Cost Reverse Directional Distance Function model

The cost reverse directional distance function model is computed by solving an input-oriented Reverse DDF model for the technical inefficiency.

As their graph counterpart presented in the Profit Reverse Directional Distance Function model, the input-oriented $RDDF$ transforms any additive measure of input technical inefficiency, $EM(I)$, such as the Russell input measure, into a single scalar measure corresponding to an input-oriented $DDF$, $\beta_I$. Therefore, given the set of $J$ firms under study, ${F}_{J}$, and their projections on the frontier, denoted by ${\hat{F}_{J}}$, the $RDDF$ assigns a new $DDF$ score to the original $EM(I)$, while keeping the same projections ${\hat{F}_{J}}$. We denote this score by $\beta_{RDDF(EM(I), F_J, \hat{F}_{J})}$. The advantage of the input-oriented RDDF is that it relates the additive and multiplicative measures of technical inefficiency because the input-oriented DDFs is equivalent to Farrell's input radial measure shown in the Cost Radial model, i.e., $\beta_I^{*}=1-\theta^{*}$.

To calculate this distance function for firm $\left( {{\mathbf{x}_o,\mathbf{y}_{o}}} \right)$ we need to determine the directional vector $\mathbf{g}= ({{\mathbf{g_{x}^-},\textbf{0}_M}})$ connecting the firm to its input-oriented projection, $\left( {{{\hat{\textbf{x}}}}_{o}},{{\textbf{y}}_{o}} \right)\in {{\hat{F}}_{J}}$. Afterwards we calculate the value of the $RDDF(I)$. However, when calculating the new scores we need to differentiate between firms that are deemed technically efficient with $EM(I)=0$ and those that are technically inefficient. The measure $EM(I)$ splits the sample of firms ${{F}_{J}}$ into two disjoint subsets: ${{F}_{E}}=\left\{ \left( {{x}_{j}},{{y}_{j}} \right)\in {{F}_{J}}:T{{I}_{E{{M(I)}}}}\left( {{x}_{j}},{{y}_{j}} \right)=0 \right\}$ and ${{F}_{J\sim E}}=\left\{ \left( {{x}_{j}},{{y}_{j}} \right)\in {{F}_{J}}:T{{I}_{E{{M(I)}}}}\left( {{x}_{j}},{{y}_{j}} \right)>0 \right\}$. Then, for the input orientation we define the directional vector joining the firm under evaluation and its projection as follows

• If $\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)\in {{F}_{J\sim E}}$, define

\[\left({\textbf{g}^{-}_{\textbf{x}_{j}}},{\textbf{g}^{+}_{\textbf{y}_{j}}} \right)=\left( \frac{{{{\hat{\textbf{x}}}}_{j}} -{{\textbf{x}}_{j}}}{T{{I}_{EM\left( I \right)}}\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)},\textbf{0}_M \right) \, \, and \, \,\beta_{RDDF(I)}^{*}=T{{I}_{E{{M}(I)}}}\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)>0.\]

• If $\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)\in {{F}_{E}}$, define

\[\beta _{RDDF(I)}^{*}=T{{I}_{EM\left( I \right)}}\left( {{\textbf{x}}_{j}},{{\textbf{y}^+}_{j}} \right)=0 \, \, and \, \,\left( {\textbf{g}^{-}_{\textbf{x}_{j}}}, \textbf{0}_M \right)= \left({{{\vec{\textbf{k}}}}_{jM}},\textbf{0}_N \right)\,\in \mathbb{R}_{++}^{M+N},\]

where ${{\vec{\textbf{k}}}_{jM}}\in \mathbb{R}_{++}^{M}$ is a vector whose units of measurement are identical to those of the firm under evaluation $\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)\in {{F}_{E}}$–making cost inefficiency units' invariant. For consistency we search for a value that yields a normalization factor whose value is equal to that associated to the underlying efficiency measure, i.e, $\sum\limits_{m=1}^{M}{w_{m}^{{}}k_{jm}^{-}}= NF_{EM(I)}$. This choice of ${{\vec{\textbf{k}}}_{jM}}$ is numerically relevant because it makes the values of normalized cost inefficiency based on the $RDDF$ and the original $EM(I)$ equivalent, and therefore their normalized allocative efficiencies can be compared to each other–their technical inefficiency values being null.

We now present the expression corresponding to the cost inefficiency measure and its decomposition associated with the $RDDF$. This results in $C{{I}_{{RDDF\left( EM\left( I \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{x}^-},{\tilde{\textbf{w}}} \right)$ = $T{{I}_{{RDDF\left( EM\left( I \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o},\mathbf{g_{x}^-}} \right)$ + $A{{I}_{RDDF\left( EM\left( I \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{x}^-}, {\tilde{\textbf{w}}} \right)$. Hence we obtain the following expressions:

\[\underbrace{\frac{\sum\limits_{m=1}^{M}{{{w}_{m}}{{x}_{om}}}-C\left( \textbf{w},{{\textbf{y}}_{o}} \right)}{\sum\limits_{m=1}^{M}{{{w}_{m}}{{g}^{-}_{om}}}}}_{\text{Norm. Cost Inefficiency}}=\underbrace{\beta _{RDDF\left( EM\left( I \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}^{*}}_{\text{Input Technical Inefficiency}}+\underbrace{A{{I}_{RDDF\left( EM\left( I \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\textbf{g}_{{{\textbf{x}}_{o}}}^{{-}}, \tilde{\textbf{w}} \right)}}_{\text{Norm. Allocative Inefficiency}}\ge 0,\]

where the efficiency score $\beta _{RDDF\left( EM\left( I \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}^{*}$ for technically inefficient firms is obtained by solving the Input-oriented Directional Distance Function model with the corresponding directional vector.

BenchmarkingEconomicEfficiency.jl offers the possibility of decomposing cost inefficiency based on the $RDDF$ considering as original measure the Russell input-oriented model.

Reference

Chapter 12 in Pastor, J.T., Aparicio, J. and Zofío, J.L. (2022) Benchmarking Economic Efficiency: Technical and Allocative Fundamentals, International Series in Operations Research and Management Science, Vol. 315, Springer, Cham.

Example

In this example we compute the cost efficiency Reverse directional distance function measure for the Russell technical inefficiency:

using BenchmarkingEconomicEfficiency

X = [2 2; 1 4; 4 1; 4 3; 5 5; 6 1; 2 5; 1.6 8];

Y = [1; 1; 1; 1; 1; 1; 1; 1];

W = [1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1];

costrddf = deacostrddf(X, Y, W, :ERG)

Cost Reverse DDF DEA Model 
DMUs = 8; Inputs = 2; Outputs = 1
Returns to Scale: VRS
Associated efficiency measure = ERG
──────────────────────────────────
       Cost  Technical  Allocative
──────────────────────────────────
1  0.0        0.0        0.0
2  0.5        0.0        0.5
3  0.5        0.0        0.5
4  0.416667   0.416667   0.0
5  0.6        0.6        0.0
6  0.25       0.166667   0.0833333
7  0.525      0.35       0.175
8  0.532609   0.4375     0.0951087
──────────────────────────────────

Estimated economic, technical and allocative efficiency scores are returned with the efficiency function:

efficiency(costrddf, :Economic)

8-element Vector{Float64}:
 0.0
 0.5
 0.5
 0.41666666666666663
 0.6
 0.24999999999999992
 0.525
 0.5326086956521738

efficiency(costrddf, :Technical)

8-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.41666666666666663
 0.6
 0.16666666666666663
 0.35
 0.43750000000000017

efficiency(costrddf, :Allocative)

8-element Vector{Float64}:
 0.0
 0.5
 0.5
 0.0
 0.0
 0.08333333333333329
 0.17500000000000004
 0.09510869565217367

deacostrddf Function Documentation

deacostrddf(X, Y, W, measure)