snqProfitEst {micEcon}R Documentation

Estimation of a SNQ Profit function

Description

Estimation of a Symmetric Normalized Quadratic (SNQ) Profit function.

Usage

snqProfitEst( priceNames, quantNames, fixNames = NULL, instNames = NULL,
   data, form = 0, base = 1, scalingFactors = NULL,
   weights = snqProfitWeights( priceNames, quantNames, data, "DW92", base = base ),
   method = ifelse( is.null( instNames ), "SUR", "3SLS" ), ... )

Arguments

priceNames a vector of strings containing the names of netput prices.
quantNames a vector of strings containing the names of netput quantities (inputs must be negative).
fixNames an optional vector of strings containing the names of the quantities of (quasi-)fixed inputs.
instNames an optional vector of strings containing the names of instrumental variables (for 3SLS estimation).
data a data frame containing the data.
form the functional form to be estimated (see details).
base the base period(s) for scaling prices (see details).
scalingFactors factors to scale prices (see details).
weights vector of weights of the prices for normalization.
method the estimation method (passed to systemfit).
... arguments passed to systemfit

Details

The Symmetric Normalized Quadratic (SNQ) profit function is defined as follows (this functional form is used if argument form equals 0):

π ( p, z ) = sum_{i=1}^{n} α_{i} p_{i} + frac{1}{2} w^{-1} sum_{i=1}^{n} sum_{j=1}^{n} β_{ij} p_{i} p_{j} + sum_{i=1}^{n} sum_{j=1}^{m} delta_{ij} p_{i} z_{j} + frac{1}{2} w sum_{i=1}^{m} sum_{j=1}^{m} gamma_{ij} z_{i} z_{j}

with π = profit, p_i = netput prices, z_i = quantities of fixed inputs, w=sum_{i=1}^{n}theta_{i}p_{i} = price index for normalization, theta_i = weights of prices for normalization, and α_i, β_{ij}, delta_{ij} and gamma_{ij} = coefficients to be estimated.
The netput equations (output supply in input demand) can be obtained by Hotelling's Lemma ( q_{i} = <=ft. partial π right/ partial p_{i} ):

x_{i} = α_{i} + w^{-1} sum_{j=1}^{n} β_{ij} p_{j} - frac{1}{2} theta_{i} w^{-2} sum_{j=1}^{n} sum_{k=1}^{n} β_{jk} p_{j} p_{k} + sum_{j=1}^{m} delta_{ij} z_{j} + frac{1}{2} theta_{i} sum_{j=1}^{m} sum_{k=1}^{m} gamma_{jk} z_{j} z_{k}

In my experience the fit of the model is sometimes not very good, because the effect of the fixed inputs is forced to be proportional to the weights for price normalization theta_i. In this cases I use following extended SNQ profit function (this functional form is used if argument form equals 1):

π ( p, z ) = sum_{i=1}^{n} α_{i} p_{i} + frac{1}{2} w^{-1} sum_{i=1}^{n} sum_{j=1}^{n} β_{ij} p_{i} p_{j} + sum_{i=1}^{n} sum_{j=1}^{m} delta_{ij} p_{i} z_{j} + frac{1}{2} sum_{i=1}^{n} sum_{j=1}^{m} sum_{k=1}^{m} gamma_{ijk} p_i z_{j} z_{k}

The netput equations are now:

x_{i} = α_{i} + w^{-1} sum_{j=1}^{n} β_{ij} p_{j} - frac{1}{2} theta_{i} w^{-2} sum_{j=1}^{n} sum_{k=1}^{n} β_{jk} p_{j} p_{k} + sum_{j=1}^{m} delta_{ij} z_{j} + frac{1}{2} sum_{j=1}^{m} sum_{k=1}^{m} gamma_{ijk} z_{j} z_{k}

The prices are scaled that they are unity in the base period or - if there is more than one base period - that the means of the prices over the base periods are unity. The argument base can be either
(a) a single number: the row number of the base prices,
(b) a vector indicating several observations: The means of these observations are used as base prices,
(c) a logical vector with the same length as the data: The means of the observations indicated as 'TRUE' are used as base prices, or
(d) NULL: prices are not scaled.
If the scaling factors are explicitly specified (argument 'scalingFactors'), the argument 'base' is ignored.

Value

a list of class snqProfitEst containing following objects:

coef a list containing the vectors/matrix of the estimated coefficients:
* alpha = α_i.
* beta = β_{ij}.
* delta = delta_{ij} (only if quasi-fix inputs are present).
* gamma = gamma_{ij} (only if quasi-fix inputs are present).
* allCoef = vector of all coefficients.
* allCoefCov = covariance matrix of all coefficients.
* stats = all coefficients with standard errors, t-values and p-values.
* liCoef = vector of linear independent coefficients.
* liCoefCov = covariance matrix of linear independent coefficients.
ela a list of class snqProfitEla that contains (amongst others) the price elasticities at mean prices and mean quantities (see snqProfitEla).
fixEla matrix of the fixed factor elasticities at mean prices and mean quantities.
hessian hessian matrix of the profit function with respect to prices evaluated at mean prices.
convexity logical. Convexity of the profit function.
r2 R^2-values of all netput equations.
est estimation results returned by systemfit.
weights the weights of prices used for normalization.
normPrice vector used for normalization of prices.
data data frame of originally supplied data.
fitted data frame that contains the fitted netput quantities and the fitted profit.
pMeans means of the scaled netput prices.
qMeans means of the scaled netput quantities.
fMeans means of the (quasi-)fixed input quantities.
priceNames a vector of strings containing the names of netput prices.
quantNames a vector of strings containing the names of netput quantities (inputs must be negative).
fixNames an optional vector of strings containing the names of the quantities of (quasi-)fixed inputs.
instNames an optional vector of strings containing the names of instrumental variables (for 3SLS estimation).
form the functional form (see details).
base the base period(s) for scaling prices (see details).
weights vector of weights of the prices for normalization.
scalingFactors factors to scale prices (and quantities).
method the estimation method.

Author(s)

Arne Henningsen

References

Diewert, W.E. and T.J. Wales (1987) Flexible functional forms and global curvature conditions. Econometrica, 55, p. 43-68.

Diewert, W.E. and T.J. Wales (1992) Quadratic Spline Models for Producer's Supply and Demand Functions. International Economic Review, 33, p. 705-722.

Kohli, U.R. (1993) A symmetric normalized quadratic GNP function and the US demand for imports and supply of exports. International Economic Review, 34, p. 243-255.

See Also

snqProfitEla and snqProfitWeights.

Examples

   data( germanFarms )
   germanFarms$qOutput   <- germanFarms$vOutput / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )

   estResult <- snqProfitEst( priceNames, quantNames, "land", data = germanFarms )
   estResult$ela   # Oh, that looks bad!

   # it it reasonable to account for technological progress
   germanFarms$time <- c( 0:19 )
   estResult2 <- snqProfitEst( priceNames, quantNames, c("land","time"), data=germanFarms )
   estResult2$ela   # Ah, that looks good!

[Package micEcon version 0.5-14 Index]