# Managing Size of Public Schools and School Boards: A Multi-Level Cost Approach Applied to Dutch Primary Education

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Model Design

- $C$ = (minimum) costs;
- $y$ = vector services produced;
- $w$ = vector of resource prices;
- $x$ = vector of resources;
- $T\left(x,y\right)$ = set of feasible combinations of services produced and resources.

- ${C}_{bs}^{\mathrm{min}}$ = minimum costs of school s of school board b;
- ${Y}_{bs}$ = vector of services of school s of school board b (e.g. number of pupils per type of training, education results);
- ${W}_{bs}$ = vector prices of resources used of school s of school board b (e.g. wage index, material price index);
- ${Z}_{bs}$ = vector environmental factors of school s of school board b (e.g. social background).

- $Ef{f}_{bs}$ = inefficiency of school s of school board b;
- ${Z}_{b}$ = attributes of board b;

#### 2.2. Functional Specification

## 3. Data

_{b}denote the number of schools governed by board $b$. Then ${y}_{b,s}$ equals zero for $n>{N}_{b}$. The largest number of schools governed by a single board in our sample equals 31.

## 4. Results

- $\tau $ = correction factor least squares standard errors of estimated parameters;
- ${\rho}_{u}$ = intra correlation of the residuals;
- $\overline{N}$ = average number of replications in the panel;

_{12}and b

_{23}may be affected in such a way that they are no longer significant at the 5% level, but they still are at the 10% level. The parameters of the number of associated schools and square of associated schools are not significant at the 5% level after correction (the square term still is at the 10% level). The hypothesis of no relationship between number of associated schools therefore cannot be rejected. The parameters estimate of the average test score and square average test score are significant at the 5% level, even after the correction, implying that the hypothesis that there is no relationship between cost and average test score must be rejected. The requirements concerning monotonicity with respect to outputs are met (positive parameters). Note that requirements regarding input prices are not relevant here, since costs are deflated by a price index number.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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VARIABLE | MEAN | STD. DEV. | MINIMUM | MAXIMUM |
---|---|---|---|---|

BOARD LEVEL (N = 723) | ||||

COST (IN MILLIONS OF EUROS) | 5.20 | 6.46 | 0.37 | 52.41 |

ENROLMENT (TOTAL) | 983.15 | 1214.64 | 45.00 | 9340.00 |

ENROLMENT (SES-1) | 903.00 | 1199.27 | 44.00 | 8347.00 |

ENROLMENT (SES-2) | 46.10 | 63.14 | 0.00 | 447.00 |

ENROLMENT (SES-3) | 34.05 | 79.56 | 0.00 | 980.00 |

NUMBER OF SCHOOLS | 4.60 | 5.53 | 1.00 | 31.00 |

SCHOOL LEVEL (N = 2601) | ||||

ENROLMENT (TOTAL) | 213.30 | 128.25 | 12.00 | 1283.00 |

ENROLMENT (SES-1) | 196.00 | 123.14 | 12.00 | 1246.00 |

ENROLMENT (SES-2) | 10.11 | 12.30 | 0.00 | 174.00 |

ENROLMENT (SES-3) | 7.19 | 16.72 | 0.00 | 205.00 |

AVERAGE TEST SCORE | 535.26 | 3.92 | 514.70 | 546.20 |

VARIABLE | PARAMETER | ESTIMATE | STD ERROR | T-VALUE | T-VALUE CORREC-TED |
---|---|---|---|---|---|

CONSTANT | a | −0.984 | 0.006 | −164.423 | −85.096 |

ENROLMENT (SES-1) | b_{1} | 0.634 | 0.006 | 110.568 | 57.224 |

ENROLMENT (SES-2) | b_{2} | 0.058 | 0.003 | 20.363 | 10.539 |

ENROLMENT (SES-3) | b_{3} | 0.128 | 0.002 | 62.498 | 32.345 |

SES-1 X SES-1 | b_{11} | 0.215 | 0.010 | 21.121 | 10.931 |

SES-1 X SES-2 | b_{12} | −0.008 | 0.002 | −3.522 | −1.823 |

SES-1 X SES-3 | b_{13} | −0.038 | 0.002 | −19.127 | −9.899 |

SES-2 X SES-2 | b_{22} | 0.023 | 0.001 | 15.694 | 8.123 |

SES-2 X SES-3 | b_{23} | 0.003 | 0.001 | 3.482 | 1.802 |

SES-3 X SES-3 | b_{33} | 0.045 | 0.001 | 44.595 | 23.080 |

NUMBER OF SCHOOLS | d_{1} | −0.017 | 0.006 | −2.790 | −1.444 |

NUMBER OF SCHOOLS X NUMBER OF SCHOOLS | d_{11} | 0.016 | 0.005 | 3.574 | 1.850 |

TEST SCORE | d_{2} | 0.100 | 0.026 | 3.883 | 2.010 |

TEST SCORE X TEST SCORE | d_{22} | 1.742 | 0.190 | 9.181 | 4.752 |

YEAR = 2011 | h_{1} | −0.019 | 0.006 | −3.182 | −3.182 |

YEAR = 2012 | h_{2} | −0.035 | 0.006 | −5.861 | −5.861 |

YEAR = 2013 | h_{3} | −0.025 | 0.006 | −4.298 | −4.298 |

YEAR = 2014 | h_{4} | 0.006 | 0.006 | 1.023 | 1.023 |

R2 | 0.99 |

OUTPUT CATEGORY | MARGINAL COST |
---|---|

SES-1 | €4253 |

SES-2 | €7830 |

SES-3 | €20,411 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Blank, J.L.T.; Niaounakis, T.K. Managing Size of Public Schools and School Boards: A Multi-Level Cost Approach Applied to Dutch Primary Education. *Sustainability* **2019**, *11*, 6662.
https://doi.org/10.3390/su11236662

**AMA Style**

Blank JLT, Niaounakis TK. Managing Size of Public Schools and School Boards: A Multi-Level Cost Approach Applied to Dutch Primary Education. *Sustainability*. 2019; 11(23):6662.
https://doi.org/10.3390/su11236662

**Chicago/Turabian Style**

Blank, Jos L. T., and Thomas K. Niaounakis. 2019. "Managing Size of Public Schools and School Boards: A Multi-Level Cost Approach Applied to Dutch Primary Education" *Sustainability* 11, no. 23: 6662.
https://doi.org/10.3390/su11236662