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

⎟⎠

⎞⎜⎝

⎛=nx

Tq

/lnλ

==> 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

Class Size at Unsubsidized Schools in Equilibrium

Class Size at Voucher Schools in Equilibrium

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

Class Size and Income

Class Size and Mother’s Schooling

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.

Scatter plot of test scores and enrollment

Histogram of Enrollment in Voucher Schools

Scatter plot of hh characteristics vs enrollment

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

1. Limited applicability of model.2. Quality variable is not well-explained or

defined. Is it perceived quality? Or is it a measure of student performance and outcomes?

3. If the latter, authors are assuming class size affects quality, which seems circular.

4. Authors show that old methods don’t work, but they don’t offer a new way to estimate effect.

5. Nevertheless, this paper does clarify the literature and point to a way forward.