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Decreasing average cost in private schools, existence of majority voting equilibrium, and a policy analysis for Turkey

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Abstract

The existing models of mixed public–private school systems usually capture only the decreasing average cost faced by public schools, whereas empirical studies find evidence of it for private schools as well. Motivated by this, an equilibrium model of a mixed public–private school system is studied in this paper, whereby private schools also face decreasing average cost over enrollment. In the model, households, heterogeneous with respect to exogenously specified income and child’s ability, choose among a public and a private school. Private school charges tuition whereas public school is free. Public school spending is financed by income tax revenue collected from all households and the tax rate is determined via majority voting. Achievement of a child depends on its ability and education spending. Under the assumptions on the parameters of the model, a joint lognormal distribution of income and ability, and a Cobb–Douglas utility, majority voting equilibrium is numerically shown to exist. The model is calibrated to match certain statistics from the 2013 Turkish data. Using the calibrated model, we compare the benchmark for a mixed public–private school system with a pure public school system to understand the impact of shutting down some of the private schools in Turkey following the July 15 coup attempt. We find that mean achievement and variance of achievement after high school is \(0.039\%\) higher and \(0.013\%\) lower respectively in a pure public school system.

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Notes

  1. Public schools facing decreasing average cost over enrollment is empirically shown in Cohn (1968), Kenny (1982), Kumar (1983) and Riew (1966).

  2. Unfortunately we don’t have data, especially average tuition, for other countries.

  3. Our results apply to the case when \(100\%\) of private schools are converted. We argue in Sect. 4 that change in mean achievement in reality is expected to be similar to what we find. Moreover, change in variance that we find is explained to be an upper bound for reality.

  4. In Nechyba (1999), a multi-district economy is studied where public education is financed by property taxes and tax rate is determined via majority voting.

  5. In our model, we ignore cases where \(\tau =0\) or \(\tau =1\).

  6. Also in previous models such as Stiglitz (1974), Epple and Romano (1996) and Glomm and Ravikumar (1998), an inverse relation between public school spending per pupil and public school enrollment exists as also noted in Gradstein et al. (2005) (page 99).

  7. Enrollment in both the public and the private school is a function of the tax rate, since households make their school choice taking as given the tax rate. The formal definitions of \(N_{u}(\tau )\) and \(N_{r}(\tau )\) are given in the next subsection.

  8. Starting from here, we will suppress income tax rate from the notation of enrollment levels.

  9. Our model assumes that public education expenditure is centrally financed without distinguishing among local school districts. Therefore, it applies mainly to countries like Turkey.

  10. We verified existence of majority voting equilibrium for 25,000 different parameter vectors. Details are available upon request.

  11. Given the set up of our model, we should ideally use score distribution data from an examination at the kindergarten level. Unfortunately, there is no such an examination in Turkey to the best of our knowledge and the OKS is the first nationwide examination students take during K-12. However, our model matches the calibration targets from the data fairly well as will be argued below.

  12. The minimum normalized score in the 2005 OKS is 0.47. We pick 0.38 as lower bound to make sure we match normalized mean and median ability in data as close as possible.

  13. In ERG (2014), the ratio of total public expenditure on education including tertiary education to nominal GDP is reported to be around \(4.3\%\). We then calculate \(2.72\%\) using other information in ERG (2014).

  14. Please see the appendix for the computational algorithm we use to solve our model.

  15. The exact values of parameters are \(\alpha =0.034741602449687\), \(\beta =0.001074989644448\), \(\theta =3.2855\), \(f_{r}=0.091202476769329\).

  16. The mean scores are computed using the data in ÖSYM (2013).

  17. Please see Fig. 2 in Glomm and Ravikumar (1998).

  18. Details are available upon request.

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Authors and Affiliations

  1. ADA University, Baku, Azerbaijan

    Muharrem Yeşilırmak

  2. CERGE-EI, Prague, Czech Republic

    Muharrem Yeşilırmak

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  1. Muharrem Yeşilırmak
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Correspondence to Muharrem Yeşilırmak.

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The author declares that he has no conflict of interest.

Additional information

At initial stages of this study, I have benefited greatly from the discussions with B. Ravikumar. I am grateful to Kuzey Yılmaz for his continuous support throughout the study. I owe special thanks to İsmail Baydur who improved my understanding of the paper. I received very helpful comments from Srihari Govindan, Kemal Yıldız, Pelin Akyol, the participants of the 2013 Public Choice Society meeting at New Orleans, 2013 Southern Economic Association meeting at Tampa and three anonymous referees. Any errors are mine.

Appendix

Appendix

The following computational algorithm is used to solve the equilibrium of our model for given values of \((\alpha ,\beta ,\theta ,f_{r})\).

  1. 1.

    Generate a lognormal joint distribution of income and ability using data.

  2. 2.

    Create a grid for tax rates in the interval (0.001, 0.99) with length 1000.

  3. 3.

    Create a grid for public school enrollment in the interval (0.01, 0.99) with length 1000. Private school enrollment can be found using (9).

  4. 4.

    For each tax rate in the grid, search over the public school enrollment grid to find a solution for (7) and (8). To find \(I_{r}(\cdot )\) for a household, compare indirect utilities given by (10) and (11).

  5. 5.

    For each tax rate and corresponding public school enrollment, find \(V(\cdot )\) defined by (4) for each household by comparing indirect utilities given by (10) and (11).

  6. 6.

    Pick an arbitrary pair of tax rates. Compare \(V(\cdot )\) under both tax rates for each household. Find the fraction of votes received by each tax rate. Do this for all pairs of tax rates.

  7. 7.

    The equilibrium tax rate is the one that gets at least \(50\%\) of the votes in any binary election against any other tax rate.

  8. 8.

    Go to fourth step above, and determine the public school enrollment that corresponds to equilibrium tax rate. This is the equilibrium public school enrollment.

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Yeşilırmak, M. Decreasing average cost in private schools, existence of majority voting equilibrium, and a policy analysis for Turkey. Rev Econ Design 22, 1–24 (2018). https://doi.org/10.1007/s10058-018-0208-1

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