Capítulo II: Propuesta de modificaciones al Reglamento para el manejo integral de
2.3 Propuesta de modificación del reglamento actual para el manejo de los desechos
The derivation of necessary programs for elasticity analysis of input sets in VRS technology with production trade-offs is given in this section. Since it is related with input sets, Assumption 4.2 is considered. Definitions, 4.4, 5.1 and 5.2 also hold for the developments in this section. Theorem 4.6 is modified to Theorem 5.10 in order to represent VRS technology. Theorems 4.7 and 4.8 are modified into Theorems 5.11 and 5.12 such that models can handle the production trade-offs. Because of the similarities, the proofs for
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Theorems 5.10, 5.11 and 5.12 are not given; related proofs to those theorems are referred in Appendix B.
Theorem 5.10. If the unit (X0,Y0)∈TVRSTO is efficient and the vector X0B has at least one
strictly positive component then Assumption 4.2 is satisfied.
If the set B contains only inputs, the input response function is defined in VRS technology with trade-offs ( TVRSTO) as follows:
ˆ β(α)=min β ≥0 (αX0A,βX 0 B,X 0 C,αY 0 A,Y 0 C)∈T VRSTO
{
}
(5.15)The input response function βˆ(α) is defined as in (5.15) and obtained as the optimal value in the following linear program in TVRSTO, where β is a variable and α is a fixed value. In this case, the trade-off coefficient matrices P and Q are divided into sub-matrices as, PA, PB,
PC, QA and QC representing the trade-off coefficients for changing (A), responding (B) and
remaining constant (C) sets of inputs and outputs.
ˆ β(α)=min β (5.16) Subject to −XAλ −PAπ ≥ −αX 0 A −XBλ −PBπ+βX 0 B≥0 −XCλ −PCπ ≥ −X 0 C YAλ+QAπ ≥ αY0 A
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λ,π ≥0, β Sign free
The right-hand (left-hand) elasticity of response of the input vector X0B with respect to marginal proportional changes of the vectors X0A and
Y0
A is defined as the right (left)
derivative of the function β(αˆ ) at α =1 in (4.11) and (4.12), respectively in Definition 4.4. The existence of one sided derivatives and elasticities with trade-offs for input sets is established in Theorem 5.11 below, which is a modification of Theorem 5.4 for the VRS technology.
Theorem 5.11. Consider any unit (X0,Y0)∈TVRSTO that satisfies Assumption 4.2. (The unit
(X0,Y0) can be either observed or unobserved.)
(a) If a proportional marginal increase of vectors X0A and
Y0
A is feasible in technology
TVRSTO, then the right-hand elasticity ρA,B
+ (X
0,Y0) exists, is finite and can be calculated as
follows: ρA,B + (X 0,Y0)=max −ν AX 0 A+µAY 0 A (5.17) Subject to −νAX 0 A−νCX 0 C +µAY 0 A+µCY 0 C+µ 0=1 −νAXA−νBXB−νCXC +µAYA+µCYC+eµ 0≤0 −νAPA−νBPB−νCPC+µAQA+µCQC ≤0 νBX 0 B=1 νA,νB,νC,µA,µC ≥0 µ0 Sign free
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(b) If a proportional marginal reduction of vectors X0A and
Y0
A is feasible in technology
TVRSTO, then the left-hand elasticity ρA,B
− (X
0,Y0) exists, is finite and can be calculated by:
ρA,B − (X 0,Y0)=min −ν AX 0 A+µAY 0 A (5.18) Subject to −νAX 0 A−νCX 0 C +µAY 0 A+µCY 0 C+µ 0=1 −νAXA−νBXB−νCXC +µAYA+µCYC+eµ 0≤0 −νAPA−νBPB−νCPC+µAQA+µCQC ≤0 νBX 0 B=1 νA,νB,νC,µA,µC ≥0 µ0 Sign free
(c) If a proportional marginal increase (reduction) of vectors X0A and
Y0
A is not feasible in
technology TVRSTO, then the objective function of above models are unbounded.
Similar to our previous developments, the use of Theorem 5.11 initially requires checking Assumption 4.2. For input sets, in the presence of production trade-offs, this can be done through solving model (5.16). However, in practice, infeasibility of linear programs (5.17) and (5.18) yields to the violation of Assumption 4.2. This is presented in Theorem 5.12, which is simply an analogue of Theorem 4.8 for the models of input sets under VRS with production trade-offs.
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Using Theorems 5.11 and 5.12, in practice, the analysis and computation of (one-sided) elasticities with production trade-offs for input sets at any unit (X0,Y0) in the given technology can be achieved by simply solving programs (5.17) and (5.18) for all the units (efficient and inefficient). The solutions to the programs can be interpreted through three cases identified in Section 5.5.
5.5. Generalizations of Elasticity Analysis with Production Trade-offs
Linear programming (LP) models to measure the elasticity of response at units on DEA frontiers can yield to three types of solutions with different interpretations; optimal solutions, unbounded solutions and infeasible solutions. These three possible cases are summarised in Chapter 4 (Section 4.3 for output sets and Section 4.7 for input sets). This framework is applicable also to both VRS and CRS technologies with production trade-offs. Consider the elasticity measure calculated for any unit (X0,Y0) through linear programs given by Theorem 5.2 for the output sets and by Theorem 5.5 for input sets in TCRSTO, or through linear programs given by Theorem 5.8 for output sets and Theorem 5.11 for input sets in TVRSTO. Three cases can be identified regarding to the solutions, briefly given below.13
Case 1. (The program has a finite optimal solution). Assuming that the unit satisfies selective radial efficiency assumption (see Assumption 4.1 in Section 4.2 for output case and Assumption 4.2 in Section 4.6 for input case), as proven for any technology with or without trade-offs included, if the LP model has a finite optimal solution, the marginal increase or reduction of the input or output vectors is feasible in the given technology and right-hand or left-hand elasticities are correctly defined as the optimum value of the program.
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13 For more specific discussions of three cases, see Section 4.3 (for output sets) and Section 4.7 (for input sets).
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Case 2. (The program has an unbounded optimal solution). Assuming that selective radial efficiency assumption is satisfied, parts (c) in Theorems 5.2, 5.5, 5.8, 5.11 imply that proportional marginal increases or reductions of vectors X0A and
Y0
A are not feasible in the
given technology. In other words, proportional marginal increases or reductions at the given unit result in leaving the boundaries of the given production possibility set. Indeed, if a proportional marginal increases or reductions were feasible, by parts (a) and (b) of Theorems 5.2, 5.5, 5.8, 5.11, the optimum value of the programs would be finite, which contradict with the case. For this reason, elasticity of response is undefined.
Case 3. (The program is infeasible). Infeasible solutions to the elasticity models indicate that the selective radial efficiency assumption (Assumption 4.1 for output case and Assumption 4.2 for input case) is not satisfied; therefore the elasticity is undefined for that unit.
Note that infeasibility can arise also because there is no strictly positive component in responding set B, since we allow zero outputs. Recall the elasticity measures for output sets under CRS explained in Section 4.2. By Theorem 4.1, we know that Assumption 4.1 is satisfied when unit is efficient and has at least one strictly positive component in responding set B. On the other hand, by Theorem 4.3, we know that Assumption 4.1 is true if and only if programs (4.7) and (4.8) are feasible. Therefore, the programs (4.7) and (4.8) are always feasible if the unit is efficient and has one strictly positive component in the responding set B. However, infeasibility of programs (4.7) and (4.8) can be related to either the inefficiency of the unit (Theorems 4.1 and 4.3) or the absence of at least one strictly positive component in responding output set B (constraints (4.7.4) and (4.8.4) are violated in this case). Above notion is also applicable to the any type of elasticity measures (with or without trade-offs