Appendix Model for Chapters 2 and 3 I
set up a simple static model of school finance and the demand for edu- cators to confirm intuitions regarding the roles of liquidity constraints, government subsidy, and government coordination in the determination of LERs. Suppose there are a finite number of public schools under a govern- ment, and that each school maximizes the per-learner output from educa- tion, given its budget constraint, without government intervention. Assume that each school can employ educators freely and that the number of learners changes exogenously at each school, for example, owing to migration and population growth.
Each school has a target LER in each period that maximizes the efficiency
in education production, yit = y* + ξit, where ξit is independent and identically distributed with zero mean and finite variance. We assume that y* is small
enough, and we ignore a negative range of yit.1 ξit could reflect transitory changes in school environments.
learning efficiency.2 The total educational outcome is defined as ei(Li, Hi)Li. Each school has its static budget constraint, qiLi + GiwHi, where qi is the school fee, Gi is a subsidy from the government, w is the wage rate (exogenous) for educators, and Hi = Hi s and Hi
a quadratic loss form: ei(Li, Hi) = 1 – (yit – —— )2 i
[ subsidized, Hi
u for two types of educators, subsidized and non- u, respectively. By definition, the budget constraint can
s + Hi
be separated into two constraints: a government subsidy constraint, Gi ≥ wHi, and a school fee constraint, qiLi ≥ wHi
u. Nonsubsidized teachers are paid only from school fees. However, since the quality of subsidized and nonsubsidized
1 The optimal level of LER can change as endowment for and technology deployment in educa- tion production vary across schools, if these factors alter the marginal productivity of educators. When other nonpersonnel inputs and LERs are substitutable, equal LERs are not necessary for
equal education output. 2 In the range of LERs below the target, the efficiency is increasing in LER. It is assumed that with positive externalities among peer learners, an increase in number of learners raises the efficiency. However, if this effect is negligible, the target level can be set arbitrarily small.
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Let ei(Li, Hi) denote an efficienty function, where Li and Hi are learners and educators in school i, respectively. I assume the efficiency function takes Li
H , where LER determines the ]
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