Page 120 - DECO403_MATHEMATICS_FOR_ECONOMISTS_ENGLISH

Unit 7: Homogeneous Function and Euler’s Theorem

6. Production function displays the elasticity of LaboXr and Capital. By the characteristics of Note
production function

LP
w
a =
PL
w
w /PP
= elasticity of labour
w /LL
K wP
b =
PK
w
w /PP
= elasticity of capital
w /KK
The expansion path of Cobb Douglas Production Function is linear homogeneous and its
passed through main point.
α β
Cobb Douglas Production Function P = AL K u
Taking log into both sides
log P = log A + a log L + b log K + log u
Differentiating with respect to L and K separately

1 wP D
PL = L ...(A)
w
1 wP E
PK = K ...(B)
w
Writing both the equation (A) and (B) again

w DP
P
MP = wL . L
L
w EP
P
and MP = wL . K
K
MP L P L
\ MP K = P K

P D /L P L
E /PK = P K

D K P L
Or . =
E L P K
Or D.KP K = E.LP L

Or D K .KP E LP L =0

Thus, Production Function is linear homogeneous and its passes through main point.

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