Test for Substitution Elasticity

The following is a brief overview of the presentation that I will be giving in my thesis meeting today.

What is Partial Elasticity?

Partial Elasticity of Substitution is the proportional change in the ratio of the use of two inputs brought about by a change in the relative price ratio for these two inputs. Since my thesis topic is concerned with a production function with multiple input factors we must use a measure that considers the multi-factor case. The main measures that I have come across my research are the Allen Partial Elasticity of Substitution (AES) and Morishima elasticity of substitution (MES).

What is AES?

AES is defined as the percentage change in the ratio of the quantity of two factors to the percentage change in their price ration allowing all other factors to adjust to their optimal level (Hitt 2005). AES tends to be usually measured via cost functions but a lot the research I have undertaken has used various production functions to measure it. As noted by Chambers (1988) AES is defined by:

Where: x_i x_j, are the inputs we wish to see if they are complements or substitutes
f_i is the first order partial derivative of our production function with respect to x_i .
F is the bordered Hessian Matrix:

F_ij is the cofactor associated with f_ji
If σ_ij > 0 than the two inputs are substitutes if σ_ij >0 then the are complements

What is MES?

MES measure the relative input changes as opposed to AES which measures the absolute input changes. MES can be re-expressed in terms of AES.

Where: σ_ij^A and σ _jj^A are Allen elasticities of substitution.

It should be noted that unlike AES which is symmetrical MES is asymmetric. Furthermore if inputs are classified as substitutes under Allen’s measure they are substitutes under Morishima’s measure. However if the inputs are classified as complements under Allen’s measure they are not necessarily compliments under Morishima’s measure. Hence it is important to note that MES has a bias towards treating the inputs as substitutes and AES is biased towards treating them as compliments.

Production Functions Used

Reviewing the literature I have found various production functions have been used in testing the elasticity of substitution for the various factor inputs. They are:

• Cobb Douglas Production Function
• Translog Production Function
• CES- Translog Production Function

The majority of the literature do not use the Cobb Douglas production function when estimate the Allen partial elasticity of substitution since its function constrains the substitution elasticities to be unity (Dewan et al. 1997). Consequently the Translog and CES-Translog production function tend be more readily used.

Statistical Tests

Since the substitution elasticities calculated are highly non-linear in their parameters the means and standard errors are approximated via various techniques these are:

• Bootstrapping
• Monte Carlo Simulation
• Dividing the sample into different sub-sample and then estimating the separate production functions for each sample.

For the first two methods standard one tailed statistical test apply. However if the latter method is applied then the distributions of substitution of elasticities for each sub-sample are then compared using distribution fee statistical tests which are:

• Median Test
• Wilcoxon Rank Test.

After my meeting I will write more in depth of any additional deliverables.