Received: 22 April 2016 Revised: 19 April 2017
Normalized CES supply systems: Replication of Klump,
McAdam, and Willman (2007)
Kenneth G. Stewart
Department of Economics, University of
Victoria, Victoria, British Columbia, Canada
Kenneth G. Stewart, Department of
Economics, University of Victoria, Victoria,
British Columbia, Canada V8W 2Y2.
The analysis of Klump, McAdam, and Willman (Review of Economics and Statistics,
2007, 89, 183–192) is replicated using alternative software. Their results are veri-
fied substantively and, in large measure, numerically. Contributions include a more
explicit consideration of the nested testing structure than has appeared previously,
and the appropriate means of imposing and testing the special case of logarithmic
growth in technology. Also, plots of the likelihood serve to emphasize that maxima
in the neighborhood of a unitary elasticity of substitution are often a spurious artifact
of the singularity of the model at this point, of which empirical researchers should
In a widely cited series of articles, Klump, McAdam, and Willman (KMW) and collaborators have shown how to use a CES
aggregate supply-side system to disentangle factor substitution and biases of technical change. In addition to providing new
estimates of the elasticity of substitution—a key parameter in many macroeconomic models—their framework permits the
study of the classic steady-state growth theorem of Uzawa (1961) that balanced growth requires technology to be solely labor
The seminal empirical contribution in this literature is Klump, McAdam and Willman (2007), which estimates a CES produc-
tion function joint with its implied factor demands. KMW introduced two innovations to the analysis that, as they demonstrate
convincingly, are critical to disentangling factor substitution and biases of technological progress. First, they draw on ear-
lier theoretical contributions to formulate their model in normalized form. Second, they use Box–Cox transformations of the
factor-specific technology parameters to distinguish long-term from short-term biases.
Given the literature that has since accumulated, the original Klump et al. (2007) estimation can now be seen to be an iconic
empirical implementation. It is therefore useful to revisit it, investigating the sensitivity of the estimation results to alternative
software. Whereas KMW used RATS, I use TSP, the numerics of which have been favorably evaluated by McCullough (1999).
As part of the replication I lay out the model's complete nested testing structure more explicitly than has been done previously
and use it to explore the incidence of multiple maxima of the likelihood function. Also, I clarify some nuances in the estimation
of these systems that may be helpful to future researchers, such as the proper imposition and testing of the special case of
logarithmic growth in technology. I also plot the log-likelihood of the system, something that does not appear to have been
done in the published literature.
This allows me to conclude more strongly than did KMW that the multiple maxima that seem
endemic to their model are of less practical importance than might appear, because they are typically an artifact of the singularity
of the system at = 1.
In their working paper, Klump, McAdam, and Willman (2004) provide scatter plots of the local minima of the log determinant of the models they estimate.
Their Graph 4.3 shows the two minima of the maintained model that correspond to the two maxima of the loglikelihood in my Figure 2.
290 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/jae J Appl Econ. 2018;33:290–296.