Reglamento actual para el manejo de los desechos peligrosos en Cuba

In this section, simple numerical examples of computation of elasticity measures in the CRS technology are considered. All examples use the same data set shown in Table 4.1 but use different sets A, B and C to illustrate various possible outcomes.

Table 4.1. The Data Set for Illustrative Example

Unit Input Output 1 Output 2 Output radial efficiency in the CRS technology

E 1 2 3 1

F 2 6 5 1

G 1 4 1 1

H 1 1 3 1

The CRS technology induced by the four observed units is shown in Figure 4.1 as the cone spanning the input axis and the rays OK, OE, OF, OG and OM. It is easy to show that all four observed units are technically efficient, that is their output radial efficiency (in the CRS technology, calculating the input radial efficiency would, of course, produce the same result) in the CRS technology is equal to 1. However, only E, F and G are (fully) efficient units. Unit H is outperformed by E and is inefficient.

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Figure 4.1. The CRS technology and One-sided Elasticities in Scenario 3

Scenario 1. Define A={input}, B={output 1} and C={output 2}. To see if the right-hand and left-hand elasticities exist at the four observed units and, in the case of their existence, calculate their values, programs (4.7) and (4.8) are solved from Theorem 4.2. For example, the right-hand elasticity at unit E is found by solving program (4.7):

εA,B + _{(E)=min 1ν} 1 Subject to 1ν1−3µ2=1 1ν1−2µ1−3µ2≥0 2ν1−6µ1−5µ2≥0 Output 2

4

3

2

1

0

₁

₂

₃

₄

2

G

K

5

5

6

L

H

E

M

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1ν1−1µ1−3µ2≥0 2µ1=1

ν₁,µ₁,µ₂≥0

The results of computations are shown in Table 4.2. For reference, the second column of this table also shows the value of output response function β(1) , which was computed independently by using model (4.4). The computation of β(1) _{is not needed in practical}

situations but it helps us understand some of the results below. Note that β(1)=1 for units E, F and G, and β(1)≠1 for unit H. Consequently, the former three units satisfy Assumption 4.1, and unit H does not.

Table 4.2. Elasticity Measures for Scenario 1 Unit _β(1) Optimal value or

diagnostics of program (4.7) Optimal value or diagnostics of program (4.8) εA,B + _ε A,B − E 1 4 Unbounded 4 Undefined F 1 1.56 2.6 1.56 2.6 G 1 1 1.16 1 1.16

H 2 Infeasible Infeasible Undefined Undefined

As discussed above, the finite optimal values of programs (4.7) and (4.8) mean that the corresponding one-sided elasticities exist and they are equal to those optimal values. These are shown in the last two columns of Table 4.2 and are consistent with part (b) of Theorem 4.5.

For unit E program (4.8) is unbounded. This means that a proportional marginal reduction of the input (which constitutes the set A) is not possible in the CRS technology under

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consideration – any such reduction would lead outside the boundaries of the technology. For this reason, the given elasticity measure is not defined at unit E.

For unit H, both programs (4.7) and (4.8) are infeasible. By Theorem 4.3 this means that unit H does not satisfy Assumption 4.1 (because, for this unit, β(1)≠1). For this reason, the notion of elasticity (for the given sets A, B and C) is not applicable to unit H.

Scenario 2. Define A={input, output 1}, B={output 2} and C=∅. The elasticity measures defined by this scenario can be calculated in two ways. First, programs (4.7) and (4.8) for this scenario may be formulated and solved. For each of the four units, both programs (4.7) and (4.8) have the same optimal value of 1. This means that all four units (including the inefficient unit H) satisfy Assumption 4.1 and the elasticity measure for this scenario is equal to 1 at each of them (the reference to one-sided elasticities can be omitted because these are equal).

Second, the same results immediately follow from Theorem 4.4, removing the need to solve programs (4.7) and (4.8). However, Theorem 4.4 is applicable only to units that satisfy Assumption 4.1, which means that the condition β(1)=1 still needs to be verified. Because units E, F and G are efficient and output 2 is strictly positive, by Theorem 4.1, Assumption 4.1 is true. Unit H is inefficient, and the required equality β(1)=1 _{can be established by} solving model (4.4).

Table 4.3 summarises the results of calculations in Scenario 2. Note that, based on Theorem 4.4, the same results are obtained in the case of standard scale elasticity defined by the following sets: A={input}, B={output 1, output 2} and C=∅.

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Table 4.3. Elasticity Measures for Scenario 2 Unit _β(1) Optimal value or

diagnostics of program (4.7) Optimal value or diagnostics of program (4.8) ε_A+_{,B ε} A,B − E 1 1 1 1 1 F 1 1 1 1 1 G 1 1 1 1 1 H 1 1 1 1 1

Scenario 3. Define A={output 2}, B={output 1} and C={input}. By solving programs (4.7) and (4.8) for the given scenario, the results shown in Table 4.4 are obtained. Note that these are consistent with part (c) of Theorem 4.5. The arrows in Figure 4.1 show the directions of marginal movements corresponding to the elasticity calculations in this scenario.

Table 4.4. Elasticity Measures for Scenario 3 Unit _β(1) Optimal value or

diagnostics of program (4.7) Optimal value or diagnostics of program (4.8) ε_A+_{,B ε} A,B − E 1 Unbounded -3 Undefined -3 F 1 -1.66 -0.55 -1.66 -0.55 G 1 -0.16 0 -0.16 0

H 2 Infeasible Infeasible Undefined Undefined

For Scenario 3, the right-hand elasticity for unit G, left-hand elasticity for unit E and both one-sided elasticities for unit F are defined with finite negative values given in Table 4.4. These elasticities represent feasible movements along the boundary of the given technology and presented in Figure 4.1 with corresponding arrows. The negative values indicate the inverse relationship since the scenario considers only outputs in both changing and responding sets. A marginal increase of output 2 at the given unit is responded by a reduction in output 1 at that unit and vice versa.

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As in Scenario 1, it is observed that the one-sided elasticities may be undefined at particular units for two reasons. Unit E has its right-hand elasticity undefined because it is not possible to increase its output 2 (set A) while keeping its input (set C) constant. Since the plane defined by OE and OK is the outside boundary of the technology, moving to the right from unit E while keeping the input constant results in leaving the boundaries of the technology. This also follows from Theorem 4.2 by noting that the objective function in program (4.7) is unbounded.

At unit H, neither one-sided elasticity is defined because, at this unit, β(1)≠1. This fact can be established by solving program (4.4) or by using Theorem 4.3 with the fact that both programs (4.7) and (4.8) are infeasible.

At unit G, the left-hand elasticity is obtained as 0, which is presented with a vertical arrow in Figure 4.1. While the input is kept constant, a reduction in output 2 is basically a vertical movement along the segment GM, which does not cause a marginal change in output 1.