3.1. Non-Radial Directional Distance Function
It is well known that desirable and undesirable outputs are jointly produced during the production process. Generally speaking, the production process strives to increase the desirable outputs while simultaneously decreasing the undesirable ones under a given input set and technology. In other words, considering the desirable outputs while ignoring the undesirable outputs to evaluate green development is misleading. Färe
et al. [
27] examined environmental technology that incorporated weak disposability of outputs and null-jointness. This framework is very useful to investigate environmental performance. Assume that
,
and
the vectors of inputs, desirable outputs and undesirable outputs respectively. Then, the production process can be described as using inputs
x to produce desirable outputs
y while also emitting undesirable environmental pollutants
b. This production technology can also be conceptually defined as
T = {(
x,
y,
b):
x can produce (
y,
b)}. Mathematically, the technology set
T is usually assumed to be a closed, bounded, and convexity set. Additionally, it must satisfy the following properties [
21,
27,
28]:
- (1)
If (x, y, b) , and b = 0, then y = 0;
- (2)
If (x, y, b) , and x' > x, then ;
- (3)
If (x, y, b) , and y' < y, then (x, y', b) ;
- (4)
If (x, y, b) , and , then .
The production technology T has been well-defined conceptually to reflect green development as it considers both the desirable and undesirable outputs, and the abatement of undesirable outputs is not free but costly in terms of a proportional reduction in desirable outputs. However, it cannot be applied in empirical analysis directly.
In fact, the directional distance function (DDF) method introduced by Chung, Färe and Grosskopf [
6] is a suitable framework to measure the above production technology
T and its properties. Originally, the DDF is defined as follows:
where
is a direction vector and it implies that the desirable outputs and the undesirable outputs can increase and decrease simultaneously. As the DDF reflects the essence of green development, it is widely used in empirical studies to evaluate environmental performance. Examples of such studies include Picazo-Tadeo
et al. [
29], Färe
et al. [
27], Watanabe and Tanaka [
30], Macpherson
et al. [
31], Yuan
et al. [
32], Li and Lin [
21] and Li and Lin [
22].
However, Equation (1) has a rigid assumption that the expansion of desirable outputs and the reduction of the undesirable outputs are at the same rate. In this sense, it is a radical efficiency measure, and when there are non-zero slacks it may overestimate the environmental performance. Another limitation of Equation (1) is that it cannot evaluate the performance of a given input [
33]. In order to overcome these limitations, Zhou, Ang and Wang [
26] presented a non-radial directional distance function (NDDF) in order to model green performance considering both the desirable and undesirable outputs. Zhou, Ang and Wang [
26] believed that NDDF relaxes the assumption of the proportional adjustments of inputs and (desirable and undesirable) outputs, and has higher discriminating power than DDF. It can be defined as follows [
26]:
where
is a vector of scaling factors which measure the distance of the real inputs/outputs from the optimal state;
denotes the weights of the inputs/outputs;
is a directional vector, while
diag implies the diagonal matrices. One of the advantages of NDDF is that
and
can be set in different ways according to the research goals. Furthermore, Model (2) is related to Fukuyama and Weber [
34] (FW hereafter) with the difference that FW did not consider bad outputs. However, as Pastor and Aparicio [
35] pointed out, the directional slacks-based measure of FW is a weighted additive-type measure in the sense that it maximizes a specific weighted sum of slacks. In fact, Model (2) is mathematically equivalent to a weighted additive-type model. Similar to DDF, NDDF also cannot be employed in empirical analysis directly.
Most studies use the piecewise convex combinations (DEA-type) of the observed data to characterize the production technology and the related distance function. As a nonparametric method, DEA constructs the best-practice frontier using linear programming. This paper applies DEA to calculate the NDDF to evaluate the green development performance of China. Specifically, each province is regarded as a DMU, and it uses capital stock (
K), labor force (
L), and energy (
E) to generate the desirable output—gross regional product (
Y). Meanwhile, we assume the production process generates four undesirable environmental pollutants as byproducts, which include waste water (
W), sulphur dioxide emission (
S), solid wastes (
D) and carbon dioxide emissions (
C). To accurately measure the green development performance, it is appropriate to assume the non-energy inputs are fixed in the NDDF model, as capital stock (
K) and labor force (
L) do not emit pollutants directly [
8]. In other words, the weight vector and the directional vector are set as
and
, respectively. The assumption of weight vector
is similar to Wang
et al. [
36], Zhang, Zhou and Kung [
8] and Lin and Du [
37], and it means that energy reduction, desirable output expansion and pollutants reduction are assigned the same importance, namely 1/3. For pollutants, the Chinese government introduced a series of regulations and measures to control the main pollutants in 2006. In other words, it is difficult to present the accurate weight for the different pollutants. Thus, we give the weight
to the pollutants. This setup also assigns the same importance to reduction of waste water (
W), sulphur dioxide emission (
S), solid wastes (
D) and carbon dioxide emissions (
C) (Note that different choices of the weight vector might lead to different performance scores. We thank a reviewer for pointing out this.). Although it is a naive choice, it is a common one and has been widely used in previous studies. Furthermore, as the effects of capital and labor are eliminated, it places emphasis on the effective consumption of energy inputs. Based on the above analysis, the value of NDDF of a specific DMU (a province in China) can be calculated by solving the following linear programming,
It is worth noting that we employ the global DEA method, which uses the whole sample to construct the best-practice frontier. That implies a single global frontier is used as a benchmark technology for all DMUs over the whole period. Thus, the estimate values can be compared over time. −
Suppose that
are the optimal solutions to Equation (3), which implies that, under the best-practice scenario, the energy input, desirable output and undesirable environmental pollutants will be
,
and
. Obviously,
means that the DMU on the best-practice frontier. Following Zhou, Ang and Wang [
26], Zhang
et al. [
38] and Lin and Du [
37], the green development performance index (GDPI) can be defined as follows:
It should be noted that the values of GDPI lie between zero and unity, and the higher the GDPI, the better the green development performance.
3.2. Green Development Growth Index and Its Decomposition
In the above section, we define an index, i.e., GDPI to measure green development performance. However, it assumes all provinces have the same level of technology, or it does not account for the technology heterogeneities across provinces. In fact, the different regions may have various technical conditions or technical constraints. For example, the eastern provinces have a relatively high level of economic development, and have advanced technologies to combat undesirable environmental pollutants. Furthermore, the GDPI index is a static index, or it cannot measure the change of green development over years. In order to investigate the changes in GDPI over time while also considering the region heterogeneity, we first combine the GDPI with a meta-frontier analysis.
Suppose
N DMUs (provinces of China in this study,
N = 30) can be divided into
h (
h = 3, including the eastern, central and western China, respectively) independent groups, and each group includes
Nh (
h = 1, 2, 3) DMUs and uses different production technologies. Thus, we can define three benchmark technology sets: a contemporaneous benchmark technology, an intertemporal benchmark technology and a global benchmark technology [
7,
39,
40].
For a given group
h, the contemporaneous benchmark technology (denoted as
) can be reflected in the following linear programming,
Essentially, constructs the best-practice frontier based on the observations for the specific group h only for that time period t. Based on calculated by Equation (5), the GDPI given by Equation (4) can be denoted as .
An intertemporal benchmark technology for a given group
h is defined as
. Obviously, the intertemporal benchmark technology develops a single best-practice frontier for a given group
h over the whole period. Because we have
h (= 3) independent groups, there are
h (= 3) distinct intertemporal technologies. Based on
, the result of GDPI can be denoted as
and can be calculated by Equation (6) and Equation (4).
Finally, a global benchmark technology can be defined as
.
is the convex envelope of all the contemporaneous technologies. It can be inferred that
constructs only one best-practice frontier based on the whole observation (including observations for all groups over the entire time period) [
41]. Obviously,
can be reflected in Equation (3), and the corresponding GDPI is denoted as
.
Similar to a global Malmquist productivity index introduced by Pastor and Lovell [
42], a new index, namely the green development growth index (GDGI), is introduced to measure the changes in GDPI over time,
Since
is calculated by a global DEA method (Equation 3), in other words there is only one global benchmark technology, and its scores can be compared over the years [
43], so there is no need to resort to the geometric mean convention when defining the global index [
41,
42].
As developed by Oh [
40] and Zhang and Choi [
7], the GDGI can be decomposed into different components as follows:
In Equation (8), the green development performance at time period
s (=
t,
t + 1) based on the contemporaneous benchmark technology (
) is measured by
and abbreviated as
. So, the
EC term in Equation (8) measures how close a DMU for a specific group moves toward the contemporaneous benchmark technology at time period
t + 1 compared to time period
t, which indicates a “catch-up” effect.
EC > (or <) 1 implies that a DMU moves close to (or far away from) the best-practice frontier at time period
t + 1 compared to time period
t.
measures the best-practice gap ratio between a contemporaneous benchmark technology and an intertemporal benchmark technology, and it changes over time measured by
BPC in Equation (8).
BPC > (or <) 1 means the contemporaneous technology frontier shifts toward (or far away from) the intertemporal technology frontier, and is also considered to reflect the innovation effect [
7].
is the technology gap ratio between an intertemporal benchmark technology and a global benchmark technology, and it changes over time measured by
TGC in Equation (8).
TGC > (or <) 1 indicates a decrease (increase) in the technology gap between the intertemporal technology for a specific group and the global technology. Thus,
TGC reflects the technology leadership effect for a given group.