2. The C-D production function is based on the empirical study of the American manufacturing
industry made by Paul H Douglass and C.w Cobb during the period 1899 to 1922. It is a linear
homogenious production Function of degree one which takes into the account of two inputs that is, Labour
and Capital, for a entire output of the manufacturing industry.
The general form of Cobb -Douglass production function can be expressed as:-
Q = AL αKβ
Here:-
Q = Output
L = Labour input
K = Capital input
α&β = Possitive parametors ( α >0 , β > 0 , α+ β =1 )
A = Technical change ( it assumed to be constant )
The equation tells that output depends directly on L &K ,and that part of output which cannot be
explained by L and K is explained by A which is the Residual often called Technical Change. And A is
assumed to be constant.
3. Properties
The following are the important properties of Cobb-Douglass production
function:-
1.Constant Returns to Scale
C – D production function exhibits constant returns to scale . To prove it ,
let us increase the quantities of L & K by λ times and output must also
increased by λ times . Then the increased output (Q*) will be ;
Q* = λQ
α+β=1 Constant returns to scale
α+β>1 Increasing returns to scale
α+β<1 Diminishing returns to scale
4. 2. The Average Product (AP) and Marginal Product (MP) of factors
The C-D production function tells that the AP & MP of factors is a function of the
ratio of the factors .
Q = ALαKβ
APL = A (K/L)β
APK = A (L/K)α
MPL = α A (K/L)β
MPK = β A (L/K )α
3. Marginal Rate of Substitution between Capital and Labour (MRS LK)
The MRS LK can be derived from the C-D production function.
MRS LK = ∂Q/∂L ÷ ∂Q /∂K
5. 4. Elasticity of factors substitution
The Elasticity of factors substitution of the C-D production function is equal to unity.
Its proof, the elasticity of substitution (es) between K&L is defined as;
σ = ( f1 × f2 )/(f12 × Q)
Here ;
f1 =MPL
f2 = MPK
f12 = cross partial derivative of L& K
∂Q = ∂L × ∂K
σ = 1
when the elasticity of substitution is unity the production function is homogenious of
degree one. That is constant returns to scale
6. 5. Euler’s theorum
The application of Euler’s theorum to distribution in an other
property of the C-D production function . If the production function
Q = f(K/L) is homogenious of degree one , then according to Euler’s
theorum
Q = L (∂Q/∂L) + K ( ∂Q/∂K )
Apply the general form of C-D production function we get Q = Q
7. 6. Factor intensity
In the C-D production function Q = ALαKβ , the factor intensity is
measured by the ratio ( α /β ). Higher the ratio , the production function is
more labour intensive and lower the ratio , the production function will be
capital intensive .
7. Efficiency of production
The coefficient ‘A’ in the C-D production function helps in
measuring the efficiency in the organisation of the factors of production. If
two firms have the same α , β , L and K but produce different quantities of
output , this difference may be due to the superior organisation of more
efficient firm as against the other. The more efficient firm will have a larger ‘
A ‘ than the other firm
8. 8. Multiplicative Function
The C-D production function is a multiplicative function . It means that if an input
has zero value , the output will also be zero . This property highlights the fact that all
inputs are necessary for production in a firm.
9. Output elasticity
It can be defined as the proportionate change in output with a given change in
input . The output elasticity of L & K can be calculated with the help of C-D production
function :-
Output elasticity of labour = ( ∂Q / ∂L ) ÷ ( L / Q )
Output elasticity of capital = ( ∂Q / ∂K ) ÷ (K / Q )
9. Criticisms
The C-D production function considers only two inputs , labour and capital
and neglects some important inputs like raw materials , which are used in
production.
In the C-D production function , the problem of measurement of capital
arises because it takes only the quantity of capital available for production .
But the full use of the available capital can be made only in periods of
fullemployment. This is unrealistic situation.
The C-D production function is criticised because it shows constant returns
to scale. But constant returns to scale are not an actuality , for either
increasing or decreasing returns to scale are applicable to production.
10. The C-D production function is based on the assumption of substitutability
of factors and neglects the complimentarity of factors.
This function is based on the assumption of perfect competitionin the
factor market which is unrealistic.
One of the weakness of C-D production function is the aggregation
problem.
11. Importance
The C-D production function has been used widely in empirical studies of manufacturing
industries and in inter-industry comparisons.
It is used to determine the relative shares of labour and capital in total output. And it is
also used to prove Euler’s theorum.
Its parametors α and β represents elasticity coefficients that are used for inter-sectoral
comparisons.
This production function is linear homogenious of degree one which shows constant
returns to scale.
If α + β >1 - increasing returns to scale
α + β <1 - decreasing returns to scale
This production function is more than two variables.
12. Conclusion
Thus the practicability of the C-D production function in the
manufacturing industry is a doubtful proposition. This is not applicable to
agriculture where for intensive cultivation , increasing the quantity of inputs
will not raise output proportionately.