Page 127 - DECO403_MATHEMATICS_FOR_ECONOMISTS_ENGLISH

Mathematics for Economists

Note
1 § 2 D ·
Or = ¨ E (p p ) ¸ p a / p b a ...(v)
ab
A © A ¹
1 § 2 D ·
Thus b = ¨ E (p p ) ¸ p a / p b ...(vi)
ab
A © A ¹
Equation (v) and (vi) displays the factor demand of a and b
Example 1: If production is in form of
α β
Q = A K L
then (A) Find out the marginal productivity of Capital (K) and Labour (L).
(B) Prove that there is elasticity of capital and labour in production function
Solution: Given production function
α β
Q = AK L ...(i)
(A) Partially differentiating from equation with respect to K and L separately

wQ D 1 E
= A KD L ...(ii)
wK
wQ
= A KLE DE 1 ...(iii)
wL
Function (ii) and (iii) displays marginal productivity of Capital (K) and Labour (L). writing this in
simple form
w DQ
α
β
MP = wK K (\ Q = AK L )
K
w EQ
And MP = wK K ·Q
L
(B) Production Elasticity of Capital
KQ K D 1 E
w
D
= · AK L
QK Q
w
K D 1 E A DK D 1 L E
D
= AK L D D
DE
AL L A DK D 1 L E
Production Elasticity of Labour
LQ L DE 1
w

= Q wL Q E · AK L
L DE 1
= DE A EKL
AKL

E D E 1
= AK L AK L E
DE 1
This way in the given production function a and b shows the production elasticity of capital and
labor.

120 LOVELY PROFESSIONAL UNIVERSITY