Class Size and Sorting in Market Equilibrium: Theory and Evidence

1. Class Size and Sorting in Market Equilibrium: Theory and Evidence Miguel Urquiola and Eric Verhoogen

2. 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

3. Outline of Presentation • Literature Review • Institutional Background • Model • Testable Implications • Data • Results • Conclusion • Critique

4. 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

5. 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

6. 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()

7. 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

8. 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

9. 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

10. 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

11. Class Size at Unsubsidized Schools in Equilibrium

12. Class Size at Voucher Schools in Equilibrium

13. 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

14. 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

15. Class Size and Income

16. Class Size and Mother’s Schooling

17. 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.

18. Scatter plot of test scores and enrollment

19. Histogram of Enrollment in Voucher Schools

20. Scatter plot of hh characteristics vs enrollment

21. 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

22. 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.