# Cobb Douglas Production Function

Cobb Douglas Production Function is the special form of a production function that shows the technological relationship between the amount of two or more inputs and the amount of output produced by these inputs.

The general form of the Cobb Douglas Production Function is:

Q=AK^α L^β N^λ T^γ…
where,
Q=Output Produced; A=Total Factor Productivity; K=Capital; L=Labour; N=Land;
T=Technology
α,β,λ,γ is Output elasticity of Capital, Labour, Land, and Technology respectively.

Now,
The general form of Cobb Douglas Production Function with two inputs, Labor and Capital, is:

Q=AK^α L^β

Properties of Cobb Douglas Production Function

1. Marginal Productivity of Inputs

Q=AK^α L^β
Diff^n  with respect to K,
dQ/dK=d(AK^α L^β )/dk
dQ/dK=AL^β×d(K^α )/dK
dQ/dK=AL^β (αK^(α-1) )
dQ/dK=α(AK^α L^β ) K^(-1) [Q=AK^α L^β]
dQ/dK=α×Q/K
dQ/dK=α×AP(K)
MP(K)=α×AP(K)

∴α=MP(K)/AP(K)

Similarly,

∴β=MP(L)/AP(L)

2. Marginal Rate of Technical Substitution (MRTS)

MRTS=-MU(L)/MU(K)
=-(β×AP(L))/(α×AP(K) )
=-(β×(Q/L))/(α×(Q/K) )
=-β/α×K/L

3. The elasticity of factor substitution (E(FS))

E(FS)=((Percentage change in (K/L)))/(Percentage change in MRTS)
=((%∆(K/L)))/(%∆MRTS)
=((∆ K/L)/(K/L))/(∆MRTS/MRTS)
=((∆ K/L)/(K/L))/((∆(β/α×K/L))/((β/α×K/L)))=1

4. Factor intensity

Factor intensity is used to compare relative factor usage.
Q=AK^α L^β
α/β>1, then it is capital intensive.
α/β<1, then it is labor-intensive,

5. Efficiency Coefficient

Q=AK^α L^β
A denotes an efficiency coefficient. The efficiency coefficient is high if the company or country is more research-oriented or invests more in ideas. A company's greater insights and ideas can use the inputs or factors of production in a more efficient way, so they produce more output or high-quality output than their competitors using the same amount of inputs.

Thus, keeping all other inputs along with their respective output elasticity constant, some firms excel much in comparison to their competitors due to the higher efficiency coefficient.

Cobb Douglas Production Function is also used to determine the Increasing Rate to Scale, Constant Rate to Scale, and Decreasing Rate to Scale.
Q=AK^α L^β-------A
In equation A,
If,

Suppose in Equation A,
Q=10K^1 L^(1/2)-----B

Suppose:
K=50; L=100 in Equation B,
Q=10×50×〖100〗^(1/2)=5000
K_1=100; L_1=200 in Equation B,
Q_1=10×100×〖200〗^(1/2)=14142.14

The input is doubled, but the output is more than double. Hence, if α+β=3/2>1  gives Increasing Return to Scale.

Similarly,

If the input is doubled, then the output is equal to double. Hence, if α+β=1  gives Constant Return to Scale.
If the input is doubled, then the output is less than double. Hence, if α+β<1  gives Decreasing Return to Scale.