A quantitative analysis of Turkish public school admission reform

Around the world, some countries rely on exam based placement into public schools whereas some others rely on home address based placement. Recently, Turkey has changed its public high school admission rule from exam based placement to home address based placement. This raised concerns regarding equality of opportunity since high quality public schools are located in rich neighborhoods. To quantify the effect of the reform, we set up a model in which households differ by exogenous income and student ability and public schools differ by exogenous teacher quality. Public schools’ expenditures are financed by income taxation where income tax rate is determined via majority voting. Households can privately supplement public spending. Each teacher determines effort for each student in her classroom. Students and teachers are matched assortatively based on student ability. Achievement depends on ability, teacher effort, and educational spending. We first show existence of a unique majority voting equilibrium and characterize the pivotal voter. After calibrating the parameters of the model, we exogenously change the matching rule between teachers and students to analyze the reform through a computational experiment. We find that variance of achievement falls by \(17.76\%\), mean achievement falls by \(51.15\%\), households’ total welfare falls by \(0.115\%\), and teachers’ total welfare rises by \(18.1\%\).

1. To endogenously capture the changes in public school spending per pupil after the reform, we rely on majority voting which is commonly used in the literature.

2. The fall in income tax rate is consistent with Turkish data after 2016 as argued in Sect. 5.

3. In Sweden, admission was based on home address initially and then it is reformed and based on exam scores.

4. The analysis provided by both papers apply equally well for any publicly provided private good which is education in our model.

5. There is a continuum of different income and ability groups in the model. The income of i’th group is denoted by \(y_{i}\) and ability of j’th group is denoted by \(b_{j}\). We use different indexes i and j for income and ability groups to indicate the assortative matching between students and teachers based on ability. As will be clarified later, we also use index j to denote the teacher quality group.

6. The parameter \(\theta \) is the inverse elasticity of substitution and \(\delta \) captures the importance of achievement for the household relative to consumption.

7. In the OECD data (Education at a Glance 2018 and 2019 Table D3.4), the average annual teacher salaries are reported for Turkey as 22, 143 USD and 24, 187 USD in 2016 and 2017 respectively. Based on this empirical evidence, a perfectly inelastic teacher quality supply assumption is more appropriate than a perfectly elastic one since latter implies no change in teacher salaries after demand shifts whereas the former implies a change.

8. We use the same notation for teacher quality and child ability since they are assortatively matched as clarified below. Moreover, the parameter \(\varphi \) controls the curvature of the utility over consumption and \(\mu \) captures the marginal disutility of effort.

9. We are not aware of any paper that finds the same result for Turkish teachers. However, we do not expect Turkish teachers to not get utility from their students’ achievements.

10. Since ability and teacher quality both belong to \(\mathfrak {R}_{++}\), then there exists a one-to-one matching between students and teachers.

11. We assume \(\tau \in [0,1)\). For technical reasons, \(\tau \) cannot be one.

12. Since measure of households is one, then non-teacher related public spending per pupil and total non-teacher related public spending are same.

13. We don’t consider class size as a factor affecting student achievement. Instead, we have teacher effort for each student which can be thought of as the time teacher invests in each student. Tamura (2001) uses teacher’s time as a proxy for inverse of class size.

14. The parameters \(\alpha \) and \(\beta \) capture the curvature of achievement over teacher’s effort and education spending respectively.

15. For technical reasons, we assume households don’t take into consideration the effect of their votes on w.

16. This parametric assumption is made to obtain a closed-form solution for the household’s problem. It should be also noted that this assumption does not imply \(\alpha \ge \beta \) or \(\alpha \le \beta \). Thus, the contribution of educational spending to achievement is not theoretically restricted to be less than or greater than the contribution of teacher’s effort to achievement.

17. We ignore households with income equal to \(\frac{\rho Y}{\gamma }\) since in a continuum setting their measure is negligible.

18. We calibrate our model for a representative city (such as İstanbul) in Turkey. Unfortunately, we don’t have specific data for İstanbul. However, we believe that aggregate data for Turkey also applies well for İstanbul since İstanbul hosts migrants from all other regions of Turkey.

19. We also calibrated our model’s parameters using 2019 Turkish data. Our results are similar. More specifically, when calibrated to 2019 data, our model implies around a \(50\%\) fall in mean achievement, \(17\%\) fall in variance of achievement, \(1\%\) fall in households’ total welfare, and \(24\%\) rise in total teachers’ welfare after the public school admission reform. More details are available from the author upon request.

20. In our model, public education is centrally financed. Therefore, it applies well to Turkey.

21. Teacher quality varies in the interval [0.01, 5].

22. We solve the equilibrium of our model for different parameter values and then pick those parameters at which model comes close to the targets. The computational algorithm used to solve our model’s equilibrium for given parameters is outlined in Appendix A.2.

23. One natural question that could come to mind is if government shut down SEC’s which constituted the largest part of total supplemental education spending, why is it included in the model? It should be noted that households could still spend on supplemental education by hiring private tutors or by purchasing education-related goods like computers for their children.

24. In our model, each public school type can be thought of as a different school district. In the benchmark, the location of a public school is not important since student ability is what matters in the admission process. In the experiment, because of address based admission, rich households use their financial power to make sure their children is admitted into high quality public schools.

25. Please see Appendix A.3 for sensitivity of our findings to calibrated parameter values.

26. Empirical studies such as Whitehurst et al. (2014) and Steinberg and Garrett (2016) find a strong positive relation between classroom composition (measured by incoming students achievement or prior year student achievement) and teacher performance controlling for nonrandom student-teacher matching.

27. The GDP data is obtained from Turkish Statistical Institute. The public high school expenditure data is obtained from ERG (2018) and ERG (2019). Vocational public high schools are also included. Moreover, between 2017 and 2019, the share of primary-secondary school spending out of GDP also fell but by a lesser amount compared to high school. We interpret the difference as being due to the high school admission reform.

28. Since mean and median incomes used in generating the income distribution is for everyone in Turkey, then teachers are also included implicitly. Thus, the households’ total welfare also includes teachers’ total welfare.

29. More precisely, variance reaches its peak when \(52\%\) of students are admitted via exam scores.

30. Income tax rate may also rise which would be inconsistent with data as reported above. So we don’t consider that case.

31. Burgess et al. (2015) note that distance to secondary school may be less highly valued by parents.

Authors and Affiliations

1. ADA University, Baku, Azerbaijan

   Muharrem Yeşilırmak

2. CERGE-EI, Prague, Czechia

   Muharrem Yeşilırmak

Corresponding author

Correspondence to Muharrem Yeşilırmak.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I received very helpful comments from the seminar participants at ADA University, TOBB-ETU University, and CERGE-EI. Any errors are mine. The author can be reached at myesilirmak@ada.edu.az

Appendix

1.1 Mathematical Proofs

Proof of Lemma 1

Let us define a function \(\Omega (e_{ij})\) as follows:

where \(N_{j}=\frac{(\lambda _{j}w)^{\varphi }b_{j}^{\alpha }}{\int \nolimits _{0}^{\infty } f(y_{i},b_{j})\mathrm{d}y_{i}}\). Since \(\alpha \in (0,1)\) and \(\mu >1\), then \(\frac{\partial ^{2} \Omega }{\partial e_{ij}^{2}}<0\). This implies \(\Omega (e_{ij})\) is strictly concave. By the definition of strict concavity, we can write:

for any \(k\in (0,1)\) and \(e_{ij}^{1}\), \(e_{ij}^{2}\) \(\in (0,\infty )\). Integrating both sides of this inequality over income levels then implies:

Therefore, the objective function in (6) is strictly concave. Therefore, there exists a unique solution. The solution can be found by taking the first order condition:

which then implies the closed-form given in Lemma 1. \(\square \)

Proof of Lemma 2

Assuming \(\beta =1-\frac{\alpha }{\mu }\), the second derivative of the objective function in problem (7) is given below:

which is strictly negative. Therefore, the objective function is strictly concave implying existence of a unique solution. To find the solution, we solve the first order condition given below:

which implies \(s_{ij}^{*}=\frac{(1-\tau )y_{i}-\kappa _{j}g}{1+\kappa _{j}\gamma }\). Since \(s_{ij}^{*}\ge 0\), then this solution applies for those households that satisfy \(y_{i}>\frac{\kappa _{j}g}{1-\tau }\). For the rest of the households, \(s_{ij}^{*}=0\). \(\square \)

Proof of Lemma 3

We show that left and right limits are same.

Since left and right limits are same, then \(V_{ij}(\tau )\) is continuous at \({\widehat{\tau }}_{ij}\). \(\square \)

Proof of Lemma 4

For \(\tau \in [0,{\widehat{\tau }}_{ij})\), we know \(V_{ij}(\tau )=\frac{\kappa _{j}^{-\theta }[\gamma (1-\tau )y_{i}+\rho \tau Y]^{1-\theta }}{(1-\theta )\gamma (1+\kappa _{j}\gamma )^{-\theta }}\).

Then \(\frac{\partial V_{ij}}{\partial \tau }=\frac{\kappa _{j}^{-\theta }[\gamma (1-\tau )y_{i}+\rho \tau Y]^{-\theta }(-\gamma y_{i}+\rho Y)}{\gamma (1+\kappa _{j}\gamma )^{-\theta }}\) which is strictly positive for \(y_{i}<\frac{\rho Y}{\gamma }\) and strictly negative for \(y_{i}>\frac{\rho Y}{\gamma }\).

Moreover, \(\frac{\partial ^{2} V_{ij}}{\partial \tau ^{2}}=\frac{-\theta \kappa _{j}^{-\theta }[\gamma (1-\tau )y_{i}+\rho \tau Y]^{-\theta -1}(-\gamma y_{i}+\rho Y)^{2}}{\gamma (1+\kappa _{j}\gamma )^{-\theta }}<0\) implying strict concavity. \(\square \)

Proof of Lemma 5

Let us first show that \({\widehat{\tau }}_{ij}<{\widetilde{\tau }}_{ij}\). Since \(y_{i}<\frac{\rho Y}{\gamma }\), then \(y_{i}^{\frac{1}{\theta }}<\left( \frac{\rho Y}{\gamma }\right) ^{\frac{1}{\theta }}\) which implies \((\rho Y)^{\frac{\theta -1}{\theta }}y_{i}<\gamma ^{\frac{-1}{\theta }}\rho Yy_{i}^{\frac{\theta -1}{\theta }}\) which implies \(\kappa _{j}(\rho Y)^{\frac{\theta -1}{\theta }}y_{i}+\gamma ^{\frac{-1}{\theta }}y_{i}^{2-\frac{1}{\theta }}<\gamma ^{\frac{-1}{\theta }}y_{i}^{2-\frac{1}{\theta }}+\kappa _{j}\gamma ^{\frac{-1}{\theta }}\rho Yy_{i}^{\frac{\theta -1}{\theta }}\). Then \(\frac{y_{i}}{y_{i}+\kappa _{j}\rho Y}<\frac{\gamma ^{\frac{-1}{\theta }}y_{i}^{\frac{\theta -1}{\theta }}}{\kappa _{j}(\rho Y)^{\frac{\theta -1}{\theta }}+\gamma ^{\frac{-1}{\theta }}y_{i}^{\frac{\theta -1}{\theta }}}\). Therefore, \({\widehat{\tau }}_{ij}<{\widetilde{\tau }}_{ij}\).

For \(\tau \in [{\widehat{\tau }}_{ij},1)\), we know \(V_{ij}(\tau )=\frac{[(1-\tau )y_{i}]^{1-\theta }}{1-\theta }+\frac{\kappa _{j}^{-\theta }(\rho \tau Y)^{1-\theta }}{\gamma (1-\theta )}\) for any household.

Then \(\frac{\partial V_{ij}}{\partial \tau }=-[(1-\tau )y_{i}]^{-\theta }y_{i}+\frac{\kappa _{j}^{-\theta }\tau ^{-\theta }(\rho Y)^{1-\theta }}{\gamma }\) for any household.

Moreover, \(\frac{\partial ^{2} V_{ij}}{\partial \tau ^{2}}=-\theta [(1-\tau )y_{i}]^{-\theta -1}(y_{i})^{2}-\frac{\theta \kappa _{j}^{-\theta }\tau ^{-\theta -1}(\rho Y)^{1-\theta }}{\gamma }<0\) implying strict concavity for any household.

Let us now prove part b of the lemma. For \(\tau \in [{\widehat{\tau }}_{ij},{\widetilde{\tau }}_{ij})\), we can write \(\tau <\frac{\gamma ^{\frac{-1}{\theta }}y_{i}^{\frac{\theta -1}{\theta }}}{\kappa _{j}(\rho Y)^{\frac{\theta -1}{\theta }}+\gamma ^{\frac{-1}{\theta }}y_{i}^{\frac{\theta -1}{\theta }}}={\widetilde{\tau }}_{ij}\). This implies \(\frac{\kappa _{j}\tau (\rho Y)^{\frac{\theta -1}{\theta }}}{\gamma ^{\frac{-1}{\theta }}}<(1-\tau )y_{i}^{\frac{\theta -1}{\theta }}\) which implies \(\frac{\kappa _{j}^{-\theta }\tau ^{-\theta }(\rho Y)^{1-\theta }}{\gamma }>(1-\tau )^{-\theta }y_{i}^{1-\theta }\). Therefore, \(\frac{\partial V_{ij}}{\partial \tau }>0\). This completes proof of the first half of part b. For the second half, since \(\tau >{\widetilde{\tau }}_{ij}\) then the reverse of above inequalities would hold implying \(\frac{\partial V_{ij}}{\partial \tau }<0\). Moreover, this analysis implies that if \(\tau ={\widetilde{\tau }}_{ij}\) then \(\frac{\partial V_{ij}}{\partial \tau }=0\).

Let us now prove part c of the lemma. Since \(\tau \ge {\widehat{\tau }}_{ij}=\frac{y_{i}}{y_{i}+\kappa _{j}\rho Y}\), then \((1-\tau )y_{i}\le \kappa _{j}\tau \rho Y\) which implies \([(1-\tau )y_{i}]^{-\theta }\ge \kappa _{j}^{-\theta }\tau ^{-\theta }(\rho Y)^{-\theta }\). This together with \(y_{i}>\frac{\rho Y}{\gamma }\) implies that \([(1-\tau )y_{i}]^{-\theta }y_{i}>\kappa _{j}^{-\theta }\tau ^{-\theta }(\rho Y)^{-\theta }\left( \frac{\rho Y}{\gamma }\right) \). Therefore, \(\frac{\partial V_{ij}}{\partial \tau }<0\). \(\square \)

1.2 Computational algorithm

We use the algorithm outlined below to solve our model’s benchmark equilibrium for a given set of parameter values.

1. 1.

   Generate joint distribution of income and ability.

2. 2.

   Generate teacher quality distribution.

3. 3.

   Guess teacher efficiency wage. Denote the guess with \(w_{g}\).

4. 4.

   Find \(z^{*}\) using Proposition 2. Make sure the assumption \(\frac{\rho Y}{\gamma }>y_{m}\) is satisfied.

5. 5.

   Find \(\tau ^{*}\) using (10).

6. 6.

   Update teacher efficiency wage using (13). Denote updated value with \(w_{u}\).

7. 7.

   Compute error as follows:

   $$\begin{aligned}error=\left| \ln \left( \frac{w_{u}}{w_{g}}\right) \right| .\end{aligned}$$

   If error is less than \(10^{-3}\), then stop. Otherwise, update \(w_{g}\) and continue from step 3 until convergence.

1.3 Sensitivity analysis

Regarding calibration, the sensitivity analysis is provided in the Tables 9 through 12 below. Each table is for a different prediction of the paper after simulating the Turkish public high school admission reform. In each table, the calibrated value for a selected parameter is varied by a certain amount (holding other parameters constant) and the resulting percentage change in a specific variable after the reform is reported. For instance, in Table 9, if the calibrated value of \(\mu _{b}\) is decreased by \(5\%\), then mean achievement falls by \(50.56\%\) after the reform. In each table, the row of \(0\%\) simply corresponds to paper’s findings under the current parameter values.

From Table 9, we see that mean achievement falls usually around \(50\%\) when parameters are varied in the interval \([-15\%,+15\%]\). Thus, our finding regarding the change in mean achievement is robust. Moreover, the change in mean achievement is most sensitive to variations in \(\mu \) and \(\alpha \).

The change in variance of achievement is qualitatively robust to variations in parameters except when \(\mu _{b}\) decreases by \(15\%\) as seen from Table 10. The change in variance is most sensitive to variations in \(\alpha \).

As seen in Table 11, when parameters are perturbed around their calibrated values, total households’ welfare again falls around \(-0.11\%\) similar to our current findings. Households’ total welfare is most sensitive to variations in the parameters \(\mu \), \(\varphi \), and \(\alpha \).

Regarding the change in teachers’ total welfare, our results are qualitatively robust to parameter variations most of the time (with some exceptions) as seen in Table 12. Total teachers’ welfare seems to be most sensitive to variations in \(\varphi \) and not sensitive to variations in teacher quality distribution parameters.

To sum up, our results are quantitatively most sensitive to parameters \(\mu \), \(\varphi \), \(\alpha \) and not sensitive at all to teacher quality distribution parameters (\(\mu _{\lambda }\) and \(s_{\lambda }\)).

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