Page 122 - DECO403_MATHEMATICS_FOR_ECONOMISTS_ENGLISH

Unit 7: Homogeneous Function and Euler’s Theorem

7.4 The Constant Elasticity Substitution (C.E.S.) Production Function Note

In the Cobb-Douglas Production Functions it has already been discussed that elasticity of substitution
is always a unit in it. Here we will discuss a function where elasticity of substitution is not required.
This is known as Constant Elasticity Substitution (C.E.S.) Production Function. This was devised by
two groups of economists. First was K J Arrow, Chenery and B S Minhas and R M Salow, whereas
second group consists M Brown, De Cani. Although they devised this function in other forms, but
result were same. First group has shown the production function as:

v
]
P = [ C   (1  )N   
(   0,  0  1,   1)
Where P = Production, C = Capital, N = Labour a = substitution parameter; g = technical efficiency
coefficient or efficiency parameter (this is considered as A of Cobb-Douglas Production Functions in
C. E.S. function); d = coefficiency of capital intensity (this is considered as a of Cobb-Douglas
Production Functions in C. E.S. function)
1–d = Labour Intensity Coefficient
v = Degree of Homogeneity

7.4.1 Properties of C.E.S. Production Function

1. If Production Function is linear homogeneous then substitution parameter a would be equal
 1 
to constant   whereas production function is P  C   (1  )N   /  provided
v
]
(
   1
   0, 0  and a > –1
Rational: According to definition elasticity of substitution
/ )/NC
/ )
log(NC (NC /
s =   log R R /R

N P
Here,  ratio of production factors and R  C  Price Ratio
C P N
Now production function
v
]
P = [ C   (1  )N   /  ...(7.1)
Partially differentiating with respect to N
P    /  1
v
[
1
]
[
N = [ v /  C  (1  )N ]   (1  )N ]
v    / 1 1)
v
(
=   [ C  (1  )N ]   [ (1  )N ] ...(7.2)

From equation 7.1

P
v
 =    [ C  (1  )N   / 
]


LOVELY PROFESSIONAL UNIVERSITY 115