23.2.1 Undesirable Congestion (UC) and Desirable Congestion (DC)

As discussed in Chapter 21, the occurrence of UC is a widely observed phenomenon in which inefficiency is identified in such a manner that an increase in some input(s) decreases some desirable output(s), without worsening other production factors, including undesirable output(s). That is an undesirable phenomenon. The straightforward expectation is that an increase in inputs leads to an increase in some components of desirable and undesirable output vectors. Such an occurrence of UC may be found in many economic activities, including business in energy sectors. See Chapter 9 that provides an illustrative example of UC in the electric power industry.

Returning to Chapter 21, we conceptually retreat our description on a possible occurrence of congestion, then connecting it to RTD and DTR. To explain the conceptual connection, this chapter presents Figure 23.1, originated from Figure 21.1. The left‐hand side of the figure exhibits the desirable output (g) on the vertical axis and the undesirable output (b) on the horizontal axis. The opposite case is found in the right‐hand side of the figure. For our visual convenience, Figure 23.1 considers only a single component of the three production factors. It is easily extendable to their multiple components in the mathematical formulations proposed in this chapter. The convex curve depicted in the figure indicates the relationship between two production factors.

On the left‐hand side of Figure 23.1, a supporting line visually specifies the shape of a production curve and a possible occurrence of UC between g and b. For example, a negative slope indicates an occurrence of UC, so being considered as “strong UC,” as discussed in Chapter 21. An occurrence of strong UC implies that an enlarged input (x), not listed in Figure 23.1 for our visual description, increases an undesirable output (b) but decreases a desirable output (g). In contrast, a positive slope implies an opposite case (i.e., no congestion), so being referred to as “no UC.” If the slope of a supporting line is zero, this chapter considers it as “weak UC,” so being between strong UC and no UC.

The right‐hand side of Figure 23.1 visually specifies the three categories (i.e., no, weak and strong) of DC whose analytical implications are directly obtained from UC, but different implications. For a description for such a difference between UC and DC, it is necessary for us to exchange between the horizontal (b to g) and vertical (g to b) coordinates from UC, as found on the right‐hand side of Figure 23.1. See Chapter 21 for an illustrative example on UC under natural disposability. See also Chapter 22 for a detailed description on how to measure MRT and RSU among production factors, with a possible occurrence of DC, under managerial disposability.

23.2.2 Returns to Damage (RTD) under Undesirable Congestion (UC)

Figure 23.2 visually specifies the visual relationship among the three types of supporting hyperplane on a production and pollution possibility set. They are closely associated with the three types of UC, which are classified into “no,” “weak” and “strong” UC. As mentioned previously, an occurrence of UC becomes a serious difficulty in the operation of energy sectors. See Chapter 9 for an illustrative example of such a difficulty in the electric power industry.

Figure 23.3 visually discusses the three types of RTD, including “positive,” “no” and “negative” RTD. An efficiency frontier consists of the six DMUs ({C}–{D}–{E}–{A}–{F}–{B}). Positive RTD, corresponding to no UC, is further separated into IRTD on {C}, CRTD on {D}, and DRTD on {E}. No RTD is found on {A}, corresponding to weak UC, and negative RTD is found on {F}, corresponding to strong UC.

Here, it is important to note that a use of point elasticity for RTD classification indicates that dg/db becomes zero on {A} and negative on {F}, where d stands for a derivative of a functional form that mathematically expresses the slope of a supporting hyperplane. The other three points {C, D and E} have a positive slope of the supporting hyperplane. Thus, Figure 23.3 indicates a mathematical linkage between strong UC and negative RTD. Such a linkage will be mathematically explored in this chapter. The identification of negative RTD is important for all industries, including energy sectors, because it indicates an occurrence of strong UC, or a capacity limit in part or all of a whole production system.

23.2.3 Damages to Return (DTR) under Desirable Congestion (DC)

Shifting our description from UC to DC, Figure 23.4 depicts a possible occurrence of DC on a production and pollution possibility set between a single desirable output (g) on the horizontal axis and a single undesirable output (b) on the vertical axis. Figure 23.4 incorporates the condition of variable DTR. As depicted in Figure 23.4, DC is classified into three categories, including no DC on {D}, weak DC on {A} and strong DC on {F}. Figure 23.5 visually specifies the relationship between a supporting hyperplane and the five types of DTR in which positive DTR is classified into three categories into increasing DTR (IDTR) on {C}, constant DTR (CDTR) on {D}, and decreasing DTR (DDTR) on {E} in addition to no DTR on {A} and negative DTR on {F}.

An important feature of Figures 23.4 and 23.5 is that they have an assumption on “byproducts” between b and g. That is, “undesirable outputs are byproducts of desirable outputs.” Hence, Figures 23.4 and 23.5 have a convex form to express an efficiency frontier. Without the assumption, an efficiency frontier for undesirable output should be structured by a concave form because the less undesirable outputs are the better in environmental assessment. See Figure 15.6, which provided a visual description on the convex and concave relationships in the three production factors. In addition, it visually discussed a rationale regarding why DEA environmental assessment is much more complicated and difficult than a conventional use of DEA. It is true that the only difference between the three production components is the existence of undesirable outputs among the production factors. However, such a difference makes the environmental assessment conceptually and analytically very different from the original use of DEA.

In Figure 23.5, the conventional use of scale elasticity for DTR classification indicates that db/dg becomes zero on {A} and negative on {F}. The derivative becomes positive at the other points ({C}, {D} and {E}). Thus, Figures 23.4 and 23.5 indicate a mathematical linkage between strong DC and negative DTR in the case that the three production factors have a single component. The visual description is extendable to the case of multiple components in the proposed formulations for DEA environmental assessment.

23.2.4 Possible Occurrence of Undesirable Congestion (UC) and Desirable Congestion (DC)

The possible occurrences of UC and DC are expressed by the following production and pollution set under the two disposability concepts:

(23.1)

and

(23.2)

Equation (23.1) incorporates a possible occurrence of UC under natural (N) disposability. Meanwhile, Equation (23.2) incorporates a possible occurrence of DC under managerial (M) disposability. The difference between the two equations can be found on the allocation of equality on desirable or undesirable outputs. Note that equality is formulated by dropping slacks in the proposed formulations. In contrast, inequality is formulated by maintaining slacks in the formulations.

Figure 23.6 visually describes a computational flow from the two disposability concepts to the measurement of RTD under natural disposability and that of DTR under managerial disposability.

23.3.1 Possible Occurrence of Undesirable Congestion (UC)

As depicted on the left‐hand side of Figure 23.6, this chapter can identify the possible occurrence of UC under natural disposability. To examine the occurrence, this chapter uses the following model that maintains equality constraints (so, no slack variable) on undesirable outputs, which is first specified in Model (21.3):

(23.3)

See a description on variables used in Model (21.3). Model (23.3) drops slack variables related to undesirable outputs (B) so that they are considered as equality constraints. The other constraints regarding inputs and desirable outputs are considered as inequality because they have slack variables in Model (23.3).

Model (23.3) has the following dual formulation:

(23.4)

See a description on variables used for Model (21.4). An important feature of Model (23.4) is that the dual variables (w_f : URS for f = 1,…, h) are unrestricted in their signs because the constraints on undesirable outputs are expressed by equality (i.e., no slack) in Model (23.3). The dual variables are often referred to as multipliers as mentioned previously.

A unified efficiency measure of the k‐th DMU under natural disposability become

(23.5)

which incorporates the possible occurrence of UC, expressed by Equation (21.5). All variables used in Equation (23.5) are determined on the optimality of Models (23.3) and (23.4). The equation within the parenthesis, obtained from the optimal objective value of Models (23.3) and (23.4), indicates the level of unified inefficiency under natural disposability. The unified efficiency, or , is obtained by subtracting the level of inefficiency from unity as specified in Equation (23.5).

Explorative Analysis: An important advantage of Model (23.4) is that it can incorporate additional information in the form of side constraints for multiplier restrictions, as discussed in Chapter 22. For example, DEA environmental assessment usually divides the observation on each production factor by a factor average to avoid the case in which the data with a large magnitude dominates the other data with a small magnitude in the computation for assessment. Such a data adjustment is important for DEA to enhance the computational reliability. See Figure 22.3. As a result of multiplier restriction, all the observations used in this chapter are unit‐less, so indicating the importance of each production factor. The data adjustment makes it possible to incorporate additional side constraints on production factors in Model (23.4) as follows:

(23.6)

(23.7)

(23.8)

See Chapter 22, which refers to the multiplier restriction as “explorative analysis.” The analysis does not need any prior information so that it can avoid subjectivity in multiplier restriction, which is widely used in conventional DEA.

An important feature of the explorative analysis, used in Chapter 22, is that Equation (22.8) incorporates the following multiplier restrictions:

The multiplier restriction is important because we are interested in the relationship between desirable and undesirable outputs as well as that between undesirable outputs. Meanwhile, this chapter is concerned with the measurement of UEN, UC and RTD under natural disposability and UEM, DC and DTR under managerial disposability. Thus, it is necessary for us to prepare multiplier restrictions on all the three production factors, as specified in Equations (23.6), (23.7) and (23.8).

Model (23.4), equipped with Equations (23.6) to (23.8), becomes as follows:

(23.9)

The degree of on the k‐th DMU is determined by

(23.10)

where all the dual variables are identified on the optimality of Model (23.9). Equation (23.10) is different from Equation (23.5) because the side constraints in Equations (23.6) to (23.8) are additionally incorporated into Model (23.9). Therefore, Models (23.4) and (23.9) produce different measures.

After computing Model (23.9), a possible occurrence of UC is determined by the following rule under the assumption that Model (23.9) produces a unique optimal solution as discussed in Chapter 7 (i.e., unique projection and a unique reference set):

1. If for some (at least one) f, then “strong UC” occurs on the k‐th DMU,
2. If for all f, then “no UC” occurs on the k‐th DMU and
3. In the others, including for some (at least one) f, then “weak UC” occurs on the k‐th DMU.

It is important to note that if for some f and for the other f ′, then both strong UC and weak UC may coexist on the k‐th DMU. In the case, this chapter considers it as an occurrence of the strong UC on the DMU.

23.3.2 RTD Measurement under the Possible Occurrence of UC

Let the dual variables of the k‐th DMU, obtained from Model (23.9), be (i = 1, 2,…, m), (r = 1, 2,…, s), (f = 1, 2,…, h) and σ* on optimality. Then, the estimated supporting hyperplane on the k‐th DMU is expressed by

(23.11)

which is determined by , where RS_k is a reference set for the k‐th DMU and . See Proposition 21.1 in Chapter 21 that discusses mathematical implications regarding the supporting hyperplane. Note that Proposition 21.1 does not consider multiplier restriction, but Equation (23.11) has the explorative analysis. Both have the same mathematical equation, but they are different in terms of their analytical implications: congestion identification in Chapter 21 versus the determination of RTD and DTR in this chapter.

The degree (Dg) of RTD, or DgRTD, under a possible occurrence of UC, on the k‐th DMU is measured by

(23.12)

As mentioned previously, this chapter assumes that the Model (23.9) has both a unique projection of an inefficient DMU onto an efficiency frontier and a unique reference set for the DMU.

Following Equation (23.12), the type of RTD on the k‐th DMU is classified by the following rule on the k‐th DMU:

1. Increasing RTD ↔ There exists an optimal solution of Model (23.9) that satisfies all (f = 1,…,h) and ,
2. Constant RTD ↔ There exists an optimal solution of Model (23.9) that satisfies all (f = 1,…,h) and ,
3. Decreasing RTD ↔ There exists an optimal solution of Model (23.9) that satisfies all (f = 1,…,h) and ,
4. Negative RTD ↔ There is an optimal solution of Model (23.9) that satisfies for at least one and
5. No RTD ↔ All other cases excluding (a) to (d).

Difference between UC and RTD: The type of UC is identified by the sign of dual variables (). The type of UC is classified into the three categories. Meanwhile, these measures related to RTD are determined by not only the sign of dual variables () but also the sign of . The type of RTD is classified into the five categories. Figure 23.7 visually classifies the type of RTD under a possible occurrence of UC.

At the end of this section, it is necessary to summarize three concerns related to Model (23.9) and Equation (23.12) as well as the proposed RTD classification. First, Model (23.9) assumes a unique solution, so implying no occurrence on multiple projections and multiple reference sets. Second, Equation (23.12) is effective on only efficient DMUs, not inefficient ones. In the case of inefficiency, Equation (23.12) needs to incorporate the projection onto an efficiency frontier by both adjusting an inefficiency score and eliminating slacks from the observed production factors, as discussed in Chapter 20. The multiplier restriction by the proposed explorative analysis is more influential, so effective, on the measurement of RTD and DTR than slack adjustment. Therefore, this chapter, like Chapter 22, does not provide a detailed description on slack adjustment. Finally, the type of RTD is determined by measuring the upper and lower bound of . The proposed approach is just an approximation method for the RTD measurement for our descriptive convenience. Thus, the proposed RTD measurement needs to be extended further by adjusting slacks under a possible occurrence of multiple solutions (e.g., multiple references and multiple projections). See Chapter 7 on how to handle such an occurrence of multiple solutions.

23.4.1 Possible Occurrence of Desirable Congestion (DC)

As depicted on the right‐hand side of Figure 23.6, this chapter can identify an occurrence of DC under managerial disposability. To examine the occurrence, this chapter uses the following model, returning to Model (21.7), which maintains equality constraints (so, no slack variable) on desirable outputs, as specified in Equation (23.2):

(23.13)

Model (23.13) drops slack variables related to desirable outputs so that they are considered as equality constraints. The other groups of constraints on inputs and undesirable outputs have slacks so that they can be considered as inequality constraints.

Model (23.13) has the following dual formulation:

(23.14)

An important feature of Model (23.14) is that the dual variables (u_r : URS for r = 1,…, s) are unrestricted in these signs because Model (23.13) does not have slacks related to desirable outputs.

A unified efficiency score, or UEM(DC) of the k‐th DMU, with a possible occurrence of DC, under managerial disposability is determined by

(23.15)

where all variables are determined on the optimality of Models (23.13) and (23.14). The equation within the parenthesis, obtained from the optimality of Models (23.13) and (23.14), indicates the level of unified inefficiency under managerial disposability. The unified efficiency is obtained by subtracting the level of inefficiency from unity.

As discussed on Model (23.9), Model (23.14) can incorporate additional information by explorative analysis that is formulated as follows:

(23.16)

The degree of is determined by

(23.17)

where all the dual variables are identified on the optimality of Model (23.16). Equation (23.17) is different from Equation (23.15) because Model (23.16) incorporates the additional side constraints in Equations (23.6) to (23.8). Thus, Equations (23.15) and (23.17) produce different measures.

After solving Model (23.16), this chapter can identify the possible occurrence of DC, or eco‐technology innovation, by the following rule under the assumption on a unique optimal solution (i. e. unique projection and a unique reference set):

1. If for some (at least one) r, then “strong DC” occurs on the k‐th DMU,
2. If for all r, then “no DC” occurs on the k‐th DMU and
3. In the others, including for some (at least one) r, then “weak DC” occurs on the k‐th DMU.

Note that if for some r and for the other r′, then the weak and strong DCs coexist on the k‐th DMU. This study considers it as the strong DC, so indicating eco‐technology innovation on undesirable outputs. If for all r. then it indicates the best case because an increase in any desirable output always decreases an amount of undesirable outputs. Meanwhile, if is identified for some r, then it indicates that there is a chance to reduce an amount of undesirable output(s). Therefore, this chapter considers the second case as an occurrence of DC.

23.4.2 DTR Measurement under the Possible Occurrence of DC

Let the dual variables of the k‐th DMU, obtained from Model (23.16), be (i = 1, 2,…, m), (r = 1, 2,…, s), (f = 1, 2,…, h) and σ *. Then, an estimated supporting hyperplane on the k‐th DMU is specified by

(23.18)

The equation is characterized by , where RS_k is a reference set of the k‐th DMU along with .

This chapter assumes that Model (23.16) has both a unique projection of an inefficient DMU onto an efficiency frontier and a unique reference set for the projected DMU. Then, the degree (Dg) of DTR, or DgDTR, is measured by

(23.19)

Consequently, the type of DTR is classified by the following rule on the k‐th DMU:

1. Increasing DTR ↔ There is an optimal solution of Model (23.16) that satisfies all (r = 1,…, s) and .
2. Constant DTR ↔ There exists an optimal solution of Model (23.16) that satisfies all (r = 1,…, s) and .
3. Decreasing DTR ↔ There is an optimal solution of Model (23.16) that satisfies all (r = 1,…, s) and .
4. Negative DTR ↔ There is an optimal solution of Model (23.16) that satisfies for at least one and
5. No DTR ↔ All other cases excluding (a) to (d).

All the concerns discussed for the measurement of RTD in Section 23.3.2 are applicable to DTR. However, it is important to note that the type of DTR is determined by measuring the upper and lower bounds of . The proposed approach for DTR is just an approximation method for the DTR measurement.

Difference between DC and DTR: The occurrence and type of DC are identified by the sign of dual variables (). The type of DC is classified into three categories. Meanwhile, these measures related to DTR are determined by not only the sign of dual variables () but also the sign of . The type of DTR is classified into five categories. Figure 23.8 visually describes an occurrence of DC and DTR classification.

23.5.1 Data

This chapter presents a data set from the National Bureau of Statistics of the People’s Republic of China (http://www.stats.gov.cn/tjsj/). The National Bureau of Statistics provides a regional data set on energy, environment and other aspects of the whole country every year. Using the data set, this chapter examines 30 provinces of China, including four well‐developed municipalities directly under the control of central government (i.e., Beijing, Shanghai, Tianjin and Chongqing) but excluding Tibet, Hong Kong and Macau because of our limited data accessibility on these three regions during 2012.

We utilize four desirable outputs: Gross Regional Product (GRP), value‐added of the primary industry, the secondary industry and the tertiary industry; three undesirable outputs: PM10, SO₂ and NO₂; five inputs: investment in energy industry, coal consumption, oil consumption, natural gas consumption and electricity consumption. Descriptive Statistics are listed in Sueyoshi and Yuan (2016b).

Data Accessibility: The supplementary section of Sueyoshi and Yuan (2016b) lists the full data set used in this chapter.

It is important to note that the three undesirable outputs are not GHG emissions, rather SO₂ and NO₂, which are acid rain gases that produce much more serious damages than GHG. This chapter clearly understands that PM 2.5 has more serious impacts than PM 10 in terms of health damage on Chinese people. However, this chapter could not access any information on PM 2.5 during the observed annual period (2012).

To control for data magnitudes, this chapter further calculates a provincially adjusted index to be used in the proposed DEA models for all variables. The index of each factor is the ratio of the actual value divided by the average of each one. Note that this chapter sets ε_s = 0.0001 in the computation of the proposed DEA models.