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Class Size and Sorting in Market Equilibrium: Theory and Evidence. Miguel Urquiola and Eric Verhoogen. Overview. Contentious literature on whether class size matters This paper’s contribution is a model in which: Households sort into schools of different quality levels
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Class Size and Sorting in Market Equilibrium: Theory and Evidence Miguel Urquiola and Eric Verhoogen
Overview • Contentious literature on whether class size matters • This paper’s contribution is a model in which: • Households sort into schools of different quality levels • Schools can choose their quality level and class size • Use data from the liberalized Chilean education market • Implications of model: • Class-size is an inverted-U function of hh income (this will bias cross-sectional estimates) • Stacking occurs at class size cap (this will invalidate regression discontinuity estimates) • Caveat: model only relevant if parents have school choice and schools can adjust prices and enrollment
Outline of Presentation • Literature Review • Institutional Background • Model • Testable Implications • Data • Results • Conclusion • Critique
Literature Review • This paper hopes to clarify the literature on the effect of class size on student performance. • Hanushek (1995, 2003) reviews many cross-sectional studies and finds no systematic effect in either developed or developing countries • Krueger (2003), Kremer (1995) argue cross-sectional estimates may be biased if hhs sort into schools & schools decide class size • Quasi-experimental approach: Angrist & Levy (1999) use regression discontinuity design based on enrollments that are multiples of class size caps • Compare 2 similiar schools whose 4th grades have 50 vs. 51 students
Institutional Background • Three types of schools in Chile’s primary school system • Public/Municipal: funded per student, can’t turn students away, max class size 45, typically low quality • Private subsidized/Voucher: same per student funding from gov’t, same class size cap, but can select students • Private unsubsidized: no gov’t funding • 40-58% of primary schools in Chile are private • Most private schools are for-profit & can charge tuition
Model: Overview • Model parents’ demand for education in a standard discrete-choice framework with quality differentiation (eg, BLP 1995) • Model unsubsidized and voucher schools as profit maximizers subject to the relevant constraints • Don’t allow for entry, exit or sector switching • Schools are heterogeneous in productivity parameter • Continuum of schools with density fu() or fv()
Model: Demand • U(p, q; ) = q – p + q = school quality, p = tuition = random match-specific utility; i.i.d. double exponential distribution = marginal willingness to pay (function of income) • Derive: s(p,q; ) = Probability hh chooses school (p,q) D(p,q) = Expected demand for school (p,q) • Monopolistic competition • Combines horizontal and vertical differentiation
Model: Quality Production Technology • Quality production technology: = school productivity, T = technological maximum class size, x is enrollment, n = # of classrooms, x/n class size ==> Complementarity of and x/n
Model: Schools’ Optimization Problem • (p, n, x; ) = (p + - c)x – nFc – Fs • p=tuition, n=# classrooms, x=enrollment, =per-student subsidy, c=variable cost, Fc= classroom fixed cost, Fs = school fixed cost • Constraints: • Enrollment cannot exceed demand: x D(p,q) • Positive integer number of classrooms • Class size cap: x/n 45 (only applies to voucher schs) • The authors’ solve for the equilibrium
Testable Implications of the Model • Testable Implication 1: There is a roughly inverted-U shaped relationship between class size and average household income in equilibrium • Testable Implication 2: Schools will stack at enrollments there are multiples of 45, implying discontinuous changes in average household income with respect to enrollment
Data • Administrative information on schools’ grade-specific enrollments and number of classrooms • Standardized testing data • Math and language performance • Student characteristics such as household income and parental schooling
Results: Inverted-U • Inverted-U shaped relationship found between income and class size at voucher schools but not unsubsidized schools ==> Cross-sectional regressions will underestimate the effects of class size among lower-income voucher schools and overstate it among higher-income ones
Results: Stacking at Class Size Cap • Voucher schools stack at enrollments that are multiples of 45. ==>Average of schools just at multiples of class size cap will be strictly less than of schools just above the multiple. ==>Since hh income is increasing in , this invalidates the regression discontinuity design.
Conclusion • Authors develop a model of endogenous household sorting and class size determination • They find that class-size is an inverted-U function of household income (which biases cross-sectional estimates) • They find that stacking occurs at class size cap (which invalidates RD estimates) • Caveat: model only applicable if parents have school choice and schools can adjust prices and enrollment
Comments • Limited applicability of model. • Quality variable is not well-explained or defined. Is it perceived quality? Or is it a measure of student performance and outcomes? • If the latter, authors are assuming class size affects quality, which seems circular. • Authors show that old methods don’t work, but they don’t offer a new way to estimate effect. • Nevertheless, this paper does clarify the literature and point to a way forward.