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\begin{document}

\author{Philip A. Viton}
\title{A Test of HTML Production in SWP4}
\date{\today{}}
\maketitle
\tableofcontents

\section{Econometric Approaches \label{econometric}}

Econometric approaches to cost efficiency estimate a parametric cost
function and then identify cost efficiency via differences between observed
cost and the minimum predicted cost. \ (A second approach is presented in
section \ref{dea}) Consider carrier $k$ in year $t$. It has a $V$-vector $%
\mathbf{z}_{kt}=(z_{kt1},\dots ,z_{ktV})$ of system characteristics and
produces the $M$-vector of outputs $\mathbf{y}_{kt}=(y_{kt1},\dots ,y_{ktM})$
from the $N$-vector of inputs $\mathbf{x}_{kt}=(x_{kt1},\dots ,x_{ktN})$
according to the transformation function $f(\mathbf{y}_{kt},\mathbf{x}_{kt};%
\mathbf{z}_{kt})=0.$ It is assumed to face the $N$ factor prices $\mathbf{w}%
_{kt}=(w_{kt1},\dots ,w_{ktN}).$ \ Its (minimum) cost function $C^{\ast }(%
\mathbf{y}_{kt},\mathbf{w}_{kt};\mathbf{z}_{kt})$ is the indirect objective
function of the problem: choose inputs $\mathbf{x}_{kt}$ to minimize cost
subject to $f$ (the production function), ie: 
\[
\begin{array}{cc}
\text{\textrm{minimize}} & \sum_{n}w_{ktn}x_{ktn} \\ 
\text{\textrm{subject to:}} & f(\mathbf{y}_{kt},\mathbf{x}_{kt};\mathbf{z}%
_{kt})=0.%
\end{array}%
\]%
If $C_{kt}$ is the \emph{observed} cost of carrier $k$ in year $t$, we allow
for a failure to attain the minimum cost by writing 
\begin{equation}
C_{kt}=C^{\ast }(\mathbf{y}_{kt},\mathbf{w}_{kt};\mathbf{z}%
_{kt})e^{\varepsilon _{kt}}  \label{costfn}
\end{equation}%
where $\varepsilon _{kt}$ is a random variable, so that 
\begin{equation}
\ln C_{kt}=\ln C^{\ast }(\mathbf{y}_{kt},\mathbf{w}_{kt};\mathbf{z}%
_{kt})+\varepsilon _{kt}  \label{linear}
\end{equation}

Because the observed cost $C_{kt}$ cannot exceed the theoretical minimum
cost $C_{kt}^{\ast },$ $\varepsilon _{kt}$ is necessarily non-negative. \ We
obtain a parametric cost function by specifying a functional form for $%
C^{\ast }(\mathbf{y}_{kt},\mathbf{w}_{kt};\mathbf{z}_{kt}).$ \ For the
remainder of this section, we shall assume that the cost function is
translog in inputs and outputs; but that system characteristics enter
linearly.\footnote{%
Cobb-Douglas models, which are nested sub-models of the translog, were also
estimated. \ In all cases they were rejected in favor of the translog.} \
Then we may write equation (\ref{linear}) as: 
\begin{eqnarray}
\ln C^{\ast }(\mathbf{y}_{kt},\mathbf{x}_{kt};\mathbf{z}_{kt}) &=&\alpha
_{0}+\sum_{i=1}^{V}\alpha _{i}z_{kti}+\sum_{n=1}^{N}\beta _{n}\ln
y_{ktn}+\sum_{m=1}^{M}\gamma _{m}\ln w_{ktm}  \label{translog} \\
&&+\frac{1}{2}\sum_{n=1}^{N}\sum_{p=1}^{N}\delta _{np}\ln y_{ktn}\ln y_{ktp}+%
\frac{1}{2}\sum_{m=1}^{M}\,\sum_{j=1}^{M}\theta _{mj}\ln w_{ktm}\ln w_{ktj} 
\nonumber \\
&&+\frac{1}{2}\sum_{n=1}^{N}\sum_{m=1}^{M}\xi _{nm}\ln w_{ktn}\ln w_{ktm} 
\nonumber
\end{eqnarray}%
The theory of cost minimization implies that $C^{\ast }(\mathbf{y}_{kt},%
\mathbf{w}_{kt};\mathbf{z}_{kt})$ is homogeneous of degree zero in input
prices, which places restrictions on the parameters of the cost function
(see, eg, \cite{spady-friedlaender:78}); in addition the parameters $\delta
,\theta $ and $\xi $ are symmetric, so that, for example $\delta
_{np}=\delta _{pn}.$

\section{The DEA Approach \label{dea}}

Data Envelopment Analysis, or DEA, see, eg \cite{bcc:84}, or \cite{fgl:94},
begins by constructing the set of feasible input-output combinations. \
(Another approach was presented in section \ref{econometric}) It then finds,
with reference to that set, the input bundle that minimizes the cost of
producing the observed output. DEA is non-parametric in that it makes no
assumption about the structure of the cost function ---\ indeed it does not
explicitly obtain the cost function at all ---\ and, being non-statistical,
it has no need for parametric distributional assumptions.\footnote{%
This is not to say that DEA cannot be given a statistical foundation: for
example, \cite{banker:93} shows that the DEA solution can be considered the
maximum-likelihood estimator of a non-parametric monotone increasing and
concave production frontier.} To construct the feasible set it begins by
assuming that the observed input-output combinations (the data) are feasible
and then adds unobserved combinations by considerations of additivity
(scale) and disposal.\footnote{%
Feasibility of the observed data is not a completely innocuous assumption.
If, for example, there are measurement errors, then the observed data may
not in fact be feasible.}

The upshot is that cost-minimizing carrier selects an input bundle $\mathbf{%
\tilde{x}}_{kt}$ to minimize total cost, $\sum_{n}\tilde{x}_{ktn}w_{ktn}=%
\mathbf{\tilde{x}}_{kt}^{\prime }\mathbf{w}_{kt}$ subject to feasibility;
and this is the solution to the non-linear program 
\begin{eqnarray*}
\text{\textrm{minimize} } &&\sum\nolimits_{n}\tilde{x}_{ktn}w_{ktn} \\
\text{subject to} &\text{:}& \\
\text{\textrm{\ }}\mathbf{y}_{ktm} &=&\mu \sum\nolimits_{l,s}r_{ls}\mathbf{y}%
_{lsm}\quad \forall m=1,2,\dots ,M \\
\lambda \mathbf{\tilde{x}}_{ktn} &=&\sum\nolimits_{l,st}r_{ls}\mathbf{x}%
_{lsn}\quad \forall n=1,2,\dots ,N \\
\sum\nolimits_{l,s}r_{ls} &=&1 \\
0 &<&\lambda \leq 1 \\
0 &<&\mu \leq 1
\end{eqnarray*}%
and we define the cost-efficiency of carrier $k$ at period $t$ by 
\[
\mathrm{CE}_{kt}=\frac{\mathbf{w}_{kt}^{\prime }\mathbf{\tilde{x}}_{kt}}{%
\mathbf{w}_{kt}^{\prime }\mathbf{x}_{kt}} 
\]%
the ratio of observed to minimal cost. \ 

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