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Revisiting the world technology frontier: a directional distance function approach

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Abstract

This paper revisits the finding of Caselli and Coleman (Amer Econ Rev 96:499–522, 2006) that poor countries are relatively more efficient in using unskilled labor whereas rich countries are more efficient in using skilled labor. The analysis is based on an approach using directional distance functions from nonparametric efficiency analysis which relies on very mild assumptions. We find that the central result of Caselli and Coleman is robust to using the nonparametric approach. The result is, however, sensitive to alternative definitions of skilled and unskilled labor, data sources and variations of the measurement approach.

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Notes

  1. See also the handbook article of Caselli (2005).

  2. In the measurement of aggregate productivity, related nonparametric methods are used in the studies of Kumar and Russell (2002) and Henderson and Russell (2005), updated by Henderson and Zelenyuk (2007) and Badunenko et al. (2013), respectively. Jerzmanowski (2007) devotes special attention to the application of data envelopment analysis to the issue of appropriate technology. All these papers also provide ample evidence on the non-neutrality of technological progress.

  3. See Temple (2012) for more about pitfalls concerning the calibration of CES production functions.

  4. Admittedly, nonparametric deterministic frontier function methods may be sensitive to outliers. We are convinced that this problem does not offset the advantages. The sample of countries analyzed here are those countries with data of presumably better quality.

  5. See Färe and Primont (1995) and Hackman (2008) for detailed expositions.

  6. In the definition and in the following ’\(\ge \)’, applied to vectors, is to be understood as a strict inequality holding for at least one component of the vector.

  7. There is a further conceptual difference to CC as pointed out by a reviewer. In CC, the \(A_{u}\) and \(A_{s}\) values are computed for each country individually using only data for the country under consideration. This means that CC are computing actual productivity levels without using the world technology frontier as a benchmark. This is generally difficult to align with the theoretical concept of a production function as defining the maximum output for given input levels. CC themselves interpret their results as ”factor-specific efficiency levels” (p. 518). The approach used in this paper rests on fewer assumptions and therefore should take the analysis closer to this concept.

  8. In practice, of course, only a limited subset of countries determines the frontier function so that the majority of the \(\lambda \) values is actually zero. The \(\lambda \)’s may also be different for each country. This allows the frontier function to be piece-wise linear and therefore local in the sense of Atkinson and Stiglitz (1969).

  9. A related method is developed by Hampf and Krüger (2015) in the context of an environmental economics application. See also Färe et al. (2013) for more on the choice of optimal directions.

  10. The formulation of the constraints shows that the solution values of the program (6) and also (7) are independent of the units of measurement of the inputs and outputs.

  11. For the computations in this paper the linear programming solver in the R-package “lpSolve” is used.

  12. In the empirical analysis we are actually not using the output and the inputs in per worker terms since this implies CRS and thus restricts the analysis in an unnecessary way. Using the output and the inputs in per worker terms (as done by CC) or in total terms is inessential under CRS, but affects the results under VRS. Therefore, we compute and discuss the results of both variants in the subsequent analysis. Moreover, since the linear programming problems (6) and (7) are units invariant the solution values for the directions are not affected by the scaling of the variables.

  13. Recall that \(A_{s}\) and \(A_{u}\) in Eq. (1) of CC multiply \(L_{s}\) and \(L_{u}\), respectively, and are therefore likewise relative measures.

  14. Recall that the program (6) maximizes the measure of inefficiency \(\beta \) jointly with determining the directions associated with this maximum. The computation is done by solving the equivalent linear program (7) as already explained.

  15. The MM-estimator used here is a robust regression estimator designed to combine the advantages of a high breakdown point (the fraction of contaminated observations in the sample that lead to an arbitrarily large deviation of the estimator) and high estimation efficiency (Yohai 1987). This goal is achieved by using an initial M-estimator searching for the regression parameters associated with the smallest robust measure of scale of the residuals (actually an S-estimator), followed by a second M-estimation which can be computed by the iteratively reweighted least squares (IRWLS) algorithm. Maronna et al. (2006, pp. 124ff.) provide a formal exposition. The implementation used in this paper is that of Yohai et al. (1991) in the R-package “robust”. Renaud and Victoria-Feser (2010) explain the kind of \(R^{2}\) measures used for the assessment of the fit of the robust regressions.

  16. Computed are nonparametric local polynomial regressions, explained in detail in Loader (1999). We apply adaptive bandwidth selection methods as described in Loader (1999, chap. 11). The computations use the functions of the R-package “locfit”.

  17. In the case of the MM-estimator there may actually be too much robustification which may lead to erratic behavior of the estimator. Koller and Stahel (2011) suggest a refinement of the MM-estimator starting with computing a design adaptive scale estimate based on the MM-residuals which is used for an additional M-estimation of the regression coefficients. This so-called SMDM-estimator is specifically designed to reach robust regression estimates with improved efficiency properties in small samples. See Koller and Stahel (2011) for a detailed exposition of the method as well as simulation results on robustness and efficiency. The SMDM-estimator is implemented in the R-package “robustbase” and readily available upon using the option setting=’KS2011’ in the lmrob command.

  18. It appears that China (CHN) has a very low level of per worker income. This may be somewhat surprising from a current perspective. In the PWT 5.6 log income per worker in China in 1988 is indeed about 7.71. In the recent version 8.0 of the PWT this value is slightly larger and about 8.26 (base year 2005). Analyzing China’s growth path over a longer time span we see that the sustained growth spurt of real GDP per worker in China started just after the year 1988 with an average annual growth rate of over 6 percent from 1989 to 2011 (using the expenditure-based real GDP measure of PWT 8.0 divided by employment to assure comparability with the real GDP per worker measure of the PWT 5.6). During this period the Chinese economy started to become increasingly oriented towards global markets. See Zhu (2012) for a more detailed account of China’s recent growth performance.

  19. Taking higher education as the split point in an analogous way leads to a significantly positive relation to \(\ln y\) under CRS and an almost flat regression line under VRS. Treating only workers with completed higher education as skilled and all others as unskilled seems to be a too restrictive definition of skills, however.

  20. Splitting at higher education again leads to a significantly positive relation to \(\ln y\) under both CRS and VRS.

  21. Hampf and Krüger (2015) suggest a different dynamic approach to direction choice.

  22. As seen below, period s may be equal to t or may differ from t.

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Acknowledgments

I am grateful to three anonymous referees for their deep and insightful comments. In particular, I am indebted to an associate editor who generously provided major suggestions during all revision rounds which were substantially improving both content and exposition. The usual disclaimer applies.

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  1. Department of Law and Economics, Darmstadt University of Technology, Hochschulstr. 1, 64289, Darmstadt, Germany

    Jens J. Krüger

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Correspondence to Jens J. Krüger.

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Appendix

Appendix

Table 7 DDF results by country with the original Caselli/Coleman data
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Krüger, J.J. Revisiting the world technology frontier: a directional distance function approach. J Econ Growth 22, 67–95 (2017). https://doi.org/10.1007/s10887-016-9136-5

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