Read up for better understanding on Class Size

1. CLASS-SIZE CAPS, SORTING, AND
THE
REGRESSION-DISCONTINUITY
DESIGN
BY MIGUEL URQUIOLA AND ERIC VERHOOGEN
PRESENTED BY
OGWUIKE CLINTON OBINNA (ADVOCATE)
AYEDEGUE TAYE PATRIC (PROSECUTOR)

2. KEYWORDS
• CLASS-SIZE CAP: this is the highest number of students
required to make up a class size. This study takes 45 students
for its Class-Size Cap.
• SORTING: this is synonymous to classifying i.e. grouping
and/or strategic selection.
DISCONTINUITY: Some sort of arbitrary jump/change thanks to a
quirk in law or nature. We’re interested in the ones that make
very similar people get very dissimilar results.

3. DISCONTINUITY EXAMPLE
•School Class Size
•Maimonides’ Rule--No more than 40
kids in a class in Israel.
•40 kids in school means 40 kids per
class. 41 kids means two classes with
20 and 21.
(Angrist & Lavy, QJE 1999)

4. MORE EXAMPLE
• Union Elections
•If employers want to unionize, NLRB holds
election. 50% means the employer doesn’t
have to recognize the union, and 50% + 1
means the employer is required to “bargain
in good faith” with the union.
(DiNardo & Lee, QJE 2004)

5. REGRESSION DISCONTINUITY
Run a regression based on a situation where you’ve got
a discontinuity.
Treat above-the-cutoff and below-the-cutoff like the
treatment and control groups from a randomization.

6. FURTHER ON RDD
• Many times, random assignment is not possible e.g:
• Universal take-ups
• Non-excludable intervention
• Treatment already assigned
• When randomization is not feasible i.e. how can we measure
implementation features of a program to measure its impact?
• The answer is QUASI-EXPERIMENTS; Regrssion Discontinuity
Design is a good example.

7. MOTIVATION
• Contentious literature on whether class size matters
• To develop a viable a model in which:
• Households sort into schools of different quality levels
• Schools can choose their quality level and class size
• Explore and to an extent evaluate liberalized Chilean education
market
• Forcast the Implications of model for reference purposes:
• 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)

8. RESEARCH QUESTIONS
•What is the effect of Class Size on the
performance of the Students?
•What is the relationship between household
income and quality of education?

9. KEY OBJECTIVE
•This paper hopes to clarify the literature
on the effect of class size on student
performance by using a Regression
Discontinuity Design.

10. DATA
• 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

11. MORE ON 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

12. MORE ON DATA
• public or municipal schools are run by roughly 300
municipalities which receive a perstudent “voucher” payment
from the central government. These schools cannot turn away
students unless demand exceeds capacity, and are limited to a
maximum class size of 45.2
In most municipalities, they are the suppliers of last resort.

13. MORE ON DATA
• private subsidized or voucher schools are independent, and
since 1981 have received
exactly the same per-student subsidy as municipal schools.3
They are also constrained to a
maximum class size of 45, but, unlike public schools, have
wide latitude regarding student
selection.

14. MORE ON DATA
• private unsubsidized schools are independent, do not accept
vouchers, receive no other
explicit subsidies, and are not bound by the class-size cap.
N.B: Parents can use the per-student voucher in any public or
private voucher school that is willing
to accept their children.

15. SUMMARY OF MODEL
• 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()

16. DEMAND
• U(P(),Q (X(),N(); ); ) =  Q ( X(),N(); ) − P() + Ε
• 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

17. • 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







nx
T
q
/
ln

18. 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 schools)
• The authors’ solve for the equilibrium

19. IMPLICATION OF THE MODEL
• TEST 1: There is a roughly inverted-U shaped relationship
between class size and average household income in
equilibrium
• TEST 2: Schools will stack at enrollments there are multiples of
45, implying discontinuous changes in average household
income with respect to enrollment

21. RESULT
•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

22. •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.

23. •The key prediction, borne out in data from
Chile’s liberalized education market, is that
schools at the class-size cap adjust prices (or
enrollments) to avoid adding an additional
classroom, which generates discontinuities in
the relationship between enrollment and
household characteristics, violating the
assumptions underlying regression-
discontinuity research designs.

24. 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

25. CRITIQUE
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 not clarify the literature and
point to a way forward.

26. MORE ON CRITIQUE
• Assumes that smaller class sizes improve school
quality and furthermore that this improvement will be
larger at higher quality schools. Writers are not
thinking about the quality that the parents pay for, not
necessarily for the quality of the output of the
students – but it seems a bit circular. The paper
doesn’t actually address if class size improves
outcomes or not!

27. BIBLOGRAPHY
• Angrist, Joshua D., and Victor Lavy. 1999. “Using Maimonides’
Rule to Estimate the Effect of Class Size
on Scholastic Achievement.” quarterly Journal of Economics,
114(2): 533–75.
• Asadullah, M. Niaz. 2005. “The Effect of Class Size on Student
Achievement: Evidence from Bangladesh.”
Applied Economics Letters, 12(4): 217–21.
• Banerjee, Abhijit V., Shawn Cole, Esther Duflo, and Leigh
Linden. 2007. “Remedying Education: Evidence from Two

28. MORE ON BIBLOGRAPHY
• Bartle, Robert G. 1976. The Elements of Real Analysis. 2nd ed.
New York: John Wiley & Sons.
Bayer, Patrick J., Robert McMillan, and Kim Reuben. 2004. “An
Equilibrium Model of Sorting in an
Urban Housing Market.” National Bureau of Economic Research
Working Paper 10865.
Bressoux, Pascal, Francis Kramarz, and Corinne Prost. 2005.
“Teachers’ Training, Class Size and Students’ Outcomes:
Evidence from Third Grade Classes in France.” Unpublished.
Browning, Martin, and Eskil Heinesen. 2003. “Class Size,
Teacher Hours and Educational Attainment.”
Centre for Applied Microeconometrics Working Paper 2003–15

29. THANK YOU FOR LISTENING