A signalling model of school grades: centralized versus decentralized examinations

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1 A signlling model of school grdes: centrlized versus decentrlized exmintions Mri De Pol nd Vincenzo Scopp Diprtimento di Economi e Sttistic, Università dell Clbri m.depol@unicl.it; v.scopp@unicl.it 1

2 The im of the pper This pper exmines the signlling vlue of individul skills of different exmintion systems, for the lbor mrket, in reltion to errors tht my ffect evlutions obtined by students. Two type of errors re possible: 1) errors tht influence student performnce; ) errors deriving from different performnce evlution stndrds. The relevnce of these errors depends on the type of exmintion system dopted. We compre on efficiency grounds decentrlized nd centrlized evlution systems in reltion to these errors. We investigte the reltionship between the optiml clss size nd evlution systems.

3 Relted Literture A number of uthors emphsize the positive consequences tht centrlised exmintions my produce on gents involved in the eductionl process. (Woessmn, 005; Bishop Woessmn, 005; Bishop 1997, 1999; Lvy, 00, 003; Glewwe, Elis, nd Kremer 003; Jürges, Richter, nd Schneider 004). In spite of the fvour encountered by centrlized exms mong economists they re strongly criticized mong techers nd pedgogicl specilists, who question their efficcy, since they undermine eductionl freedom nd the pedgogicl discretion tht is supposed to be necessry to del with heterogeneity mong students. An importnt issue, which we nlyse in this pper, is represented by the effect produced by different evlution systems on mesurement errors ffecting students evlutions t exms nd the use of these evlutions s signl of effective skills. 3

4 Orgniztion - We present simple model showing how the signlling vlue of school grdes on the lbor mrket is influenced by the ccurcy of evlution systems. - We nlyse the effect tht more precise evlution systems produce on student effort nd welfre. - We compre dvntges nd disdvntges of centrlized nd decentrlized evlution systems. - We discuss the reltionship between clss size nd evlution systems. - Conclusions 4

5 The model We ssume tht individuls re risk-neutrl nd live for two periods: in the first period they go to school, sustining the cost of effort, nd in the second period they enter the lbor mrket, obtining wge W. No discounting. Individuls re identicl in every respect except their bility tht is distributed ccording to probbility density function with men nd vrince. Students ttend school nd ttin n eductionl qulifiction with n evlution of their skills mde by schools. [1] vi = ei + i + ε i + i η vr ( ε ) =, vr( η ) =, Cov ( e, η) = Cov( e, ε ) = Cov(, η) = Cov(, ε ) = Cov( e, η) = Cov( ε, η) = 0 ε η ( v) = + ε + η Vr. E ( v) e + = 5

6 Lbour mrket We ssume tht the output q i produced by n individul in the lbour mrket is relted to his skills deriving from his innte bility nd from the effort provided during the period he ws ttending school. Therefore, we suppose tht skills re equl to s i = ei + i. Output is then relted to skills ccording to the following production function: [] qi = πsi where π is productivity prmeter. Firms re not ble to observe neither individuls bilities neither the effort they provided in the eductionl process, but only observe the evlution v i obtined by ech student. Firms py wge relted to individul expected productivity. Firms seek to infer the effective skills of workers, on the bsis of the evlution v i. 6

7 They solve typicl signl extrction problem nd therefore form n expecttion of workers skills E( si vi ) = β 0 + β1vi, where the prmeters β 0 nd β 1 cn be estimted using the stndrd OLS formule. It follows tht the two prmeters β 0 nd β 1 re given by the following expressions: [3] β Cov ( s, v) Cov( e +, e + + ε + η) 1 = = β0 = E( s) β1e( v) Vr() v Vr( e + + ε + η) Given our ssumptions on vrince nd covrince of vribles, it is possible to show Cov s v = Vr =. Therefore: tht ( ) ( ), [4] β1 = + ε + η β = ( e + )( ) 0 1 β 1 the wge pid by employers is: = π ( β 0 + β ) W 1 v i 7

8 Student behviour The lifetime individul utility function tkes the following simple form: [7] E( U ) = E( W ) i i γei Since π E( e + v) = π ( β + β v) W = 0 1, the expected utility of student i with bility i who provides level of effort e i nd obtins n evlution v i is equl to: i [8] E( U ) = E( W ) c( e ) = π [ β + β E( v )] = π [ β + β ( + e )] i i i 0 1 i γe 0 1 i i γei Students decide the level of effort which mximizes their utility function, tking s given how the mrket rewrds effort. [9] e πβ = 1 γ e πβ π1 = 1 < γ γ 8

9 Substituting eq. [4] in eq. [9] we obtin: [10] e π = γ + ε + η The optiml level of effort increses when ε nd 9 η decrese (the evlution system is less ffected by stochstic vribles) since employers, receiving better signl of students skills, re willing to py higher wge premium on the grde ttined t school. Moreover, if the vrince of bilities is higher, then β 1 increses, positively ffecting the effort provided by students. Evlution is more importnt when the vribility in bilities is higher. In ddition, when heterogeneity in individul bilities is high the effect produced by shocks is less relevnt nd the signl provided by schools is more informtive. In the ppendix we show tht, ssuming v i = iei + ε + η, low bility students rect less compred to high bility students to signl improvements.

10 The effects of the evlution system ccurcy on student welfre In this section we evlute whether students welfre improves when the evlution system dopted by the school system becomes more precise. Considering the optiml level of effort written s: * = e πβ 1 γ, the student s expected utility cn be [11] E ( ) πβ1 U i = π β0 + β1 i + γ ( πβ ) 1 γ Deriving the individul expected utility with respect to rerrngements, we obtin: ( ) E U β1 π [1] = π + 1 β i 1 γ ( ) ( ) =, fter some ε + η 10

11 Individul utility reduces when increses, when: ~ π β i β1 since 1 < 0 γ [13] > = ( 1 ) It follows tht students whose bility is bove the threshold ~ re negtively ffected by less ccurte evlution systems, while students with bilities below this threshold re positively ffected. The threshold vlue ~ decreses when productivity increses. In highly productive economic systems lso individuls with reltively low bilities prefer more ccurte school performnce evlution systems. Less ccurte evlution systems produce, in fct, two effects: 1) they led to more eglitrin py structure; ) they reduce effort nd, s consequence, reduce the totl output produced in the economy. When π is high the wge reduction deriving from the lower level of effort tends to counterblnce the positive effect tht low bility individuls obtin from low vlue of β 1. 11

12 A comprison between centrlized nd decentrlized evlution systems With centrlized evlution system students re evluted ccording to common stndrd nd η = 0. On the other hnd, we ssume tht due to very high dministrtion costs, this type of exm is undertken only t the end of the eductionl process nd shocks ffecting students my influence their performnce t exms. It follows tht the vrince of evlutions wrded by the centrlized system is equl to Vr ( v) = + ε. When evlution is t decentrlized level, delegted to techers, it is possible to evlute student performnce lrge number of times, which we denote with n. Effects deriving from stochstic vribles relted to student performnce re reduced, but different techers dopt different evlution methods. The vrince of evlution v i is equl to: ε η () v = + + Vr. n 1

13 Compring the expected utility of individul i under centrlized exmintion system with his expected utility under decentrlized evlution system, we obtin tht for i =, centrlized evlution system is preferred when: + γ C D π β β 1 1 > C D 1 1 [14] ( β β ) 1 0 C D Since β 1 nd β 1, the term in squre brckets is lwys positive, condition [14] 1 < holds when C D 1 β 1 1 < β >, tht is when ε η > ε. n ε The difference ε increses when ε increses, implying tht the dvntge of n decentrlized systems is higher when shocks ffecting student performnce hve higher vrince. Then, these systems my hve greter dvntges for students enrolled in primry school, since their performnce is usully more influenced by emotionl fctors. 13

14 Clss size nd decentrlized evlution systems Generlly techers evlutions re considered costless. Nevertheless, if techers fce very lrge clsses it my result difficult for them to judge students on the bsis of dily interctions, prticiption to work-clss, etc. When clss size increses the cost of evluting students increses nd techers my ssign grdes on the bsis of lower number of evlutions. The centrl uthority is ble to define clss size, but it is not ble to define the number of evlutions tht students hve to undertke. We nlyze the choice of the clss-size by policy mker who tkes into ccount techers behviour. We model this choice s sequentil gme in which, in the first stge, the policy-mker sets the clss-size, while in the second stge, techers decide how mny evlutions to undertke. 14

15 15 Let us ssume school system with N students, C clsses nd C techers, where C N S = is the size of ech clss. Techers mximize the following utility function: ( ) ns S E W U T = [15] ns S n n n U T = γ π π η ε Deriving [15] with respect to n we obtin the optiml number of evlutions: [16] [ ] η ε ε γ π + = S S n n reduces when S increses; increses when ε increses nd decreses when η increses.

16 16 We nlyze the choice of the optiml clss size when the policy-mker ims to mximize the wge obtined by students on the lbour mrket net of school costs, D, due to the wges pid to techers nd to the rentl vlue of the cpitl ssocited to ech clssroom. The socil welfre function, considering the wge obtined by the representtive student with bility net of cost per student, is given by: [5] S D S n V = γ π π η ε ) * (

17 Substituting n* in the socil welfre function nd deriving V with respect to S, we obtin the optiml clss size: 1 = + η [16] ( ) 3 D 1 S π ε γ When the vrince ε increses the optiml clss size reduces. When η increses it is optiml to define lrger clsses since grdes wrded by the school system re not good signl of students bilities nd re scrcely rewrded on the lbour mrket. When the cost of eduction per student increses the optiml clss size increses. When the productivity of skills is higher it is optiml to reduce clss size. 17

18 In this nlysis, for the ske of simplicity, we hve neglected the direct effect of clss size on the humn cpitl ccumultion process. In more generl frmework, considering this spect, it is possible to show tht in eductionl systems bsed on decentrlized evlutions, the optiml clss size is smller compred to systems bsed on centrlized evlutions. While in centrlized evlution systems the optiml clss size only depends on the mrginl benefit deriving from smller clsses in terms of student chievement nd mrginl costs relted to higher expenditures for wges nd rentl cpitl, under decentrlized evlution systems clss size lso ffects how informtive evlutions re of individul skill. Lrger clsses my reduce the frequency of evlutions undertken by techers nd worsen the informtive vlue of evlutions. Insted, this effect does not ply ny role in the definition of optiml clss size in centrlized exmintion systems. 18

19 Concluding Remrks - We hve shown tht more precise evlution systems, being ssocited to higher rectivity of wges to school grdes, induce n higher level of student effort. Low bility students tend to rect less compred to high bility students. - Wheres high bility individuls strictly prefer more precise evlution systems, low bility individuls my prefer less precise evlutions. - When lbor productivity increse lso individuls with reltively low bility prefer more precise evlution systems. - We hve used our frmework to compre costs nd benefits of centrlized vs. decentrlized evlutions. The performnce of the decentrlized evlution system improves when the mesurement errors deriving from shocks hitting students re prticulrly importnt (for exmple, when students re very young). - We hve shown tht there is reltionship mong clss size nd evlution systems. 19

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