On the allocation of resources for secondary education schools

Transcription

1 On the allocaton of resources for secondary educaton schools C. Haelermans, K. De Wtte and J. Blank TIER WORKING PAPER SERIES TIER WP /08

2 On the allocaton of resources for secondary educaton schools Carla Haelermans* ψ, Krstof De Wtte* α and Jos L.T. Blank ψ * Top Insttute for Evdence Based Educaton Research, Maastrcht Unversty, the Netherlands ψ Centre for Innovatons and Publc Sector Effcency Studes, Delft Unversty of Technology, the Netherlands α Faculty of Busness and Economcs, KU Leuven, Belgum 0 September 20 Abstract Ths paper studes the optmal allocaton of resources n terms of school management, teachers, supportng employees and materals n secondary educaton schools. We use a flexble budget constraned output dstance functon model to estmate both techncal and allocatve effcency scores for 448 Dutch secondary schools between 2002 and The results ndcate that the average techncal effcency amounts to about 89 percent, mplyng that, wth the gven resources, schools could mprove students educatonal performance by eleven percent. In terms of allocatve effcency, we estmate a 0.38 percent overutlzaton of teachers whle management and supportng personnel are underutlzed. The outcomes ndcate that, despte the sgnfcant varaton among schools, the average school s close to the optmal allocaton of teachers. JEL-classfcaton I2, C33, D24, O3 Key words Secondary Educaton; Productvty; Allocatve Effcency; Output Dstance Functon. * ψ Correspondng author: Carla.Haelermans@maastrchtunversty.nl PO Box 66, 6200 MD Maastrcht We would lke to thank Wm Groot, Henrëtte Maassen van den Brnk, TIER-semnar partcpants and EWEPA 20 partcpants for useful comments. The usual caveat apples.

3 . Introducton Teacher shortage s a serous ssue n both European countres (European Commsson, 2006) and the Unted States (Cooley Nchols et al., 2008; Ingersoll & Perda, 200). For example, n the Netherlands, the expected shortage of teachers n secondary educaton for the next 0 years s about 500 full tme equvalents (FTE) per year (Mnstry of Educaton, 20b). Fgure presents the gap between the outflow and nflow of teachers. Smlar patterns are detected for other European countres, lke Germany and the UK (e.g., Kng, 2009; Lpsett, 2008; Santago, 2002). The ncreasng teacher shortage s due to four reasons. Frst, on the supply sde of teachers: teachng became less attractve for youngsters as the jobs socal status dmnshed over tme (Hargreaves et al., 2006; Hargreaves et al., 2007). Second, teachng became a more female occupaton whch s consdered to combne easly wth a famly. Consequently, more teachers work part-tme (e.g., 0.5 FTE) (Mnstry of Educaton, 20a). Thrd, the agng of the baby boomers (the cohort born after World War II) accelerates the outflow (Mnstry of Educaton, 20b). Fourth, at the demand sde: an ncreasng attenton to ndvdual counselng and teachng (partcularly for youngsters from low soco-economc status and for students wth lower educatonal attanments) ncreased the demand for teachers (Mnstry of Educaton, 20b). Fgure The predcted dfference between nflow and outflow of teachers Number of FTE Year outflow nflow (Source of used data: CentER data/ecorys/qqq Delft, 2006) 2

4 Ths paper analyzes whether alternatve school resources, such as supportng personnel, school management and materals, can be - to a lmted extend - substtuted for teachers. In addton, t dscusses the scope for performance mprovements. Indeed, tghtenng budget constrants and ncreasng demands force schools to spend resources n terms of employees, management and materal n the most productve way. Much s at stake. The ncreasng requrements for schools (e.g., more pupl counselng, addtonal extra-currculum actvtes, use of school buldngs durng weekends and summer holdays) nduce pressure on the resources whch are already n place. Productvty becomes an ssue n schools (see Ball & Goldman, 997; Mnstry of Educaton, 20a, p. 98). Wth the nsghts at hand, resources can be reallocated among schools such that hgher educatonal attanments can be obtaned. Although n polcy documents the teacher shortage s referred to as an overall phenomenon (Mnstry of Educaton, 20b), t s of course clear that some subjects are more lable to a shortage of teachers than others (e.g., Ingersoll & Perda, 200; Mangrubang, 2005). We make abstracton of ths heterogenety for two reasons. Frst, our data nclude the total amount of teachers per school and do not allow us to make a dstncton n the dfferent teachers for the dfferent subjects. Second, even f t were possble to make ths dstncton n the data, the amount of teachers per subject per school s very small, whch makes the varaton n the amount of teachers even smaller. The latter makes t mpossble to study the teacher shortage per subject. In lne wth prevous lterature (Fred et al., 2008) performance s consdered n two complementng ways: techncal and allocatve effcency. The former denotes the relatve rato between the avalable resources and the outcomes. Relatve to a producton fronter, t measures the output shortfall of schools gven the avalable resources. The latter enrches the techncal effcency estmatons by prce nformaton. The measure of allocatve effcency yelds nsghts n the under- or over-utlzaton of school resources. Allocatve effcency n secondary educaton has been largely overlooked. The avalable allocatve effcency studes mostly consder hgher educaton (Cherchye & Vanden Abeele, 2005; Johnes & Johnes, 2009; Soares de Mello et al., 2006; Tauer et al., 2007). Some rare exceptons are Banker et al. (2004) and Grosskopf et al. (997, 200) who both study school dstrcts n Texas. Ths paper examnes performance n Dutch schools. Studyng Dutch schools s attractve and nsghtful for three reasons. Frst, standardzed performance measures of Dutch students make educatonal attanments well comparable. Second, there s nformaton on student achevement, whch compares the educatonal career of a student (both n terms of school track and retentons) wth the educaton track predcted for a student at the end of prmary 3

5 educaton. Thrd, Dutch schools receve a yearly lump sum budget from the government, whch s at the dscreton of the school such that, wthn the exstng legal framework, the allocaton of ths budget among the several resources s the decson of the school (Stb. 963; 40). Therefore, a sgnfcant heterogenety n hred resources, n terms of management, teachers, supportng personnel and materal use, s observed. Note that the lump sum budget excludes (large and dscontnuous) payments on housng nfrastructure (the latter are provded by the muncpal and central government). From a methodologcal perspectve, we argue that schools are maxmzng educatonal attanments wth a gven budget. From ths startng pont, one can derve an economc model whch explans the performance and the optmal usage of resources. As we do not observe the relatonshp between nputs and outputs (Yatchew, 998), we rely on sem-parametrc models whch avod a specfcaton bas. In partcular, techncal and allocatve effcency are estmated aganst a fronter consstng of best practce observatons. We examne the potental ncrease n educatonal attanments under budget constrants. Formally, ths corresponds to a so-called budget restrcted ndrect output dstance functon (IOD) model (Grosskopf, 997; Blank, 2009). The remander of the paper s structured as follows: Secton 2 outlnes the methodology and the model. Secton 3 descrbes that data and data sources whle Secton 4 dscusses the results. Some concludng remarks are presented n Secton Methodology Economc model We start from the presumpton that schools, wth an exogenous budget, are maxmzng the educatonal attanments of students. In a sense, schools are tryng to become best practces n educatng students, and, thus, to reach the fronter of the producton functon wth ther budget. The best practce fronter can be determned along varous ways (see Fred et al., 2008 for a dscusson). Ths paper apples the rather strct Corrected Ordnary Least Squares (COLS) method n whch the OLS regresson s shfted to the observaton(s) wth the most favorable rato of outputs to nputs. 2 The(se) observatons are consdered as best practces and obtan an effcency score λ equal to 00. Ineffcent observatons le below ths best practce fronter and obtan an effcency score λ smaller than 00. The score (00- λ) can be nterpreted as the percentage output ncrease f the observaton would perform as effcent as 2 The strct assumpton on effcency s convenent n the settng at hand as t provdes a lower bound on the effcency estmatons. In other words, by no other assumpton, neffcency wll be hgher. 4

6 ts best practce. If n the effcency estmaton the observaton s further constraned by the total cost of nputs, a so-called ndrect output dstance (IOD) functon s estmated (for a detaled dscusson, see Färe and Prmont, 995). We apply an IOD functon whch s smlar to the model of Grosskopf et al. (997). Schools are producng multple outputs (see below) under budget constrants, whch can be thought of as exogenous to the schools. The determnstc model can be wrtten n short hand as follows (note that the full budget restrcted output dstance functon ncludng the full share equatons, s presented n Appendx ): w ln( IOD) = f (ln y,ln,ln z) () C where IOD stands for the ndrect output dstance functon wth an a pror determned functonal form denoted by f (e.g., Cobb-Douglas, Translog, Fourer), y s the vector of outputs, w denotes the vector of prces of the used resources, C s the vector of costs and z s a vector capturng fxed nputs or school characterstcs (reflectng observed and unobserved heterogenety). Several (mld) propertes, requred for a meanngful estmaton, hold for the model. It s straghtforward to verfy (see, e.g. Coell et al., 2005; Färe & Prmont, 995) that the model s non-decreasng and quas-concave n C w. Condtonal ndependence (cfr. a separablty condton) s mposed to the model n that nformaton on one nput does not yeld nformaton on other nputs. The costs shares are mmedately derved from the IOD by Roy s dentty, n partcular, as the frst dervatve wth respect to costs (see Färe & Prmont, 995, p. 92). They correspond to: S δ ln IOD y, w δ ln = C w C, z n w δ ln IOD y,, z C wn δ ln C (2) Includng costs shares (and thus estmatng an IOD functon nstead of an output dstance functon) s convenent as t allows us to estmate the optmal amount of dfferent types of 5

7 costs (e.g., personnel costs, management costs, supportng personnel and materal use). The comparson between the actual costs and the optmal costs yelds allocatve effcency scores. 3 The ndrect output dstance functon and the cost share equatons provde a set of equatons from whch the parameters can be estmated. In practce, the system of equatons s estmated by system OLS 4. Intutvely, ths corresponds to a least squares estmaton wth multvarate regresson systems under a jont densty. 5 To foster further applcatons, the TSP code s avalable upon request. Model specfcaton As typcal n (sem-)parametrc analyss, Equaton () requres an a pror specfcaton of the producton functon f. A survey by Yatchew (998) clearly ndcates that economc theory hardly ever provdes a precse specfcaton of the functonal form of producton functons. Ths has recently been confrmed for educatonal settngs by Rothsten (200). To avod a specfcaton bas and to allow for some flexble model characterstcs t seems approprate to test the choce of the functonal form. Therefore, we start from estmatng a Cobb-Douglas producton functon, and systematcally add nteracton terms such that we estmate a Translog and Fourer specfcaton. The Fourer cost functon adds sne (sn) and cosne (cos) terms to the Translog model, whch n turn adds quadratc terms to the Cobb-Douglas (CD) specfcaton. The Fourer cost functon does not only allow for lnear relatonshps as n CD and non-lnear causaltes as n Translog, but for almost nfntely flexble relatonshps. The Fourer specfcaton s thus a generalzaton of these models. A large advantage of the Fourer functon s that t s essentally unbased and obtans accurate fnte parameter approxmatons (Gallant, 98; and see Gallant, 984 for vsual representatons) 6. Whle CD and Translog specfcatons only explot the domnatng trend n the data, and thus provde only a local approxmaton for the unknown functon, the Fourer model estmates a global alternatve as t explots the varablty over the whole range of data. In fact, the Fourer specfcaton provdes a framework to estmate a parametrc functon wth a flexblty comparable to a nonparametrc approach (Gallant, 98, 984; Kuenzle, 2005). It s possble to nclude several grades (denoted by N) n the Fourer functon. N denotes the number of sne and cosne terms ncluded. N s determned statstcally. The grade of the gonometrc terms s establshed by testng on the bass of an F-test whether the sum of squared resduals (SSR) 3 From a practcal perspectve, an addtonal advantage of ncludng cost shares s that the estmatons are more effcent as more nformaton s added to the model (Blank, 200; Greene, 2008; Grosskopf et al., 997). 4 Ths leads to smlar results compared to usng SUR as estmaton method. 5 Note that one share equaton n the output dstance functon model has to be dropped because the shares add up to one causng the varance-covarance matrx of the error terms to be sngular. 6 Ths has also been tested usng Monte Carlo Smulatons (Chalfant & Gallant, 985). 6

8 dffers sgnfcantly between two values of N. As such, t tres to capture the true relatonshp between costs, and nputs and outputs. For nstance, testng whether the SSR for N = 0 and N = dffers sgnfcantly, reveals whether a Fourer functon of grade, whch ncludes sn w, cos w, sn( y ),cos( y ), sn( z ) and cos( z ), s statstcally preferred to a functon C C of grade 0,.e. a Translog model. Comparng N = and N = 2 makes clear whether a Fourer functon of grade 2, whch ncludes sn w w w w, sn 2, cos and cos 2, sn(2 y ), C C C C sn(2 y ), cos( y ) and cos(2 y ) et cetera, s preferred to a Fourer of grade. We start from the Cobb-Douglas specfcaton and gradually test the robustness of the results by dfferent specfcatons. The results (see below) pont out that the Fourer specfcaton of grade 2 should be preferred n the settng at hand. Ths specfcaton s ncluded n Appendx. Observed and unobserved heterogenety Observed heterogenety s captured by ncludng exogenous characterstcs n the model (denoted by z). The exogenous characterstcs cannot be nfluenced by polcy or school management and these varables are drectly ncluded n the output dstance functon and share equatons as the z-varables. Includng or excludng exogenous characterstcs nfluences the poston of the fronter such that they have to be taken nto account n determnng the optmal composton of resources and student attanment. Ths s graphcally presented n Fgure 2. To capture unobserved heterogenety, we explot the panel data structure of the data by ncludng school type, year and regon fxed effects n the estmaton. We observe 9 dfferent school types (rangng from pre-vocatonal educaton tll pre-unversty educaton), 6 years, and 8 regons. Regon fxed effects are ncluded as school nspectorate argues the large dsperson n school attanments across regons (Inspecte van het Onderwjs, 20). The fxed effects are estmated for the output dstance functon but not for the share equatons (as the share equatons are a dervatve of the output dstance functon wth respect to w, and C the fxed effects do not appear n the output dstance functon as an nteracton term wth w ). C 7

9 Fgure 2 The producton fronter wth and wthout envronmental varables Y Observed producton functon z-varables Producton fronter wthout exogenous nfluences Outputs x,y Inputs X 3. Data and model specfcaton The data To estmate the techncal and allocatve effcency of schools, we apply the sem-parametrc IOD model to a representatve balanced panel dataset of 448 Dutch secondary schools coverng the school years 2002/03 to 2007/08. For convenence the school year 2002/03 s referred to as 2002, and so on. The data are obtaned from several exstng admnstratve data sets from the Mnstry of Educaton, the Educaton Inspectorate and Statstcs Netherlands. Producton Educatonal producton s commonly defned as some knd of measure of educatonal attanments n knowledge and sklls (Wenger, 2000). We use two measures for educatonal producton: the average student central examnaton grades per school and the average student achevement each year durng secondary educaton. 7 The former serves as a proxy for attanments, whle the latter for qualty. Wth respect to the frst, every graduatng student undertakes ths test at the end of secondary educaton for the subjects the student s regstered. The exams are graded n a double blnd way and the school and the teacher of the students cannot drectly nfluence the 7 A robustness check wth only central examnaton grades as output and number students and student achevement as control varables yelds very smlar results. 8

10 outcome of the exams. 8 The average grade per school s based on the average of all central examnaton grades for all students n all subjects. Note that the level of analyss s the school and not the subject or the student. Our results wll therefore not make a concluson on teacher shortage n ndvdual subjects. To proxy the qualty of the educatonal process, we nclude a composte number of Educaton Inspectorate: average aggregated corrected achevement. Ths number compares the educatonal track of a student n a gven year wth the educaton track predcted for a student at the end of prmary educaton. An aggregated corrected achevement of denotes that all students are exactly n the year and level of educaton they are predcted to be accordng to prmary educaton test results. Both output measures are multpled by the total number of students, as more expenses should ether lead to hgher performance of the current number of students or the same performance for a hgher number of students. Summary statstcs are provded n Table. Table shows that the average central examnaton grade amounts to 6.4 on a 0 pont scale. A 5.5 s suffcent to pass the subject. The grades range from 5.4 to 7.2. Despte the larger varaton, the standard error s relatvely low. Student achevement has an average of and the total number of students s on average,744 per school. Resources School resources are summarzed along four relatvely homogeneous categores: () management personnel, (2) teachng personnel, (3) supportng personnel and (4) materal supples. Captal n the sense of housng nfrastructure s not accounted for due to data constrants. For the human resources (the frst three groups), data are avalable on full tme equvalents (FTE) and costs. In partcular, the costs correspond to the costs per FTE per year. The total average number of FTE per school equals 63, consstng for 80% out of teachers, 4% of supportng personnel and the remanng 6% of managng personnel. The total costs of a school are about 2 mllon euro per year. The majorty of the costs are spend on teachng personnel, followed by materals, managng personnel and, fnally, supportng personnel. Besdes teachng, a teacher has some management and admnstratve dutes. The dfferent tasks wthn one functon are not offcally reported. We assume that there s a homogeneous dstrbuton of these dfferent tasks wthn one functon, both between teachers wthn one school and across schools. 8 School mght be able to ndrectly nfluence exam results on an abstract level as they may, for example, be teachng to the test. However, as the exams are natonally developed and cannot drectly be nfluenced by the school, we assume ndependence of these results. 9

11 The wage of teachers, school management and supportng personnel s, a pror, smlar for all schools n the Netherlands, as a teacher wth a gven experence and a certan age would earn a smlar wage across all schools. Therefore, we expect prces to be comparable and use the reported expenses spend on the dfferent personnel types per school. For the prce of materals the consumer prce ndex provded by Statstcs Netherlands s used as a proxy. As we use school years whereas the prce ndex s measured n calendar years, we have to recalculate the value of the prce ndex to ft our school year data. Control varables (z-varables) Some nfluences are exogenous to the school and create heterogenety across schools. These varables nclude the number of schools per governng body, the degree of competton, the share of students from a dsadvantaged neghborhood, degree of urbanzaton, students qualty, total number of students (whch s endogenous to school qualty n de long run but can be assumed exogenous n the short run), management and teacher experence and the number of locatons per school. The latter reflects the geographcal spread of a school over dfferent stes. Ths mght lead, on the one hand, to addtonal costs due to extra travel tme (for teachers), or on the other hand, to scale economes (e.g., n management). All these varables nfluence the effcency of a school but cannot be nfluenced by the school management. The number of schools per governng body may, on the one hand, reflect scale economes thanks to partcpaton n a schools network and shared servces; on the other hand, t captures the harms of a centralzed bureaucracy. The average number of schools per governng body s almost 6 wth a mnmum of and a maxmum of 3. The latter s one specfc governng body whch manages many schools. Prevous research has shown that competton fosters productvty (Abbott & Doucoulagos, 2009; Mllmet & Coller, 2008) and that urbanzaton s related to productvty (Naper, 200). Competton s estmated as the number of schools per 0,000 nhabtants n a muncpalty and has an average value of The share of students from a dsadvantaged neghborhood amounts, on average, to 2.7%. Furthermore, we see that the number of locatons per school s 2 on average, but ranges from to 0. We observe managers wth an average experence of 7 years. The average experence of teachers s 2 years, whle the average amount of part tme teachers s 26%. 0

12 Table Summary statstcs (for 2007) Mean Standard devaton Mnmum Maxmum Central examnaton grade Student achevement Number of FTE management personnel Number of FTE teachng personnel Number of FTE supportng personnel Total FTE Total costs (n 000 euro) Management personnel costs per FTE (n euro) Teachng personnel costs per FTE (n euro) Supportng personnel costs per FTE (n euro) Competton Number of schools per governng body Share of students from a dsadvantaged neghborhood Degree of urbanzaton Total number of students per school Average years of experence managers Average years of experence teachers Percentage part tme teachers Number of locatons per school Results Before estmatng the results, we attempt to reveal the functonal form of the IOD functon by examnng the grade of the Fourer functon. We estmate the followng model optons: the Cobb-Douglas (result: degree of freedom (df) =4, log lkelhood (logl) =20532), Translog (df=07, logl=264), Translog Fourer grade (df=35, logl=23524) and Translog Fourer grade 2 (df=64, logl=25225). The log lkelhood scores ndcate that the Translog Fourer model of grade 2 s the best model. The remander of the paper apples a Translog Fourer functon of grade 2 (.e., ncludng sn w w and cos C w, sn( y ) and cos( y ), sn( z ) and cos( z )), sn 2 and C C w cos 2, sn(2 y ) and cos(2 y ) and sn(2 z ) and cos(2 z )) as functonal form. C

13 The full estmaton results are presented n Appendx 2. 9 More than half of the parameters are sgnfcant at the 5 percent level, the r-squares of the cost shares are reasonable (rangng up to 4 percent) and all cost varables have a postve coeffcent. The latter ndcates that the monotoncty condton s satsfed. The concavty condton s also satsfed as all elastcty s of substtuton are negatve and the Hessan s negatve defnte. 0 Techncal effcency Table 2 presents the average techncal effcency scores for Dutch secondary schools for However, because of smlarty of the results and for the sake of space we wll frst start dscussng the results for Note that n our settng, by constructon, effcency ( dong thngs rght ) s closely related to effectveness ( dong the rght thngs ). Indeed, our output varable corresponds to average pupl test scores and average student achevement. Therefore, n most cases the most effcent school also has a hgh average central examnaton grade. Wth respect to effcency, a devaton (λ) from 00 denotes that the school can mprove ts output by (00-λ) percent f t would use ts gven resources as effcent as ts best practce n The average techncal effcency score for the 448 schools s 88.6%, mplyng that wth the same resources they could mprove educatonal performance by about.4%. 75% of the schools could mprove the educatonal attanments of the students by % (note that ths s controlled for observed heterogenety among schools, as well as resources; see before) 2. We observe a mnmum effcency score of 75% whch ndcates that the least performng Dutch school could mprove the educatonal attanments of ts students by 25% f the school would use ts resources as effcent as ts best practce. The standard devaton of the scores amount to 2.4%, whch ndcates that, on average, Dutch secondary educaton schools are well comparable and do not dffer much n terms of effcency. The results for other years are smlar to the outcomes for For 2005, the lowest average effcency score s observed. Ths could be due to the large varance n student achevement n Moreover, n 2005 there was a relatve large jump n costs, compared to the years before and the years after. To our best knowledge, there s no evdent explanaton for the latter two devatons. The hghest average effcency s observed n In ths year, 9 Although the statstcal analyss ponts out that the Fourer functon of grade 2 s the preferred functonal form, a robustness test wth Cobb-Douglas and Translog specfcaton delvered comparable results. Ths ponts to robust results. 0 The Hessan has been evaluated for every observaton separately. Although ths does not necessarly need to hold. 2 Runnng the analyss wthout fxed effects shows that the relatvely hgh effcency scores are not related to the ncluson of the fxed effects. 2

14 schools could mprove ther performance by about 9% f they would perform as effcent as the best n class. Table 2 -Techncal effcency scores of Dutch secondary educaton schools n percentages (Model: Fourer grade 2) Average Standard devaton Mnmum nd quartle Medan th quartle Maxmum Allocatve effcency Whle techncal effcency does not account for prces and costs, allocatve effcency scores do. Although all schools observe the same exogenous prces, a pror, t can be expected that not all schools use the optmal nput mx. In ths case, hgher output can be obtaned by reallocatng the nputs. As schools dffer on resources, student populaton, locaton etc., the optmal nput mx wll be dfferent for every school. The results are presented n Table 3 and Table 4. A postve sgn denotes an overutlzaton of the nput (.e., allocatve effcency could be mproved by dmnshng ths varable), whle a negatve sgn ndcates underutlzaton of the nput. On average, the share of management and supportng personnel s lower than the optmum. On the other hand, schools use more teachers and materal than would be optmal. The devaton from the optmum for materal and supportng personnel s sgnfcantly dfferent from zero at the % level. The devaton from the optmum for management and teachng personnel are not sgnfcant, not even at the 5% level. In the followng descrpton and nterpretaton of the results we focus on the results of the medan n From Table 3, we observe an average overutlzaton of teachers of 0.38%. Ths average hdes a sgnfcant dsperson across the dstrbuton. The latter s also vsble n Fgure 3. As the average school hres 3 FTE teachers, an overutlzaton of 0.38% ndcates that the same or hgher test scores could be reached wth 30 FTE teachers. Ths amounts to 3 Results for the other years are avalable upon request from the correspondng author. They are not ncluded n the paper to reduce ts length. 3

15 an average cost reducton of 47,000 euro. It s nterestng to note that the 0.38% overutlzaton n FTE n 2007 amounts to about 250 FTE on a total of 64,000 FTE (Mnstry of Educaton, 20a), whch s about one thrd of the teacher gap n 2007 that has been predcted n the report of CentER data, ECORYS and QQQ Delft (2006). In the latter report, the teacher shortage n 2007 s estmated on about,000 on a total of almost 62,000 FTE (.6%). 4 Ths mght ndcate that, for some schools, part of the predcted teacher shortage could be tackled by better allocaton of resources. However, the average school operates close to the optmum wth respect to the number of teachers. Furthermore, there s an underutlzaton of 0.60% of managers. Ths denotes that the medan school could ncrease ts management structure from 9 FTE managers to 9. FTE. Ths small average underutlzaton hdes a sgnfcant dsperson across the dstrbuton. Whle the medan school has an underutlzaton of management, the top 25% of schools have a (large) overutlzaton. These correspond to the larger schools compared wth the total sample. Although prevous research ndcates a suffcent number of managers at Dutch secondary schools (Blank et al., 2007), our results show that many schools have ether an underutlzaton (bottom 50%) or overutlzaton (top 25%) of managers. Ths msmatch can be explaned by the constant warnng for an upcomng shortage, not only for teachers but also for managers (CentER data/ecorys/qqq Delft, 2006). Ths has been confrmed by sector organzatons (Sectorbestuur Onderwjsarbedsmarkt, 2009; Van Velden, 2002). Another reason mght be the ncreasng number of the so-called mddle managers and organzatonal unts (Odenthal et al., 2007; Smkns, 2000). Interestngly, schools have.03% too few supportng personnel than would be optmal, and 0.48% too much materals than would be optmal. The underutlzaton of supportng personnel s rather surprsng gven ts low costs and gven ts effectveness (e.g. Groom & Rose, 2005). 4 Ths stuaton was predcted for an economc recesson, whch was the case n

16 Table 3 Allocatve effcency n percentages (data for 2007) Average Standard devaton Mnmum Frst Quntle Medan Thrd Quntle Maxmum Devaton from optmal share - management Devaton from optmal share - teachng personnel Devaton from optmal share - supportng personnel Devaton from optmal share - materal * Sgnfcantly dfferent from zero at 0% level. ** Sgnfcantly dfferent from zero at 5% level. *** Sgnfcantly dfferent from zero at % level. Table 4 Medan of allocatve effcency n percentages Devaton from optmal share - management Devaton from optmal share - teachng personnel Devaton from optmal share - supportng personnel Devaton from optmal share - materal Other output for all years as presented n Table 3 s avalable upon request 5

17 Despte the crtcal teacher shortage, t s nterestng to observe that some schools use too many teachers and too lttle of the other resources. At one end, there s a school whch should hre 4.7% more teachers (.e., from 70 FTE to 80 FTE) to mprove the educatonal attanments of ts students. At the other end, a school hres 2.% too many teachers. Also, the medan school hres too many teachers, although schools are very close to the optmum. It seems that the teacher shortage n the Netherlands may be decreased by reallocatng teachers across schools and/or by relyng on alternatve resources. Not all tasks n a school have to be fullflled by teachers. For example, supervson of a group of students n self-study, the support and coachng of student projects or supervson durng lunch tme can be undertaken by supportng personnel (see e.g. Blatchford & Sumpner, 998). Fgure 3 Over/Underutlzaton of teachers (n %) Trends n allocatve effcency The nter-temporal trends n resource utlzaton allow us to explore the under and overutlzaton over tme. Teacher shortage s already an ssue for some tme (e.g. CentER data/ecorys/qqq Delft, 2006; Muncpalty of Amsterdam, 2003; Nels, 2004; Stjnen, 2003). Nevertheless, we observe n Fgure 3 that the medan school n the sample has an overutlzaton of teachers between 2002 and In those years, a better allocaton of teachers over schools could have decreased the teacher shortage for the bottom quartle of schools. Also n ths nter-temporal perspectve, the numbers are on a smlar level as the predctons made by CentER data, ECORYS and QQQ Delft (2006). Furthermore, the resultng numbers from our study are on the same level as the numbers from the report of the Mnstry of Educaton (20b), n whch a yearly shortage of about 500 FTE was predcted. 6

18 Only between 2005 and 2006 a medan underutlzaton emerged, whch s altered n A possble reason for ths sudden change n trend could be that schools and muncpaltes antcpated on the teacher shortage, as ths was already an ssue for a longer tme (Muncpalty of Amsterdam, 2003). Control varables n the model The estmaton results of the exogenous varables are presented n Table 5. They ndcate the correlaton between the control varables and educatonal performance or cost shares. In the IOD model, a postve (negatve) sgn ndcates an unfavorable (favorable) nfluence on performance. From Table 5 we observe that three control varables have a sgnfcant nfluence on performance, namely competton, years of experence teachers, and number of locatons. An ncrease n competton decreases ether student performance or costs. Ths contrasts prevous lterature (see for example Bradley et al., 999; Mllmet & Coller, 2008), but can be ntutve as schools wth less competton need to nvest less n outperformng the other schools and therefore costs are lower. However, also the ncentve for excellence s lower, whch n turn mght lead to lower student performance. An ncrease n share of part tme teachers ether decreases student performance or decreases costs. Ths s n lne wth prevous lterature (e.g. Jacoby, 2006). The fndng on urbanzaton s smlar to prevous lterature. Naper (200) shows that schools wth many students from rural areas tend to be less effcent, ndcatng the mportance of ncludng both the degree of urbanzaton and a regon varable. The number of locatons and the number of schools per governng body ncrease ether student performance or costs. An addtonal locaton results n hgher costs, as s acknowledged by the fundng system of Dutch secondary educaton, whch allocates addtonal money to schools wth extra locatons (Stcrt. 2008; 40). Furthermore, both an extra locaton and more schools per governng body normally means a larger number of students and thereby an ncrease n scale. Ths postve relatonshp between scale and school performance has also been stressed n lterature (e.g., Chakraborty et al., 2000; Johnes et al., 2005). The number of locatons represents the scale of the school. 7

19 Table 5 Estmaton results control varables n the model Varable Estmate Standard Error T- statstc P-value Competton [.000] Number of schools per governng body Percentage students from dsadvantaged neghborhood [.049] [.246] Percentage urbanzaton [.049] Number of locatons per [.000] school Years of experence teachers [.927] Share part tme teachers [.04] 5. Concluson Ths paper studes the optmal allocaton of resources n secondary educaton. We use a semparametrc effcency model wth budget constrants, n partcular the budget restrcted ndrect fourer output dstance functon, to estmate both techncal and allocatve effcency scores for 448 Dutch secondary schools between 2002 and Tests for the parametrc specfcaton of the output dstance functon ndcate that the fourer functon of grade 2 s to be preferred. Four types of resources are dstngushed: management, teachers, supportng personnel and materal use. Wth the gven resources, schools are supposed to maxmze student performance as proxed by central examnaton grades. We control for heterogenety among schools by ncludng control varables, e.g., nformaton on the share of students from dsadvantaged neghborhoods. The results ndcate that the average techncal effcency score for the 448 schools amounts to 89%, mplyng that wth the same resources schools could mprove educatonal performance by about %. Moreover, we observe an overutlzaton of teachers by 0.38% (whch s about 250 FTE (Mnstry of Educaton, 20a)) and an underutlzaton of other personnel resources n The absolute numbers of our results correspond to the predctons made by CentER data, ECORYS and QQQ Delft (2006), who predcted a teacher shortage of about,000 FTE n 2007 on a total of about 62,000 teachers. Our results are also partly n lne wth the numbers from the report of the Mnstry of Educaton (20b). From an ntertemporal perspectve and n contrast to prevous fndngs, the results ndcate that between 2002 and 2004 there was an overutlzaton of teachers. In those years, a better allocaton of teachers between schools could have decreased the teacher shortage for the bottom quartle of schools. Despte the sgnfcant heterogenety across schools, ths artcle 8

20 argues that effcency gans are possble n some secondary educaton schools by reallocatng resources. Appendx Productvty Model To estmate productve effcency, we estmate an output dstance functon and a number of cost share equatons. The functonal form of the producton functon has been estmated as a Fourer functon of grade 2. The model s algebracally presented as: m n = n' W ln( IOD ) a b ln( Y ) c ln d = = C = 2 0 ln( Z ) = j= ln( Y )ln( Y ) 2 W ln C + W j ln C + 2 m m n n n ' b j j + cj = j= n' = j= d j ln( Z )ln( Z j ) + m n = j= e j W j ln( Y )ln C + n' n = j= f j ln( Z W j )ln C N N W W + λ *sn(ln + 2 *cos(ln ) λ = C = C + n' m = j= g j ln( Z )ln( Y j ) N + λ *sn(ln( Y )) + λ = N * cos(ln( Y )) 3 4 = N + λ *sn(ln( Z )) + λ = *cos(ln( Z 5 6 = N )) N N W W + λ *sn(ln 2 ) + λ2 * cos(ln 2 ) = C = C (A) N + λ *sn(ln(2y )) + λ = N = *cos(ln(2y )) 3 4 = N + λ *sn(ln(2z )) + λ6 *cos(ln(2z FE + µ 5 = N )) + 9

21 where IOD = Indrect Output Dstance functon C = total costs Y = output/producton-ndcator ( =,.., m) W = prce used resource ( =,.., n) Z = envronmental varable ( =,.., n ) FE = fxed effects on school type, year and regon µ = randomly dstrbuted error term ao, b, c, d, bj, cj, d j, ej, f j, g, λ, λ2, λ3, λ4, λ5, λ6, λ, λ2, λ3, λ4, λ5 and λ6 are the estmated parameters Roy s dentty s used to derve the optmal cost shares (see Färe & Prmont, 995, p. 92). The optmal cost shares are the followng: S o = T T j n j= (A2) Wth: T = c + n j= c j W j ln C + m e ln( Y ) + n' j j j= j= f j ln( Z j ) + λ + λ 2 + λ + λ 2 + ε (A3) The output dstance functon s specfed as the Fourer grade 2 functon and the share equatons are derved from t. Homogenety of degree one n prces and symmetry s mposed by puttng constrants on some of the parameters to be estmated. These constrants are the n b = followng: n =, b = 0, j and e = 0, j. = j n = j 20

22 Appendx 2 Estmaton Results Fnal Model Parameter Estmate Standard error T- statstc P-value A0 Constant [.000] B Producton ndcator [.000] B2 Producton ndcator [.000] B Producton ndcator * producton ndcator [.650] B2 Producton ndcator * producton ndcator [.650] B22 Producton ndcator 2 * producton ndcator [.650] C prce management personnel / costs [.94] C2 prce teachng personnel / costs [.008] C3 prce supportng personnel / costs [.857] C4 prce materal / costs [.000] C C2 C3 C4 C22 C23 C24 C33 C34 prce management personnel / costs * prce management personnel / costs prce management personnel / costs * prce teachng personnel / costs prce management personnel / costs * prce supportng personnel / costs prce management personnel / costs * prce materal / costs prce teachng personnel / costs * prce teachng personnel / costs prce teachng personnel / costs * prce supportng personnel / costs prce teachng personnel / costs* prce materal / costs prce supportng personnel / costs * prce supportng personnel / costs prce supportng personnel / costs* prce materal / costs [.228] [.987] [.275] [.232] [.220] [.232] [.227] [.695] [.269] C44 prce materal / costs* prce materal / costs [.225] E E2 E3 producton ndcator * prce management personnel / costs producton ndcator * prce teachng personnel / costs producton ndcator * prce supportng personnel / costs [.249] [.239] [.24] E4 producton ndcator * prce materal / costs [.229] E2 E22 E23 producton ndcator 2 * prce management personnel / costs producton ndcator 2 * prce teachng personnel / costs producton ndcator 2 * prce supportng personnel / costs [.249] [.239] [.24] E24 producton ndcator 2 * prce materal / costs [.229] D Competton [.000] D2 Number of schools per governngbody [.049] D3 percentage dsadvantaged students [.246] 2

23 D4 percentage urbanzaton [.049] D7 number of locatons per school [.000] D8 average experence teachers [.927] D9 share of part tme workers [.04] D Competton*Competton [.000] D2 D3 Competton*Number of schools per governngbody Competton*percentage dsadvantaged students [.200] [.055] D4 Competton*percentage urbanzaton [.0] D7 Competton*number of locatons per school [.006] D8 Competton*average experence teachers [.605] D9 Competton*share of part tme workers [.000] D22 D23 D24 D27 Number of schools per governngbody*number of schools per governngbody Number of schools per governngbody*percentage dsadvantaged students Number of schools per governngbody*percentage urbanzaton Number of schools per governngbody*number of locatons per school [.807] [.87] [.869] [.544] D28 D29 D33 D34 D37 D38 D39 Number of schools per governngbody*average experence teachers Number of schools per governngbody*share of part tme workers percentage dsadvantaged students*percentage dsadvantaged students percentage dsadvantaged students*percentage urbanzaton percentage dsadvantaged students*number of locatons per school percentage dsadvantaged students*average experence teachers percentage dsadvantaged students*share of part tme workers [.02] [.344] [.422] [.885] [.484] [.0] [.456] D44 percentage urbanzaton [.338] D47 D48 D49 percentage urbanzaton*number of locatons per school percentage urbanzaton*average experence teachers percentage urbanzaton*share of part tme workers [.774] [.748] [.07] D77 number of locatons per school [.000] D78 D79 number of locatons per school*average experence teachers number of locatons per school*share of part tme workers [.065] [.462] D88 average experence teachers [.948] 22

24 D89 D99 average experence teachers*share of part tme workers share of part tme workers*share of part tme workers [.057] [.032] G producton ndcator *competton [.00] G2 producton ndcator 2 *competton [.774] G2 G22 G3 G32 G4 G42 G7 G72 G8 G82 G9 G92 producton ndcator *Number of schools per governngbody producton ndcator 2 *Number of schools per governngbody producton ndcator *percentage dsadvantaged students producton ndcator 2 *percentage dsadvantaged students producton ndcator *percentage urbanzaton producton ndcator 2 *percentage urbanzaton producton ndcator *number of locatons per school producton ndcator 2 *number of locatons per school producton ndcator *average experence teachers producton ndcator 2 *average experence teachers producton ndcator *share of part tme workers producton ndcator 2 *share of part tme workers F prce management personnel / costs*competton [.033] [.07] [.293] [.420] [.486] [.754] [.086] [.00] [.538] [.427] [.984] [.829] [.366] F2 prce teachng personnel / costs* competton [.325] F3 prce supportng personnel / costs*competton [.25] F4 prce materal / costs* competton [.585] F2 F22 F23 F24 F3 F32 F33 F34 prce management personnel / costs*number of schools per governngbody prce teachng personnel / costs*number of schools per governngbody prce supportng personnel / costs*number of schools per governngbody prce materal / costs*number of schools per governngbody prce management personnel / costs *percentage dsadvantaged students prce teachng personnel / costs *percentage dsadvantaged students prce supportng personnel / costs *percentage dsadvantaged students prce materal / costs *percentage dsadvantaged students F4 prce management personnel / costs*percentage urbanzaton [.27] [.27] [.27] [.27] [.29] [.29] [.29] [.29] [.24] 23

25 F42 F43 prce teachng personnel / costs*percentage urbanzaton prce supportng personnel / costs*percentage urbanzaton [.23] [.24] F44 prce materal / costs*percentage urbanzaton [.24] F7 F72 F73 F74 F8 F82 F83 F84 F9 F92 F93 F94 prce management personnel / costs*number of locatons per school prce teachng personnel / costs*number of locatons per school prce supportng personnel / costs*number of locatons per school prce materal / costs*number of locatons per school prce management personnel / costs*average experence teachers prce teachng personnel / costs*average experence teachers prce supportng personnel / costs*average experence teachers prce materal / costs*average experence teachers prce management personnel / costs *share of part tme workers prce teachng personnel / costs *share of part tme workers prce supportng personnel / costs *share of part tme workers prce materal / costs *share of part tme workers [.000] [.000] [.000] [.000] [.22] [.27] [.28] [.29] [.232] [.230] [.230] [.233] M sn * prce management personnel / costs [.842] M2 cos*prce management personnel / costs [.000] M2 sn * prce teachng personnel / costs [.033] M22 cos*prce teachng personnel / costs [.000] M3 sn * prce supportng personnel / costs [.908] M23 cos*prce supportng personnel / costs [.65] M4 sn * prce materal / costs [.93] M24 cos*prce materal / costs [.000] M3 sn * producton ndcator [.000] M4 cos *producton ndcator [.000] M32 sn * producton ndcator [.000] M42 cos *producton ndcator [.000] M5 sn * competton [.000] M6 cos * competton [.000] M52 sn * number of schools per governng body [.030] M62 cos * number of schools per governng body [.822] M53 sn * percentage dsadvantaged students [.08] M63 cos * percentage dsadvantaged students [.49] 24

26 M54 sn * percentage urbanzaton [.328] M64 cos * percentage urbanzaton [.457] M57 sn * number of locatons per school [.000] M67 cos * number of locatons per school [.000] M58 sn * average experence teachers [.956] M68 cos * average experence teachers [.767] M59 sn * share of part tme workers [.04] M69 cos * share of part tme workers [.035] M sn * 2* prce management personnel / costs [.04] M2 cos*2*prce management personnel / costs [.000] M2 sn * 2*prce teachng personnel / costs [.00] M22 cos*2*prce teachng personnel / costs [.000] M3 sn * 2*prce supportng personnel / costs [.884] M23 cos*2*prce supportng personnel / costs [.399] M4 sn * 2*prce materal / costs [.703] M24 cos*2*prce materal / costs [.000] M3 sn * 2*producton ndcator [.60] M4 cos *2*producton ndcator [.000] M32 sn * 2*producton ndcator [.000] M42 cos 2**producton ndcator [.000] M5 sn * 2*competton [.000] M6 cos * 2*competton [.97] M52 sn * 2*number of schools per governng body [.027] M62 cos * 2*number of schools per governng body [.903] M53 sn * 2*percentage dsadvantaged students [.07] M63 cos * 2*percentage dsadvantaged students [.000] M54 sn * 2*percentage urbanzaton [.264] M64 cos *2* percentage urbanzaton [.88] M57 sn * 2*number of locatons per school [.000] M67 cos * 2*number of locatons per school [.000] M58 sn * 2*average experence teachers [.8] M68 cos * 2*average experence teachers [.938] M59 sn * 2*share of part tme workers [.056] M69 cos * 2*share of part tme workers [.045] L2 school type * year= [.836] L3 school type * year= [.759] L4 school type * year= [.76] L5 school type * year= [.002] L6 school type * year= [.004] L2 school type 2 * year= [.049] 25