Page 118 - DECO403_MATHEMATICS_FOR_ECONOMISTS_ENGLISH

Unit 7: Homogeneous Function and Euler’s Theorem

2. Production function is homogeneous and of a degree: Note
If Production function is homogeneous and of a degree, it means that production will come
under constant formula.
α β
Production function = P = AL K u
Taking log of two sides log P = log A + α log L + β log K + log u
Partially differentiating with respect to L and K separately

w 1 P D
PL = L ...(i)
w
1 P E
w
And = (ii)
PK L
w
Writing (i) and (ii) further
w P
L = αP ...(iii)
w L

w P
K = βP ...(iv)
w L
Adding equation (iii) and (iv)
L ∂P + K ∂P = P α + P = P α β)
+
(
β
∂P ∂L
3. If Production function is homogeneous and of a degree, then elasticity of substitution will
β
α
always be equal to unit. If production function is P = AL K u, where α + β = 1 we know that
elasticity of substitution
= σ = Change in ration of factor’s quantity/ % change in price ratio of factor
/
KL
w KL K w(/ )/(/ ) L (/ )/KL
∴ α =
w L /P K )/P L /P K w(P R /R
Where K/L = ratio of factor quantity
R = P P = Price ratio of factor
L
K
wK
We know that rate of marginal substitute technique =
wL
wK MP L P L
∴ = R
wL MP K P K

w P w / L
In other words R =
w P w / K

α β
Our production function P = AL L u
Differentiating with respect to L and K separately
wP D E
wL = EAL K u

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