In the last decades, demographic change coupled with new and expensive medical innovations have put most health care systems in developed countries under financial pressure. Therefore, ensuring efficient service provision is essential for a sustainable health care system. This paper investigates the performance of regional health care services in six West European countries between 2005 and 2014. We apply a stochastic metafrontier model to capture the different conditions in the health care systems in the countries within the European Union. By means of this approach, it is possible to detect performance differences in the European health care systems subject to different conditions and technologies relative to the potential technology available. The results indicate that regional deprivation plays a key role for the efficiency of health care provision. Furthermore, a pooled model which assumes a similar technology for all countries cannot sufficiently account for differences between countries. Surprisingly, the Scandinavian regions lag behind other regions with respect to the metafrontier. JEL Classification: C23, D61, I12, I18, R10 Keywords: Health production, Health efficiency, Stochastic frontier analysis, Metafrontier analysis Background Several attempts have been made to analyse cross- Demographic and social changes likely rise the burden country differences in the performance of health care of chronic diseases, like cancer, cardiovascular diseases, services. A pioneer work is by Evans et al. . The and diabetes. Along with new and expensive treatment authors analyse the efficiency of 191 countries based on options this puts the health care systems around the world data provided by the World Health Organisation (WHO). under financial pressures. As a result of an increasing Greene  challenged the work by Evans et al.  number of patients health care budgets in many countries demonstrating that considerable unobserved heterogene- inflate. Already, health expenditure grows more rapidly ity due to cultural and economic characteristics leads to than the economy in many countries  comprising the an underestimation of systematic health care differences. sustainability of health care funding. Accordingly, Greene  respecified the model introduc- The increasing demand for new and innovative treat- ing fixed effects parameters. His ‘true’ fixed effects model ments and an increasing request for better value for is able to distinguish between unobservable cross-country money for patients at constant budgets raised the topic heterogeneity not associated with inefficiency and ineffi- of value based care [31, 32]. In this sense, Porter  ciency itself. For the European Union, few studies have diagnoses a need to restructure the delivery of care to analysed the performance of health care systems. For obtain sustainable health care budgets. Along these lines, instance, Asandului et al.  analyse the efficiency of a the evaluation of the allocation and utilisation of medical cross-section of 30 European countries by means of a Data resources is likely an important lever for the assessment of Envelopment Analysis. They show that some countries the performance of health care systems. lie on the production frontier while most are under- neath. The analysis of the regional variation of health Correspondence: email@example.com outcomes and performance of health care services at University of Goettingen, Humboldtallee 3, 37073 Göttingen, Germany © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Schley Health Economics Review (2018) 8:11 Page 2 of 11 the national level has recently gained increasing interest Methods ([11, 17, 20, 29, 33], for example). In this section, we describe the empirical model to assess So far, very little attention has been paid to the evalua- the stochastic health frontier and metafrontier approach. tion of performance differences in a cross-country context Moreover, we introduce the data and the variables. at the regional level. This would increase the understand- ing of underlying factors of cross-country performance The stochastic frontier model differences as many countries are faced by regional vari- The stochastic frontier analysis (SFA) has become a com- ations in health outcomes and the availability of medical mon approach to asses production potentials and ineffi- infrastructure. Even though, national governments decide ciencies in the production of goods and services in, for in general on health policy measures municipalities and example, farms, firms and hospitals. In contrast to stan- other regional level bodies are responsible for the provi- dard non-parametric approaches which treat any devia- sion of medical services further highlighting the need for tion from a production function as inefficiency , the a regional level analysis. Furthermore, potential sources of parametric SFA model allows to differentiate random inefficiency such as an over or under use of services are deviations from the efficient frontier and inefficiency. likely located at the regional level . Against this back- We base the formalisation of a one-step SFA model on ground, the aim of this study is to evaluate the efficiency an approach proposed by Wang and Schmidt . By and performance of health care services at the regional quantifying production frontier parameters and exoge- level in (Western) Europe. For this purpose, we apply a nous influences on inefficiency simultaneously, we avoid stochastic metafrontier model [4, 28]. The metafrontier the risk of biased estimation results inherent in two-step model is essentially a two step approach. In the first step, approaches based on estimated efficiency scores . country specific frontiers are estimated. In a second step, the metafrontier production function, which is a deter- The health frontier ministic parametric function enveloping the individual In order to evaluate performance differences in the frontiers, is calculated. This approach has the advantage regional provision of health care services, we define the that cross-country differences in the utilised technol- European regions as producers of health (DMUs). In a ogy are taken into account. The general frontier model strict sense, the regions themselves do not convert inputs assumes that all producers use the same technology. Based into outputs. However, they can be considered as (health) on this, one would assume that in the health care set- producers in a wider sense as they provide a framework ting all health care systems are subject to the same rules for the provision of health care services . The pooled and regulations. Bos and Schmiedel relatethistothe SFA model for health production in region i at time t benchmark paradox. In the European framework, this is reads as not the case as the design of the European health care systems is heterogeneous. In general, though, the health y = x β + v − u,(1) care systems in the European Union have access to similar it it it it health care inputs and the same technology which is taken v ∼ N (0, ω ),(2) it into account by means of the metafrontier. Moreover, it is + 2 u ∼ N (0, σ ) i = 1, ... , N, t = 1, ... , T.(3) it possible to distinguish between the regional efficiency in relation to the country’s own frontier and the metafron- In (1), y is the log output and x is a K-dimensional it it tier. For the analysis, we combine regional administrative vector of the log input factors of region i at time t.Stochas- information from six European countries (Austria, France, tic deviations from the health frontier are captured by Germany, Italy, Scandinavia and Spain) for an extended v , which is conventionally assumed to be normally dis- it time frame covering the period from 2005 to 2014. tributed with zero mean and variance ω . The inefficiency To preview some results, we find that regional depri- term u follows a half normal distribution with mean it vation plays a key role for the efficiency of health zero and variance σ . To include demographic and socio- care provision. Furthermore, we show that the efficiency economic characteristics which might influence the indi- scores of the health care systems in a pooled model, vidual inefficiency of health care provision in each region, which assumes a common technology for all countries, the model fulfills the so-called scaling property  differ from the efficiency scores with respect to the metafrontier. σ = exp z δ.(4) it The next section describes the stochastic frontier model and the metafrontier approach and introduces the data. In (4), z is a R-dimensional vector of the individual it Empirical results are discussed in the Section Results explanatory variables of the variance of the inefficiency and discussion. The last section concludes. The appendix term u for each country j. By virtue of the scaling prop- it provides a detailed description of the data (Appendix A). erty, the shape of the distribution of the inefficiencies u it Schley Health Economics Review (2018) 8:11 Page 3 of 11 is the same for all regions . This is intuitively appeal- To compare the efficiency scores of the countries across ing, as in general all regions have the same possibilities different technology sets (frontiers), the technical effi- to reach the efficiency frontier. Demographic and socio- ciency with respect to the metafrontier can be calculated economic characteristics shape the deviations from the according to health frontier. The model in (1)-(4)isestimated by x β + v − u y (j) it(j) it(j) it(j) it(j) means of Maximum-Likelihood methods. ∗ TE = = ,(7) it(j) ∗ ∗ In efficiency analyses the main interest lies on the esti- y x β it it mation of technical efficiency which is the ratio of the ∗ ∗ where y is the output on the metafrontier. TE is the it it(j) observed output and the maximum feasible output on the ratio of the observed output of region i at time t to the production frontier . It is measured as metafrontier output. y x β + v − u The ratio of the output of the prodcution function for it it it it TE = = ,(5) it country j relative to the potential output of the metafron- y ˜ x β + v it it it tier for a given set of input variables is the metatechnology where y ˜ is the maximum feasible output which lies on the it ratio (MTR) which is given by production frontier. TE it(j) MTR =.(8) it(j) Metafrontier model TE it(j) The general (pooled) stochastic frontier model assumes The MTR captures the difference between the produc- homogeneous technologies for all individuals. In case of a tivity between the group and the metatechnology (the comparison of health care systems across countries, this technology available to all countries). Figure 1 illustrates assumptions is far fetched. Even though, similar inputs the concept of the metafrontier graphically. The produc- and technologies are available to all the usage pattern, tion model is set to a single input - single output frame- however, differs within the European countries due to work with three convex country frontiers labelled Country different rules and regulations. A common production 1, 2, and 3. The metafrontier envelopes the three coun- frontier cannot sufficiently account for these differences. try frontiers. It is assumed to be a deterministic smooth In a metafrontier approach, it is possible to evaluate how function with values no less than the individual coun- efficient each country works in producing health and to try functions. Point A indicates a point of production compare the productivity and efficiency across nations of region i at time t. The figure illustrates the technical without assuming similar technologies. The metafrontier efficiency of region i with respect to its group frontier approach is essentially a two step procedure developed by (0A/0B) and the respective technical efficiency TE of it Battese et al.  and O’Donnell et al. . In a first step, region i to the metafrontier (0B/0C). Further, it shows the the group-specific (country-specific) frontiers are esti- distance between the respective country’s production set mated by means of a stochastic frontier model as in (1)- and the metafrontier (0B/0C) [24, 28]. (4) for each country separately. Accordingly, the estimated model parameters β and δ of the pooled SFA model in (1) j j Data and variables and (4)changeto β and δ for each country j.Inasec- Region-specific data on mortality, health care infrastruc- ond step, a metafrontier is enveloped over the individual ture and other characteristics are extracted from Euro- frontiers. Battese et al.  show that the metafrontier opti- stat online database. Annual data cover the period from mization can be solved by a linear programming problem 2005 to 2014. The panel is unbalanced due to the non- for log-linear production functions according to availability of data for certain regions and years. The N T overall number of observations in the study is 1149 (see ∗ j min L ≡ (x β − x β ) Appendix A for a detailed description of the data). We it it (6) i=1 t=1 base the regional analysis on the NUTS-2 (Nomenclature ∗ j s.t. x β ≥ x β . des unités territoriales statistiques) regions. it it In (6), β is a vector of the metafrontier parameters and Health production β is a vector of the estimated country specific stochas- It is hardly possible to directly measure the health of the tic frontier parameters. As the β are assumed to be fixed population. The health status of the population can merely in the linear programming problem, (6) is equivalent to be approximated by measures such as life expectancy, min L ≡ x ¯ β ,where x ¯ is a vector of the means of mortality or morbidity. As specific measures of morbid- all input variables for all observations . The stan- ity are not available at regional level, we use an age and dard errors for the metafrontier parameters can either be sex standardised mortality rate (SMR) to resemble the obtained by bootstrap or simulation. population’s health status (see Appendix A for detailed Schley Health Economics Review (2018) 8:11 Page 4 of 11 Fig. 1 Metafrontier model. The graph shows the group specific frontiers for three groups (Country 1 to 3) and the metafrontier. Own presentation based on O’Donnell et al.  information on the construction of the standardised mor- regions. The North of Spain is characterised by a higher tality rate). The standardised mortality rate takes differ- supply of outpatient care. An eyeball inspection of Panel ences in the age and sex distribution of the population (c) reveals the highest densities of hospital beds in Fin- into account. By means of the standardisation, we cal- land. The supply of inpatient care exhibit a North-South culate an indicator of the population’s health status that gradient in Italy, Spain and Sweden with higher densities reflects the number of deaths that would have occurred in the North of each country. Furthermore, the South of if the European regions would have the same age and sex France and Austria and the North-East of Germany are composition. To account for the health status of the pop- characterised by slightly higher densities of hospital beds. ulation, we consider the inverse of the SMR. Panel (a)of We further include the population density (popdens)to Fig. 2 shows the spatial distribution of the average SMR. capture the degree of urbanisation of the region. Empir- An visual inspections reveals a slight North-South gradi- ical evidence has shown a relationship between health ent of the SMR for most countries with higher mortality and the population density of a region. However, a simple rates in the South. In Italy and Spain the mortality rates rural/urban differentiation does not sufficiently describe appear to be lower in the South of the respective coun- the relationship between health and location (see for tries. Further, the SMR diverges between the regions in instance Fassio et al. and Adair). East and West Germany. Socio-economic and demographic profiles Health care infrastructure Besides the health care infrastructure demographic and The inclusion of health care inputs corresponds to the socio-economic factors play an important role for the pro- related literature [2, 6, 17, 23]. In our stylized specifica- duction of health and health outcomes . In a health tion of the country specific SFA model, we concentrate production framework those demographic and socio- on the number of physicians (doctors)and thenumberof economic factors can illustrate the utilisation of the health hospital beds (beds) both per 100,000 inhabitants as input care infrastructure . Due to varying regional utilisa- variables representing the outpatient and inpatient sector, tion structures, inefficiencies in the provision of health respectively as no data on the utilisation of health care ser- care services may arise. Possible sources of inefficiency vices is available at such a high spatial resolution. Panels include inaccurate and unnecessary medical treatments (b)and (c)ofFig. 2 display the average regional distri- due to a lack of understanding and an over or under bution of physicians and hospital beds, respectively. The use of medical services [17, 20]. To control for differ- spatial distribution of physicians does not exhibit a clear ent patterns in the utilisation of and the access to health spatial pattern in Austria, Germany, Italy and Scandinavia. care services we include the GDP per capita in (national) In France, the number of physicians is relatively low in the purchasing power parities, education, the share of the North. The area around Paris is an exception with a higher elderly and the population density in the inefficiency supply of physicians in comparison to the surrounding scaling function (as z in (4)) similar to Herwartz and it Schley Health Economics Review (2018) 8:11 Page 5 of 11 Fig. 2 Health care output and inputs. The figure presents the spatial distribution for the average standardised mortality rate (a) and physician density per 100,000 inhabitants (b) and the number of hospital beds per 10,000 inhabitants (c) Schley . The relationship between health and income and population’s health status due to different conditions is based on several factors. On the one side, income dif- of life both at the individual and at the population level ferences are directly related to differences in individual’s . Empirical evidence has shown that regional (and Schley Health Economics Review (2018) 8:11 Page 6 of 11 individual) deprivation increases the risk of poor health Results and discussion . As health care can be seen as a luxury good , In the following, we discuss estimation and inferential regional deprivation does not only describe a direct link results for the group specific stochastic frontier models between health and income. Additionally, regional depri- and the metafrontier model. First, we discuss the relation- vation might account for access barriers to medical ser- ship between the health care infrastructure and overall vices . We therefore include the GDP per capita (gdp health. Second, we examine the extend to which demo- pc) to describe regional deprivation. graphic and socio-economic characteristics shape devi- The empirical literature has shown that a positive rela- ations from the health frontier. Third, we analyse the tionship between health and education exists [1, 12]. regional distribution of efficiency scores and how effi- Higher education is related to a healthier life style which ciency levels change with respect to the metafrontier includes a healthy diet and exercises. Further, higher estimation. If not mentioned otherwise, the discussion of education is likely related to an improved understand- estimation results refers to the nominal 5% significance ing of medical treatments. To approximate education, we level. include the proportion of employees with a university degree as share of all employees (education). Both income Elasticities of health care service provision and education can be seen to describe the access to and Table 2 documents the estimation results for the SFA the utilisation of the regional health care services. To model for pooled and group specific frontiers and the account for differences in the utilisation patterns associ- metafrontier model. Comparing the results of the group ated with age, we include the share of the elderly (age65) specific frontiers and the pooled model, which analy- as older age is related to an intensified need of medical ses the data for all countries simultaneously, shows that treatment. applying a joint model for all countries does not suffi- Additionally, we include the population density (pop- ciently capture the differences in health care production dens) in the inefficiency scaling function to control for across the countries as the model parameters differ across factors influencing the efficiency of service production models. To check the appropriateness of the metafron- based on the location. As all countries offer a general tier model, we apply a Likelihood Ratio (LR) test to test coverage of the population by means of statutory health if technological differences between the countries are sta- insurances or a taxed based National Health Services we tistically significant. Particularly, the test statistic for the do not control for price related access barriers. LR test is LR = 1214.8 and χ distributed with 50 degrees Table 1 displays means, pooled standard deviations as of freedom. We can therefore reject the null hypothesis of well as within and between standard deviations. Cross- identical group frontiers. sectional heterogeneity across different NUTS 2 regions The results of the group frontiers indicate a positive is more pronounced than the time heterogeneity. The relationship between the number of physicians and health stochastic frontier model for the production of health care outcomes. Solely, the estimated effect for doctors care in the European regions is estimated for all variables in Italy is negative. Somewhat surprising, the number in the production function and gdp p.c. and popdens in of hospital beds negatively relates to population’s health. logarithmic form. This counter-intuitive effect might relate to an inappro- priate distribution of health care infrastructure. Similarly, Herwartz and Schley  find a negative association Table 1 Descriptive Statistics. The table documents descriptive between the supply of hospital beds and health outcomes statistics for the 125 NUTS-2 regions from 2005 to 2014. The in the German districts. Noticing the negative connec- overall number of observations is 1149. In the second column the tion between inpatient care and health outcomes is in pooled sample means are reported, the third column contains line with the supply-sensitivity of medical care. Accord- the unbiased pooled standard deviations. In the last two columns ingly, the supply and availability of medical resources the between and within sample standard deviations are influences its utilisation . In other words, in regions presented, respectively with an increased level of inpatient care the hospital Mean SD Between SD Within SD admission rates are relatively higher with likely adverse SMR 5.71 0.69 0.61 0.34 effects on health . In line with this, Fisher et al.  doctors 370.23 77.71 62.72 44.83 find for the US that increased regional mortality rates are associated with a relatively high level of health care beds 600.19 236.59 235.98 29.12 expenditure. popdensity 311.13 684.23 660.49 26.45 At first glance, the negative association of the popula- gdp p.c. 27647.00 6787.16 6556.47 1674.96 tion density and health seems counter-intuitive. However, education 24.73 7.54 7.49 2.40 as others have shown (see for instance [10, 16]) a low age65 0.19 0.03 0.03 0.01 population density relates to an improved quality of life. Schley Health Economics Review (2018) 8:11 Page 7 of 11 Table 2 Stochastic frontier and metafrontier estimation results (t-statistics in parentheses). This tables documents the estimation results from the regression model in (1)-(4) using data from 2005 to 2014. The second columns shows the regression results for a pooled model for all countries. The last column gives the metafrontier parameter results as in (6). The t-statistics for the metafrontier parameters are based on simulated standard errors (simulation with 500 replications) Pooled Austria France Germany Italy Scandinavia Spain Metafrontier Output elasticities (x ) it Intercept 0.013 -0.042 0.034 0.01 0.053 -0.04 0.065 0.152 (1.42) (-3.63) (4.18) (1.62) (5.68) (-5.91) (4.05) (11.31) ln(doctors) 0.092 0.261 0.465 0.219 -0.07 0.539 0.161 0.125 (4.11) (3.90) (12.33) (5.59) (-2.44) (11.31) (4.54) (3.46) ln(beds) -0.109 -0.275 -0.551 -0.258 -0.149 -0.213 -0.148 -0.209 (-13.67) (-4.91) (-16.46) (-5.75) (-3.46) (-8.62) (-2.88) (-6.99) ln(popdens) -0.006 -0.071 -0.049 -0.058 -0.034 -0.079 -0.016 -0.035 (-1.47) (-8.25) (-6.53) (-7.19) (-3.17) (-8.41) (-2.21) (-4.19) Effects on inefficiency (z ) it Intercept -1.49 -0.134 3.421 -1.193 1.455 -8.408 1.119 (-1.87) (-0.07) (3.73) (-1.36) (1.69) (-1.72) (1.47) ln(gdp p.c.) -1.83 -5.878 -2.449 -3.344 -1.38 -1.765 -1.056 (-4.2) (-4.43) (-2.97) (-4.78) (-3.48) (-0.67) (-1.21) education -0.031 -0.019 -0.089 -0.128 -0.39 -0.139 -0.12 (-2.2) (-0.34) (-3.29) (-6.04) (-5.20) (-2.10) (-3.69) age65 -4.26 -25.972 -22.513 2.791 5.303 23.964 0.303 (-1.03) (-2.60) (-4.00) (0.63) (1.03) (1.18) (0.06) ln(popdens) 0.564 -0.937 0.343 -0.451 0.092 -1.954 0.212 (4.03) (-1.65) (1.73) (-1.70) (0.54) (-4.00) (2.16) σ 0.994 0.932 0.953 0.943 0.966 0.924 0.984 2 2 γ = σ /σ 0.318 0.92 0.882 0.72 0.846 0.731 0.842 log-likelihood 968.757 138.016 350.592 525.984 228.379 153.657 179.541 no of observations 1149 90 218 368 177 116 180 A note of caution is in order regarding the interpretation medical services, we examine the relationships of demo- of the empirical results due to the potential of estima- graphic and socio-economic variables and the efficiencies tion bias as a result of reversed causality. For instance, of health care service provision. Interestingly, many esti- more health care services could possibly be available in mated coefficients in the medium panel of Table 2 are regions with a higher need, i.e. poorer population’s health. significant. Nevertheless, the health care sector is a highly regulated Income positively relates to the efficient provision of market in which fundamental market mechanisms might medical services. The estimated coefficients attached to fail . For example, the regional planning of health care gdp p.c. lack statistical significance in Scandinavia and services is based on allocation formulas in Germany which Spain. This result is intuitively appealing as low income do not or only implicitly take morbidity into account . families are likely confronted with access barriers result- Moreover, it is noteworthy that regulators do not know ing in a lower utilisation of and satisfaction with the mortality and morbidity rates at the time of structural health care system . Furthermore, we diagnose a pos- planning weakening the potentials of endogeneity bias itive relationship between the proportion of university (see also ). graduates in the overall number of employees and effi- ciency. The effect lacks statistical significance only in The effect of socio-economic factors on the efficiency of Austria. Education helps to improve the execution of medical treatments and might enhance the utilisation of health care services preventive care which might reduce inefficiencies in the In order to identify how the access to and the utilisation of the health care systems shapes the efficient provision of health care sector. Additionally, as empirical literature Schley Health Economics Review (2018) 8:11 Page 8 of 11 has shown, people with higher educational achievements country-specific frontiers cannot be compared across might have a lower burden of disease due to healthier groups as they are calculated with respect to different lifestyle choices [1, 12]. Initially, the positive associa- technologies . tion of the number of senior citizens and the perfor- For the sampled countries, the technical efficiency mance of health care systems in Austria and France seems scores for the pooled model range from 0.6518 in Ciudad somewhat surprising. One would expect that more co- Autónoma de Ceuta, Spain and 0.9985 in Övre Norrland, morbidities linked with old age would likely decrease Sweden with an overall average of 0.9696. In the pooled the efficient provision of health care services. Similarly, model the regions in Scandinavia perform the best while EibichandZiebarth find a direct positive correlation the Italian regions exhibit relatively low efficiency scores. between the share of the elderly and well-being at the indi- These results are intuitively appealing as the Scandi- vidual level possibly related to the provision of improved navia social security systems enjoy an excellent reputation. health care resources in regions with an older population. Panel (a)ofFig. 3 shows the regional distribution of the The relationship between the population density and average technical efficiency scores of the pooled model. inefficiency depends on the specified country frontier. We A visual inspection reveals a limited regional variation of diagnose a negative association between efficiency and the efficiency across countries. population density in the pooled model and the model The average efficiency scores of the country-specific for Spain. In Scandinavia, the relationship is reversed. The frontiers are slightly lower compared to the pooled model. effects lack significance for all other countries. The relatively high group-specific efficiencies indicate Taken together, the results provide important insights that the regional health care systems use the available into how demographic and socio-economic factors shape resources efficiently. the efficient provision of health care services. The results In the next step, we compare the MTR and the TE highlight that allocation rules for medical infrastructure scores with respect to the metafrontier (TE ). The MTR should take those factors into account (see for instance measures how close the country-specific frontier is to the Smith  for the case of the UK). Furthermore, reducing metafrontier. Higher (lower) values of the MTR imply access barriers possibly increases the efficiency of health a smaller (larger) technology gap between the coun- care service provision by promoting the utilisation of pre- try specific individual frontier and the metafrontier. The ventive care. Additionally, raising the awareness in health MTR ranges from 0.7071 (Sjælland, Denmark) to unity care personnel (i.e. physicians and nurses) for the needs (Övre Norrland in Denmark, Länsi-Suomi in Finland, of specific (deprived) population groups likely decreases and Basilicata in Italy). The MTR values equal to unity inefficiency by possibly improving the communication indicate that the individual country frontiers are tan- leading to a better understanding of medical treatments gent to the metafrontier. The TE range from 0.5719 in (see also Herwartz and Schley ). Ciudad Autónoma de Ceuta, Spain to 0.9979 in Länsi- Suomi, Finland. The overall mean is 0.8375. Panel (b) Metafrontier estimates The different parameter estimates in the group specific Table 3 Technical efficiency (TE) and metatechnology ratio frontiers indicate that differences in the production tech- (MTR) for group frontiers and metafrontier. This table documents nology of the respective health care systems exist. To the average technical efficiencies for the respective countries for investigate if those country specifics trigger differences in the pooled model in the first column, the average efficiencies for the efficiency of the health care provision, we analyse the the group specific models in the second column, the average MTR in the third column and the average TE with respect to the health care systems by means of a metafrontier approach. metafrontier (TE ) in the rightmost column for 2005 to 2014 The last row of Table 2 reports the parameter estimates and t-statistics based on simulated standard errors (see TE TE MTR TE Battese et al. ). In line with the parameter estimates (pooled model) (country specific) attached to the elasticities of the pooled model and the Austria 0.9700 0.9403 0.8668 0.8146 group frontiers, the results reveal a positive association France 0.9746 0.9461 0.8938 0.8462 of physicians and health. We find a negative relationship Germany 0.9669 0.9577 0.8785 0.8412 between hospital beds and the populations density and Italy 0.9583 0.9377 0.9097 0.8552 health. Scandinavia 0.9867 0.9662 0.8563 0.8249 Regional distribution of efficiency scores Denmark 0.9725 0.9997 0.7883 0.7881 Table 3 documents the average efficiency scores for Finland 0.9933 0.9904 0.8883 0.8795 the pooled model, the country-specific frontiers, the Sweden 0.9911 0.9421 0.8772 0.8234 MTR and the technical efficiencies with respect to Spain 0.9592 0.9207 0.8959 0.8244 the metafrontier. Note that the efficiency scores of the Schley Health Economics Review (2018) 8:11 Page 9 of 11 a b Fig. 3 Spatial distribution of the average technical efficiency scores for the pooled model (a)and theTE with respect to the metafrontier (b) of Fig. 3 displays the regional distribution of the effi- The results show that a single European stochastic ciency scores with respect to the metafrontier. The graph frontier model cannot sufficiently capture the hetero- highlights some interesting regional pattern with higher geneous conditions of health care provision in Europe. efficiencies in the North of Finland, Italy and Spain and The comparison of the spatial distribution of the effi- the South of France and Germany. Somewhat surpris- ciency scores from a pooled European model assum- ingly, the MTR results hint at a large technology gap in ing a homogeneous technology hints at significant Scandinavia. In combination with high country-specific efficiency differences across countries. The Scandina- efficiency scores, the results indicate that the regions vian countries achieve on average the highest efficiency in Scandinavia would profit from rising the production scores in a pooled model. Surprisingly, the regions in potentials as the productivity with respect to their own Scandinavia lag behind other regions with respect to frontier is already very high. Similar to the results of oth- the metafrontier highlighting potentials for rising the ers (see for instance Joumard and Nicq ) this might productivity in these regions by, for instance, easing highlight failing market mechanisms within a highly reg- some regulatory burdens. The relatively lower efficiency ulated health care system. For Italy and Spain, which on scores in Italy and Spain hint at substantial oppor- average perform the worst with respect to the pooled tunities to rise the populations’ health by improving (and country specific) frontiers, the results imply that a the management of and the access to the available better management of and an improved access to and resources. utilisation of the available resources could likely lead to Endnotes efficiency gains. For a more detailed discussion of the minimization One can interpret the metafrontier as highlighting long- run production potentials . Accordingly, the relatively problem in (6)see Batteseetal.  and O’Donnell et al. low TE hint at substantial scopes of improvement in the . regional provision of health care services in Europe. For Germany, we approximate the number of physi- cians and hospital beds by the respective number on Conclusion federal state level as no information on NUTS-2 level is The health care systems around Europe are faced by challenges regarding the demographic change along with available. expensive medical innovations. In regard of those devel- The test statistic for the LR test is opments an efficient use of the scarce financial resources LR =−2 ln (L(H0)) − ln (L(H1)) , where ln (L(H0)) is [ ] is necessary. Based on the different health care systems in the log-likelihood value of a stochastic frontier model Europe, the European governments adopt different strate- estimated by pooling the data for all countries, and gies to deal with this and to guarantee an efficient use of the financial resources. ln (L(H1)) is the sum of the log-likelihood values of the This paper analyses the efficiency of health service pro- country specific stochastic frontier models. The test vision across several European countries by means of a statistic is χ distributed. The degrees of freedom are the stochastic metafrontier approach. The application of a difference between the number of parameters estimated metafrontier has the advantage that it is possible to dis- in the individual stochastic frontier models and the tinguish between regional efficiency in relation to the country’s own frontier and an European metafrontier. number of parameters estimated in the pooled model. Schley Health Economics Review (2018) 8:11 Page 10 of 11 The point estimates of technical efficiency for region Competing interests The authors declare that they have no competing interests. i at time t are defined according to Battese and Coelli  Publisher’s Note as TE = E(exp(−u | ) where is the composite error it it it it Springer Nature remains neutral with regard to jurisdictional claims in term of the production function, i.e. = v − u . it it it published maps and institutional affiliations. Author details Appendix A: Data description Sample Received: 1 December 2017 Accepted: 17 May 2018 The data cover the period 2005 to 2014. We analyse the health care systems at the NUTS-2 (Nomenclature des References unités territoriales statistiques). The data is mainly drawn 1. Aka B, Dumont J-C. Health, education and economic growth: Testing for from the Eurostat database. For Germany, the infras- long-run relationships and causal links in the United States. 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Health Economics Review – Springer Journals
Published: May 31, 2018
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