# CES production function estimation

Introduction

There are different ways of estimating the parameters of a production function. For example, single-equation and system equation techniques are both possible. Another difference among methods is that estimable forms may either involve direct or indirect estimation of the parameters. For instance, rather than estimate the production function as is, one may instead estimate the derived demand functions or, say, use other relations such as factor shares. Also to be considered are the underlying economic assumptions; such possibilities being perfect competition, constant returns to scale, profit maximization, etc.

Question

Bearing the above in mind, my question is what is the standard (accepted?) method of estimating normalized CES (constant elasticity of substitution) production functions?

For information, a normalized CES production function can be written as

$$Y = Y_{0} \lbrace \pi_{0} K_{0}^{\frac{1-\sigma}{\sigma}}(K_{t}\cdot e^{\gamma_{K}(t-t_{0})})^{\frac{\sigma-1}{\sigma}} + (1-\pi_{0})N_{0}^{\frac{1-\sigma}{\sigma}}(N_{t}\cdot e^{\gamma_{N}(t-t_{0})})^{\frac{\sigma-1}{\sigma}}\rbrace$$

following the notation found in Klump, McAdam, and Willman (2011, p.22).

Importantly, what data is required to estimate the parameters of this particular production function? And, what is the exact procedure involved? Demonstrations and references are most welcome!

Reference:

Rainer Klump, Peter McAdam, and Alpo Willman, The normalized nested CES production function, theory and empirics, ECB Working Paper Series, No. 1294. (Feb., 2011), 2011.

• Please clarify which of the symbols in the production function are to be estimated, based on data on which other symbols. – Alecos Papadopoulos Feb 9 '16 at 1:35
• @AlecosPapadopoulos Well, that's actually part of my question! Obviously, $\sigma$ is the chief parameter to be estimated. To my understanding, direct estimates of this parameter can be obtained by estimating the derived marginal profit-maximization conditions. That's one option. The symbols with subscript zero are calibrated by, say, taking averages of the data (I guess?). The gamma's are growth rates of the data (again, guess). The data is subscripted by the letter t. A time trend is denoted t (or some other measure of technical progress?). – Graeme Walsh Feb 9 '16 at 1:47
• @GraemeWalsh, the function needs to be raised to the power $\frac{\sigma}{\sigma-1}$ – london May 3 '16 at 16:31

@GraemeWalsh, another option is to use nonlinear least squares to estimate the elasticitiy of substitution, $\sigma$. This is what you want to estimate. The rest is either assumption or data. If you assume the technology is directed i.e. induced, you should use Box-Cox transformation of technology growth so that factor augmenting technological change grows at varying rates over time.

Values of $\gamma$ depend on the average growth rate of output and how you want the technological progress to be augmented. If you think labour-augmentation was strong, then choose a value that is close to actual average rate of output growth for $\gamma_N$, something smaller for the growth rate of the capital augmenting tech growth rate $\gamma_K$.

$\pi$s are constant share parameter.I do not know much about normaisation except that it should be chosen with reference to a common time period. You could normalise all the variables and parameters using the first period, or any other. Alternatively, use non-linear methods of estimation. You could try using micEconCES package in R.

The paper Beyond Cobb-Douglas: Estimation of a CES Production Function with Factor Augmenting Technology should be of use here:

Both the recent literature on production function identification and a considerable body of other empirical work on firm expansion assume a Cobb-Douglas production function. Under this assumption, all technical differences are Hicks neutral. I provide evidence from US manufacturing plants against Cobb-Douglas and present an alternative production function that better fits the data. A Cobb Douglas production function has two empirical implications that I show do not hold in the data: a constant cost share of capital and strong comovement in labor productivity and capital productivity (revenue per unit of capital). Within four digit industries, differences in cost shares of capital are persistent over time. Both the capital share and labor productivity increase with revenue, but capital productivity does not. A CES production function with labor augmenting differences and an elasticity of substitution between labor and capital less than one can account for these facts. To identify the labor capital elasticity, I use variation in wages across local labor markets. Since the capital cost to labor cost ratio falls with local area wages, I strongly reject Cobb-Douglas: capital and labor are complements. Now productivity differences are no longer neutral, which has implications on how productivity affects firms’ decisions to expand or contract. Non neutral technical improvements will result in higher stocks of capital but not necessarily more hiring of labor. Specifying the correct form of the production function is more generally important for empirical work, as I demonstrate by applying my methodology to address questions of misallocation of capital.

In this paper, the first order conditions for capital and labor of the CES production function under cost minimization imply:

$$log(rk/wl) = -(1-\sigma)log(w/r) + (1-\sigma)log B + \sigma log (\alpha / (1-\alpha))$$

This doesn't seem to require any special non-linear estimation technique.

• What you noted above is just a drop in the ocean in the literature on the topic. There are of course different estimation methods and one can choose any depending on what is convenient, or follow the literature or come up with another method. The issue he is referring to here is the normaisation of the variables. With the non-linear estimation methods, you do not have to normalise. Anyway, following paper gives a good discussion of estimation methods cran.r-project.org/web/packages/micEconCES/vignettes/CES.pdf – london May 4 '16 at 21:31