cesCalc: Calculate CES function

In micEconCES: Analysis with the Constant Elasticity of Substitution (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 * x1^(-rho) + ( 1 - delta ) * x2^(-rho) )^(-nu/rho)

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

y = gamma * exp( lambda * t ) * ( sum(i=1 to N) delta_i * x_i^(-rho) )^(-nu/rho)

where in the latter case sum(i=1 to N) delta_i = 1.

In both cases, the elesticity of substitution is s = 1 / ( 1 + rho ).

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

y = gamma * exp( lambda * t ) * [ delta * ( delta_1 * x_1^(-rho_1) + ( 1 - delta_1 ) * x_2^(-rho_1) )^(rho / rho_1) + ( 1 - delta ) * x_3^(-rho) ]^(-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 ) * [ delta ( delta_1 * x_1^(-rho_1) + ( 1 - delta_1 ) * x_2^(-rho_1) )^(rho / rho_1) + ( 1 - delta ) ( delta_2 * x_3^(-ρ_2) + ( 1 - delta_2 ) * x_4^(-rho_2) )^(rho / rho_2) ]^(-nu / rho)

Usage

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

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).

Value

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

Author(s)

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.

See Also

cesEst.

Examples

   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 ) )

Example output

Loading required package: minpack.lm
Loading required package: DEoptim
Loading required package: parallel

DEoptim package
Differential Evolution algorithm in R
Authors: D. Ardia, K. Mullen, B. Peterson and J. Ulrich


If you have questions, suggestions, or comments regarding one of the 'micEcon' packages, please use a forum or 'tracker' at micEcon's R-Forge site:
https://r-forge.r-project.org/projects/micecon/
Warning message:
In nls.lm(par = start, fn = residFun, data = data, jac = jac, yName = yName,  :
  lmder: info = -1. Number of iterations has reached `maxiter' == 50.

 [1] 1150.129 1189.522 1213.680 1224.445 1224.445 1240.227 1273.760 1276.645
 [9] 1237.746 1281.499 1315.982 1468.405 1529.469 1582.718 1549.750 1567.549
[17] 1649.292 1724.152 1834.960 1937.390
Warning message:
In nls.lm(par = start, fn = residFun, data = data, jac = jac, yName = yName,  :
  lmder: info = -1. Number of iterations has reached `maxiter' == 50.

 [1] 1056.404 1101.576 1119.665 1154.728 1154.728 1164.593 1185.412 1214.062
 [9] 1168.251 1244.544 1280.481 1556.668 1633.060 1668.616 1616.492 1603.359
[17] 1698.034 1776.320 1890.527 1994.320

micEconCES documentation built on Jan. 7, 2021, 3:01 p.m.