Measuring and Improving Eco-efficiency

Abstract

As energy and environmental issues continue to generate social concerns, firms are pushed to alleviate their energy and environmental externalities from economic activities and network operations while pursuing economic interests. A comprehensive performance evaluation involving both productive and environmental factors is therefore desired by firms to aggregate multiple aspects into condensed information to facilitate decision-making. In this study, we develop a new eco-efficiency measure for energy-intensive economic activities, which integrates productive and environmental factors in the presence of undesirable outputs. The proposed measure has the following merits in comparison with a majority of existing ones that have been widely used in energy- and environment-related areas. It guarantees Pareto-Koopmans eco-efficiency because only non-dominated entities are measured as eco-efficient. It also captures any actual fluctuations in inputs, desirable outputs, and undesirable outputs, as it proves to be strongly monotonic in all input and output dimensions. It has a positive significance for promoting undesirable output abatement, because it stimulates scale improvement for entities in pursuit of economies of scale. It also provides great flexibility of improvement for individual inputs and outputs to achieve eco-efficiency. The marginal abatement costs of undesirable outputs are estimated to facilitate managing the potential tradeoff between undesirable output abatement and desirable output production. We also extend it to weighted eco-efficiency measures for specific situations wherever necessary. Its practicability and comparison with existing models are illustrated by an empirical example.

Appendices

Appendix 1. Proof of Theorem 1

Suppose that the original data (X_j, Y_j, U_j) are replaced by $\left({\hat{X}}_j,{\hat{Y}}_j,{\hat{U}}_j\right)=\left(\alpha {X}_j,\beta {Y}_j,\gamma {U}_j\right),\alpha, \beta, \gamma >0$ for all of the observations. Without loss of generality, suppose that $\left({\theta}_{mo}^{\ast },{\varphi}_{no}^{\ast },{\delta}_{lo}^{\ast },{\lambda}_j^{\ast },\forall m,n,l,j\right)$ is an optimal solution to (4) when the o^th entity (X_o, Y_o, U_o) is being observed. Then, for any input m(∈{1, 2, …, M}), we have ${\sum}_{j=1}^J{\lambda}_j^{\ast }{x}_{mj}\le {\theta}_{mo}^{\ast }{x}_{mo}$. It follows that $\alpha {\sum}_{j=1}^J{\lambda}_j^{\ast }{x}_{mj}\le {\alpha \theta}_{mo}^{\ast }{x}_{mo}$ and ${\sum}_{j=1}^J{\lambda}_j^{\ast}\left(\alpha {x}_{mj}\right)\le {\theta}_{mo}^{\ast}\left(\alpha {x}_{mj}\right)$, indicating that ${\sum}_{j=1}^J{\lambda}_j^{\ast }{\hat{x}}_{mj}\le {\theta}_{mo}^{\ast }{\hat{x}}_{mo}$. Similarly, for any desirable output n(∈{1, 2, …, N}) and undesirable output l(∈{1, 2, …, L}), we have ${\sum}_{j=1}^J{\lambda}_j^{\ast }{\hat{y}}_{nj}\le {\varphi}_{no}^{\ast }{\hat{y}}_{no}$ and ${\sum}_{j=1}^J{\lambda}_j^{\ast }{\hat{u}}_{lj}\le \left(1-{\delta}_{lo}^{\ast}\right){\hat{u}}_{lo}$. Because α, β, and γ are arbitrary positive values, it can be concluded that $\left({\theta}_{mo}^{\ast },{\varphi}_{no}^{\ast },{\delta}_{lo}^{\ast },{\lambda}_j^{\ast },\forall m,n,l,j\right)$ is also an optimal solution to (4) when $\left({\hat{X}}_o,{\hat{Y}}_o,{\hat{U}}_o\right)$ is being observed after the replacement. As a result, we have E(X_o, Y_o, U_o) = E(αX_o, βY_o, γU_o). □

Appendix 2. Proof of Theorem 2

As defined in (4), 0 ≤ θ_mo ≤ 1, φ_no ≥ 1, 0 ≤ δ_lo ≤ 1, ∀ m, n, l. With ρ_o as the objective of (4), it is easy to obtain that 0 ≤ E(X_o, Y_o, U_o) ≤ 1. According to (12), we derive the benchmark target $\left({\tilde{X}}_o,{\tilde{Y}}_o,{\tilde{U}}_o\right)$ for (X_o, Y_o, U_o). Because no output can be produced if no input is consumed, we have ${\tilde{X}}_o>0$ for the benchmark target. Therefore, ${\theta}_{mo}^{\ast }>0$ for at least one input m(∈{1, 2, …, M}). As a result, we have 0 < E(X_o, Y_o, U_o) ≤ 1. □

Appendix 3. Proof of Theorem 3

Suppose that (X_o, Y_o, U_o) is dominated by (X_k, Y_k, U_k) in Ω. Then, we have X_k ≤ X_o, Y_k ≥ Y_o, and U_k ≤ U_o with at least one strict inequality. Without loss of generality, suppose that X_k < X_o and $\left({\theta}_{mk}^{\ast },{\varphi}_{nk}^{\ast },{\delta}_{lk}^{\ast },{\lambda}_j^{\ast },\forall m,n,l,j\right)$ is an optimal solution to (4) when (X_k, Y_k, U_k) is being observed. It follows that there must be at least one input m(∈{1, 2, …, M}) for which x_mk < x_mo. Accordingly, there exists a feasible solution $\left({\theta}_{1k}^{\ast },\dots, {\theta}_{mo},\dots, {\theta}_{Mk}^{\ast },{\varphi}_{1k}^{\ast },\dots, {\varphi}_{Nk}^{\ast },{\delta}_{1k}^{\ast },\dots, {\delta}_{Lk}^{\ast },{\lambda}_1^{\ast },\dots, {\lambda}_J^{\ast}\right)$ to (4) when (X_o, Y_o, U_o) is being observed, where ${\theta}_{mo}{x}_{mo}={\theta}_{mk}^{\ast }{x}_{mk}$ and thus ${\theta}_{mo}<{\theta}_{mk}^{\ast}\le 1$, indicating that E(X_o, Y_o, U_o) < 1. Similarly, if Y_k > Y_o and/or U_k < U_o, it can also be concluded that E(X_o, Y_o, U_o) < 1. This completes the only if part.

Conversely, if E(X_o, Y_o, U_o) < 1, then we have ${\theta}_{mo}^{\ast}\le 1,{\varphi}_{no}^{\ast}\ge 1,{\delta}_{lo}^{\ast}\le 1,\forall m,n,l$ with at least one strict inequality, implying that its benchmark target $\left({\tilde{X}}_o,{\tilde{Y}}_o,{\tilde{U}}_o\right)$ defined by (12) must dominate (X_o, Y_o, U_o). Therefore, the if part is established. □

Appendix 4. Proof of Theorem 4

Suppose that $\left({\theta}_{mo}^{\ast },{\varphi}_{no}^{\ast },{\delta}_{lo}^{\ast },{\lambda}_j^{\ast },\forall m,n,l,j\right)$ and $\left({\theta}_{mk}^{\ast },{\varphi}_{nk}^{\ast },{\delta}_{lk}^{\ast },{\lambda^{\prime}}_j^{\ast },\forall m,n,l,j\right)$ are the optimal solutions to (4) when the two entities (X_o, Y_o, U_o) and (X_k, Y_k, U_k) involved below are being observed, respectively. Let ${\rho}_o^{\ast }$ and ${\rho}_k^{\ast }$ be the optimal values of (4) when they are being observed, respectively.

First, consider two entities, (X_o, Y_o, U_o) and (X_k, Y_k, U_k), which differ from each other in only one input m(∈{1, 2, …, M}), i.e., x_mk = x_mo + a, (a > 0). We must show that E(X_o, Y_o, U_o) > E(X_k, Y_k, U_k). Let ${\theta}_{mk}{x}_{mk}={\theta}_{mo}^{\ast }{x}_{mo}$; we have ${\theta}_{mk}<{\theta}_{mo}^{\ast}\le 1$ and $\left({\theta}_{1o}^{\ast },\dots, {\theta}_{mk},\dots, {\theta}_{Mo}^{\ast },{\varphi}_{1o}^{\ast },\dots, {\varphi}_{No}^{\ast },{\delta}_{1o}^{\ast },\dots, {\delta}_{Lo}^{\ast },{\lambda}_1^{\ast },\dots, {\lambda}_J^{\ast}\right)$ is a feasible solution to (4) when (X_k, Y_k, U_k) is being observed, with the corresponding value of the objective ${\rho}_k\ge {\rho}_k^{\ast }$. Because ${\theta}_{mk}<{\theta}_{mo}^{\ast }$, we also have ${\rho}_k<{\rho}_o^{\ast }$. As a result, ${\rho}_k^{\ast }<{\rho}_o^{\ast }$ and thus E(X_o, Y_o, U_o) > E(X_k, Y_k, U_k).

Next, consider two entities, (X_o, Y_o, U_o) and (X_k, Y_k, U_k), which differ from each other in only one desirable output n(∈{1, 2, …, N}), i.e., y_nk = y_no + a, (a > 0). We must show that E(X_o, Y_o, U_o) < E(X_k, Y_k, U_k). Let ${\varphi}_{nk}^{\ast }{y}_{nk}={\varphi}_{no}{y}_{no}$; it follows that $1\le {\varphi}_{nk}^{\ast }<{\varphi}_{no}$ and $\left({\theta}_{1k}^{\ast },\dots, {\theta}_{Mk}^{\ast },{\varphi}_{1k}^{\ast },\dots, {\varphi}_{no},\dots, {\varphi}_{Nk}^{\ast },{\delta}_{1k}^{\ast },\dots, {\delta}_{Lk}^{\ast },{\lambda^{\prime}}_1^{\ast },\dots, {\lambda^{\prime}}_J^{\ast}\right)$ are a feasible solution to (4) when (X_o, Y_o, U_o) is being observed, with the corresponding value of the objective ${\rho}_o\ge {\rho}_o^{\ast }$. Because ${\varphi}_{nk}^{\ast }<{\varphi}_{no}$, we obtain ${\rho}_o<{\rho}_k^{\ast }$. As a result, ${\rho}_k^{\ast }>{\rho}_o^{\ast }$ and hence E(X_o, Y_o, U_o) < E(X_k, Y_k, U_k).

Finally, consider two entities, (X_o, Y_o, U_o) and (X_k, Y_k, U_k), which differ from each other in only one undesirable output l(∈{1, 2, …, L}), i.e., u_lk = u_lo + a, (a > 0). We must show that E(X_o, Y_o, U_o) > E(X_k, Y_k, U_k). Let $\left(1-{\delta}_{lk}\right){u}_{lk}=\left(1-{\delta}_{lo}^{\ast}\right){u}_{lo}$; we have ${\delta}_{lk}>{\delta}_{lo}^{\ast}\ge 0$ and $\left({\theta}_{1o}^{\ast },\dots, {\theta}_{Mo}^{\ast },{\varphi}_{1o}^{\ast },\dots, {\varphi}_{No}^{\ast },{\delta}_{1o}^{\ast },\dots, {\delta}_{lk},\dots, {\delta}_{Lo}^{\ast },{\lambda}_1^{\ast },\dots, {\lambda}_J^{\ast}\right)$ is a feasible solution to (4) when (X_k, Y_k, U_k) is being observed, with the corresponding value of the objective ${\rho}_k\ge {\rho}_k^{\ast }$. Because ${\delta}_{lk}>{\delta}_{lo}^{\ast }$, it follows that ${\rho}_k<{\rho}_o^{\ast }$. Therefore, we have ${\rho}_k^{\ast }<{\rho}_o^{\ast }$ and E(X_o, Y_o, U_o) > E(X_k, Y_k, U_k). □

Appendix 5. Proof of Theorem 5

• (i) Suppose that $\left({\theta}_{mo}^{\ast },{\varphi}_{no}^{\ast },{\delta}_{lo}^{\ast },{\lambda}_j^{\ast },\forall m,n,l,j\right)$ and $\left({\theta}_{mo}^{\alpha \ast },{\varphi}_{no}^{\alpha \ast },{\delta}_{lo}^{\alpha \ast },{\lambda}_j^{\alpha \ast },\forall m,n,l,j\right)$ are the optimal solutions to (4) when (X_o, Y_o, U_o) and (αX_o, Y_o, U_o) are being observed, respectively. Let ${\rho}_o^{\ast }$ and ${\rho}_o^{\alpha \ast }$ be the optimal values of (4) when they are being observed, respectively. Because $\alpha \ge \underset{m}{\max}\left\{{\theta}_{mo}^{\ast}\right\}$, we have ${\theta}_{mo}^{\ast }/\alpha \le 1$ for any m(∈{1, 2, …, M}). Then, it can be concluded that $\left({\theta}_{mo}^{\ast }/\alpha, {\varphi}_{no}^{\ast },{\delta}_{lo}^{\ast },{\lambda}_j^{\ast },\forall m,n,l,j\right)$ is a feasible solution to (4) when (αX_o, Y_o, U_o) is being observed, with the corresponding value of the objective ${\rho}_o^{\alpha}\ge {\rho}_o^{\alpha \ast }$. Because

$$ {\rho}_o^{\alpha }=\frac{\frac{1}{M}\sum \limits_{m=1}^M\frac{\theta_{mo}^{\ast }}{\alpha }}{\frac{1}{N}\sum \limits_{n=1}^N{\varphi}_{no}^{\ast }+\frac{1}{L}\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}=\frac{1}{\alpha}\cdot \frac{\frac{1}{M}\sum \limits_{m=1}^M{\theta}_{mo}^{\ast }}{\frac{1}{N}\sum \limits_{n=1}^N{\varphi}_{no}^{\ast }+\frac{1}{L}\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}=\frac{\rho_o^{\ast }}{\alpha }, $$

we have ${\rho}_o^{\alpha \ast}\le {\rho}_o^{\ast }/\alpha$ and thus E(αX_o, Y_o, U_o) ≤ (1/α)E(X_o, Y_o, U_o).

• (ii) Suppose that $\left({\theta}_{mo}^{\beta \gamma \ast },{\varphi}_{no}^{\beta \gamma \ast },{\delta}_{lo}^{\beta \gamma \ast },{\lambda}_j^{\beta \gamma \ast },\forall m,n,l,j\right)$ is an optimal solution to (4) when (X_o, βY_o, γU_o) is being observed. Let ${\rho}_o^{\beta \gamma \ast }$ be the optimal value of (4) when it is being observed. Because $\beta \le \underset{n}{\min}\left\{{\varphi}_{no}^{\ast}\right\}$ and $\gamma \ge 1-\underset{l}{\min}\left\{{\delta}_{lo}^{\ast}\right\}$, we have ${\varphi}_{no}^{\ast }/\beta \ge 1$ for any n(∈{1, 2, …, N}) and $\left(1-{\delta}_{lo}^{\ast}\right)/\gamma \le 1$ for any l(∈{1, 2, …, L}). As a result, $\left({\theta}_{mo}^{\ast },{\varphi}_{no}^{\ast }/\beta, \left(1-\left(1-{\delta}_{lo}^{\ast}\right)/\gamma \right),{\lambda}_j^{\ast },\forall m,n,l,j\right)$ is a feasible solution to (4) when (X_o, βY_o, γU_o) is being observed, with the corresponding value of the objective ${\rho}_o^{\beta \gamma}\ge {\rho}_o^{\beta \gamma \ast }$. In addition, from

$$ \left\{\begin{array}{l}\frac{L}{\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}-1\le \gamma \le 1, if\ \frac{\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}{L}\ge \frac{1}{2},\\ {}1\le \gamma \le \frac{L}{\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}-1, if\ \frac{\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}{L}\le \frac{1}{2},\end{array}\right. $$

it can be deduced that

$$ \frac{1}{L}\sum \limits_{l=1}^L\left(1-\frac{1-{\delta}_{lo}^{\ast }}{\gamma}\right)\ge \frac{1}{L}\sum \limits_{l=1}^L{\gamma \delta}_{lo}^{\ast }. $$

Thus, we have

$$ {\rho}_o^{\beta \gamma}=\frac{\frac{1}{M}\sum \limits_{m=1}^M{\theta}_{mo}^{\ast }}{\frac{1}{N}\sum \limits_{n=1}^N\frac{\varphi_{no}^{\ast }}{\beta }+\frac{1}{L}\sum \limits_{l=1}^L\left(1-\frac{1-{\delta}_{lo}^{\ast }}{\gamma}\right)}\le \frac{\frac{1}{M}\sum \limits_{m=1}^M{\theta}_{mo}^{\ast }}{\frac{1}{N}\sum \limits_{n=1}^N\frac{\varphi_{no}^{\ast }}{\beta }+\frac{1}{L}\sum \limits_{l=1}^L{\gamma \delta}_{lo}^{\ast }}\le \sigma \cdot \frac{\frac{1}{M}\sum \limits_{m=1}^M{\theta}_{mo}^{\ast }}{\frac{1}{N}\sum \limits_{n=1}^N{\varphi}_{no}^{\ast }+\frac{1}{L}\sum \limits_{l=1}^L{\delta}_{lo}^{\ast }}={\sigma \rho}_o^{\ast }, $$

where σ = max {β, 1/γ}.

It follows that ${\rho}_o^{\beta \gamma \ast}\le {\sigma \rho}_o^{\ast }$ and hence E(X_o, βY_o, γU_o) ≤ σE(X_o, Y_o, U_o).

Appendix 6. Proof of Corollary 1

By combining the proofs of (i) and (ii) in Theorem 5, Corollary 1 is immediately established. □