Package 'micEconSNQP'

Title: Symmetric Normalized Quadratic Profit Function
Description: Tools for econometric production analysis with the Symmetric Normalized Quadratic (SNQ) profit function, e.g. estimation, imposing convexity in prices, and calculating elasticities and shadow prices.
Authors: Arne Henningsen
Maintainer: Arne Henningsen <[email protected]>
License: GPL (>= 2)
Version: 0.6-10
Built: 2024-11-01 11:43:21 UTC
Source: CRAN

Help Index

Predictions from an SNQ profit function

Description

Returns the predicted values, their standard errors and the confidence limits of prediction for an Symmetric Normalized Quadratic (SNQ) profit function.

Usage

## S3 method for class 'snqProfitEst'
predict( object, newdata = object$data,
   se.fit = FALSE, se.pred = FALSE, interval = "none", level = 0.95,
   useDfSys = TRUE, ... )

## S3 method for class 'snqProfitImposeConvexity'
predict( object, newdata = object$data,
   se.fit = FALSE, se.pred = FALSE, interval = "none", level = 0.95,
   useDfSys = TRUE, ... )

Arguments

object

an object of type snqProfitEst or snqProfitImposeConvexity.
newdata

data frame in which to predict.
se.fit

logical. Return the standard error of the fitted values?
se.pred

logical. Return the standard error of prediction?
interval

Type of interval calculation ("none", "confidence" or "prediction").
level

confidence level.
useDfSys

logical. Use the degrees of freedom of the whole system (and not the degrees of freedom of the single equation) to calculate the confidence intervals.
...

currently not used.

Details

The variance of the fitted values (used to calculate the standard errors of the fitted values and the "confidence interval") is calculated by Var[E[y0]−y^0]=x0  Var[b]  x0′Var[E[y^0]-\hat{y}^0]=x^0 \; Var[b] \; {x^0}'
The variances of the predicted values (used to calculate the standard errors of the predicted values and the "prediction intervals") is calculated by Var[y0−y^0]=σ^2+x0  Var[b]  x0′Var[y^0-\hat{y}^0]=\hat{\sigma}^2+x^0 \; Var[b] \; {x^0}'

Value

predict.snqProfitEst and predict.snqProfitImposeConvexity return a dataframe that contains the predicted profit and for each netput the predicted quantities (e.g. "quant1" ) and if requested the standard errors of the fitted values (e.g. "quant1.se.fit"), the standard errors of the prediction (e.g. "quant1.se.pred"), and the lower (e.g. "quant1.lwr") and upper (e.g. "quant1.upr") limits of the confidence or prediction interval(s).

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.

Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Macmillan.

Gujarati, D. N. (1995) Basic Econometrics, Third Edition, McGraw-Hill.

Kmenta, J. (1997) Elements of Econometrics, Second Edition, University of Michigan Publishing.

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

snqProfitEst, snqProfitCalc and predict

Examples

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time      <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )
   estResult <- snqProfitEst( priceNames, quantNames, c("land","time"), data=germanFarms )
   predict( estResult )
   predict( estResult, se.fit = TRUE, se.pred = TRUE, interval = "confidence" )
}

Print output of estimated SNQ profit function

Description

This function prints a summary estimation results of a symmetric normalized quadraric (SNQ) profit function.

Usage

## S3 method for class 'snqProfitEst'
print( x, ... )

Arguments

x

an object of class snqProfitEst.
...

arguments passed to print.

Author(s)

Arne Henningsen

See Also

snqProfitEst

Examples

## Not run: library( systemfit )
if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   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 )
   print( estResult )
}

Residuals of an SNQ profit function

Description

Extract the residuals from the estimation of a Symmetric Normalized Quadratic (SNQ) profit function.

Usage

## S3 method for class 'snqProfitEst'
residuals( object, scaled = TRUE, ... )

## S3 method for class 'snqProfitImposeConvexity'
residuals( object, scaled = TRUE, ... )

Arguments

object

an object of type snqProfitEst or snqProfitImposeconvexity.
scaled

logical. Return scaled quantities?
...

currently not used.

Value

residuals.snqProfitEst and residuals.snqProfitEst return a dataframe that contains the residuals for each netput and the profit.

Author(s)

Arne Henningsen

See Also

snqProfitEst, snqProfitImposeConvexity and residuals

Examples

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time      <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )
   estResult <- snqProfitEst( priceNames, quantNames, c("land","time"), data=germanFarms )
   residuals( estResult )
   residuals( estResult, scaled = FALSE )
}

Calculations with the SNQ Profit function

Description

Calculation of netput quantities and profit with the Symmetric Normalized Quadratic (SNQ) Profit function.

Usage

snqProfitCalc( priceNames, fixNames, data, weights,
      scalingFactors = rep( 1, length( weights ) ), coef,
      quantNames = NULL, form = 0 )

Arguments

priceNames

a vector of strings containing the names of netput prices.
fixNames

an optional vector of strings containing the names of the quantities of (quasi-)fix inputs.
data

a data frame containing the data.
weights

vector of weights of the prices for normalization.
quantNames

optional vector of strings containing the names of netput quantities.
scalingFactors

factors to scale prices (and quantities).
coef

a list containing the coefficients alpha, beta, delta and gamma.
form

the functional form to be estimated (see snqProfitEst).

Value

a data frame: the first n columns are the netput quantities, the last column is the profit.

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

snqProfitEst and snqProfitWeights.

Examples

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time      <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )
   fixNames <- c( "land", "time" )

   estResult <- snqProfitEst( priceNames, quantNames, fixNames, data = germanFarms )
   snqProfitCalc( priceNames, fixNames, estResult$data, estResult$weights,
      estResult$scalingFactors, estResult$coef )
}

Price Elasticities of SNQ Profit function

Description

Calculates the Price Elasticities of a Symmetric Normalized Quadratic (SNQ) profit function.

Usage

snqProfitEla(  beta, prices, quant, weights,
   scalingFactors = rep( 1, length( weights ) ),
   coefVcov = NULL, df = NULL )

Arguments

beta

matrix of estimated β\beta coefficients.
prices

vector of netput prices at which the elasticities should be calculated.
quant

vector of netput quantities at which the elasticities should be calculated.
weights

vector of weights of prices used for normalization.
scalingFactors

factors to scale prices (and quantities).
coefVcov

variance covariance matrix of the coefficients (optional).
df

degrees of freedom to calculate P-values of the elasticities (optional).

Value

a list of class snqProfitEla containing following elements:

ela

matrix of the price elasticities.
vcov

variance covariance matrix of the price elasticities.
stEr

standard errors of the price elasticities.
tval

t-values of the price elasticities.
pval

P-values of the price elasticities.

Note

A price elasticity is defined as

Eij=∂qiqi∂pjpj=∂qi∂pj⋅pjqiE_{ij} = \frac{ \displaystyle \frac{ \partial q_i }{ q_i } } { \displaystyle \frac{ \partial p_j }{ p_j } } = \frac{ \partial q_i }{ \partial p_j } \cdot \frac{ p_j }{ q_i }

Thus, e.g. Eij=0.5E_{ij}=0.5 means that if the price of netput j (pjp_j) increases by 1%, the quantity of netput i (qiq_i) will increase by 0.5%.

Author(s)

Arne Henningsen

See Also

snqProfitEst.

Examples

# just a stupid simple example
snqProfitEla( matrix(101:109,3,3), c(1,1,1), c(1,-1,-1), c(0.4,0.3,0.3) )

# now with real data
if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput   / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time      <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )

   estResult <- snqProfitEst( priceNames, quantNames, c("land","time"), data=germanFarms )

   estResult$ela  # price elasticities at mean prices and mean quantities

   # price elasticities at the last observation (1994/95)
   snqProfitEla( estResult$coef$beta, estResult$data[ 20, priceNames ],
      estResult$data[ 20, quantNames ], estResult$weights,
      estResult$scalingFactors )
}

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). If argument weights is not specified, argument base is also used to obtain the weights for normalizing prices (see snqProfitWeights).
scalingFactors

a vector of factors to scale prices (see details).
weights

a vector of weights for normalizing prices.
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)=∑i=1nαipi+12w−1∑i=1n∑j=1nβijpipj+∑i=1n∑j=1mδijpizj+12w∑i=1m∑j=1mγijzizj\pi \left( p, z \right) = \sum_{i=1}^{n} \alpha_{i} p_{i} + \frac{1}{2} w^{-1} \sum_{i=1}^{n} \sum_{j=1}^{n} \beta_{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\pi = profit, pip_i = netput prices, ziz_i = quantities of fixed inputs, w=∑i=1nθipiw=\sum_{i=1}^{n}\theta_{i}p_{i} = price index for normalization, θi\theta_i = weights of prices for normalization, and αi\alpha_i, βij\beta_{ij}, δij\delta_{ij} and γij\gamma_{ij} = coefficients to be estimated.
The netput equations (output supply in input demand) can be obtained by Hotelling's Lemma (qi=∂π/∂piq_{i} = \left. \partial \pi \right/ \partial p_{i}):

xi=αi+w−1∑j=1nβijpj−12θiw−2∑j=1n∑k=1nβjkpjpk+∑j=1mδijzj+12θi∑j=1m∑k=1mγjkzjzkx_{i} = \alpha_{i} + w^{-1} \sum_{j=1}^{n} \beta_{ij} p_{j} - \frac{1}{2} \theta_{i} w^{-2} \sum_{j=1}^{n} \sum_{k=1}^{n} \beta_{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 θi\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)=∑i=1nαipi+12w−1∑i=1n∑j=1nβijpipj+∑i=1n∑j=1mδijpizj+12∑i=1n∑j=1m∑k=1mγijkpizjzk\pi \left( p, z \right) = \sum_{i=1}^{n} \alpha_{i} p_{i} + \frac{1}{2} w^{-1} \sum_{i=1}^{n} \sum_{j=1}^{n} \beta_{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:

xi=αi+w−1∑j=1nβijpj−12θiw−2∑j=1n∑k=1nβjkpjpk+∑j=1mδijzj+12∑j=1m∑k=1mγijkzjzkx_{i} = \alpha_{i} + w^{-1} \sum_{j=1}^{n} \beta_{ij} p_{j} - \frac{1}{2} \theta_{i} w^{-2} \sum_{j=1}^{n} \sum_{k=1}^{n} \beta_{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}

Argument scalingFactors can be used to scale prices, e.g. for improving the numerical stability of the estimation (e.g. if prices are very large or very small numbers) or for assessing the robustness of the results when changing the units of measurement. The prices are multiplied by the scaling factors, while the quantities are divided my the scaling factors so that the monetary values of the inputs and outputs and thus, the profit, remains unchanged. If argument scalingFactors is NULL, argument base is used to automatically obtain scaling factors so that the scaled prices are unity in the base period or - if there is more than one base period - that the means of the scaled prices over the base periods are unity. 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 length equal to the number of rows of the data set that is specified by argument data: The means of the observations indicated as 'TRUE' are used as base prices, or
(d) NULL: prices are not scaled. If argument base is NULL, argument weights must be specified, because the weights cannot be calculated if the base period is not specified. An alternative way to use unscaled prices is to set argument scalingFactors equal to a vector of ones (see examples below).
If the scaling factors are explicitly specified by argument scalingFactors, argument base is not used for obtaining scaling factors (but it is used for obtaining weights if argument weights is not specified).

Value

a list of class snqProfitEst containing following objects:

coef

a list containing the vectors/matrix of the estimated coefficients:
* alpha = αi\alpha_i.
* beta = βij\beta_{ij}.
* delta = δij\delta_{ij} (only if quasi-fix inputs are present).
* gamma = γij\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

R2R^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

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   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 better!
   
   # estimation with unscaled prices
   estResultNoScale <- snqProfitEst( priceNames, quantNames, c("land","time"), 
    data = germanFarms, scalingFactors = rep( 1, 3 ) )
   print( estResultNoScale )
   
   # alternative way of estimation with unscaled prices
   estResultNoScale2 <- snqProfitEst( priceNames, quantNames, c("land","time"), 
      data = germanFarms, base = NULL, 
      weights = snqProfitWeights( priceNames, quantNames, germanFarms ) )
   all.equal( estResultNoScale[-20], estResultNoScale2[] )

   # please note that the SNQ Profit function is not invariant
   # to units of measurement so that different scaling factors 
   # result in different estimates of elasticities:
   all.equal( estResult2$ela, estResultNoScale$ela )
}

Fixed Factor Elasticities of SNQ Profit function

Description

Calculates the Fixed Factor Elasticities of a Symmetric Normalized Quadratic (SNQ) profit function.

Usage

snqProfitFixEla( delta, gamma, quant, fix, weights,
   scalingFactors = rep( 1, length( weights ) ) )

Arguments

delta

matrix of estimated δ\delta coefficients.
gamma

matrix of estimated γ\gamma coefficients.
quant

vector of netput quantities at which the elasticities should be calculated.
fix

vector of quantities of fixed factors at which the elasticities should be calculated.
weights

vector of weights of prices used for normalization.
scalingFactors

factors to scale prices (and quantities).

Note

A fixed factor elasticity is defined as

Eij=∂qiqi∂zjzj=∂qi∂zj⋅zjqiE_{ij} = \frac{ \displaystyle \frac{ \partial q_i }{ q_i } } { \displaystyle \frac{ \partial z_j }{ z_j } } = \frac{ \partial q_i }{ \partial z_j } \cdot \frac{ z_j }{ q_i }

Thus, e.g. Eij=0.5E_{ij}=0.5 means that if the quantity of fixed factor j (zjz_j) increases by 1%, the quantity of netput i (qiq_i) will increase by 0.5%.

Author(s)

Arne Henningsen

See Also

snqProfitEst and snqProfitEla.

Examples

# just a stupid simple example
snqProfitFixEla( matrix(1:6/6,3,2 ), matrix(4:1/4,2 ), c(1,1,1), c(1,1),
   c(0.4,0.3,0.3) )

# now with real data
if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput   / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time      <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )
   fixNames <- c( "land", "time" )

   estResult <- snqProfitEst( priceNames, quantNames, fixNames, data=germanFarms )

   estResult$fixEla  # price elasticities at mean quantities of netputs
                     # and fixed factors

   # fixed factor elasticities at the last observation (1994/95)
   snqProfitFixEla( estResult$coef$delta, estResult$coef$gamma,
      estResult$data[ 20, quantNames ], estResult$data[ 20, fixNames ],
      estResult$weights, estResult$scalingFactors )
}

SNQ Profit function: Hessian matrix

Description

Returns the Hessian (substitution) matrix of a Symmetric Normalized Quadratic (SNQ) Profit Function.

Usage

snqProfitHessian( beta, prices, weights,
   scalingFactors = rep( 1, length( weights ) ) )

Arguments

beta

matrix of the betabeta coefficients.
prices

vector of netput prices at which the Hessian should be calculated.
weights

vector of weights of prices for normalization.
scalingFactors

factors to scale prices (and quantities).

Author(s)

Arne Henningsen

See Also

snqProfitEst, snqProfitEla and snqProfitHessianDeriv.

Examples

# just a stupid simple example
snqProfitHessian( matrix(101:109,3,3), c(1,1,1), c(0.4,0.3,0.3) )

# now with real data
if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput   / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )

   estResult <- snqProfitEst( priceNames, quantNames, c("land","time"), data=germanFarms )

   estResult$hessian  # the Hessian at mean prices and mean quantities

   # Hessian at the last observation (1994/95)
   snqProfitHessian( estResult$coef$beta, estResult$data[ 20, priceNames ],
      estResult$weights, estResult$scalingFactors )
}

SNQ Profit function: Derivatives of the Hessian

Description

Returns the matrix of derivatives of the vector of linear independent values of the Hessian with respect to the vector of the linear independent coefficients.

Usage

snqProfitHessianDeriv( prices, weights, nFix = 0, form = 0 )

Arguments

prices

vector of netput prices at which the derivatives should be calculated.
weights

vector of weights for normalizing prices.
nFix

number of (quasi-)fix inputs.
form

the functional form to be estimated (see snqProfitEst).

Author(s)

Arne Henningsen

See Also

snqProfitHessian.

Examples

# just a stupid simple example
snqProfitHessianDeriv( c(1,2,3),c(0.4,0.3,0.3) )

# now with real data
if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput   / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )

   estResult <- snqProfitEst( priceNames, quantNames, c("land","time"), data=germanFarms )

   snqProfitHessianDeriv( estResult$pMean, estResult$weights, 2 )
}

Imposing Convexity on a SNQ Profit function

Description

Imposing Convexity on a Symmetric Normalized Quadratic (SNQ) Profit function.

Usage

snqProfitImposeConvexity( estResult, rankReduction = 0,
   start = 10, optimMethod = "BFGS", control = list( maxit=5000 ),
   stErMethod = "none", nRep = 1000, verbose = 0 )

Arguments

estResult

object returned by snqProfitEst.
rankReduction

an integer specifying the reduction of the rank of the β\beta matrix.
start

starting values of the triangular Cholesky matrix.
optimMethod

method to be used by optim.
control

list of control parameters passed to optim.
stErMethod

method to compute standard errors, either 'none', 'resample', 'jackknife' or 'coefSim' (see details).
nRep

number of replications to compute the standard errors if stErMethod is either 'resample' or 'coefSim'.
verbose

an integer idicating the verbose level.

Details

The procedure proposed by Koebel, Falk and Laisney (2000, 2003) is applied to impose convexity in prices on an estimated symmetric normalized quadratic (SNQ) profit function.
The standard errors of the restricted coefficients can be either calculated by bootstrap resampling ('resampling'), jackknife ('jacknife') or by simulating the distribution of the unrestricted coefficients using its variance covariance matrix ('coefSim').

Value

a list of class snqProfitImposeConvexity containing the same objects as an object of class snqProfitEst and additionally the objects:

mindist

object returned by optim.
sim

results of the simulation to obtain the standard errors of the estimated coefficients.

Author(s)

Arne Henningsen

References

Koebel, B., M. Falk and F. Laisney (2000), Imposing and Testing Curvature Conditions on a Box-Cox Cost Function. Discussion Paper No. 00-70, ZEW, Mannheim, https://madoc.bib.uni-mannheim.de/515/1/dp0070.pdf.

Koebel, B., M. Falk and F. Laisney (2003), Imposing and Testing Curvature Conditions on a Box-Cox Cost Function. Journal of Business and Economic Statistics, 21, p. 319-335.

See Also

snqProfitEst.

Examples

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   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  # Note: it is NOT convex in netput prices
   estResultConvex <- snqProfitImposeConvexity( estResult )
   estResultConvex  # now it is convex
}

Shadow Prices of a SNQ Profit function

Description

Calculates the shadow prices of a Symmetric Normalized Quadratic (SNQ) profit function.

Usage

snqProfitShadowPrices( priceNames, fixNames, estResult = NULL,
   data = estResult$data, weights = estResult$weights,
   scalingFactors = estResult$scalingFactors,
   coef = estResult$coef, form = estResult$form )

Arguments

priceNames

a vector of strings containing the names of netput prices.
fixNames

an optional vector of strings containing the names of the quantities of (quasi-)fix inputs.
estResult

object returned by snqProfitEst.
data

a data frame containing the data.
weights

vector of weights of prices used for normalization.
scalingFactors

factors to scale prices (see details).
coef

a list containing the coefficients (at least delta and gamma).
form

the functional form to be estimated (see details).

Author(s)

Arne Henningsen

See Also

snqProfitEst, snqProfitCalc and snqProfitEla.

Examples

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   germanFarms$qOutput   <- germanFarms$vOutput   / germanFarms$pOutput
   germanFarms$qVarInput <- -germanFarms$vVarInput / germanFarms$pVarInput
   germanFarms$qLabor    <- -germanFarms$qLabor
   germanFarms$time      <- c( 0:19 )
   priceNames <- c( "pOutput", "pVarInput", "pLabor" )
   quantNames <- c( "qOutput", "qVarInput", "qLabor" )
   fixNames <- c( "land", "time" )

   estResult <- snqProfitEst( priceNames, quantNames, fixNames, data = germanFarms )

   snqProfitShadowPrices( priceNames, fixNames, estResult )
}

SNQ Profit function: Weights of prices for normalization

Description

Returns a vector of weights to normalize prices on a Symmetric Normalized Quadratic (SNQ) Profit function.

Usage

snqProfitWeights( priceNames, quantNames, data, method = "DW92", base = 1 )

Arguments

priceNames

a vector of strings containing the names of netput prices.
quantNames

a vector of strings containing the names of netput quantities.
data

a data frame containing the data.
method

the method to determine the weights (see details).
base

the base period(s) for scaling prices (see details).

Details

If argument method is 'DW92' the method of Diewert and Wales (1992) is applied. They predetermine the weights by

θi=∣x‾i∣pi0∑i=1n∣x‾i∣pi0\theta_{i} = \frac{ \displaystyle \left | \overline{x}_{i} \right| p_{i}^{0} }{ \displaystyle \sum_{i=1}^{n} \left| \overline{x}_{i} \right| p_{i}^{0}}

Defining the scaled netput quantities as x~it=xit⋅pi0\widetilde{x}_{i}^{t} = x_{i}^{t}\cdot p_{i}^{0} we get following formula:

θi=∣x~‾i∣∑i=1n∣x~‾i∣\theta_{i} = \frac{ \displaystyle \left| \overline{ \widetilde{ x } }_{i} \right|}{ \displaystyle \sum_{i=1}^{n} \left| \overline{ \widetilde{ x } }_{i} \right|}

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.

Author(s)

Arne Henningsen

See Also

snqProfitEst.

Examples

if( requireNamespace( 'micEcon', quietly = TRUE ) ) {
   data( germanFarms, package = "micEcon" )
   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" )
   snqProfitWeights( priceNames, quantNames, germanFarms )
}