allocative.efficiency: Allocative efficiency estimation

In snfa: Smooth Non-Parametric Frontier Analysis

Description

Fits frontier to data and estimates technical and allocative efficiency

Usage

allocative.efficiency(X, y, X.price, y.price, X.constrained = NA,
  H.inv = NA, H.mult = 1, model = "br", method = "u",
  scale.constraints = TRUE)

Arguments

X	Matrix of inputs
y	Vector of outputs
X.price	Matrix of input prices
y.price	Vector of output prices
X.constrained	Matrix of inputs where constraints apply
H.inv	Inverse of the smoothing matrix (must be positive definite); defaults to rule of thumb
H.mult	Scaling factor for rule of thumb smoothing matrix
model	Type of frontier to use; "br" for boundary regression, "sf" for stochastic frontier
method	Constraints to apply; "u" for unconstrained, "m" for monotonically increasing, and "mc" for monotonically increasing and concave
scale.constraints	Boolean, whether to scale constraints by their average value, can help with convergence

Details

This function estimates allocative inefficiency using the methodology in McKenzie (2018). The estimation process is a non-parametric analogue of Schmidt and Lovell (1979). First, the frontier is fit using either a boundary regression or stochastic frontier as in Racine et al. (2009), from which technical efficiency is estimated. Then, gradients and price ratios are computed for each observation and compared to determine the extent of misallocation. Specifically, log-overallocation is computed as

\log≤ft(\frac{w_i^j}{p_i}\right) - \log≤ft(φ_i\frac{\partial f(x_i)}{\partial x^j}\right),

where φ_i is the efficiency of observation i, \partial f(x_i) / \partial x^j is the marginal productivity of input j at observation i, w_i^j is the cost of input j for observation i, and p_i is the price of output for observation i.

Value

Returns a list with the following elements

y.fit	Estimated value of the frontier at X.fit
gradient.fit	Estimated gradient of the frontier at X.fit
technical.efficiency	Estimated technical efficiency
log.overallocation	Estimated log-overallocation of each input for each observation
X.eval	Matrix of inputs used for fitting
X.constrained	Matrix of inputs where constraints apply
H.inv	Inverse smoothing matrix used in fitting
method	Method used to fit frontier
scaling.factor	Factor by which constraints are multiplied before quadratic programming

References

\insertRef

AignerLovellSchmidtsnfa

\insertRefMcKenziesnfa

\insertRefParmeterRacinesnfa

\insertRefSchmidtLovellsnfa

Examples

data(USMacro)

USMacro <- USMacro[complete.cases(USMacro),]

#Extract data
X <- as.matrix(USMacro[,c("K", "L")])
y <- USMacro$Y

X.price <- as.matrix(USMacro[,c("K.price", "L.price")])
y.price <- rep(1e9, nrow(USMacro)) #Price of $1 billion of output is $1 billion

#Run model
efficiency.model <- allocative.efficiency(X, y,
                                         X.price, y.price,
                                         X.constrained = X,
                                         model = "br",
                                         method = "mc")

#Plot technical/allocative efficiency over time
library(ggplot2)

technical.df <- data.frame(Year = USMacro$Year,
                          Efficiency = efficiency.model$technical.efficiency)

ggplot(technical.df, aes(Year, Efficiency)) +
  geom_line()

allocative.df <- data.frame(Year = rep(USMacro$Year, times = 2),
                            log.overallocation = c(efficiency.model$log.overallocation[,1],
                                                   efficiency.model$log.overallocation[,2]),
                            Variable = rep(c("K", "L"), each = nrow(USMacro)))

ggplot(allocative.df, aes(Year, log.overallocation)) +
  geom_line(aes(color = Variable))

#Estimate average overallocation across sample period
lm.model <- lm(log.overallocation ~ 0 + Variable, allocative.df)
summary(lm.model)
  

snfa documentation built on May 1, 2019, 9:16 p.m.