2. What is Cobb-Douglas Production
Function?
During 1900–1947, Charles Cobb and Paul
Douglas formulated and tested the Cobb–
Douglas production function through various
statistical evidence.
Q = b0 X Y
b1
b2
The Cobb–Douglas functional form of
production functions is widely used to represent
the relationship of an output and two inputs.
3. Question 7.2
Production Function Estimation. WashingtonPacific, Inc., manufactures and sells lumber,
plywood, veneer, particle board, medium-density
fiber board, and laminated beams. The company
has estimated the following multiplicative
production function for basic lumber products in
the Pacific Northwest market:
Q = b0 L K E
b1
b2
b3
Q = output,
L = labor input in worker hours,
K = capital input in machine hours and
E = energy input in BTUs (British Thermal Unit)
4. Each of the parameters of this model
was estimated by regression analysis
using monthly data over a 3-years
period. Coefficient estimation results
were as follows:
ˆ
ˆ
ˆ
ˆ
b0 = 0.9; b1 = 0.4; b2 = 0.4; b3 = 0.2
The standard error estimates for each
coefficient are
σ b 0 = 0.6; σ b1 = 0.1; σ b 2 = 0.2; σ b 3 = 0.1
5. Question 1. Estimate the effect on
output of a 1% decline in worker hours
(holding K and E constant)
Given,
Q = b0 L K E
b1
b2
b3
Take the first derivation with respect to
worker hours (L)
Q = 0L K
b
b1
b2
E
b3
6. ∂
Q
1
=b0b1 Lb1 − K b2 E b3
∂
L
∂
Q
1
=b1b0 Lb1 K b2 E b3 L−
∂
L
∂
Q
1
=b1QL−
∂
L
∂
Q
Q
∂
Q
∂
L
=b1
=b1 *
Q
L
∂
L
L
∂
Q
∂
Q L
= 0.4( − .01)
0
*
=b1
Q
∂
L Q
∂
Q
= − .004 = − .4%
0
0
∂
Q ∂
L
Q
b1 =
÷
Q
L
7. Question 2 . Estimate the effect on output
of a 5% reduction in machine hours
availability accompanied by a 5% decline in
energy input (holding L constant)
Solution: From part A it is clear that,
∂Q
= b2 (∆K / K ) + b3 (∆E / E )
Q
∂Q
= 0.4(−0.05) + 0.2(−0.05)
Q
∂Q
= −0.03 = −3%
Q
ˆ
b0 = 0.9
ˆ
b = 0 .4
1
ˆ
b2 = 0.4
ˆ
b3 = 0.2
8. Question 3. Estimate the returns to scale for
this production system.
Solution:
In case of Cobb Douglas production function,
the returns to scale are determined by
summing up exponents because:
Q =b0 L K
b1
b2
E
b3
hQ =b0 ( kL) ( kK )
b1
b2
hQ =k
b1 + 2 + 3
b
b
b0 L K
hQ =k
b1 + 2 + 3
b
b
Q
b1
( kE )
b2
E
b3
b3
9. Thus, summing up the value of the
exponents, we get,
b1 + b2 + b3 = 0.4 + 0.4 + 0.2 = 1
hQ = k Q
1
h=k
1
This indicates constant returns to
scale estimation.
11. Conclusion
Returns
to Scale is the quantitative change
in output of a firm or industry resulting from
a proportionate increase in all inputs.
Adding
the value of the exponents, we can
determine the returns to scale of a
production function.
