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Consistent estimation of technical and allocative efficiencies for a semiparametric stochastic cost frontier with shadow input prices

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Abstract

Conventional parametric stochastic cost frontier models are likely to suffer from biased inferences due to misspecification and the ignorance of allocative efficiency (AE). To fill up the gap in the literature, this article proposes a semiparametric stochastic cost frontier with shadow input prices that combines a parametric portion with a nonparametric portion and that allows for the presence of both technical efficiency (TE) and AE. The introduction of AE and the nonparametric function into the cost function complicates substantially the estimation procedure. We develop a new estimation procedure that leads to consistent estimators and valid TE and AE measures, which are proved by conducting Monte Carlo simulations.

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Notes

  1. Robinson (1988) showed that the parametric estimators are consistent at the parametric rate of N−1/2, while the nonparametric estimators converge at a slower rate than N−1/2, where N denotes the sample size.

  2. It can be shown that these estimators are consistent and asymptotically normal.

  3. The two-step estimation procedures of Kumbhakar and Lovell (2000) are found to give consistent estimates of AE when the cost function takes the translog form. The TE estimates in general perform not as well as AE estimates, due mainly to the badly performed distribution parameter estimates. Inferences on TE scores using a small sample are doubtful.

  4. Note that the objective function of (3-2) is initially expressed as \( W^{ *} X \) and the choice vector is X. Since parameter b emerges in the constraint of F(·,·), together with X, it is convenient to transform the objective function into \( \left( {W^{ *} /b} \right)bX \). This is equivalent to treat \( W^{ * } /b \) as the new input prices and bX the new choice vector.

  5. Term g t decreases at an increasing rate if γ > 0, increases at an increasing rate if γ < 0, or stays constant if γ = 0.

  6. There is a concern with the referee’s suggestion that equation (3-13) would be plus an extra term \( E(\ln \hat{G}_{nt} |\ln Y_{nt} ) - E(\ln G_{nt} |\ln Y_{nt} ) \), which is non-zero. We can examine how fast it converges to zero as either N or T grows by simulations. The results reveal that the bias measures are small in all of the (N, T) combinations. In addition, the biases decrease as either N or T grows, with the exception of the case of (N, T) = (50, 20). This leads us to conclude that the extra term \( E(\ln \hat{G}_{nt} |\ln Y_{nt} ) - E(\ln G_{nt} |\ln Y_{nt} ) \) does converge rapidly.

  7. To save space, the results for the case of N = 50 are not shown, but available upon request from the authors.

  8. We also check whether the other two regularity conditions are satisfied, that is, a cost function is concave in input prices and the marginal cost should be positive. The model now is specified with an output (y) and two inputs (w1, w2) for simplicity. The result presents that most of the simulated outcomes meet the requirements, although the last condition performs a little worse for smaller sample. We conclude that vast majority of the simulated results satisfy the regularity properties.

  9. We agree with the referee’s opinion that measures of scale economies (SE) and cost elasticity (CE) are important topics particularly in conventional performance analysis. Evidence is found that the simulated estimates of the SE would accurately predict the scope of the true SE and the predictability rises as the sample size increases. Since the CE of outputs is the reciprocal of SE, its measure has very similar performance to the SE. Viewed from this angle, our modeling appears to provide satisfactory results.

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Authors and Affiliations

  1. Department of Money and Banking, National Chengchi University, Taipei, Taiwan

    Tai-Hsin Huang, Chien-Hsiu Lin & Ming-Tai Chung

  2. School of Health Care Administration, Taipei Medical University, Taipei, Taiwan

    Kuan-Chen Chen

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  1. Tai-Hsin Huang
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  2. Kuan-Chen Chen
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  3. Chien-Hsiu Lin
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  4. Ming-Tai Chung
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Correspondence to Tai-Hsin Huang.

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Huang, TH., Chen, KC., Lin, CH. et al. Consistent estimation of technical and allocative efficiencies for a semiparametric stochastic cost frontier with shadow input prices. J Prod Anal 41, 307–320 (2014). https://doi.org/10.1007/s11123-012-0316-9

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