allocative.efficiency {snfa}

Allocative efficiency estimation

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\left(\frac{w_i^j}{p_i}\right) - \log\left(\phi_i\frac{\partial f(x_i)}{\partial x^j}\right),

where \phi_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

Aigner D, Lovell CK, Schmidt P (1977). “Formulation and estimation of stochastic frontier production function models.” Journal of econometrics, 6(1), 21–37.

McKenzie T (2018). “Semi-Parametric Estimation of Allocative Inefficiency Using Smooth Non-Parametric Frontier Analysis.” Working Paper.

Racine JS, Parmeter CF, Du P (2009). “Constrained nonparametric kernel regression: Estimation and inference.” Working paper.

Schmidt P, Lovell CK (1979). “Estimating technical and allocative inefficiency relative to stochastic production and cost frontiers.” Journal of econometrics, 9(3), 343–366.

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)