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Cobb-Douglas Cobb-Douglas Production Production
FunctionFunction
Pasakorn S. 5520212001Nabduan D. 5520212002
Ata K. 552022009
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.
210
bb YXbQ =
The Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output and two inputs.
Question 7.2 Production Function Estimation. Washington-Pacific, 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 = output,L = labor input in worker hours,K = capital input in machine hours andE = energy input in BTUs (British Thermal Unit)
3210
bbb EKLbQ =
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:
The standard error estimates for each coefficient are
2.0ˆ;4.0ˆ;4.0ˆ;9.0ˆ3210 ==== bbbb
1.0;2.0;1.0;6.0 3210 ==== bbbb σσσσ
Question 1. Estimate the effect on output of a 1% decline in worker hours (holding K and E constant)
Given,
Take the first derivation with respect to worker hours (L)
3210
bbb EKLbQ=
3210
bbb EKLbQ =
L
L
Q
Qb
bQ
L
L
QL
Qb
L
Q
QLbL
Q
LEKLbbL
Q
EKLbbL
Q
bbb
bbb
∂÷∂=
=∂∂
=∂∂
=∂∂
=∂∂
=∂∂
−
−
−
1
1
1
11
101
110
*
321
321
%4.0004.0
)01.0(4.0
*1
−=−=∂
−=∂
∂=∂
Q
Q
Q
Q
L
Lb
Q
Q
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,
%303.0
)05.0(2.0)05.0(4.0
)/()/( 32
−=−=∂
−+−=∂
∆+∆=∂
Q
Q
Q
Q
EEbKKbQ
Q
2.0ˆ
4.0ˆ
4.0ˆ
9.0ˆ
3
2
1
0
=
=
=
=
b
b
b
b
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:
QkhQ
EKLbkhQ
kEkKkLbhQ
EKLbQ
bbb
bbbbbb
bbb
bbb
321
321321
321
0
0
3210
)()()(
++
++
=
=
=
=
Thus, summing up the value of the exponents, we get,
This indicates constant returns to scale estimation.
1
1
321 12.04.04.0
kh
QkhQ
bbb
==
=++=++
GraphhQ = kn.f(X.Y.Z)
Constantn = 1 h = kIncreasingn > 1 h > k decreasing n < 1 h < k
returns-to-scale estimation
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.
ConclusionConclusion