CHAPTER 3 Literature Review
3.4 Limitations of previous research
After reviewing the current researches for transit efficiency analysis, it was found that these literature have some limitations.
First, it is especially unfortunate that only a handful of frontier studies have focused on the effects of privatization and regulatory changes (De Borger et al., 2002).
Second, most of the extant literature restrict their analysis to the use of technical efficiency (TE) rather than allocative efficiency (AE), due to
1. lack of price information, and
2. allocative efficiency assumes cost minimizing.
Third, a large strain of previous studies on measuring firm’s efficiency are typically conducted without taking into account undesirable outputs which may not be freely or costlessly disposable.
Fourth, few DEA studies relating to transportation organizations which engage in various activities (services) deal with the allocation problem of shared inputs in a proper way.
Fifth, the perishability of the transportation service produced, and the fact that only a proportion of the service produced are actually consumed is often neglected in transit performance measures.
The potential impact of this omission is two-fold. First, by not including these important issues in an efficiency analysis, the transit system’s primary activities are not being fully knowledged. The analysis is thus carried out by using a conventional model, this could lead to less demanding than by employing a refined model. Second, by including these issues in measure of transit efficiency, the accuracy and representation of the efficiency analysis for providing performance targets for inputs and outputs and for identifying benchmark operating practices can be improved.
CHAPTER 4
The Basic Data Envelopment Analysis (DEA) Model
The purpose of performance measurement is to compare behavior of organization over time, across space, or both. Furthermore, benchmarking comparisons can be made within a sector or across sectors, comparisons can be limited to the national level or may have an international character etc (De Borger et al., 2002). The focus in this performance evaluation of Taiwanese bus industry is on issue of efficiency and effectiveness. This chapter begins by distinguishing between (technical) efficiency and productivity－two related concepts that tend to be confused in daily usage, and which lie at the foundation of efficiency analysis. The discussion here closely follows the presentation found in Coelli et al. (1998) and De Borger et al. (2002).
4.1 Efficiency and productivity
The terms, efficiency and productivity, are often used interchangeably. But this is unfortunate because they are not precisely the same things. Though the concepts are related, in general, productivity can be thought of as being a broader concept than efficiency. Both concepts can be related to a production function which is the primitive (in the single output case) representing the transformation of inputs to output. From a conceptual viewpoint, efficiency measurement can be classified into the frontier and non-frontier approaches and from an implementation perspective, into parametric and non-parametric. These are discussed below. The productivity of a firm is defined as the ratio of the output(s) that it produces to the input(s) that it uses.
productivity = outputs/inputs (4.1)
y
A
O x
C
B F′
To illustrate the distinction between the terms, it is useful to consider a simple production process in which a single input (x) is used to produce a single output ( y ). The line O ′F in Figure 4.1 represents a production frontier which may be used to define the relationship between the input and the output. The production frontier represents the maximum output attainable from each input level. Hence it reflects the current state of technology in the industry. Firms in that industry operate either on that frontier, if they are technically efficient, or beneath the frontier if they are not technically efficient. Point A represents an inefficient point whereas points B and C represent efficient points. A firm operating at point A is inefficient because technically it could increase output to the level associated with the point B without requiring more input.
Figure 4.1 Production Frontiers and Technical Efficiency
Figure 4.1 also illustrates the concept of a feasible production set which is the set of all input-output combinations that are feasible. This set consists of all points between the production frontier, O ′F , and the x-axis (inclusive of these bounds). The points along the production frontier define the efficient subset of this feasible production set. The primary advantage of the set representation of a production technology is made clear when discussing multi-input/multi-output production and the use of distance functions below.
x A
y
O
C B
F′0
F′1
optimal scale
To illustrate further the distinction between technical efficiency and productivity, Figure 4.2 is utilized. In this figure, a ray through the origin is used to measure productivity at a particular data point. The slope of this ray is y x and hence provides a measure of productivity. If the firm operating at point A were to move to the technically efficient point B, the slope of the ray would be greater, implying higher productivity at point B. However, by moving to the point C, the ray from the origin is at a tangent to the production frontier and hence defines the point of maximum possible productivity. This latter movement is an example of exploiting scale economies. The point C is the point of (technically) optimal scale.
Operation at any other point on the production frontier results in lower productivity.
Figure 4.2 Productivity, Technical Efficiency, Technical change and Scale Economies
From this discussion, a conclusion can be drawn that a firm may be technically efficient, but may still be able to improve its productivity by exploiting scale economies. Given that changing the scale of operations of a firm can often be difficult to achieve quickly, technical efficiency and productivity can be given short-run and long-run interpretations.
In addition, when referring to productivity, it is total factor productivity (TFP) that is referred to. It is a productivity measure involving all factors of production. One of the popular
measure is the Malmquist TFP index (See Caves et al. 1982 for detail) Other traditional measures of productivity, such as labour productivity in a factory, fuel productivity in power stations, and land productivity (yield) in farming, are what is known as partial measures of productivity. These partial productivity measures can provide a misleading indication of overall productivity when considered in isolation.
There is a variety of methods that can be derived for measuring performance based on the concepts of efficiency and effectiveness. In public agencies, efficiency should be considered separately from effectiveness. Efficiency is the relationship between inputs and outputs of what is referred to as “productive” or “technical” efficiency in the economic literature. Effectiveness, on the other hand, refers to the use of outputs to achieve objectives, or service consumption. However, in addition to efficiency and effectiveness, relationships also exist between the efficiency and effectiveness criteria. For example, the frequency of service (an efficiency criterion) affects riders’ waiting time (an effectiveness criterion).
4.2 Frontier methodologies
To estimate production or cost frontiers, methods have been developed for analyzing time series, cross-section or panel data. Once frontiers have been estimated, productivity changes can directly be derived from shifts in the frontier over time. Technical inefficiency estimates are readily available as well, as is illustrated below.
Existing approaches to reconstruct production frontiers can be usefully distinguished along the lines below (A general survey is found in Lovell (1993). Färe et al. (1994) overviewed non-parametric methods, while Greene (1997) surveyed parametric frontiers).
1. Parametric versus non-parametric frontier specifications:
(1) The parametric approach assumes that the boundary of the production possibility set can be represented by a particular functional form with constant parameters.
(2) The non-parametric approach imposes minimal regularity axioms on the production possibility set and directly constructs a piecewise technology on the sample.
2. Deterministic versus stochastic frontier specifications:
(1) Stochastic methods make explicit assumptions with respect to the stochastic nature of the data by allowing for measurement error.
(2) Deterministic methods take all observations as given and implicitly assume that these observations are exactly measured.
Combining these distinctions yields a four-way classification, as illustrated in Table 4.1.
Since the literature on stochastic non-parametric frontiers is still burgeoning (recent proposals include resampling (bootstrap), chance constrained programming, etc.) and no consensus has yet emerged, this issue was not pursued here (Grosskopf 1996). The three other cells of Table 4.1 are introduced as follows.
Table 4.1 Taxonomy of Frontier Methodologies
Measurement error
Functional form Deterministic Stochastic
Parametric Corrected OLS, etc. frontiers with explicit assumptions (exponential, half-normal, etc.) for the TE distributions
Non-parametric FDH,DEA-type models, etc. resampling; chance constrained programming, etc.
Source: De Borger et al. (2002).
First, the early literature often used deterministic parametric frontier methods. However, given that they combine the most restrictive assumptions (deterministic and parametric) they are no longer very popular (Lovell 1993).
Second, the popular parametric frontier approach (sometimes referred to as the econometric frontier approach) is the stochastic frontier approach (SFA) which is based on the econometric regression theory. The SFA (see Aigner and Chu, 1968, Aigner et al. 1997, and Meeusen and van den Broeck, 1977, for details) specifies a functional form (e.g. translog or
Cobb-Douglas) for the cost, profit, or production relationship among inputs, outputs, and environmental factors, and allows for random error. Both the inefficiencies and the random errors are assumed to be orthogonal to the input, output, or environmental variables specified in the estimating equation.
Third, the popular nonparametric method is the Data Envelopment Analysis (DEA) technique which is based on the mathematical programming approach. The DEA put relatively little structure on the specification of the best practice frontier. DEA is a linear programming technique where the set of best practice or frontier observations are those for which no other decision-making unit (DMU) or linear combination of units has as much or more of every output (given inputs) or as little or less of every input (given outputs). The DEA frontier is formed as the piecewise linear combinations that connect the set of these best practice observations, yielding a convex production possibilities set. As such, DEA does not require the explicit specification of the form of the underlying production relationship. DEA permits efficiency to vary over time and makes no prior assumption regarding the form of the distribution of inefficiencies across observations except that observations that are not dominated are 100% efficient.
The nonparametric approaches impose less structure on the frontier. The nonparametric approaches, however, do not allow for random error owing to weather, strikes, luck, data problems, or other measurement errors. If random error exists, measured efficiency may be confounded with these random deviations from the true efficiency frontier. As well, statistical inference and hypothesis tests cannot be conducted for the estimated efficiency scores.
4.3 Relative merits and drawbacks of the methods
The parametric approach of stochastic production frontiers and the nonparametric approach of data envelopment analysis presented in the previous sections, along with their
specific research models identified for carrying out this study, possess their own strengths and weaknesses. What is interesting and useful for this study is that they are mostly complementary; i.e., the weaknesses of one approach are oftentimes the strengths of the other approach.
The following Table 4.2 summarize the comparative merits and potential drawbacks associated with each approach.
Table 4.2 Comparison between DEA and SFA Approaches
SFA approach DEA approach
Strengths
♦ DEA assumes all deviations from the frontier are due to inefficiency. If any noise is present (e.g., due to measurement error, weather, strikes, etc.), then this may influence the placement of the DEA frontier and hence the measurement of productive efficiency more than would be the case with stochastic production frontiers; and
♦ tests of hypotheses regarding the existence of productive inefficiency and the structure of the production technology can be performed without difficulty in a stochastic frontier approach.
♦ it doesn’t need to specify a distribution for the inefficiency part;
♦ it doesn’t need to specify a functional form for the production process;
♦ it can accommodate multiple outputs; and
♦ it can include environment factors in the model.
Weaknesses
♦ the need to specify a statistical distribution for the inefficiency component;
♦ the need to specify a functional form for the production frontier; and
♦ it is more difficult to accommodate multiple outputs.
♦ measurement errors and other noise may influence the shape and position of the envelopment surface, and hence the derived scores of productive efficiency;
♦ when one has few observations and many inputs or outputs, many of the firms will appear on the DEA frontier, with perfect scores of one (curse of dimensionality).
DEA provides a comprehensive picture and evaluation of organizational performance, without the constraints and assumptions of the SFA. By Simultaneously handling multiple inputs and outputs without making judgments on their relative importance, and by not requiring the specification of a functional form for the input-output relationship, DEA offers a more complete examination of performance. Since it is difficult to find a commonly agreement upon functional form relating inputs consumed to outputs produced, the multidimensional nature of the bus transit industry makes it an idea application area for DEA.
4.4 Distance functions
The frontier efficiency measures which the present author focuses mainly on in this dissertation, are based on the concept of distance functions for multi-input and multi-output technology. One may specify both input and output distance functions. Distance functions describe multi-input and multi-output production technology without the need to specify a behaviorual objective (such as profit-maximization or cost-minimization). An input distance function characterizes the production technology by looking at a minimal proportional contraction of the input vector, given an output vector. An output distance function considers a maximal proportional expansion of the output vector, given an input vector.
4.4.1 Input and output distance functions
Following Fare and Primont (1995), the notation x and y is used to denote a non-negative 1K× input vector and a non-negative M×1 output vector, respectively. The technology set is then defined as
( ) {
x y xS= , : can produce y
}
(4.2) That is, the set of all input-output vectors( )
x,y , such that x can produce y .Given some assumptions, the input and output distance functions are defined on the input set, L
( )
y , and output set, P( )
x , which are assumed to satisfy some properties and are redefined from the production technology set S, respectively, as:Input distance function Output distance function
( )
x y{ ( ) ( )
x L y}
di , =max ρ: ρ ∈ (4.3) d0
( )
x,y =min{
δ :(
y δ)
∈P( )
x}
(4.4) Two inputs, x1 and x2, and an output vector, y , are used to illustrate the input distance function in a two-dimensional diagram as shown in Figure 4.3. Here the input set,( )
yL , is the area bounded from below by the isoquant, Isoq-L
( )
y . The value of the distanceA
x1
C B
O
d(x,y) = 1
x1A
x2A
x2
Isoq - L(y) OA/OB = d(x,y) > 1
L(y)
C
O y1
A
P(x) y2
y1A
y2A
B
d0 (x,y) = 1 PPS - P(x) OA/OB < 1 = d0 (x,y)
function for the point, A , which defines the production point where firm A uses x1A of input 1 and x2A of input 2, to produce the output vector, y , is equal to the ratio
OB
=OA
p .
Figure 4.3 Input Distance Function and Input Requirement Set
The output distance function may be illustrated using the production possibility curve concept in a two-dimensional space as in Figure 4.4. Assuming two outputs, y1 and y2 are produced using the input vector, x. Thus, the production possibility set, P
( )
x , is the area bounded by the production possibility frontier, PPS−P( )
x , and y1 and y2 axes. The value of the distance function for the firm using input level x to produce the outputs defined by the point A is equal to the ratio δ =OA OB. Points B and C are on the production possibility surface and hence would have distance function values equal to 1.Figure 4.4 Output Distance Function and Output Requirement Set
4.5 Efficiency measurement concepts
Much of the discussion here draws on Farrell’s (1957) original ideas, Coelli (1996), and Coelli et al (1998). Modern efficiency measurement begins with Farrell (1957) who drew upon the work of Debreu (1951) and Koopmans (1951) to define a simple measure of firm efficiency that could account for multiple inputs. He identified two components of a firm’s efficiency: technical efficiency (TE), which reflects the ability of a firm to obtain maximal output from a given set of inputs, and price (allocative) efficiency (AE), which reflects the ability of a firm to use the inputs in optimal proportions, given their respective prices. These two measure are then combined to derive a measure of overall (total) economic efficiency (EE).
4.5.1 Input-oriented and output-oriented measures
Farrell illustrated his ideas using a simple example involving firms that use two inputs (x1 and x2) to produce a single output ( y ), under the assumption of constant returns to scale (CRS). Specifically, it addresses the question: “By how much can input quantities be proportionally reduced without changing the output quantities produced ?” Knowledge of the unit isoquant of the fully efficient firm, represented by S ′S in Figure 4.5, permits the measurement of technical efficiency. Note that the technical efficiency here is the inverse of the input distance function as defined in Figure 4.3.
If the input price ratio, represented by the line AA ′ in Figure 4.5 is known, allocative efficiency may also be calculated.
The output-orientated technical efficiency measure addresses the question: “By how much can output quantities be proportionally expanded without altering the input quantities used?” Output-orientated measures are then discussed by considering the case where production involves two outputs (y1 and y2) and a single input (x). If assuming constant
Figure 4.5 Input-oriented Technical and Allocative Efficiencies
eturn to scale, the technology can be represented by a unit production possibility frontier in two dimensions. As shown in Figure 4.6 where the line ZZ ′ is the unit production possibility frontier. If one has price information then one can draw the isorevenue line D ′D , and define the allocative efficiency.
Figure 4.6 Output-oriented Technical and Allocative Efficiencies
The input-oriented and output-oriented measures are defined as:
Input-oriented Output-oriented OP
OQ
TEi = TE0 =OA OB
OQ OR
AEi = AE0 =OB OC
OP OR AE
TE
EEi = i× i = (4.5) EE0 =TE0 ×AE0 =OA OC (4.6) P
O
Q A
R
x1/y x2/y S
S′
Q′
A′
D′
Z′
B′
y1/x O
Z
A
D C
B y2/x
All the above efficiency measures lie between zero and one. A value of one indicates the firm is fully technically (or allocatively or economically) efficient. It is also worth noting the following points about all the efficiency measures discussed above. All are measured along a ray from the origin to the observed production point. Hence they hold the relative proportions of inputs (or outputs) constant. One advantage of these radial efficiency measures is that they are unit invariant. That is, changing the units of measurement (e.g., measuring vehicle age in kilo-meters instead of years) will not change the value of the efficiency measure. The Farrell input-and output-orientated technical efficiency measures can be shown to be equal to the input and output distance functions discussed in Shephard (1970).
4.6 Basic concept of DEA method
Broadly speaking, DEA evaluates efficiency by reference to the best use of resources that other comparable production units are observed to make, the so-called “best-practice”
frontier.
Data envelopment analysis (DEA) involves the use of linear programming methods to construct a non-parametric piecewise surface (or frontier) over all data, so as to be able to calculate efficiencies relative to this surface. The DEA is based on the piecewise-linear convex hull approach to frontier estimation originally proposed by Farrell (1957). Charnes, Cooper and Rhodes (1978) proposed a so called CCR model that had an input orientation and assumed constant returns to scale (CRS). Banker, Charnes and Cooper (1984) extended the model by imposing a variable returns to scale (VRS) assumption, usually termed BCC model. Following is a description of the basic DEA model.
DEA can be thought of as an empirical specification of activity analysis, which is itself a piecewise-linear specialization of the general notion of a production possibility set or technology.
All DEA models begin with the idea of input and output bundles, which are assumed to satisfy the following four assumptions. First, they are non-negative. Second, there is at least one feasible input-output pair, that is, the production possibility set S is non-empty. Third, the production possibility set S (and hence the input set, L
( )
y , and the output set, P( )
x ) contain their boundaries. This implies that all three sets contain efficient input-output combinations, or that the production frontier exists. Finally, in order to produce strictlyAll DEA models begin with the idea of input and output bundles, which are assumed to satisfy the following four assumptions. First, they are non-negative. Second, there is at least one feasible input-output pair, that is, the production possibility set S is non-empty. Third, the production possibility set S (and hence the input set, L