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Cobb-Douglas Production Function (C-D Function)

The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas between 1927 and 1947.  It is the empirical production function that shows that the output can be expressed as a multiple of inputs. If there are two inputs, K and L, then the general form of the C-D production function is given as

Q = AKαLβ ————– (i)

Where,

  • Q = Output
  • A = Efficiency parameter which measures the technological efficiency of production, and A > 0.
  • α = Partial output elasticity of capital α >0
  • β = Partial output elasticity of labour, β >0
  • α = 0.1 ⇒ if 1% labour is increased, capital will increase by 10%.
  • β = 0.1 ⇒ if 1% of capital is increased, labour will increase by 10%.
Basic Properties of the Cobb-Douglas Production Function

1) C-D production function is log-linear:

We have, Q = AKαLβ 

Taking the log of both sides

LogQ = Log A +α Log K + β Log L, which is in linear form now.

2) C-D production function is homogeneous of degree of α + β

  • If α + β = 1, it shows constant returns to scale
  • If α + β > 1, it shows increasing returns to scale
  • If α + β < 1, it shows decreasing returns to scale

3) The marginal physical product of K and L of the C-D production function is positive

i.e. MPPK > 0, MPPL > 0

We have, Q = AKαLβ 

Now, MPPK  = ðQ/ðK = ð(AKαLβ )/ðK

= ALβ( ðKα/ðK)

= ALβ α . K α-1

= α. AL β.K α K-1

= α. Q/K > 0

Similarly, for MPPL = β.(Q/L) > 0

4) The Isoquants of the Cobb-Douglas production function are downward sloping and convex to the origin.

i.e.dK/dL<0 and for convexity d2K/dL2>0.

i.e., Rate of change of the slope of the isoquant is positive.

5) In a special case, when α + β =1, the  C-D production function shows diminishing returns to each input

i.e. MPPL > 0, MPPK > 0 and ðMPPL /ðL <0 and ðMPPK/ðK < 0

6) In a special case, when α + β =1, the C-D production function satisfies Euler’s theorem. i.e., if both K and L are paid according to their marginal product, the total product is exhausted or finished.

i.e. K.MPPK + L.MPPL  = Q

or K. α Q/K + L.β.Q/L = Q

or, α Q + β.Q = Q

or, Q (α+β) = Q

or, Q.1 = Q (α+β = 1)

∴ Q = Q,  proved!

7) Elasticity of substitution (σ) of the C-D production function is unity or 1

σ is defined as the ratio of the proportionate change in the factor ratio (K/L) and proportionate change in the marginal rate of substitution between the factors (L and K).

i.e σ = {d(K/L)/K/L} / {d.

MRTSLK/MRTSLK} ———— (i)

Now, MRTSLK = slope of isoquant = – (α/β.K/L)

d.MRTSLK = d(- α/β.K/L)

= – α/β .d(K/L)

Substituting the value of MRTSLK and d.MRTSLK in equation (i)

We have,

σ = {d(K/L)/K/L} / (- α/β.K/L)/- (α/β.K/L)

σ =  1

i.e., the elasticity of substitution of the C-D production function is 1, which means that the producer substitutes the relatively cheaper input at the same rate if the other input becomes relatively expensive at the same rate. i.e,. If the input price PL/PK increases by 1%, then K/L also increases by 1%.

Uses/Importance of C-D Production Function

The C-D production function is the most commonly used empirical production function at both the firm level and the macro level. Its common uses are

  1. To estimate the production function of a firm or economy, as per the objective of a study.
  2. To find efficiency gains or technological progress in production.
  3. To develop and explain the economic growth model, such as the neo-classical or Solow-Swan growth model, the neo-classical or endogenous growth model, etc.
  4. To design an appropriate policy by the government based on the partial output elasticity of the inputs and efficiency parameters.
  5. To design an appropriate strategy for the firm based on the estimated result by C-D production function.
  6. To examine the effectiveness of any policy intervention

Why is the C-D production function preferred?

There are different production functions used for empirical purpose such as the C-D production function, Constant Elasticity of Substitution (CES) production function, Leontief production function, etc. However, the C-D production function is more commonly used in empirical works due to the following reasons.

  1. C-D production function is log-linear which makes this production function easy to estimate and interpret.
  2. This model can be extended to any number of inputs as per the requirements of the study. i.e.Q = A.x1a1 x2 a2 x3 a3 …….xnan      a1+a2+a3 ….. + an is the degree of homogeneity. x1,x2,x3 ….. xn is input factor.
  3. C-D production function satisfies the properties of a well-behaved production function. i.e., isoquants are downward sloping and convex to the origin, and the marginal physical product of the inputs is positive and diminishing.