cesCalc: Calculate CES function

Description

Calculate the endogenous variable of a ‘Constant Elasticity of Substitution’ (CES) function.

The original CES function with two explanatory variables is

$$y = \gamma \: \exp ( \lambda \: t ) \: ( \delta \: x_1^{-\rho} + ( 1 - \delta ) \: x_2^{-\rho} ) ^{-\frac{\nu}{\rho}}$$

and the non-nested CES function with $N$ explanatory variables is

$$y = \gamma \: \exp ( \lambda \: t ) \: \left( \sum_{i=1}^N \delta_i \: x_i^{-\rho} \right) ^{-\frac{\nu}{\rho}}$$

where in the latter case $\sum_{i=1}^N \delta_i = 1$.

In both cases, the elesticity of substitution is $s = \frac{1}{ 1 + \rho }$.

The nested CES function with 3 explanatory variables proposed by Sato (1967) is

$$y = \gamma \: \exp ( \lambda \: t ) \: \left[ \delta \: \left( \delta_1 \: x_1^{-\rho_1} + ( 1 - \delta_1 ) x_2^{-\rho_1} \right)^{\frac{\rho}{\rho_1}} + ( 1 - \delta ) x_3^{-\rho} \right]^{-\frac{\nu}{\rho}}$$

and the nested CES function with 4 explanatory variables (a generalisation of the version proposed by Sato, 1967) is

$$y = \gamma \: \exp ( \lambda \: t ) \: \left[ \delta \cdot \left( \delta_1 \: x_1^{-\rho_1} + ( 1 - \delta_1 ) x_2^{-\rho_1} \right)^{\frac{\rho}{\rho_1}} + ( 1 - \delta ) \cdot \left( \delta_2 \: x_3^{-\rho_2} + ( 1 - \delta_2 ) x_4^{-\rho_2} \right)^{\frac{\rho}{\rho_2}} \right]^{-\frac{\nu}{\rho}}$$

Usage

cesCalc( xNames, data, coef, tName = NULL, nested = FALSE, rhoApprox = 5e-6 )

Value

A numeric vector with length equal to the number of rows of the data set specified in argument data.

Arguments

xNames: a vector of strings containing the names of the explanatory variables.
data: data frame containing the explanatory variables.
coef: numeric vector containing the coefficients of the CES: if the vector is unnamed, the order of the coefficients must be $\gamma$, eventuelly $\lambda$, $\delta$, $\rho$, and eventually $\nu$ in case of two expanatory variables, $\gamma$, eventuelly $\lambda$, $\delta_1$, ..., $\delta_N$, $\rho$, and eventually $\nu$ in case of the non-nested CES with $N>2$ explanatory variables, $\gamma$, eventuelly $\lambda$, $\delta_1$, $\delta$, $\rho_1$, $\rho$, and eventually $\nu$ in case of the nested CES with 3 explanatory variables, and $\gamma$, eventuelly $\lambda$, $\delta_1$, $\delta_2$, $\delta$, $\rho_1$, $\rho_2$, $\rho$, and eventually $\nu$ in case of the nested CES with 4 explanatory variables, where in all cases the $\nu$ is only required if the model has variable returns to scale. If the vector is named, the names must be "gamma", "delta", "rho", and eventually "nu" in case of two expanatory variables, "gamma", "delta_1", ..., "delta_N", "rho", and eventually "nu" in case of the non-nested CES with $N>2$ explanatory variables, and "gamma", "delta_1", "delta_2", "rho_1", "rho_2", "rho", and eventually "nu" in case of the nested CES with 4 explanatory variables, where the order is irrelevant in all cases.
tName: optional character string specifying the name of the time variable ($t$).
nested: logical. ; if FALSE (the default), the original CES for $n$ inputs proposed by Kmenta (1967) is used; if TRUE, the nested version of the CES for 3 or 4 inputs proposed by Sato (1967) is used.
rhoApprox: if the absolute value of the coefficient $\rho$, $\rho_1$, or $\rho_2$ is smaller than or equal to this argument, the endogenous variable is calculated using a first-order Taylor series approximation at the point $\rho$ = 0 (for non-nested CES functions) or a linear interpolation (for nested CES functions), because this avoids large numerical inaccuracies that frequently occur in calculations with non-linear CES functions if $\rho$, $\rho_1$, or $\rho_2$ have very small values (in absolute terms).

Author

Arne Henningsen and Geraldine Henningsen

References

Kmenta, J. (1967): On Estimation of the CES Production Function. International Economic Review 8, p. 180-189.

Sato, K. (1967): A Two-Level Constant-Elasticity-of-Substitution Production Function. Review of Economic Studies 43, p. 201-218.

Examples

Run this code

   data( germanFarms, package = "micEcon" )
   # output quantity:
   germanFarms$qOutput <- germanFarms$vOutput / germanFarms$pOutput
   # quantity of intermediate inputs
   germanFarms$qVarInput <- germanFarms$vVarInput / germanFarms$pVarInput


   ## Estimate CES: Land & Labor with fixed returns to scale
   cesLandLabor <- cesEst( "qOutput", c( "land", "qLabor" ), germanFarms )

   ## Calculate fitted values
   cesCalc( c( "land", "qLabor" ), germanFarms, coef( cesLandLabor ) )


   # variable returns to scale
   cesLandLaborVrs <- cesEst( "qOutput", c( "land", "qLabor" ), germanFarms,
      vrs = TRUE )

   ## Calculate fitted values
   cesCalc( c( "land", "qLabor" ), germanFarms, coef( cesLandLaborVrs ) )

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