Dynamic sustainability assessment of poverty alleviation in China: evidence from both novel non-convex global two-stage DEA and Malmquist productivity index

Abstract

Performance evaluation of poverty alleviation (PA) is one of the important tools to promote the effective linkage between PA and rural revitalization in China. This paper exposes the internal structure of PA and specifies its input–output process as a two-stage series system consisting of public investment and PA processes. Based on the centralized and decentralized management mechanisms, we propose non-convex global two-stage data envelopment analysis (DEA) models and non-convex global Malmquist productivity indices (GMPIs) to measure the dynamic efficiency of each PA system and its changes. We select 22 provinces (regions) in China as decision making units (DMUs) to illustrate the validity of the proposed models. The results show that: (i) the global PA efficiencies of the overall system (subsystems) are increasing over time, and most of them have already been at a high efficiency in the recent period; (ii) there is a certain inconsistency in the performance development trends of the overall system and subsystems, particularly in terms of the non-convex GMPI, local efficiency change (LEC), and best practice change (BPC).

Appendices

Appendix

The proof of Theorem 1

Proof

For the definition of ${P}_{NC}^{G}$, let $P=\{{P}^{1}\cup {P}^{2}\cup ,\dots ,{\cup P}^{T}\}$, we then have.

$$P = \bigcup\limits_{{t = 1}}^{T} {\left\{ {\left( {X,W,Y} \right)\left| {\begin{array}{*{20}l} {\sum\limits_{{j = 1}}^{n} \lambda _{j}^{t} X_{j}^{t} \le X,\sum\limits_{{j = 1}}^{n} {\lambda _{j}^{t} } W_{j}^{t} \ge W,} \hfill \\ {\sum\limits_{{j = 1}}^{n} {\lambda _{j}^{t} } Z_{j}^{t} \ge \sum\limits_{{j = 1}}^{n} {\mu _{j}^{t} } Z_{j}^{t} ,\sum\limits_{{j = 1}}^{n} {\mu _{j}^{t} } Y_{j}^{t} \ge Y,} \hfill \\ {\sum\limits_{{j = 1}}^{n} {\lambda _{j}^{t} } = 1,\sum\limits_{{j = 1}}^{n} {\mu _{j}^{t} } = 1,} \hfill \\ {\lambda _{j}^{t} ,\mu _{j}^{t} \ge 0,j = 1, \cdots ,n.} \hfill \\ \end{array} } \right.} \right\}}$$

(A1)

In this case, we only need to prove that $P$ and ${P}_{NC}^{G}$ defined in Theorem 1 are equivalent. To this end, this paper illustrates this conclusion in two aspects.

(i) Suppose that an arbitrary given DMU $\left(\widehat{X}, \widehat{W},\widehat{Y}\right)\in P$. This implies that there exists at least one time period ${t}_{0}\in \left\{1,\cdots ,T\right\}$ such that $\left(\widehat{X}, \widehat{W},\widehat{Y}\right)\in {P}_{{t}_{0}}$ holds. Therefore, we obtain

$$\left\{\begin{array}{l}{\sum }_{j=1}^{n}{\lambda }_{j}^{{t}_{0}}{X}_{j}^{{t}_{0}}\le \widehat{X}, \sum_{j=1}^{n}{\lambda }_{j}^{{t}_{0}}{W}_{j}^{{t}_{0}}\ge \widehat{W},\\ \sum_{j=1}^{n}{\lambda }_{j}^{{t}_{0}}{Z}_{j}^{{t}_{0}}\ge \sum_{j=1}^{n}{\mu }_{j}^{{t}_{0}}{Z}_{j}^{{t}_{0}}, \sum_{j=1}^{n}{\mu }_{j}^{{t}_{0}}{Y}_{j}^{{t}_{0}}\ge \widehat{Y},\\ \sum_{j=1}^{n}{\lambda }_{j}^{{t}_{0}}=1, \sum_{j=1}^{n}{\mu }_{j}^{{t}_{0}}=1,\\ {\lambda }_{j}^{{t}_{0}}, {\mu }_{j}^{{t}_{0}}\ge 0,j=1,\cdots ,n.\end{array}\right.$$

(A2)

In addition, we make the following settings for ${P}_{NC}^{G}$ in Theorem 1: ${\kappa }^{t}$ is equal to 1 when $t$ is equal to ${t}_{0}$, otherwise ${\kappa }^{t}$ is equal to 0. Then, ${P}_{NC}^{G}$ can be simplified as

$$P_{{NC}}^{G} = \left\{ {\left( {X,W,Y} \right)\left| {\begin{array}{*{20}l} {\sum\limits _{{j = 1}}^{n} \lambda _{j}^{{t_{0} }} X_{j}^{{t_{0} }} \le X,\sum\limits_{{j = 1}}^{n} {\lambda _{j}^{{t_{0} }} } W_{j}^{{t_{0} }} \ge W,} \hfill \\ {\sum\limits_{{j = 1}}^{n} {\lambda _{j}^{{t_{0} }} } Z_{j}^{{t_{0} }} \ge \sum\limits_{{j = 1}}^{n} {\mu _{j}^{{t_{0} }} } Z_{j}^{{t_{0} }} ,\sum\limits_{{j = 1}}^{n} {\mu _{j}^{{t_{0} }} } Y_{j}^{{t_{0} }} \ge Y,} \hfill \\ {\sum\limits_{{j = 1}}^{n} {\lambda _{j}^{{t_{0} }} } = 1,\sum\limits_{{j = 1}}^{n} {\mu _{j}^{{t_{0} }} } = 1,} \hfill \\ {\lambda _{j}^{{t_{0} }} ,\mu _{j}^{{t_{0} }} \ge 0,j = 1, \cdots ,n.} \hfill \\ \end{array} } \right.} \right\}$$

(A3)

The above results show that $\left(\widehat{X}, \widehat{W},\widehat{Y}\right)\in {P}_{NC}^{G}$ holds, i.e., $P\subset {P}_{NC}^{G}$ is true here.

(ii) For an arbitrary given DMU $\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{W} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{Y} } \right)$$\in {P}_{NC}^{G}$, there exists $t={\tau }_{0}\in \left\{1,\cdots ,T\right\}$ such that ${\kappa }^{{\tau }_{0}}$=1 and satisfies ${ \kappa }^{t}=0$ when $t\ne {\tau }_{0}$. It indicates that $\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{W} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{Y} } \right)$$\in {P}_{{\tau }_{0}}$. We then obtain that $\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{W} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{Y} } \right)$$\in P$ due to ${P}_{{\tau }_{0}}\subset P$. Then, the conclusion of ${P}_{NC}^{G}\subset P$ is proven.

Finally, incorporating the conclusions in (i) and (ii), it is shown that $P={P}_{NC}^{G}$. Similarly, we prove that ${P}_{NC}^{G,1}$ and ${P}_{NC}^{G,2}$ can also be written as Sets (5) and (6), respectively. At this point, we conclude that Theorem 1 is true. □

The DEA models for comparison

When decision-makers do not consider the internal structure of the system and treats the PA system as a “black box”, the following traditional BCC model is obtained.

$${D}_{BCC}({X}_{0}^{\tau }, {{W}_{0}^{\tau },Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}{\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }{X}_{j}^{\tau }\le {X}_{0}^{\tau }-\theta {g}_{X}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{W}_{j}^{\tau }\ge {W}_{0}^{\tau }+\theta {g}_{W}^{\tau },\\ \sum_{j=1}^{n}{\mu }_{j}^{\tau }{Y}_{j}^{\tau }\ge {Y}_{0}^{\tau }+\theta {g}_{Y}^{\tau },\\ {\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }=1,{\lambda }_{j}^{\tau }\ge 0, j=1,\cdots ,n.\end{array}\right.$$

(A4)

When decision-makers focus only on the production technology for a given time period in measuring the PA efficiency, the following local two-stage DEA models with different management mechanisms are obtained according to Sets (1)–(3).

$${D}^{\tau }({X}_{0}^{\tau }, {{W}_{0}^{\tau },Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}{\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }{X}_{j}^{\tau }\le {X}_{0}^{\tau }-\theta {g}_{X}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{W}_{j}^{\tau }\ge {W}_{0}^{\tau }+\theta {g}_{W}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{Z}_{j}^{\tau }\ge \sum_{j=1}^{n}{\mu }_{j}^{\tau }{Z}_{j}^{\tau },\\ \sum_{j=1}^{n}{\mu }_{j}^{\tau }{Y}_{j}^{\tau }\ge {Y}_{0}^{\tau }+\theta {g}_{Y}^{\tau },\\ {\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }=1, \sum_{j=1}^{n}{\mu }_{j}^{\tau }=1,\\ {\lambda }_{j}^{\tau },{\mu }_{j}^{\tau }\ge 0, j=1,\cdots ,n.\end{array}\right.$$

(A5)

$${D}^{\tau ,1}({X}_{0}^{\tau }, {W}_{0}^{\tau }, {Z}_{0}^{\tau },{Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}{\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }{X}_{j}^{\tau }\le {X}_{0}^{\tau }-\theta {g}_{X}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{W}_{j}^{\tau }\ge {W}_{0}^{\tau }+\theta {g}_{W}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{Z}_{j}^{\tau }\ge {Z}_{0}^{\tau }+\theta {g}_{Z}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{Z}_{j}^{\tau }\ge \sum_{j=1}^{n}{\mu }_{j}^{t}{Z}_{j}^{t},\\ \sum_{j=1}^{n}{\mu }_{j}^{\tau }{Y}_{j}^{\tau }\ge {Y}_{0}^{\tau }, \\ {\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }=1, \sum_{j=1}^{n}{\mu }_{j}^{\tau }=1,\\ {\lambda }_{j}^{\tau }, {\mu }_{j}^{\tau }\ge 0, j=1,\cdots ,n.\end{array}\right.$$

(A6)

$${D}^{\tau ,2}({{X}_{0}^{\tau },{W}_{0}^{\tau },Z}_{0}^{\tau }, {Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}{\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }{X}_{j}^{\tau }\le {X}_{0}^{\tau },\\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{W}_{j}^{\tau }\ge {W}_{0}^{\tau }, \\ \sum_{j=1}^{n}{\lambda }_{j}^{\tau }{Z}_{j}^{\tau }\ge \sum_{j=1}^{n}{\mu }_{j}^{\tau }{Z}_{j}^{\tau }, \\ {\sum }_{j=1}^{n}{\mu }_{j}^{\tau }{Z}_{j}^{\tau }\le {Z}_{0}^{\tau }-\theta {g}_{Z}^{\tau },\\ \sum_{j=1}^{n}{\mu }_{j}^{\tau }{Y}_{j}^{\tau }\ge {Y}_{0}^{\tau }+\theta {g}_{Y}^{\tau },\\ {\sum }_{j=1}^{n}{\lambda }_{j}^{\tau }=1, \sum_{j=1}^{n}{\mu }_{j}^{\tau }=1, \\ {\lambda }_{j}^{\tau },{\mu }_{j}^{\tau }\ge 0, j=1,\cdots ,n.\end{array}\right.$$

(A7)

In addition, we also allow that the convex global two-stage DEA models obtained from the convex combination of production technologies at different time periods. Similarly, the following models are constructed under different management mechanisms.

$${D}_{C}^{G}({X}_{0}^{\tau }, {{W}_{0}^{\tau },Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}\sum_{t=1}^{T}{\sum }_{j=1}^{n}{\lambda }_{j}^{t}{X}_{j}^{t}\le {X}_{0}^{\tau }-\theta {g}_{X}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{W}_{j}^{t}\ge {W}_{0}^{\tau }+\theta {g}_{W}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{Z}_{j}^{t}\ge \sum_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}{Z}_{j}^{t},\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}{Y}_{j}^{t}\ge {Y}_{0}^{\tau }+\theta {g}_{Y}^{\tau },\\ {\sum }_{t=1}^{T}{\sum }_{j=1}^{n}{\lambda }_{j}^{t}=1, {\sum }_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}=1,\\ {\lambda }_{j}^{t},{\mu }_{j}^{t}\ge 0, t=1,\cdots ,T, j=1,\cdots ,n.\end{array}\right.$$

(A8)

$${D}_{C}^{G,1}({X}_{0}^{\tau }, {W}_{0}^{\tau }, {Z}_{0}^{\tau },{Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}\sum_{t=1}^{T}{\sum }_{j=1}^{n}{\lambda }_{j}^{t}{X}_{j}^{t}\le {X}_{0}^{\tau }-\theta {g}_{X}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{W}_{j}^{t}\ge {W}_{0}^{\tau }+\theta {g}_{W}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{Z}_{j}^{t}\ge {Z}_{0}^{\tau }+\theta {g}_{Z}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{Z}_{j}^{t}\ge \sum_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}{Z}_{j}^{t},\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}{Y}_{j}^{t}\ge {Y}_{0}^{\tau }, \\ {\sum }_{t=1}^{T}{\sum }_{j=1}^{n}{\lambda }_{j}^{t}=1, {\sum }_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}=1,\\ {\lambda }_{j}^{t}, {\mu }_{j}^{t}\ge 0, t=1,\cdots ,T, j=1,\cdots ,n.\end{array}\right.$$

(A9)

$${D}_{C}^{G,2}({{X}_{0}^{\tau },{W}_{0}^{\tau },Z}_{0}^{\tau }, {Y}_{0}^{\tau })=\mathit{max}~\theta$$

$$s.t.\left\{\begin{array}{l}\sum_{t=1}^{T}{\sum }_{j=1}^{n}{\lambda }_{j}^{t}{X}_{j}^{t}\le {X}_{0}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{W}_{j}^{t}\ge {W}_{0}^{\tau }, \\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\lambda }_{j}^{t}{Z}_{j}^{t}\ge \sum_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}{Z}_{j}^{t}, \\ \sum_{t=1}^{T}{\sum }_{j=1}^{n}{\mu }_{j}^{t}{Z}_{j}^{t}\le {Z}_{0}^{\tau }-\theta {g}_{Z}^{\tau },\\ \sum_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}{Y}_{j}^{t}\ge {Y}_{0}^{\tau }+\theta {g}_{Y}^{\tau },\\ {\sum }_{t=1}^{T}{\sum }_{j=1}^{n}{\lambda }_{j}^{t}=1, {\sum }_{t=1}^{T}\sum_{j=1}^{n}{\mu }_{j}^{t}=1, \\ {\lambda }_{j}^{t},{\mu }_{j}^{t}\ge 0, t=1,\cdots ,T, j=1,\cdots ,n.\end{array}\right.$$

(A10)