Abstract:
Most production functions are often nonlinear in parameters and difficult to linearise. The major shortcomings of the classical approach in estimating such parameters are varying complex inter- parametric relationship, parameters estimates and their standard errors taking infeasible values in the process of iteration. Even when the initial values are carefully selected, the parameters of such models remain difficult to estimate often yielding ambiguous results owing to their specification in the models. This work therefore, proposed Bayesian estimators (BE) for estimating the parameters of nonlinear production function which ameliorate the problems of Classical Approach.
The Cobb-Douglas (CD) and Constant Elasticity of Substitution (CES) production functions are nonlinear in parameters. The CD production function with Additive Error (CDAE), CD Multiplicative Error (CDME) and (CES) were considered. Three possible specifications of Returns to Scale (RS) parameters were also investigated for CD namely: Constant RS (CRS, ), Increasing RS (IRS, ) and Decreasing RS (DRS, ), where and are the output elasticities of Capital and Labour, respectively. For CES, the behaviour of the substitution parameter was chosen such that the Elasticity of Substitution was ( ). The three BE: CDAE, CDME and CES were obtained using independent Normal Gamma prior. The posterior estimates was obtained using metropolis- within- Gibbs sampler. The performance of Bayesian approach (BA) and CA was compared using a Monte Carlo study with sample sizes (N = 50, 100, 150, 250 and 500) each in 10,000 iterations, considering the minimum Numerical Standard Error (NSE) or Standard Error (SE) as the criteria for BE and CA, respectively.
The derived BE were and for CDAE, and for CDME and and for CES. For CA, the estimates of of CDAE and CDME were 0.841800, 0.148060, 0.014130, 0.007060 and 0.702530, 0.186370, 0.118600, 0.070790 for CRS, respectively. Those for IRS were 0.916189, 0.201102, 0.017134, 0.008889 and 0.816900, 0.419520, 0.128530, 0.091390, respectively. For DRS, they were 0.521822, 0.126701, 0.014130, 0.008448 and 0.618900, 0.179160, 0.109110, 0.068120, respectively. However, for BA, the estimates of of CDAE and CDME were 0.759519, 0.132379, 0.000120, 0.000116 and 0.845861, 0.176499, 0.000114, 0.000103 for CRS, respectively. Those for IRS were 0.897875, 0.204861, 0.000118, 0.000114 and 0.894150, 0.227022, 0.000116, 0.000102, respectively. For DRS, they were 0.68244, 0.170446, 0.000121, 0.000114 and 0.644050, 0.143532, 0.000116, 0.000101, respectively. For CA, the CES estimates of and SE when were 0.014610, 2221.7626, when were -0.206960, 190.593100 and when were 0.995520, 0.849600, while the estimates of BA for and NSE when were 0.893575, 0.000817, when were 0.503699, 0.001117 and when were 1.462442, 0.001215. Thus, BE performed better than CA in all the specifications of production functions with minimum value in NSE.
The Bayesian estimators outperformed classical approach for the production functions considered, thus making them more appropriate in handling nonlinear production functions regardless of the error specification.
