concepts comprehensively covered in Introductory Microeconomics
course of Semester 1. In connection with the theory, the present unit
discussed Mathematical treatment of the concepts like Total Product,
Marginal Product, Average Product, Law of Variable Proportion, Output
Elasticity of a factor, Marginal rate of technical substitution, Elasticity of
Substitution, Producer’s equilibrium, etc. Followed by this, the concepts of
Homogeneous and Homothetic functions were touched upon. For a
homogenous function, we learnt that the value of the function when all its
arguments are multiplied by positive number m equals mk times the value of
the function with its original arguments. Monotonic transformation of this
function is the Homothetic function. Thus, all the Homogeneous functions
are Homothetic but converse is not true.
Various types of production functions, viz. Linear (Perfect Substitutes),
Leontief (Perfect Compliments), Cobb Douglas, and CES production function
were subsequently discussed. Text explained how a CES production function
as a general function approaches a Leontief or a Linear production function
for different values of ρ, which is referred to as the substitution parameter
�
with the relation given by, σ = ���, where � represents elasticity of
substitution between two factors of production. Unit concluded with a brief
127
Production and Cost discussion about the impact of technological improvement on production
function or an isoquant.

5.7 REFERENCES
1) Koutsoyiannis, A.(1979). Modern Microeconomics, Macmillan;
Macmillan; New York Chapters 3 and 4, page 67-148.
2) Bhardwaj, R.S. (2005). Mathematics for economics and business, Excel
Books.
3) Henderson, M.J. (2003). MicroEconomic Theory A Mathematical
Approach Tata McGrawl-Hill Publiching Company Limited New Delhi.
4) Varian, H.R. (2010). Intermediate Microeconomics, A Modern Approach,
W.W.Norton & Company New York.

5.8 ANSWERS OR HINTS TO CHECK YOUR PROGRESS

EXERCISES
Check Your Progress 1
1) Due to the operation of diminishing marginal returns, marginal product
begins to decline at some point, but for some range though diminishing
it remains greater than average product. Only when in its diminishing
phase marginal product becomes less than average product, average
product starts declining. That is why marginal product curve cuts the
average product curve at the latter’s highest point.
Marginal product continues to diminish after it is equal to the maximum
average product but remains positive which causes the total output to
continue increasing. Only when marginal product becomes zero, the
total product reaches its maximum level. As a result, total output
continues to increase even after the maximum average product point
and begins to decline only when marginal product becomes negative.
2) a) MPL = 30 − 2L
MPL is 0 at L = 15
� (��� )
b) = −2 < 0
��

Yes, the production function shows diminishing marginal productivity of

labour.
c) – 12L3 + 540L2 – 7200L + 27000; – 3000
Hint: TR = AR × X = (1200 – 3X). X = 1200X – 3X2
Now, MRPL = MR . MPL
� (��) ��
= .
�� ��
128
 = (1200 – 6X) . (30 – 2L) Production Function with
One and More Variable
= [1200 − 6 × (50 + 30L − L� )] (30 − 2L) Inputs

= – 12L3 + 540L2 – 7200L + 27000

d) Since MRPL is negative it is not profitable to employ more
labourers rather it would be profitable to employ less labourers.
3. a) Yes, degree 0
b) No
c) Yes, degree 1
d) Yes, degree 6
e) No
Check Your Progress 2
� �.�� � �.��
1) a) MPL= 0.75 � � � ; MPK = 0.25 ���

b) Law of diminishing returns to variable factor would hold when

���� ����
< 0 and < 0, that is, when marginal product of a factor
�� ��
declines with increase in employment of that factor keeping
constant the employment of other factor of production.
���� K0.25 ����
Since = −0.25 × 0.75� 1.25 � < 0 and = −0.75 ×
�� L ��
L0.75
0.25� � < 0, therefore here law of diminishing returns to
K1.75
variable factor holds true.
c) To show Q = MPL. L + MPK. K (the Product Exhaustion theorem)
� �.�� � �.��
Consider R.H.S, MPL. L + MPK. K ⟹ 0.75 � � � . L + 0.25 ��� .K

⟹ 0.75K0.25L0.75 + 0.25K0.25L0.75
⟹ K0.25L0.75 (0.75 + 0.25)
⟹ L0.75K0.25, which equals Q (the L.H.S).
�
d) MRTSLK = 3� �
�

e) σ=1
K K
Hint: σ L L
MRTS / MRTS
K
Substituting value of MRTS = 3
L

129
Production and Cost K/L K/L
Elasticity of substitution = 1
K K
3 3
L L

2) a) MPL = αLρ – 1 [αLρ + (1 − α)Kρ ] 1/ρ – 1

�
Hint: MPL = � � [αLρ + (1 − α)Kρ ] 1/ρ – 1 ρ αLρ – 1 ⟹ αLρ – 1 [αLρ + (1 −
�
α)Kρ ] 1/ρ – 1
MPK = (1 – α) Kρ – 1 [αLρ + (1 − α)Kρ ] 1/ρ – 1
� � ���
b) MRTSLK = ��� ���

c) Yes, the function is Homothetic since MRTSLK depends upon factor

�
proportion ���.

130
 Cost Function
UNIT 6 COST FUNCTION
Structure
6.0 Objectives
6.1 Introduction
6.2 Cost Minimisation
6.2.1 Graphical Approach for Cost Minimisation
6.2.2 Expansion Path
6.2.3 Analytical Approach for Cost Minimisation
6.3 Conditional Factor Demand Function
6.4 Cost Function
6.4.1 Properties of a Cost Function
6.4.2 Average and Marginal Cost Functions
6.4.3 Relationship between AC and MC Function
6.5 Short-run and Long-run Cost Functions
6.5.1 Short-run Cost Function
6.5.2 Long-run Cost Function
6.6 Let Us Sum Up
6.7 References
6.8 Answers or Hints to Check Your Progress Exercises

6.0 OBJECTIVES
After going through this unit, you should be able to:
• state the concept of cost minimisation;
• graphically and analytically approach the problem of cost minimisation;
• explain and derive conditional factor demand functions as a solution
of the constrained optimisation problem of cost minimisation;
• subsequently derive the cost function as a function of factor prices
and output;
• analyse average cost and marginal cost functions, along with the
relationship between them; and
• discuss the concept of short-run and the long-run cost functions.

6.1 INTRODUCTION
A production activity is undertaken for earning profits, and the producer
decides how much input to use to minimise its costs and maximise its
profits. Profits are given by the difference between the revenue earnings
from and the costs incurred during the production process. Costs, be it
131
Production and Cost implicit or explicit, are the expenses incurred by the producer for
undertaking the production of goods or services. Explicit costs are the out of
pocket expenses which the producer makes payment for, like paying for raw
materials, salaries and wages of staff employed, packaging and distribution
expenses, etc. On the other hand, by implicit it simply means the implied or
the opportunity cost of the self-owned inputs used by the producer in the
production process, like opportunity cost of entrepreneurial skills of the
entrepreneur, self-owned building used as office for business operations,
etc. Economic profits are calculated using both, the explicit as well as the
implicit costs. The optimal output of the firm is decided by maximisation of
profits or by minimisation of costs incurred. The present unit is an attempt
to analyse the approach of cost minimisation.
Unit begins with explaining the concept of cost minimisation. It proceeds
with the sub-sections discussing the graphical and the analytical approach
for cost minimisation. Subsequently, the concept of conditional input/factor
demand functions will be introduced and plugging these optimal values, cost
function will be derived. We will then derive the algebraic expression of
average cost and the marginal cost functions from the cost functions. The
unit also covers a mathematical proof of the relationship between the AC
and the MC curve, which was already covered in Introductory
Microeconomics course of Semester 1 (BECC-101). Towards the end, the
concepts of variable and fixed factors of production, and consequently the
short-run and the long-run cost functions have been discussed.

6.2 COST MINIMISATION

By cost minimisation it simply means to produce a specified output at the
minimum cost. This in turn results from employment of a mix of factors or
inputs so that the desired level of output is produced at the least cost.
Consider a production function given by Q = f (K, L), where Q is the output
produced by employing inputs K and L at given per unit factor prices r and w,
respectively. Then total cost of producing a specified output Q will be given
by:
C = Lw + Kr
Now cost minimisation would result in the following constrained
optimisation problem:
Min Lw + Kr
s.t. Q = f (L, K)
Here, the problems entails finding out the cheapest way to produce a given
level of output (Q) by a firm employing inputs K and L at the given inputs
prices and a technology relationship f (L, K). The optimal solution of the
above constrained minimisation problem will be given by (L*, K*) such that
for all (L, K) satisfying Q = f (L, K), we will have L*w + K*r ≤ Lw + Kr.

132
 Cost Function
6.2.1 Graphical Approach for Cost Minimisation
Recall the concept of Producer’s equilibrium we discussed in Unit 7 of your
Introductory Microeconomics course of Semester 1 (BECC-101). A producer
attains equilibrium by minimising the cost of producing output. This in turn
involves employing a particular factor combination at the given factor prices
and the input-output technological relationship. Fig. 6.1 represents such an
optimal factor combination.

C� Q
�0, � A′′
r
C�
�0, � A′ F
r
C� A
�0, �
r

K1 E
K (Capital)

G
Q′

O L1 B B′ B′′
L (Labour)

Fig. 6.1: Cost minimisation combination of factors

Lines AB, A′B′ and A′′B′′ represent Isocost lines. An isocost line is a locus of
various combinations of factor inputs (here K and L) that yield the same total
cost (C) for the firm. The equation of the isocost line is given by,
� �
C = Lw + Kr ⟹ K = − L
� �
�
This is a linear equation with slope , a constant measuring the cost of one
�
factor of production in terms of the other factor. For different values of C in
the above equation, we get different isocost lines. In Fig. 6.1, A′′B′′
� �
representing total cost C1 is given by, K = � − L. Similarly, outlays
� �
represented by A′B′ and AB are C2 and C3, respectively. Given the factor
prices, a higher outlay results in an outward parallel shift of the isocost line,
thus, to the north-east, higher isocost lines correspond to higher levels of
cost. In the above figure we have C1 > C2 > C3.
Curve QQ′ is the isoquant giving various combinations of factor inputs that
yield the same level of output (Q). Fig. 6.1 shows an isoquant representing a
given output level of output (let say Q*). Cost minimisation exercise involves
minimising the total cost of producing a given level of output (here Q*). For
instance, consider three possible factor combinations denoted by points E, F
and G giving different cost of producing output level Q*. Point E provides
factor combination producing output Q* at the cost of C3, while F and G
133
Production and Cost represents factor combinations producing output level Q* at the cost of C2
and C1, respectively. Among these possibilities, point E provides the least
cost (= C3) factor combination to the firm, as we know both C1 and C2 are
higher than C3 (C1 > C2 > C3). Graphically at point E, slope of the isoquant
given by the Marginal Rate of Technical Substitution (MRTSLK— the rate at
which two factors can be substituted with each other in the production of a
constant level of output) equals the slope of the isocost line. Point E, the
tangency point between isoquant and the isocost line gives us the optimal
combination of factors of production, i.e. OL1 amount of labour and OK1
amount of capital. Symbolically, at E we have,
� ��� � ��
MRTSLK = or = (as we know MRTSLK = �� � )
� ��� � �

6.2.2 Expansion Path

How does the cost-minimising factor combination changes as the output
production increases, keeping constant the factor prices?— an expansion
path answers such a question. Given factor prices, a firm can determine cost
minimising combination of factors for every level of output following the
�
rule MRTSLK = , that is the tangency between isoquants and isocost lines.
�
The expansion path is nothing but a locus of such optimal factor
combinations as the scale of production expands. In Fig. 6.2, expansion path
OE is determining minimum cost combinations of labor (L) and capital (K) at
each level of output. As output expands, (represented by higher isoquants
Q3 > Q2 > Q1) total cost of production rises as well, which is depicted in the
figure by parallel rightward shifted isocost lines. The cost minimising factor
combination occurs at point E1, E2, E3, the locus of which, is called the
expansion path.
Remember
The equation of the expansion path is determined by the cost minimisation
�
rule, MRTSLK = .
�
K (Capital)

E3
E2
Q3
E1
Q2
Q1

O L (Labour)
134 Fig. 6.2: Expansion Path
 Cost Function
6.2.3 Analytical Approach for Cost Minimisation
Analytically, optimal combination of factors employed can be ascertained by
finding the solution of the following constrained optimisation problem:
Min Lw + Kr
s.t. Q* = f (L, K)
where Q* is the stipulated level of output produced.
We proceed solving the above problem by finding the Lagrangian function:
ℒ = Lw + Kr + λ [Q* – f (L, K)]
The first-order optimisation conditions are:
�ℒ ��
��
= 0 ⟹ w − λ ���� = 0 ⟹ w = λ MP� 1)
�ℒ ��
��
= 0 ⟹ r − λ ���� = 0 ⟹ r = λ MP� 2)
�ℒ
��
= 0 ⟹ Q∗ − f(L, K) = 0 ⟹ Q∗ = f(L, K) 3)

From (1) and (2) we get

��� �
= 4)
��� �

Same as is given by the tangency of the isocost and the isoquant. Thus, cost
minimisation requires equality between the MRTSLK and factor price ratio.
Equation (4) can also be written as
��� ���
=
� �
The above equation implies that a producer minimises cost of production
when marginal output generated by the last monetary unit spent on each
factor is equal.
Now, Equations (3) and (4) can be solved to arrive at the solution of the
��� �
cost-minimisation problem. That is, we solve = and Q∗ = f(L, K) to
��� �
get optimal inputs (L*, K*). L* and K* represent the amount of labour and
capital factors needed to be employed at the given prices of w and r
respectively, so as to produce output level Q at the minimum cost. This
minimum cost will then be given by
C = L*w + K*r
Example 1
� �
Consider the production function Q = L� K � where output Q is produced
using factors L and K. Given per unit factor prices of L and K as Rs 10 and
Rs. 5, respectively, find the expression for minimum cost of producing
output Q.

135
Production and Cost Solution
We need to find solution (L*, K*) of the following constrained optimisation
problem:
Min 10L + 5K
� �
s.t. Q = L� K �
From the first order condition of the cost minimisation, we get the following
equation:
−1 2
�
��� � L 3 K3 ��
�
= ⇒ 2 −1 =
��� � � �
L3 K 3
�
�
⇒ � = 2 ⇒ K = 2L , substituting this relation in our production
function, we get the two conditional demand functions for input K and L as
follows:
� �
L* = Q�
√�
3
K* = √2 Q4
Therefore, the cost function is given by
3 �
1
C = 10 �√2 Q4 � + 5�√2 Q� �
�
= 10 √2 Q�

Check Your Progress 1

1) Determine the equation for the expansion path of a production function
given by Q = LK2, where Q stands for output produced using factors K
and L priced at Rs. 10 and Rs. 15, respectively.
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
2) Production of a good is represented by the following technological
relationship

Q = 10√KL
Where K and L are the factor inputs and Q is the output. Given per unit
price of factor K as Rs. 3 and that of factor L as Rs. 12, what will be the
minimum cost of producing 1000 units of this good?
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
136
 Cost Function
6.3 CONDITIONAL FACTOR DEMAND FUNCTION
As we saw in Sub-section 6.2.3, the solution of the cost minimisation
problem
Min Lw + Kr
s.t. Q* = f (L, K)
gives us L* and K* for the given values of w, r and Q*. That is, we arrive at
the optimum level of factors employed in the production process for the
given output level Q* and factor prices w and r. Any change in any one or all
of these parameters will give us different values of L* and K*. Thus, solution
to the cost minimisation problem involves finding L* and K* as a function of
w, r and Q*. Symbolically, we get
L* (w, r, Q*)
K* (w, r, Q*)
which are known as the conditional factor demand functions as a solution of
the constrained cost minimisation problem for the given production
function, output level Q* and factor prices as w and r.
You might wonder why the word “conditional” before “factor demand
function”. The reason for this is— a factor demand function specifies profit
maximising levels of factor employment at given unit factor price, when
output level is free to be chosen, whereas a conditional factor demand
function gives the cost minimising level of input employment at given unit
factor prices, to produce a given level of output. That is, factor employment
is conditional upon the output level to be produced. So along the
conditional input demand function of L or K the output remains constant at
Q*. Therefore any increase (or decrease) in Q* will be accompanied by a
outward (or inward) shift of conditional demand function of L or K.
Conditional input demand function for labour L (or capital K) is always
negatively sloped with respect to its own price w (or r in case of capital K).
Conditional input demand function captures only the substitution effect
(similar concept like compensated or Hicksian demand functions which you
have done in Consumer theory) and therefore is less elastic as compared to
ordinary input demand function (a function of prices of inputs and output).

6.4 COST FUNCTION

A cost function is derived using production function and factor prices,
assuming the rational of minimisation of the cost of production by the
producer. On inserting conditional demand functions, i.e. L* (w, r, Q*) and
K* (w, r, Q*) in our expression for total cost, C = Lw + Kr, we arrive at the
cost function C (w, r, Q*).
C (w, r, Q*) = L*(w, r, Q*) w + K* (w, r, Q*) r
A function of factor prices and output, the cost function C (w, r, Q*) gives
the minimum cost of producing a specific level of output (Q*) for some given 137
Production and Cost factor prices. You may note that optimisation has already been taken care of
during construction of conditional factor demand functions, thus cost
function gives the optimised solution to the producer regarding how much
to employ of a factor in the production process.
Till now we have been considering a production function with two factor
inputs (L and K), but in reality there can be more than two inputs in the
production process. Let there be “n” factor inputs represented by vector X =
(x1, x2, x3,…. xn) used in the production of output Q, such that our production
function becomes, Q = f (x1, x2, x3,…. xn). Corresponding to factor vector X =
(x1, x2, x3,…. xn), let factor price vector be W = (w1, w2, w3,…., wn). Then, the
conditional factor demand function for factor i (where i = 1, 2,.…, n) will be
represented by xi = (W, Q), total cost function by C (W, Q).

6.4.1 Properties of a Cost Function

Let us now discuss some properties of the cost function:
1) Cost function is non-decreasing in factor prices, that is, considering two
factor price vectors W′ and W, so that W′ ≥ W, then C (W′, Q) ≥ C (W, Q).
Also, the function is strictly increasing in at least one factor price.
2) Cost function is non-decreasing in output. That is, Q′ ≥ Q, then C (W, Q′)
≥ C (W, Q) for W > 0. That is, increasing production increases the cost of
production.
3) Cost function is homogeneous of degree 1 in factor prices. That is, a
simultaneous change in all factor prices by a certain proportion (let say
by λ, where λ > 0), changes the cost of production by the same
proportion (λ). Symbolically,
C (λ W, Q) = λ C (W, Q) for W, Q, λ > 0
Similarly it can be shown that the conditional factor demand functions (L
and K) are homogeneous of degree zero.
4) Cost function is concave in factor prices. Symbolically,
C (t W + (1 ‒ t) W, Q) ≥ t C (W, Q) + (1 ‒ t) C (W, Q) for t ∈[0,1]
5) Shephard’s Lemma: If a cost function C (W, Q) is differentiable at (W, Q)
and wi > 0 for i = 1,2,..,n, then a conditional factor demand function for
factor i, that is, xi (W, Q) is given by
��(�,�)
xi (W, Q) =
���

This lemma allows us to obtain conditional factor demand functions as

partial derivatives of the cost function.

Example 2
� �
For the production function Q = L� K � , where Q is the output, K and L the
production factors with per unit prices as r and w, respectively, determine
the cost function. Check the homogeneity condition and Shephard’s Lemma.
138
Solution Cost Function

Constrained optimisation problem is given by:

Min Lw + Kr
� �
s.t. Q* = L� K �
We solve the above problem by Lagrangian method to arrive at the
following condition pertaining to cost minimisation
−2 1
�
��� � L 3 K3 �
�
= ⇒ 1 −2 =
��� � � �
L3 K 3
�
� � �
⇒ = ⇒ K = L , substituting this relation in our production
� � �
function, we get our conditional factor demand functions:
1
� r 2
L* (w, r, Q*) = Q � � �
w
1
� w 2
K* (w, r, Q*) = Q � � �
r
Therefore, cost function is given by
C (w, r, Q) = L*(w, r, Q*) w + K* (w, r, Q*) r

1 1
� r 2 � w 2
∗�
= w�Q � � � + r�Q∗ � � � �
w r

= 2�Q∗ � wr

In order to check the homogeneity condition, let input prices (w and r)

increase by proportion λ. Therefore the cost function becomes:

C (λw, λr, Q*) = 2�Q∗ � (λw)(λr) = 2λ�Q∗ � wr = λ C (w, r, Q*)

Hence the cost function is homogeneous of degree one. You may also check
the homogeneity condition for the conditional factor demand functions of L
and K.
In order to check for Shephard’s Lemma we differentiate the cost function
with respect to the input price w (and r):

�
��(�,�,�∗ ) ����∗ � �� � �
� �
∗� ∗�
��
= ��
=2× ×Q r=Q �� � ⟹ Conditional input
���∗ � ��
demand function for Labour L.

139
Production and Cost �
��(�,�,�∗ ) ����∗ � �� � �
� �
∗�
��
= ��
= 2× ×Q w= Q∗ � � � � ⟹ Conditional
���∗ � ��
input demand function for capital K.

6.4.2 Average and Marginal Cost Functions

Average Cost Function
Average cost is defined as the cost per unit of output produced. Average
cost function, a function of input vector W and output Q, is derived from the
total cost function as follows:
� (�,�)
AC (W, Q) =
�

The function will determine the minimum per unit cost of producing a
specific level of output, given the factor prices.
Marginal Cost Function
Marginal cost is the addition to total cost as an additional unit of output is
produced. A function of input vector W and output Q, marginal cost function
is derived from total cost function as a partial derivative of it with respect to
the output:
�� (�,�)
MC (W, Q) =
��

It simply determines the minimum addition to the total cost from producing
an additional unit of output, given the factor prices.
Example 3
Given a total cost function, C = 100Q + wrQ2, find the average and marginal
cost functions.
Solution
� (�,�)
Average cost function (AC) =
�

���� �����
= = 100 + wrQ
�
�� (�,�)
Marginal cost function (MC) =
��

������ ����� �
= = 100 + 2wrQ
��

6.4.3 Relationship between AC and MC Function

We present below the relationship between the AC and the MC function.
� (�,�)
We know, AC =
�
140
Rate of change of AC with respect to Q will be given by, Cost Function

� C (W,Q)
� �
�� Q
�� ’(�,�)��(�,�)
⟹
��
C’(W,Q) C(W,Q) � �(�,�)
⟹ − ⟹ � �C’(W, Q) − �
Q Q2 �

�
⟹ � �MC − AC�

From the above result it follows that, for Q > 0:

i) Slope of AC curve or rate of change of AC will be positive, that is
�
(AC) > 0, as long as MC > AC, or in other words, as long as MC curve
��
lies above AC curve.
�
ii) Slope of AC curve or rate of change of AC will be zero, that is �� (AC) =
0, when MC = AC, or in other words when MC curve intersects AC curve.
This happens at the minimum point of the AC curve.
iii) Slope of AC curve or rate of change of AC will be negative, that is
�
(AC) < 0, when MC < AC, or in other words when MC curve lies below
��
AC curve.
Fig. 6.3 represents such a relationship.
AC/MC

MC
AC

O Output

Fig. 6.3: Relationship between AC and MC curve

6.5 SHORT-RUN AND LONG-RUN COST FUNCTIONS

The distinction between the short-run and the long-run is done in terms of
the fixed and variable factors of production. Fixed factors of productions are
inputs in the production process that do not change with the level of output,
generally comprising large capital assets, office building, machinery, etc. On
the other hand, variable factors are inputs that change with the level of
141
Production and Cost output. In the short-run firms may face some constraints in expanding or
contracting their inputs so there exist some fixed factors at some
predetermined levels along with some variable factors, whereas in the long-
run all factors become variable. Let us now try to incorporate the distinction
of short-run and long-run with the concept of cost function.
Let XV represent a vector of variable inputs used in the production process,
and XF be the vector of fixed inputs. Similarly, vector WV and WF be the price
vectors of variable and fixed factors, respectively.

6.5.1 Short-run Cost Function

In the short-run, we just discussed, exist some fixed factors (XF) which are
given, and some variable factors (XV) which vary with the output level Q.
Earlier, in Section 6.4, we derived cost function as a function of given
parameters W and Q, now for the short-run cost function, XF will be
considered as another factor which is given in the short-run. Considering
presence of both, the variable and the fixed factors in the production
process, the short-run cost function will be given by:
CS (W, Q, XF) =WV XV (W, Q, XF) + WFXF
Where, XV (W, Q, XF) is the conditional variable factor demand function,
which in general depends upon the value of the given fixed factor XF.
On the basis of the short-run cost function, we can further define the
following:
�� (�,�,�� )
Short-run average cost (SAC) function =
�
��� (�,�,�� )
Short-run marginal cost (SMC) function =
��

Short-run variable cost (SVC) function = WV XV (W, Q, XF)

�� �� (�,�,�� )
Short-run average variable cost (SAVC) function =
�

Short-run fixed cost (SFC) function = WFXF

�� ��
Short-run average fixed cost (SAFC) function =
�

Remember:
If Q = 0, SVC = 0 as XV = 0, but SFC = WFXF i.e. when output production
reduces to nil, short-run variable cost also reduces to nil as employment of
variable inputs reduces to zero, but short-run fixed costs are still needed to
be incurred regardless of the output level.

6.5.2 Long-run Cost Function

In the long-run, all factors become variable (XV) that is they vary with output
Q. Since there are no fixed factors (XF), and all factors are variable (XV), we
can write vector XV as vector X (our factor vector) with corresponding factor
142 price vector given by W. Then, we get the following long-run cost function:
 CL (W, Q) =W X (W, Q) Cost Function

which is the cost function we discussed in Section 6.4. From the above
function, we can derive the following:
�� (�,�)
Long-run average cost (LAC) function =
�
��� (�,�)
Long-run marginal cost (LMC) function =
��

Check Your Progress 2

1) For the following total cost functions, find the AC, MC, AVC and AFC
functions.
a) C = 5Q3
b) C = 10 + 7Q2
c) C = Q3 – 4Q2 + 10Q + 10
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
2) Given the technological relationship between output Q and inputs L and
K priced at the per unit rate of w and r, respectively
�
Q = (L� + K � )�
a) Find the conditional factor demand functions L*(w, r, Q) and
K*(w, r, Q)
b) Derive the expression for the Cost function C (w, r, Q).
��(�,�,�)
c) Check for Shephard’s Lemma, that is check whether =
��
��(�,�,�)
L*(w, r, Q) and = K*(w, r, Q)
��
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
3) Use Shephard’s lemma to derive factor demand functions from the
following cost functions:
a) C = Q2 (7w3 + 5r3)
b) C = 2 Q2 √wr
where K and L are the two factors of production with per unit prices r
and w, respectively.
………………………………………………………………………………………………………………
……………………………………………………………………………………………………………… 143
Production and Cost
6.6 LET US SUM UP
The objective of profit maximisation explains the rationale behind cost
minimisation by a producer of a good or a service. Costs incurred to carry
out production include both, the explicit as well as the implicit costs. The
present unit dealt with minimisation of such costs. For this, two approaches
were explained— the graphical and the analytical. Graphical approach
requires tangency between the isoquant curve and the isocost line.
Analytical approach on the other hand involved solving the constrained
optimisation problem of cost minimisation. We adopted the Lagrangian
method for solving our optimisation problem, given the technological
relationship between the factors and the output (i.e. the production
function) and the factor prices.
On the basis of the optimisation exercise, the conditional factor demand
functions— giving the optimum level of factors employed in the production
process for the given output level Q and factor prices w and r, were derived.
These functions were then employed to derive the cost function— a
function of factor prices and output. By linking the production function with
the factor prices, a cost function gives the minimum cost of producing a
specific level of output, given the factor prices. Certain properties of a cost
function were also touched upon. An important one being the Shephard’s
lemma, as per which a differentiable cost function can be used to derive
conditional factor demand functions.
Subsequently, average cost and marginal cost functions were derived from
the cost function. Average cost is the per unit cost of production, while
marginal cost is the addition to cost from producing an additional unit.
Further, the relationship between the AC and MC curve, which you have
already studied in your Introductory Microeconomics course of Semester 1
(BECC-101), was mathematically established. Towards the end, the criterion
distinguishing the short-run cost function from the long-run cost function
was set in place by bringing in the picture the fixed and variable factors of
production. Short-run cost function is composed of the cost incurred on
both, the variable and the fixed factors. Long-run cost function, on the other
hand is composed of the cost incurred on only the variable factors, as in the
long-run all factors become variable. You must be clear with this by now—
short-run and long-run are differentiated on the basis of the presence and
absence of fixed factors of production, respectively.

6.7 REFERENCES
1) Hal R. Varian, (2010). Intermediate Microeconomics, a Modern
Approach, W.W. Norton and company / Affiliated East- West Press
(India), 8th Edition.
2) Shephard, Ronald W, (1981). Cost and Production Functions, Springer-
Verlag Berlin Heidelberg.
3) Nicholson, W., & Snyder, C. (2008). Microeconomic theory: Basic
principles and extensions. Mason, Ohio: Thomson/South-Western.
144
 Cost Function
6.9 ANSWERS OR HINTS TO CHECK YOUR PROGRESS
EXERCISES
Check Your Progress 1
1) Equation for expansion path is given by, K = 3L
��� �
Hint: Use cost minimisation condition ���
= �
, where MPL = K2, MPK =
2LK, w = 15 and r = 10.
2) Minimum cost = Rs. 1200
� �
Hint: Production function Q = 10√KL can be written as 10K � L�
� ��
��� �
Use cost minimisation condition ���
= �
, where MPL = 5K � L � , MPK
�� �
=5K L , w = 12 and r = 3 to arrive at the relation K = 4L. Then use this
� �

relation in production function Q = 10√KL , where Q = 1000 to get L =

50 and K = 200. Minimum cost thus equals 12(50) + 3(200) = 1200.
Check Your Progress 2
1) a) AC = 5Q2, MC = 15Q2, AVC = 5Q2, AFC = 0
�� ��
b) AC = �
+ 7Q, MC = 14Q, AVC = 7Q, AFC = �
��
c) AC = Q2 – 4Q + 10 + �
, MC = 3Q2 – 8Q + 10, AVC = Q2 – 4Q + 10,
��
AFC = �
�
�����
2) a) L*(w, r, Q) = �
� � �
����� ����� �

�
�����
K*(w, r, Q) = �
� � �
����� � ���� �

Hint: Use the cost-minimisation condition,

1
� ρ ρ ρ−1 ρ−1
��� � (L +K ) ρL �
�
= ⇒ 1 =
��� � � ρ ρ ρ−1 ρ−1 �
(L +K ) ρK
�

� ��� �
⇒ � � =
� �
1
w ρ−1
⇒ L = K�r� 1)

From our production function and Equation (1), we get

145
Production and Cost � � � �
� � ��� � ���
Q = (L + K � )� ⇒ Q = �
�K � � � � �
+K ⇒Q =K � �
�� � � + 1�

� � �
��� � ���� � ����
� � �
⇒Q =K � � � ⇒ K* = �
� � �
���� ��� ���
�� �� �

Substituting value of K*in Equation (1), we get

�
� ����
L* = �
� � �
�� ��� ����� �

���
� �
�
b) C (w, r, Q) = Q �w ��� + r ��� �

Hint: Insert values of L* and K* in the cost function given by,

C (w, r, Q) = L*(w, r, Q) w + K* (w, r, Q) r
3) a) L*(w, r, Q) = 21 Q2w2 and K*(w, r, Q) = 15 Q2r2
��(�,�,�) ��(�,�,�)
Hint: ��
= L*(w, r, Q) = 21 Q2w2 and ��
= K*(w, r, Q) =
15 Q2r2
� �
b) L*(w, r, Q) = Q� �� and K*(w, r, Q) =Q� � �

146