School of Economics | The CES Production Function
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## 05 Feb The CES Production Function

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The below mentioned article provides a close view on the CES Production Function.

Arrow, Chenery, Minhas and Solow in their new famous paper of 1961 developed the Constant Elasticity of Substitution (CES) function. This function consists of three variables Q, С and L, and three parameters A, and.

It may be expressed in the form:

Q = A [aC+ (l-α)L-1/θ

where Q is the total output, С is capital, and L is labour. A is the efficiency parameter indicating the state of technology and organisational aspects of production.

It shows that with technological and/or organisational changes, the efficiency parameter leads to a shift in the production function, α (alpha) is the distribution parameter or capital intensity factor coefficient concerned with the relative factor shares in the total output, and θ (theta) is the substitution parameter which determines the elasticity of substitution.

And A > 0; 0 < a < 1; > -1.

Its Properties:

The CES production function possesses the following properties:

1. The CES function is homogenous of degree one. If we increase the inputs С and L in the CES function by n-fold, output Q will also increase by n-fold.

Thus like the Cobb-Douglas production function, the CES function displays constant returns to scale.

2. In the CES production function, the average and marginal products in the variables С and L are homogeneous of degree zero like all linearly homogeneous production functions.

3. From the above property, the slope of an isoquant, i.e., the MRTS of capital for labour can be shown to be convex to the origin.

4. The parameter (theta) in the CES production function determines the elasticity of substitution. In this function, the elasticity of substitution,

σ = 1/ 1 + θ

This shows that he elasticity of substitution is a constant whose magnitude depends on the value of the parameter θ. If α =0, then a = 1. If θ = ∞, then a =0. If θ = -0, then a =∞. This reveals that when a = 1, the CES production function becomes the Cobb-Douglas production function.

If Q < 0, then a = – 0; and if в < ∞, then a < 1. Thus the isoquants for the CES production function range from right angles to straight lines as the elasticity substitution ranges from 0 to.

5. As a corollary of the above, if L and С inputs are substitutable ∞ for each other an increase in С will require less of L for a given output. As a result, the MP of L will increase. Thus, the MP of an input will increase when the other input is increased.

Merits of C.E.S. Production Function:

The CES function has the following merits:

1. CES function is more general.

2. CES function covers all types of returns.

3. CES function takes account of a number of parameters.

4. CES function takes account of raw materials among its inputs.

5. CES function is very easy to estimate.

6. CES function is free from unrealistic assumptions.

CES function vs. CD function:

There are some fundamental differences between the CES function and the CD production function:

1. The CD function is based on the observation that the wage rate is a constant proportion of output per head. On the other hand, the CES function is based on the observation that output per head is a changing proportion of wage rate.

2. The CES production function is based on larger parameters than the CD production function and as such allows factors to be either substitutes or complements. The CD function is, on the other hand, based on the assumption of substitutability of factors and neglects the complementarity of factors. Thus the CES function has wider scope and applicability.

3. The CES production function can be extended to more than two inputs, unlike the CD function which is applicable to only two inputs.

4. In the CES function, the elasticity of substitution is constant but not necessarily equal to unity. It ranges from 0 to ∞. But the CD function is related to elasticity equal to unity. Thus the CD function is a special case of the CES function.

5. The CES function covers constant, increasing and decreasing returns to scale, while the CD function relates to only constant returns to scale.

Limitations of CES Production Function:

But the CES function has certain limitations:

1. The CES production function considers only two inputs. It can be extended to more than two inputs. But it becomes very difficult and complicated mathematically to use it for more than two inputs.

2. The distribution parameter or capital intensity factor coefficient, α is not dimensionless.

3. If data are fitted to the CES function, the value of the efficiency parameter A cannot be made independent of 0 or of the units of Q, С and L.

4. If the CES function is used to describe the production function of a firm, it cannot be used to describe the aggregate production function of all the firms in the industry. Thus it involves the problem of aggregation of production function of different firms in the industry.

5. It suffers from the drawback that elasticity of substitution between any part of inputs is the same which does not appear to be realistic.

6. In estimating the parameters of CES production function, we may encounter a large number of problems like choice of exogenous variables, estimation procedure and the problem of multi-collinear ties.

7. There is little possibility of identifying the production function under technological change.

Conclusion:

Despite these limitations, the CES production function is useful in its application to prove Euler’s theorem, to exhibit constant returns to scale, to show that average and marginal products of С and L are homogeneous of degree zero, and to determine the elasticity of substitution.