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

Thank You