WPS8563
Policy Research Working Paper 8563
Upping the Ante
The Equilibrium Effects of Unconditional Grants
to Private Schools
Tahir Andrabi
Jishnu Das
Asim I Khwaja
Selcuk Ozyurt
Niharika Singh
Development Economics
Development Research Group
August 2018
Policy Research Working Paper 8563
Abstract
This paper tests for financial constraints as a market failure oligopoly model with capacity constraints and endoge-
in education in a low-income country. In an experimental nous quality: greater financial saturation crowds-in quality
setup, unconditional cash grants are allocated to one pri- investments. The findings of higher social surplus in the
vate school or all private schools in a village. Enrollment setting of all private schools, but greater private returns in
increases in both treatments, accompanied by infrastructure the setting of one private school underscore the importance
investments. However, test scores and fees only increase in of leveraging market structure in designing educational
the setting of all private schools along with higher teacher subsidies.
wages. This differential impact follows from a canonical
This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the
World Bank to provide open access to its research and make a contribution to development policy discussions around the
world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/research. The authors
may be contacted at jdas1@worldbank.org @worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Upping the Ante: The Equilibrium E↵ects of
Unconditional Grants to Private Schools
By Tahir Andrabi, Jishnu Das, Asim I Khwaja, Selcuk Ozyurt, and
Niharika Singh ⇤
JEL Codes: I25; I28; L22; L26; O16
Keywords: Private schools, Financial innovation, Educational
Achievement, Education Markets, Return to Capital, SMEs
⇤ Pomona College; Development Research Group, World Bank; Harvard University; Sa-
banci University; and Harvard University. Email: tandrabi@pomona.edu; jdas1@worldbank.org;
akhwaja@hks.harvard.edu; ozyurt@sabanciuniv.edu; and niharikasingh@g.harvard.edu. We thank
Narmeen Adeel, Christina Brown, Asad Liaqat, Benjamin Safran, Nivedhitha Subramanian, and Fa-
had Suleri for excellent research assistance. We also thank seminar participants at Georgetown, UC
Berkeley, NYU, Columbia, University of Zurich, BREAD, NBER Education Program Meeting, Harvard-
MIT Development Workshop, and the World Bank. This study is registered in the AEA RCT Registry
with the unique identifying number AEARCTR-0003019. This paper was funded through grants from
the Aman Foundation, Templeton Foundation, National Science Foundation, Strategic Impact Evalua-
tion Fund (SIEF) and Research on Improving Systems of Education (RISE) with support from UK Aid
and Australian Aid. We would also like to thank Tameer Microﬁnance Bank (TMFB) for assistance
in disbursement of cash grants to schools. All errors are our own. The ﬁndings, interpretations, and
conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent
the view of the World Bank, its Executive Directors, or the countries they represent.
1
Government intervention in education is often predicated on market failures.1
However, addressing such failures does not require government provision. This
recognition has allowed alternate schooling models that separate the ﬁnancing
and provision of education by the state to emerge. These range from vouchers
in developing countries (Hsieh and Urquiola, 2006; Muralidharan et al., 2015;
Barrera-Osorio et al., 2017) to charter schools in the United States (Hoxby and
Rocko↵, 2004; Hoxby et al., 2009; Angrist et al., 2013; Abdulkadiro˘ glu et al.,
2016) and, more recently, to public–private partnership arrangements with pri-
vate school chains (Romero et al., 2017). One key consideration is that the impact
of these interventions is mediated by the underlying market structure. Yet, es-
tablishing the causal impact of such policies on schools and understanding how
the impact is mediated by program design and the prevailing market structure is
challenging.
The rise of private schooling in low- and middle-income countries o↵ers an
opportunity to map policies to school responses by designing market-level inter-
ventions that uncover and address underlying market failures. In previous work,
we have leveraged “closed” education markets in rural Pakistan to identify labor
and informational market failures and evaluated interventions that ameliorate
them and improve education outcomes (Andrabi et al., 2013, 2017).2 In addi-
tion to these failures, data from our longitudinal study of rural schooling markets
and interviews with school owners suggest that private schools also lack access to
ﬁnancing, with few external funding sources outside their own families.
Here, we present results from an experiment that alleviated ﬁnancial constraints
for private schools in rural Pakistan. We study how this intervention a↵ects
educational outcomes and how variations in intervention design interact with
market structure. Speciﬁcally, our experiment allocates an unconditional cash
grant of Rs.50,000 ($500 and 15 percent of the median annual revenue for sample
schools) to each treated (private) school from a sample of 855 private schools in
266 villages in the province of Punjab, Pakistan. We assign villages to a control
group and one of two treatment arms: In the ﬁrst treatment, referred to as the
‘low-saturation’ or L arm, we o↵er the grant to a single, randomly assigned,
private school within the village (from an average of 3.3 private schools). In
the second treatment, the ‘high-saturation’ or H arm, all private schools in the
experimentally assigned village are o↵ered the Rs.50,000 grant.
The motivation for this experimental design is twofold. First, it helps examine
whether limited ﬁnancial access hinders private school quality and expansion.
Even if private schools lack access to ﬁnance, it is not immediately clear that the
1 Examples include credit market failures for households (Carneiro and Heckman, 2002), the lack
of long-term contracting between parents and children (Jensen, 2012), and the social externalities from
education (Acemoglu and Angrist, 2000).
2 Private sector primary enrollment shares are 40 percent in countries like India and Pakistan and
28 percent in all LMIC combined with signiﬁcant penetration in rural areas (Baum et al., 2013; Andrabi
et al., 2015). Because villages are “closed”— children attend schools in the village and schools in the
village are mostly attended by children in the village— it is both easier to deﬁne markets and to isolate
the impact of interventions on a schooling market as a whole.
2
results from the small and medium enterprises (SME) literature will extend to
education (Banerjee and Duﬂo, 2012; de Mel et al., 2012).3 Second, our design
allows us to assess whether the nature of ﬁnancing— in our case, the extent of
market saturation with unconditional grants— a↵ects equilibrium outcomes. This
saturation design is motivated by our previous research documenting the role of
market competition in determining supply-side responses (Andrabi et al., 2017)
as well as concerns that the return on funds may be smaller if all ﬁrms in the
market receive ﬁnancing (Rotemberg, 2014). Intervening experimentally in this
manner thus presents a unique opportunity to better understand school reactions
to changes in access to ﬁnance and link them to models of ﬁrm behavior and
ﬁnancial access in the literature on industrial organization.
We start with two main results. First, the provision of the grant leads to
greater expenditures in both treatment arms with no evidence that treated schools
in either arms used the grant to substitute away from more expensive forms
of capital, such as informal loans to the school owner’s household. Following
Banerjee and Duﬂo (2012), this suggests the presence of credit constraints in our
setting. It also conﬁrms that the money was used to make additional investments
in the school even though the cash grants were unconditional.
Second, school responses di↵er across the two treatment arms. In the L arm,
treated (Lt ) schools enroll an additional 22 children, but there are no average
increases in test scores or fees. We do not detect any impact on untreated (Lu )
private schools in this arm. In the H arm, enrollment increases are smaller at
9 children per school. Unlike the L arm however, test scores improve by 0.22
standard-deviation for children in these schools, accompanied by an increase in
tuition fees by Rs.19 (8 percent of baseline fees). Revenue increases among H
schools therefore reﬂect both an increase in enrollment and in fees. Even so,
revenue increases in the H arm still fall short relative to that in Lt schools:
Although we cannot reject equal revenue increases in Lt and H schools, the point
estimates for the former are consistently larger.
Our theoretical framework highlights why Lt schools expand capacity while H
schools improve test scores (with smaller capacity expansion). We ﬁrst extend
the canonical model of Bertrand duopoly competition with capacity constraints
due to Kreps and Scheinkman (1983) to allow for vertically di↵erentiated ﬁrms.
Then, using the same rationing rule, whereby students are allocated to the schools
that produce the highest value for them, we prove that expanding ﬁnancial access
to both ﬁrms in the same market is more likely to lead to quality improvements.
Here, ‘more likely’ implies that the parameter space under which quality improve-
ments occur as an equilibrium response is larger in the H relative to L arm.
3 Despite better access to ﬁnance, parents may be unable to discern and pay for quality improvements;
school owners themselves may not know what innovations increase quality; alternate uses of such funds
may give higher returns; or bargaining within the family may limit how these funds can be used to improve
schooling outcomes (de Mel et al., 2012). Alternatively, ﬁnancial constraints may be exacerbated in the
educational sector with fewer resources that can be used as collateral, social considerations that hinder
collection and enforcement, and outcomes that are multi-dimensional and di cult to value for lenders.
3
The key intuition is as follows: When schools face capacity constraints, they
make positive proﬁts even when they provide the same quality. This is the fa-
miliar result that Bertrand competition with capacity constraints recovers the
Cournot equilibrium (Kreps and Scheinkman, 1983). If only one school receives
an additional grant, it behaves like a monopolist on the residual demand from
the capacity constrained school: The (untreated) credit-constrained school cannot
react by increasing investments since these reactions require credit. The treated
school now faces a trade-o↵ between increasing revenue by bringing in additional
children or increasing quality. While the former brings in additional revenue
through children who were not in the school previously, the latter increases rev-
enues from children already enrolled in the school. To the extent that the school
can increase market share without poaching from other private schools, it will
choose to expand capacity as it can increase enrollment without triggering a price
war that leads to a loss in proﬁts. In this model, Lt schools should increase en-
rollment, but not beyond the point where they would substantially ‘poach’ from
other private schools and must rely instead on primarily attracting children from
public schools or those not currently attending school. We indeed ﬁnd increases
in enrollment in Lt schools without a discernible decline in the enrollment of Lu
schools.
On the other hand, if both schools receive the grant money, neither school can
behave like the residual monopolist and this makes it more likely that they invest
in quality. The logic is as follows. If both schools attempt to increase capacity
equally, this makes a price war more likely, leading to a low-payo↵ equilibrium.
There are only two ways around this adverse competitive e↵ect: schools must ei-
ther increase the overall size of the market or must retain some degree of market
power in equilibrium. Investing in quality allows for both as the overall revenue
in the market increases, and schools can relax market competition through (ver-
tical) product di↵erentiation. Investments in quality thus protect positive proﬁts,
although these are not as high as in the L case.4
The model assumes that schools know how to increase quality but are respond-
ing to market constraints in choosing not to do so. This is consistent with our
previous work showing that low cost private schools are able to improve test scores
without external training or inputs (Andrabi et al., 2017). How they choose to
do so is of independent interest for estimates of education production functions.
We therefore further empirically investigate changes in school inputs to shed light
on the channels through which schools are able to attract more students or raise
test scores. We ﬁnd that Lt schools invest in desks, chairs and computers. Mean-
while, while H schools invest in these items as well, they also spend money on
upgrading classrooms, on libraries, and on sporting facilities. More signiﬁcantly,
the wage bill in H schools increased, reﬂecting increased pay for both existing
4 In equilibrium, all schools in a village may invest in quality if the cost of quality investment is
su ciently small and the schools’ existing capacities are su ciently close to their Cournot optimal
capacities.
4
and new teachers. Bau and Das (2016) show that a 1 standard deviation increase
in teacher value-added increases student test scores by 0.15sd in a similar sam-
ple from Punjab, and, in the private sector, this higher value-added is associated
with 41% higher wages. A hypothesis consistent with the test score increases in H
schools is that schools used higher salaries to retain and recruit higher value-added
teachers.
Given the di↵erent responses under the two treatment arms, it is natural to ask
which one is more socially desirable. Accurate welfare estimates require strong
assumptions, but we can provide suggestive estimates. While school owners see
a large increase in their proﬁts under the L arm, this is comparable to the esti-
mated gain in welfare that parents obtain under the H arm, driven by test score
improvements. If, in addition, we factor in that society at large may value test
score gains over and above parental valuations, then the H treatment is more
socially desirable. Higher weights to teacher salaries compared to owner proﬁts
strengthen this conclusion further.
This analysis highlights a tension between market-based and socially preferred
outcomes. Left to the market, a private ﬁnancier would prefer to ﬁnance a single
school in each village; the H arm however is preferable for society. A related policy
question is then whether the government would want to subsidize the private
sector to lend in a manner that multiple schools receive loans in the same village.
To the extent that a lender is primarily concerned with greater likelihood of
default and using the fact that school closures were 9 percentage points lower in
the L arm, a plausible form of this subsidy is a loan-loss guarantee for private
investors. We estimate that the expected cost of such a guarantee is a third of the
gain in consumer surplus suggesting that such a policy may indeed be desirable.
Interestingly, this also implies that the usual “priority sector” lending policies
need to be augmented with a “geographical targeting” subsidy that rewards the
market for increasing ﬁnancial saturation in a given area–the density of coverage
matters.
Our paper contributes to the literatures on education and on SMEs, with a focus
on ﬁnancial constraints to growth and innovation. In education, e↵orts to improve
test scores include direct interventions in the production function; improvements
in allocative e ciency through vouchers or school matching algorithms; and struc-
turing partnerships to select privately operated schools using public funding.5 As
a complement to this literature, we have focused on the impact of policies that
alter the overall operating environments for schools, leaving school inputs and en-
rollment choices to be determined in equilibrium. Such policies, especially when
5 McEwan (2015), Evans and Popova (2015), and JPAL (2017), provide reviews of the ‘production
function’ approach (the causal impact of changing speciﬁc school, teacher, curriculum, parent or student
inputs in the education production function) to improving test scores. Recent studies with considerable
promise tailor teaching to the level of the child rather than curricular standards— see Banerjee et al.
(2017) and Muralidharan et al. (2016). Examples of approaches designed to increase allocative e ciency
include a literature on vouchers (see Epple et al. (2015) for a critical review) and school matching
algorithms (Abdulkadiro˘ glu et al., 2009; Ajayi, 2014; Kapor et al., 2017).
5
they address market failures, are increasingly relevant for education with the rise
of market-based providers, where ﬂexibility allows schools to respond to changes
in the local policy regime.6 In two previous papers, we have shown that these
features permit greater understanding of the role of teacher availability (Andrabi
et al., 2013) as well as information about school performance for private school
growth and test scores (Andrabi et al., 2017).
Closest to our approach of evaluating ﬁnancing models for schools are two recent
papers from Liberia and Pakistan. In Liberia, Romero et al. (2017) show that a
PPP arrangement brought in 7 school operators, each of whom managed several
schools with evidence of test-score increases, albeit at costs that were higher than
business-as-usual approaches. In Pakistan, Barrera-Osorio et al. (2017) study
a program where new schools were established by local private operators using
public funding on a per-student basis. Again, test scores increased. Further,
decentralized input optimization came close to what a social welfare maximizing
government could achieve by tailoring school inputs to local demand. However,
these interventions are not designed to exploit competitive forces within markets.
Viewed through this lens, our contributions are twofold. First, we extend our
market-level interventions approach to the provision of grants to private schools
and track the e↵ects of this new policy on test scores and enrollment. Second, we
conﬁrm that the speciﬁc design of subsidy schemes matters (Epple et al., 2015)
in the context of a randomized controlled trial, and show that these design e↵ects
are consistent with (an extension of) the theory of oligopolistic competition with
credit constraints. In doing so, we are able to directly isolate the link between
policy and school level responses.7
Our paper also contributes to an ongoing discussion in the SME literature on
how best to use ﬁnancial instruments to engender growth. Previous work from the
SME literature consistently ﬁnds high returns to capital for SMEs in low-income
countries (Banerjee and Duﬂo, 2012; de Mel et al., 2008, 2012; Udry and Anagol,
2006). A more recent literature raises the concern that these returns may be
“crowded out” when credit becomes more widely available if these returns are due
to diversion of proﬁts from one ﬁrm to another (Rotemberg, 2014). We are able to
extend this literature to a service like education and simultaneously demonstrate
a key trade-o↵ between low and high-saturation approaches. While low-saturation
infusions may lead SMEs to invest more in capacity and increase market share
at the expense of other providers, high-saturation infusions can induce ﬁrms to
o↵er better value to the consumer and e↵ectively grow the size of the market by
6 Private schools in these markets face little (price/input) regulation, rarely receive public subsidies
and, optimize based on local economic factors. Public school inputs are governed through an adminis-
trative chain that starts at the province and includes the districts. While we can certainly see changes
in locally controlled inputs (such as teacher e↵ort), it is harder for government schools to respond to
local policy shocks with a centralized policy change. In Andrabi et al. (2018), we examine the impact of
similar grants to public schools, which addresses government rather than market failures.
7 Isolating the causal link between policies and educational improvements that is due to school re-
sponses (as opposed to compositional changes) has proven di cult. Large-scale policies usually change
how children sort across schools, making it di cult to ﬁnd an appropriate control group for the policy.
6
“crowding in” innovations and increasing quality. That the predictions of our
experiment are consistent with a canonical model of ﬁrm behavior establishes
further parallels between the private school market and small enterprises. Like
these enterprises, private schools cannot sustain negative proﬁts, obtain revenue
from fee paying students, and operate in a competitive environment with multiple
public and private providers. We have shown previously that, with these features,
the behavior of private schools can be approximated by standard economic models
in the ﬁrm literature (Andrabi et al., 2017). If the returns to alleviating ﬁnancial
constraints for private schools are as large as those documented in the literature
on SMEs, the considerable learning from the SME literature becomes applicable
to this sector as well (Beck, 2007; de Mel et al., 2008; Banerjee and Duﬂo, 2012).
The remainder of the paper is structured as follows: Section I outlines the
context; Section II presents the theoretical framework; Section III describes the
experiment, the data, and the empirical methodology; Section IV presents and
discusses the results; and Section V concludes.
I. Setting and Context
The private education market in Pakistan has grown rapidly in the last three
decades. In Punjab, the largest province in the country and the site of our study,
the number of private schools increased from 32,000 in 1990 to 60,000 in 2016
with the fastest growth taking place in rural areas of the province. In 2010-11,
38% of all enrollments among children between the ages of 6 and 10 was in private
schools (Nguyen and Raju, 2014). These schools operate in environments with
substantial school choice and competition; in our study district, 64% of villages
have at least one private school, and within these villages there is a median of
5 (public and private) schools (NEC, 2005). Our previous work has shown that
these schools are not just for the wealthy; 18 percent of the poorest third sent
their children to private schools in villages where they existed (Andrabi et al.,
2009). One reason for this success is better learning. While absolute levels of
learning are below curricular standards across all types of schools, test scores of
children enrolled in private schools are 1 standard deviation higher than for those
in public schools, which is a di↵erence of 1.5 to 2.5 years of learning (depending
on the subject) by Grade 3 (Andrabi et al., 2009). These di↵erences remain large
and signiﬁcant after accounting for selection into schooling using the test score
trajectories of children who switch schools (Andrabi et al., 2011).
A second reason for this success is that private schools have managed to keep
their fees low; in our sample, the median private school reports a fee of Rs.201 or
$2 per month, which is less than half the daily minimum wage in the province. We
have argued previously that the ‘business model’ of these private schools relies
on the local availability of secondary school educated women with low salaries
and frequent churn (Andrabi et al., 2008). In villages that have a secondary
school for girls, there is a steady supply of such potential teachers, but also
frequent bargaining between teachers and school owners around wage setting—
in the teacher market, a 1sd increase in teacher value-added is associated with a
7
41% increase in wages (Bau and Das, 2016). A typical teacher in our sample is
female, young and unmarried, and is likely to pause employment after marriage
and her subsequent move to the marital home. An important feature of this
market is that the occupational choice for teachers is not between public and
private schools: Becoming a teacher in the public sector requires a college degree,
and an onerous and highly competitive selection process as earnings are 5-10
times as much as private school teachers and applicants far outweigh the intake.
Accordingly, transitions from public to private school teaching and vice versa are
extremely rare.
Despite their successes in producing higher test-scores at low costs, once a
village has a private school, future quality improvements appear to be limited.
We have collected data through the Learning and Educational Achievement in
Pakistan Schools (LEAPS) panel for 112 villages in rural Punjab, each of which
reported a private school in 2003. Over ﬁve rounds of surveys spanning 2003 to
2011, test scores remain constant in “control” villages that were not exposed to
any interventions from our team. Furthermore, there is no evidence of an increase
in the enrollment share of private schools or greater allocative e ciency whereby
more children attend higher quality schools. This could represent a (very) stable
equilibrium, but could also be consistent with the presence of systematic con-
straints that impede the growth potential of this sector.
This study focuses on one such constraint: access to ﬁnance. This focus on
ﬁnance is driven, in part, by what school owners themselves tell us. In our survey
of 800 school owners, two-thirds report that they want to borrow, but only 2%
report any borrowing for school related loans.8 School owners wish to make a
range of investments to improve school performance as well as their revenues and
proﬁts. The most desired investments are in infrastructure, especially additional
classrooms and furniture, which owners report as the primary means of increasing
revenues. While also desirable, school owners ﬁnd raising revenues through better
test scores and therefore higher fees a somewhat riskier proposition. Investments
like teacher training that may directly impact learning are thought to be risky as
they may not succeed (the training may not be e↵ective or a trained teacher may
leave) and even if they do, they may be harder to demonstrate and monetize.
The Pakistani educational landscape therefore presents an active and compet-
itive educational marketplace, but one where schools may face signiﬁcant con-
straints, including ﬁnancial, that may limit their growth and innovation. This
setting suggests that alleviating ﬁnancial constraints may have positive impacts
on educational outcomes; whether these impacts arise due to infrastructure or
pedagogical improvements depends on underlying features of the market and the
competitive pressure schools face.
8 This is despite the fact that school owners are highly educated and integrated with the ﬁnancial
system: 65 percent have a college degree; 83 percent have at least high school education; and 73 percent
have access to a bank account.
8
II. Theoretical Framework
Our theoretical exercise consists of two parts that shed light on the market
level impacts of an increase in ﬁnancial resources. First, we introduce credit
constrained ﬁrms and quality into the canonical Kreps and Scheinkman (1983)
framework (henceforth KS).9 Schools in our model are willing to increase their
capacities or qualities (to charge higher fees) but are credit constrained beyond
their initial capital. Second, we introduce comparative static exercises through
the provision of unconditional grants and study the equilibrium with varying
degrees of ﬁnancial saturation. Our approach of extending a canonical model
disciplines the theory exercise and provides us with a robust conceptual framework
to conduct empirical analysis and interpret ﬁndings.
A. Setup
Two identical private schools, indexed by i = 1, 2, choose whether to invest
in capacity, xi 0, or quality, qt , where t 2 {H, L} is high or low quality.
High quality is conceptualized as investments that allow schools to o↵er better
quality/test scores and charge higher prices, such as specialty infrastructure (e.g.
library or sports facility) or higher-quality teachers. Low quality investments,
such as basic infrastructure (desks or chairs) or basic renovations, allow schools
to retain or increase enrollment but do not change existing students’ willingness
to pay.
SCHOOLS: Each school i maximizes ⇧i = (pi c)xe i + Ki rxi wt subject to
rxi + wt Ki and xe i x i , where x e is the enrollment, p is the price of school i
i i
per seat, c is the constant marginal cost for a seat, r is the ﬁxed cost for a seat, wt
is the ﬁxed cost for quality type, and Ki is the amount of ﬁxed capital available
to the school. Schools face the same marginal and ﬁxed costs for investments.
The ﬁxed cost for low quality is normalized to 0, and so w is the ﬁxed cost of
delivering high quality.10
STUDENTS: There are T students each of whom demands only one seat. Each
student j has a taste parameter for quality ✓j and maximizes utility U (✓j , qt , pi ) =
✓j qt pi by choosing a school with quality qt and fee pi . The value of the outside
option is zero for all students, and students choose to go to school as long as U 0.
We initially assume students are homogeneous with ✓ = 1. Later, we show our
results hold when the model is extended to allow for consumer heterogeneity.
TIMING: The investment game has three stages. In the ﬁrst stage, schools
simultaneously choose their capacity and quality. After observing these choices,
schools simultaneously choose their prices in the second stage. Demand is realized
in the ﬁnal stage. Standard allocation rules are assumed.11
9 KS (1983) develop a model of ﬁrm behavior under binding capacity commitments. In their model,
the Cournot equilibrium is recovered as the solution to a Bertrand game with capacity constraints.
10 Alternative parameterizations for the proﬁt function including allowing for school heterogeneity,
will naturally lead to di↵erent sets of equilibrium outcomes. However, our main results, which are
concerned with the comparisons between the H and L treatments, will remain una↵ected as long as
parameterizations do not vary by treatment arm. We discuss this point further at the end of this section.
11 We assume: (i) The school o↵ering the higher surplus to students serves the entire market up to
9
B. Equilibrium Analysis
We ﬁrst examine the subgame perfect Nash equilibrium (NE) of this investment
game at baseline and then assess how the equilibrium changes in the L arm where
only one school receives a grant K > 0, and in the H arm where both schools
receive the same grant K . The receipt of grants is common knowledge among all
schools in a given market.
An Example
Prior to the full analysis, consider the following example to build intuition for
the pricing decisions of schools. Suppose that the ﬁxed cost of quality is w = 8;
the cost of expanding capacity by one unit is r = 1; and, there are 30 (identical)
consumers who value qL at $3 and qH at $5. The marginal cost of each enrolled
student is c = 0.
Capacity constrained schools and student homogeneity suggests the existence
of an uncovered market in the baseline equilibrium. That is, there are students
willing to attend a (private) school at the prevailing price but cannot do so because
schools do not have the capacity to accommodate these students.12 Without loss
of generality (WLOG), we assume that in the baseline, schools produce low quality
and cannot seat more than 10 students each. Therefore, the size of the uncovered
market is N = 10. Both schools charge $3 and earn a proﬁt of $3 per child for a
total proﬁt of $30. Given capacity constraints, decreasing the price only lowers
school proﬁts.
In the L arm, a single school receives $9, which it can spend on expanding
capacity by 9 units or increasing quality and expanding capacity by 1 unit. Com-
paring proﬁts establishes that capacity expansions are favored with a proﬁt of
$57.13
In the H arm, each of the two schools receives $9. First, consider the subgame
where both schools invest in capacity so that the overall market capacity expands
to 38, which is more than the 30 children in the village. In this subgame, there is
no pure strategy NE. In the mixed strategy equilibrium, schools will randomize
between $3 and $ 3319 (⇡ $1.74) with a continuous and atomless probability distri-
bution and obtain an (expected) proﬁt of $33.14 However, the subgame where
both schools invest in capacity is not consistent with equilibrium in the full game,
its capacity and the residual demand is met by the other school; (ii) If schools set the same price and
quality, market demand is split in proportion to their capacities as long as their capacities are not met;
(iii) If schools choose di↵erent qualities but o↵er the same surplus, then the school o↵ering the higher
quality serves the entire market up to its capacity and the residual demand is met by the other school.
12 These rationed students may instead enroll in public schools in the village, an outside option in this
model, or not attend any school at all.
13 If the school expands capacity, it enrolls 9 more children for a total proﬁt of 19 ⇥ 3=$57. In contrast,
if it invests in quality it receives (10 + 1) ⇥ 5=$55.
14 To see why, note that $3 is not an equilibrium price since a school can deviate by charging $3 ✏ and
enrolling 19 children while the other school obtains the residual demand of 30 19 = 11. Alternatively,
$0 is not an equilibrium price either— deviating to $0 + ✏ with an enrollment of 11 yields a positive
proﬁt as the other school cannot enroll more than 19 children. To derive the mixed strategy equilibrium,
schools must be indi↵erent between any two prices in the support of the mixing distribution. Suppose
one school charges $3. Given that the mixing distribution is atomless, the price of the other school must
be lower. Therefore, the school that charges $3 is price undercut for sure and it will obtain the residual
10
where schools can also choose quality. Speciﬁcally, if one school deviates and in-
vests $8 in quality and $1 in an additional chair instead, then schools could serve
the entire market of 30 children without a price war and the deviating school
would charge $5 for a total proﬁt of $55, which is higher than $33.
The possibility of a price war thus compels schools to not spend the entire grant
on capacity expansion when the size of the uncovered market is ‘small.’ Now
consider the case where each school buys 5 additional chairs, serves 15 students,
and keeps the remaining $4. In this case, equilibrium dictates that each school
should charge a price of $3 and achieve proﬁt of $49. However, investing in
5 additional chairs is also not consistent with equilibrium because one of the
schools would proﬁtably deviate and invest in quality and one additional chair
for a proﬁt of $55. Therefore, when the size of the uncovered market is su ciently
small, at least one of the schools will switch to quality investments instead of a
partial expansion in capacity. In fact, the only equilibrium in this case is such that
one school expands quality with a proﬁt of $55 and the other expands capacity
with a proﬁt of $57. If the uncovered market size had been less than 10, then
both schools investing in quality would be consistent with equilibrium because
the school that deviates cannot fully utilize the grant to avoid price competition
with a rival o↵ering higher quality.
Full Analysis
Consider ﬁrst the baseline scenario. As before, WLOG, we consider the case
where schools produce low quality initially. It is straightforward to show that
in the unique baseline equilibrium, schools enroll the same number of students,
M
2 (where M < T refers to the covered market and N = T M is the size of
the uncovered market) and charge the same price p = qL , extract full consumer
surplus and earn positive proﬁts. Schools do not lower prices since they cannot
meet the additional demand.
Now consider the impact of the grants. When schools receive additional ﬁnanc-
ing, they can increase capacity at the risk of price competition or increase quality
at a (possibly) higher cost. Our previous example illustrates the tension between
these two strategies. Two key parameters inﬂuence the investment strategies of
schools, the cost of quality, w, and the size of the uncovered market, N . When
both w and N are very low, schools prefer to invest in quality in both treatment
arms. For su ciently high values of w, schools in both treatments prefer to invest
in capacity as long as N is quite large. As N decreases, schools will invest in ca-
pacity as long as increasing revenues through new students is more rewarding than
increasing revenues among existing students through higher quality and prices,
but spend less of their grants to escape from price competition. At a threshold
level of N , at least one of the schools switches to quality investment instead of
demand of 11 children and a proﬁt of $33. Now consider a lower bound, y , of the mixing distribution.
Suppose one school charges y . Then it must be the case that it price undercuts the other school and
obtains a demand of 19. But the school must be indi↵erent between charging $3 and charging y , which
implies that $33 = 19 ⇥ y , or y ⇡ 1.74.
11
a partial expansion in capacity. This threshold for N decreases as w increases,
suggesting a negative relationship between the two. We formally prove these
claims for both treatment arms and characterize the wN space where quality
investment by at least one school is consistent with equilibrium.
Because the schools are credit constrained, they cannot a↵ord high quality if
its cost is greater than the grant size. Therefore, we are concerned with the part
of the wN space where quality investment is feasible, i.e. w K . We also
parameterize the size of the grant, K , to be neither ‘too small’ nor ‘too large.’
In particular, we assume that K is large enough such that investing in quality
is not always the optimal action but small enough so that rate of return of each
investment is positive.15
Figure 1a: Low-saturation Treatment Figure 1b: High-saturation Treatment
N N
2K 2K
r r
H
EL EH
K K
r r
L
w⇤ Kw w⇤ Kw
Theorem 1. The shaded regions EL and EH in Figure 1 represent the set of
parameters in wN space where there exists an equilibrium of the investment game
in the low and high-saturation treatment, respectively, such that (at least one)
treated school invests in quality.
All the proofs are presented in Appendix A1. Suppose that the size of the
uncovered market is su ciently large such that the Lt school cannot cover it even
if it spends the entire grant on capacity, i.e. K/r N . If this school increases
capacity, then the gain in proﬁts is equal to the return on each new student
times the number of new students, (qL c) K r . If it increases quality instead,
then the gain in proﬁts is equal to the sum of (i) increase in return on existing
students from the higher price times the number of existing students and (ii) the
return from higher quality to each new student times the number of new students,
( qH qL ) M2 + ( qH c) K r w . Therefore, investing in capacity is more proﬁtable if
⇣ ⌘
15 We ¯ where k = Mr qH qL ¯ = M
suppose that k < K < k 2 ql c
and k 2
(q H qL ). If the inequality
K
k < K does not hold, then the revenue from capacity investment, r
c), is lower than revenue from
(q L
quality (only) investment, M2
( q H q L ), and thus, quality investment is always optimal. The rate of
return from capacity investment is positive because we assume qL c r > 0. Finally, K < k ¯ implies
that rate of return from quality (only) investment is always positive. This assumption is not essential
for our results, and in Appendix A1, we show how equilibrium sets would change if we relax it.
12
the former ⇤
⇣ term ⌘ than the latter, yielding the condition w > w where
⌘⇣is greater
w⇤ = r qqH qL
H c
M K
2 + r . However, if the size of the uncovered market is smaller,
in particular N < K r , then spending the entire grant on additional capacity implies
that the treated school must steal some students from the rival school, resulting
in a price war and lower payo↵s. In order to avoid lower payo↵s, the treated
school will partially invest in capacity. The line L indicates the parameters w
and N that equate the treated school’s proﬁt from quality investment to its proﬁt
from partial capacity investment.16
On the other hand, schools will never engage in a price war in the H arm as
long as the uncovered market size is large enough, so that schools cannot cover it
even if both spend the entire grant on capacity, i.e. 2r K
N . Therefore, for these
values of N , equilibrium predictions will be no di↵erent than the L arm. However,
when N is less than 2r K
, spending the entire grant on additional capacity implies
that the school must steal some students from the rival school, resulting again
in a price war. The constraint indicating the indi↵erence between proﬁt from
quality investment and from partial capacity investment, the line H in Figure 1b,
is much farther out because now both schools can invest in capacity, and hence
price competition is likely even for higher values of the uncovered market size,
N .17 The next result is self-evident from the last two ﬁgures and thus provided
with no formal proof.
Corollary 1 (Homogeneous Consumers). If the treated school in the low-
saturation treatment invests in quality, then there must exist an equilibrium in
the high-saturation treatment that at least one school invests in quality. However,
the converse is not always true.
C. Generalization of the Model and Discussion
Consumer Heterogeneity
Now, we extend our analysis by incorporating consumer heterogeneity in will-
ingness to pay. We assume that students’ taste parameter for quality ✓j is uni-
formly distributed over [0, 1], resulting in a downward sloping demand curve.
Speciﬁcally, if the schools’ quality and price are q and p, respectively, then de-
mand is D(p) = T (1 p q ). Unlike the case with homogeneous consumers, there are
never students who would like to enroll in a school at the existing price but are
rationed out— prices always rise to ensure that the marginal student is kept at her
reservation utility. Nevertheless, our previous intuition will carry forward. The
driving force for our results in the homogeneous case was the tension between the
uncovered market and the schools’ actual capacities; in the heterogeneous case,
the role of the uncovered market is played by the schools’ Cournot best response
capacities, akin to KS (1983).
In the formal exposition in Appendix A2, we maintain the entire KS framework,
⇣ ⌘ ⇣ ⌘
16 More M K w M
formally, L represents the line (qH c) 2
+ r
= (qL c) 2
+N +K N r.
17 More M K w M K
formally, H represents the line (qH c)( 2 + r
) = ( qL c)( 2 + N r
) N r.
13
including their rationing rule, and prove two results. We ﬁrst show that if schools
can choose quality, there always exists a pure strategy NE.18 We then prove that,
as in the case of homogeneous consumers, if both schools invest in capacity in the
H arm, this makes capacity expansion beyond the Cournot best response levels
more likely, thereby increasing the likelihood of price competition. It is thus more
likely that (at least one) treated school in the H arm will invest in quality. Using
this intuition, we prove a version of Theorem 1 under a mild set of parameter
restrictions discussed in Appendix A2.
Theorem 2 (Heterogeneous Consumers). If the treated school in the low-
saturation treatment invests in quality, then there must exist an equilibrium in the
high-saturation treatment where at least one school invests in quality. However,
the converse is not always true.
Potential Extensions
There are a number of other plausible modiﬁcations that could be made to the
model. For instance, we could introduce risk-averse owners who are insurance
(rather than credit) constrained, or introduce a degree of altruism in the proﬁt
function to allow for school owners who intrinsically care about the number of
children in school. We can also allow quality to be a continuous variable and
also move beyond our static setting to introduce dynamic considerations such as
over-investment to deter entry. These modiﬁcations potentially change the set
of parameters supporting equilibria where (at least one) treated school invests in
quality. However, our theorems will remain unchanged as long as these changes
a↵ect the schools’ proﬁt functions symmetrically in each treatment arm. In this
case, the risk of price competition will still be higher in the H arm, and thus
quality investment will still be more likely in H than the L arm.
On the other hand, adjustments to the model that generate asymmetric pa-
rameterization of the proﬁt function in each treatment arm may alter our main
results. For example, if school owners have the ability to collectively a↵ect the
market size or input prices (e.g. higher competition among schools may raise
teachers’ salaries), then the return or cost of an investment would be di↵erent in
each treatment arm, which may meaningfully change our results. Given that the
total resources available in a village vary across treatment arms, we assess this
possibility further in Section IV.A and show that it is not empirically salient in
our case.
To summarize, our model provides insights on how schools in the two treatment
arms respond to a relaxation of credit constraints, either by increasing revenue
18 The intuition follows from the nature of the proﬁt function. The mixed strategy equilibrium in the
KS game is due to discontinuities in the proﬁt function. When both ﬁrms produce the same quality, if
one price undercuts the other, then it takes all consumers up to its capacity and sees a discontinuous
jump in proﬁts. When ﬁrms are di↵erentiated in quality, proﬁts always change smoothly as the marginal
consumer’s valuation distribution is atomless. If all consumers are homogeneous as before however, even
with di↵erentiated quality, the smoothness in consumer demand vanishes and we again ﬁnd no pure
strategy equilibria in the game.
14
from existing consumers or expanding market share and risking price competi-
tion. Our main result is that we are more likely to observe higher enrollment in
treated schools in the L arm and higher quality (and increased fees) in the H arm.
Moreover, private proﬁts will be higher for Lt schools. Although, conceptually, a
test of the theory can be based on variation in the size of the uncovered market
and the cost of quality investments, these are not observed in the data. Therefore,
we focus attention in our empirical results on the di↵erence in impact between
low and high-saturation villages.
III. Experiment, Data and Empirical Methods
A. Experiment
Our intervention tests the impact of increasing ﬁnancial access for schools for
outcomes guided by theory (revenue, expenditures, enrollment, fees and quality
captured as test scores) and assesses whether this impact varies by the degree of
ﬁnancial saturation in the market. Our intervention has three features: (i) it is
carried out only with private schools where all decisions are made at the level of
the school;19 (ii) we vary ﬁnancial saturation in the market by comparing villages
where only one (private) school receives a grant (L arm) versus villages where
all (private) schools receive grants (H arm); and (iii) we never vary the grant
amount at the school level, which remains ﬁxed at Rs.50,000.
Randomization Sample and Design.— Our sampling frame is deﬁned as all
villages in the district of Faisalabad in Punjab province with at least 2 private
or NGO schools; 42 percent (334 out of 786) of villages in the district fall in
this category. Based on power calculations using longitudinal LEAPS data, we
sampled 266 villages out of the 334 eligible villages with a total of 880 schools, of
which 855 (97%) agreed to participate in the study.
Table 1 presents summary statistics from our sample at the village (Panel A)
and the private school level (Panel B). The median village has 2 public schools, 3
private schools and 416 children enrolled in private schools. The median private
school has 140 enrolled children, charges Rs. 201 in monthly fees, and reports
a monthly revenue of Rs. 26,485. Monthly variable costs are Rs. 16,200 and
annual ﬁxed costs are Rs. 33,000, for an annual proﬁt of Rs. 90,420. The range
of outcome variables is quite large. Relative to a mean of 164 students, the
5th percentile of enrollment is 45 compared to 353 at the 95th percentile of the
distribution. Similarly, fees range from Rs. 81 (5th percentile) to Rs. 503 (95th
percentile) and monthly revenues from Rs. 4,943 to Rs. 117,655. The kurtosis,
a measure of the density at the tails, is 17 for annual ﬁxed expenses and 51
for revenues relative to a kurtosis of 3 for a standard normal distribution. Our
decision to include all schools in the market provides external validity, but has
implications for precision and mean imbalance, both of which we discuss.
19 This excludes public schools, which cannot charge fees and lack control over hiring and pedagogic
decisions. In Andrabi et al. (2018), we study the impact of a parallel experiment with public schools
between 2004 and 2011. It also excludes 5 (out of close to 900) private schools that were part of a larger
school chain with schooling decisions taken at the central o ce rather than within each school.
15
We use a two-stage stratiﬁed randomization design where we ﬁrst assign each
village to one of three experimental groups and then schools within these villages
to treatment. Stratiﬁcation is based on village size and village average revenues,
as both these variables are highly auto-correlated in our panel dataset (Bruhn
and McKenzie, 2009). Based on power calculations, 3 7 of the villages are assigned
2
to the L arm, and 7 to the H arm and the control group; a total of 342 schools
across 189 villages receive grant o↵ers (see Appendix Figure C1). In the second
stage, for the L arm, we randomly select one school in the village to receive the
grant o↵er; in the H arm, all schools receive o↵ers; and, in the control group, no
schools receive o↵ers.
The randomization was conducted through a public computerized ballot in La-
hore on September 5, 2012, with third-party observers (funders, private school
owners and local NGOs) in attendance. The public nature of the ballot and
the presence of third-party observers ensured that there were no concerns about
fairness; consequently, we did not receive any complaints from untreated schools
regarding the assignment process. Once the ballot was completed, schools re-
ceived a text message informing them of their own ballot outcome. Given village
structures, information on which schools received the grant in the L arm was not
likely to have remained private, so we assume that the receipt of the grant was
public information.
Intervention.— We o↵er unconditional cash grants of Rs.50,000 (approximately
$500 in 2012) to every treated school in both L and H arms. The size of the grant
represents 5 months of operating proﬁts for the median school and reﬂects both
our overall budget constraint and our estimate of an amount that would allow for
meaningful ﬁxed and variable cost investments. For instance, the median wage
for a private school teacher in our sample is Rs. 24,000 per year; the grant thus
allows the school to hire 2 additional teachers a year. Similarly, the costs of desks
and chairs in the local markets range from Rs. 500 to Rs. 2,000, allowing the
school to purchase 25-100 additional desks and chairs.
We deliberately do not impose any conditions on the use of the grant apart from
submission of a (non-binding) business plan (see below). School owners retain
complete ﬂexibility over how and when they spend the grant and the amount
they spend on schooling investments with no requirements of returning unused
funds. As we show below, most schools choose not to spend the full amount in the
ﬁrst year and the total spending varies by the treatment arm. Our decision not to
impose any conditions follows our desire to provide policy-relevant estimates for
the simplest possible design; the returns we observe therefore provide a ‘baseline’
for what can be achieved through a relatively ‘hands-o↵ ’ approach to private
school ﬁnancing.
Grant Disbursement.— All schools selected to receive grant o↵ers are visited
three times. In the ﬁrst visit, schools choose to accept or reject the grant o↵er: 95
16
percent (325 out of 342) of schools accept.20 School owners are informed that they
must (a) complete an investment plan to gain access to the funds and may spend
these funds on items that would beneﬁt the school and (b) be willing to open a
one-time use bank account for cash deposits. Schools are given two weeks to ﬁll
out the plan and must specify a disbursement schedule with a minimum of two
installments. In the second visit, investment plans are collected and installments
are released according to desired disbursement schedules.21 A third and ﬁnal
disbursement visit is conducted once at least half of the grant amount has been
released. While schools are informed that failure to spend on items may result in
a stoppage of payments, in practice, as long as schools provide an explanation of
their spending or present a plausible account of why plans changed, the remainder
of the grant is released. As a result, all 322 schools receive the full amount of the
grant.
Design Confounders.— If the investment plan or the temporary bank account
a↵ected decision making, our estimates will reﬂect an intervention that bundles
cash with these additional features. We discuss the plausibility of these channels
in Section IV.A below and use additional variation and tests in our experiment
to show that any contribution of these mechanisms to our estimated treatment
e↵ects are likely small. In Section IV.A, we also discuss that the treatment unit in
a saturation experiment is a design variable; in our case, this unit could have been
either the village (total grants are equalized at the village level) or the school. We
chose the latter to compare schools in di↵erent treatment arms that receive the
same grant. Consequently, in the H arm, with a median of 3 private schools, the
total grant to the village is 3 times as large as to the L arm. Observed di↵erences
between these arms could therefore reﬂect the equilibrium e↵ects of the total
inﬂow of resources into villages, rather than the degree of ﬁnancial saturation.
Again, using variation in village size, we show in section IV.A that this is unlikely
to be a concern since our results remain qualitatively the same when we compare
villages with similar per-capita grant inﬂow.
B. Data Sources
Between July 2012 and November 2014, we conducted a baseline survey and ﬁve
rounds of follow-up surveys. In each follow-up round, we survey all consenting
schools in the original sample and any newly opened schools.22
Our data come from three di↵erent survey exercises, detailed in Appendix C.
20 Reasons for refusal include anticipated school closure; unwillingness to accept external funds; or a
failure to reach owners despite multiple attempts.
21 At this stage, 3 schools refused to complete the plans and hence do not receive any funds. Our ﬁnal
take-up is therefore 94% (322 out of 342 schools), with no systematic di↵erence between the L and H
arms.
22 There are 31 new school openings two years after baseline: 3 public and 28 private schools. 13
new private schools open in H villages, 10 in the L villages, and 5 in control villages. Given these small
numbers, we omit these schools from our analysis. Even though the overall number of school openings is
low, we ﬁnd that H villages report a higher fraction of new schools relative to control, though this e↵ect
is small at an increase of 2%. Our main results remain qualitatively similar if we include these schools
in our analyses with varying assumptions on their baseline value.
17
We conduct an extended school survey twice, once at baseline and again 8 months
after treatment assignment in May 2013 (Round 1 in Appendix Figure C2), col-
lecting information on school characteristics, practices and management, as well
as household information on school owners. In addition, there are 4 shorter follow-
up rounds every 3-4 months that focus primarily on enrollment, fees and revenues.
Finally, children are tested at baseline and once more, 14 months after treatment
(Round 3). During the baseline, we did not have su cient funds to test every
school and therefore administered tests to a randomly selected half of the sample
schools. We also never test children in public schools. At baseline, this decision
was driven by budgetary constraints and in later rounds we decided not to test
children in public schools because our follow-up surveys showed enrollment in-
creases of at most 30 children in treatment villages. Even if we were to assume
that these children came exclusively from public schools, this suggests that public
school enrollment across all grades declined at most 2-3% on average. This e↵ect
seemed too small to generate substantial impacts on public school quality.23
C. Regression Speciﬁcation
We estimate intent-to-treat (ITT) e↵ects using the following school-level spec-
iﬁcation:24
t u
Yijt = ↵s + t + 1 Hijt + 2 Lijt + 3 Lijt + Yij 0 + ✏ijt
Yijt is an outcome of interest for a school i in village j at time t, which is mea-
sured in at least one of ﬁve follow-up rounds after treatment. Hijt , Lt u
ijt , and Lijt
are dummy variables for schools assigned to high-saturation villages, and treated
and untreated schools in low-saturation villages respectively. We use strata ﬁxed
e↵ects, ↵s , since randomization was stratiﬁed by village size and revenues, and t
are follow-up round dummies, which are included as necessary. Yij 0 is the baseline
value of the dependent variable, and is used whenever available to increase preci-
sion and control for any potential baseline mean imbalance between the treated
and control groups (see discussion in section III.D). All regressions cluster stan-
dard errors at the village level and are weighted to account for the di↵erential
probability of treatment selection in the L arm as unweighted regressions would
assign disproportionate weight to treated (untreated) schools in smaller (larger)
L villages relative to schools in the control or H arms (see Appendix B). Our
coe cients of interest are 1 , 2 , and 3 , all of which identify the average ITT
e↵ect for their respective group.
23 Another option would have been to test those students at baseline whom we expected to be marginal
movers due to the treatment and see their gains from the switch. Detecting marginal movers ex-ante
however is di cult especially given that churn is not uncommon in this setting.
24 We focus on ITT e↵ects and do not present other treatment e↵ect estimates since take-up is near
universal at 94 percent.
18
D. Validity
Balance.— Appendix Table D1 presents tests for baseline di↵erences in means
and distributions as well as joint tests of signiﬁcance across experimental groups
at the village (Panel A) and at the school level (Panel B). At the village level,
covariates are balanced across the three experimental groups (H , L and Control),
and village level variables do not jointly predict village treatment status for the
H or L arm.
Balance tests at the school level involve four experimental groups: Lt and Lu
schools; schools in the H arm; and untreated schools in control. Panel B shows
comparisons between control and each of the three treatment groups (cols 3-5)
and between the H and Lt schools (col 6), our other main comparison of interest.
5 out of 32 univariate comparisons (Panel B, cols 3-6) show mean imbalance at
p-values lower than 0.10— a fraction slightly higher than what we may expect
by random chance. If this imbalance leads to di↵erential trends beyond what can
be accounted for through the inclusion of baseline variables in the speciﬁcation,
our results for the Lt schools may be biased (Athey and Imbens, 2017). Despite
this mean imbalance however, our distributional tests are always balanced (Panel
B, colss 7-9), and, furthermore, covariates do not jointly predict any treatment
status. Nevertheless, we conduct a number of robustness checks in Appendix D
and show that the mean imbalance we observe is largely a function of heavy(right)-
tailed distributions arising from the inclusion of all schools in our sample and
trimming our data eliminates the imbalance without qualitatively changing our
treatment e↵ects (see Appendix Tables D2 and D3).
Attrition.— Schools may exit from the study either due to closure, a treat-
ment e↵ect of interest that we examine in Section IV.A, or due to survey refusals.
Survey completion rates in any given round are uniformly high (95% for rounds
1-4 and 90% for round 5), with only 14 schools refusing all follow-up surveys
(7 control, 5 H , and 2 Lu ). Nevertheless, since round 5 was conducted 2 years
after baseline, we implemented a randomized procedure for refusals, where we
intensively tracked half of the schools who refused the survey in round 5 for an
interview. We apply weights to the data from this round to account for this
intensive tracking (see Appendix B for details). In regressions, we ﬁnd that Lt
schools are less likely to attrit relative to control in every round (Appendix Ta-
ble D4, Panel A). For other experimental groups, attrition is more idiosyncratic.
Despite this di↵erential attrition, baseline characteristics of those who refuse sur-
veying at least once do not vary by treatment status in more than 2 (of 21) cases,
which could occur by random chance (Appendix Table D4, Panel B).25 We check
robustness to attrition using inverse probability weights in Appendix Table D5,
discussed in greater detail in section IV.A, and ﬁnd that our results are una↵ected
25 Comparing characteristics for the at-least-once-refused set is a more conservative approach than
looking at the always-refused set since the former includes idiosyncratic refusals. There are 14 schools
in the always-refused set however making inference di cult; nevertheless, when we do consider this set,
one signiﬁcant di↵erence emerges with lower enrollment in Lu relative to control schools.
19
by this correction.
IV. Results
In this section, we present results on the primary outcomes of interest, inves-
tigate potential channels of impact, and discuss the implications and potential
welfare impact of our ﬁndings.
A. Main Results
Expenditures and Revenues
We ﬁrst present evidence that the grant increased school expenditures; this is
of independent interest as school and household ﬁnances are fungible and school
owners had considerable leeway in how the grant could be spent. Table 2, column
1, shows that school ﬁxed expenditures increased for Lt and H schools relative to
control in the ﬁrst year after treatment; the magnitudes as a fraction of the grant
amount in the ﬁrst year were 61% for Lt and 70% for the H schools. Fixed costs
primarily include infrastructure-related investments, such as upgrading rooms or
new furniture and ﬁxtures; spending on these items is consistent with self-reported
investment priorities in our baseline data.
The fact that schools increase their overall expenditures despite the grant being
(e↵ectively) unconditional suggests that school investments o↵er better returns
relative to other investment options. While consistent with the presence of credit
constraints, investing in the school could also reﬂect the lower (zero) cost of ﬁ-
nancing through a grant. In this context, Banerjee and Duﬂo (2012) suggest a
test to directly establish the presence of credit constraints. Suppose that ﬁrms
borrow from multiple sources. When cheaper credit (i.e. a grant) becomes avail-
able, if ﬁrms are not credit constrained, they should always use the cheaper credit
to pay o↵ more expensive loans. In fact, they should draw down the expensive
loans to zero if credit is freely available. In Appendix Table E1, we examine
data on borrowing for school and household accounts of school owner households.
While there is limited borrowing for investing in the school, over 20% of school
owner households do borrow (presumably for personal reasons). Yet, we ﬁnd no
statistically signiﬁcant declines in borrowing at the school or household level as
a result of our intervention.
We now consider whether these expenditure changes a↵ected school revenues.
Since schools may not always be able to fully collect fees from students, we use two
revenue measures: (i) posted revenues based on posted fees and enrollment (cols
2-4), calculated as the sum of revenues expected from each grade as given by the
grade-speciﬁc monthly tuition fee multiplied by the grade-level enrollment; and
(ii) collected revenues as reported by the school (cols 5-7).26 To obtain the latter
measure, we inspected the school account books and computed revenues actually
collected in the month prior to the survey.27 While this measure captures revenue
26 Posted revenues are available for rounds 1,2, and 4, and collected revenues are available from rounds
2-5. We use baseline posted revenues as the control variable in all revenue regressions.
27 Over 90% of schools have registers for fee payment collection, and for the remainder, we record
20
shortfalls due to partial fee payment, discounts and reduced fees under exceptional
circumstances, it may not adjust appropriately for delayed fee collection.
First, there are substantial posted revenue increases in all treated schools. Col-
umn 2 shows that schools in the H arm gain Rs.5,484 (p=0.12) each month
while Lt schools gain Rs.10,665 (p=0.03) a month. Annual revenue increases
(twelve times the reported monthly coe cient estimates) compare favorably to
the Rs.50,000 grant amount for the returns on investment. In contrast, we never
ﬁnd any signiﬁcant change in revenues among Lu schools, with small coe cients
across all speciﬁcations. Second, the impact on collected revenues is similar for H
schools (Rs.4,400 with p=0.22), but is smaller (Rs.7,924, p=0.09) for Lt schools
(col 5). One explanation for this di↵erence could be that marginal new children
pay lower (than posted) fees in Lt schools. We examine this in more detail later
(Table 3) when we decompose our revenue impacts into enrollment and school
fees. Third, the results are large but often imprecise due to the high variance in
the revenue distribution (the distribution is highly skewed with a skewness of 5.6
and kurtosis of 51.2); precision increases however when we either top-code the
data, assigning the 99th percentile value to the top 1% of data, or drop the top
1 percent of data (cols 3 & 6 and cols 4 & 7, respectively), and our results are
signiﬁcant at conventional levels. We (still) cannot reject equality of coe cients
across the treatment arms of the intervention.28
Enrollment and Fees
Table 3 considers the impact of the grant on the two main components of
(posted) school revenue— school enrollment and fees— to shed light on the sources
of revenue changes and whether they di↵er across treatment arms.
Our ﬁrst result is that school enrollment increased in Lt and H schools, where
enrollment is measured across all grades in a given school and coded as zero if a
school closed. Columns 1-3 examine enrollment impacts, annually in columns 1-2
and pooling across the two treatment years in column 3. In the ﬁrst year, the
Lt schools enroll 19 additional children, representing a 12 percent increase over
baseline enrollment. This compares to an average increase of 9 children for H
schools (p=0.10). These gains are sustained and even higher in the second year
(col 2); the pooled estimate thus gives an overall increase of 22 children for Lt
schools (col 3). Appendix Table E2 shows that these gains are not grade-speciﬁc
with signiﬁcant positive e↵ects of 11-18 percent over baseline enrollment across
the grade distribution. We never observe an average impact on Lu schools, which
is consistent with our theory prediction: Schools should not increase capacity
beyond the point where they decrease the enrollment of their competitors, as this
can trigger severe price competition leading to lower proﬁts for all schools.
Part of the higher enrollment among Lt schools is due to a reduction in the
self-reported fee collections.
28 In this analysis, we assign a zero value to a school once it closes down. If instead, we restrict our
analysis to schools that remain open throughout the study with the caveat that these estimates partially
reﬂect selection, we still observe revenue impacts though they are smaller in magnitude, especially for Lt
schools. We discuss this further in Section IV.A when we break down the sources of revenue impacts.
21
number of school closures. Over the period of our experiment, 13.7 percent of the
schools in the control group closed. As column 4 shows, Lt schools were 9 percent-
age points less likely to close over the study period. We ﬁnd no average impact on
school closure for H or Lu schools relative to control. Although fewer school clo-
sures naturally imply higher enrollments for the average school (given that closed
schools are assigned zero enrollment), we emphasize that there were enrollment
gains among the schools that remained open throughout the study: Column 5
restricts the analysis sample to open schools only, and still shows higher enroll-
ment for H and Lt schools, though magnitudes for the latter are naturally smaller
for the latter relative to Column 3 (11.6 children, p=0.13). Conditioning on a
school remaining open without accounting for the selection into closure implies
that enrollment gains are likely biased downwards, as schools that closed tend to
have fewer children at baseline. This suggests that Lt schools not only staved o↵
closure, but also beneﬁted through investments that increased enrollment among
open schools.
Understanding where this enrollment increase came from would have required
us to track over 100,000 children in these villages over time. Even with this
tracking, it would not have been possible to separately identify the children who
moved due to the experiment from regular churn. However, to the extent that
there is typically more entry at lower grades and greater drop-out in higher grades,
the fact that we see similar increase in both these grade levels suggests that both
new student entry (in lower grades) and greater retention (in higher grades) are
likely to have played a role.29
Unlike enrollment, which increased in both treatment arms, fees increased only
among H schools as seen in Table 3, columns 6-8. Average monthly tuition fees
across all grades in H schools is Rs.19 higher than control schools, an increase of
8 percent relative to the baseline fee (col 8). These magnitudes are similar across
the two years of the intervention. Appendix Table E4 also shows that all grades
experienced fee increases, with e↵ect sizes ranging from 8-12% of baseline fee. As
higher grades have higher baseline fees, there is a hint of greater absolute increases
for grades 6 and above, but small sample sizes preclude further investigation of
this di↵erence. In sharp contrast, we are unable to detect any impact on school
fees for either Lt or Lu schools. Consequently, we reject equality of coe cients
between H and Lt at a p-value of 0.02 (col 8).
These results use posted (advertised) fees, but actual fees paid by parents may
be di↵erent as collection rates may be below 100%. As we found previously, the
impacts on posted and collected revenues were similar for H schools, but not for
Lt schools, suggesting that collected fees may have been lower in these schools. We
29 While noisier and limited to the tested grades, we can track enrollment using data on the tested
children. Doing so in Appendix Table E3, we ﬁnd that Lt schools have a higher fraction of children who
report being newly enrolled in round 3, measured as attending their contemporaneous school for fewer
than 18 months from the date of treatment assignment (col 2). The data do not however allow us to
distinguish whether these children switched from other (public) schools in the village or were not-enrolled
at baseline but re-enrolled as a consequence of the treatment.
22
conﬁrm this in column 9 by computing collected fees as collected revenues divided
by school enrollment. These estimates are less precise than for posted fees, but
suggest that fees increased by Rs.29 in H schools (p=0.14) and decreased by Rs.8
(p=0.54) among Lt schools.30
Treated schools therefore respond to the same amount of cash grant in di↵erent
ways depending on the degree of ﬁnancial saturation in their village. Consistent
with the predictions of our model, the main increase in revenue for Lt schools
comes from marginal children who may otherwise have not been in school, whereas
over half of the revenue increase among schools in H schools is from higher fees
charged to inframarginal children (which, as we examine below, likely reﬂects
increases in school quality).
Test Scores
We now examine whether increases in school revenues are accompanied by
changes in school quality, as measured by test scores. To assess this, we use
subject tests administered in Math, English and the vernacular, Urdu, to chil-
dren in all schools 16 months after the start of the intervention (near the end of
the ﬁrst school year after treatment).31 We graded the tests using item response
theory, which allows us to equate tests across years and place them on a common
scale (Das and Zajonc, 2010). Appendix C provides further details on testing,
sample and procedures.
Columns 1 to 4 in Table 4 present school level test score impacts (unweighted
by the number of children in the school) and column 5 presents the impact at the
child level. While the latter is relevant for welfare computations, the school level
scores ensure comparability with our other (school level) outcome variables. To
improve precision, we include the baseline test score where available.32
Test score increases for H schools are comparably high in all subjects with co-
e cients ranging from 0.19sd in English (p=0.04) to 0.11sd in Urdu (p=0.12).
Averaged across subjects, children in H schools gain an additional 0.16sd, rep-
resenting a 42% additional gain relative to the (0.38sd) gain children in control
schools experience over the same 16-month period. In contrast, and consistent
with the school fee results, there are no detectable impacts on test scores for
schools in the L relative to control. Given this pattern, we also reject a test of
30 This decline is consistent with our theory given heterogeneous consumer preferences over school
quality. With a downward sloping demand curve, schools would have to decrease their fees to bring in
more children as they increase capacity.
31 As discussed previously, budgetary considerations precluded testing the full sample at baseline, so
we instead randomly chose half our villages for testing. In the follow-up round however, an average of 23
children from at least two grades were tested in each school, with the majority of tested children enrolled
in grades 3-5; in a small number of cases, children from other grades were tested if enrollment in these
grades was zero. In tested grades, all children were administered tests and surveys regardless of class
size; the maximum enrollment in any single class was 78 children.
32 Since we randomly tested half our sample at baseline, we replace missing values with a constant
and an additional dummy variable indicating the missing value. In Appendix Table E5, we show that
alternate speciﬁcations that either exclude baseline controls (cols 1-4) or include additional controls
(cols 5-8) do not a↵ect our results, with similar point estimates but a reduction in precision in some
speciﬁcations.
23
equality of coe cients between H and Lt schools at p-value 0.07 (col 4). Finally,
column 5 shows that child level test score impacts are higher at 0.22sd, suggesting
that gains are higher in larger schools.
Given that enrollment increases across all grades and H schools see an addi-
tional enrollment of 9 children or 5% of baseline enrollment, compositional e↵ects
would have to be unduly large to drive these e↵ects. To formally assess this
claim, we ﬁrst restrict the sample to those children who were in the same school
throughout our study, which includes 90% of all children in the follow-up round.
Average school level and child level test score increases for this restricted sample
are 0.14sd (p=0.09) and 0.24sd (p=0.01) for the H arm, respectively (Appendix
Table E6, col 4).33
One may also believe that test score increases reﬂect a change in the composition
of peers. Although we cannot rule out such peer e↵ects, we note that Lt schools
gain more children but show no learning gains. Moreover, a school in the H
arm attracts an average of at most 1 new child into a tested grade average of
13 children. The peer e↵ects from this single child would have to be very large
to induce the changes we see and is unlikely given the typical magnitude of such
e↵ects in the literature (Sacerdote, 2011).
Finally, we tested at most two grades per school. Therefore, we cannot directly
examine whether children across all grades in the school have higher test scores
due to our treatment. Instead, we make two points: (i) average fees are higher
across all grades in H schools and insofar as fee increases are sustained through
test score increases, this suggests that test score increases likely occurred across all
grades; and (ii) if we examine test scores gains in the two tested grades separately,
we still observe positive (if imprecise) test score improvements in H schools for
each grade.
Robustness and Further Results
Our preferred explanation for the reduced form results— especially the dif-
ferential results between the treatment arms— relies on the strategic returns to
investing in quality when ﬁnancial saturation in markets is high. We now ex-
amine factors in our design and analysis that could potentially confound this
interpretation.
Investment Plan.— Our intervention required every treated school to submit
an investment plan before any disbursement could take place. It is not obvious
how this requirement, by itself, could lead to the di↵erential treatment e↵ects we
observe, particularly as the experimental literature on business plans seldom ﬁnds
signiﬁcant e↵ects (McKenzie, 2017). Moreover, our process was designed to be
minimally invasive and e↵ectively non-binding as schools could propose any plan
and change it at any time as long as they informed us.34 Nevertheless, consider the
two following channels of impact. The plan could either have forced school owners
33 If stayers were positively selected in terms of their baseline test scores, this result would be biased
upwards; in fact, stayers have lower test scores at baseline in the H relative to control.
34 Schools could propose investments with private value as long as they could argue it beneﬁted the
24
to consider new investments or, perhaps, the act of submission itself notionally
committed school owners to a course of action. We can show neither of these
channels is salient by drawing on three separate sources of (proposed and actual)
school investments: (a) pre-treatment proposed investment questions from the
baseline survey; (b) investment plan data; and (c) investments as reported in
the follow-up surveys. First, the correlation in proposed investments between (a)
and (b) is high, suggesting that simply asking schools about investment plans is
unlikely to explain our treatment e↵ects since (a) is asked of both treatment and
control schools and (a) provides similar information to (b). Second, it also does
not seem that (b) was particularly binding as the correlation between investments
in (b) and (c) is low. Schools do not seem to have treated the business plan as a
commitment device; instead, owners appeared to have ﬁnalized school investments
after disbursement. Thus, it is unlikely that the submission of investment plans
induced the kinds of large e↵ects we document here, and even less likely that it
induced di↵erential e↵ects between the treatment arms.
Bank Account.— In order to receive the grant funds, school owners had to open
a one-time use bank account with our banking partner. This begs the question:
Could this account opening have driven the e↵ects we observe? In our sample,
73% of school owner households already had bank accounts at baseline and this
fraction is balanced across treatment arms. Further, in Appendix Table E7, we
use an interaction between treatment and baseline bank account availability to
check whether our pattern of treatment e↵ects is driven by previously unbanked
households. We detect no statistically signiﬁcant di↵erential impact by baseline
bank account status.
Village level Resources.— Given our design preference for school level compar-
isons, the grant amount was the same for all schools regardless of treatment arm.
Therefore, grant per capita in a L village is necessarily always lower than in a
H village, holding constant village size. To investigate whether this di↵erence
in overall resource availability at the village level can explain our results, we use
baseline variation in village size to additionally control for the per-capita grant
size in each village. If per-capita grant size is an omitted variable that is corre-
lated with treatment saturation and driving our results, we should ﬁnd that the
additional inclusion of this variable drives the di↵erence in our treatment coe -
cients to zero. We therefore replicate our base speciﬁcations including per-capita
grant size as an additional control in Appendix Table E8, columns 1-3. We ﬁnd
that the qualitative pattern of our core results on enrollment, fees and test scores
is unchanged. Lt schools see higher enrollment on average, while H schools ex-
perience higher fees and test scores on average. While we lose precision in the H
arm, we cannot reject that these coe cients are identical to our base speciﬁcation.
This suggests that alternative explanations based on equilibrium e↵ects from an
school or spend the money on previously planned investments, thereby e↵ectively using the grant for
personal uses. They could also propose changes to their plans at any time during the disbursement.
25
increase in overall resources at the village level are unlikely.
Attrition:— As discussed previously, attrition in our data never exceeds 5% in
the ﬁrst year and 10% in the second year of the study, and baseline characteristics
of attriters are similar across treatment groups (Appendix Table D4, Panel B).
Although attrition is higher in the second year of treatment, recall that wherever
available our ﬁrst and second year estimates are similar (Table 3). This suggests
that any bias from increased attrition in the second year is likely small. Further-
more, our results are robust to using inverse probability weights to account for
higher attrition (see Appendix Table D5).
B. Channels
In this section, we consider potential channels of impact by examining changes
in school investments as a result of the grants. We ﬁrst look at overall ﬁxed and
variable costs and then focus on the main components of each— infrastructure
and teacher costs.
Overall Fixed and Variable Costs
Table 5 presents the average impacts of the intervention on (annualized) ﬁxed
and variable costs. Fixed costs represent annual investments, usually before the
start of the school year, for school infrastructure (furniture, ﬁxtures, classroom
upgrades) or educational materials (textbooks, school supplies); variable costs are
recurring monthly expenses on teacher salaries, the largest component of these
expenses, and non-teaching sta↵ salaries, utilities, and rent. Columns 1-4 include
closed schools in the regressions assigning them zero costs once closed; cols 5-6
sum costs over the years; and cols 7-8 restrict the sample to schools that were
open throughout the study period.
To facilitate comparisons, column 1 repeats the regression presented in Table 2,
Column 1. Whereas in the ﬁrst year, H schools spend Rs.34,950 and Lt schools
spend Rs.30,719 more than control schools on ﬁxed costs, by the second year,
there is no detectable di↵erence in ﬁxed costs between the treated and control
schools (col 3). On the other hand, annualized variable costs are higher among H
schools and increase over time, though these estimates are imprecisely measured
at p-values of 0.20 (col 2 and 4). Cumulatively over two years, ﬁxed costs are
higher in all treated schools (col 5 and 7), but variable costs are higher only in H
schools (col 6 and 8). Therefore, if we consider open schools only, we cannot reject
equality of coe cients between the treated groups for ﬁxed costs (col 7), but can
reject equality in variable costs at a p-value of 0.02 (col 8). Since teacher salaries
comprise 75 percent of variable costs, H schools were likely spending more on
teachers after the intervention leading us to further investigate this in Table 7.
Infrastructure
For treated schools, infrastructure constitutes the largest fraction of ﬁxed costs,
and although we cannot reject that the magnitudes are the same, H schools spend
Rs. 6,209 more on average than Lt schools (Table 6, column 1). Table 6 also pro-
vides evidence that spending on infrastructure components di↵ers by treatment
26
saturation. While we cannot reject equality of coe cients for H and Lt compar-
isons, relative to Lt schools, H schools purchase fewer desks and chairs (cols 2
and 3); are more likely to report increased access to computers, library and sports
facilities (cols 4-6); and report a higher number of upgraded classrooms (col 7).35
There are no further e↵ects in year 2 (Appendix Table E9), which is consistent
with most schools choosing to front-load their investments at the beginning of
the school year immediately after they received the grant. If we are willing to
assume that libraries, computers and better classrooms contribute to learning,
these patterns are quite consistent with a focus on capacity expansion (desks
and chairs) among Lt schools and a greater emphasis on quality improvements
among H schools.36 This di↵erential emphasis becomes clearer once we focus on
teachers.
Teachers
Table 7 shows that variable costs increase by Rs.3,145 per month among H
schools, but not in Lt schools, which if anything show a negative coe cient (col-
umn 1). This 12% increase in costs is in large part due to the signiﬁcantly higher
wage bill for teachers in H relative to Lt schools (p = 0.05, column 2). There
is no signiﬁcant average increase in the number of teachers employed at a school
(col 3); however, there is an increase in the number of new teachers in H schools
suggesting the presence of teacher churn (col 4). There are signiﬁcant di↵erences
in remuneration with greater monthly pay for teachers in H schools relative to
control. This pay di↵erential emerges both for newly hired teachers (column 6)
and (to a slightly lesser degree) for existing teachers (column 7). The increase in
teacher wages is consistent with school owners increasing salaries to attract or re-
tain better teachers as previous evidence shows that in Pakistani private schools,
a 1sd increase in teacher value added is associated with 41% higher wages (Bau
and Das, 2016).
C. Discussion
Our results present a consistent narrative in terms of the use of grant funds, the
subsequent impacts, and the channels through which these impacts are realized.
Lt schools invest primarily in increasing capacity with no average changes in test
scores, and, as a result, bring in more children while collecting slightly lower fees
per child. On the other hand, H schools raise test scores and fees, with a smaller
increase in capacity. These di↵erent strategies are reﬂected in schools’ choice of
ﬁxed and variable investments, with H schools more focused on teacher hiring,
remuneration and retention. These results are also consistent with the predictions
of our model. As long as increasing capacity does not impinge on the enrollment
of existing private schools (and it appears not to have done so), Lt schools act as
35 A standard desk accommodates 2 students implying that 12 additional students can be seated in
H schools, and 18 students in Lt school; these numbers are similar in magnitude to the enrollment gains
documented earlier.
36 While additional facilities could justify increasing prices, the per-student availability of desks and
chairs in Lt schools was arguably the same, although there is an increase in the availability of computers.
27
monopolists on the residual demand from other schools. This option is no longer
available when all schools receive the funds, as capacity enhancements among all
schools will trigger a price war. The only option then is to expand the size of the
market through quality investments and this is indeed what we observe in the
data.
Welfare Comparisons.— The di↵erential responses between the low and high-
saturation arms naturally raise the question of whether the public sector has a
role to play in this ﬁnancing model for private schools, which depends on the
computed beneﬁts of the intervention for di↵erent groups. Since estimating de-
mand curves requires household choice data (which we do not have), we use the
experimental estimates together with a linear parameterization of the demand
curve to compute the gains that accrue to schools owners, parents, teachers and
children. Considering child test scores beyond the parental consumer surplus cal-
culation allows us to incorporate the idea that there may be social externalities
from learning gains beyond the direct beneﬁts to parents. The key intuition driv-
ing our comparison is that when quality remains the same, gains in consumer
surplus are concentrated among inframarginal consumers, as the welfare gains
from new, ‘marginal’, enrollees is small given they are indi↵erent between at-
tending the school or not prior to the intervention. However, on the producer
side, gains in ﬁrm proﬁts depend entirely on new enrollment among marginal
consumers. Consequently, when schools expand enrollment without increasing
quality, increases in proﬁts can be substantial even as the change in consumer
surplus is small. When schools improve quality and quantity, consumers accrue
the beneﬁts of higher quality and an implicit decline in price at the higher quality
required to bring in new students.
We start by considering the exact policy analogue to our experiment, where
a government decides to give unconditional grants to private schools but faces a
budget constraint. With a total grant budget of PKR 150K, it can either provide
(i) PKR 50K to one school each in three villages (L treatment), or (ii) PKR
50K to each of the three schools in one village (H treatment). The table below
shows welfare computations for Lt and H schools, giving monetary returns for
the ﬁrst three beneﬁciary groups and test score increases for children; we omit
consideration of Lu schools in these calculations given the lack of any detectable
impacts. The monetary returns are monthly, while the test score increases are
from a snapshot in time 16 months after treatment. We should emphasize that
these calculations, especially for consumer surplus are necessarily speculative and
often require strong assumptions.37 Details of the computations are provided in
37 For consumer surplus computations, we assume that (a) the demand curve can be approximated
as linear and (b) regardless of quality, demand at zero price in the village is ﬁxed at an upper-bound,
which follows in our case from the assumption of ‘closed’ markets. For test score increases, we examine
the overall standard deviation increase from the grant. As Dhaliwal et al. (2013) discuss, this assumes
that gains across students are perfectly substitutable and returns are linear. Finally, we use point
estimates from our experiment regardless of statistical signiﬁcance. We could alternatively only consider
statistically signiﬁcant estimates and assume 0 values for statistically insigniﬁcant coe cients. While
28
Appendix F. For school owners and teachers, the calculations are standard and
In PKR Standard Deviations
Group Owners Teachers Parents Children
Lt 10,918 -2,514 4,080 61.1
H 5,295 8,662 7,560 117.2
are based on the monthly variable proﬁts (the estimated impacts on collected
revenues minus variable costs) and the teacher wage bill, respectively. Turning to
consumer surplus, recall that there is no change in quality for Lt schools, but there
is a decline in collected fees and an increase in enrollment. Following standard
welfare computations, the ﬁrst order gain of these changes are realized among
those already enrolled. In the H arm, since both quality and prices increase, we
compute the consumer surplus increase along the new demand curve at higher
quality. Finally, the last column in the table shows the total increase in test scores
for children in the village.38
These estimates highlight the tension between the two treatment arms. While
the L arm is substantially better in terms of school owner proﬁtability, social
returns (including parents, teachers and children) are likely higher in the H arm.
Viewed as a policy of providing unconditional grants, the H arm o↵ers favorable
(learning) returns relative to other educational RCTs as well.39 If we believe that
educational interventions should primarily focus on learning with limited weight
on school owner proﬁts, the H approach is clearly preferable.
Policy Response.— Thus far, we have evaluated a policy of a grant, but our
estimates of ﬁnancial return suggest that lending should be privately proﬁtable
in both the low or the high-saturation model. Speciﬁcally, our ﬁnancial returns
calculations give an internal rate of return (IRR) of 61-83% for Lt schools and
12%-32% for H schools for 2-year and 5-year scenarios (see Appendix F).40 As
interest rates on loans to this sector range from 15-20%, the IRR almost always
exceeds the market interest rate: Lt schools would be able to pay back a Rs.50,000
doing so does not qualitatively alter our results, we prefer the approach taken.
38 While test score increases for children already in private schools at baseline are captured by our
treatment e↵ects, we also need to account for test score increases that may have been experienced by
newly enrolled children. Since this cannot be identiﬁed from the data, we assume test score gains of
0.33sd for new children, which is the gain for children switching from government to private schools in
Punjab (Andrabi et al., 2017).
39 This represents a gain of 7.8sd for every $100 invested in H and a gain of 4.1sd for Lt schools.
Relative to the literature (JPAL, 2017), these are highly cost-e↵ective interventions— the median test
score gain in the literature is 2.3sd per $100.
40 For the 2-year scenario we use actual returns estimated over the two year period and then assume
no further returns accrue thereafter and any assets accumulated are resold at 50% value. For the 5-year
scenario we assume the revenue impact lasts for 5 years and is zero thereafter and any assets have 0 value
at the end of the period.
29
loan in 1.5 years whereas H schools would take four years. Even though returns in
both treatment arms pass a market interest rate threshold, from the perspective
of an investor, investing using a low-saturation approach is more desirable.
The above calculus suggests that left to the market, a monopolist lender will
favor the L approach as long as the (village level) ﬁxed costs of ﬁnancing are not
too large. If a government or a social planner prefers the H approach instead, we
can ask what level of subsidy would make the private lender indi↵erent between
the two approaches. Our results point towards a loan-loss guarantee for banks,
which would encourage greater market saturation by mitigating the higher default
risk from the H approach (as the rate of school closures is 1% for Lt schools
compared to 8% for the H schools).
We show in Appendix F that a loan loss guarantee of Rs.17,363 over a two
year period for a total loan value of Rs.150K would make banks indi↵erent be-
tween the two approaches.41 To evaluate this policy, we compare the subsidy
to the additional consumer surplus generated from the H approach, which is
Rs.41,760 a year, computed as the di↵erence in consumer surplus between the
two arms ([Rs.7,560-Rs.4,080]*12). Thus, such a policy passes the test required
for a Pigouvian subsidy— households should themselves to be willing to o↵er such
a loan-loss guarantee, with gains for both ﬁrms and households. Interestingly, this
policy also di↵ers somewhat from standard “priority sector” lending policies in
that the subsidy is not based on a sectoral preference per se but rather on the
“density/saturation” of the ﬁnancial o↵ering by a lender.
V. Conclusion
Alleviating ﬁnancial constraints of (private) schools by providing unconditional
grants leads to signiﬁcants gains in enrollment and/or learning. In addition, vary-
ing the design of the ﬁnancial infusion through the degree of market saturation
a↵ects the margins of improvement. Consistent with theory, when all schools in a
given market receive grants, they have a greater incentive to invest in quality to
avoid a price war by competing over the same set of students. Further, and con-
sistent with the emphasis on capacity versus quality, in low-saturation villages,
schools invest in basic infrastructure or on capacity-focused investments, while
schools in high-saturation villages invest in both capacity and quality-focused in-
vestments. Most starkly, these schools invest more in teachers by paying higher
salaries. Alleviating credit constraints for a wider set of market participants thus
“crowds-in” higher quality service provision.
Our estimates suggest that the ﬁnancial returns to investing in the low-cost
(private) educational sector are large and above normal market lending rates,
41 This calculation makes the conservative assumption that schools that shut down will not pay back
any of their loan. In practice, from ongoing work, we note that default in the case of school closure is
never 100%. Moreover, the ﬁrst instance of default, missing a cycle of payment, in this sector typically
occurs about 7 months after loan disbursement, and even then owners often end up partially repaying
the remaining loan amount. Furthermore, even if school owners decide to close the school, they will often
continue to pay back the loan. The risks to the lender are therefore quite minimal.
30
especially in the low-saturation case. This raises questions about why ﬁnancial
players have not entered this sector. We maintain this is yet another market failure
as lenders perceive this market to be risky. These concerns may be legitimate—
after all, even if schools make money, they may choose not to repay their loans.
However, in an ongoing collaboration with a micro-ﬁnance provider where we
extend loans to private schools, our preliminary results show that lending to this
sector is working well with relatively high take-up and very low default rates.
Yet, even when one is able to catalyze the private sector to start lending in
this space, there remains the question of ﬁnancial saturation. Barring cost of
delivery considerations, for a monopolist ﬁnancial intermediary seeking to max-
imize returns the decision is quite straight-forward–— invest in single schools
using the low-saturation approach. Indeed, this approach to venture funding is
what we typically see for larger players in the education sector worldwide, whether
through investments in franchises or in single schools. Surprisingly, our approach,
which selected a school at random led to higher IRR than the typical approach
of picking a franchise or single school. Existing ﬁnancial models can also enable
the emergence of monopolies. Already in our data, we ﬁnd that schools in low-
saturation villages increase revenues only through increases in market share and
although we do not explicitly model this (we do not have an empirical counter-
part as our grant size is small relative to market revenue), it is straightforward to
construct situations where a low-saturation approach wipes out the competition.
In contrast, in the high-saturation villages, while school level ﬁnancial return is
lower, we observe large test score gains across all children enrolled in the village
and, as we suggest above, potentially higher social gains. Thus, a government
seeking to enhance child learning may favor the latter approach because it helps
crowd-in more investments in quality that beneﬁt students. This is not a new
trade-o↵— governments can always alleviate market constraints in a way that
allows select providers to ﬂourish and grow rapidly or in a manner that enhances
rather than curtails competition. Ultimately, this is a judgment call that each
government will need to make and will critically depend on the nature of market
competition, market demand, and the production function facing providers. Our
work emphasizes that the educational marketplace is remarkably similar to other
sectors in this regard, with arguably greater social and long-term consequences.
31
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Table61:6Baseline6Summary6Statistics6
(1) (2) (3) (4) (5) (6) (7) (8)
Standard6
Variable Mean 5th6pctl 25th6pctl Median 75th6pctl 95th6pctl Deviation N
Panel&A:&Village&level&Variables
Number6of6public6schools 2.45 1.0 2.0 2.0 3.0 5.0 1.03 266
Number6of6private6schools 3.33 2.0 2.0 3.0 4.0 7.0 1.65 266
Private6enrollment 523.52 149.0 281.0 415.5 637.0 1,231.0 378.12 266
Panel&B:&&Private&School&level&Variables&
Enrollment 163.6 45.0 88.0 140.0 205.0 353.0 116.0 851
Monthly6fee6(PKR) 238.4 81.3 150.0 201.3 275.0 502.5 166.1 851
Monthly6revenue6(PKR) 40,181.1 4,943.0 13,600.0 26,485.0 44,400.0 117,655.0 54,883.9 850
Monthly6variable6costs6(PKR) 25,387.0 3,900.0 9,400.0 16,200.0 27,200.0 79,000.0 30,961.1 848
Annual6fixed6expenses6(PKR) 78,860.9 0.0 9,700.0 33,000.0 84,000.0 326,000.0 136,928.2 837
School6age6(No6of6years) 8.3 0.0 3.0 7.0 12.0 19.0 6.7 852
Number6of6teachers 8.2 3.0 5.0 7.0 10.0 17.0 4.8 851
Monthly6teacher6salary6(PKR) 2,562.8 1,000.0 1,500.0 2,000.0 2,928.5 5,250.0 3,139.5 768
Number6of6enrolled6children6in6 13.1 1.0 5.0 10.0 18.0 34.5 11.7 420
tested6grade
Number6of6tested6children 11.7 1.0 4.0 9.0 16.0 31.5 10.6 420
Average6test6score O0.21 O1.24 O0.59 O0.22 0.15 0.84 0.64 401
Notes:
a)6This6table6displays6summary6statistics6for6the62666villages6(Panel6A)6and6the68556private6schools6(Panel6B)6in6our6sample.
b)6These6baseline6data6come6from6two6sources:6school6surveys6administered6to6the6full6sample6(8556schools),6and6child6tests
administered6to6half6of6the6sample6(4206schools).6Any6missing6data6are6due6to6school6refusals,6child6absences6or6zero6enrollment
in6the6tested6grades6at666schools.
Table 2—Expenditures and Revenues
Fixed Costs (annual) Overall Posted Revenues (monthly) Overall Collected Revenues (monthly)
(1) (2) (3) (4) (5) (6) (7)
Year 1 Full Top Coded 1% Trim Top 1% Full Top Coded 1% Trim Top 1%
High 34,950.4*** 5,484.4 5,004.5* 4,771.6** 4,400.0 4,642.0* 3,573.4*
(9,915.1) (3,532.4) (2,602.0) (2,203.3) (3,589.0) (2,413.2) (1,933.3)
Low Treated 30,719.2** 10,665.6** 9,327.2** 8,254.0** 7,923.7* 6,991.8** 5,399.5*
(11,883.9) (4,882.8) (3,976.0) (3,711.7) (4,623.2) (3,252.5) (2,896.0)
Low Untreated 5,086.9 -549.8 -684.5 328.7 494.4 430.9 737.6
(10,107.9) (2,750.1) (2,345.6) (1,887.7) (2,560.2) (2,225.9) (1,711.9)
Baseline 0.2*** 1.0*** 1.0*** 0.9*** 0.8*** 0.9*** 0.7***
(0.0) (0.1) (0.1) (0.1) (0.1) (0.1) (0.1)
R-Squared 0.11 0.65 0.65 0.58 0.55 0.62 0.53
Observations 794 2,459 2,459 2,423 3,214 3,214 3,166
# Schools (Rounds) 794 (1) 832 (3) 832 (3) 820 (3) 831 (4) 831 (4) 820 (4)
Mean Depvar 78,860.9 40,181.0 38,654.1 36,199.2 30,865.0 30,208.8 27,653.0
Test pval (H=0) 0.00 0.12 0.06 0.03 0.22 0.06 0.07
Test pval (Lt =0) 0.01 0.03 0.02 0.03 0.09 0.03 0.06
Test pval (Lt =H) 0.73 0.35 0.32 0.37 0.52 0.52 0.55
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table examines annual ﬁxed costs and monthly revenues. The dependent variable in column 1 is annual ﬁxed costs
in year 1, which includes spending on infrastructure and educational supplies. The remaining columns look at overall
monthly revenues pooled across years 1 and 2. Cols 2-4 consider posted revenues, deﬁned as the sum of revenues expected
from each grade based on enrollment and posted fees. Cols 5-7 consider collected revenues, deﬁned as revenues actually
collected from all students at the school. Both revenue measures are coded as 0 once a school closes. Top coding of the
data assigns the value at the 99th percentile to the top 1% of data. Trimming top 1% of data assigns a missing value to
data above the 99th pctl. Both top coding and trimming are applied to each round of data separately.
b) Regressions are weighted to adjust for sampling and tracking where necessary, include strata and round ﬁxed e↵ects,
with standard errors clustered at village level. The number of observations may vary across columns as data are pooled
across rounds and not all outcomes are measured in every round. We thus also report the unique number of schools and
rounds in each regression; any variation in the number of schools arises from attrition or missing values for some
variables. The mean of the dependent variable is its baseline value or the follow-up control mean.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact for high (H=0)
and low treated (Lt =0) schools, or whether we can reject equality of coe cients between high and low treated (Lt =H) schools.
35
Table 3—School Enrollment and Fees (Monthly)
Enrollment (All) Closure Enrollment (Open) Posted Fees Collected Fees
(1) (2) (3) (4) (5) (6) (7) (8) (9)
Year 1 Year 2 Overall Overall Overall Year 1 Year 2 Overall Per Child
High 8.86 9.12 9.01 -0.02 8.95* 17.68** 21.04** 18.83** 29.48
(5.38) (7.99) (6.04) (0.03) (5.10) (7.63) (10.27) (7.88) (20.15)
Low Treated 18.83*** 26.02*** 21.80*** -0.09*** 11.57 1.93 -2.51 0.51 -7.69
(7.00) (10.01) (7.73) (0.03) (7.63) (7.93) (9.43) (7.48) (12.42)
Low Untreated -0.31 1.00 0.31 -0.03 -2.43 0.07 -0.38 -0.00 3.37
(5.09) (7.23) (5.51) (0.03) (5.41) (6.24) (9.13) (6.49) (10.45)
Baseline 0.78*** 0.72*** 0.75*** 0.73*** 0.83*** 0.82*** 0.83*** 0.63***
(0.04) (0.06) (0.05) (0.05) (0.04) (0.04) (0.04) (0.04)
R-Squared 0.69 0.53 0.62 0.05 0.63 0.71 0.73 0.72 0.14
Observations 2,454 1,605 4,059 855 3,599 1,563 749 2,312 2,949
# Schools (Rounds) 827 (3) 826 (2) 836 (5) 855 (1) 742 (5) 796 (2) 749 (1) 800 (3) 782 (4)
Mean Depvar 163.6 163.6 163.6 0.1 171.5 238.1 238.1 238.1 238.1
Test pval (H=0) 0.10 0.25 0.14 0.60 0.08 0.02 0.04 0.02 0.14
Test pval (Lt =0) 0.01 0.01 0.01 0.01 0.13 0.81 0.79 0.95 0.54
t
Test pval (L =H) 0.15 0.10 0.10 0.04 0.72 0.06 0.01 0.02 0.08
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table examines school enrollment and average monthly tuition fees across all grades. Columns 1-3 look at enrollment in
year 1 and 2, and overall across the two years of the study, respectively. Enrollment is 0 once a school closes down. Col 4
examines closure rates two years after treatment. Col 5 repeats col 3 restricting the sample to schools that remain open
throughout the study. Cols 6-8 show e↵ects on monthly tuition fees charged in year 1 and 2 and overall, respectively. Tuition
fees are averaged across all grades taught at the school, and are coded as missing for closed schools. Col 9 shows collected
fees per child, and is constructed by dividing monthly collected revenues by enrollment in each round.
b) Regressions are weighted to adjust for sampling and tracking where necessary and include strata and round ﬁxed e↵ects, with
standard errors clustered at village level. The number of observations may vary across columns as data are pooled across rounds
and not all outcomes are measured in every round. We thus also report the number of schools and round for each regression; any
variation in the number of schools arises from attrition or missing values for some variables. The mean of the dependent
variable is its baseline value or the follow-up control mean.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact for high (H=0) and low
treated (Lt =0) schools, or whether we can reject equality of coe cients between high and low treated (Lt =H) schools.
36
37
Table 4—Test Scores
School level Child level
(1) (2) (3) (4) (5)
Math English Urdu Avg Avg
High 0.16* 0.19** 0.11 0.15* 0.22**
(0.09) (0.09) (0.08) (0.09) (0.09)
Low Treated -0.07 0.08 -0.08 -0.03 0.10
(0.11) (0.11) (0.11) (0.10) (0.10)
Low Untreated 0.03 0.06 0.01 0.03 0.01
(0.08) (0.08) (0.07) (0.07) (0.08)
Baseline 0.27** 0.43*** 0.25** 0.36*** 0.63***
(0.11) (0.08) (0.12) (0.12) (0.05)
R-Squared 0.18 0.14 0.13 0.16 0.21
Observations 725 725 725 725 12,613
# Schools (Rounds) 725 (1) 725 (1) 725 (1) 725 (1) 719 (1)
Mean Depvar -0.21 -0.18 -0.24 -0.21 -0.19
Test pval (H=0) 0.08 0.05 0.18 0.07 0.02
Test pval (Lt =0) 0.50 0.43 0.45 0.79 0.33
Test pval (Lt =H) 0.03 0.33 0.07 0.07 0.24
Notes: * p<0.10, ** p<0.05, *** p<0.01
a) This table examines impacts on school and child level test scores. Columns 1-3
construct school test scores by averaging child scores for a given subject from a
given school; Col 4 shows the average score (across all subjects) for the school.
Col 5 shows the average (across all subjects) score at the child level. We tested
two grades at endline between grades 3-6, and grade 4 at baseline. In columns
1-4, we use all available test scores, and child composition may be di↵erent
between baseline and endline.
b) Regressions are weighted to adjust for sampling and include strata ﬁxed e↵ects,
with standard errors clustered at village level. We include a dummy variable
for the untested sample at baseline across all columns and replace the baseline
score with a constant. Since the choice of the testing sample at baseline was
random, this procedure allows us to control for baseline test scores wherever
available. The number of observations and schools are the same since test scores
are collected once after treatment. The number of schools is lower than the full
sample in round 3 due to attrition (39 schools refused surveying), closure (57
schools closed down), zero enrollment in the tested grades (9 schools), and
missing values for the remaining schools. The mean of the dependent variable is
the test score for those tested at random at baseline.
c) The bottom panel shows p-values from tests that either ask whether we can
reject a zero average impact for high (H=0) and low treated (Lt =0) schools, or
whether we can reject equality of coe cients between high and low treated
(Lt =H) schools.
Table 5—Fixed and Variable Costs (Annual)
Year 1 Year 2 Cumulative Cumulative (Open Only)
(1) (2) (3) (4) (5) (6) (7) (8)
Fixed Variable Fixed Variable Fixed Variable Fixed Variable
High 34,950.4*** 26,108.5 2,560.1 34,961.9 39,202.0*** 72,241.5* 42,570.5*** 103,181.5**
(9,915.1) (20,508.3) (6,868.1) (27,985.1) (10,792.0) (38,049.5) (11,866.0) (40,227.5)
Low Treated 30,719.2** -8,133.1 6,207.0 13,943.1 42,630.4*** 26,609.9 38,353.5** 1,154.6
(11,883.9) (25,486.1) (9,063.6) (20,355.2) (14,199.2) (38,284.8) (15,018.8) (39,812.1)
Low Untreated 5,086.9 1,402.7 4,992.3 2,656.0 10,509.8 34,854.1 9,595.2 33,530.2
(10,107.9) (17,596.0) (7,904.8) (19,907.5) (11,732.7) (33,815.7) (12,814.7) (34,829.3)
Baseline 0.2*** 0.9*** 0.0* 0.9*** 0.2*** 1.1*** 0.2*** 1.1***
(0.0) (0.1) (0.0) (0.1) (0.0) (0.1) (0.0) (0.1)
R-Squared 0.11 0.71 0.05 0.60 0.10 0.56 0.09 0.57
Observations 794 817 768 777 837 842 745 747
# Schools (Rounds) 794 (1) 817 (1) 768 (1) 777 (1) 837 (1) 842 (1) 745 (1) 747 (1)
Mean Depvar 78,860.9 304,644.2 78,860.9 304,644.2 78,860.9 304,644.2 82,453.9 319,550.0
Test pval (H=0) 0.00 0.20 0.71 0.21 0.00 0.06 0.00 0.01
Test pval (Lt =0) 0.01 0.75 0.49 0.49 0.00 0.49 0.01 0.98
Test pval (Lt =H) 0.73 0.23 0.67 0.42 0.81 0.28 0.78 0.02
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table looks at the treatment impact on annualized ﬁxed and variable costs. Annualized ﬁxed costs include spending on
infrastructure or educational materials and supplies; annualized variable costs include recurring expenses— teaching and
non-teaching sta↵ salaries, utilities and rent. Columns 1-2 show these costs for year 1, and cols 3-4 for year 2. Closed schools
are coded as having 0 costs in cols 1-4. Cols 5-6 show cumulative ﬁxed and variable costs across the two years of the study, i.e.
instead of pooling, these columns sum data across rounds. Cols 7-8 repeat cols 5-6 restricting to those schools that remain open
throughout the experiment.
b) Regressions are weighted to adjust for sampling and tracking where necessary and include strata ﬁxed e↵ects, with
standard errors clustered at village level. The number of observations and unique schools are the same since we either show one
round of data (cols 1-4) or show cumulative costs across rounds (cols 5-8). Observations vary across year 1 and 2 due to
attrition and missing values for some schools. The mean of the dependent variable is its baseline value.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact for high (H=0) and
low treated (Lt =0) schools, or whether we can reject equality of coe cients between high and low treated (Lt =H) schools.
38
39
Table 6—School Infrastructure (Year 1)
Spending Number purchased Facility present (Y/N) Other
(1) (2) (3) (4) (5) (6) (7)
Amount Desks Chairs Computers Library Sports # Rooms Upgraded
High 25,460.31*** 5.97*** 3.76*** 0.20*** 0.11*** 0.10** 0.70***
(8,787.82) (1.63) (1.40) (0.05) (0.04) (0.04) (0.26)
Low Treated 19,251.19** 8.71*** 6.13** 0.17*** -0.03 -0.03 0.47
(8,702.52) (2.45) (2.76) (0.06) (0.05) (0.04) (0.40)
Low Untreated -1,702.36 1.31 0.87 0.04 -0.03 0.02 0.16
(8,376.89) (1.40) (1.19) (0.04) (0.04) (0.03) (0.26)
Baseline 0.09*** 0.10* 0.12* 0.26*** 0.32*** 0.23*** 0.71***
(0.03) (0.05) (0.07) (0.04) (0.04) (0.05) (0.06)
R-squared 0.06 0.09 0.08 0.20 0.20 0.11 0.57
Observations 798 810 811 822 822 822 822
# Schools (Rounds) 798 (1) 810 (1) 811 (1) 822 (1) 822 (1) 822 (1) 822 (1)
Mean Depvar 57,258.48 14.59 10.92 0.39 0.35 0.19 6.36
Test pval (H=0) 0.00 0.00 0.01 0.00 0.01 0.02 0.01
Test pval (Lt =0) 0.03 0.00 0.03 0.01 0.58 0.49 0.24
Test pval (Lt =H) 0.50 0.31 0.45 0.60 0.01 0.01 0.59
Notes: * p<0.10, ** p<0.05, *** p<0.01
a) This table examines outcomes relating to school infrastructure using data from round 1. Column 1 is the
annual (ﬁxed) expenditure on infrastructure– e.g. furniture, ﬁxtures, or facilities. Columns 2-3 refer to the
number of desks and chairs purchased; columns 4-6 are dummy variables for the presence of particular school
facilities; and column 7 measures the number of rooms upgraded from temporary to permanent or semi-permanent
classrooms. Closed schools take on a value of 0 in all columns.
b) Regressions are weighted to adjust for sampling and include strata ﬁxed e↵ects, with standard errors
clustered at the village level. The number of observations and unique schools are the same since we use one
round of data. Observations may vary across year 1 and 2 due to attrition and missing values. The mean of the
dependent variable is its baseline value.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact for
high (H=0) and low treated (Lt =0) schools, or whether we can reject equality of coe cients between high and
low treated (Lt =H) schools.
40
Table 7—Teacher Salaries and Composition
School Costs (monthly) Teacher Roster Teacher Salaries (monthly)
(1) (2) (3) (4) (5) (6) (7)
Total Wage Bill Total Num New All New Existing
High 3,147.48* 2,741.83* 0.42 0.46** 519.52** 580.05** 492.01*
(1,894.67) (1,510.50) (0.32) (0.18) (257.94) (265.80) (284.29)
Low Treated -1,127.41 -838.26 0.32 0.27 -175.63 -89.45 -223.10
(1,716.66) (1,520.25) (0.33) (0.24) (273.11) (406.49) (246.45)
Low Untreated -302.25 65.14 0.25 0.25 194.48 89.47 253.39
(1,374.56) (1,106.67) (0.29) (0.18) (202.53) (236.07) (201.69)
Baseline 0.88*** 0.85*** 0.77***
(0.07) (0.08) (0.05)
R-Squared 0.69 0.63 0.50 0.19 0.20 0.23 0.20
Observations 1,470 1,470 1,590 1,645 11,725 3,903 7,818
# Schools (Rounds) 797 (2) 797 (2) 816 (2) 840 (2) 802 (2) 723 (2) 793 (2)
Mean Depvar 25,387.0 19,491.2 6.7 2.0 2,676.6 2,665.5 2,681.9
Test pval (H=0) 0.10 0.07 0.19 0.01 0.05 0.03 0.08
Test pval (Lt =0) 0.51 0.58 0.33 0.25 0.52 0.83 0.37
Test pval (Lt =H) 0.05 0.05 0.78 0.45 0.04 0.13 0.04
Notes: * p<0.10, ** p<0.05, *** p<0.01
a) This table looks at impacts on teacher salaries and composition from the intervention. The dependent
variable in column 1 is monthly variable costs, which includes utilities, rent, teaching and
non-teaching sta↵ salaries, over two years of the experiment. Column 2 shows the impact on the teaching
salary component of variable costs. Data used in the ﬁrst two columns are from school survey data. The
remaining columns use teacher level data from the teacher roster. Columns 3-4 collapse data at the school
level to understand changes in teacher composition; cols 5-7 decompose teacher salaries by employment
status at the school before and after treatment. Whether a teacher is new or existing is determined by
their start date at the school relative to baseline. Closed schools are coded as missing in all columns,
except cols 3-4 where they are coded as 0.
b) Regressions are weighted to adjust for sampling and tracking where necessary and include strata
and round ﬁxed e↵ects, with have standard errors clustered at village level. The number of observations
may vary across columns as data are pooled across rounds and not all outcomes are measured in every
round. We thus also report the unique number of schools and rounds in each regression; any variation in
the number of unique schools arises from attrition or missing values for some variables. The mean of the
dependent variable is the baseline value or the follow-up control mean.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact
for high (H=0) and low treated (Lt =0) schools, or whether we can reject equality of coe cients
between high and low treated (Lt =H) schools.
ONLINE APPENDIX
Upping the Ante: The Equilibrium Eﬀects of Unconditional
Grants to Private Schools
T. Andrabi, J. Das, A.I. Khwaja, S. Ozyurt, and N. Singh
Contents
A Theory 2
A.1 Homogeneous Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
A.2 Generalization of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
B Weighting of average treatment eﬀects with unequal selection probabilities 23
B.1 Saturation Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
B.2 Tracking Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
C Sampling, Surveys and Data 25
D Balance and Attrition 30
D.1 Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
D.2 Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
E Additional Results 37
F Private and Social Returns Calculations 48
F.1 Welfare Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
F.2 IRR and Loan-loss guarantee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1
A Theory
A.1 Homogeneous Demand
Suppose that schools choose x1 , x2 0 and q1 , q2 2 {qH , qL } in the ﬁrst stage and p1 , p2 in
the second stage. Let si be school i’s surplus, that is si = qi pi . Therefore, school i’s proﬁt
function is:
8
M
>
< (pi c)(xi + 2 ) rxi wt + K, if [si > sj ] or [si = sj and qi > qj ]
M
⇧i = ( p i c )( N x j + 2 ) rx i w t + K, if [si < sj ] or [si = sj and qi < qj ]
>
: (pi c) (M/2+xi )T rxi wt + K,
M + xi + xj if si = sj and qi = qj
Deﬁne nH = K r w and nL = K r to be the additional capacity increase that schools can
aﬀord under high and low technologies, respectively. Note that feasibility requires that xi nL
and xi nH whenever qi = qH . One can easily verify that if the schools’ capacity choices x1
and x2 are such that x1 + x2 N , then in the pricing stage, school i picks pi = qi . Let µ be a
¯]. Then for notational simplicity, we use µ
probability density function with support [p, p ˆ(p) for
¯] to denote µ({p}). Before proving the main results, we prove the following result,
any p 2 [p, p
which applies to both low (L) and high-saturation (H ) treatments.
Proposition A. Suppose that the schools’ quality choices are q1 , q2 2 {qH , qL } and capacity
choices are x1 , x2 0 with x1 , x2 N + M2 and x1 + x2 > N . Then in the (second) pricing
stage, there exists no pure strategy equilibrium. However, there exists a mixed strategy equilibrium
1 , µ2 ), where for i = 1, 2, µi is
( µ⇤ ⇤ ⇤
i , qi ], satisfying c < pi < qi , and
(i) a probability density function with support [p⇤ ⇤
(ii) atomless except possibly at qi , that is µ
ˆ⇤i (p) = 0 for all p 2 [pi , qi ).
⇤
(iii) Furthermore, µ
ˆ⇤ µ⇤
1 (p1 )ˆ 2 (p2 ) = 0 for all p1 2 [p1 , q1 ] and p2 2 [p2 , q2 ] satisfying q1
⇤ ⇤ p1 =
q 2 p2 .
Proof of Proposition A. Because no school alone can cover the entire market, i.e., xi < N + M 2 ,
p1 = p2 = c cannot be an equilibrium outcome. Likewise, given that the schools compete in a
Bertrand fashion and total capacity, M + x1 + x2 , is greater than total demand, M + N , showing
that there is no pure strategy equilibrium is straightforward, and left to the readers.
However, by Theorem 5 of Dasgupta and Maskin (1986), the game has a mixed-strategy
equilibrium: The discontinuities in the proﬁt functions ⇧i (p1 , p2 ) are restricted to the price
couples where both schools oﬀer the same surplus, that is {(p1 , p2 ) 2 [c, qH ]2 |q1 p1 = q2 p2 }.
Lowering its price from a position c < q1 p1 = q2 p2 qH , a school discontinuously increases
its proﬁt. Hence, ⇧i (p1 , p2 ) is weakly lower semi-continuous. ⇧i (p1 , p2 ) is also clearly bounded.
Finally, ⇧1 + ⇧2 is upper semi-continuous because discontinuous shifts in students from one
school to another occur where either both schools derive the same proﬁt per student (when
q1 = q2 ) or the total proﬁt stays the same or jumps per student because the higher quality
school steals the student from the low quality school and charges higher price (when q1 6= q2 ).
Thus, by Theorem 5 of Dasgupta and Maskin (1986), the game has a mixed-strategy equilibrium.
Suppose that (µ⇤ 1 , µ2 ) is a mixed-strategy equilibrium of the pricing stage. Let p
⇤ ¯i be the
supremum of the support of µ⇤ i , so p
¯ i = inf { p 2 [ c, qi ] | p 2 supp ( µ ⇤ )}. Likewise, let p⇤ be
i i
the inﬁmum of the support of µ⇤ i . Deﬁne s(pi , qi ) to be the surplus that school i oﬀers, so
s(pi , qi ) = qi pi . We will prove the remaining claims of the proposition through a series of
Lemmata.
1 , q1 ) = s(p2 , q2 ) and pi > c for i = 1, 2.
Lemma A1. s(p⇤ ⇤ ⇤
2
Proof. Note that the claim turns into the condition p⇤ 1 = p2 > c when q1 = q2 . To show
⇤
1 , q1 ) = s(p2 , q2 ), suppose for a contradiction that s(p1 , q1 ) 6= s(p2 , q2 ). Assume, without
s ( p⇤ ⇤ ⇤ ⇤
loss of generality, that s(p1 , q1 ) > s(p2 , q2 ). For any p1
⇤ ⇤ p1 in the support of µ⇤
⇤
1 satisfying
s(p1 , q1 ) s(p1 , q1 ) > s(p2 , q2 ), player 1 can increase its expected proﬁt by deviating to a price
⇤ ⇤
p01 = p1 + ✏ satisfying s(p1 , q1 ) > s(p2 , q2 ). This is true because by slightly increasing its price
0 ⇤
from p1 to p1 school 1 keeps its expected enrollment the same. This opportunity of a proﬁtable
0
deviation contradicts with the optimality of equilibrium. The case for s(p⇤ 1 , q1 ) < s(p2 , q2 ) is
⇤
symmetric. Thus, we must have s(p1 , q1 ) = s(p2 , q2 ).
⇤ ⇤
Showing that p⇤ i > c for i = 1, 2 is straightforward: Suppose for a contradiction that pi = c
for some i, so school i is making zero proﬁt per student it enrolls. However, because no school
can cover the entire market, i.e., xj < M 2 + N , school i can get positive residual demand and
positive proﬁt by picking a price strictly above c, contradicting the optimality of equilibrium.
Deﬁnition 1. Let [ai , bi ) be a non-empty subset of [c, qi ] for i = 1, 2. Then [a1 , b1 ) and [a2 , b2 )
are called surplus-equivalent if s(a1 , q1 ) = s(a2 , q2 ) and s(b1 , q1 ) = s(b2 , q2 ).
Lemma A2. Let [ai , bi ) be a non-empty subset of [c, qi ] for i = 1, 2. If [a1 , b1 ) and [a2 , b2 ) are
1 ([a1 , b1 )) = 0 if and only if µ2 ([a2 , b2 )) = 0.
surplus equivalent, then µ⇤ ⇤
Proof. Take any two such intervals and suppose, without loss of generality, µ⇤ 1 ([a1 , b1 )) = 0.
That is, [a1 , b1 ) is not in the support of µ⇤
1 . Therefore, for any p 2 [ a ,
2 2b ) , player 2’s expected
enrollment does not change by moving to a higher price within this set [a2 , b2 ). However, player
2 receives a higher proﬁt simply because it is charging a higher price per student. Hence,
optimality of equilibrium implies that player 2 should never name a price in the interval [a2 , b2 ),
implying that µ⇤ 2 ([a2 , b2 )) = 0.
Lemma A3. If pi 2 (c, qi ] for i = 1, 2 with s(p1 , q1 ) = s(p2 , q2 ), then µ
ˆ⇤ µ⇤
1 (p1 )ˆ 1 ( p2 ) = 0 .
Proof. Suppose for a contradiction that there exists some p1 and p2 as in the premises of this
claim such that µˆ⇤ µ⇤
1 (p1 )ˆ 1 (p2 ) > 0. Because µ
ˆ⇤1 (p1 ) > 0, player 2 can enjoy the discrete chance of
price-undercutting his opponent. That is, there exists suﬃciently small ✏ > 0 such that player
2 gets strictly higher proﬁt by naming price p2 ✏ rather than price p2 . This contradicts the
optimality of the equilibrium.
Lemma A4. Equilibrium strategies must be atomless except possibly at p ¯i . More formally,
suppose that s(¯ pi , q i ) pj , qj ) where i, j 2 {1, 2} and j 6= i, then for any k 2 {1, 2} and
s(¯
¯j , it must be the case that µ
p 2 [c, qH ], satisfying p 6= p k (p) = 0.
ˆ⇤
Proof. Suppose without loss of generality that k = 1 and suppose for a contradiction that
ˆ⇤
µ ¯j }. Therefore, there must exist suﬃciently small ✏ > 0 and
1 (p) > 0 for some p 2 [c, qH ] \ {p
> 0 such that for all p2 2 I ⌘ [q2 s(p, q1 ), q2 s(p, q1 ) + ✏) player 2 prefers to name a price
p2 instead of p2 and enjoy the discrete chance of price-undercutting his opponent. Therefore,
the optimality of the equilibrium strategies suggests that µ⇤ 2 (I ) = 0. Because the intervals
[p, p + ✏) and I are surplus-equivalent, Lemma A2 implies that we must have µ⇤ 1 ([p, p + ✏)) = 0,
contradicting µ ˆ 1 ( p ) > 0.
⇤
Lemma A5. s(¯ ¯i = qi for i = 1, 2.
p2 , q2 ) = 0, and thus p
p1 , q1 ) = s(¯
Proof. To show s(¯ p2 , q2 ) suppose for a contradiction that s(¯
p1 , q1 ) = s(¯ p1 , q1 ) 6= s(¯p 2 , q2 ) .
Suppose, without loss of generality, that s(¯ p1 , q1 ). Therefore, by Lemma A4 we
p2 , q2 ) > s(¯
have µ⇤2 ([¯ ˜2 )) = 0 where p
p2 , p ˜2 ⌘ q2 s(¯ p1 , q1 ), and by Lemma A2 µ⇤ 1 ([˜ ¯1 )) = 0 where
p1 , p
˜1 ⌘ q1 s(¯
p p2 , q2 )). In fact, there must exist some small ✏ > 0 such that µ1 ([˜ ⇤ p1 ✏ , p¯1 )) = 0.
The last claim is true because player 1 prefers to deviate from any p 2 [˜ ˜1 ] to price p
p1 ✏ , p ¯1 since
the change in proﬁt, ⇧1 (p, p2 ) ⇧1 (¯ p1 , p2 ) is equal to (p c)µ ([p, p
⇤ ˜1 ])x1 (¯ p1 c)(T x2 ) < 0
as ✏ converges zero. Because the sets [¯ p2 ✏ , p ˜2 ) and [˜
p1 ✏ , p ¯1 ) are surplus-equivalent and
3
µ⇤1 ([˜
p1 ✏ , p¯1 )) = 0, Lemma A2 implies that µ⇤ p2 ✏ , p
2 ([¯ ˜2 )) = 0, contradicting that p ¯2 is the
supremum of the support of µ2 . Thus, s(¯
⇤ p1 , q1 ) = s(¯ p2 , q2 ) must hold.
To show that s(¯ pi , qi ) = 0 for i = 1, 2, assume for a contradiction that s(¯ p 2 , q2 ) >
p1 , q1 ) = s(¯
0. By Lemma A3 we know that µ ˆ⇤ p
(¯ )ˆ
µ
1 1 1 2
⇤ (¯
p ) = 0 . Suppose, without loss of generality, that
ˆ1 (¯
µ ⇤ p1 ) = 0. Therefore, player 2 can proﬁtable deviate from price p ¯2 to price q2 : the deviation
does not change player 2’s expected enrollment, but it increases expected proﬁt simply because
player 2 is charging a higher price per student it enrolls. This contradicts with the optimality
of the equilibrium, and so we must have s(¯ pi , qi ) = 0 for i = 1, 2.
Lemma A6. For each i 2 {1, 2}, p i , and there exists no p, p with pi < p < p < qi such
¯i > p⇤ 0 ⇤ 0
i ([p, p ]) = 0.
that µ⇤ 0
Proof. If p i for some i, that is player i is playing a pure strategy, then player j can proﬁtably
¯i = p⇤
deviate from qj by price undercutting its opponent, contradicting the optimality of equilibrium.
Next, suppose for a contradiction that there exists p, p0 with p⇤ i < p < p < qi such that
0
µ⇤i ([p, p ]) = 0. By Lemma A2, there exists pj , pj that are surplus equivalent to p, p , respectively,
0 0 0
and µj ([pj , pj ]) = 0. Then the optimality of equilibrium and Lemma A4 implies that there
⇤ 0
exists some ✏ > 0 such that µ⇤ i ([p ✏, p0 ]) = 0. This is true because instead of picking a price in
[p ✏, p], school i would keep expected enrollment the same and increase its proﬁt by picking a
higher price p0 . Repeating the same arguments will eventually yield the conclusion that we have
µ⇤i ([pi , p ]) = 0, contradicting the assumption that pi is the inﬁmum of the support of µi .
⇤ 0 ⇤ ⇤
For the rest of the proofs, we use ⇧t to denote the proﬁt of a school that picks quality
t 2 {H, L}. Let ⇧DevH denote the deviation proﬁt of a school that deviates from high to low
quality (once the other school’s actions are ﬁxed). Similarly, ⇧Dev
L denotes the deviation proﬁt
of a school that deviates from low to high quality.
Proof of Theorem 1 (Low-Saturation Treatment) . Suppose that (only) school 1 receives
the grant. Because the schools are symmetric, this does not aﬀect our analysis. There are
four exhaustive cases we must consider for the low-saturation treatment and all these cases are
summarized in the following ﬁgure:
N
2K ¯
r as K % k
Case 1 Case 3 & 4
)
N
/2 +
qL )(M
K
r
Case 2
N L
r+
w
(qH
=
Case 3 K
w=
k/r
Case 4
w⇤ K w
Case 1: K N r (or equivalently nL N ): There would be no price competition among the
schools whether school 1 invests in capacity or quality. Therefore, ⇧H = (qH c) M K w
2 + r
and ⇧L = (qL c) M 2 + r . Thus, there is an equilibrium where school 1 invests in quality if
K
and only if ⇧H ⇧L , implying w w⇤ .
4
Case 2: K w N r < K (or equivalently nH N < nL ): If school 1 invests in quality,
then ⇧H = (qH c) M K w
2 + r . But if it invests in capacity, then its optimal choice would be
x1 = N (as we formally prove below) and proﬁt would be ⇧L = (qL c) M 2 +N +K N r.
Claim: If school 1 invests in capacity, then its optimal capacity choice x1 is such that x1 = N .
Proof. Suppose for a contradiction that x1 = N + e where e > 0. In the mixed strategy
equilibrium of the pricing stage, each school i randomly picks a price over the range [p⇤ i , qL ]
h a probability measure µi . School 1’s
with i proﬁt functions are given by ⇧1 (qL , µ2 ) = (qL
ˆ2 (M/2+x1 )(M +N )
µ
c) M + x1 ˆ2 ) 2 + N + K rx1 , where µ
+ (1 µ M
ˆ2 = µ ˆ2 (qL ), and ⇧1 (p⇤
1 , µ2 ) =
h c)(x1 + M/2) + K rx1 . However,i school 2’s proﬁt functions are ⇧2 (qL , µ1 ) = (qL
⇤
( p1
ˆ1 (M/2)(M +N )
µ
c) M + x1 + (1 µ ˆ1 ) M 2 +N x1 , where µ ˆ1 (qL ) and ⇧2 (p⇤
ˆ1 = µ ⇤
2 , µ 1 ) = ( p2 2 ).
c)( M
In equilibrium both schools oﬀer the same surplus, and so p1 = p2 holds. Moreover, be-
⇤ ⇤
cause each school i is indiﬀerent between qL and p⇤ i we must have ⇧1 (qL , µ2 ) = ⇧1 (p1 , µ2 ) and
⇤
2 , µ1 ). We can solve these equalities for µ
⇧2 (qL , µ1 ) = ⇧2 (p⇤ ˆ1 and µ ˆ2 . However, we know that
in equilibrium we must have µ ˆ2 = 0, then it is easy to see that ⇧1 (qL , µ2 ) de-
ˆ2 = 0. If µ
ˆ1 µ
creases with x1 (or e), and thus optimal capacity should be x1 = N . However, µ ˆ1 = 0 yields
ˆ2 = 4(e+N )(
µ e +M +N )
M2
< 0 , contradicting with the optimality of equilibrium because we should
have µˆ2 0. Thus, school 1’s optimal capacity is x1 = N .
Therefore, school 1 selects high quality if and only if ⇧H ⇧L , which implies
w M K
( qL c r ) N + ( qH c) ( qH qL ) + ( qh c r) .
r 2 r
The last condition gives us the line L. Drawing the line L on wN space implies that the
N intercept is greater than K/r and the w intercept is greater than K whenever K < k ¯.
Moreover, when w = w , N takes the value K/r and when w = K , N takes a value which is
⇤
less than K/r because K > k .
M r (qH q L )
Case 3: 2 (q L c ) Nr < K w (or equivalently k/r N < nH )
Claim: If school 1 invests in quality, then its optimal capacity choice x1 is such that x1 = N .
Proof. Suppose for a contradiction that x1 = N + e where e > 0. This time school 1 randomly
picks a price over the range [p⇤ 1 , qH ] with a probability measure µ1 and school 2 randomly
picks a price over the range [p2 , qL ] ⇥
⇤ with a probability measure µ2 . Schools’ ⇤ proﬁt functions
are given by ⇧1 (qH , µ2 ) = (qH c) µ ˆ2 M 2 + x 1 + (1 µ
ˆ 2 ) ( M/ 2 + N ) + K rx1 w and
⇧1 (p1 , µ2 ) = (p1 c)(x1 + 2 ) + K rx1 w for school 1 and ⇧2 (qL , µ1 ) = (qL c)( M
⇤ ⇤ M
2 +N x1 )
and ⇧2 (p⇤ ,
2 1µ ) = ( p ⇤
2 c )( M
2 ) for school 2.
This time equilibrium prices must satisfy qH p⇤ 1 = qL 2 . Solving this equality along
p⇤
with ⇧1 (qL , µ2 ) = ⇧1 (p1 , µ2 ), and ⇧2 (qL , µ1 ) = ⇧2 (p2 , µ1 ) implies that either µ
⇤ ⇤ ˆ2 = 0, and thus
⇧1 (qL , µ2 ) decreases with x1 and the optimal capacity should be x1 = N , or µ ˆ 2 0.
ˆ1 = 0 and µ
q H qL 2(qL c)(e+N )
However, solving for µ ˆ2 implies that µ ˆ 2 = qH c M ( qH c) which is less than zero for all
e > 0 whenever kr N . This contradicts with the optimality of the equilibrium, and thus
school 1’s optimal capacity is x1 = N .
Therefore, school 1’s proﬁt is ⇧H = (qH c)( M 2 + N) + K w N r if it invests in quality
and ⇧L = (qL c)( 2 + N ) + K N r if it invests in capacity. Therefore, investing in quality is
M
optimal if and only if w (qH qL )( M ¯
2 + N ) which holds for all N and w as long as K < k .
5
r (q H q L )
Case 4: N r < M 2 (qL c) (or equivalently N r < k ): In this case, school 1 prefers to select
x1 > N and start a price war. This is true because the proﬁt maximizing capacity (derived
from the proﬁt function ⇧H calculated in the previous case) is greater than N , and so price
competition ensues. Therefore, school 1’s proﬁt function is strictly greater than (qH c)( M 2 +
N ) + K w N r if it invests in quality. However, if school 1 invests in capacity, then as we
proved in the second case school 1 prefers to choose its capacity as N , and thus its proﬁt would
be ⇧L = (qL c)( M 2 + N) + K N r. Therefore, school 1 prefers to invest in quality as long as
the ﬁrst term is greater than or equal to ⇧L , implying that w (qH qL )( M 2 + N ) which is less
than K because K < k . ¯
Proof of Theorem 1 (High-Saturation Treatment) . There are four exhaustive cases we
must consider for the high-saturation treatment and all these cases are summarized in the fol-
lowing ﬁgure:
N
K
(q H + q L 2 c) M (q q )
r
+ 2(q H c L
r)
(q L c r) L
| {z } ¯
as K % k
>2
w4 (Case 4)
Case 1
¯
as K % k
2K w3 (Case 3)
r
Case 2
Nr
+w
Ca=2
N
se K
r+
K 3
2w
r
Case 4
=
H(case 2)
2K
w⇤ K K
(q H + q L 2c ) M (q q )
+ 2(q H c L
w
(q L c r) L r)
| {z }
>1
Case 1: Suppose that 2K N r (or equivalently, 2nL N ): Because the uncovered market
is large, price competition never occurs in this case. Therefore, ⇧H = (qH c)( M 2 + r ) and
K w
⇧L = (qL c)( M 2 + r ). Moreover, ⇧H
K Dev = (q
L 2 + r ) and ⇧L
c)( M K Dev = (q
H 2 + r ).
c)( M K w
To have an equilibrium where one school invests in high quality and the other invests in low
quality, we must have ⇧H ⇧Dev H = ⇧L and ⇧L ⇧Dev L = ⇧H implying that w = w⇤ , which is
less than K because k < K . To have an equilibrium where both schools pick the high quality,
we must have ⇧H ⇧Dev H , implying w w . Hence, there exists an equilibrium where at least
⇤
one school invests in quality if and only if w w⇤ .
Case 2: Suppose that 2K w N r < 2K (or equivalently, nL + nH N < 2nL ): Because
we still gave nH + nH N , there exists an equilibrium where (H, H ) is an equilibrium outcome
for all values of w w⇤ . Now, consider an equilibrium where only one school, say school 1,
invests in high quality, and so (H, L) is the outcome. In this case nL + nH N and no price
competition occurs, so ⇧H = (qH c)( M 2 + r ) and ⇧L = (qL
K w
2 + r ). Moreover,
c)( M K
⇧Dev
L = (qH c)( M 2 + r ) because the other school has picked nH and 2nH < N . However,
K w
if school 1 deviates to low quality and picks quantity higher than nL , price competition ensues.
First we prove that it is not optimal for school 1 to pick a large capacity if it deviates to L.
6
Claim: Consider an equilibrium strategy where both schools invest in capacity only and x2 =
nL . Then school 1’s optimal capacity choice x1 is such that x1 = N nL .
Proof. Suppose for a contradiction that x1 = N nL + e where e > 0. In the mixed strategy
i , qL ] with a probability measure
equilibrium each school i randomly picks a price over the range [p⇤
µi and we have
✓ ◆
ˆ2 (M/2 + x1 )(M + N )
µ M
⇧1 ( q L , µ 2 ) = ( q L c) + (1 ˆ2 )
µ +N x2 +K rx1 (1)
M + x1 + x2 2
and
⇧1 ( p ⇤ ⇤
1 , µ2 ) = ( p1 c)(x1 + M/2) + K rx1 (2)
ˆ2 = µ2 ({qL }). Moreover,
where µ
✓ ◆
ˆ1 (M/2 + x2 )(M + N )
µ M
⇧2 ( q L , µ 1 ) = ( q L c) + (1 ˆ1 )
µ +N x1 +K rx2 (3)
M + x1 + x2 2
and
⇧2 ( p ⇤ ⇤
2 , µ1 ) = ( p2 c)(M/2 + x2 ) + K rx2 (4)
where µ ˆ1 = µ1 ({qL }). In equilibrium we have p⇤ 1 = p2 , ⇧1 (qL , µ2 ) = ⇧1 (p1 , µ2 ), and ⇧2 (qL , µ1 ) =
⇤ ⇤
ˆ2 = 0, then ⇧1 (qL , µ2 ) decreases with x1 , and thus the optimal capacity
⇧2 (p2 , µ1 ). Moreover, if µ
⇤
should be x1 = N x2 . Therefore, we must have µ ˆ1 = 0. Solving for µ ˆ2 0, and then solving
@ ⇧1 (qL , µ2 )/@ e = 0 implies
K N M r + 2K
e= .
r 2 4(qL c)
Because N (2K w)/r, e is less than or equal to K r w M 4(qL c) , which is negative because
r+2K
K < w, contradicting with the initial assumption that e > 0.
Therefore, if school 1 deviates to low quality, then its payoﬀ is ⇧Dev
H = (qL c)( M 2 +N
K
r ) N r . Thus, there is an equilibrium with one school investing in quality and other investing in
capacity if and only if ⇧L ⇧Dev L and ⇧H ⇧Dev H , which implies the following two inequalities:
w w⇤ and
M r ( q H q L ) ( q H + q L 2c ) K N r ( qL c r )
w + .
2(qH c) qH c qH c
The last condition gives us the line H. Drawing the line H on wN space implies that the
N -intercept is greater than 2K/r because qH +qL 2c
qL c r > 2 and the w -intercept is bigger than K
because qH q+ qL 2c
H c
> 1. However, when w = K , H gives the value of M (q H q L ) K ( qL c)
2(qL c r ) + r (qL c r) for
N which is strictly greater than K/r. However, it is less than or greater than 2K/r depending
on whether M r (q H q L )
2(qL c 2r) is greater or less than K/r . That is, for suﬃciently small values of K , H
lies above 2K/r. However, it is easy to verify that H always lies above K/r.
Case 3: Suppose that 2K 2w N r < 2K w (or equivalently, 2nH N < nL + nH ):
Note that for all values of w w⇤ there exists an equilibrium where (H, H ) is an equilibrium
outcome. This is true because ⇧H is the same as the one we calculated in Case 1 in the proof
of Theorem 1 (Low-saturation Treatment) but ⇧Dev H is much less.
If (H, L) is an equilibrium outcome, then the optimal capacity for school 2 is x2 = N x1 .
The reason for this is that if it ever starts a price war (i.e., a mixing equilibrium), then school 2
will only get the residual demand when it picks the price of qL , implying that its payoﬀ will be
a decreasing function of x2 as long as x2 > N x1 . On the other hand, because schools’ proﬁts
increase with their capacity, as long as there is no price competition, the school 1’s optimal
capacity choice will be x1 = nH = K r w . Thus, in an equilibrium where (H, L) is the outcome,
7
the proﬁt functions are ⇧H = (qH c) M K w
2 + r and ⇧L = (qL c) M 2 +N
K w
r +
K w
K r N r . If school 2 deviates to high quality, then its deviation payoﬀ is ⇧L =
Dev
(qH c) 2 + N K r w because 2nH N . Now we prove that it is not optimal for school 1
M
to deviate to L and pick a large capacity that will ensue price competition.
Claim: Consider an equilibrium strategy where both schools invest in capacity only and x2 =
N nH . Then school 1’s optimal capacity choice x1 is such that x1 = nH .
Proof. Suppose for a contradiction that x1 = nH + e where e > 0. In the mixed strategy
equilibrium schools’ proﬁt functions are given by Equations 1-4 of Case 2. Once again, solving
ˆ2 = 0, then
1 = p2 , ⇧1 (qL , µ2 ) = ⇧1 (p1 , µ2 ), and ⇧2 (qL , µ1 ) = ⇧2 (p2 , µ1 ) imply that if µ
p⇤ ⇤ ⇤ ⇤
⇧1 (qL , µ2 ) decreases with x1 , and so the optimal capacity should be x1 = N x2 . Therefore,
we must have µ ˆ1 = 0. Solving for µ ˆ2 0, and then solving @ ⇧1 (qL , µ2 )/@ e = 0 implies
N (qL c r) w(2qL 2c r) K (2qL 2c r) Mr
e= + .
2(qL c) 2r(qL c) r ( ql c ) 4(qL c)
| {z }
e1
N (2qL 2c r) (2qL 2c r )
which is strictly less than zero because e1 2
w
r + 2 ( qL c) and it is less than K
r (q L c )
because we are in the region where w + N r < 2K . However, e < 0 contradicts with our initial
assumption.
Therefore, x1 = nH is the optimal choice for school 1 if it deviates to low quality, and thus
we have ⇧Dev
H = ( qL c ) M2 + r
K w
+ w. To have an equilibrium outcome (H, L) we must have
⇧q ⇧q for each q 2 {H, L}. Equivalently,
Dev
✓ ◆
w M K
( qL c r ) N + ( qH + qL 2c r) ( qH qL ) + 2K
r 2 r
and ✓ ◆
M K w
( qH qL ) + ( qH qL + r ) .
2 r r
It is easy to verify that the ﬁrst inequality holds for all w w⇤ and N 0. The second inequal-
ity implies w (( q H q L )r
qH qL +r) 2 + r ⌘ w which is strictly higher than K whenever K k .
M K 3 ¯
Case 4: Suppose that N r < 2K 2w (or equivalently, N < 2nH ): We will prove, for all
parameters in this range, that there exists an equilibrium where both schools invest in quality
and x1 = x2 = N/2. For this purpose, we ﬁrst show that school 1’s best response is to pick
x1 = N/2 in equilibrium where both schools invest in quality and x2 = N/2. Suppose for
a contradiction that school 1 picks x1 = N/2 + e where e > 0. Then in the mixed strategy
equilibrium of the pricing stage, each school i randomly picks a price over the range [p⇤i , qH ]
with a probability measure µi and the proﬁt functions are given by
" ✓ ◆#
ˆ2 ( M
µ 2 + x1 )(M + N ) M
⇧1 ( q H , µ 2 ) = ( q H c ) ˆ2 )
+ (1 µ + N x2 + K rx1 w
M + x1 + x2 2
where µˆ2 = µ2 ({qH }) and ⇧1 (p⇤ ⇤
1 , µ 2 ) = ( p1 c)(x1 + M/2) +K rx1 w. On the other hand,
" ✓ ◆#
ˆ1 ( M
µ 2 + x2 )(M + N ) M
⇧2 ( q H , µ 1 ) = ( q H c ) + (1 µ ˆ1 ) + N x1 + K rx2 w
M + x1 + x2 2
8
where µ ˆ1 = µ1 ({qH }) and ⇧2 (p⇤ ⇤
2 , µ1 ) = ( p2 c)(x2 + M/2) + K rx2 w.
Once again, solving p1 = p2 , ⇧1 (qH , µ2 ) = ⇧1 (p⇤
⇤ ⇤
1 , µ2 ), and ⇧2 (qH , µ1 ) = ⇧2 (p2 , µ1 ) imply
⇤
that if µ ˆ2 = 0, then ⇧1 (qL , µ2 ) decreases with x1 , and so the optimal capacity should be
x1 = N x2 . Therefore, we must have µ ˆ1 = 0. Solving for µ2 0 yields µ ˆ2 = 4e( (e + M + N )
M + N )2
which is clearly negative for all values of e > 0, yielding the desired contradiction. Therefore,
school 1’s optimal capacity choice is x1 = N x2 = N/2.
In equilibrium with (H, H ) and xi = N/2 for i = 1, 2, proﬁt function is ⇧H = (qH
M +N
c) 2 +K w N 2 . However, if a school deviates to low quality, then its optimal capacity
r
choice would still be N/2 because entering into price war is advantageous for the opponent,
making proﬁt of the deviating school a decreasing function of its own capacity (beyond N/2).
Therefore, ⇧DevH = ( qL c ) M + 2
N
+K 2 . Thus, no deviation implies that w (qH
Nr
qL ) M +N
⌘ w 4 which holds for all w ¯ and N
k 0. That is, for all the parameters of
2
interest, (H, H ) is an equilibrium outcome.
9
A.2 Generalization of the Model
Suppose that each of T students has a taste parameter for quality ✓j that is uniformly distributed
over [0, 1] and rest of the model is exactly the same as before. Therefore, if the schools have
p
quality q and price p, then demand is D(p) = T (1 q ). We adopt the rationing rule of
Kreps and Scheinkman (1983), henceforth KS. In what follows, we ﬁrst characterize the second
stage equilibrium prices (given the schools’ quality and capacity choices), and thus calculate
the schools’ equilibrium payoﬀs as a function of their quality and capacity. We do not need to
characterize equilibrium prices when the schools’ qualities are the same because they are given
by KS. For that reason, we will only provide the equilibrium prices when schools’ qualities are
diﬀerent. After the second stage equilibrium characterization, we prove, for a reasonable set
of parameters, that if the treated school in the L arm invests in quality then at least one of
the schools in the H arm must invest in quality. We prove this result formally for the case
w = K , which signiﬁcantly reduces the number of cases we need to consider. Therefore, even
when the cost of quality investment is very high, quality investment in the H arm is optimal if
it is optimal in the L arm. There is no reason to suspect that our result would be altered if the
cost of quality investment is less than the grant amount, and thus we omit the formal proof for
w < K . To build intuition, consider the following modiﬁcation of the example in the main text
to 10 consumers, A to J , who value low quality in descending order:
Consumers A B C D E F G H I J
Value for low quality 10 9 8 7 6 5 4 3 2 1
where A values low quality at $10 and J at $1. Following KS, the rationing rule allocates
consumers to schools in order of maximal surplus.1 Fix the capacity of the ﬁrst school at 2 and
let the capacity of the second school increase from 1 to 6. As School 2’s capacity increases from
1 to 5, equilibrium prices in the second stage drop from $8 to $4 as summarized in the next
table:2
Capacity of School 2 1 2 3 4 5 6
NE prices 8 7 6 5 4 mixed
The reason for the existence of pure strategy equilibrium prices is provided by Proposition 1 of
KS that the schools’ unique equilibrium price is the market clearing price whenever both schools’
capacity is less than or equal to their Cournot best response capacities.3 But, once school 2’s
capacity increases to 6, there is no pure strategy NE.4 The threat of mixed strategy equilibrium
prices forces schools to not expand their capacities beyond their Cournot optimal capacities.5
Equilibrium Prices when Qualities are the Same
Following this basic intuition, when both schools’ qualities are the same in the ﬁrst stage, we
are in the KS world, where the schools’ optimal capacity choices will be equal to their Cournot
1
Suppose that both schools have a capacity of 2 and school 1 charges $7 and School 2 charges $9. Then, the
rationing rule implies that consumers A and B will choose School 1 since they obtain a higher surplus by doing
so and consumer C is rationed out of the market.
2
For example, the equilibrium price is $8 when School 2 capacity is 1 because if school 1 charges more than
$8, given the rationing rule, A derives maximal surplus from choosing school 2 and School 1’s enrollment declines
to 1. A lower price also decreases proﬁts since additional demand cannot be met through existing capacity.
3
Given that school 1’s capacity is 2, school 2’s Cournot best response capacity is both 4 and 5 (if only integer
values are allowed).
4
Now p = $3 is no longer a NE, since school 2 can increase proﬁts by charging $4 and serving 5 students
rather than charging $3 and enrolling 6 students. But, $4 is not a NE either, since $4 ✏ will allow 6 students
to enroll for a proﬁt just below $4 ⇥ 6 = 24.
5
In our example, suppose now that schools can also oﬀer high quality, which doubles consumer valuation (A
values low quality at $10 but high quality at $20). Now, when School 1 has a capacity of 2 and school 2 has a
capacity of 6, in an equilibrium where school 2 chooses high quality, school 1 charges $3 and caters to consumers
G and H and school 2 charges $9 and caters to consumers A through F .
10
quantity choices in the absence of credit constraint. However, if schools are credit constrained,
then they will choose their capacities according to their capital up to their Cournot capacity.
In the Cournot version of our model, when schools’ quantities are x1 and x2 , the market
price is P (x1 + x2 ) = q (1 x1 + x2 ). Therefore, the best response function for school with no
capacity cost is
B (y ) = arg max{xT P (x + y )}
0 x 1 y
which implies that
1 y
B (y ) = .
2
According to Proposition 1 of KS, if xi B (xj ) for i, j = 1, 2 and i 6= j , then a subgame
equilibrium is for each school to name price P (x1 + x2 ) with probability one. The equilib-
rium revenues are xi P (x1 + x2 ) for school i. However, if xi xj and xi > B (xj ), then
the price equilibrium is randomized (price war) and school i’s expected revenue is R(xj ) =
B (xj )P (B (xj ) + xj ), i.e., school i cannot fully utilize its capacity, and school j ’s proﬁt is some-
x
where between [ xj i
R(xj ), R(xj )].
Equilibrium Prices when Qualities are Diﬀerent
Suppose that one school has quality qH and the other school has quality qL . Let xH and xL
denote these schools’ capacity choices and pH and pL be their prices, where p pH
qL qH . The next
L
ﬁgure summarizes students’ preferences as a function of their taste parameter ✓ 2 [0, 1].
Figure 1: Student’s preferences over the space of taste parameter
students willing to go to High
students willing to go to (any) school
pL pH pH pL
0 qL qH qH qL
1
prefer Low to High prefer High to Low
pH pL
Therefore, demand for the high quality school is DH = 1 qH qL and enrollment is eH =
⇣ ⌘
min xH , 1 p H pL
qH qL . Demand for the low quality school is
(
pH pL pL pH pL
qH q L qL , if xH 1 qH qL
DL =
1 pqL
L
xH , otherwise,
⇣ ⇣ ⌘⌘
and enrollment of the low quality school is eL = min xL , max p H
qH
pL
qL
pL
qL , 1 pL
qL xH .
Best response prices: Next, we ﬁnd the best response functions for the schools given their
ﬁrst stage choices, qH , qL , xH and xL . The high quality school’s proﬁt is pH eH which takes its
maximum value at pH = qH q2 L +p L
. Therefore, the best response price for the high quality
q H qL + p L
school is PH (pL ) = 2 whenever the school’s capacity does not fall short of the demand
at these prices, i.e. pL (qH qL )(2xH 1). Otherwise, i.e. pL > (qH qL )(2xH 1), we have
PH (pL ) = pL + (1 xH )(qH qL ). To sum,
⇢ qH qL +p L
2 , if pL (qH qL )(2xH 1)
PH ( p L ) =
pL + (1 xH )(qH qL ), otherwise.
⇣ the low quality
Now, given xH , xL and pH , we ﬁnd the best response price for ⌘school, pL .
pH pL pH pL pL
We know that if xH 1 qH qL , then the enrollment is eL = min xL , qH qL qL . However,
11
⇣ ⌘
if xH < 1 p H pL
qH q L , then the enrollment is e L = min xL , 1 pL
qL xH . Therefore, the proﬁt
functions are as follows:
pH pL
1) xH 1 qH qL
pH pL pL
(i) If xL < qH q L qL , then eL = xL , and so ⇧L = pL xL .
⇣ ⌘
pH pL pL pH pL pL pH pL pL
(ii) If xL qH q L qL , then eL = qH q L qL , and so ⇧L = pL qH qL qL .
pH pL
2) xH < 1 qH qL
pL
(i) If xL < 1 qL xH , then eL = xL , and so ⇧L = pL xL .
⇣ ⌘
pL pL pL
(ii) If xL 1 qL xH , then eL = 1 qL xH , and so ⇧L = pL 1 qL xH .
Proﬁt maximizing pL ’s yield the following best response function:
8 p H qL
> 2q H , if xH 1 p H pL
qH qL and pH 2x L ( q H qL )
>
>
< pH qL xL qL (qH qL ) , if x 1 p H pL
qH H qH qL and pH > 2x L ( q H qL )
PL ( p H ) = (1 xH )qL
>
>
> 2 , if xH < 1 p H pL
qH qL and xH + 2x L 1
: pH pL
qL (1 xL xH ), if xH < 1 qH qL and xH + 2x L < 1
Finding Optimal Prices: Solving the best response functions simultaneously implies working
out the following eight cases:
Case 1: Consider the parameters satisfying
pL ( q H qL )(2xH 1) (5)
qH q L +p L
so that the best response function for the high quality school is PH (pL ) = 2 . We need
to consider the following four sub-cases:
Case 1.1: Consider the parameters satisfying
pH pL
xH 1 (6)
qH qL
pH 2x L ( q H qL ) (7)
p H qL
so that the best response function for the low quality school is PL (pH ) = 2q H . Solving the best
response functions simultaneously yields
qL ( qH qL )
pL =
4q H q L
2qH (qH qL )
pH =
4q H q L
2 qH qH
Therefore, the inequalities (5) and (6) yield xH 4 qH q L and equation (7) yields xL 4q H qL .
Case 1.2: Consider the parameters satisfying
pH pL
xH 1 (8)
qH qL
pH > 2x L ( q H qL ) (9)
12
p H q L xL q L ( q H q L )
so that the best response function for the low quality school is PL (pH ) = qH .
Solving them simultaneously yields
qL ( qH qL )(1 2xL )
pL =
2qH qL
( qH qL )(qH qL xL )
pH =
2q H q L
Therefore, the inequalities (5) and (8) yield qH qL xL + (2qH qL )xH and equation (9) yields
qH
x L < 4q H qL .
Case 1.3: Consider the parameters satisfying
pH pL
xH < 1 (10)
qH qL
1 x H + 2x L (11)
(1 xH )qL
so that the best response function for the low quality school is PL (pH ) = 2 . Solving
them simultaneously yields
(1 x H ) qL
pL =
2
qH qL qL (1 xH )
pH = +
2 4
The inequality (10) yields xH < 42qqHH qL
3qL and the inequality (5) yields xH
2q H q L
4qH 3qL , which can-
not be satisﬁed simultaneously. Therefore, there cannot exist an equilibrium for the parameter
values satisfying inequalities (5), (10) and (11).
Case 1.4: Consider the parameters satisfying
pH pL
xH < 1 (12)
qH qL
1 > xH + 2x L (13)
so that the best response function for the low quality school is PL (pH ) = qL (1 xH xL ) .
Solving them simultaneously yields
pL = qL (1 xH xL )
qH qL ( x L + x H )
pH =
2
The inequality (12) yields xH < q2 H q L xL
qH qL and the inequality (5) yields xH
q H q L xL
2qH qL , which can-
not be satisﬁed simultaneously. Therefore, there cannot exist an equilibrium for the parameter
values satisfying inequalities (12), (13) and (5).
Case 2: Now, consider the parameters satisfying
pL > ( q H qL )(2xH 1) (14)
so that the best response function for the high quality school is PH (pL ) = pL +(1 xH )(qH qL ) .
We need to consider the following four sub-cases:
13
Case 2.1: Consider the parameters satisfying
pH pL
xH 1 (15)
qH qL
pH 2x L ( q H qL ) (16)
p H qL
so that the best response function for the low quality school is PL (pH ) = 2q H . Solving the best
response functions simultaneously yields
qL ( qH qL )(1 xH )
pL =
2q H q L
2qH (qH qL )(1 xH )
pH =
2q H q L
2q H
Therefore, the inequalities (14), (15), and (16) yield xH < 4q H q L , xH xH , and qH xH +(2qH
qL )xL qH respectively.
Case 2.2: Consider the parameters satisfying
pH pL
xH 1 (17)
qH qL
pH > 2x L ( q H qL ) (18)
p H q L xL q L ( q H q L )
so that the best response function for the low quality school is PL (pH ) = qH .
Solving them simultaneously yields
pL = qL (1 xH xL )
pH = (1 x H ) qH x L qL
Therefore, the inequalities (14), (17), and (18) yield qH > xL qL + xH (2qH qL ) , x H xH , and
qH xH + (2qH qL )xL < qH respectively.
Case 2.3: Consider the parameters satisfying
pH pL
xH < 1 (19)
qH qL
1 x H + 2x L (20)
(1 xH )qL
so that the best response function for the low quality school is PL (pH ) = 2 . Solving
them simultaneously yields
(1 x H ) qL
pL =
2
qL
pH = (1 xH )(qH )
2
The inequality (19) yields xH < xH implying that there cannot exist an equilibrium for the
parameter values satisfying inequalities (14), (19), and (20).
14
Case 2.4: Consider the parameters satisfying
pH pL
xH < 1 (21)
qH qL
1 > xH + 2x L (22)
so that the best response function for the low quality school is PL (pH ) = qL (1 xH xL ) .
Solving them simultaneously yields
pL = qL (1 xH xL )
pH = (1 x H ) qH qL x L
The inequality (21) yields xH < xH implying that there cannot exist an equilibrium for the
parameter values satisfying inequalities (14), (21), and (22).
Summary of the Equilibrium: The equilibrium prices can be summarized in the following
picture where
2q H qH
Region 1: Parameters satisfy xH 4q H q L and xL 4q H q L . Equilibrium prices are
q L (q H q L ) 2 q H (q H q L )
pL = 4 qH q L and pH = 4qH qL . Therefore, enrollment and revenue (per student)
2 (q
4q H H qL )
of the high quality school are eH = 4q2 qH
H qL
and ⇧ H = (4qH qL )2
. Note that this is not
the proﬁt function of the high quality school, and so the cost of choosing capacity xH and
high quality are excluded.
qH
Region 2: Parameters satisfy xL < 4q H q L and qL xL + (2qH qL ) x H qH . Equilibrium
qL (qH qL )(1 2xL ) (qH qL )(qH qL xL )
prices are pL = and pH =
2q H q L 2q H q L . Therefore, enrollment and
q H q L xL
revenue (per student) of the high quality school are eH = and ⇧H = (qH
2q H q L
2
q H q L xL )
qL ) ((2 q H q L )2
.
2q H
Region 3: Parameters satisfy xH < 4 qH q L and qH xH + (2qH qL )xL qH . Equilibrium
qL (qH qL )(1 xH ) 2qH (qH qL )(1 xH )
prices are pL = 2q H q L and pH = 2 qH q L . Therefore, enrollment and
2qH (qH qL )(1 xH )xH
revenue of the high quality school are eH = xH and ⇧H = 2 qH q L . Moreover,
⇣ ⌘
pH pL pL qH qL (qH qL )(1 xH )2
the proﬁt of the low quality school is ⇧L = pL q H q L qL = (2qH qL )2
.
Region 4: Parameters satisfy qH xH + (2qH qL )xL < qH and qL xL + (2qH qL )xH < qH .
Equilibrium prices are pL = qL (1 xH xL ) and pH = (1 xH )qH xL qL . Enrollment
and revenue of the high quality school are eH = xH and ⇧H = xH [(1 xH )qH xL qL ].
Enrollment and revenue of the low quality school are eL = xL and ⇧L = pL xL = qL (1
xH xL ) xL .
15
xH
1
Region 2 Region 1
qH
2q H q L
2q H
4q H q L
1
2 Region 3
Region 4
qH 1 qH 1 qH xL
4q H qL 2 2q H q L qL
The First Stage Equilibrium: Quality and Capacity
Now we consider the ﬁrst stage equilibrium strategies. In the baseline, we still assume that
schools do not have enough capital to adopt high quality, and thus both schools are of low
2 . Therefore, the baseline market
quality. Moreover, the schools’ initial capacity is x1 = x2 = M
price is P (M ) = qL (1 M ). We make the following two assumptions regarding the size of the
covered market, M :
Assumption 1: 2 T M . That is, total private school enrollment is at least 2.
⇣ ⌘
M 1 r
Assumption 2: 2 3 1 qL .
K 2q H
r + 2 4q H qL .
M
Assumption 3: T
If the second assumption does not hold, then the treated school in the L arm would prefer
not to increase its capacity. This assumption implies that schools do not have enough capital to
pick their Cournot optimal capacities at baseline. If the third assumption does not hold, then
the treated school can increase its capacity to the level where it can cover more than half of
the market. We impose these three assumptions simply because parameters that do not satisfy
them seem irrelevant for our sample. We also like to note the following observations that help
us to pin down what the equilibrium prices will be when schools’ quality choices are diﬀerent.
Observation 1: x1 = x2 = M 2 satisfy the constraint qH x1 +x2 (2qH qL ) < qH if assumption
2 holds.
Observation 2: 4q2 qH
H qL
>1 2 qH
2 , and so 2 < 4qH qL .
M
Therefore, the schools would be in Region 4 with their baseline capacities. If school 1
receives a grant and invests in quality and capacity, then the schools either stay in Region 4,
i.e. school 1 picks its quality such that xH , xL satisﬁes the constraints of Region 4, or move to
Region 2. However, the next result shows that schools will always stay in Region 4, both in the
H and L arms, if the schools’ quality choices are diﬀerent.
Lemma 1. Both in low and high saturation treatment, if schools’ quality choices are diﬀerent,
then their equilibrium capacities xL and xH must be such that both qH xH + xL (2qH qL ) < qH
and qL xL + xH (2qH qL ) < qH hold.
Proof. Whether it is the low or high saturation treatment, suppose that school 1 receives the
grant and invests in higher quality while school 2 remains in low quality. We know by assump-
tion 3 that school 1’s ﬁnal capacity will never be above 2qH /(4qH qL ). Therefore, schools’
equilibrium capacities xH and xL will be in Region 4 or in Region 3. Next, we show that school
2 will never pick its capacity high enough to move Region 3 even if it can aﬀord it.
16
School 2’s proﬁt, if it picks x such that x + M
2 and xH remains in Region 4, is
M M
⇧L = T q L ( x + )(1 xH x) + K T rx.
2 2
The ﬁrst order conditions imply that the optimal (additional) capacity is 1 xH2 r/qL M
2 or
less if the grant is not large enough to cover this additional capacity. On the other hand, the
qH (1 xH )
capacity school 2 needs to move to Region 3, xL , must satisfy xL 2qH qL , which is strictly
higher x + 2 . Therefore, given school 1’s choice, school 2’s optimal capacity will be such that
M
schools remain in Region 4.
On the other hand, if school 2 could pick the capacity required to move into Region 3, the
proﬁt maximizing capacity would be qH (1 xH )
2qH qL because school 2’s proﬁt does not depend on its
capacity beyond this level. Therefore, the proﬁt under this capacity level would be
✓ ◆
T qH (1 xH ) qL (qH qL )(1 xH ) M
⇧3 = r Tr .
2q H q L 2qH qL 2
However, if school 2 picks x and remains in Region 4, then its proﬁt would be
✓ ◆2
4 T qL r M
⇧ = 1 xH Tr .
2 qL 2
The diﬀerence yields
T (2qH r + qL2 (1 x H ) qL r ) 2
⇧3 ⇧4 = <0
4qL (2qH qL ) 2
implying that school 2 prefers to choose a lower capacity and remain in Region 4 even if it can
choose a higher capacity.
Theorem 2. If the treated school in the low saturation treatment invests in quality, then there
must exist an equilibrium in the high saturation treatment where at least one school invests in
quality. However, the converse is not always true.
Proof. We prove our claim for w = K .
Low saturation treatment: If school 1 invests in quality its proﬁt is
✓ ◆
H TM M M
⇧Low = 1 qH qL
2 2 2
⇣ ⌘
However, if school 1 invests in capacity, then its optimal capacity choice is xl = 1 2 1
3M
2
r
qL
and proﬁt is
8 h i
> (2 M )2 (2 3M ) r2
< K +T 16 q L 4 r + 4q L , if xl min T K M
r , B( 2 )
L
⇧Low = K M K
if T
K l M
>
:
T qL T r + 2 1 M T r , r < x B( 2 )
T qL B ( M 2 )+ 2
M
1 M B( M 2 ) +K 2 ), if B ( 2 ) < min x , T r
T rB ( M M l K
High saturation treatment with (H, L) Equilibrium: We are trying to create an equilibrium
⇣ only one school
where at least one school invests in high quality. In an equilibrium where ⌘ invests
in quality, the low quality school’s optimal capacity choice is x = 2 1
l 1 3M
2
r
qL and proﬁt
17
is
8 h i
> (2 M )2 (2 3M ) r2
< K + T 16 q L 4 r + 4q L , if xl min K M
T r , B( 2 )
⇧L = K M K
if K l M
> T qL T r M
+ 2 1 Tr M , T r < x B( 2 )
(H,L)
:
T qL B ( 2 ) + M 2 1 B( M 2 ) M +K T rB ( M
2 ) , if B( M l K
2 ) < min x , T r
On the other hand, the high quality school’s equilibrium proﬁt is
✓ ◆
TM M
⇧H(H,L) = 1 qH x L qL
2 2
where 8 M
< 2 + xl , if xl min T K M
r , B( 2 )
M K
xL = 2 +T r, if T
K l M
r < x B( 2 )
: M
2 + B ( 2 ), if B ( 2 ) < min xl , T
M M K
r
Deviation payoﬀs from (H, L): If the low type deviates to high quality, then we are back in
KS world, and thus its (highest) deviation payoﬀ will be
bL TM
⇧ (H,L) = (1 M ) qH .
2
However, if the high quality school deviates to low quality, then we are again in KS world.
Thus,
⇣ given that the⌘ other school’s capacity is xL , deviating school’s optimal capacity is x
b=
1
2 1 M xL qL and optimal proﬁt is
r
8 h i
(1 xL )2 2
>
< K +T 4 qL (1 xL M )
2 r + 4rqL , if x
b min K
T r , B ( xL )
bH
⇧(H,L) = K M
> T qL T r + 2 1 M2 xL T K
r , if T
K
r (2 M )2 (2 3M ) r2
< K +T 16 q L 4 r + 4q L , if xl min T K M
r , B( 2 )
⇧b (H,H ) = K M K
> T qL T r + 2 1 T r M , if T
K l
r < x B( 2 )
M
:
T qL B ( M2 )+ 2
M
1 B( M 2 ) M + K T rB ( M 2 ), if B ( 2 ) < min x , T r
M l K
Note the following:
K M K
Claim 1. If xl < min T r , B( 2 ) , then x
b < min T r , B ( xL ) .
l
Proof. Assume that xl satisﬁes the above inequality. Then xL = M 2 + x , B ( xL ) = B ( 2 )
l M
2,
x
l
and xb = 2 , which is less than T r . Moreover, x
x K
b < B (xL ) because x < B ( 2 ), and thus the
l M
desired result.
K
Claim 2. If Tr < xl B ( M
2 ), then either x
b < min K
T r , B ( xL ) or K
Tr b B ( xL ) .
B (xL ) and B (xL ) < B ( 2 ) < T r , and thus the desired result.
have x M K
K M K
Lemma 1. Suppose that xl min T r , B ( 2 ) and x
b min T r , B (xL ) . If the treated school in
the low saturation treatment invests in quality, then there is an equilibrium in the high saturation
treatment such that at least one school invests in quality.
Proof. For the given parameter values we know that the optimal capacity of the low quality
l
school in low saturation treatment is xl , and thus xL = M 2 + x and x
l b= x 2 . Assume that the
treated school in the low saturation treatment invests in quality. Then we must have
⇧L
⇧H Low
Low
⇥ ⇤ h i
(2 M )2 (2 3M ) r2
or equivalently, T 2
M
1 M 2 q H
M
2 q L K + T 16 q L 4 r + 4qL . We need to
show that either (H, L) or (H, H ) is an equilibrium outcome. Equivalently, we need to prove
that either the inequalities in (1) or (2) below hold:
(1) Both the low and high quality schools do not deviate from (H, L), i.e.,
⇧L
(H,L)
bL
⇧ (H,L) and ⇧H
(H,L)
bH
⇧ (H,L) .
h i
M )2 2
Equivalently, K + T (2 16 qL (2 43M )
r + 4rqL
TM
2 (1 M )qH and
⇥ ⇤ h i
(1 xL )2 2
TM
2 1 M
2 qH x L qL K +T 4 qL (1 xL M )
2 r qL hold.
+ 4r
(2) Alternatively, the schools do not deviate from (H, H ), that is
⇧(H,H ) b (H,H )
⇧
h i
M )2 r2
or equivalently, TM
2 (1 M ) qH K + T (2 16 qL (2 3M )
4 r + 4q L .
Note that if ⇧L bL
(H,L) < ⇧(H,L) , then the inequality in (2) holds, and so we have an equilibrium
where both schools pick high quality. Inversely, if the inequality in (2) does not hold, then
⇧L(H,L)
bL
⇧ (H,L) , i.e., the low quality school does not deviate from (H, L). If we show that the
high quality school also doesn’t deviate from (H, L), then we complete our proof. Because ⇧H Low
⇧LLow , showing ⇧(H,L)
H ⇧H
Low
bH
⇧ (H,L) ⇧LLow would prove that the second inequality in (1) holds
as well. Therefore, we will prove that ⇧H bH T M qL l b H
⇧H ⇧L 2 x + ⇧(H,L) ⇧Low 0.
L
Low (H,L) + ⇧(H,L) Low =
T M qL l b H T qL l r
x + ⇧(H,L) ⇧L
Low = x 2 + 3M + x l
2 4 qL
✓ ◆
T qL l r 3 3M 1 3M r
= x + since xl = 1
4 2 qL 2 2 2 2 qL
T qL l r 1 3M r
x since 1 by Assumption 2
4 2 qL 2 2 qL
< 0.
Thus, either (H, L) or (H, H ) is an equilibrium outcome.
19
K M K
Lemma 2. Suppose that T r < x B ( 2 ) and x
l b min T r , B (xL ) . If the treated school in
low saturation treatment invests in quality, then there is an equilibrium in the high saturation
treatment such that at least one school invests in quality.
Proof. For the given parameter values we know that the optimal capacity of the low quality
school is xl is greater than T
K
R , and thus xL = 2 + T r . Moreover, because x
M K K
b < min T r , B ( xL )
holds, we have x < 2T r . Assume that the treated school in the low saturation treatment invests
l 3K
in quality. Then we must have
⇧HLow ⇧LLow
⇥ ⇤
or equivalently, 2 TM
1 M
2 qH
M
2 qL T qL TK M
r + 2 1 M T K
r . Then we need to
show that either (H, L) or (H, H ) is an equilibrium. Equivalently, we need to show that either
the inequalities in (1) or (2) below hold:
(1) Both the low and high quality schools do not deviate from (H, L), i.e.,
⇧L bL
⇧ and ⇧H bH .
⇧
(H,L) (H,L) (H,L) (H,L)
Equivalently, T qL T
K
r + 2
M
1 Mh T K
r
TM
2 (1 M )qH and i
⇥ ⇤ (1 x ) 2 2
TM
2 1 M 2 qH x L qL K +T 4
L
qL (1 xL M )
2 r + 4r
qL hold.
(2) Alternatively, the schools do not deviate from (H, H ), that is
⇧(H,H ) b (H,H )
⇧
or equivalently, TM
2 (1 M ) qH T qL K
Tr + M
2 1 M K
Tr .
Note that if ⇧L bL
(H,L) < ⇧(H,L) , then the inequality in (2) holds, and so we have an equilibrium
where both schools pick high quality. Inversely, if the inequality in (2) does not hold, then
⇧L b L , i.e., the low quality school does not deviate from (H, L). If we show that the
⇧
(H,L) (H,L)
high quality school also doesn’t deviate from (H, L), then we complete our proof. Because ⇧H Low
bH
⇧LLow , showing ⇧(H,L) Low would prove that the second inequality in (1) holds
H ⇧H ⇧ ⇧L
Low (H,L)
as well. Therefore, we will prove that ⇧H bH KM qL bH
⇧H ⇧L 2r + ⇧(H,L) ⇧Low 0.
L
Low (H,L) + ⇧(H,L) Low =
KM qL bH T 3K 5 K 2 qL
+⇧ (H,L) ⇧L
Low = (2r (2 3 M ) qL ) 2 + (2r (2 3 M ) qL ) + 2
2r 16qL 4r } 4r T
| {z } | {z
= T q L (x l )2 3KqL xl
r
✓ ◆
KqL Tr l 2 5K
= (x ) 3 xl +
r K 4T r
✓ ◆
KqL Tr l 2 5 K
(x ) 3xl + xl since < xl
r K 4 Tr
✓ ◆
KqL Tr l 2 7 l
= (x ) x
r K 4
✓ ◆
KqL 3 7 l 3K
( xl ) 2 x since xl <
r 2 xl 4 2T r
< 0.
Thus, either (H, L) or (H, H ) is an equilibrium outcome.
K M K
Lemma 3. Suppose that T r < x B ( 2 ) and T r < x
l b B (xL ). If the treated school in the
low saturation treatment invests in quality, then there is an equilibrium in the high saturation
treatment such that at least one school invests in quality.
20
Proof. Assume that the treated school in the low saturation treatment invests in quality. Then
we must have
⇧HLow ⇧LLow
⇥ ⇤
or equivalently, 2TM
1 M
2 qH
M
2 qL T qL TK M
r + 2 1 M T K
r . Then we need to
show that either (H, L) or (H, H ) is an equilibrium. Equivalently, we need to show that either
the inequalities in (1) or (2) below hold:
(1) Both the low and high quality schools do not deviate from (H, L), i.e.,
⇧L
(H,L)
bL
⇧ (H,L) and ⇧H
(H,L)
bH
⇧ (H,L) .
⇥ ⇤
Equivalently, T qL K
Tr + M
2 1 M T K
r
TM
2 (1 M )qH and TM
2 1 M
2 qH xL qL
K M K
T qL T r + 2 1 M xL Tr hold.
(2) Alternatively, the schools do not deviate from (H, H ), that is
⇧(H,H ) b (H,H )
⇧
or equivalently, TM
2 (1 M ) qH T qL K
Tr + M
2 1 M K
Tr .
Same as before if we show that the high quality school doesn’t deviate from (H, L), i.e.,
bH KM qL bH
⇧H
Low ⇧H
(H,L) + ⇧(H,L) ⇧LLow = 2r +⇧ (H,L) ⇧LLow 0, then we complete our proof.
✓ ◆✓ ◆
KM qL bH KM qL K M K
+⇧ (H,L) ⇧L
Low = + T qL +
2r 2r Tr 2 Tr
✓ ◆
KqL M K M
=
r 2 Tr 2
< 0.
Thus, either (H, L) or (H, H ) is an equilibrium outcome.
Lemma 4. Suppose that B ( M K
2 ) < min T r , x
l and B (x ) < min K , x
L T r b . If the treated school
in the low saturation treatment invests in quality, then there is an equilibrium in the high satu-
ration treatment such that at least one school invests in quality.
Proof. For the given parameter values B ( M
2 )= 2
1
4 , xL = 2 + B ( 2 ), and B (xL ) = 2 B ( 2 ).
M M M 1 M
⇥ saturation
Assume that the treated school in the low treatment
⇤ invests in quality. Then we must
have ⇧Low ⇧Low or equivalently, 2
H L TM
1 M
2 qH
M
2 qL T qL B ( M M
2 )+ 2 1 M B( M 2 ) +
K T rB ( 2 ). Then we need to show that either (H, L) or (H, H ) is an equilibrium. Equivalently,
M
we need to show that either the inequalities in (1) or (2) below hold:
(1) Both the low and high quality schools do not deviate from (H, L), i.e., ⇧L bL
⇧
(H,L) (H,L)
and ⇧H ⇧b H . Equivalently, T qL B ( M ) + M 1 M B ( M ) + K T rB ( M )
(H,L) (H,L) 2 2 2 2
TM
(1
2 ⇥ M ) q H and ⇤
TM
2 1 M 2 qH x L qL T qL B ( x L ) + M
2 (1 M xL B (xL )) + K T rB (xL ) hold.
b (H,H ) or equiva-
(2) Alternatively, the schools do not deviate from (H, H ), that is ⇧(H,H ) ⇧
lently, 2 (1 M )qH T qL B ( 2 ) + 2 1 M B ( 2 ) + K T rB ( M
TM M M M
2 ).
Same as before if we show that the high quality school doesn’t deviate from (H, L), i.e.,
21
⇧H ⇧H bH
(H,L) + ⇧(H,L) ⇧L
Low =
T M qL M bH Low 0, then we complete our proof.
⇧L
Low 2 B( 2 ) +⇧ (H,L)
" #
T M qL M bH T M qL B ( M ) T rB ( M ) T qL B ( M ) M B( M )
B( ) + ⇧ (H,L) ⇧L
Low =
2
+ 2
+ 2
+ 2
1
2 2 2 2 2 2 2
✓ ◆
T B( M2
) 11M 3
= r + qL
2 8 4
✓ ◆
M 1 r
< 0 since < 1 by Assumption 2.
2 3 qL
Thus, either (H, L) or (H, H ) is an equilibrium outcome.
Finally, the converse of the claim is not necessarily true because ⇧H
Low ⇧H bH
(H,L) + ⇧(H,L) ⇧L
Low
is strictly negative. That is, there are many parameters in which at least one school invests in
quality in the high saturation treatment, but the treated school invests only in capacity in the
low saturation treatment.
22
B Weighting of average treatment eﬀects with unequal selection
probabilities
B.1 Saturation Weights
Our experimental design is a two-stage randomization. First, villages are assigned to one of three
groups: Pure Control; High-saturation, H ; and Low-saturation, L; based on power calculations,
3
7 of the villages are assigned to the L arm, and 7 each to the H arm and the control group.
2
Second, in the L arm, one school in each village is further randomly selected to receive a grant
oﬀer; meanwhile, all schools in H and no school in control villages receive grant oﬀers. This
design is slightly diﬀerent from randomization saturation designs that have been recently used
to measure spillover eﬀects (see Crépon et al., 2013; Baird et al., 2016) since the proportion
of schools that receive grant oﬀers is not randomly assigned within L villages. Instead, since
we are interested in examining what happens when a single school is made the grant oﬀer, the
proportion of schools within L villages assigned to treatment depends on village size at the time
of treatment; this changes the probability of selection into treatment for all schools in these
villages. For instance, if a L village has 2 schools, then probability of treatment is 0.5 for a given
school, whereas if the village has 5 schools, the selection probability reduces to 0.20.
While this consideration does not aﬀect the estimates for the H arm, the impact for schools
in the L arm need to adjust for this diﬀerential selection probability. This can be done fairly
simply by constructing appropriate weights for schools in the L villages. Not doing so would
overweight treated schools in small villages and untreated schools in large villages. Following
the terminology in Baird et al. (2016), we refer to the weights given below as saturation weights,
sg where g represents the treatment group:
• shigh = scontrol = 1
• slowtreated = B , where B is the number of private schools in the village
B
• slowuntreated = B 1
To see why weighting is necessary, consider this example. Assume we are interested in the
following unweighted simple diﬀerence regression: Yij = ↵ + Tij + ✏ij , where i indexes a school
in village j ; Tij is a treatment indicator that takes value 1 for a treated school in L villages
and 0 for all control schools. That is, we are only interested in the diﬀerence in outcomes
between low-treated and control schools. Without weighting, our treatment eﬀect is the usual
= [E (T T 0 )] 1 E (T Y ).
If instead we were to account for the diﬀerential probability of selection of the low-treated
schools, we would weight these observations by B and control observations by 1. This weighting
transforms the simple diﬀerence regression as follows: Y ˜
ij = ↵˜ + 0T ˜ ij , and our 0 =
ij + ✏˜
p
˜ ˜
[E (T T )] E (T Y ), where T and Y are obtained by multiplying through by Bj where Bj is
0 1 ˜ ˜ ˜ ˜
the weight assigned to the low-treated observation based on village size. Note that the bias from
not weighting is therefore more severe as village size increases. However, since our empirical
village size distribution is quite tight (varying only between 1 and 9 private schools), in practice,
weighting does not make much of a diﬀerence to our results.
While we must account for weights to address the endogenous sampling at the school level
in the low-saturation treatment, we do not need weights to account for the unequal probability
of village level assignment in the ﬁrst stage since this assignment is independent of village
characteristics. Nevertheless, if we were to do so, our results are nearly identical. The weights
in this case would be as follows:
7
• shigh = scontrol = 2
• slowtreated = 7
3B
7 B
• slowuntreated = 3B 1
23
B.2 Tracking Weights
In addition to the saturation weights, tracking weights are required to account for the random-
ized intensive tracking procedure used in round 5. These weights are only used for regressions
containing data from round 5; regressions using data from rounds 1-4 only require saturation
weights. We implemented this randomized tracking procedure in order to address attrition con-
cerns, which we expected to be more severe two years after treatment. We describe below the
details of the procedure and specify the tracking weights for round 5 data.
In round 5, 60 schools do not complete surveys despite being operational. We randomly
select half of these schools to be intensively tracked, i.e. our enumerators make multiple visits to
these schools to track down the respondent, and, if necessary, survey the respondents over the
phone or at non-school premises. These eﬀorts increase our round 5 survey completion rate from
88 to 94 percent. To account for the additional attention received by this tracked subsample, we
assign a weight of 2 if the school was selected to be part of the intensively tracked subsample,
and 0 if it was not.
24
C Sampling, Surveys and Data
Sampling Frame
Villages: Our sampling frame includes any village with at least two non-public schools, i.e.
private or NGO, in rural areas of Faisalabad district in the Punjab province. The data come
from the National Education Census (NEC) 2005 and are veriﬁed and updated during ﬁeld visits
in 2012. There are 334 eligible villages in Faisalabad, comprising 42 percent of all villages in
the district; 266 villages are chosen from this eligible set to be part of the study based on power
calculations.
Schools: Our intervention focuses on the impact of untied funding to non-public schools. The
underlying assumption here is that a school owner or manager has discretion over spending in
their own school. If instead the school is part of a network of schools and is centrally managed,
as is the case for certain NGO schools in the area, then it is often unclear how money is allocated
across schools in the network. Therefore, we decided to exclude schools in our sample where we
could not obtain guarantees from oﬃcials that the money would be spent only on the randomly
selected schools. In practice, this was a minor concern since it only excluded 5 schools (less than
1 percent of non-public schools) across all 266 villages from participation in the study. The ﬁnal
set of eligible schools for participation in the study was 880.
Study Sample
All eligible schools that consented to participate across the 266 villages are included in the ﬁnal
randomization sample for the study. This includes 822 private and 33 NGO schools, for a total
of 855 schools; there were 25 eligible schools (about 3 percent) that refused to participate in
either the ballot or the surveys. The reasons for refusals were: impending school closure, lack
of trust, survey burden, etc. Note that while the ballot randomization included all 855 schools,
the ﬁnal analysis sample has 852 schools (unbeknownst to us 1 school had closed down by the
time of the ballot and the other 2 were actually refusals that were mis-recorded by ﬁeld staﬀ ).
Appendix Figure C1 summarizes the number of villages and schools in each experimental group.
Power Calculations
We use longitudinal LEAPS data for power calculations and were able to compare power un-
der various randomization designs. Given high auto-correlation in school revenues, we chose a
stratiﬁed randomization design, which lowers the likelihood of imbalance across treatment arms
and increases precision since experimental groups are more comparable within strata than across
strata (Bruhn and McKenzie, 2009). The sample size was chosen so that the experiment had 90
percent power to detect a 20 percent increase in revenue for H schools, and 78 percent power
for the same percentage increase in revenue for Lt schools (both at 5% signiﬁcance level).
Survey Instruments
We use data from a range of surveys over the project period. We outline the content and the
respondents of the diﬀerent surveys below. For the exact timing of the surveys, please refer to
Appendix Figure C2.
Village Listing: This survey collects identifying data such as school names and contact num-
bers for all public and private schools in our sampling frame.
25
School Survey Long: This survey is administered twice, once at baseline in summer 2012
and again after treatment in the ﬁrst follow-up round in May 2013. It contains two modules:
the ﬁrst module collects detailed information on school characteristics, operations and priorities;
and the second module collects household and ﬁnancial information from school owners. The
preferred respondent for the ﬁrst module is the operational head of the school, i.e. the individual
managing day-to-day operations; if this individual was absent the day of the survey, either the
school owner, the principal or the head teacher could complete the survey. For the second
module, the preferred respondent was either the legal owner or the ﬁnancial decision-maker of
the school. In practice, the positions of operational head or school owner are often ﬁlled by the
same individual.
School Survey Short: This survey is administered quarterly between October 2013 and
December 2014, for a total of four rounds of data. Unlike the long school survey, this survey
focuses on our key outcome variables: enrollment, fees, revenues and costs. The preferred
respondent is the operational head of the school, followed by the school owner or the head
teacher. Please consult Appendix Figure C3 to see the availability of outcomes across rounds.
Child Tests and Questionnaire: We test and collect data from children in our sample
schools twice, once at baseline and once after treatment in follow-up round 3. Tests in Urdu,
English and Mathematics are administered in both rounds; these tests were previously used and
validated for the LEAPS project (Andrabi et al., 2002). Baseline child tests are only administered
to a randomly selected half of the sample (426 schools) in November 2012. Testing is completed
in 408 schools for over 5000 children, primarily in grade 4.6 If a school had zero enrollment in
grade 4 however, then the preference ordering of grades to test was grade 3, 5, and then 6.7 A
follow-up round of testing was conducted for the full sample in January 2014. We tested two
grades between 3 and 6 at each school to ensure that zero enrollment in any one grade still
provided us with some test scores from every school. From a roster of 20,201 enrolled children
in this round, we tested 18,376 children (the rest were absent). For children tested at baseline,
we test them again in whichever grade they are in as long as they were enrolled at the same
school. We also test any new children that join the baseline test cohort. In the follow-up round,
children also complete a short survey, which collects family and household information (assets,
parental education, etc.), information on study habits, and self-reports on school enrollment.
Teacher Rosters: This survey collects teacher roster information from all teachers at a school.
Data include variables such as teacher qualiﬁcations, salary, residence, tenure at school and in
the profession. It was administered thrice during the project period, bundled with other surveys.
The ﬁrst collection was combined with baseline child testing in November 2012, and hence data
was collected from only half of the sample. Two follow-up rounds with the full sample took place
in May 2013 (round 1) and November 2014 (round 5).
Investment Plans: These data are collected once from the treatment schools as part of the
disbursement activities during September-December 2012. The plans required school owners
to write down their planned investments and the expected increase in revenues from these
investments— whether through increases in enrollment or fees. School owners also submitted a
desired disbursement schedule for the funds based on the timing of their investments.
6
The remaining schools had either closed down (2), refused surveying (10) or had zero enrollment in the
tested grades at the time of surveying (6). The number of enrolled children is 5611, of which 5018 children are
tested; the remaining 11% are absent.
7
97 percent of schools (394/408) had positive enrollment in grade 4.
26
Data Deﬁnitions
The table below lists, deﬁnes and provides the data source for key variables in our empirical
analysis. Group A are variables measured at the village level; Group B at the school level; and
Group C at the teacher level.
Variable Description Survey
Source
Group A: Village Level
Grant per capita Grant amount per private school going child in School
treatment villages. For L villages, this is Rs
50,000/total private enrollment, and for H villages,
this equals (50,000*# of private schools in vil-
lage)/total private enrollment. Control schools are
assigned a value of 0.
Group B: School Level
Closure An indicator variable taking the value ‘1’ if a school School
closed during the study period
Refusal An indicator variable taking the value ‘1’ if a school
refused a given survey
Enrollment School enrollment in all grades, veriﬁed through School
school registers. Coded as 0 after school closure.
Fees Monthly tuition fees charged by the school averaged School
across all grades.
Posted Revenues Sum of revenues across all grades obtained by mul- School
tiplying enrollment in each grade by the monthly
fee charged for that grade. Coded as 0 after school
closure.
Collected Rev- Self-reported measure on total monthly fee collec- School
enues tions from all enrolled students. Coded as 0 after
school closure unless otherwise speciﬁed.
Test Scores Child test scores in English, Math and Urdu, are av- Child
eraged across enrolled children to generate school- tests
level test scores in these subjects. Tests are graded
using item response theory (IRT), which appropri-
ately adjusts for the diﬃculty of each question and
allows for comparison across years in standard de-
viation units.
Stayer A stayer is a child who self-reports being at the Child
same school for at least 18 months in round 3. survey
Fixed Costs Sum of spending on infrastructure (construc- School
tion/rental of a new building, additional classroom,
furniture and ﬁxtures), educational materials, and
other miscellaneous items in a given year. Data is
collected at the item level, e.g. furniture, equip-
ment, textbooks etc. Coded as 0 after school clo-
sure.
Continued on next page
27
Continued from previous page
Variable Description Survey
Source
Variable Costs Sum of spending on teacher salaries, non-teaching School
staﬀ salaries, rent and utilities for a given month.
Coded as 0 after school closure.
Sources of school Indicator variables for whether school items were School
funding (Y/N) purchased through (i) self-ﬁnancing- school fees or
owner’s own household income, or (ii) credit- loans
from a bank or MFI
Household bor- Indicator variables for borrowing behavior of the School
rowing (Y/N) school owner’s household: whether household ever owner
borrowed from any sources, formal sources (e.g.
bank, MFI) and informal (e.g. family, friend, pawn-
shop, moneylender) sources.
Household bor- Value of total borrowing in PKR by the owner School
rowing: Loan household from any source for any purpose. owner
value
Group C: Teacher Level
Teacher salaries Monthly salary collected for each teacher present Teacher
during survey. roster
Teacher start date YYYY-MM at which the teacher started her tenure Teacher
at the school. This allows us to tag a teacher as roster
a newly arrived or an existing teacher relative to
treatment date.
28
Appendix(Figure(C1:(Sample(Details
Sample Control( HighDsaturaEon ( LowDsaturaEon (
776villages,666 756villages,666666 1146villages,666666
2496schools6 2286schools6 3786schools6
Treatment+Offers
06schools6 2286schools6 1146schools6
Take0up 06schools6 2136schools6 1096schools6
Appendix(Figure(C2:(Project(Timeline
2012 2013 2014
Round 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Baseline6Survey
Baseline6Child6Testing6
Randomization6Ballot
Disbursements
Round61
Round62
Round636
Round64
Round656
Notes:6Rounds61T36correspond6to6the6first6school6year6and6rounds646and656refer6to6the6second6school6year6after6treatment.6A6school6
year6in6this6region6is6typically6from6AprilTMarch,6with6a6three6month6break6for6summer6between6JuneTAugust.6
Appendix(Figure(C3:(Data(Availability(by(Survey(Rounds
Outcome Baseline Round61 Round62 Round63 Round64 Round65
Enrollment ✓ ✓ ✓ ✓ ✓ ✓
Fees ✓ ✓ ✓ ✓ ✓ ✓
Posted6Revenues ✓ ✓ ✓ ✓
Collected6Revenues ✓ ✓ ✓ ✓
Expenditures6 ✓ ✓ ✓
Test6Scores* ✓ ✓
Teacher6variables* ✓ ✓ ✓
Notes:6This6table6shows6data6availablity6in6each6round6for6key6outcomes.6Different6modules
are6administered6in6different6rounds6based6on6cost6and6attrition6concerns.6Variables6with6a
star6marking6are6only6collected6for6half6of6the6sample6at6baseline.6See6Appendix6C6for6details.
D Balance and Attrition
In this section, we discuss and address issues of experimental balance and attrition in detail.
D.1 Balance
As noted in the main text of the paper, our randomization is always balanced in distributional
tests across the village and school level. While there is no mean imbalance at the village level
in univariate comparisons, we do detect mean imbalance in a few comparisons between the Lt
schools and schools in H and control. This imbalance is primarily driven by the skewness (heavy
right tail) of some of our covariates. To see this, recall that our randomization is stratiﬁed by
village size and average revenue and takes place in two stages, ﬁrst at the village level and then
at the school level. While stratiﬁcation helps in reducing the ex-ante probability of imbalance
at the village level, it does not automatically guarantee the same for school level regressions.
Instead, the source of imbalance for the Lt group is related to distributional skewness and the
sample sizes we realize as a result of our design. Because only 1 school in a low-saturation
village is oﬀered a grant, there are 114 Lt schools in comparison with 228 H and 249 control
schools. The smaller sample size for the Lt group increases the likelihood that the distributional
overlap for a given covariate between the Lt group and the H or control group may have uneven
mass, especially in the tails of the distribution. It is therefore reassuring that though we may
have mean imbalance in comparisons with the Lt group, the Kolmogorov-Smirnov (K-S) tests
in Appendix Table D1 show that we cannot reject that the covariate distributions are the same
for comparisons between Lt and other groups. Nevertheless, we conduct two types of additional
analyses, presented below, to address any concerns arising from the detected imbalance.
First, we conduct simulations to see whether we still observe mean covariate imbalance when
we randomly select data from 1 school in the control or H arm to compare with our Lt sample.
The thought experiment here is as follows: Assume we only had money to survey 1 school in each
experimental group, but the treatment condition remained the same (i.e. all schools are treated
in H ; 1 school in L; and no schools in control). Our school level balance regressions would now
only use data from the surveyed schools. Since these sample sizes are more comparable, the
likelihood of imbalance is now lower. Indeed, when we run 1000 simulations of this procedure,
we ﬁnd no imbalance on average using this approach between either Lt and control, or Lt and
H schools. This approach can also be applied to estimate our treatment eﬀects, and we ﬁnd
that our key results are quite similar in magnitudes though we lose some precision due to the
smaller sample sizes. This exercise lends support to the idea that the mean imbalance at the
school level does not reﬂect a randomization failure but rather issues of covariate overlap in
group distributions.
Second, we assess the robustness of our results by trimming the right tails, top 2%, of the
imbalanced variables and re-running the balance and main outcome regressions. The previous
analysis provides justiﬁcation for undertaking these approaches as a way to understand our
treatment eﬀects. Appendix Table D2 shows our balance regressions with trimmed baseline
variables. There is no average imbalance for enrollment or fees in comparisons between Lt versus
control; we observe some imbalance at the 10% level for H vs Lt schools for fees. However,
observing 3 out of 32 imbalanced tests at the 10% level may occur by random chance. Our
outcome regressions using trimmed baseline data in Appendix Tables D3 are also nearly identical
to the tables in the main text. Together, these tests reveal that the limited imbalance we detect
does not pose any noteworthy concerns for our results.
D.2 Attrition
Even though we have high survey completion rates throughout the study, we do observe some
diﬀerential response rates between the Lt and control schools (see Appendix Table D4). It is not
30
surprising that treated schools, especially in the L arm, may be more willing to participate in
surveys given the sizable nature of the cash grant they received. Here, we check robustness of our
results to this (small) diﬀerential attrition using predicted attrition weights. The procedure is as
follows: We calculate the probability of refusal (in any follow-up round) given treatment variables
and a set of covariates using a probit model, and use the predicted values to construct weights.8
The weight is the inverse probability of response (1 prob(attrition)) 1 , and is simply multiplied
to the existing saturation weight. This procedure gives greater weight to those observations that
are more likely to refuse in a subsequent round.
Appendix Table D5 shows our key regressions using attrition weights. Given the low levels
of attrition, our results, unsurprisingly, are similar in magnitudes and signiﬁcance to tables in
the main text.
8
The probit model reveals that only our treatment variable has any predictive power for attrition.
31
Table8D1:8Randomization8Balance
(1) (2) (3) (4) (5) (6) (7) (8) (9)
Panel&A:&Village&level&variables
Control8 Tests8of8difference K-S8Test8p-values
N Mean H-C=0 L-C=0 H-L=0 H=C L=C H=L
Number8of8public8schools 266 2.5 0.011 0.010 0.001 0.95 1.00 1.00
[0.95] [0.95] [0.99]
Number8of8private8schools 266 3.3 0.021 0.162 -0.141 1.00 1.00 0.99
[0.85] [0.16] [0.18]
Private8enrollment 266 523.5 -23.549 11.202 -34.750 0.28 0.86 0.30
[0.51] [0.71] [0.29]
Average8monthly8fee8(PKR) 266 232.1 12.668 -12.855 25.523 0.46 0.85 0.57
[0.41] [0.20] [0.07]
Average8test8score 133 -0.222 -0.013 0.031 -0.044 0.27 0.51 0.35
[0.88] [0.75] [0.57]
Overall8Effect:8p-value 0.95 0.96 0.99
Panel&B:&Private&school&level&variables
Tests8of8difference K-S8Test8p-values
Control8
N Mean H-C=0 8Lt8-C8=0 8Lu8-C8=0 8H-Lt8=0 H=C 8Lt=C H=Lt
Enrollment 851 163.6 -3.9 -18.9 0.9 15.0 0.18 0.69 0.90
[0.66] [0.07] [0.91] [0.17]
Monthly8fee88(PKR) 851 238.1 24.1 -32.3 -10.7 56.4 0.94 0.42 0.24
[0.15] [0.02] [0.35] [0.00]
Annual8expenses88(PKR) 837 78860.9 21,559.2 -16,659.5 -5,747.2 38,218.7 0.58 0.88 0.57
[0.13] [0.15] [0.60] [0.01]
Monthly8expenses8(PKR) 848 25387.0 2,692.7 -2,373.7 2,280.1 5,066.3 0.81 0.82 0.94
[0.32] [0.43] [0.28] [0.16]
Infrastucture8index8(PCA) 835 -0.008 0.073 0.308 -0.074 -0.235 0.22 0.40 0.27
[0.64] [0.17] [0.56] [0.33]
School8age8(in8years) 852 8.3 0.028 0.296 0.220 -0.268 0.98 0.73 0.61
[0.96] [0.69] [0.70] [0.72]
Number8of8teachers 851 8.2 0.015 -0.408 0.242 0.423 1.00 0.95 0.81
[0.97] [0.39] [0.48] [0.37]
Average8test8score 401 -0.210 -0.054 0.160 -0.052 -0.214 0.55 0.39 0.11
[0.53] [0.18] [0.61] [0.05]
Overall8Effect:8p-value 0.83 0.28 0.24 0.33
Notes:8*8p<0.1,8**8p<0.05,8***8p<0.01
a)8This8table8shows8randomization8checks8at8the8village8and88private8school8level,8Panel8A8and8B8respectively,8for8key8variables8in8our
study.8Across8both8panels,8column818shows8number8of8observations8and8col828shows8the8control8mean.8Panel8A,8cols83-58and8Panel8B,
3-68show8tests8of8differences--8regression8coefficients8and8p-values8in8square8brackets--88between8experimental8groups.8Panel8A,
cols86-8,8and8Panel8B,8cols87-98show8p-values8from8Kolmogorov-Smirnov8(K-S)8tests8of8equality8of8distributions.8In8the8bottom8row,
we8report8p-value8from8a8test8asking8whether8variables8jointly8predict8treatment8status8for8each8group.
b)88All8regressions8include8strata8fixed8effects.8Panel8A8regressions8have8robust8standard8errors.8Panel8B8regressions8are8weighted8to
adjust8for8sampling8and8have8clustered8errors8at8the8village8level.
c)8All8variables8are8defined8in8Appendix8C.8There8are8fewer8observations8for8test8scores8since8half8of8the8sample8was8tested8at8baseline.
Table.D2:.Randomization.Balance,.Trimmed.Sample
(1) (2) (3) (4) (5) (6)
Control. Tests.of.difference
Private(school(level(variables N Mean H*C=0 .Lt.*C.=0 .Lu.*C.=0 .H*Lt.=0
Enrollment 836 154.1 *5.7 *13.8 *2.0 8.1
[0.39] [0.14] [0.77] [0.35]
Monthly.fee..(PKR) 834 221.6 2.5 *20.3 *8.4 22.8
[0.81] [0.13] [0.38] [0.07]
Annual.expenses..(PKR) 821 65441.7 5,875.8 *5,477.6 *4,902.8 11,353.3
[0.53] [0.60] [0.57] [0.32]
Monthly.expenses.(PKR) 832 22293.5 1,061.4 *2,774.9 2,720.2 3,836.4
[0.49] [0.14] [0.10] [0.05]
Infrastucture.index.(PCA) 819 *0.141 0.077 0.133 *0.012 *0.056
[0.41] [0.31] [0.88] [0.69]
School.age.(No.of.years) 836 7.9 *0.191 0.615 0.171 *0.806
. . [0.69] [0.40] [0.74] [0.25]
Number.of.teachers 834 7.7 *0.045 *0.290 0.316 0.245
[0.88] [0.44] [0.31] [0.47]
Average.test.score 393 *0.242 *0.020 0.074 *0.029 *0.095
[0.81] [0.48] [0.75] [0.34]
Overall.Effect:.p*value 0.85 0.47 0.94 0.38
Notes:.*.p<0.1,.**.p<0.05,.***.p<0.01
a).This.table.reproduces.Table.D1,.Panel.B,.using.trimmed.data.to.assess.whether.mean.imbalance.in.Table.D1,.Panel.B,.is.
driven.by.large.values.in.the.right.tails..The.trimming.procedure.makes.the.top.2%.of.baseline.values.missing.for.each.variable.
Column.1.shows.the.number.of.observations,.and.col.2.shows.the.control.mean..The.remaining.columns.show.tests.of.
difference.**.regression.coefficients.and.p*values.in.square.brackets**.between.groups..In.the.bottom.row,.we.report.p*values
from.a.test.asking.whether.variables.jointly.predict.treatment.status.for.each.group.
b)..Regressions.are.weighted.to.adjust.for.sampling.and.include.strata.fixed.effects,.with.clustered.standard.errors.at.the.village.level.
c).All.variables.are.defined.in.Appendix.C..There.are.fewer.observations.for.test.scores.since.half.of.the.sample.was.tested.at
baseline.
Table D3: Main Outcomes, Trimmed Sample
(1) (2) (3)
Enrollment Fees Score
High 10.50* 13.20* 0.154*
(5.73) (7.20) (0.08)
Low Treated 24.01*** -1.49 0.005
(7.39) (7.42) (0.10)
Low Untreated -2.16 -1.58 0.033
(5.44) (6.10) (0.07)
Baseline 0.78*** 0.75*** 0.473***
(0.04) (0.04) (0.09)
R-Squared 0.52 0.58 0.19
Observations 3985 2272 720
# Schools (Rounds) 821 (5) 786 (3) 720 (1)
Mean Depvar 154.13 221.58 -0.24
Test pval (H=0) 0.07 0.07 0.07
Test pval (Lt = 0) 0.00 0.84 0.96
Test pval (Lt = H) 0.07 0.06 0.13
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table reproduces our results using baseline variables
trimmed at the top 2% as controls; the trimming procedure drops
the top 2% of the baseline measure of the dependent variable
from the regression. Columns 1-3 show impacts on enrollment,
fees and test-scores.
b) Regressions are weighted to adjust for sampling and tracking
as necessary and include strata and round ﬁxed eﬀects, with
clustered standard errors at the village level. The number of
observations may vary across columns as data are pooled across
rounds and not all outcomes are measured in every round. We
thus also report the number of schools and round for each
regression; any variation in the number of schools arises from
attrition or missing values for some variables.
c) The bottom panel shows p-values from tests that either ask
whether we can reject a zero average impact for high (H=0) and
low treated (Lt =0) schools, or whether we can reject
equality of coeﬃcients between high and low treated (Lt =H)
schools.
34
Table4D4:4Differential4Attrition
(1) (2) (3) (4) (5)
Control High Low4Treated Low4Untreated N
Panel&A:&Differential&Survey&Attrition&
Round41 0.059 @0.032 @0.044** @0.035* 824
(0.02) (0.02) (0.02)
Round42 0.052 @0.028 @0.045** @0.031 806
(0.02) (0.02) (0.02)
Round43 0.087 @0.063*** @0.079*** @0.038 798
(0.02) (0.02) (0.02)
Round44 0.054 @0.030 @0.054*** @0.029 781
(0.02) (0.02) (0.02)
Round45 0.126 @0.084*** @0.106*** @0.030 758
(0.02) (0.02) (0.03)
Always4refused 0.033 @0.007 @0.033** @0.025* 758
(0.02) (0.01) (0.01)
Panel&B:&Differential&Baseline&Characteristics&for&Attriters&(At&least&once&refused)&by&Treatment&Status&
Enrollment 191.4 8.4 6.4 @33.0* 79
(44.68) (28.77) (18.74)
Monthly4fee4(PKR) 257.5 @28.5 @47.5 37.2 79
(60.78) (42.46) (50.90)
Annual4fixed4expenses4(PKR) 103745.0 55017.7 20106.0 @49684.0 77
(90071.94) (26347.19) (39480.86)
Monthly4variable4costs4(PKR) 31768.8 7830.1 44448.2 @4501.2 79
(19060.95) (31225.62) (9184.26)
Infrastructure4index 0.062 0.536 1.140 @0.192 78
(0.39) (0.74) (0.36)
School4age4(No4of4years) 8.8 6.3* @3.47 0.59 79
(3.64) (2.79) (2.62)
Number4of4teachers 9.7 1.01 @0.61 @0.81 79
(2.59) (0.94) (0.79)
Notes:4*4p<0.1,4**4p<0.05,4***4p<0.01
a)4This4table4examines4differential4attrition,4defined4as4refusal4to4participate4in4follow@up4surveying,4across4experimental4groups,4and4
assesses4whether4attriters4have4systematically4different4baseline4characteristics4across4groups.4Panel4A4tests4for4differential4attrition
in4each4follow@up4round4(1@5)4and4across4all4rounds.4Only4144schools4refuse4surveying4in4every4follow@up4round.4Panel4B4restricts4to
attriters4to4look4for4any4differences4in4baseline4characteristics4by4treatment.4Since4doing4this4exercise4on4144schools4would4not4be
informative,4we4conservatively4define4an4attriter4to4be4any4school4that4refuses4surveying4at4least4once4after4treatment4(794schools).
b)4All4regressions4include4strata4fixed4effects4and4are4weighted4to4adjust4for4sampling,4with4clustered4standard4errors4at4the4village
level.4The4number4of4observations4in4Panel4A4is4declining4over4time4because4closed4schools4are4coded4as4missing4in4these4regressions.
Table D5: Main Outcomes, using Attrition-predicted Weights
(1) (2) (3)
Enrollment Fees Score
High 8.71 25.69*** 0.17*
(5.55) (7.88) (0.09)
Low Treated 16.73** 5.47 -0.04
(7.19) (7.86) (0.11)
Low Untreated 0.91 6.30 0.06
(5.27) (6.40) (0.07)
Baseline 0.77*** 0.82*** 0.37***
(0.04) (0.04) (0.11)
R-Squared 0.62 0.71 0.16
Observations 3878 2230 706
# Schools (Rounds) 797 (5) 769 (3) 706 (1)
Mean Depvar 163.64 238.13 -0.21
Test pval (H=0) 0.12 0.00 0.05
Test pval (Lt = 0) 0.02 0.49 0.72
Test pval (Lt = H) 0.24 0.01 0.05
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table checks whether our results are robust to accounting
for diﬀerential attrition using the inverse probability weighting
technique. In addition to using saturation or tracking weights, we
now weight all regressions with attrition-predicted weights. This
procedure is described in detail in Appendix D. Cols 1-3 show
impacts on enrollment, fees, and test scores with these weights.
b) Regressions are weighted to adjust for sampling, tracking
where necessary, and attrition, and include strata and round ﬁxed
eﬀects, with standard errors clustered at the village level.
The number of observations may vary across columns as data are
pooled across rounds and not all outcomes are measured in every
round. We thus also report the number of schools and rounds for
each regression; any variation in the number of schools arises
from attrition or missing values for some variables.
c) The bottom panel shows p-values from tests that either ask
whether we can reject a zero average impact for high (H=0) and low
treated (Lt =0) schools, or whether we can reject equality of
coeﬃcients between high and low treated (Lt =H) schools.
36
E Additional Results
This section includes additional tables referenced in the main text.
37
Table E1: Credit Behavior (Year 1)
School funding sources (Y/N) HH borrowing (Y/N) HH loan value
(1) (2) (3) (4) (5) (6)
Self-ﬁnanced Credit Any Formal Informal Any
High -0.007 0.002 -0.010 0.020 -0.033 1,063.0
(0.01) (0.01) (0.05) (0.02) (0.05) (15,092.8)
Low Treated -0.0004 -0.006 -0.039 0.010 -0.053 17,384.2
(0.01) (0.01) (0.05) (0.02) (0.05) (29,982.8)
Low Untreated -0.002 -0.011 -0.005 0.035* -0.055 13,611.9
(0.01) (0.01) (0.04) (0.02) (0.04) (21,581.8)
Baseline 0.078 -0.017 0.080** 0.208*** 0.003 0.064*
(0.09) (0.01) (0.04) (0.05) (0.04) (0.03)
R-Squared 0.03 0.02 0.04 0.14 0.02 0.03
Observations 795 795 784 784 784 784
# Schools (Rounds) 795 (1) 795 (1) 784 (1) 784 (1) 784 (1) 784 (1)
Mean Depvar 0.99 0.02 0.23 0.02 0.21 44,782.7
Test pval (H=0) 0.48 0.88 0.83 0.23 0.47 0.94
Test pval (Lt = 0) 0.97 0.68 0.45 0.64 0.27 0.56
Test pval (Lt =H) 0.53 0.56 0.60 0.65 0.69 0.60
Notes: * p<0.10, ** p<0.05, *** p<0.001
a) This table looks at credit behavior of school owners in year 1 to understand whether the treatment
simply acted as a substitute for other types of credit. Data for columns 1-2 are from the school survey
and from the school owner survey for cols 3-6. The dependent variables in col 1-2 are indicators for
whether a school reports ﬁnancing school expenditures through fees or owner income or through a formal
or informal ﬁnancial institution, respectively. Col 3 reports whether the household of the school owner
has ever borrowed any money for any reason. Cols 4-5 disaggregate this household borrowing into formal
and informal sources. Col 6 examines total borrowing by the owner’s household for any reason. If the
owner household did not borrow, the loan value is coded as 0. Schools that closed or refused surveying
are coded as missing for credit behavior.
b) Regressions are weighted to adjust for sampling and include strata and round ﬁxed eﬀects, with
standard errors clustered at the village level. The number of observations and unique schools are the
same since we use one round of data. Observations may vary across columns due to attrition and missing
values. The mean of the dependent variable is the follow-up control mean.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact
for high (H=0) and low treated (Lt =0) schools, or whether we can reject equality of coeﬃcients
between high and low treated (Lt =H) schools.
38
Table E2: Enrollment by Grades
(1) (2) (3) (4) (5)
Lower than 1 1 to 3 4 to 5 6 to 8 9 to 12
High 3.11 2.49 1.57 1.82 1.36
(2.15) (2.05) (1.11) (1.55) (1.15)
Low Treated 6.51** 8.81*** 2.85** 4.33** 3.73
(2.52) (2.57) (1.27) (2.04) (2.45)
Low Untreated 1.31 1.78 1.32 0.63 -1.29
(1.95) (1.83) (1.06) (1.48) (1.29)
Baseline 0.59*** 0.73*** 0.70*** 0.62*** 0.78***
(0.06) (0.05) (0.03) (0.04) (0.10)
R-Squared 0.38 0.54 0.59 0.57 0.65
Observations 3,334 3,420 3,420 3,420 3,420
# Schools (Rounds) 852 (4) 855 (4) 855 (4) 855 (4) 855 (4)
Mean Depvar 49.89 53.68 28.15 23.10 8.22
Test pval (H=0) 0.15 0.22 0.16 0.24 0.24
Test pval (Lt = 0) 0.01 0.00 0.03 0.03 0.13
Test pval (Lt =H) 0.17 0.01 0.28 0.20 0.39
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table disaggregates school enrollment into grade bins to examine the
source of enrollment gains over the two years of the study. Data from rounds
1-4 are used since grade-wise enrollment was not collected in round 5. All
grades in closed schools are assigned 0 enrollment.
b) Regressions are weighted to adjust for sampling and include strata and round
ﬁxed eﬀects, with standard errors clustered at village level. We report the
number of observations and the unique number of schools and rounds in each
regression; the number of unique schools may be fewer than the full sample due
to attrition or missing values for some variables. The mean of the dependent
variable is its baseline value.
c) The bottom panel shows p-values from tests that either ask whether we can
reject a zero average impact for high (H=0) and low treated (Lt =0) schools, or
whether we can reject equality of coeﬃcients between high and low treated (Lt =H)
schools.
39
Table E3: Enrollment Decomposition Using Year 1 Child Data
(1) (2)
Enrollment % New
High 0.348 0.025*
(0.702) (0.015)
Low Treated 0.776 0.056**
(0.740) (0.024)
Low Untreated -0.382 0.024
(0.706) (0.017)
Baseline 0.641***
(0.048)
R-Squared 0.61 0.04
Observations 765 711
# Schools (Rounds) 765 (1) 711 (1)
Mean Depvar 14.69 0.07
Test pval (H=0) 0.62 0.10
Test pval (Lt = 0) 0.30 0.02
Test pval (Lt =H) 0.56 0.21
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table examines changes in child enrollment status.
The dependent variables are from tested children in round 3.
Col 1 is the number of children enrolled in grade 4, and
col 2 is the fraction of those children who newly enroll in
the school after treatment. Enrollment status is determined
based on child self-reports; any child who reports joining
the school fewer than 18 months before are considered new.
b) Regressions are weighted to adjust for sampling and
include strata and round ﬁxed eﬀects, with standard errors
clustered at village level. The number of observations and
schools is the same in this table since we survey children
just once. Observations may be lower than the full sample
due to missing values for some variables. The mean of the
dependent variable is its baseline value or the follow-up
control mean.
c) The bottom panel shows p-values from tests that either
ask whether we can reject a zero average impact for high
(H=0) and low treated (Lt =0) schools, or whether we can
reject equality of coeﬃcients between high and low treated
(Lt =H) schools.
40
Table E4: Monthly Tuition Fees by Grades
(1) (2) (3) (4) (5)
Lower than 1 1 to 3 4 to 5 6 to 8 9 to 12
High 14.43 21.22* 19.38 36.87** 142.64**
(10.49) (12.12) (12.54) (17.75) (66.98)
Low Treated -4.85 -3.22 -8.05 -18.75 88.64
(5.39) (6.39) (8.04) (12.58) (78.69)
Low Untreated 2.33 4.23 -1.06 -2.44 -68.85
(4.59) (6.21) (6.54) (11.24) (54.93)
Baseline 0.83*** 0.75*** 0.79*** 0.67*** 0.47***
(0.05) (0.05) (0.04) (0.06) (0.13)
R-Squared 0.64 0.60 0.59 0.57 0.48
Observations 2,277 2,278 2,240 1,485 360
# Schools (Rounds) 789 (3) 789 (3) 773 (3) 542 (3) 144 (3)
Mean Depvar 169.89 207.82 237.43 319.88 425.94
Test pval (H=0) 0.17 0.08 0.12 0.04 0.04
Test pval (Lt = 0) 0.37 0.61 0.32 0.14 0.26
Test pval (Lt =H) 0.08 0.05 0.04 0.00 0.53
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table averages monthly tuition fees by grade bins to assess whether
fee changes occur in speciﬁc grades. Fees for closed schools or schools
that do not oﬀer certain grade levels are coded as missing.
b) Regressions are weighted to adjust for sampling and include strata and
round ﬁxed eﬀects, with standard errors clustered at village level. We
report the number of observations and the unique number of schools and
rounds in each regression. Higher grades have fewer school observations
because fewer schools oﬀer those grade levels and hence post tuition
fees. These observations are subsequently coded as missing. In contrast,
in Table E2, enrollment in higher grades is coded as 0 if a school does
not oﬀer those grades. The pattern of results in Table E2 stay the same
if we restrict its sample to the sample in this table. The mean of the
dependent variable in all regressions is its baseline value.
c) The bottom panel shows p-values from tests that either ask whether we
can reject a zero average impact for high (H=0) and low treated (Lt =0)
schools, or whether we can reject equality of coeﬃcients between high and
low treated (Lt =H) schools.
41
Table E5: School Test Scores, Diﬀerent Controls
No controls Additional controls
(1) (2) (3) (4) (5) (6) (7) (8)
Math Eng Urdu Avg Math Eng Urdu Avg
High 0.155 0.181* 0.115 0.150 0.157* 0.185* 0.108 0.151*
(0.105) (0.102) (0.092) (0.096) (0.093) (0.094) (0.088) (0.088)
Low Treated -0.066 0.108 -0.059 -0.006 -0.0832 0.069 -0.087 -0.038
(0.122) (0.114) (0.114) (0.111) (0.106) (0.104) (0.102) (0.0981)
Low Untreated 0.021 0.055 0.007 0.028 0.005 0.046 -0.024 0.007
(0.091) (0.091) (0.081) (0.083) (0.078) (0.082) (0.077) (0.074)
Baseline 0.373*** 0.457*** 0.312*** 0.433***
(0.077) (0.064) (0.01) (0.086)
R-Squared 0.08 0.06 0.08 0.08 0.27 0.20 0.21 0.24
Observations 732 732 732 732 722 722 722 722
# Schools (Rounds) 732 (1) 732 (1) 732 (1) 732 (1) 722 (1) 722 (1) 722 (1) 722 (1)
Mean Depvar -0.21 -0.18 -0.24 -0.21 -0.21 -0.18 -0.24 -0.21
Test pval (H=0) 0.14 0.08 0.21 0.12 0.09 0.05 0.22 0.08
Test pval (Lt = 0) 0.59 0.34 0.60 0.96 0.43 0.51 0.40 0.70
Test pval (Lt =H) 0.07 0.52 0.13 0.16 0.02 0.27 0.05 0.05
Notes: * p<0.10, ** p<0.05, *** p<0.001
a) This table conducts robustness checks on our school test score results. School test scores are
generated by averaging child average (across all subjects) test scores for a given school. Columns
1-4 are the same regressions as Table 4, Columns 1-4, but without any baseline controls. Columns
5-8 repeat these regressions with additional controls, which include the baseline score, percentage
of students in speciﬁc grades and percentage female. Test scores are averaged across all children
in a given school separately for each round, and child composition is hence diﬀerent across rounds.
b) Regressions are weighted to adjust for sampling and include strata ﬁxed eﬀects, with standard
errors clustered at village level. We include a dummy variable for the untested sample at baseline
across all columns and replace the baseline score with a constant. Since the testing sample was
chosen randomly at baseline, this procedure allows us to control for baseline test scores wherever
available. The number of observations and the unique number of schools are the same since test
scores are only collected once after treatment. The number of schools is lower than the full sample
due to attrition and zero enrollment in some schools in the tested grades. The mean of the dependent
variable is the test score for those schools tested at random at baseline.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average
impact for high (H=0) and low treated (Lt =0) schools, or whether we can reject equality of
coeﬃcients between high and low treated (Lt =H) schools.
42
Table E6: Test Scores, Stayers Only
School level Child level
(1) (2) (3) (4) (5)
Math Eng Urdu Avg Avg
High 0.150 0.191* 0.120 0.132* 0.235**
(0.093) (0.098) (0.085) (0.077) (0.094)
Low Treated -0.114 0.054 -0.090 -0.034 0.095
(0.115) (0.111) (0.111) (0.089) (0.108)
Low Untreated 0.031 0.055 0.015 0.016 0.002
(0.077) (0.084) (0.071) (0.063) (0.083)
Baseline Score 0.279** 0.429*** 0.365*** 0.337*** 0.637***
(0.135) (0.118) (0.109) (0.098) (0.049)
R-Squared 0.17 0.13 0.15 0.17 0.21
Observations 720 720 720 720 11,676
# Schools (Rounds) 720 (1) 720 (1) 720 (1) 720 (1) 711 (1)
Mean Depvar -0.21 -0.21 -0.21 -0.21 -0.18
Test pval (H=0) 0.11 0.05 0.16 0.09 0.01
Test pval (Lt = 0) 0.32 0.62 0.42 0.71 0.38
Test pval (Lt =H) 0.02 0.21 0.06 0.06 0.19
Notes: * p<0.10, ** p<0.05, *** p<0.001
a) This table examines whether our school test score results are driven by compositional
changes. As before, school test scores are generated by averaging child average (across
all subjects) test scores for a given school. We repeat all of the regressions in Table
4, but only include all children who report being at the same school for at least 1.5
years.
b) Regressions are weighted to adjust for sampling and include strata ﬁxed eﬀects,
with standard errors clustered at village level. We include a dummy variable for the
untested sample at baseline across all columns and replace the baseline score with a
constant. Since the testing sample was chosen randomly at baseline, this procedure allows
us to control for baseline test scores wherever available. The number of observations and
the unique number of schools are the same since test scores are only collected once after
treatment. The number of schools is lower than the full sample due to attrition and zero
enrollment in some schools in the tested grades. The mean of the dependent variable is
the test score for those tested at random at baseline.
c) The bottom panel shows p-values from tests that either ask whether we can reject a
zero average impact for high (H=0) and low treated (Lt =0) schools, or whether we can
reject equality of coeﬃcients between high and low treated (Lt =H) schools.
43
Table E7: Main Outcomes, Interacted with Baseline Availability of Bank Account
(1) (2) (3)
Enrollment Fees Score
High 6.93 18.28* 0.118
(7.36) (10.09) (0.10)
Low Treated 21.85** -1.76 0.021
(10.35) (10.14) (0.13)
Low Untreated -0.49 0.75 0.005
(6.86) (8.14) (0.08)
High*NoBankAct 7.55 2.09 0.110
(10.72) (15.12) (0.16)
Low Treated*NoBankAct 0.05 6.98 -0.133
(14.41) (14.93) (0.22)
Low Untreated*NoBankAct 2.93 -2.91 0.091
(11.63) (13.60) (0.15)
HH does not have bank act -1.13 -0.77 -0.102
(7.42) (10.01) (0.11)
Baseline 0.75*** 0.83*** 0.35***
(0.05) (0.04) (0.11)
R-Squared 0.62 0.72 0.17
Observations 4,059 2,312 725
# Schools (Rounds) 836 (5) 800 (3) 725 (1)
Mean Depvar 163.64 238.13 -0.21
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table examines whether our results are driven by
baseline access to bank accounts in school owner households.
Cols 1-3 reproduce our key results adding an interaction
with a dummy variable for whether the owner’s household
does not have a bank account with treatment indicators.
The primary coeﬃcients of interest are the three
interaction terms with the treatment groups, which tell us
whether treated schools where the owner did not have access
to a bank account at baseline beneﬁted more from treatment.
b) Regressions are weighted to adjust for sampling and
tracking and include strata and round ﬁxed eﬀects, with
standard standard errors clustered at village level. The
number of observations may vary across columns as data are
pooled across rounds and not all outcomes are measured in
every round. We thus also report the number of schools and
rounds for each regression, and any remaining variation in
the number of schools arises from attrition or missing
values for variables. The mean of the dependent variable is
its baseline value or the follow-up control mean.
44
Table E8: Main Outcomes, controlling for Grant size per capita
(1) (2) (3)
Enrollment Fees Score
High -2.714 10.764 0.227
(10.605) (12.677) (0.165)
Low Treated 18.050** -2.128 -0.004
(8.345) (8.197) (0.110)
Low Untreated -3.310 -2.431 0.055
(6.245) (7.383) (0.083)
Grant per capita 0.031 0.022 -0.0002
(0.020) (0.024) (0.0004)
Baseline 0.760*** 0.826*** 0.359***
(0.047) (0.037) (0.114)
R-Squared 0.62 0.72 0.17
Observations 4,059 2,312 725
# Schools (Rounds) 836 (5) 800 (3) 725 (1)
Mean Depvar 163.64 238.13 -0.21
Test pval (H=0) 0.80 0.40 0.17
Test pval (Lt = 0) 0.03 0.80 0.97
Test pval (Lt =H) 0.03 0.21 0.10
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table repeats our main results with an additional
village level control variable, grant amount per capita.
This control variable captures whether our results are
driven by total resources provided to a village. It is
constructed by adding the total amount of funding
received by treatment villages, which is 50,000 PKR for
low-saturation villages and a multiple of 50,000 PKR
based on the number of private schools in high-saturation
villages.
b) Regressions are weighted to adjust for sampling and
tracking where necessary and include strata ﬁxed eﬀects,
with standard errors clustered at village level. The number
of observations may vary across columns as data are pooled
across rounds and not all outcomes are measured in every
round. We thus also report the number of schools and round
for each regression. Any remaining variation in the number
of schools arises from attrition or missing values for some
variables. The mean of the dependent variable is its
baseline value.
c) The bottom panel shows p-values from tests that either
ask whether we can reject a zero average impact for high
(H=0) and low treated (Lt =0) schools, or whether we
can reject equality of coeﬃcients between high and low
treated (Lt =H) schools.
45
Table E9: School Infrastructure (Year 2)
Spending Number purchased Facility present (Y/N) Other
(1) (2) (3) (4) (5) (6) (7)
Amount (PKR) Desks Chairs Computers Library Sports # Rooms Upgraded
High 606.00 0.56 1.16 0.06 -0.00 0.05* 0.24
(6537.56) (1.39) (0.83) (0.05) (0.03) (0.03) (0.37)
Low Treated 353.44 -0.92 0.84 0.14** 0.00 0.02 0.31
(7911.96) (1.44) (0.54) (0.06) (0.03) (0.03) (0.36)
Low Untreated 1497.67 -1.46 0.28 -0.02 0.02 0.02 0.08
(7029.37) (1.28) (0.38) (0.04) (0.03) (0.03) (0.33)
Baseline 0.04 0.08** 0.01 0.31*** 0.02 0.07* 0.74***
(0.03) (0.04) (0.02) (0.05) (0.03) (0.04) (0.05)
R-Squared 0.05 0.08 0.04 0.16 0.04 0.11 0.51
Observations 770 746 780 784 784 784 784
# Schools (Rounds) 770 (1) 746 (1) 780 (1) 784 (1) 784 (1) 784 (1) 784 (1)
Mean Depvar 57258.48 14.59 10.92 0.39 0.35 0.19 6.36
Test pval (H=0) 0.93 0.68 0.16 0.26 1.00 0.06 0.52
Test pval (Lt = 0) 0.96 0.53 0.12 0.03 0.95 0.46 0.39
Test pval (Lt =H) 0.97 0.32 0.74 0.21 0.95 0.44 0.86
Notes: * p<0.10, ** p<0.05, *** p<0.01
a) This table examines outcomes relating to school infrastructure using data from round 5 only. Column 1 is the
annual ﬁxed expenditure on infrastructure– e.g. furniture, ﬁxtures, or facilities. Columns 2-3 refer to the
number of desks and chairs purchased. Columns 4-6 are dummy variables for the presence of particular school
facilities. Finally, column 7 measures the number of rooms upgraded from temporary to permanent or semi-permanent
classrooms. Closed schools take on a value of 0 in all columns.
b) Regressions are weighted to adjust for sampling and include strata ﬁxed eﬀects, with standard errors
clustered at the village level. The number of observations and unique schools are the same since we only use one
round of data. The mean of the dependent variable is its baseline value.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact for
high (H=0) and low treated (Lt =0) schools, or whether we can reject equality of coeﬃcients between high
and low treated (Lt =H) schools.
46
Table E10: Revenues, excluding Closed schools
Overall Posted (monthly) Overall Collected (monthly)
(1) (2) (3) (4) (5) (6)
Full Top Coded 1% Trim Top 1% Full Top Coded 1% Trim Top 1%
High 5,471.4 4,872.2* 4,543.6** 4,748.8 4,775.2** 3,593.5*
(3,432.9) (2,498.8) (2,094.2) (3,482.7) (2,425.1) (1,871.3)
Low Treated 8,589.9* 7,287.7* 6,271.1* 5,600.5 4,747.5 3,191.9
(4,988.8) (4,032.3) (3,742.7) (4,804.2) (3,349.9) (2,964.8)
Low Untreated -1,239.5 -1,434.3 -405.0 -119.6 -298.1 6.9
(2,843.0) (2,378.4) (1,847.0) (2,753.9) (2,364.5) (1,765.4)
Baseline Posted Revenues 1.0*** 1.0*** 0.9*** 0.8*** 0.9*** 0.7***
(0.1) (0.1) (0.1) (0.1) (0.1) (0.1)
R-Squared 0.66 0.67 0.61 0.57 0.64 0.56
Observations 2,312 2,312 2,276 2,948 2,948 2,900
# Schools (Rounds) 800 (3) 800 (3) 788 (3) 781 (4) 781 (4) 770 (4)
Mean Depvar 40,181.0 38,654.1 36,199.2 30,865.0 30,208.8 27,653.0
Test pval (H=0) 0.11 0.05 0.03 0.17 0.05 0.06
Test pval (Lt = 0) 0.09 0.07 0.10 0.24 0.16 0.28
Test pval (Lt =H) 0.57 0.57 0.65 0.87 0.99 0.89
Notes: * p<0.1, ** p<0.05, *** p<0.01
a) This table repeats Table 2, Columns 2-7, to look at monthly posted and collected revenues dropping schools
once they close down. Columns 1-3 consider posted revenues, deﬁned as the sum of revenues expected from each
grade based on enrollment and posted fees. Cols 4-6 consider collected revenues, deﬁned as self-reported
revenues actually collected from all students at the school. Top coding of the data assigns the value at the
99th percentile to the top 1% of data. Trimming top 1% of data assigns a missing value to data above the 99th
pctl. Both top coding and trimming are applied to each round of data separately.
b) Regressions are weighted to adjust for sampling and tracking where necessary and include strata and round
ﬁxed eﬀects, with standard errors clustered at village level.The number of observations may vary across
columns as data are pooled across rounds and not all outcomes are measured in every round. We thus also report
the unique number of schools and rounds in each regression. Any remaining variation in the number of schools
arises from missing values for some variables. The mean of the dependent variable is its baseline value or the
follow-up control mean.
c) The bottom panel shows p-values from tests that either ask whether we can reject a zero average impact for
high (H=0) and low treated (Lt =0) schools, or whether we can reject equality of coeﬃcients between high
and low treated (Lt =H) schools.
47
F Private and Social Returns Calculations
In this section, we describe our calculations from Section 4 in the main text as well as show IRR
calculations. Note that this exercise is necessarily suggestive since a complete welfare calculus
is beyond the scope of this paper. We document changes for four beneﬁciary groups from our
intervention: school owners, teachers, parents and children.
Note that for these calculations, we take all point estimates seriously even if they are not
statistically signiﬁcant or precise. We use these estimates to compare gains from a total grant
of PKR 150K under two diﬀerent ﬁnancial saturations— the L arm where we give PKR 50K to
one school in three villages, and the H arm where each school in one village receives PKR 50K.
We now proceed by looking at each beneﬁciary group separately.
F.1 Welfare Calculations
Summary of calculations: We reproduce the table from the main text below for reference.
In Rupees Standard Deviations
Group Owners Teachers Parents Children
Lt 10,918 -2,514 4,080 61.1
H 5,295 8,662 7,560 117.2
School Owners: We consider net collected revenues, subtracting variable costs from actual
collected revenues, as the monthly gains for school owners. Closed schools are considered missing
in these calculations (diﬀerent from Table 2) because we are interested in the gains for school
owners rather than the average impact on schools. That is, we implicitly assume that owners
who close down their school make (on the margin) a similar amount to what they did before
closing the school. Imputing a zero revenue value would be a less plausible and more extreme
assumption.
Using Table Table E10, col 5, monthly collected revenues for Lt are Rs.4,748 and Rs.4,775
for H schools. Variable costs are computed using estimates from Table 5, col 6– the cumulative
eﬀect is divided by 24 (12 months per year over 2 years of the intervention) for a monthly increase
of Rs.1,109 for Lt and Rs.3,010 for H schools. Thus, we have a monthly proﬁt of Rs.3,639 for
Lt and Rs.1,765 for H schools. Multiplying by 3 gives us the owner estimates in table above.
Teachers: We use changes in the teacher wage bill to understand how the intervention aﬀected
the teacher market. Recall from Table 7 that we do not observe signiﬁcant overall changes in
number of teachers employed by schools, but do observe teacher churn in the H arm. Under
the assumption that this churn arises simply from switches in employment status for teachers,
we can use these estimates of wage gains to compute changes in teacher welfare. We see that
the average monthly wage bill in H increases by Rs.2,742 relative to control and decreases by
Rs.838 for the Lt schools (Table 7, Column 2). We simply multiply these coeﬃcients by 3, and
ﬁnd that teachers in H increase their overall income by Rs.8,226, while teachers in Lt over three
villages decrease their overall income by Rs.2,514.
Parents: Calculating consumer surplus requires some strong assumptions on the demand func-
tion. These assumptions include: (i) the demand curve can be approximated as linear; and (ii)
there is an upper bound to demand at zero price because of the reasonable assumption of ‘closed’
markets in our context.
48
Since quality does not change in the L arm, our treatment eﬀects arise from a movement
along the demand curve, as in Appendix Figure F1, Panel A. We derive this linear demand curve
using two points from our experiment— the baseline price-enrollment (PQ) combination of (238,
164), denoted by (P0 , Q0 ) in the ﬁgure, and the Lt PQ-combination, denoted by (PL , QL ). Since
collected fees drop by Rs 8 (Table 3, Col 9) and enrollment increases by 12 children (Table 3,
col 5), the Lt PQ-combination is PL =230Rs and QL =176. Hence, our linear demand curve is
Q = 521 1.5P .
From Appendix Figure F1, Panel A, we can calculate the baseline consumer surplus, the
triangle CS0 , and the additional surplus gain in Lt from movement down the demand curve,
represented by the dotted quadrilateral region. This additional surplus is calculated as the
diﬀerence in areas of the two triangles generated by the baseline and Lt PQ-combinations and
equals Rs.1,360. For a total 150K in grants across three villages, the increase in CS is therefore
Rs.4,080. The increase in consumer surplus in Lt is largely driven by the fee reduction faced
by the inframarginal children; the newly enrolled, ‘marginal,’ children were just at the cusp of
indiﬀerence before the intervention and so their gains are quite small.
For the H arm, we see test score gains accompanied by fee increases. This implies a
movement of the demand curve. Given our earlier assumption of an upper bound on demand
arising from closed markets, an increase in quality pivots our baseline demand curve outward,
as in Appendix Figure F1, Panel B. We use our H estimates to obtain this new demand curve.
Since collected fees increase by Rs 29 and enrollment by 9 children, our pivoted linear demand
curve is Q = 521 1.3P . The consumer surplus from this new demand curve is Rs.11,485;
relative to the baseline consumer surplus, this represents an increased surplus of Rs.2,520 per
school. The total consumer surplus increase from grant investment of RS.150K is thus Rs 7,560.
Children: We measure beneﬁt to children in terms of test score gains. Conceptually, there
are two types of children we need to consider: (i) children that remain at their baseline schools,
and (ii) children that newly enroll at the school.
As seen in Appendix Table E6, the H arm dramatically improves test scores for already
enrolled children. In particular, considering a total baseline enrollment of 492 children from 3
schools, our H child test score gains of 0.22 sd (Table 4, Col 5) suggest a total increase of 108.2
sd. In comparison, the total gain in Lt is substantially lower at 49.2sd, even if we take the
(statistically insigniﬁcant) 0.1sd coeﬃcient at face value.
For newly enrolled children, we rely on our previous work, Andrabi et al. (2017), showing
test score gains of 0.33sd for children who switch from public to private schools.9 In H villages,
this leads to a total test score gain of 8.9 standard deviations as each of the three schools gains 9
children (0.33sd*9*3). For the Lt sample, each school gains 12 children (Table 3, Col 5), which
means a total increase of 36 children across 3 villages, and a total test score increase of 11.9sd
(0.33*36).
Summing the gains for already and newly enrolled children, we obtain a total sd gain of
117.2 for H and 61.1 for L approaches.
These calculations assume that test score gains accrue to children across all grades, which
may be reasonable given that fee increases are observed across grades (Appendix Table E4).
Using the same method, if we instead restrict to the tested children in grades 3-5, we obtain a
total increase of 31sd in H compared with a 18.2sd increase in Lt .
9
Our current study was not designed to estimate the eﬀects for newly enrolled children since it would have
been enormously expensive to test all enrolled children in each public and private school in the village, and
identifying marginal movers for testing at baseline is a diﬃcult, if not impossible, task.
49
Appendix(Figure(F1:(Consumer(Surplus(
P
CS0
P0
PL
CSL
Q0 QL
Q
t
Panel&A:!Consumer!surplus!at!baseline,!CS0,!and!in!L !from!movement
along!demand!curve
P
CSH
PH
P0
CS0
Q0 QH
Q
Panel&B:!Consumer!surplus!in!H!after!a!pivot!of!the!demand!curve!
F.2 IRR and Loan-loss guarantee
The welfare calculations show the tension between private and social returns posed by the two
ﬁnancing treatments. We will now compute the internal rate of return (IRR) directly, and see
whether lenders would be willing to lend to schools in this sector.
We conduct two types of IRR calculations and then assess whether schools would be able
to pay back a Rs.50,000 loan at 15% interest rate based on the IRR. We begin by calculating:
(i) Returns over a 2 year period with resale of assets at 50% value at the end of the term; and
(ii) Returns over a 5 year period with no resale of assets. We still use the same estimates of
collected revenues and costs as for the welfare calculations, but now also consider ﬁxed costs for
assets purchased in year 1 (Table 5, Col 1). With these assumptions, we ﬁnd returns between
61-83% for Lt and between 12-32% for H schools.
These rates of return are above or just around market interest rates in Pakistan, which
range from 15-20%, suggesting that this may be a proﬁtable lending sector. If we were to oﬀer
our grant as a RS 50,000 loan at 15% interest rate, it would take a Lt school 1.5 years to pay
oﬀ the loan and a H school 4 years to pay oﬀ their loan.
The higher rates of return coupled with the lower chance of default (Table 3, Col 4) may
lead the lender to prefer L over the H approach, unless the ﬁxed costs of visiting three villages
(versus one) is much higher. A social planner who cares about child test scores may however
prefer the H approach. To incentivize the H approach, the social planner could subsidize the
lender based on the expected losses from defaults in a manner that makes the lender indiﬀerent
between the L and H approaches.
We calculate this subsidy amount as follows. We ﬁrst note that closure rates are diﬀerential
across the Lt and H groups by 7pp (Table 3, col 4). The closure rate in Lt group is 1% and 8%
for the H group. If we assume that closed schools would default on their loans completely, then
we can estimate the expected loss that would make a lender indiﬀerent. The expected loss for a
given school in Lt group is Rs.613, while it is Rs.6400 for a H school. For every Rs.150K given
out in loans, the social planner would need to subsidize the lender by Rs.17,363 over a two year
period of the loan to make them indiﬀerent between the two approaches. This subsidy compares
favorably to the annual consumer surplus gain estimated to be Rs.41,760 higher ([Rs.7,560-
Rs.4,080]*12) in the H arm as compared to the L arm.
51
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