In this article, we explain the major properties of the Cobb-Douglas production function

Introduction to Cobb Douglas Production Function

The Cobb-Douglas production function is an empirical production function developed by Charles W. Cobb (American Mathematician) and Paul H. Douglas (American Economist) based on empirical studies of various manufacturing industries of the USA. This production function was published in American Economic Review in 1928 in the form of an article A Theory of Production. By the name of mathematician C.W. Cobb and economist P.H. Douglas, this production function is termed Cobb-Douglas production function. Here we explain the major properties of the Cobb-Douglas production function.

The mathematical form of this production function can be expressed as

Q=A.K^αL^β 

Where

Q = The level of output produced in a year,

K = Capital (machine, equipment, and buildings),

L = Labour,

A = It is an index of technology or efficiency parameter also called total factor productivity and is positive, α and β are positive parameters of the production function which measures output elasticities of capital and labor, respectively.  These values are constants and are determined by the available state of technology.  

It is confirmed by the C-D production function that it is a multiplicative production function which implies that both the factors of production are essential to produce goods and services. It means, if the amount of one of the factors of production is zero, no output can be produced. (i.e. if K = 0, Q = 0 and if L = 0, Q = 0).  

Another important thing to note is that originally it was found that the sum of exponents of the C-D production function was equal to one. That is α+β is one. From further research and analysis, it was generalized and found that the sum of exponents (α+β) could be equal to one, more than one, and less than one.

Major Properties/Features of the Cobb-Douglas Production Function

Here are some of the essential/major properties/features of the C-D production function.

1. The C-D Production Function Can be Used to Measure the Returns to Scale

The C-D production function can be used in the calculation of the nature of returns to scale. The sum of the powers/exponents of factors in Cobb-Douglas production function, that is α+β measures the returns to scale. Therefore,

• If α+β=1, it exhibits constant returns to scale (CRS)
• If α+β>1, it exhibits increasing returns to scale (IRS)
• If α+β<1, it exhibits decreasing returns to scale (DRS)

2. The Factor Intensity (A Relative Importance of Factor in Production Process)

In the C-D production function, the factor intensity is computed by taking the between ratio α and β (ratio between exponent of capital and exponent of labor) as

If α/β>1, there is the use of the capital-intensive technique of production (capital is more vital in the manufacturing process), and

If α/β<1, there is the use of the labor-intensive technique of production (labor is more vital in the manufacturing process)

3. Average Physical Productivity of Inputs

The average product is the total output per unit of input. The average products of capital and labor (AP_K and AP_L) can be computed below.

AP_L=Q/L=AK^αL^β/L= AK^αL^β.L^-1 = AK^αL^β-1

AP_K=Q/K=AK^αL^β/K= AK^αL^β.K^-1 = AK^α-1L

4. The Marginal Product of an Input Can be Expressed in terms of its Average Product

Marginal physical productivity/marginal products/marginal productivities of the inputs in the case of Cobb-Douglas production function can be expressed in terms of their average physical productivity/average productivities. It can be justified below.

A marginal product is a change in total output due to a one-unit change in the use of a particular input. The partial derivative of the C-D production function measures the marginal product of its factor inputs. Therefore,

Marginal Product of Labour (MP_L)= ∂Q/ ∂L or ΔQ/ΔL

MP_L=∂Q/∂L=∂(AK^αL^β)/∂L = AK^α∂(L^β)/∂L = βAK^αL^β-1= βAK^αL^β/L =βQ/L = β.AP_L

Similarly,

Marginal Product of Capital (MP_K)= ∂Q/ ∂K or ΔQ/ΔK

MP_K=∂Q/∂K=∂(AK^αL^β)/∂K = A L^β ∂(K^α)/∂K = αAK^α-1L^β= αAK^αL^β/K = αQ/K = α.AP_K  

5. Output Elasticities

The powers of labor and capital (that are β and α) in the C-D production function measure output elasticities of labor (L) and capital (K) respectively. The output elasticity of a factor shows the percentage change in output due to a given percentage change in the number of factor inputs.

As we know that,

Q=A.K^αL^β 

Output elasticity of Labour= Exponents of labor in the C-D production function=β

The output elasticity of labor is given by ΔQ/ΔL×L/Q

And ΔQ/ΔL×L/Q= β.Q/L×L/Q = β

Similarly,

The output elasticity of Capital=Exponents of capital in the C-D production function=α

The output elasticity of capital is given by ΔQ/ΔK×K/Q

And ΔQ/ΔK×K/Q = α.Q/K×K/K = α

Therefore, the output elasticity of labor is β and the output elasticity of capital is α. These two are the power of labor and capital respectively in the standard Cobb-Douglas production function. So, α and β in the C-D production function represent the coefficient of output elasticities of capital and labor, respectively. The value of α can be inferred as one percentage increase in capital will result in an increase in output by α percentage. Similarly, the value of β can be understood as a one percent increase in labor will increase the output by β percentage.

6. Marginal Rate of Technical Substitution (MRTS)

The marginal rate of technical substitution (MRTS) in the theory of production that measures the degree of substitutability/interchangeability/exchangeability between factors inputs. The MRTS in the case of the C-D production function can be expressed in terms of the ratio between labor and capital. It is given below.

MRTS_{L, K} = dK/dL= MP_L/MP_K= (β.Q/L)/(α.Q/K) = β/α (K/L)

And

MRTS _{K, L}= dL/dK= MP_K/MP_L= (α.Q/K)/(β.Q/L) = α /β (L/K)

7. Elasticity of Factor Substitution is Equal to Unity

The elasticity of substitution (σ) of the C-D production function is defined as the percentage change in the capital-labor ratio divided by the percentage change in the marginal rate of technical substitution and is equal to unity. It is given as

σ = (Percentage change in K/L)/ (Percentage change in MRTS _{L, K})

= {d(K/L)/(K/L)}/ {d (MRTS _{L, K})/ (MRTS _{L, K})}

={d(K/L)/(K/L)}/ {d (β/α. K/L)/ (β/α. K/L)}  = 1

The value of elasticity of substitution (σ) can lie between zero and infinity. The larger the value of σ, the greater the possibility of substitution between factor inputs. In the limiting case if σ=0, two inputs must be used in a fixed proportion and they are complementing to each other; in this case, the production isoquants are right-angles or L-shaped. In another limiting case, if σ tends to infinity, two inputs are perfectly substituted, and the isoquants are straight lines (linear) touching both axes. This indicates that a given amount of product can be produced by using only capital or only labor or by an infinite combination of K and L.

8. The Efficiency of Production

In the C-D production function, the efficiency of production can be measured by coefficient A.

• If the value of A is higher, there is a higher degree of efficiency of production
• If the value of A is lower, there is a lower degree of efficiency of production

For Example

Case of Firm Everest: if 100= 100 K^αL^β

Case of Firm Mount: if 1000= 200 K^αL^β

If the value of K, L, α, and β are the same, then the firm Everest is less efficient than the firm Mount as the value of ‘A’ in the firm Mount is greater than the value of ‘A’ in the firm Everest (200>100).

9. Linear Form of Cobb-Douglas Production Function

C-D Production function can be made linear using logarithm and the level of output can be estimated based on regression analysis.

As we know

 Q=A.K^αL^β …………. (i)  

Applying log on both sides of equation (i) we get

Log Q = Log (A. K^αL^β)

Or Log Q = Log A + long K^α+ Log L^β

Or Log Q= Long A + α Log K + β Log L

Conclusion

The cobb-Douglas production function is based on the empirical studies of several manufacturing industries of the USA made by C.W. Cobb and P.H. Douglas. This empirical production function was published in the American Economic Review in 1928 A.D. The Cobb-Douglas Production Function is one of the most applied production functions around economics research. In this article, we explained the major properties of the Cobb-Douglas production function.