Evaluating eco-efficiency with data envelopment analysis: an analytical reexamination

Abstract

This paper reexamines the unintended consequences of the two widely cited models for measuring environmental efficiency—the hyperbolic efficiency model (HEM) and directional distance function (DDF). I prove the existence of three main problems: (1) these two models are not monotonic in undesirable outputs (i.e., a firm’s efficiency may increase when polluting more, and vice versa), (2) strongly dominated firms may appear efficient, and (3) some firms’ environmental efficiency scores may be computed against strongly dominated points. Using the supply-chain carbon emissions data from the 50 major U.S. manufacturing companies, I empirically compare these two models with a weighted additive DEA model. The empirical results corroborate the analytical findings that the DDF and HEM models can generate spurious efficiency estimates and must be used with extreme caution.

Appendices

Appendix A: Proofs of theorems

Proof of Theorem 2

For the proof, it suffices to show that a DMU originally deemed inefficient can become efficient with a sufficient increase in any b _k . We first show the proof for the non-monotonicity of the DDF model. Consider $(X_{j},Y_{j},B_{j})\in\Re_{+}^{m+s+p}$ for j=1,…,n. Without loss of generality, we consider DMU n and suppose $\beta ^{*}_{n}=\mathit{DDF}(X_{n},Y_{n},B_{n}|g^{Y},g^{B})>0$ for a directional vector $(g^{Y},g^{B})\in \Re^{s+p}_{++}$. We will show that DMU n can become efficient in the DDF model after a sufficient increase in one of its undesirable output b _np . By construction in optimality it must hold that $\lambda_{1}^{*} B_{1}+ \lambda_{2}^{*}B_{2}+\cdots+ \lambda_{n}^{*} B_{n}=B_{n}-\beta^{*}_{n}g^{B}$, where $\lambda_{1}^{*},\lambda_{2}^{*},\ldots,\lambda_{n}^{*},\beta^{*}$ are the optimal solution to (3); namely, $B_{n}-\beta^{*}_{n}g^{B}$ is in the convex cone generated by vectors B ₁ to B _n . To formalize the feasible region for (2), define the convex cones generated by the B vector of all except for DMU n’s as C _∖n ={B|λ ₁ B ₁+⋯+λ _n−1 B _n−1,λ _i ≥0,i=1,…,n−1} and the one including B _n as C ^∗(B _n )={B|λ ₁ B ₁+⋯+λ _n−1 B _n−1+λ _n B _n ,λ _i ≥0,i=1,…,n}.

Suppose DMU n now increases its b _np to $b_{np}^{*}$ (so $b_{np}^{*}\gg b_{np}$) until the new output vector $B_{n}^{*}$ satisfies: (i) $B_{n}^{*}\notin C_{\setminus n}$, and (ii) $(B_{n}^{*}-\epsilon g^{B})\notin C^{*}(B_{n}^{*})$ for all ϵ>0 such that $(B_{n}^{*}-\epsilon g^{B})\geqq 0$. There must exist a $b_{np}^{*}$ satisfying condition (i) because B _j is strictly positive for all j and therefore $C_{\setminus n}\subset\Re_{++}^{p}$. We next show that there must also exist a $b_{np}^{*}$ satisfying condition (ii). Suppose $(B_{n}^{*}-\epsilon g^{B})\in C^{*}(B_{n}^{*})$. Then according to Farkas’ Lemma, condition (ii) corresponds to the following condition (iii): there does not exist a vector d∈ℜ^p such that $\{d'B_{j}\ge0\ \mathrm{for\ all}\ j\neq n\}$ ∧ $\{d'B_{n}^{*}\ge0\}$ ∧ $\{d'(B_{n}^{*}-\epsilon g^{B})<0\}$. However, we can further increase $b_{np}^{*}$ to $b_{np}^{**}$ (so $b_{np}^{**}\gg b_{np}^{*}\gg b_{np}$) and construct a vector d that is component-wise positive except for the pth component (i.e., d _p <0), where $d_{p}\ge-(\sum_{k=1}^{p-1} b_{jk})/b_{jp}$ for j=1,…,n−1, $d_{p}\ge-(\sum_{k=1}^{p-1} b_{nk}^{*})/b_{np}$, and $d_{p}< -(\sum_{k=1}^{p-1}b_{nk}-\epsilon g^{B}_{k})/(b_{np}^{**}-\epsilon g^{B}_{p})$. So we can always find such a d and $b_{np}^{**}$ that meet condition (iii). Denote $B_{n}^{**}$ as B _n with the pth component replaced by $b_{np}^{**}$. By Farkas’ Lemma, it then holds that $(B_{n}^{**}-\epsilon g^{b})\notin C^{*}(B^{**}_{n})$ and hence condition (ii) is satisfied.

Given that $B^{**}_{n}$ satisfies conditions (i) and (ii), the only feasible (and optimal) solution to (3) is $\mathit{DDF}(X_{n},Y_{n},B_{n}^{**}|g^{Y},g^{B})=\beta^{**}_{n}=0$. So DMU n is considered efficient since $\beta^{**}_{n}=0$. Thus DDF is non-monotonic in undesirable outputs, given that β ^∗>0. The proof for HEM can be analogously constructed by replacing g ^B with a common radial contraction factor θ for undesirable outputs and therefore its proof is omitted. □

Proof of Theorem 3

Part (i): Denote the optimal solution to DDF′ as β ^∗>0 and $\lambda_{j}^{*}$, j=1,…,n, and optimal slack variables as $(S^{Y},S^{B})=(s_{1}^{y},\ldots,s_{s}^{y},s_{1}^{b},\ldots,s_{p}^{b})\in\Re^{s}_{+}\times\Re ^{p}_{+}$. Then by Definition 1, the optimal solution to DDF′ must satisfy: $\sum_{j=1}^{n}\lambda_{j}^{*} x_{ji} \le x_{qi}$ for i=1,…,m; $\sum_{j=1}^{n}\lambda_{j}^{*} y_{jr} = y_{qr}+\beta ^{*}g_{r}^{y}+s_{r}^{y}$ for r=1,…,s; and $\sum_{j=1}^{n}\lambda_{j}^{*} b_{jk} = b_{qk}-\beta^{*}g_{k}^{b}-s_{k}^{b}$ for k=1,…,p.

Now let $(\tilde{g}^{Y},\tilde{g}^{B})=(\beta^{*}g^{Y}+S^{Y},\beta^{*}g^{B}+S^{B})$ be the new directional vector of DDF. It must hold that $\mathit{DDF} ((X_{q},Y_{q},B_{q}) |\tilde{g}^{Y},\tilde{g}^{B} )=1$, which is a sufficient condition that DMU q is not output efficient. In addition, the original optimal value DDF((X _q ,Y _q ,B _q )|g ^Y,g ^B)=0 implies that DMU q is not in the relative interior of f(X) defined in (2). Hence DMU $q\in\widetilde{E}$.

Part (ii): DDF((X _q ,Y _q ,B _q )|g ^Y,g ^B)>0 indicates that DMU q is in the relative interior of f(X _q ). This then excludes the possibility that DMU q∈E or DMU $q\in\widetilde{E}$. In addition, DDF((X _q ,Y _q ,B _q )|g ^Y,g ^B)≠DDF _f ((X _q ,Y _q ,B _q )|g ^Y,g ^B) suggests that the projection points for DMU q under DDF and DDF _f are different. Therefore, from part (i) of the theorem, we can conclude that the projection points of DMU q under DDF include DMUs in $\widetilde{E}$. Therefore DMU $q\in\widetilde{NE}$. □

Proof of Theorem 5

The proof is basically constructed by showing that, for an inefficient DMU, we can always find a directional vector that leads to a projection point that dominates that of the HEM. Define $g:\Re_{++}\rightarrow\Re ^{s+p}_{++}$, where g(γ)=(γY,1/γB) to be a function in the HEM model that projects (B,Y) to the boundary of f _w (X) in (2). To begin, we first show that the function g is convex with respect to the componentwise inequality “≽” in $\Re^{s+p}_{+}$ (i.e., the generalized convexity induced by $\Re^{s+p}_{+}$). We proceed by comparing the following two functions for λ∈[0,1] and γ ₁,γ ₂≥1: ${ \lambda g(\gamma_{1})+(1-\lambda)g(\gamma_{2})}=\lambda(\gamma_{1} Y,\frac {1}{\gamma_{1}} B)+ (1-\lambda) (\gamma_{2} Y,\frac{1}{\gamma_{2}} B)= ((\lambda\gamma_{1} + (1-\lambda)\gamma_{2})Y,(\frac{\lambda}{\gamma_{1}} + \frac{1-\lambda}{\gamma_{2}}) B )$, and $g(\lambda\gamma_{1}+(1-\lambda )\gamma_{2})= ((\lambda\gamma_{1}+(1-\lambda)\gamma_{2})Y,\frac{1}{\lambda \gamma_{1}+(1-\lambda)\gamma_{2}}B )$.

From the first equation, we obtain $(\frac{\lambda}{\gamma_{1}} + \frac {1-\lambda}{\gamma_{2}})B= \frac{\lambda\gamma_{2}+(1-\lambda)\gamma _{1}}{\gamma_{1}\gamma_{2}} B$. To prove g is convex with respect to “≽”, we thus need to show ${ \frac{\lambda\gamma _{2}+(1-\lambda)\gamma_{1}}{\gamma_{1}\gamma_{2}} \ge\frac{1}{\lambda(\gamma _{1})+(1-\lambda)\gamma_{2}}}$, or equivalently $\frac{(\lambda\gamma _{2}+(1-\lambda)\gamma_{1})({\lambda(\gamma_{1})+(1-\lambda)\gamma_{2}})}{\gamma _{1}\gamma_{2}} \ge1$. Let $\zeta=\frac{(\lambda\gamma_{2}+(1-\lambda)\gamma _{1})({\lambda(\gamma_{1})+(1-\lambda)\gamma_{2}})}{\gamma_{1}\gamma_{2}}$. Observe that the two points (λγ ₂+(1−λ)γ ₁) and (λγ ₁+(1−λ)γ ₂) are symmetric with respect to $\frac{1}{2}(\min\{\gamma_{1},\gamma_{2}\}+\max\{\gamma_{1},\gamma_{2}\})$ for all λ in [0,1]. Thus we can express ζ as:

$$\begin{aligned} \zeta&=\frac{ (\frac{1}{2}(\min\{\gamma_1,\gamma_2\}+\max\{\gamma _1,\gamma_2\})+\delta ) (\frac{1}{2}(\min\{\gamma_1,\gamma_2\} +\max\{\gamma_1,\gamma_2\})-\delta )}{\max\{\gamma_1,\gamma_2\}\min\{ \gamma_1,\gamma_2\}} \\ &=\frac{\frac{1}{4}(\min\{\gamma_1,\gamma_2\}+\max\{\gamma_1,\gamma_2\} )^2-\delta^2}{\max\{\gamma_1,\gamma_2\}\min\{\gamma_1,\gamma_2\}}, \end{aligned}$$

(A.1)

where $\delta\in[0, \frac{1}{2}(\max\{\gamma_{1},\gamma_{2}\}-\min\{\gamma _{1},\gamma_{2}\})]$ is a scalar contingent on λ; also observe that γ ₁ γ ₂=max{γ ₁,γ ₂}min{γ ₁,γ ₂}. Now if we set δ to its upper bound $\frac{1}{2}(\max\{\gamma _{1},\gamma_{2}\}-\min\{\gamma_{1},\gamma_{2}\})$, we would obtain ζ=1. Since ζ is strictly decreasing in d, the preceding result implies that ζ≥1 as intended.

We have shown that g is convex because λg(γ ₁)+(1−λ)g(γ ₂)≽g(λγ ₁+(1−λ)γ ₂) for λ in [0,1], γ ₁≥1, and γ ₂≥1. Because g is positive, convex and continuously differentiable for γ≥1, then for (Y _q ,B _q ), it must hold that g(γ ₁)≽(Y _q ,B _q )+∇g(1)′(γ ₁−1), where ∇g(γ) is the tangent vector of g defined as $[\frac{\partial y_{1}}{\partial\gamma },\ldots,\frac{\partial y_{s}}{\partial\gamma},\frac{\partial b_{1}}{\partial \gamma},\ldots,\frac{\partial b_{p}}{\partial\gamma}]$. Setting (g ^Y,g ^B)=∇g(1)′(γ ₁−1) as the directional vector for (Y _q ,B _q ), by the strong disposability for Y _q and convexity, it hold that $(Y_{q}^{\star},B_{q}^{\star})=(Y_{q},B_{q}) +\theta\nabla g(1)' (\gamma _{1}-1)\in f_{w}(X_{q})$ if $g(\gamma')=(Y_{q}',B_{q}')$ is feasible, where $Y_{q}'=Y_{q}^{\star}$. Furthermore, $(Y_{q}',B_{q}')$ is dominated by $(Y_{q}^{\star},B_{q}^{\star})$ in B. Note that $\widetilde{E}$ and $\widetilde{NE}$ can be non-empty given (g ^Y,g ^B) by Theorem 4. By the above dominance relationship just stated, $\widetilde{E}$ and $\widetilde{NE}$ can also be non-empty under g. □

Appendix B: A modified RAM model for environmental efficiency

One important issue for implementing the weighted additive model is that we must specify weights. This is particular a problem as DEA models are known as a weight-free approach and do not require subjective weight assignments. Chen and Delmas (2013) use the DMU’s own outputs to normalize the output improvements and then calculate environmental efficiency as the average normalized score. This approach has a potential limitation in that different DMUs would be based its own production but miss information about distributions of different outputs across the entire sample, which may carry significant practical implications. Some studies assign weights based on the sample statistics, such as the range adjusted measure (RAM) model proposed by Cooper et al. (1999):

$$\begin{aligned} \begin{aligned} & \mathrm{max}\quad \varGamma=\frac{1}{s+p} \Biggl(\sum _{r=1}^ss^{+}_r/R^+_{r}+ \sum_{k=1}^ps^{-}_k/R^-_{p} \Biggr) \\ &\mathrm{s.t.}\ \ \quad \sum_{j=1}^n \lambda_j x_{ji} \le x_{qi},\quad i=1,\ldots,m \\ &\phantom{\max\quad } \sum_{j=1}^n \lambda_j y_{jr} = y_{qr}^* + s^{+}_r, \quad r=1, \ldots,s \\ &\phantom{\max\quad } \sum_{j=1}^n \lambda_j b_{jk} = b_{qk}^* - s^{-}_k,\quad k=1, \ldots,p \\ &\phantom{\max\quad } \lambda_j \ge0, \quad j=1,\ldots,n \\ &\phantom{\max\quad } s^{+}_r, s^{-}_k\ge0, \quad r=1, \ldots,s; \ k=1,\ldots,p, \end{aligned} \end{aligned}$$

(A.2)

where $R^{+}_{r}$ is the range of the rth desirable output and $R^{-}_{p}$ is the range of the pth undesirable output. Note that the RAM model can also incorporate slacks variables for inputs. For the purpose of the current paper, we focus on the output-oriented RAM model. For the economic intuition behind the RAM model, see Cooper et al. (1999) for an excellent exposition of the rationale behind the additive efficiency model and its use to measure allocative, technical, and overall inefficiencies.

We propose a model based on the concept from the RAM model, as Cooper et al. (1999) point out that the RAM-type of efficiency models come with a number of desirable properties, including (i) the efficiency score is bounded in [0,1], (ii) the model is unit invariant, (iii) the model is strongly monotonic in slacks, and (iv) the model is translation invariant under the variable returns-to-scale technological assumption (Banker et al. 1984). However, we find using ranges as the normalizing factors problematic, and choose to use other normalizing variables instead of ranges in the original model. For example, it is stated in Cooper et al. (1999) that 0≤Γ≤1, where a zero value indicates efficiency and a value of one indicates full efficiency. As the slacks are usually much lower in magnitude than their corresponding ranges, the efficiency scores obtained from the original RAM model tends to be low in both magnitude and variation (Cooper et al. 1999; Steinmann and Zweifel 2001). Therefore the RAM scores cannot effectively differentiate the performance of different DMUs. Furthermore, if we observe extremely inefficient firms that makes certain $R^{+}_{r}$ and/or $R^{-}_{p}$ larger. These extremely inefficient firms may be those that produce lower than minimal observed desirable outputs but higher than maximum observed undesirable outputs at a fixed input level. The efficiency scores of all the other firms may decrease markedly, and most firms would appear more efficient although the efficient frontier remains unaltered. As it is not uncommon to observe “heavy polluters” in applications, using ranges or other dispersion measures of outputs do not seem appropriate. Also note that if a weighted additive model is used, the disposability assumption on undesirable outputs will not have any impact on the resultant efficiency scores (e.g., Theorem 1).

Another problem of using ranges is that ranges cannot reveal the relative magnitude of the output. For example, suppose we obtain for a particular DMU that its slack for an output is 5 and the corresponding range for that output is 50. The managerial implication of this output slack for this DMU may be quite different if the maximum and minimum of the output are respectively 10 and 60 rather than 500 and 550, for example. As the main purpose of the normalizing factors are to obtain unit invariance, we opt for using the median of outputs to replace the range used in the objective function of (A.2), which is more robust than ranges or averages as the basic statistical properties of these measures. We call our efficiency measure based on median the “Median Adjusted Measure” (MAM). The MAM score then has an intuitive interpretation as the average of slacks compared to the sample median of the corresponding output variables. Note that one may designate the normalizing parameters in the original range adjusted model in other ways; see, e.g., Cooper et al. (2011) for a comprehensive discussion.

Table 6 Environmental efficiency scores and classifications for the major 50 manufacturing firms: the VRS case

Appendix C: VRS model under the weak disposability assumption

The following discussion is taken from Kuosmanen (2005) and Kuosmanen and Podinovski (2009); see also Kuosmanen and Kazemi Matin (2011) for a updated summary of the development in the VRS model with WDA. The classical Shephard’s WDA production model under variable returns-to-scale is $f_{vrs}(X) := \{(Y,B): \sum_{j=1}^{n} \lambda_{j} x_{ji} \le x_{i},\;i=1,\ldots,m;\ \mu\sum_{j=1}^{n} \lambda_{j} y_{jr} \ge y_{r}, \;r=1,\ldots,s;\ \mu\sum_{j=1}^{n} \lambda_{j} b_{jk} = b_{k},\; k=1,\ldots,p;\ \sum_{j=1}^{n} \lambda_{j}=1;\ \lambda_{j} \ge0, \;j=1,\ldots ,n;\;0\le\mu\le1\}$. The constraint that makes λ _j ’s sum up to one to express the variable returns-to-scale condition (Banker et al. 1984). The μ variable is meant to reflect that the output space of Ω _vrs is the convex combination of the DMU’s output vector proportionally scaled down by the ratio μ. Kuosmanen (2005) generalizes Shephard’s model by allowing each DMU to contribute a different value of μ (i.e., 0≤μ ₁≤1,…,0≤μ _j ≤1, and thus each DMU can assume a different scale down factor).

The Kuosmanen technology set also rectify the problem that Ω _vrs is not convex. Like the Shephard’s model, the Kuosmanen’s VRS model is nonlinear, too. However, Kuosmanen (2005) shows the VRS formulation can be converted into an equivalent linear form: $f_{vrs}^{TK}(X) := \{(Y,B): \sum_{j=1}^{n} z_{j} x_{ji} \le x_{i},\; i=1,\ldots,m;\ \sum_{j=1}^{n} z_{j} y_{jr} \ge y_{r}, \;r=1,\ldots,s;\ \sum_{j=1}^{n} (z_{j}+\nu_{j}) b_{jk} = b_{k},\;k=1,\ldots,p;\ \sum_{j=1}^{n} (z_{j}+\nu_{j})=1;\ z_{j}, \nu_{j} \ge0, \;j=1,\ldots,n\}$.

Table 6 shows the efficiency scores under the VRS assumption for firms that also appeared in the application presented earlier in this article. As did in that application, we calculate the HAM, DDF, HEM scores for the fifty firms, and the MEM scores of their projection targets using the DDF and HEM models. Finally, the last two columns display the efficiency class of the firms when either DDF or HEM is used for efficiency evaluation.