Henningsen and Henningsen 2011

I read this paper because I needed to estimate a Constant Elasticity of Substitution (CES) production function. Typically, when estimating Total Factor Productivity (TFP), economists prefer the Cobb-Douglas function because it’s easy to linearize and its parameters have useful real-world interpretations. Unfortunately, for structural models like Computable General Equilibrium (CGE), estimating with Cobb-Douglas has the drawback that the substitution parameter is always 1. No modern CGE model uses a substitution elasticity of 1.

Unfortunately, CES estimation is quite difficult. Linearization is possible using Taylor series and its extensions, but if you want to use it for economic models, many restrictions are needed. The alternative is non-linear estimation, which comes in various algorithmic flavors. Plus, I needed step-by-step instructions for implementing it computationally.

{% assign author = site.data.article[page.author] %} {{ author.name }} {{ author.year}} is the perfect paper for me. They developed an R package called micEconCES. This package can perform CES estimation and documents the various algorithms available. They also survey the history of CES estimation from when CES was first proposed by Arrow and colleagues through to recent developments. Comprehensive stuff.

A quick refresher on CES: in general, the CES function takes the form:

where x represents factors of production (capital, labor, materials, etc.). Ideally, we want different substitution parameters for each pair of factors. {{ author.name }} {{ author.year}} uses an approach pioneered by Sato (1967) employing nested CES. For example, with 3 factors of production, instead of putting all factors into model \ref{1} with a single substitution parameter $\rho$, Sato structures it as follows:

With this nested CES method, we can have multiple substitution parameters, including the substitution between $x_1$ and $x_2$ (given by ${\rho/\rho_1}$), and the parameter between the nested $x_1$ and $x_2$ and $x_3$. Depending on your needs as a researcher, the model can be configured. If you’re satisfied with a single $\rho$ for all factors, you can use model \ref{1} directly. Sometimes we feel that among 3 factors, two have a different substitution relationship with the third. For instance, if your factors are capital, skilled labor, and unskilled labor, you might use model \ref{2} where $x_1=skilled\ labor, x_2=unskilled\ labor$, and $x_3 = capital$.