Model formulation - The Case Study 1 - 汽車客運業績效評估之研究

CHAPTER 5 The Case Study 1

5.2 Model formulation

5.2.1 The hyperbolic graph measure

As indicated by Fielding (1987), cost effectiveness measures the relationship between input and consumed service such as passenger-kilometer or passenger trips. It is concerned with demand-side relationships. Assume that one is interested in cost effectiveness, this requires that one simultaneously adjusts input and output quantities, since one wishes to increase output quantities and decrease input quantities concurrently. A technology with a

hyperbolic graph efficiency approach which seeks the maximum simultaneous equiproportionate expansion for the output and contraction for the inputs is modeled here.

In contrast to radial contractions or expansions of observed data, the model introduced here is a hyperbolic path to the frontier of technology.

To measure technical efficiency, one can follow Fare et al’s (1985, 1994) hyperbolic Farrell measure. Let y∈R^M⁺ and x∈R^N⁺ denote respectively vectors of outputs produced and inputs employed by an individual station which represents a decision making unit (DMU) in this study. The graph reference set, T

( ) ( )

x,y =

{

x,y :x can producey

}

, satisfies constant returns to scale (CRS) and the strong disposability of inputs and outputs.

The hyperbolic graph measure of technical efficiency is defined as:

( )

x y

{ (

x y

)

}

F_g , =min λ: λ , λ ∈ . (5.1) The term graph is indicated by the subscript g . If F_g

( )

x,y =1, then the firm operates on the frontier of T ,

( )

x y , while F_g

( )

x,y <1 indicates that the firm operates inside

( )

x y

T , . Following Fare et al. (1985, 1994) the input-oriented Farrell measure is defined as

( )

x y

{ (

x y

)

}

F_i , =min λ: λ , ∈ , with the input measure F_i

( )

x,y equal to the square of the hyperbolic measure F_g

( )

x,y (see Fare et al. (2002) for more details in such a relationship).

The reciprocal of F_i

( )

x,y serves as the output-oriented Farrell measure F_o

( )

x,y if and only if technology exhibits CRS. Thus, CRS implies

(

^Fg

( )

^x^,^y

)

² = ^Fi

^{( )}

^x^,^y ^, (5.2)

( ( ) ) _{( )}

y x y F

x F

g ,

, ² = 1 (5.3)

Those stations that are able to minimize the proportional or scaling factor λ, relative to other stations, are considered perfectly efficient and are thus found on the “best practice frontier” with a λ-value of “1”. Those firms which are less efficient are found at some distance from the frontier, with that distance being the basis for measuring their inefficiency;

the λ-value of these stations will be “less than 1”. Figure 5.1 illustrates the case with only one input and one output. Given observed

(

x ,j yj

)

, Fg

(

xj,yj

)

simultaneously expands

y and contracts j x at the same rate, following the hyperbolic path shown in the Figure _j

5.1. In contrast, the Fi

(

xj,yj

)

measure of input technical efficiency contracts x _j following the horizontal path to the graph. The Fo

(

xj,yj

)

measure of output technical

efficiency expands y to the graph, holding _j x fixed, i.e., following the vertical path to _j the graph.

Figure 5.1 Comparison of Hyperbolic, Input, and Output Measures of Technical Efficiency

5.2.2 “Return to the dollar” and the hyperbolic graph measure

If one knows the prices of the inputs w and the outputs _s p , then one can measure _s overall efficiency as follows:

(

x y w p

)

(

x y w p

) (

F x y w p

)

O_g , , , = _g , , , ⋅ _g , , ,

, (5.4) where A_g

( )

⋅ denotes the allocative efficiency. It is difficult to know all of the input and output prices of each station in detail, but it is easier to obtain the observed revenue and observed cost. Thus, the hyperbolic measure is related to “return to the dollar” which can be seen as the dual to the hyperbolic technical efficiency measure (Färe et al., 2002).

y x Fi(xj, j)⋅ j

)) , ( , ) , (

(Fg xj yj ⋅xj yj Fg xj y_j

y x y Fo( j, _j)⋅ _j

) , ( yx T

) , (xj yj

x y

Let w∈R^N⁺⁺ represent the set of strictly positive input prices and p∈R^M⁺⁺output prices. With this additional information, the maximum profit can be defined as follows:

(

p,w

)

=max

{

py−wx:

( )

x,y ∈T

}

π , (5.5) Thus, one can

(

p,w

)

≥ py−wx

π _{for all}(x,y)∈T , (5.6)

since

(

^xFg(^x,^y),^y ^Fg(^x,^y)∈^T

)

, one has

(

p,w

)

≥ py F_g(x,y)−F_g(x,y)wx

π (5.7) However, the maximum feasible profit π^* is equal to zero.

Following Färe et al. (2002) the dual relationship between the hyperbolic graph measure and “return to the dollar” is defined as:

(

( )

^x^{, y}

)

wx py

≤ g (5.8) In order to derive a graph measure analog of the “return to the dollar”, one must introduce Georgeson-Roegen’s “return to the dollar” measure py wx , where py represents observed revenue and wx stands for observed cost. There is no need to know input or output prices in calculating the “return to the dollar”.

Following Farrell (1957) the allocative efficiency can be defined as a residual, i.e. the value of the allocative efficiency measure is

(

( )

^x¹^,^y

)

wx AE py

= (5.9)

thus

TE wx AE

py = ⋅ . (5.10)

The technical efficiency component TE equals

(

^Fg

( )

^x^{, y}

)

², given an indication of how well resources are being managed, since it represents the gap between the DMU and the best

practice frontier. This takes values between zero and one. The allocative measure AE gives an indication of the appropriateness of the mix of inputs. On the other hand, AE can also be rearranged as

are technically efficient quantities of outputs and inputs, respectively. If prices pˆ and ^wˆ support yˆ and ^xˆ, then ^AE⁼ ^p^ˆ^y^ˆ ^w^ˆ^x^ˆ⁼¹, The term AE represents the short-run prices p deviating from their optimal long-run values pˆ with respect to the short-run prices w that deviate from their optimal long-run values wˆ. Since AE is computed in reference to the “return to the dollar” which may take a value bigger or smaller than one, AE may also take values smaller or bigger than one. If the value of AE is greater than one, then it implies the ability to distort output prices at higher rates than input prices (Fare et al. 2002).

Since this is an index based on discrete time, each station here will have an index for each year. This entails calculating the hyperbolic efficiency incorporating an environmental variable as well as using linear programming methods. In the interest of simplicity, however, neither the complete picture of the complexity inherent in the station efficiency problem being modeled and conveyed in Figure 5.1, nor the working of constraints expressing the existence of an environmental variable (non-discretionary input), are depicted in the graph.

Nonetheless, such constraints are used in this modeling effort in order to “control” for the service area population of the station effort faced by each station. Since contraction does not take place along these dimensions, comparisons of “like-to-like”, in the form of service area population, are facilitated by the model.

The hyperbolic measure of technical efficiency,F_g

( )

x,y , for each DMU is calculated by solving the following linear program:

k corresponding to the level of efficiency. One can assume that there are k =1 L, ,K stations with p=1 L, ,P environmental variables which use n=1 L, ,N inputs to produce m=1 L, ,M outputs. z is an intensity variable. _k

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Model formulation

CHAPTER 5 The Case Study 1

5.2 Model formulation

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