Abstract
While Benefit-of-the-Doubt (BoD) models, as used in the context of composite indicator construction, seek to aggregate outputs only, they are formally equivalent to an input-oriented Data Envelopment Analysis (DEA) model in multiplier form. Strictly read, this introduces conceptually ambiguous results, as input adjustments have no real significance in the context at hand. At heart of this ambiguity lies a double interpretation of what is essentially a binding constraint in BoD’s underlying linear-fractional program. Moreover, there is a direct, reciprocal relation between the BoD-model and an earlier output-oriented DEA model introduced by Lovell et al. (Eur J Oper Res 87:507–518, 1995) for similar purposes. Although these models are essentially similar, I also show that there are instances (i.e. when adding additional weight restrictions) in which the results of one alternative are easier to communicate. The models surveyed in this paper are complemented with a short application on human development data for females living in Brussels municipalities.
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Notes
A log-linear aggregation yields the multiplicative BoD-model. Many other variants, e.g. including additional weight bounds, looking for a common set of BoD weights etc. are further possible but are not considered here. Also, for expositional purposes all models in this paper are presented without an explicit treatment of slacks (which e.g. accounts for a minor difference between model (4) below and the model appearing in Lovell et al. 1995).
The original Charnes et al. (1978) DEA-model accordingly builds on a deterministic axiom (any observed input–output combination is an element of the true production set), on (‘strongly’), free disposability assumptions (if any input/output combination is an element of the production set, then input/output combinations with at least as much inputs and/or at most as much outputs are also elements of that set), on convexity assumptions (given any two elements of the production set, any linear combination of these two elements also is an element of that set) and on the (‘constant returns to scale’) assumption that any input/output combination belonging to the production set can be proportionally rescaled and will in that manner still identify a feasible input/output combination. Several later variants of this seminal model can be understood as building on a different set of axioms.
That is, there are in fact an infinite number of optimal solutions to (III), as any optimal weighting vector \((u^{*} ,v^{*} )\) implies the existence of an equally optimal vector \((\alpha u^{*},\alpha v^{*}),\alpha > 0\). By choosing among those the solution for which the denominator of the objective function in (III) equals one, that denominator vanishes and one arrives at the linear program.
Unlike (6), this is a variable returns to scale model. Lovell and Pastor (1999) however demonstrate its equivalence with an output-oriented constant returns to scale model with a unitary input.
In particular, this implies \(\tilde{w}_{i0}^{*} /\tilde{w}_{k0}^{*} = w_{i0}^{*} /w_{k0}^{*} \forall i,k = 1, \ldots ,I\), i.e. equal pairwise optimal trade-offs, as well as equality of the optimal proportional shares \(\tilde{w}_{i0}^{*} Y_{i0} /\left( {\sum\nolimits_{i = 1}^{I} {\tilde{w}_{i0}^{*} Y_{i0} } } \right) = w_{i0}^{*} Y_{i0} /\left( {\sum\nolimits_{i = 1}^{I} {w_{i0}^{*} Y_{i0} } } \right)\forall i = 1, \ldots ,I\).
Since the denominator needs to equal 1 in the optimum to go from fractional Model (6) to its LP-counterpart (5), model (6) has a constraint ‘set’ different from the one employed in (1).
Care should of course be taken to set the max and min-operators at the right places. In this case, if one would proceed wrongly and start from \(\mathop {\hbox{min} }\limits_{w} \left( {{{\sum\nolimits_{i = 1}^{I} {w_{i0} Y_{i0} } } \mathord{\left/ {\vphantom {{\sum\nolimits_{i = 1}^{I} {w_{i0} Y_{i0} } } {\left( {\mathop {\hbox{max} }\limits_{{Y_{ij} }} \sum\nolimits_{i = 1}^{I} {w_{i0} Y_{ij} } } \right)}}} \right. \kern-0pt} {\left( {\mathop {\hbox{max} }\limits_{{Y_{ij} }} \sum\nolimits_{i = 1}^{I} {w_{i0} Y_{ij} } } \right)}}} \right)\), one actually is dealing with the model of Athanassoglou (2015), who combines least favorable weights with the identification of best-practices.
See statistics.brussels.
The exact procedure to find this estimated share is similar to the one used by UNDP (see their technical note, UNDP, 2016). The value of the wage gap used is 0.8 (i.e. the average gross wage gap in Belgium). Changing this value to 0.9 (the wage gap adjusted for labor time etc) leaves the municipality rankings unaltered.
For life expectancy, the maximal value was set at 86.1 years (i.e. the maximum value in the sample) and the minimum was set at 20 as is done by UNDP. The maximum (worst possible) share of pupils with significant grade retention is set at 50%, the minimum at 5%. Maximum female per capita income is 20.000 euro, the minimum is set at 300 euro.
See e.g. Qizilbash (1997) for an application of the Borda method to HDI country data.
Note that in Table 2 we exploit the intimate relation between the results of these two envelopment models as discussed in the previous section. In the table we opt to show the results of model (7), which has the convenience that all λ j -variables add up to 1 by construction, rather than those of model (6), which are the same up to a scalar transformation.
Note that opposite, but otherwise qualitatively similar results would be generated when looking for endogenous ‘worst possible’ weights, i.e. when applying model (12) to the data of Table 2. That is, the emphasis would largely be on the educational dimension, and there would be one municipality (Sint-Joost-ten-Node) that would have a major role in identifying worst practices.
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Van Puyenbroeck, T. On the Output Orientation of the Benefit-of-the-Doubt-Model. Soc Indic Res 139, 415–431 (2018). https://doi.org/10.1007/s11205-017-1734-x
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DOI: https://doi.org/10.1007/s11205-017-1734-x