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 below.
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 implemented a nonlinear least squares model with a CES production function and multiplicative error terms. The implementation was inspired by Erwin Kalvelagen’s work (2007) on econometric modeling in GAMS.
Nonlinear Least Squares with CES Production Function
This demo provides two data input options for variable estimation and reports regression statistics based on a Constant Elasticity of Substitution (CES) production function with multiplicative error terms. The reported statistics include estimators, standard errors, t-statistics and p-values (against non-significant coefficients assumption) at the estimated point. For the best results, we recommend using Firefox for this interactive case study.
Back to the Nonlinear Least Squares with Constant Elasticity of Substitution Production Function case study.
Option 1: Data in a text file
Users who have access to the data needed in the estimation should create a text file with the data, for example, the capital, labor, and production data collected in Mizon (1977). See mizon_ces_data.txt. User-provided data files must satisfy the following restrictions:
- The first column of the data file must be a column of output indexed by Q, denoting the quantity of output.
- The second and subsequent columns of the data file that contain input data may not contain any negative or zero input values. The last column of the data file must be a column of input indexed by v_end and users are free to index any other input column as they wish.
The estimated variables in the CES model are indexed by phi for scale factor, b_1 for value share of input 1, b_2 for value share of input 2, b_3 for output elasticity of input 3, …, etc, and rho for substitution parameter. Note that the value share of the last input is calculated as a residual, $(1 – \sum^{m-1}_{n=1}b_n)$.
Users then can download a sample GAMS model file, ces_md_txt.gms (CES model with multiplicative disturbance, text input), and modify it to solve their own estimation problems. Users should specify their own set definitions (sets “t” and “m” in the sample), include their own table of data (as described above), and run the modified model to obtain the estimation results.
Option 2: Data in a GAMS data exchange (gdx) file
Users who have access to the data in a GAMS data exchange (gdx) file can use one of the following two methods.
- Method 1: Solve using the NEOS solver
Users can click on the “Solve with NEOS” button to find estimation results based on the default gdx file, i.e., the file with the capital, labor, and production data collected in Mizon (1977). See mizon_ces.gdx. Alternatively, users can upload their own data by clicking on the button next to “Upload GDX File” and then “Solve with NEOS”. User-provided gdx files must satisfy the same restrictions as listed above in Option 1.
Clicking on the “Reset” button will clear the solution.
- Method 2: Calculate the regression statistics locally
Users who have access to GAMS can download the GAMS model file ces_md_gdx.gms and solve the model locally with the following command:
- “gams ces_md_gdx –in=mydata”
where mydata.gdx is a data file provided by the user. The gdx file must satisfy the same restrictions as in Option 1.
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.