Evaluating the Efficiency and Productivity of Colombian Criminal Justice

Nicolás Enrique Valencia Santiago; Camilo Almanza Ramírez

doi:10.1515/rle-2021-0082

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Evaluating the Efficiency and Productivity of Colombian Criminal Justice

Nicolás Enrique Valencia Santiago und Camilo Almanza Ramírez

Veröffentlicht/Copyright: 7. November 2022

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Abstract

In this paper, Data Envelopment Analysis (DEA) is used to calculate the technical efficiency and the Malmquist Total Factor Productivity Index (MPI) of the municipal (JM) and circuit courts (JC) of the ordinary jurisdiction of the Colombian criminal justice system, from 2012 to 2016. The results show an average technical inefficiency of 16, 9% for de JCs, and 17.3% for JMs. Additionally, we find a total factor productivity (TFP) decline for these courts of 24 and 44%, respectively. Although both components of the MPI registered average values lower than one, the decrease in TFP is mainly driven by the decline in the technical change component.

Keywords: criminal justice; efficiency; productivity; DEA; Malmquist Index

JEL Classification: C67; D2; H11; K4; K14

Corresponding authors: Camilo Almanza Ramírez, PhD, Fundación Universidad del Norte, Barranquilla, Colombia, E-mail: almanza@uninorte.edu.co; and Nicolás Enrique Valencia Santiago, MSc, Barranquilla, Colombia, E-mail: nienvasa@gmail.com

Acknowledgments

Acknowledgements to certified translator Iván Yunis for the English translatiom. Acknowledgements to Luis Abdenago Chaparro Galan, Administrative Director of the Labor Division of the Judiciary and Nina Puentes Mora, of the Registry, Labor History Control, and Correspondence Group of the Labor Division of the Judiciaty for her comments on the implications of this kind of analysis on the Colombian judicial system.

Appendix

Formally, consider a productive process where each j ∈ J agent uses x ∈ R + N inputs to produce y ∈ R + M outputs. The technology of production can be represented by the following set:

(1) P x = y : x c a n p r o d u c e y

Set P x maintains every vector of output y that can be produced with the vector of inputs x. It is assumed that P x is non-empty, convex, and compact, and satisfies the assumption of free disposability of outputs,^[14] which is, for every y′ ≤ y, both y′ and y belong to P x .

For each x ∈ R + N , the frontier (isoquant) of set P x , i.e., the subset of all combination of outputs that can be produced using exactly the vector of inputs x, is defined by the following:

(2) δ P x = y ∈ R + M : y ∈ P ( x ) , θ y ∉ P ( x ) i f θ > 1

Following (Fare et al. (1994)), a technology of reference was constructed for each period with x ∈ R + N inputs used by j DMUs (Courts) for producing y ∈ R + M outputs, under Constant Returns to Scale (CRS), and Free Disposability of Outputs (FDO), by the following:

(3) P x | C R S , F D O = y : y m ≤ ∑ j = 1 J z j y j m , m = 1,2 , … , M ; ∑ j = 1 J z j x j n ≤ x n n = 1,2 , … N ⋅ z ∈ R + j ,

where z_j are intensity variables that allow the model to generate the linear combinations of input and output set for every DMU, while inequalities y m ≤ ∑ j = 1 J z j y j m , and x n ≥ ∑ j = 1 J z j model free disposability of inputs and outputs, respectively.

The output-based technical efficiency, i.e. the ability of a DMU to produce the maximum output with a given set of inputs, is measured as the maximum possible expansion of the vector of outputs y ∈ R + M , until the frontier of the output set, without changing the number of used inputs. That is:

(4) E F o t C R S , F D O = M a x θ { θ : θ y ∈ P ( x ) }

y ∈ P(x) if and only if the conditions established in (3) are satisfied, technical efficiency under the assumption of CRS, and FDO of Court j for year t can be calculated by solving the following linear problem:

(5) E F o t C R S , F D O = M a x z , θ θ ⋅ s ⋅ a θ y j m ≤ ∑ j = 1 J z j y j m , m = 1,2 , … , M x n ≥ ∑ j = 1 J z j x j n n = 1,2 , … N . z j ∈ R + J j = 1,2 , … , J

A DMU might be technically efficient, but operate at a suboptimal size, that is, showing scale inefficiency. Scale Efficiency ( E F o E ) is measured as the quotient between technical efficiency under Constant Returns to Scale and technical efficiency under Variable Returns to Scale (VRS).^[15] To obtain the measure of technical efficiency under variable returns to scale, the restriction z j ∈ R + J in (3) and (5) is changed to ∑ j = 1 n z j = 1 .

To determine the source of inefficiency it is also required to measure technical efficiency under Non-increasing Returns to Scale (NIRS), which is achieved by replacing the restriction z j ∈ R + J in (3) and (5) with ∑ j = 1 n z j ≤ 1 . The DMU operate at an optimal scale if and only if E F o E = 1 . If E F o E < 1 , inefficiency of scale exists, and this result is due to:

Increasing Returns to Scale (IRS) if the efficiency measure under non-increasing returns is different from the efficiency measure under variable returns. This is:
E F o t N I R S , F D O ≠ E F o t V R S , F D O
Decreasing Returns to Scale (DRS) if the Efficiency measure under NISR is not different from the Efficiency measure under variable returns. This is:
E F o t N I R S , F D O = E F o t V R S , F D O

The Output-Based Malmquist Productivity Index is defined as the geometric average of the two indices based on period-t and period-t + 1 levels of technology, as follows:
(6) I M o t + 1 x t + 1 , y t + 1 , x t , y t | C R S , F D O = C E F 0 t + 1 x C T 0 t + 1 1 / 2
where Efficiency Change, C E F o T , is:
(7) C E F o t + 1 = E F o t ( x t , y t | C , F ) E F o t + 1 ( x t + 1 , y t + 1 | C , F ) ,

And Technical Change, CT_o, is:

(8) C T o t + 1 = E F o t + 1 ( x t + 1 , y t + 1 | C , F ) E F o t ( x t + 1 , y t + 1 | C , F ) x E F o t + 1 ( x t , y t | C , F ) E F o t ( x t , y t | C F ) 1 / 2

Following Fare et al. (1994), Technical Efficiency Change (7) is also decomposed into two elements:

(9) C E F o t + 1 = P E T t + 1 x E E S t + 1 ,

where Pure Technical Efficiency, PET, is:

(10) P E T t + 1 = E F o t ( x t , y t | V , F ) E F o t + 1 ( x t + 1 , y t + 1 | V , F )

and the Scale Efficiency Change, EES, is:

(11) E E S t + 1 = E F o t ( x t , y t | C , F ) E F o t + 1 ( x t , y t | V , F )

To calculate TFP, Equations (6)–(11), the following linear problems must be solved:

(12) E F o t C R S , F D O = M a x z , θ θ ⋅ s ⋅ a θ y j m t + 1 ≤ ∑ j = 1 J z j y j m , m = 1,2 , … , M x j n t + 1 ≥ ∑ j = 1 J z j x j n n = 1,2 , … N . z j ≥ 0 j = 1,2 , … , J

And:

(13) E F o t + 1 C R S , F D O = M a x z , θ θ ⋅ s ⋅ a θ y j m t ≤ ∑ j = 1 J z j y j m , m = 1,2 , … , M x j n t ≥ ∑ j = 1 J z j x j n n = 1,2 , … N . z j ≥ 0 j = 1,2 , … , J

It can be stated that productivity increases or decreases if IM is greater or less than one. In the same direction, there are gains (loss) in technical efficiency and technical progress (Regress) if the corresponding components CET and CT are greater (less) than one.

Comparing the levels of the components CET and CT, it is possible to identify the main source of gain or loss of the observed productivity of a DMU. If CET > CT, for example, it is more likely that gains in productivity are due to improvements in Technical Efficiency. If the opposite occurs, the increase in productivity is conferred to Technical Progress.

Because the technical efficiency change is decomposed in PET and EES, if the first has a higher value than the second, then the DMU’s CET’s improvements are more likely due to what happened in that component. On the other hand, if the relation between PET and EES is the opposite, the reason of the increase in productivity changes in the same direction.

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Published Online: 2022-11-07

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https://doi.org/10.1515/rle-2021-0082

Schlagwörter für diesen Artikel

criminal justice; efficiency; productivity; DEA; Malmquist Index