Example 2: Nonlinear Least Squares with Constant Elasticity of Substitution Production Function

Case Study Contents

Problem Statement

In this example, we estimate the unknown parameters of Constant Elasticity of Substitution (CES) production functions with either additive or multiplicative error terms using Mizon (1977). The CES production function is a neoclassical production function that displays constant elasticity of substitution. In other words, the production technology has a constant percentage change in input proportions (for example, labor and capital) due to a percentage change in marginal rate of technical substitution.

Mathematical Formulation

This example uses the same optimization model as that in Example 1, except that the optimization is restricted by some CES functions rather than the Cobb-Douglas production function. We introduce two different forms of CES functions. The first one, which we call the MD model, has Multiplicative Disturbance (MD). The standard form with two inputs is

$$q_t = \phi[\beta k^{\rho}_t + (1 - \beta)l^{\rho}_t]^{\frac{1}{\rho}}\exp(\mu_t);$$

The second one, which we call the AD model, has Additive Disturbance (AD). The standard form is

$$q_t = \phi[\beta k^{\rho}_t + (1 - \beta)l^{\rho}_t]^{\frac{1}{\rho}} + \mu_t,$$

where:

  • $\phi \equiv$ scale parameter, $\phi$ > 0
  • $\beta \equiv$ value share of capital, $\beta \in (0,1)$
  • $\rho \equiv$ the substitution parameter, $\rho \in (-\infty,1)$, and $\rho \neq 0$
  • $k_t$, $l_t$ and $\mu_t$ stand for capital, labor and error term at observation $t$.

And we define $\sigma = \frac{1}{1-\rho}$ as the elasticity of substitution in these production functions.

As its name suggests, the CES production function exhibits constant elasticity of substitution between capital and labor. Leontief, linear and Cobb–Douglas production functions are special cases of the CES production function. That is, if the substitution parameter $\rho$ equals one, we have a linear or perfect substitutes production function; if $\rho$ approaches zero in the limit, we get the Cobb–Douglas production function; as $\rho$ approaches negative infinity, we get the Leontief or perfect complements production function.

In more general cases with more than two inputs, we denote the inputs as $V_{1}, V_{2}, \dots, V_{m}$, $m > 2$. Then the fitting constraints in the MD model are:

\[q_t = \phi[\beta_1 V_{1,t}^{\rho} + \beta_2 V_{2,t}^{\rho} + \dots + (1 - \sum_n\beta_n)V_{m,t}^{\rho}]^{\frac{1}{\rho}}\exp(\mu_t),\]

and, in the AD model, they are:
\[q_t = \phi[\beta_1 V_{1,t}^{\rho} + \beta_2 V_{2,t}^{\rho} + \dots + (1 - \sum_n\beta_n)V_{m,t}^{\rho}]^{\frac{1}{\rho}} + \mu_t.\]

We then need to estimate an unknown vector of $\theta = (\phi, \beta_{1}, \beta_{2},\dots, \beta_n, \dots, \beta_{m-1}, \rho)'$ with $m + 1$ elements, where $n = 1, 2, \dots, m-1$ and $m$ is the total number of inputs. Both the standard and general form of the CES production function in the MD model are included in the demo of example 2.

Similar to Example 1, we estimate the unknown vector of parameters $\theta$ by minimizing the Sum of Squared Errors (SSE) subject to the CES production function having either additive (AD) and multiplicative (MD) error terms. We report estimator of unknown vector $\hat\theta$, estimated variances based on the diagonal elements of the covariance matrix $\hat{V}_{\hat{\theta}}$, and t-statistics as well as p-values as we did in Example 1.

Demo

We have implemented a nonlinear least squares model with a CES production function and multiplicative error terms. Click here to experiment with the demo of Example 2. The implementation was inspired by Erwin Kalvelagen's work (2007) on econometric modeling in GAMS.

Models

Multiplicative Disturbance

A printable version of the nonlinear least squares model with the multiplicative error terms is here: ces_md_gdx.gms for gdx input and ces_md_txt.gms for text input.

Additive Disturbance

A printable version of the nonlinear least squares model with the additive error terms is here: ces_ad_gdx.gms for GAMS data exchange (gdx) format input and ces_ad_txt.gms for text input.

References

  • Mizon, Grayham E. 1977. Inferential Procedures in Nonlinear Models: An Application in a UK Industrial Cross Section Study of Factor Substitution and Returns to Scale. Econometrica 45(5), 1221-1242.
  • Kalvelagen, Erwin. 2007. Least Squares Calculations with GAMS. Available for download at http://www.amsterdamoptimization.com/pdf/ols.pdf.
  • Greene, William. 2011. Econometrics Analysis, 7th ed. Prentice Hall, Upper Saddle River, NJ.

Optimization Category (Linear Programing, Integer, MIP and etc.):