1 Introduction
“In summary, if the underlying DEA problem represents a form of ‘production process’, then ‘inputs’ and ‘outputs’ can often be more clearly identified. The resources used or required are usually the inputs and the outcomes are the outputs.
If, however, the DEA problem is a general benchmarking problem, then the inputs are usually the ‘less-the-better’ type of performance measures and the outputs (…) the ‘more the-better’ type (…). DEA then can be viewed as a multiple-criteria evaluation methodology where DMUs are alternatives, and the DEA inputs and outputs are two sets of performance criteria where one set (inputs) is to be minimized and the other (outputs) to be maximized.”
2 Standards, pitfalls, and extensions of DEA
2.1 Oriented radial measurement of efficiency
DMU |
x
|
y
| CCR | BCC | SE | Σλ |
---|---|---|---|---|---|---|
O&I | O/I | O/I | O/I | |||
A | 4 | 2 | 50 | inf/150 | 50 | 67/33 |
B | 6 | 6 | 100 | 120/111 | 100 | 100 |
C | 8 | 8 | 100 | 109/113 | 100 | 100 |
D | 12 | 10 | 83 | 125/inf | 83 | 150/125 |
E | 10 | 4 | 40 | 44/50 | 90/80 | 125/67 |
F | 6 | 1 | 17 | 17/67 | 100/25 | 100/17 |
2.2 Non-oriented additive measurement of efficiency
DMU |
x
|
y
| cRTS | vRTS | SE | Σλ |
---|---|---|---|---|---|---|
A | 4 | 2 | 50 | 100 | 50 | 33 |
B | 6 | 6 | 100 | 100 | 100 | 100 |
C | 8 | 8 | 100 | 100 | 100 | 100 |
D | 12 | 10 | 83 | 100 | 83 | 150 |
E | 10 | 4 | 40 | 40 | 100 | 100 |
F | 6 | 1 | 17 | 17 | 100 | 100 |
2.3 Measurement of balance or specialization
“(T)he predominantly application of Farrell efficiency measures in DEA surely has an historical background. However, there are also practical reasons to justify their consideration. Compared to other approaches like, e.g., the slack-based measure of Tone (2001), it is easier for practitioners to interpreting them, i.e., to understanding the concept of radially measured efficiency scores. Furthermore, Farrell efficiency corresponds particularly well with findings in incentive theory. Imposing proportional changes, which is the very essence of Farrell, can under some circumstances be shown to be the optimal response of a principal who lacks information about the relative costs of different activities of his agents …”
DMU |
x
1
|
y
1
|
y
2
|
θ
|
θ
b
|
β
|
---|---|---|---|---|---|---|
A | 1 | 2 | 10 | 71 | 20 | 29 |
B | 1 | 8 | 14 | 100 | 81 | 81 |
C | 1 | 6 | 8 | 61 | 61 | 100 |
D | 1 | 10 | 12 | 94 | 94 | 100 |
E | 1 | 12 | 12 | 100 | 100 | 100 |
F | 1 | 10 | 8 | 79 | 79 | 100 |
G | 1 | 6 | 4 | 46 | 46 | 100 |
H | 1 | 14 | 6 | 100 | 69 | 69 |
I | 1 | 8 | 3 | 57 | 34 | 60 |
3 DEA applications to welfare evaluation and some common pitfalls
-
Selection from available welfare studies: By far, the most frequently used procedure for selecting the relevant DEA indicators is the reference to already established welfare indices and rankings (e.g., Jablonsky 2004; Murias et al. 2006; Lee et al. 2006; Bougnol et al. 2010). Thereby, sub-indicators are often used, or their underlying indices are adopted directly as DEA indicators. In principle, this approach offers the advantage of data sets not needing to be elaborately collected and of their presumably having been checked for errors during their initial use. However, the use of index numbers can constitute a problem if volume measures are also used at the same time (Dyson et al. 2001).
-
Own selection via normatively selected attribute groups of welfare: Some authors are inclined to select indicators first by (normatively) forming groups of attributes or partial aspects of welfare. Individual indicators are then assigned to these attribute groups or partial aspects which reflect them or are associated with them; for example, Ramanathan (2006: 158) identifies the groups “economic, educational and health attributes”; Li and Ma (2011) similarly determine four categories of welfare to derive relevant indicators. In this procedure, however, it is questionable and is barely justified as to how precisely the relevant attribute groups should be determined.
-
Own selection without substantiation or reference to the literature: Often, justifications for selecting indicators are totally absent (e.g., Malul et al. 2009; Jahanshaloo et al. 2011). Some authors argue that there is already general consensus on the relevant indicators in the literature (e.g., Dominguez-Serrano and Blancas 2011: 485) and, therefore, that they do not need to be justified any further. This often leads to difficulties in the interpretability of the models or of the results. Sabermahani et al. (2013), for example, use the employment rate as input, to be minimized in their radial input-oriented model; this is hardly justifiable without due explanation.
4 DEA case study for the Prosperity Quintet
4.1 The Prosperity Quintet for the countries of the European Union
-
Gross Domestic Product per capita (GDP/POP in €/cap): It stands for the purely economic dimension and the material level of prosperity.
-
80/20 ratio (HIC/LIC measured in %): It covers the socio-economic dimension and represents the income distribution in a national economy. The incomes of the upper fifth of the population are considered in relation to those of the lower fifth, where the calculation of income refers to the equivalized disposable income per household according to the OECD standard.
-
Social exclusion rate (SER in %): It shows the social coherence and indicates the proportion of individuals interviewed (15 years or older) who perceive themselves to be excluded from society.
-
Ecological footprint per capita in relation to availableglobal biocapacity of a human being (EFC/BC in %): It represents the ecological dimension. The ecological footprint contained in the numerator measures the areas of land and water which an inhabitant requires, on average, for the production of goods and services consumed by him or her as well as the absorption of emissions generated in the process, including the areas required by imports. The resulting total area is then considered in relation to the total biocapacity that can be regenerated by the ecosystem and which is available to a person on average (at present, 1.8 global hectares).
-
Public debt rate (PDR = DEB/GDP in %): It is used as a barometer for the credit financing of a national economy and is calculated as the gross public debt of a country in relation to the GDP. This expresses the extent to which material wealth is generated at the expense of future generations and thus limiting the future viability of a country.
EU-27 | GDP/POP (€/cap) | 80/20-ratio (%) | SER (%) | EFC/BC (%) | PDR (%) | Evaluation |
---|---|---|---|---|---|---|
A-Austria | 32200 | 420 | 7.5 | 289 | 73 | +++ −+ |
B-Belgium | 29630 | 391 | 15.2 | 406 | 100 | ++−−− |
BG-Bulgaria | 3700 | 611 | 17.9 | 161 | 19 | −−−+* |
CY-Cyprus | 17365 | 467 | 20.7 | 244 | 85 | −+−++ |
CZ-Czech Republic | 11389 | 350 | 14.1 | 261 | 46 | −+−−* |
D-Germany | 30070 | 429 | 9.0 | 250 | 82 | +++ ++ |
DK-Denmark | 37263 | 451 | 6.6 | 422 | 46 | +++ −* |
E-Spain | 20300 | 716 | 7.9 | 239 | 84 | −−+++ |
EST-Estonia | 9400 | 544 | 12.3 | 278 | 10 | −−−−* |
F-France | 27600 | 454 | 15.9 | 272 | 90 | ++−−− |
FIN-Finland | 31100 | 370 | 4.3 | 311 | 53 | +++ −* |
GB-Great Britain | 30400 | 533 | 11.9 | 250 | 90 | +−−+− |
GR-Greece | 14900 | 662 | 15.6 | 261 | 157 | −−−−− |
H-Hungary | 8809 | 395 | 9.4 | 167 | 79 | −+++ + |
I-Italy | 22800 | 552 | 6.5 | 244 | 127 | −−++− |
IRL-Ireland | 37100 | 465 | 9.8 | 317 | 118 | +++ −− |
L-Luxembourg | 62600 | 415 | 18.7 | 833 | 21 | ++−−* |
LT-Lithuania | 8100 | 534 | 10.3 | 228 | 41 | −−++* |
LV-Latvia | 6800 | 657 | 14.0 | 211 | 41 | −−−+* |
M-Malta | 13500 | 395 | 11.1 | 244 | 72 | −+−++ |
NL-Netherlands | 32833 | 360 | 3.7 | 328 | 71 | +++ −+ |
P-Portugal | 14200 | 589 | 9.9 | 244 | 124 | −−++− |
PL-Poland | 8500 | 494 | 14.8 | 228 | 56 | −+−+* |
RO-Romania | 4400 | 637 | 8.2 | 133 | 38 | −−++* |
S-Sweden | 35536 | 374 | 8.8 | 300 | 38 | +++−* |
SK-Slovakia | 9409 | 372 | 8.3 | 194 | 52 | −+++* |
SLO-Slovenia | 15000 | 342 | 4.6 | 228 | 54 | −+++* |
Ø-EU 27 | 23200 | 500 | 10.5 | 250 | 85 |
4.2 Standard DEA results for the prosperity quintet
EU-27 | Rating | CCR | BCC | SE | Σλ |
---|---|---|---|---|---|
O&I | O/I | O/I | O/I | ||
A-Austria | 4 | 97 | 98/99 | 99/98 | 95/92 |
B-Belgium | 2 | 71 | 72/91 | 99/78 | 103/73 |
BG-Bulgaria | 2* | 20 | inf/127 | 20 | 51/10 |
CY-Cyprus | 3 | 59 | 60/85 | 98/70 | 98/57 |
CZ-Czech Republic | 2* | 37 | inf/102 | 37 | 90/33 |
D-Germany | 5 | 101 | 115/106 | 100 | 100 |
DK-Denmark | 4* | 105 | 109/106 | 100 | 100 |
E-Spain | 3 | 71 | 81/93 | 88/77 | 89/63 |
EST-Estonia | 1* | 42 | inf/192 | 42 | 40/17 |
F-France | 2 | 84 | 85/91 | 100/93 | 107/90 |
FIN-Finland | 4* | 104 | inf/112 | 100 | 100 |
GB-Great Britain | 2 | 101 | 101 | 100 | 100 |
GR-Greece | 0 | 47 | 47/71 | 99/66 | 104/49 |
H-Hungary | 4 | 43 | inf/109 | 43 | 67/29 |
I-Italy | 2 | 81 | 99 | 81 | 80/65 |
IRL-Ireland | 3 | 98 | 102/104 | 98 | 111/109 |
L-Luxembourg | 3* | 318 | inf | 100 | 100 |
LT-Lithuania | 3* | 30 | 36/90 | 82/33 | 79/24 |
LV-Latvia | 2* | 27 | 35/82 | 76/33 | 74/20 |
M-Malta | 3 | 46 | 53/89 | 87/52 | 94/43 |
NL-Netherlands | 4 | 123 | inf/134 | 100 | 100 |
P-Portugal | 2 | 48 | 49/79 | 98/61 | 98/47 |
PL-Poland | 3* | 31 | 36/83 | 87/37 | 84/26 |
RO-Romania | 3* | 27 | inf/134 | 27 | 50/14 |
S-Sweden | 4* | 128 | inf/128 | 100 | 100 |
SK-Slovakia | 4* | 40 | inf/105 | 40 | 73/29 |
SLO-Slovenia | 4* | 60 | inf/118 | 60 | 73/44 |
4.3 Pitfalls of DEA application to the Prosperity Quintet
5 DEA case study for a basic welfare set
5.1 A set of six basic welfare indicators
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High income per capitay1 (HIC): average income of the population’s quintile with the highest incomes.
-
Middle income per capitay2 (MIC): average income of the three quintiles with middle incomes.
-
Low income per capitay3 (LIC): average income of the population’s quintile with the lowest incomes.
-
Social exclusion ratex1 (SER): proportion of individuals interviewed (15 years or older) who perceive themselves to be excluded from society.
-
Ecological footprint per capitax2 (EFC): areas of land and water which an inhabitant requires on average for the production of goods and services consumed by that person as well as the absorption of emissions generated in the process.
-
Debt level per capitax3 (DBC = DEB/POP): average gross debt of a country per inhabitant.
Prosperity Quintet | Backward calculation using basic welfare indicators |
---|---|
GDP/POP (in €/cap) |
\(0.2y_{1} + 0.6y_{2} + 0.2y_{3}\)
|
80/20 ratio (in %) |
\(\frac{{y_{1} }}{{y_{3} }}\)
|
SER (in %) |
\(x_{1}\)
|
EFC/BC (in %) |
\(\frac{{x_{2} }}{1.8}\)
|
PDR (in %) |
\(\frac{{x_{3} }}{{0.2y_{1} + 0.6y_{2} + 0.2y_{3} }}\)
|
EU-27 | POP (103cap) | y1 HIC (€/cap) | y2 MIC (€/cap) | y3 LIC (€/cap) | x1 SER (%) | x2 EFC (gha/cap) | x3 DBC (€/cap) |
---|---|---|---|---|---|---|---|
A-Austria | 8443 | 58765 | 29409 | 14007 | 7.5 | 5.2 | 23635 |
B-Belgium | 11095 | 52096 | 27577 | 13320 | 15.2 | 7.3 | 29482 |
BG-Bulgaria | 7327 | 7456 | 3275 | 1221 | 17.9 | 2.9 | 685 |
CY-Cyprus | 862 | 34539 | 14964 | 7395 | 20.7 | 4.4 | 14758 |
CZ-Czech Republic | 10505 | 20121 | 10355 | 5757 | 14.1 | 4.7 | 5221 |
D-Germany | 81844 | 55535 | 27291 | 12943 | 9.0 | 4.5 | 24652 |
DK-Denmark | 5574 | 67327 | 34689 | 14920 | 6.6 | 7.6 | 17083 |
E-Spain | 46196 | 41412 | 18101 | 5786 | 7.9 | 4.3 | 17093 |
EST-Estonia | 1294 | 18659 | 8303 | 3431 | 12.3 | 5.0 | 949 |
F-France | 65328 | 54510 | 23828 | 12006 | 15.9 | 4.9 | 24895 |
FIN-Finland | 5401 | 55203 | 28457 | 14928 | 4.3 | 5.6 | 16483 |
GB-Great Britain | 63256 | 61560 | 26296 | 11552 | 11.9 | 4.5 | 27360 |
GR-Greece | 11290 | 30098 | 13286 | 4545 | 15.6 | 4.7 | 23378 |
H-Hungary | 9932 | 15972 | 8008 | 4048 | 9.4 | 3.0 | 6970 |
I-Italy | 59394 | 44688 | 20406 | 8094 | 6.5 | 4.4 | 28956 |
IRL-Ireland | 4583 | 70676 | 33205 | 15211 | 9.8 | 5.7 | 43630 |
L-Luxembourg | 525 | 115497 | 56549 | 27857 | 18.7 | 15.0 | 13250 |
LT-Lithuania | 3004 | 15998 | 7169 | 2997 | 10.3 | 4.1 | 3297 |
LV-Latvia | 2042 | 14518 | 5757 | 2210 | 14.0 | 3.8 | 2768 |
M-Malta | 418 | 24503 | 12263 | 6210 | 11.1 | 4.4 | 9733 |
NL-Netherlands | 16730 | 57892 | 30067 | 16072 | 3.7 | 5.9 | 23354 |
P-Portugal | 10542 | 30104 | 11928 | 5112 | 9.9 | 4.4 | 17551 |
PL-Poland | 38538 | 16575 | 7523 | 3358 | 14.8 | 4.1 | 4726 |
RO-Romania | 21356 | 8690 | 3982 | 1364 | 8.2 | 2.4 | 1663 |
S-Sweden | 9483 | 60350 | 33725 | 16153 | 8.8 | 5.4 | 13561 |
SK-Slovakia | 5404 | 16262 | 8805 | 4371 | 8.3 | 3.5 | 4897 |
SLO-Slovenia | 2055 | 25125 | 14175 | 7350 | 4.6 | 4.1 | 8115 |
5.2 Standard DEA results for the basic welfare set
EU-27 | CCR | BCC | SE | Σλ |
---|---|---|---|---|
O&I | O/I | O/I | O/I | |
A-Austria | 99 | 100 | 99 | 95 |
B-Belgium | 62 | 72/64 | 85/96 | 144/89 |
BG-Bulgaria | 64 | inf/157 | 64 | 55/36 |
CY-Cyprus | 67 | 72/82 | 93/81 | 85/57 |
CZ-Czech Republic | 61 | 76/83 | 80/73 | 39/24 |
D-Germany | 102 | 114/106 | 100 | 100 |
DK-Denmark | 101 | 104/106 | 100 | 100 |
E-Spain | 81 | 92/96 | 89/84 | 85/69 |
EST-Estonia | 202 | inf/209 | 100 | 100 |
F-France | 88 | 89/91 | 99/97 | 105/93 |
FIN-Finland | 113 | 123/116 | 100 | 100 |
GB-Great Britain | 111 | 111/112 | 100 | 100 |
GR-Greece | 50 | 50/70 | 100/71 | 99/49 |
H-Hungary | 49 | 89/96 | 56/51 | 51/25 |
I-Italy | 86 | 108/103 | 86 | 77/67 |
IRL-Ireland | 104 | 110/118 | 100 | 100 |
L-Luxembourg | 162 | 195/inf | 100 | 100 |
LT-Lithuania | 54 | 69/83 | 78/65 | 34/18 |
LV-Latvia | 56 | 73/82 | 77/69 | 37/21 |
M-Malta | 53 | 57/76 | 93/69 | 72/38 |
NL-Netherlands | 125 | inf/151 | 100 | 100 |
P-Portugal | 56 | 60/79 | 94/71 | 88/49 |
PL-Poland | 49 | 60/71 | 81/69 | 35/17 |
RO-Romania | 55 | inf/152 | 55 | 25/14 |
S-Sweden | 130 | 135/133 | 100 | 100 |
SK-Slovakia | 56 | 79/90 | 72/63 | 36/21 |
SLO-Slovenia | 76 | inf/128 | 76 | 54/41 |
6 Extended DEA case study for the basic welfare set
6.1 Measurement of income balance
Country | CCR | BBC-O | SBM-cRTS | SBM-vRTS | ||||
---|---|---|---|---|---|---|---|---|
θ
|
β
|
θ
|
β
|
θ
|
β
|
θ
|
β
| |
A-Austria | 99 | 95 | 100 | 96 | 99 | 81 | 100 | 85 |
B-Belgium | 62 | 98 | 72 | 99 | 49 | 97 | 49 | 97 |
BG-Bulgaria | 64 | 86 | 100 | 100 | 47 | 78 | 100 | 78 |
CY-Cyprus | 67 | 86 | 72 | 88 | 42 | 95 | 47 | 95 |
CZ- Czech Republic | 61 | 87 | 76 | 87 | 44 | 95 | 60 | 95 |
D-Germany | 100 | 98 | 100 | 100 | 100 | 78 | 100 | 96 |
DK-Denmark | 100 | 89 | 100 | 90 | 100 | 83 | 100 | 83 |
E-Spain | 81 | 56 | 92 | 58 | 51 | 74 | 66 | 71 |
EST-Estonia | 100 | 100 | 100 | 100 | 100 | 84 | 100 | 84 |
F-France | 88 | 94 | 89 | 93 | 57 | 96 | 64 | 92 |
FIN-Finland | 100 | 100 | 100 | 100 | 100 | 96 | 100 | 96 |
GB-Great Britain | 100 | 87 | 100 | 89 | 100 | 58 | 100 | 69 |
GR-Greece | 50 | 66 | 50 | 67 | 28 | 77 | 31 | 76 |
H-Hungary | 49 | 100 | 89 | 99 | 39 | 99 | 70 | 99 |
I-Italy | 86 | 75 | 100 | 78 | 57 | 82 | 100 | 63 |
IRL-Ireland | 100 | 92 | 100 | 94 | 100 | 67 | 100 | 67 |
L-Luxembourg | 100 | 100 | 100 | 100 | 100 | 98 | 100 | 98 |
LT-Lithuania | 54 | 79 | 69 | 87 | 39 | 86 | 63 | 86 |
LV-Latvia | 56 | 65 | 73 | 72 | 33 | 79 | 55 | 78 |
M-Malta | 53 | 100 | 57 | 100 | 44 | 99 | 50 | 99 |
NL-Netherlands | 100 | 100 | 100 | 100 | 100 | 95 | 100 | 95 |
P-Portugal | 56 | 71 | 60 | 72 | 36 | 85 | 43 | 85 |
PL-Poland | 49 | 82 | 60 | 87 | 34 | 90 | 49 | 90 |
RO-Romania | 55 | 68 | 100 | 100 | 34 | 77 | 100 | 74 |
S-Sweden | 100 | 100 | 100 | 100 | 100 | 94 | 100 | 94 |
SK-Slovakia | 56 | 94 | 79 | 93 | 46 | 95 | 73 | 95 |
SLO-Slovenia | 76 | 89 | 100 | 100 | 69 | 91 | 100 | 91 |
6.2 Slack-based welfare measurement
DEA model | Prosperity Quintet: targets | Eff.-score | Scale | Benchmarks | ||||
---|---|---|---|---|---|---|---|---|
GDP/POP (€/cap) | 80/20-ratio (%) | SER (%) | EFC/BC (%) | PDR (%) | θ (%) |
\(\mathop \sum \nolimits \lambda_{j}\)
| ||
Original data | 29630 | 391 | 15.2 | 406 | 100 | – | – | – |
BCC-O (5) | 40987 | 391 | 10.8 | 406 | 43 | 72 | 1.00 | 0.10 IRL; 0.20 L; 0.71 S |
BCC-I (5) | 29630 | 357 | 3.9 | 310 | 68 | 91 | 1.00 | 0.82 NL; 0.18 SLO |
CCR-O (6) | 48279 | 391 | 12.8 | 406 | 53 | 62 | 1.44 | 0.53 D; 0.91 S |
CCR-I (6) | 29822 | 391 | 7.9 | 251 | 53 | 62 | 0.89 | 0.32 D; 0.56 S |
SBM-cRTS (6) | 48039 | 374 | 11.9 | 406 | 38 | 49 | 1.35 | 1.35 S |
SBM-vRTS (6) | 35536 | 374 | 8.8 | 300 | 38 | 49 | 1.00 | 1.00 S |
SBM-niRTS (6) | 30675 | 374 | 7.6 | 259 | 38 | 49 | 0.86 | 0.86 S |