metadata: authors: 0: author: Andrabi,Tahir
1: author: Das,Jishnu
2: author: Khwaja,Asim Ijaz
3: author: Ozyurt,Selcuk
4: author: Singh,Niharika
count: World
docna: 0: docna: Upping the ante: the equilibrium effects of
            unconditional grants to private schools
docty: Policy Research Working Paper
subtopic: Educational Sciences,Economics of Education,Education Finance,Effective Schools and Teachers,Educational Institutions & Facilities,Governance Diagnostic Capacity Building,Macroeconomic Management,Economic Forecasting,Armed Conflict
teratopic: Governance,Education,Macroeconomics and Economic Growth,Conflict and Development
abstracts: cdata!: This paper tests for financial
            constraints as a market failure in education in a low-income
            country. In an experimental setup, unconditional cash grants
            are allocated to one private school or all private schools
            in a village. Enrollment increases in both treatments,
            accompanied by infrastructure investments. However, test
            scores and fees only increase in the setting of all private
            schools along with higher teacher wages. This differential
            impact follows from a canonical oligopoly model with
            capacity constraints and endogenous quality: greater
            financial saturation crowds-in quality investments. The
            findings of higher social surplus in the setting of all
            private schools, but greater private returns in the setting
            of one private school underscore the importance of
            leveraging market structure in designing educational subsidies.
repnme: repnme: Upping the ante: the equilibrium effects of
            unconditional grants to private schools
display_title: Upping the ante: the equilibrium
            effects of unconditional grants to private schools
listing_relative_url: /research/2018/08/30362313/upping-ante-equilibrium-effects-unconditional-grants-private-schools
topicv3: Teachers Management m1326119 1523,Education m1327048 470,Market Failures m1327149 1068,Private Education m1327271 1490,Political Violence and War m1327310 2462
publishtoextweb_dt: 2018-08-21T00:00:00Z
historic_topic: Governance,Education,Macroeconomics and Economic Growth,Conflict and Development
no_of_pages: 96
author_keys: 0: author_key: 645376
1: author_key: 439480
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model_extractions: 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 in education in a low-income country. In an experimental setup, unconditional cash grants are allocated to one private school or all private schools in a village. Enrollment increases in both treatments, accompanied by infrastructure investments. However, test scores and fees only increase in the setting of all private schools along with higher teacher wages. This differential impact follows from a canonical oligopoly model with capacity constraints and endogenous quality: greater financial saturation crowds-in quality investments. The findings of higher social surplus in the setting of all private schools, but greater private returns in the setting of one private school underscore the importance of leveraging market structure in designing educational subsidies.

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<sup>_⇤_</sup>

**_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; Sabanci 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 Fahad Suleri for excellent research assistance. We also thank seminar participants at Georgetown, UC Berkeley, NYU, Columbia, University of Zurich, BREAD, NBER Education Program Meeting, HarvardMIT 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 Evaluation 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.<sup>1</sup> 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 private school chains (Romero et al., 2017). One key consideration is that the impact of these interventions is mediated by the underlying market structure. Yet, establishing 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 interventions 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).<sup>2</sup> In addition 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

> 1Examples 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).

> 2Private 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).<sup>3</sup> 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 ( _L_<sup>_t_</sup> ) 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 ( _L_<sup>_u_</sup> ) 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 _L_<sup>_t_</sup> schools: Although we cannot reject equal revenue increases in _L_<sup>_t_</sup> and _H_ schools, the point estimates for the former are consistently larger.

Our theoretical framework highlights why _L_<sup>_t_</sup> 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 improvements occur as an equilibrium response is larger in the _H_ relative to _L_ arm.

> 3Despite 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 difficult 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 familiar 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 revenues 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, _L_<sup>_t_</sup> schools should increase enrollment, 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 _L_<sup>_t_</sup> schools without a discernible decline in the enrollment of _L_<sup>_u_</sup> 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 either 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 (vertical) product di↵erentiation. Investments in quality thus protect positive profits, although these are not as high as in the _L_ case.<sup>4</sup> The model assumes that schools know how to increase quality but are responding 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 _L_<sup>_t_</sup> schools invest in desks, chairs and computers. Meanwhile, 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

> 4In equilibrium, all schools in a village may invest in quality if the cost of quality investment is sufficiently small and the schools’ existing capacities are sufficiently 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 sample 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 estimated 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 efficiency through vouchers or school matching algorithms; and structuring partnerships to select privately operated schools using public funding.<sup>5</sup> 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 enrollment choices to be determined in equilibrium. Such policies, especially when

> 5McEwan (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 efficiency 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.<sup>6</sup> 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.<sup>7</sup> 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

> 6Private 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 administrative 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.

> 7Isolating the causal link between policies and educational improvements that is due to school responses (as opposed to compositional changes) has proven difficult. Large-scale policies usually change how children sort across schools, making it difficult 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 efficiency whereby more children attend higher quality schools. This could represent a (very) stable equilibrium, but could also be consistent with the presence of systematic constraints 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.<sup>8</sup> 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 competitive educational marketplace, but one where schools may face significant constraints, 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.

> 8This 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).<sup>9</sup> 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_ ) _x_<sup>_e_</sup> _i_<sup>+</sup><sup>_Ki −rxi −wt_subjectto</sup> _rxi_ + _wt  Ki_ and _x_<sup>_e_</sup> _i_<sup>_xi_,where</sup><sup>_xe_</sup> _i_<sup>istheenrollment,</sup><sup>_pi_isthepriceofschool</sup><sup>_i_</sup> 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.<sup>10</sup> 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_ ) = _✓jqt − 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.<sup>11</sup>

> 9KS (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.

> 10Alternative 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.

> 11We 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.<sup>12</sup> 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. Comparing profits establishes that capacity expansions are favored with a profit of $57.<sup>13</sup> 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 $<sup><u>33</u></sup> 19<sup>(</sup><sup>_⇡_$1</sup><sup>_._74)withacontinuousandatomlessprobabilitydistri-</sup> bution and obtain an (expected) profit of $33.<sup>14</sup> 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. 12These rationed students may instead enroll in public schools in the village, an outside option in this model, or not attend any school at all.

> 13If 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.

> 14To 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 invests $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 sufficiently 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, _<u>M</u>_<sup>_M<T_referstothecoveredmarketand</sup><sup>_N_=</sup><sup>_T−M_isthesizeof</sup> 2<sup>(where</sup> 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 financing, 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 sufficiently 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 capacity 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.<sup>15</sup> <!-- Start of picture text --> Figure 1a: Low-saturation Treatment Figure 1b: High-saturation Treatment<br>N N<br>2 K 2 K<br>r r<br>H<br>EL EH<br>K K<br>r r<br>L<br>w ⇤ K w w ⇤ K w<br><!-- End of picture text -->

**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 sufficiently large such that the _L_<sup>_t_</sup> 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_ )<sup>_<u>K</u>_</sup> _r_<sup>.Ifitincreasesqualityinstead,</sup> 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_ )<sup>_<u>M</u>_</sup> 2<sup>+ (</sup><sup>_qH −c_)</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> . Therefore, investing in capacity is more profitable if

> 15We suppose that _<u>k</u> < K < k_<sup> ̄</sup> where _<u>k</u>_ =<sup>_<u>Mr</u>_</sup> 2 ⇣ _<u>qHql−−qcL</u>_ ⌘ and _k_<sup> ̄</sup> =<sup>_<u>M</u>_</sup> 2<sup>(</sup><sup>_qH−qL_).Iftheinequality</sup> _<u>k</u> < K_ does not hold, then the revenue from capacity investment,<sup>_<u>K</u>_</sup> _r_<sup>(</sup><sup>_qL −c_),islowerthanrevenuefrom</sup> quality (only) investment, _<u>M</u>_ 2<sup>(</sup><sup>_qH−qL_),andthus,qualityinvestmentisalwaysoptimal.Therateof</sup> return from capacity investment is positive because we assume _qL − c − r >_ 0. Finally, _K < k_<sup> ̄</sup> 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 is greater than the latter, yielding the condition _w > w_<sup>_⇤_</sup> where _w_<sup>_⇤_</sup> = _r_ ⇣ _<u>qqHH−−qcL</u>_ ⌘⇣ _<u>M</u>_ 2<sup>+</sup><sup>_<u>K</u>_</sup> _r_ ⌘. However, if the size of the uncovered market is smaller, in particular _N <_<sup>_<u>K</u>_</sup> _r_<sup>, then spending the entire grant on additional capacity implies</sup> 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.<sup>16</sup> 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.<sup><u>2</u></sup> _r_<sup>_<u>K</u>N_.Therefore, for these</sup> values of _N_ , equilibrium predictions will be no di↵erent than the _L_ arm. However, when _N_ is less than<sup><u>2</u></sup><sup>_<u>K</u>_spendingtheentiregrantonadditionalcapacityimplies</sup> _r_<sup>,</sup> 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_ .<sup>17</sup> 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 lowsaturation 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 willingness to pay. We assume that students’ taste parameter for quality _✓j_ is uniformly distributed over [0 _,_ 1], resulting in a downward sloping demand curve. Specifically, if the schools’ quality and price are _q_ and _p_ , respectively, then demand is _D_ ( _p_ ) = _T_ (1 _−_<sup>_<u>p</u>_</sup> _q_<sup>).Unlike the case with homogeneous consumers, there are</sup> 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,

> 16More formally, **L** represents the line ( _qH − c_ ) <u>⇣</u> _<u>M</u>_ 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> ⌘ = ( _qL − c_ ) <u>⇣</u> _<u>M</u>_ 2<sup>+</sup><sup>_N_</sup> ⌘ + _K − Nr_ .

> 17More formally, **H** represents the line ( _qH − c_ )( _<u>M</u>_ 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> ) = ( _qL − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N−_</sup><sup>_<u>K</u>_</sup> _r_<sup>)</sup><sup>_−Nr_.</sup>
_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.<sup>18</sup> 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 lowsaturation 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_ parameterization 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

> 18The 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 competition. 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 _L_<sup>_t_</sup> 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;<sup>19</sup> (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.

> 19This 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 office 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,<sup><u>3</u></sup> 7<sup>ofthevillagesareassigned</sup> to the _L_ arm, and<sup><u>2</u></sup> 7<sup>tothe</sup><sup>_H_armandthecontrolgroup;atotalof342schools</sup> 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 Lahore 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 received 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.<sup>20</sup> 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.<sup>21</sup> 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.<sup>22</sup> Our data come from three di↵erent survey exercises, detailed in Appendix C.

> 20Reasons for refusal include anticipated school closure; unwillingness to accept external funds; or a failure to reach owners despite multiple attempts.

> 21At 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.

> 22There 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), collecting information on school characteristics, practices and management, as well as household information on school owners. In addition, there are 4 shorter followup 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 sufficient 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 increases 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.<sup>23</sup> _C._ **_Regression Specification_** We estimate intent-to-treat (ITT) e↵ects using the following school-level spec-<sup>24</sup> _Yijt_ is an outcome of interest for a school _i_ in village _j_ at time _t_ , which is measured in at least one of five follow-up rounds after treatment. _Hijt_ , _L_<sup>_t_</sup> _ijt_<sup>,and</sup><sup>_Lu_</sup> _ijt_ 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 _δt_ 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 precision and control for any potential baseline mean imbalance between the treated and control groups (see discussion in section III.D). All regressions cluster standard 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 coefficients of interest are _β_ 1, _β_ 2, and _β_ 3, all of which identify the average ITT e↵ect for their respective group.

> 23Another 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 difficult especially given that churn is not uncommon in this setting.

> 24We 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: _L_<sup>_t_</sup> and _L_<sup>_u_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 treatment 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 _L_<sup>_u_</sup> ). 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 _L_<sup>_t_</sup> schools are less likely to attrit relative to control in every round (Appendix Table D4, Panel A). For other experimental groups, attrition is more idiosyncratic. Despite this di↵erential attrition, baseline characteristics of those who refuse surveying 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).<sup>25</sup> 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

> 25Comparing 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 difficult; nevertheless, when we do consider this set, one significant di↵erence emerges with lower enrollment in _L_<sup>_u_</sup> relative to control schools.
_19_ by this correction.

# **IV. Results**

In this section, we present results on the primary outcomes of interest, investigate 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 financing 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 available, 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).<sup>26</sup> To obtain the latter measure, we inspected the school account books and computed revenues actually collected in the month prior to the survey.<sup>27</sup> While this measure captures revenue

> 26Posted 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.

> 27Over 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. Column 2 shows that schools in the _H_ arm gain Rs.5,484 (p=0.12) each month while _L_<sup>_t_</sup> schools gain Rs.10,665 (p=0.03) a month. Annual revenue increases (twelve times the reported monthly coefficient 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 _L_<sup>_u_</sup> schools, with small coefficients 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 _L_<sup>_t_</sup> schools (col 5). One explanation for this di↵erence could be that marginal new children pay lower (than posted) fees in _L_<sup>_t_</sup> 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 coefficients across the treatment arms of the intervention.<sup>28</sup>

# 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_u_</sup> 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 _L_<sup>_t_</sup> schools is due to a reduction in the

> self-reported fee collections.

> 28In 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 _L_<sup>_t_</sup> 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, _L_<sup>_t_</sup> schools were 9 percentage points less likely to close over the study period. We find no average impact on school closure for _H_ or _L_<sup>_u_</sup> schools relative to control. Although fewer school closures 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 enrollment for _H_ and _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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.<sup>29</sup> 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 _L_<sup>_t_</sup> or _L_<sup>_u_</sup> schools. Consequently, we reject equality of coefficients between _H_ and _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> schools, suggesting that collected fees may have been lower in these schools. We

> 29While 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> schools.<sup>30</sup> 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 _L_<sup>_t_</sup> 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 children in all schools 16 months after the start of the intervention (near the end of the first school year after treatment).<sup>31</sup> 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.<sup>32</sup> Test score increases for _H_ schools are comparably high in all subjects with coefficients 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, representing 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

> 30This 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.

> 31As 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.

> 32Since 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 coefficients between _H_ and _L_<sup>_t_</sup> 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 additional 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).<sup>33</sup> 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 _L_<sup>_t_</sup> 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 differential results between the treatment arms— relies on the strategic returns to investing in quality when financial saturation in markets is high. We now examine 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.<sup>34</sup> Nevertheless, consider the two following channels of impact. The plan could either have forced school owners

> 33If 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.

> 34Schools 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 comparisons, 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 correlated with treatment saturation and driving our results, we should find that the additional inclusion of this variable drives the di↵erence in our treatment coefficients 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. _L_<sup>_t_</sup> schools see higher enrollment on average, while _H_ schools experience higher fees and test scores on average. While we lose precision in the _H_ arm, we cannot reject that these coefficients 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. Furthermore, 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 _L_<sup>_t_</sup> 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 coefficients 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 _L_<sup>_t_</sup> schools (Table 6, column 1). Table 6 also provides evidence that spending on infrastructure components di↵ers by treatment
_26_ saturation. While we cannot reject equality of coefficients for _H_ and _L_<sup>_t_</sup> comparisons, relative to _L_<sup>_t_</sup> 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).<sup>35</sup> 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 _L_<sup>_t_</sup> schools and a greater emphasis on quality improvements among _H_ schools.<sup>36</sup> 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 _L_<sup>_t_</sup> schools, which if anything show a negative coefficient (column 1). This 12% increase in costs is in large part due to the significantly higher wage bill for teachers in _H_ relative to _L_<sup>_t_</sup> 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 retain 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. _L_<sup>_t_</sup> 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), _L_<sup>_t_</sup> schools act as

> 35A standard desk accommodates 2 students implying that 12 additional students can be seated in _H_ schools, and 18 students in _L_<sup>_t_</sup> school; these numbers are similar in magnitude to the enrollment gains documented earlier.

> 36While additional facilities could justify increasing prices, the per-student availability of desks and chairs in _L_<sup>_t_</sup> 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 highsaturation 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 demand 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 calculation allows us to incorporate the idea that there may be social externalities from learning gains beyond the direct benefits to parents. The key intuition driving 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 attending 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 _L_<sup>_t_</sup> and _H_ schools, giving monetary returns for the first three beneficiary groups and test score increases for children; we omit consideration of _L_<sup>_u_</sup> 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.<sup>37</sup> Details of the computations are provided in

> 37For 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 coefficients. While
_28_ Appendix F. For school owners and teachers, the calculations are standard and

|||In PKR||Standard Deviations|
|---|---|---|---|---|
|Group|Owners|Teachers|Parents|Children|
|_L_<sup>_t_</sup>|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 _L_<sup>_t_</sup> 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.<sup>38</sup> 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.<sup>39</sup> 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 _L_<sup>_t_</sup> schools and 12%-32% for _H_ schools for 2-year and 5-year scenarios (see Appendix F).<sup>40</sup> As interest rates on loans to this sector range from 15-20%, the IRR almost always exceeds the market interest rate: _L_<sup>_t_</sup> schools would be able to pay back a Rs.50,000 doing so does not qualitatively alter our results, we prefer the approach taken.

> 38While 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).

> 39This 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.

> 40For 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 _L_<sup>_t_</sup> 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 between the two approaches.<sup>41</sup> 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, varying 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 consistent 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 investments. 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,

> 41This 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 maximize 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 lowsaturation villages increase revenues only through increases in market share and although we do not explicitly model this (we do not have an empirical counterpart 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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|||Table1:Ba|selineSumm|aryStatistics|||||
|---|---|---|---|---|---|---|---|---|
|Variable|(1)<br>Mean|(2)<br>5thpctl|(3)<br>25thpctl|(4)<br>Median|(5)<br>75thpctl|(6)<br>95thpctl|(7)<br>Standard<br>Deviation|(8)<br>N|
|**_PanelA:VillagelevelVariables_**|||||||||
|Numberofpublicschools|2.45|1.0|2.0|2.0|3.0|5.0|1.03|266|
|Numberofprivateschools|3.33|2.0|2.0|3.0|4.0|7.0|1.65|266|
|Privateenrollment|523.52|149.0|281.0|415.5|637.0|1,231.0|378.12|266|
|**_PanelB:PrivateSchoollevelVari_**|**_ables_**||||||||
|Enrollment|163.6|45.0|88.0|140.0|205.0|353.0|116.0|851|
|Monthlyfee(PKR)|238.4|81.3|150.0|201.3|275.0|502.5|166.1|851|
|Monthlyrevenue(PKR)|40,181.1|4,943.0|13,600.0|26,485.0|44,400.0|117,655.0|54,883.9|850|
|Monthlyvariablecosts(PKR)|25,387.0|3,900.0|9,400.0|16,200.0|27,200.0|79,000.0|30,961.1|848|
|Annualfixedexpenses(PKR)|78,860.9|0.0|9,700.0|33,000.0|84,000.0|326,000.0|136,928.2|837|
|Schoolage(Noofyears)|8.3|0.0|3.0|7.0|12.0|19.0|6.7|852|
|Numberofteachers|8.2|3.0|5.0|7.0|10.0|17.0|4.8|851|
|Monthlyteachersalary(PKR)|2,562.8|1,000.0|1,500.0|2,000.0|2,928.5|5,250.0|3,139.5|768|
|Numberofenrolledchildrenin|13.1|1.0|5.0|10.0|18.0|34.5|11.7|420|
|testedgrade|||||||||
|Numberoftestedchildren|11.7|1.0|4.0|9.0|16.0|31.5|10.6|420|
|Averagetestscore|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 Revenu|es (monthly)|
|---|---|---|---|---|---|---|---|
||(1)<br>Year 1|(2)<br>Full|(3)<br>Top Coded 1%|(4)<br>Trim Top 1%|(5)<br>Full|(6)<br>Top Coded 1%|(7)<br>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 (_L_<sup>_t_</sup>=0)|0.01|0.03|0.02|0.03|0.09|0.03|0.06|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.
Table 3—School Enrollment and Fees (Monthly)

||En|rollment (A|ll)|Closure|Enrollment (Open)||Posted Fee|s|Collected Fees|
|---|---|---|---|---|---|---|---|---|---|
||(1)<br>Year 1|(2)<br>Year 2|(3)<br>Overall|(4)<br>Overall|(5)<br>Overall|(6)<br>Year 1|(7)<br>Year 2|(8)<br>Overall|(9)<br>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)<br>|0.10|0.25|0.14|0.60|0.08|0.02|0.04|0.02|0.14|
|Test pval (_L_<sup>_t_</sup>=0)<br>|0.01|0.01|0.01|0.01|0.13|0.81|0.79|0.95|0.54|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.
_37_ Table 4—Test Scores

|||Schoo|l level||Child level|
|---|---|---|---|---|---|
||(1)<br>Math|(2)<br>English|(3)<br>Urdu|(4)<br>Avg|(5)<br>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**<br>|0.43***<br>|0.25**<br>|0.36***<br>|0.63***<br>|
||(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 (_L_<sup>_t_</sup>=0)<br>|0.50|0.43|0.45|0.79|0.33|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.
Table 5—Fixed and Variable Costs (Annual)

||Year|1|Ye|ar 2|Cumul|ative|Cumulative|(Open Only)|
|---|---|---|---|---|---|---|---|---|
||(1)<br>Fixed|(2)<br>Variable|(3)<br>Fixed|(4)<br>Variable|(5)<br>Fixed|(6)<br>Variable|(7)<br>Fixed|(8)<br>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)<br>|0.00|0.20|0.71|0.21|0.00|0.06|0.00|0.01|
|Test pval (_L_<sup>_t_</sup>=0)|0.01|0.75|0.49|0.49|0.00|0.49|0.01|0.98|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.
_39_ Table 6—School Infrastructure (Year 1)

||Spending|Number|purchased|Facility|present (Y|/N)|Other|
|---|---|---|---|---|---|---|---|
||(1)<br>Amount|(2)<br>Desks|(3)<br>Chairs|(4)<br>Computers|(5)<br>Library|(6)<br>Sports|(7)<br># 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***<br>(0.03)|0.10*<br>(0.05)|0.12*<br>(0.07)|0.26***<br>(0.04)|0.32***<br>(0.04)|0.23***<br>(0.05)|0.71***<br>(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 (_L_<sup>_t_</sup>=0)|0.03|0.00|0.03|0.01|0.58|0.49|0.24|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.
_40_ Table 7—Teacher Salaries and Composition

||School Cost|s (monthly)|Teach|er Roster|Teache|r Salaries (m|onthly)|
|---|---|---|---|---|---|---|---|
||(1)<br>Total|(2)<br>Wage Bill|(3)<br>Total|(4)<br>Num New|(5)<br>All|(6)<br>New|(7)<br>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***<br>(0.07)|0.85***<br>(0.08)|0.77***<br>(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 (_L_<sup>_t_</sup>=0)|0.51|0.58|0.33|0.25|0.52|0.83|0.37|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =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 efects with unequal selection probabilities**<br>B.1<br>Saturation Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|**23**<br>23|
|B.2<br>Tracking Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|24|
|**C Sampling, Surveys and Data**|**25**|
|**D Balance and Attrition**|**30**|
|D.1 Balance<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|30|
|D.2 Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|30|
|**E Additional Results**|**37**|
|**F Private and Social Returns Calculations**|**48**|
|F.1<br>Welfare Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|48|
|F.2<br>IRR and Loan-loss guarantee<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . .|51|

1
# **A Theory**

## **A.1 Homogeneous Demand**

Suppose that schools choose _x_ 1 _, x_ 2 _≥_ 0 and _q_ 1 _, q_ 2 _2 {qH , qL}_ in the first stage and _p_ 1 _, p_ 2 in the second stage. Let _si_ be school _i_ ’s surplus, that is _si_ = _qi − pi_ . Therefore, school _i_ ’s profit function is: Define _nH_ =<sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> and _nL_ =<sup>_<u>K</u>_</sup> _r_<sup>tobetheadditionalcapacityincreasethatschoolscan</sup> 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 _x_ 1 and _x_ 2 are such that _x_ 1 + _x_ 2 _ N_ , then in the pricing stage, school _i_ picks _pi_ = _qi_ . Let _μ_ be a ˆ probability density function with support [ _<u>p,</u>_ ̄ _p_ ]. Then for notational simplicity, we use _μ_ ( _p_ ) for any _p 2_ [ _<u>p,</u>_ ̄ _p_ ] to denote _μ_ ( _{p}_ ). Before proving the main results, we prove the following result, which applies to both low ( _L_ ) and high-saturation ( _H_ ) treatments.

**Proposition A.** _Suppose that the schools’ quality choices are q_ 1 _, q_ 2 _2 {qH , qL} and capacity choices are x_ 1 _, x_ 2 _≥_ 0 _with x_ 1 _, x_ 2 _ N_ +<sup>_<u>M</u>_</sup> 2<sup>_andx_1+</sup><sup>_x_2</sup><sup>_>N.Theninthe(second)pricing_</sup> _stage, there exists no pure strategy equilibrium. However, there exists a mixed strategy equilibrium_ ( _μ_<sup>_⇤_</sup> 1<sup>_, μ⇤_</sup> 2<sup>)</sup><sup>_,wherefori_= 1</sup><sup>_,_2</sup><sup>_,μ⇤_</sup> _i_<sup>_is_</sup>

( _i_ ) _a probability density function with support_ [ _p_<sup>_⇤_</sup> _i_<sup>_, qi_]</sup><sup>_,satisfyingc < p⇤_</sup> _i_<sup>_< qi,and_</sup> ( _ii_ ) _atomless except possibly at qi, that is μ_ ˆ<sup>_⇤_</sup> _i_<sup>(</sup><sup>_p_) = 0</sup><sup>_forallp 2_[</sup><sup>_p⇤_</sup> _i_<sup>_, qi_)</sup><sup>_._</sup>

- ˆ

- ( _iii_ ) _Furthermore, μ_<sup>_⇤_</sup> 1<sup>(</sup><sup>_p_1)ˆ</sup><sup>_μ⇤_</sup> 2<sup>(</sup><sup>_p_2)=0</sup><sup>_forallp_1</sup><sup>_2_[</sup><sup>_p⇤_</sup> 1<sup>_, q_1]</sup><sup>_andp_2</sup><sup>_2_[</sup><sup>_p⇤_</sup> 2<sup>_, q_2]</sup><sup>_satisfyingq_1</sup><sup>_−p_1=</sup> _q_ 2 _− p_ 2 _._

**_Proof of Proposition A_ .** Because no school alone can cover the entire market, i.e., _xi < N_ +<sup>_<u>M</u>_</sup> 2<sup>,</sup> _p_ 1 = _p_ 2 = _c_ cannot be an equilibrium outcome. Likewise, given that the schools compete in a Bertrand fashion and total capacity, _M_ + _x_ 1 + _x_ 2, 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_ ( _p_ 1 _, p_ 2) are restricted to the price couples where both schools offer the same surplus, that is _{_ ( _p_ 1 _, p_ 2) _2_ [ _c, qH_ ]<sup>2</sup> _|q_ 1 _− p_ 1 = _q_ 2 _− p_ 2 _}_ . Lowering its price from a position _c < q_ 1 _− p_ 1 = _q_ 2 _− p_ 2 _ qH_ , a school discontinuously increases its profit. Hence, ⇧ _i_ ( _p_ 1 _, p_ 2) is weakly lower semi-continuous. ⇧ _i_ ( _p_ 1 _, p_ 2) 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 _q_ 1 = _q_ 2) 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 _q_ 1 = _q_ 2). Thus, by Theorem 5 of Dasgupta and Maskin (1986), the game has a mixed-strategy equilibrium.

Suppose that ( _μ_<sup>_⇤_</sup> 1<sup>_, μ⇤_</sup> 2<sup>)isamixed-strategyequilibriumofthepricingstage.Let</sup><sup>_p_ ̄</sup><sup>_i_bethe</sup> supremum of the support of _μ_<sup>_⇤_</sup> _i_<sup>,so</sup><sup>_p_ ̄</sup><sup>_i_=inf</sup><sup>_{p2_[</sup><sup>_c, qi_]</sup><sup>_|p2supp_(</sup><sup>_μ⇤_</sup> _i_<sup>)</sup><sup>_}_.</sup> Likewise, let _p_<sup>_⇤_</sup> _i_<sup>be</sup> the infimum of the support of _μ_<sup>_⇤_</sup> _i_<sup>.</sup> 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.

**Lemma A1.** _s_ ( _p_<sup>_⇤_</sup> 1<sup>_, q_1) =</sup><sup>_s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2)</sup><sup>_andp⇤_</sup> _i_<sup>_> cfori_= 1</sup><sup>_,_2</sup><sup>_._</sup>

2
_Proof._ Note that the claim turns into the condition _p_<sup>_⇤_</sup> 1<sup>=</sup><sup>_p⇤_</sup> 2<sup>_>c_when</sup><sup>_q_1=</sup><sup>_q_2.</sup> To show _s_ ( _p_<sup>_⇤_</sup> 1<sup>_, q_1)=</sup><sup>_s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2),supposeforacontradictionthat</sup><sup>_s_(</sup><sup>_p⇤_</sup> 1<sup>_, q_1)=</sup><sup>_s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2).Assume,without</sup> loss of generality, that _s_ ( _p_<sup>_⇤_</sup> 1<sup>_, q_1)</sup><sup>_>s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2).Forany</sup><sup>_p_1</sup><sup>_≥p⇤_</sup> 1<sup>inthesupportof</sup><sup>_μ⇤_</sup> 1<sup>satisfying</sup> _s_ ( _p_<sup>_⇤_</sup> 1<sup>_, q_1)</sup><sup>_≥s_(</sup><sup>_p_1</sup><sup>_, q_1)</sup><sup>_> s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2),player1canincreaseitsexpectedprofitbydeviatingtoaprice</sup> _p_<sup>_0_</sup> 1<sup>=</sup><sup>_p_1 +</sup><sup>_✏_satisfying</sup><sup>_s_(</sup><sup>_p0_</sup> 1<sup>_, q_1)</sup><sup>_>s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2).Thisistruebecausebyslightlyincreasingitsprice</sup> from _p_ 1 to _p_<sup>_0_</sup> 1<sup>school1keepsitsexpectedenrollmentthesame.Thisopportunityofaprofitable</sup> deviation contradicts with the optimality of equilibrium. The case for _s_ ( _p_<sup>_⇤_</sup> 1<sup>_, q_1)</sup><sup>_<s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2)is</sup> symmetric. Thus, we must have _s_ ( _p_<sup>_⇤_</sup> 1<sup>_, q_1) =</sup><sup>_s_(</sup><sup>_p⇤_</sup> 2<sup>_, q_2).</sup> Showing that _p_<sup>_⇤_</sup> _i_<sup>_> c_for</sup><sup>_i_= 1</sup><sup>_,_2isstraightforward:Supposeforacontradictionthat</sup><sup>_pi_=</sup><sup>_c_</sup> 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 <_<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N_,school</sup><sup>_i_cangetpositiveresidualdemandand</sup> 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_ [ _a_ 1 _, b_ 1) _and_ [ _a_ 2 _, b_ 2) _are called surplus-equivalent if s_ ( _a_ 1 _, q_ 1) = _s_ ( _a_ 2 _, q_ 2) _and s_ ( _b_ 1 _, q_ 1) = _s_ ( _b_ 2 _, q_ 2) _._ **Lemma A2.** _Let_ [ _ai, bi_ ) _be a non-empty subset of_ [ _c, qi_ ] _for i_ = 1 _,_ 2 _. If_ [ _a_ 1 _, b_ 1) _and_ [ _a_ 2 _, b_ 2) _are surplus equivalent, then μ_<sup>_⇤_</sup> 1<sup>([</sup><sup>_a_1</sup><sup>_, b_1)) = 0</sup><sup>_ifandonlyifμ⇤_</sup> 2<sup>([</sup><sup>_a_2</sup><sup>_, b_2)) = 0</sup><sup>_._</sup> _Proof._ Take any two such intervals and suppose, without loss of generality, _μ_<sup>_⇤_</sup> 1<sup>([</sup><sup>_a_1</sup><sup>_, b_1))=0.</sup> That is, [ _a_ 1 _, b_ 1) is not in the support of _μ_<sup>_⇤_</sup> 1<sup>.Therefore,forany</sup><sup>_p2_[</sup><sup>_a_2</sup><sup>_, b_2),player2’sexpected</sup> enrollment does not change by moving to a higher price within this set [ _a_ 2 _, b_ 2). 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 [ _a_ 2 _, b_ 2), implying that _μ_<sup>_⇤_</sup> 2<sup>([</sup><sup>_a_2</sup><sup>_, b_2)) = 0.</sup> _Proof._ Suppose for a contradiction that there exists some _p_ 1 and _p_ 2 as in the premises of this claim such that _μ_ ˆ<sup>_⇤_</sup> 1<sup>(</sup><sup>_p_1)ˆ</sup><sup>_μ⇤_</sup> 1<sup>(</sup><sup>_p_2)</sup><sup>_>_0.Because</sup><sup>_μ_ˆ</sup><sup>_⇤_</sup> 1<sup>(</sup><sup>_p_1)</sup><sup>_>_0,player2canenjoythediscretechanceof</sup> price-undercutting his opponent. That is, there exists sufficiently small _✏>_ 0 such that player 2 gets strictly higher profit by naming price _p_ 2 _− ✏_ rather than price _p_ 2. 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, qi_ ) _≥ s_ ( ̄ _pj, qj_ ) _where i, j 2 {_ 1 _,_ 2 _} and j_ = _i, then for any k 2 {_ 1 _,_ 2 _} and_  ̄ ˆ _p 2_ [ _c, qH_ ] _, satisfying p 6_ = _pj, it must be the case that μ_<sup>_⇤_</sup> _k_<sup>(</sup><sup>_p_) = 0</sup><sup>_._</sup>

_Proof._ Suppose without loss of generality that _k_ = 1 and suppose for a contradiction that _μ_ ˆ<sup>_⇤_</sup> 1<sup>(</sup><sup>_p_)</sup><sup>_>_0forsome</sup><sup>_p2_[</sup><sup>_c, qH_]</sup><sup>_\ {p_ ̄</sup><sup>_j}_.Therefore,theremustexistsufficientlysmall</sup><sup>_✏>_0and</sup> _δ >_ 0 such that for all _p_ 2 _2 I ⌘_ [ _q_ 2 _− s_ ( _p, q_ 1) _, q_ 2 _− s_ ( _p, q_ 1) + _✏_ ) player 2 prefers to name a price _p_ 2 _− δ_ instead of _p_ 2 and enjoy the discrete chance of price-undercutting his opponent. Therefore, the optimality of the equilibrium strategies suggests that _μ_<sup>_⇤_</sup> 2<sup>(</sup><sup>_I_)=0.</sup> Because the intervals [ _p, p_ + _✏_ ) and _I_ are surplus-equivalent, Lemma A2 implies that we must have _μ_<sup>_⇤_</sup> 1<sup>([</sup><sup>_p, p_+</sup><sup>_✏_)) = 0,</sup> contradicting _μ_ ˆ<sup>_⇤_</sup> 1<sup>(</sup><sup>_p_)</sup><sup>_>_0.</sup>

**Lemma A5.** _s_ ( ̄ _p_ 1 _, q_ 1) = _s_ ( ̄ _p_ 2 _, q_ 2) = 0 _, and thus p_  ̄ _i_ = _qi for i_ = 1 _,_ 2 _._

_Proof._ To show _s_ ( ̄ _p_ 1 _, q_ 1) = _s_ ( ̄ _p_ 2 _, q_ 2) suppose for a contradiction that _s_ ( ̄ _p_ 1 _, q_ 1) = _s_ ( ̄ _p_ 2 _, q_ 2). Suppose, without loss of generality, that _s_ ( ̄ _p_ 2 _, q_ 2) _> s_ ( ̄ _p_ 1 _, q_ 1). Therefore, by Lemma A4 we have _μ_<sup>_⇤_</sup> 2<sup>([ ̄</sup><sup>_p_2</sup><sup>_,_ ̃</sup><sup>_p_2))=0where</sup><sup>_p_ ̃2</sup><sup>_⌘q_2</sup><sup>_−s_( ̄</sup><sup>_p_1</sup><sup>_, q_1),andbyLemmaA2</sup><sup>_μ⇤_</sup> 1<sup>([ ̃</sup><sup>_p_1</sup><sup>_,_ ̄</sup><sup>_p_1))=0where</sup> _p_ ̃1 _⌘ q_ 1 _− s_ ( ̄ _p_ 2 _, q_ 2)). In fact, there must exist some small _✏>_ 0 such that _μ_<sup>_⇤_</sup> 1<sup>([ ̃</sup><sup>_p_1</sup><sup>_−✏,_ ̄</sup><sup>_p_1))=0.</sup> ̄ The last claim is true because player 1 prefers to deviate from any _p 2_ [ ̃ _p_ 1 _−✏,_ ̃ _p_ 1] to price _p_ 1 since the change in profit, ⇧1( _p, p_ 2) _−_ ⇧1( ̄ _p_ 1 _, p_ 2) is equal to ( _p − c_ ) _μ_<sup>_⇤_</sup> ([ _p,_ ̃ _p_ 1]) _x_ 1 _−_ ( ̄ _p_ 1 _− c_ )( _T − x_ 2) _<_ 0 as _✏_ converges zero. Because the sets [ ̄ _p_ 2 _− ✏,_ ̃ _p_ 2) and [ ̃ _p_ 1 _− ✏,_ ̄ _p_ 1) are surplus-equivalent and 3
_μ_<sup>_⇤_</sup> 1<sup>([ ̃</sup><sup>_p_1</sup><sup>_−✏,_ ̄</sup><sup>_p_1))=0,LemmaA2impliesthat</sup><sup>_μ⇤_</sup> 2<sup>([ ̄</sup><sup>_p_2</sup><sup>_−✏,_ ̃</sup><sup>_p_2))=0,contradictingthat</sup><sup>_p_ ̄2isthe</sup> supremum of the support of _μ_<sup>_⇤_</sup> 2<sup>.Thus,</sup><sup>_s_( ̄</sup><sup>_p_1</sup><sup>_, q_1) =</sup><sup>_s_( ̄</sup><sup>_p_2</sup><sup>_, q_2)musthold.</sup> To show that _s_ ( ̄ _pi, qi_ ) = 0 for _i_ = 1 _,_ 2, assume for a contradiction that _s_ ( ̄ _p_ 1 _, q_ 1) = _s_ ( ̄ _p_ 2 _, q_ 2) _>_ ˆ 0. By Lemma A3 we know that _μ_<sup>_⇤_</sup> 1<sup>( ̄</sup><sup>_p_1)ˆ</sup><sup>_μ⇤_</sup> 1<sup>( ̄</sup><sup>_p_2)=0.Suppose,withoutlossofgenerality,that</sup> ˆ _μ_<sup>_⇤_</sup> 1<sup>( ̄</sup><sup>_p_1)=0.Therefore,player2canprofitabledeviatefromprice</sup><sup>_p_ ̄2toprice</sup><sup>_q_2:thedeviation</sup> 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 > p_<sup>_⇤_</sup> _i_<sup>_,andthereexistsnop, p0withp⇤_</sup> _i_<sup>_<p<p0<qisuch_</sup> _that μ_<sup>_⇤_</sup> _i_<sup>([</sup><sup>_p, p0_]) = 0</sup><sup>_._</sup>

_Proof._ If _p_ ̄ _i_ = _p_<sup>_⇤_</sup> _i_<sup>for some</sup><sup>_i_, that is player</sup><sup>_i_is playing a pure strategy, then player</sup><sup>_j_can profitably</sup> deviate from _qj_ by price undercutting its opponent, contradicting the optimality of equilibrium. Next, suppose for a contradiction that there exists _p, p_<sup>_0_</sup> with _p_<sup>_⇤_</sup> _i_<sup>_<p<p0<qi_suchthat</sup> _μ_<sup>_⇤_</sup> _i_<sup>([</sup><sup>_p, p0_]) = 0.By Lemma A2, there exists</sup><sup>_pj, p0_</sup> _j_<sup>that are surplus equivalent to</sup><sup>_p, p0_, respectively,</sup> and _μ_<sup>_⇤_</sup> _j_<sup>([</sup><sup>_pj, p0_</sup> _j_<sup>])=0.ThentheoptimalityofequilibriumandLemmaA4impliesthatthere</sup> exists some _✏>_ 0 such that _μ_<sup>_⇤_</sup> _i_<sup>([</sup><sup>_p −✏, p0_]) = 0.Thisistruebecauseinsteadofpickingapricein</sup> [ _p − ✏, p_ ], school _i_ would keep expected enrollment the same and increase its profit by picking a higher price _p_<sup>_0_</sup> . Repeating the same arguments will eventually yield the conclusion that we have _μ_<sup>_⇤_</sup> _i_<sup>([</sup><sup>_p⇤_</sup> _i_<sup>_, p0_]) = 0,contradictingtheassumptionthat</sup><sup>_p⇤_</sup> _i_<sup>istheinfimumofthesupportof</sup><sup>_μ⇤_</sup> _i_<sup>.</sup> For the rest of the proofs, we use ⇧ _t_ to denote the profit of a school that picks quality _t 2 {H, L}_ . Let ⇧<sup>_Dev_</sup> _H_ denote the deviation profit of a school that deviates from high to low quality (once the other school’s actions are fixed). Similarly, ⇧<sup>_Dev_</sup> _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:

<!-- Start of picture text --> Case 1<br><!-- End of picture text --> <!-- Start of picture text --> N<br>2 K<br>r as K % k ̄<br>Case 1<br>Case 3 & 4<br>K<br>r<br>Case 2<br>L<br>Case 3<br>k/r<br>Case 4<br>w ⇤ K w<br>Case 1: K  Nr (or equivalently nL  N ): There would be no price competition among the<br>schools whether school 1 invests in capacity or quality. Therefore, ⇧ H = ( qH − c ) � M 2 + K− r w �<br>and ⇧ L = ( qL − c ) � M 2 + K r �. Thus, there is an equilibrium where school 1 invests in quality if<br>and only if ⇧ H ≥ ⇧ L , implying w  w ⇤ .<br>Nr<br> +<br> w<br>=<br> K<br>)<br> N<br>2 +<br>M/<br>)(<br>L<br>q<br>−<br>H<br>q<br>= (<br>w<br><!-- End of picture text --> 4
**Case 2:** _K − w  Nr < K_ (or equivalently _nH  N < nL_ ): If school 1 invests in quality, then ⇧ _H_ = ( _qH − c_ ) � _<u>M</u>_ 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> �. But if it invests in capacity, then its optimal choice would be _x_ 1 = _N_ (as we formally prove below) and profit would be ⇧ _L_ = ( _qL − c_ ) � _<u>M</u>_ 2<sup>+</sup><sup>_N_</sup> � + _K − Nr_ .

**Claim:** _If school 1 invests in capacity, then its optimal capacity choice x_ 1 _is such that x_ 1 = _N ._

_Proof._ Suppose for a contradiction that _x_ 1 = _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_<sup>_⇤_</sup> _i_<sup>_, qL_]</sup> with a probability measure _μi_ . School 1’s profit functions are given by ⇧1( _qL, μ_ 2) = ( _qL −_ ˆ _c_ ) h _<u>μ</u>_ 2( _M/_ 2 _M_ <u>++</u> _x_ 1 _x_ <u>)(1</u> _M_ <u>+</u> _N_ <u>)</u> + (1 _− μ_ ˆ2) � _<u>M</u>_ 2<sup>+</sup><sup>_N_</sup> �<sup>i</sup> + _K − rx_ 1, where _μ_ ˆ2 = _μ_ ˆ2( _qL_ ), and ⇧1( _p_<sup>_⇤_</sup> 1<sup>_, μ_2)=</sup> ( _p_<sup>_⇤_</sup> 1<sup>_−c_)(</sup><sup>_x_1+</sup><sup>_M/_2) +</sup><sup>_K−rx_1.However,school2’sprofitfunctionsare⇧2(</sup><sup>_qL, μ_1)=(</sup><sup>_qL−_</sup> ˆ _c_ ) h _<u>μ</u>_ 1( _MM/_ 2+)( _xM_ 1 <u>+</u> _N_ <u>)</u> + (1 _− μ_ ˆ1) � _<u>M</u>_ 2<sup>+</sup><sup>_N−x_1</sup> �<sup>i</sup> , where _μ_ ˆ1 = _μ_ ˆ1( _qL_ ) and ⇧2( _p_<sup>_⇤_</sup> 2<sup>_, μ_1) = (</sup><sup>_p⇤_</sup> 2<sup>_−c_)(</sup><sup>_<u>M</u>_</sup> 2<sup>).</sup> In equilibrium both schools offer the same surplus, and so _p_<sup>_⇤_</sup> 1<sup>=</sup><sup>_p⇤_</sup> 2<sup>holds.Moreover,be-</sup> cause each school _i_ is indifferent between _qL_ and _p_<sup>_⇤_</sup> _i_<sup>wemusthave⇧1(</sup><sup>_qL, μ_2)=⇧1(</sup><sup>_p⇤_</sup> 1<sup>_, μ_2)and</sup> ⇧2( _qL, μ_ 1) = ⇧2( _p_<sup>_⇤_</sup> 2<sup>_, μ_1).Wecansolvetheseequalitiesfor</sup><sup>_μ_ˆ1and</sup><sup>_μ_ˆ2.However,weknowthat</sup> ˆ ˆ ˆ in equilibrium we must have _μ_ 1 _μ_ 2 = 0. If _μ_ 2 = 0, then it is easy to see that ⇧1( _qL, μ_ 2) deˆ creases with _x_ 1 (or _e_ ), and thus optimal capacity should be _x_ 1 = _N_ . However, _μ_ 1 = 0 yields ˆ _μ_ 2 = _−_<sup>4(</sup><sup>_e_</sup><sup><u>+</u></sup><sup>_N_</sup><sup><u>)(</u></sup> _M_<sup>_e_</sup><sup><u>+2</u></sup><sup>_M_</sup><sup><u>+</u></sup><sup>_N_</sup><sup><u>)</u></sup> _<_ 0, contradicting with the optimality of equilibrium because we should ˆ have _μ_ 2 _≥_ 0. Thus, school 1’s optimal capacity is _x_ 1 = _N_ .

Therefore, school 1 selects high quality if and only if ⇧ _H ≥_ ⇧ _L_ , which implies 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_<sup> ̄</sup> . Moreover, when _w_ = _w_<sup>_⇤_</sup> , _N_ takes the value _K/r_ and when _w_ = _K_ , _N_ takes a value which is less than _K/r_ because _K > k_ <u>.</u>

**Claim:** _If school 1 invests in quality, then its optimal capacity choice x_ 1 _is such that x_ 1 = _N ._

_Proof._ Suppose for a contradiction that _x_ 1 = _N_ + _e_ where _e >_ 0. This time school 1 randomly picks a price over the range [ _p_<sup>_⇤_</sup> 1<sup>_, qH_]withaprobabilitymeasure</sup><sup>_μ_1andschool2randomly</sup> picks a price over the range [ _p_<sup>_⇤_</sup> 2<sup>_, qL_]withaprobabilitymeasure</sup><sup>_μ_2.Schools’profitfunctions</sup> are given by ⇧1( _qH , μ_ 2) = ( _qH − c_ ) ⇥ _μ_ ˆ2 � _<u>M</u>_ 2<sup>+</sup><sup>_x_1</sup> � + (1 _− μ_ ˆ2) ( _M/_ 2 + _N_ )⇤ + _K − rx_ 1 _− w_ and ⇧1( _p_<sup>_⇤_</sup> 1<sup>_, μ_2) = (</sup><sup>_p⇤_</sup> 1<sup>_−c_)(</sup><sup>_x_1 +</sup><sup>_<u>M</u>_</sup> 2<sup>)+</sup><sup>_K −rx_1</sup><sup>_−w_for school 1 and ⇧2(</sup><sup>_qL, μ_1) = (</sup><sup>_qL −c_)(</sup><sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N −x_1)</sup> and ⇧2( _p_<sup>_⇤_</sup> 2<sup>_, μ_1) = (</sup><sup>_p⇤_</sup> 2<sup>_−c_)(</sup><sup>_<u>M</u>_</sup> 2<sup>)forschool2.</sup> This time equilibrium prices must satisfy _qH − p_<sup>_⇤_</sup> 1<sup>=</sup><sup>_qL−p⇤_</sup> 2<sup>.Solvingthisequalityalong</sup> with ⇧1( _qL, μ_ 2) = ⇧1( _p_<sup>_⇤_</sup> 1<sup>_, μ_2),and⇧2(</sup><sup>_qL, μ_1)=⇧2(</sup><sup>_p⇤_</sup> 2<sup>_, μ_1)impliesthateither</sup><sup>_μ_ˆ2=0,andthus</sup> ˆ ˆ ⇧1( _qL, μ_ 2) decreases with _x_ 1 and the optimal capacity should be _x_ 1 = _N_ , or _μ_ 1 = 0 and _μ_ 2 _≥_ 0. ˆ ˆ However, solving for _μ_ 2 implies that _μ_ 2 =<sup>_<u>q</u>_</sup> _q_<sup>_<u>H</u>_</sup> _H_<sup>_−_</sup> _−_<sup>_<u>q</u>_</sup> _c_<sup>_<u>L</u>−_2(</sup><sup>_<u>q</u>_</sup> _M_<sup>_<u>L−</u>_</sup> ( _q_<sup>_c_</sup> _H_<sup><u>)(</u></sup> _−_<sup>_e_</sup><sup><u>+</u></sup> _c_ )<sup>_N_</sup><sup><u>)</u></sup> which is less than zero for all _e >_ 0 whenever _<u>kr</u>  N_ . This contradicts with the optimality of the equilibrium, and thus school 1’s optimal capacity is _x_ 1 = _N_ .

Therefore, school 1’s profit is ⇧ _H_ = ( _qH − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N_) +</sup><sup>_K −w −Nr_ifitinvestsinquality</sup> and ⇧ _L_ = ( _qL − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N_) +</sup><sup>_K −Nr_ifitinvestsincapacity.Therefore,investinginqualityis</sup> optimal if and only if _w _ ( _qH − qL_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N_)whichholdsforall</sup><sup>_N_and</sup><sup>_w_aslongas</sup><sup>_K<k_ ̄.</sup> 5
<u>(</u> _<u>qH</u> −qL_ <u>)</u> **Case 4:** _Nr <_<sup>_<u>Mr</u>_</sup> 2 ( _qL−c_ ) (or equivalently _Nr <_ _<u>k</u>_ <u>):</u> In this case, school 1 prefers to select _x_ 1 _> 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_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup> _N_ ) + _K − w − Nr_ 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_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N_) +</sup><sup>_K −Nr_.Therefore,school1preferstoinvestinqualityaslongas</sup> the first term is greater than or equal to ⇧ _L_ , implying that _w _ ( _qH − qL_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N_)whichisless</sup> than _K_ because _K < k_<sup> ̄</sup> .

**_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 following figure:

<!-- Start of picture text --> as K % k ̄<br>w 4 (Case 4)<br>Case 1<br>as K % k ̄<br>w 3 (C ase 3)<br>Case 2<br>Case 4<br>H<br>Nr<br> +<br> w<br>= 2<br>Case K<br>3<br>Nr<br> + 2<br>w<br>= 2<br>K<br><!-- End of picture text -->

**Case 1:** Suppose that 2 _K  Nr_ (or equivalently, 2 _nL  N_ ): Because the uncovered market is large, price competition never occurs in this case. Therefore, ⇧ _H_ = ( _qH − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> ) and ⇧ _L_ = ( _qL − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K</u>_</sup> _r_<sup>).Moreover,⇧</sup><sup>_Dev_</sup> _H_ = ( _qL − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K</u>_</sup> _r_<sup>)and⇧</sup><sup>_Dev_</sup> _L_ = ( _qH − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> ). To have an equilibrium where one school invests in high quality and the other invests in low quality, we must have ⇧ _H ≥_ ⇧<sup>_Dev_</sup> _H_ = ⇧ _L_ and ⇧ _L ≥_ ⇧<sup>_Dev_</sup> _L_ = ⇧ _H_ implying that _w_ = _w_<sup>_⇤_</sup> , which is less than _K_ because _<u>k</u> < K_ . To have an equilibrium where both schools pick the high quality, we must have ⇧ _H ≥_ ⇧<sup>_Dev_</sup> _H_ , implying _w  w_<sup>_⇤_</sup> . Hence, there exists an equilibrium where at least one school invests in quality if and only if _w  w_<sup>_⇤_</sup> .

**Case 2:** Suppose that 2 _K − w  Nr <_ 2 _K_ (or equivalently, _nL_ + _nH  N <_ 2 _nL_ ): Because we still gave _nH_ + _nH  N_ , there exists an equilibrium where ( _H, H_ ) is an equilibrium outcome for all values of _w  w_<sup>_⇤_</sup> . 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_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> ) and ⇧ _L_ = ( _qL − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K</u>_</sup> _r_<sup>).Moreover,</sup> ⇧<sup>_Dev_</sup> _L_ = ( _qH − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> ) because the other school has picked _nH_ and 2 _nH < N_ . However, 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 x_ 2 = _nL. Then school 1’s optimal capacity choice x_ 1 _is such that x_ 1 = _N − nL._

_Proof._ Suppose for a contradiction that _x_ 1 = _N − nL_ + _e_ where _e >_ 0. In the mixed strategy equilibrium each school _i_ randomly picks a price over the range [ _p_<sup>_⇤_</sup> _i_<sup>_, qL_] with a probability measure</sup> _μi_ and we have and ˆ where _μ_ 2 = _μ_ 2( _{qL}_ ). Moreover, and ˆ where _μ_ 1 = _μ_ 1( _{qL}_ ). In equilibrium we have _p_<sup>_⇤_</sup> 1<sup>=</sup><sup>_p⇤_</sup> 2<sup>, ⇧1(</sup><sup>_qL, μ_2) = ⇧1(</sup><sup>_p⇤_</sup> 1<sup>_, μ_2), and ⇧2(</sup><sup>_qL, μ_1) =</sup> ⇧2( _p_<sup>_⇤_</sup> 2<sup>_, μ_1).Moreover, if</sup><sup>_μ_ˆ2= 0, then ⇧1(</sup><sup>_qL, μ_2) decreases with</sup><sup>_x_1, and thus the optimal capacity</sup> ˆ ˆ should be _x_ 1 = _N − x_ 2. Therefore, we must have _μ_ 1 = 0. Solving for _μ_ 2 _≥_ 0, and then solving _@_ ⇧1( _qL, μ_ 2) _/@e_ = 0 implies Because _N ≥_ (2 _K − w_ ) _/r_ , _e_ is less than or equal to _−_<sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> _−_<sup>_<u>Mr</u>_</sup> 4( _qL_<sup><u>+2</u></sup> _−_<sup>_<u>K</u>_</sup> _c_ )<sup>,whichisnegativebecause</sup> _K < w_ , contradicting with the initial assumption that _e >_ 0.

Therefore, if school 1 deviates to low quality, then its payoff is ⇧<sup>_Dev_</sup> _H_ = ( _qL − c_ )(<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_N−_</sup> _<u>Kr</u>_<sup>)</sup><sup>_−Nr_.Thus, there is an equilibrium with one school investing in quality and other investing in</sup> capacity if and only if ⇧ _L ≥_ ⇧<sup>_Dev_</sup> _L_ and ⇧ _H ≥_ ⇧<sup>_Dev_</sup> _H_ , which implies the following two inequalities: _w ≥ w_<sup>_⇤_</sup> and The last condition gives us the line **H** . Drawing the line **H** on _wN −_ space implies that the _N_ -intercept is greater than 2 _K/r_ because<sup>_<u>qH</u>_</sup> _qL_<sup><u>+</u></sup> _−_<sup>_<u>qL</u>_</sup> _c−_<sup>_−_</sup> _r_<sup>2</sup><sup>_c_</sup> _>_ 2 and the _w_ -intercept is bigger than _K_ because<sup>_<u>qH</u>_</sup> _q_<sup><u>+</u></sup> _H_<sup>_<u>q</u>_</sup> _−_<sup>_<u>L−</u>_</sup> _c_<sup>2</sup><sup>_c_</sup> _>_ 1. However, when _w_ = _K_ , **H** gives the value of<sup>_M_</sup> 2( _q_<sup><u>(</u></sup> _L_<sup>_<u>qH</u>_</sup> _−_<sup>_−_</sup> _c−_<sup>_<u>qL</u>_</sup> _r_ )<sup><u>)</u>+</sup> _rK_ ( _qL_ <u>(</u> _<u>q−Lc−−cr</u>_ <u>))</u><sup>for</sup> _N_ which is strictly greater than _K/r_ . However, it is less than or greater than 2 _K/r_ depending on whether<sup>_Mr_</sup> 2( _qL_<sup><u>(</u></sup><sup>_<u>q</u>_</sup> _−_<sup>_<u>H</u>_</sup> _c_<sup>_−_</sup> _−_<sup>_<u>q</u>_</sup> 2<sup>_<u>L</u>_</sup> _r_ )<sup><u>)</u>isgreaterorlessthan</sup><sup>_K/r_.Thatis,forsufficientlysmallvaluesof</sup><sup>_K_,</sup><sup>**H**</sup> lies above 2 _K/r_ . However, it is easy to verify that **H** always lies above _K/r_ .

**Case 3:** Suppose that 2 _K −_ 2 _w  Nr <_ 2 _K − w_ (or equivalently, 2 _nH  N < nL_ + _nH_ ): Note that for all values of _w  w_<sup>_⇤_</sup> 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 ⇧<sup>_Dev_</sup> _H_ is much less.

If ( _H, L_ ) is an equilibrium outcome, then the optimal capacity for school 2 is _x_ 2 = _N − x_ 1. 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 _x_ 2 as long as _x_ 2 _> N − x_ 1. 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 _x_ 1 = _nH_ =<sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> . Thus, in an equilibrium where ( _H, L_ ) is the outcome, 7
the profit functions are ⇧ _H_ = ( _qH − c_ ) � _<u>M</u>_ 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> � and ⇧ _L_ = ( _qL − c_ ) � _<u>M</u>_ 2<sup>+</sup><sup>_N−_</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> � + _K − r_ � _N −_<sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> �. If school 2 deviates to high quality, then its deviation payoff is ⇧<sup>_Dev_</sup> _L_ = ( _qH − c_ ) � _<u>M</u>_ 2<sup>+</sup><sup>_N−_</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> � because 2 _nH  N_ . Now we prove that it is not optimal for school 1 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 x_ 2 = _N − nH . Then school 1’s optimal capacity choice x_ 1 _is such that x_ 1 = _nH ._

_Proof._ Suppose for a contradiction that _x_ 1 = _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 _p_<sup>_⇤_</sup> 1<sup>=</sup><sup>_p⇤_</sup> 2<sup>,⇧1(</sup><sup>_qL, μ_2)=⇧1(</sup><sup>_p⇤_</sup> 1<sup>_, μ_2),and⇧2(</sup><sup>_qL, μ_1)=⇧2(</sup><sup>_p⇤_</sup> 2<sup>_, μ_1)implythatif</sup><sup>_μ_ˆ2=0,then</sup> ⇧1( _qL, μ_ 2) decreases with _x_ 1, and so the optimal capacity should be _x_ 1 = _N − x_ 2. Therefore, ˆ ˆ we must have _μ_ 1 = 0. Solving for _μ_ 2 _≥_ 0, and then solving _@_ ⇧1( _qL, μ_ 2) _/@e_ = 0 implies which is strictly less than zero because _e_ 1 __ � 2 _<u>wr</u>_<sup>+</sup><sup>_<u>N</u>_</sup> 2 � <u>(2</u> _<u>q</u>_ ( _<u>LqL−−</u>_ 2 _cc−_ ) _r_ <u>)</u> and it is less than<sup>_<u>K</u>_</sup> _r_ <u>(2</u> _<u>q</u>_ ( _<u>LqL−−</u>_ 2 _cc−_ ) _r_ <u>)</u> because we are in the region where _w_ + _Nr <_ 2 _K_ . However, _e <_ 0 contradicts with our initial assumption.

Therefore, _x_ 1 = _nH_ is the optimal choice for school 1 if it deviates to low quality, and thus _<u>M</u>_ we have ⇧<sup>_Dev_</sup> _H_ = ( _qL − c_ ) � 2<sup>+</sup><sup>_<u>K−</u>_</sup> _r_<sup>_<u>w</u>_</sup> � + _w_ . To have an equilibrium outcome ( _H, L_ ) we must have ⇧ _q ≥_ ⇧<sup>_Dev_</sup> _q_ for each _q 2 {H, L}_ . Equivalently, and It is easy to verify that the first inequality holds for allity implies _w _ (( _qqHH−−qqLL_ +) _rr_ ) � _<u>M</u>_ 2<sup>+</sup><sup>_<u>K</u>_</sup> _r_ � _⌘ w_<sup>3</sup> which is strictly _w ≥_ higher _w_<sup>_⇤_</sup> andthan _N ≥K_ 0.wheneverThe second inequal- _K  k_<sup> ̄</sup> .

**Case 4:** Suppose that _Nr <_ 2 _K −_ 2 _w_ (or equivalently, _N <_ 2 _nH_ ): We will prove, for all parameters in this range, that there exists an equilibrium where both schools invest in quality and _x_ 1 = _x_ 2 = _N/_ 2. For this purpose, we first show that school 1’s best response is to pick _x_ 1 = _N/_ 2 in equilibrium where both schools invest in quality and _x_ 2 = _N/_ 2. Suppose for a contradiction that school 1 picks _x_ 1 = _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_<sup>_⇤_</sup> _i_<sup>_, qH_]</sup> with a probability measure _μi_ and the profit functions are given by

8
ˆ where _μ_ 1 = _μ_ 1( _{qH }_ ) and ⇧2( _p_<sup>_⇤_</sup> 2<sup>_, μ_1) = (</sup><sup>_p⇤_</sup> 2<sup>_−c_)(</sup><sup>_x_2 +</sup><sup>_M/_2) +</sup><sup>_K −rx_2</sup><sup>_−w_.</sup> Once again, solving _p_<sup>_⇤_</sup> 1<sup>=</sup><sup>_p⇤_</sup> 2<sup>,⇧1(</sup><sup>_qH, μ_2)=⇧1(</sup><sup>_p⇤_</sup> 1<sup>_, μ_2),and⇧2(</sup><sup>_qH, μ_1)=⇧2(</sup><sup>_p⇤_</sup> 2<sup>_, μ_1)imply</sup> ˆ that if _μ_ 2 = 0, then ⇧1( _qL, μ_ 2) decreases with _x_ 1, and so the optimal capacity should be ˆ ˆ _x_ 1 = _N − x_ 2. Therefore, we must have _μ_ 1 = 0. Solving for _μ_ 2 _≥_ 0 yields _μ_ 2 = _−_<sup>4</sup><sup>_e_</sup> (<sup><u>(</u></sup> _M_<sup>_e_</sup><sup><u>+</u></sup> +<sup>_M_</sup> _N_<sup><u>+</u></sup> )<sup>2</sup><sup>_N_</sup><sup><u>)</u></sup> which is clearly negative for all values of _e >_ 0, yielding the desired contradiction. Therefore, school 1’s optimal capacity choice is _x_ 1 = _N − x_ 2 = _N/_ 2.

In equilibrium with ( _H, H_ ) and _xi_ = _N/_ 2 for _i_ = 1 _,_ 2, profit function is ⇧ _H_ = ( _qH − c_ ) � _<u>M</u>_ <u>+2</u> _<u>N</u>_ � + _K − w −_<sup>_<u>Nr</u>_</sup> 2<sup>.However,ifaschooldeviatestolowquality,thenitsoptimalcapacity</sup> 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, ⇧<sup>_Dev_</sup> _H_ = ( _qL − c_ ) � _<u>M</u>_ <u>+2</u> _<u>N</u>_ � + _K −_<sup>_<u>Nr</u>_</sup> 2<sup>.Thus,nodeviationimpliesthat</sup><sup>_w_(</sup><sup>_qH−_</sup> _<u>M</u>_ <u>+</u> _<u>N</u> qL_ ) � 2 � _⌘ w_<sup>4</sup> which holds for all _w  k_<sup> ̄</sup> and _N ≥_ 0. That is, for all the parameters of interest, ( _H, H_ ) is an equilibrium outcome.

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# **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 quality _q_ and price _p_ , then demand is _D_ ( _p_ ) = _T_ (1 _−_<sup>_<u>p</u>_</sup> _q_<sup>).</sup> 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.<sup>1</sup> 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:<sup>2</sup>

|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.<sup>3</sup> But, once school 2’s capacity increases to 6, there is no pure strategy NE.<sup>4</sup> The threat of mixed strategy equilibrium prices forces schools to not expand their capacities beyond their Cournot optimal capacities.<sup>5</sup>

# **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

> 1Suppose 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.

> 2For 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.

> 3Given 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).

> 4Now _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.

> 5In 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_ .

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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 _x_ 1 and _x_ 2, the market price is _P_ ( _x_ 1 + _x_ 2) = _q_ (1 _− x_ 1 + _x_ 2). Therefore, the best response function for school with no capacity cost is which implies that According to Proposition 1 of KS, if _xi  B_ ( _xj_ ) for _i, j_ = 1 _,_ 2 and _i_ = _j_ , then a subgame equilibrium is for each school to name price _P_ ( _x_ 1 + _x_ 2) with probability one. The equilibrium revenues are _xiP_ ( _x_ 1 + _x_ 2) 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 somewhere between [<sup>_x_</sup> _x_<sup>_<u>j</u>_</sup> _i_<sup>_R_(</sup><sup>_xj_)</sup><sup>_, R_(</sup><sup>_xj_)].</sup>

# **Equilibrium Prices when Qualities are**

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<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_</sup><sup>_<u>p</u>_</sup> _qH_<sup>_<u>H</u>_.Thenext</sup> 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|(any) school<br>students willing to go to High|
|---|---|
|0<br>_pL_<br><br>_pH_<br>|1<br>_pH−pL_|
|_qL_<br>_qH_|_qH−qL_|
|prefer Low to High|prefer High to Low|

Therefore, demand for the high quality school is _DH_ = 1 _−_<sup>_<u>p</u>_</sup> _q_<sup>_<u>H</u>_</sup> _H_<sup>_−_</sup> _−_<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_andenrollmentis</sup><sup>_eH_=</sup> min ⇣ _xH ,_ 1 _−_<sup>_<u>p</u>_</sup> _q_<sup>_<u>H</u>_</sup> _H_<sup>_−_</sup> _−_<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_</sup> ⌘. Demand for the low quality school is

**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_ =<sup>_<u>qH</u>−q_</sup> 2<sup>_<u>L</u>_</sup><sup><u>+</u></sup><sup>_<u>pL</u>_</sup> . Therefore, the best response price for the high quality school is _PH_ ( _pL_ ) =<sup>_<u>qH</u>−q_</sup> 2<sup>_<u>L</u>_</sup><sup><u>+</u></sup><sup>_<u>pL</u>_</sup> whenever the school’s capacity does not fall short of the demand at these prices, i.e. _pL _ ( _qH − qL_ )(2 _xH −_ 1). Otherwise, i.e. _pL >_ ( _qH − qL_ )(2 _xH −_ 1), we have _PH_ ( _pL_ ) = _pL_ + (1 _− xH_ )( _qH − qL_ ). To sum,

Now, given _xH , xL_ and _pH_ , we find the best response price for the low quality school, _pL_ . We know that if _xH ≥_ 1 _−_<sup>_<u>p</u>_</sup> _q_<sup>_<u>H</u>_</sup> _H_<sup>_−_</sup> _−_<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_,thentheenrollmentis</sup><sup>_eL_= min</sup> ⇣ _xL,_<sup>_<u>p</u>_</sup> _q_<sup>_<u>H</u>_</sup> _H_<sup>_−_</sup> _−_<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>−_</sup><sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_</sup> ⌘. However, 11
if _xH <_ 1 _−_<sup>_<u>p</u>_</sup> _q_<sup>_<u>H</u>_</sup> _H_<sup>_−_</sup> _−_<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_,thentheenrollmentis</sup><sup>_eL_=min</sup> ⇣ _xL,_ 1 _−_<sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>−xH_</sup> ⌘. Therefore, the profit functions are as follows: maximizing _pL_ ’s yield the following best response function:

**Finding Optimal Prices:** Solving the best response functions simultaneously implies working out the following eight cases:

**Case 1:** Consider the parameters satisfying

so that the best response function for the high quality school is _PH_ ( _pL_ ) =<sup>_<u>qH</u>−q_</sup> 2<sup>_<u>L</u>_</sup><sup><u>+</u></sup><sup>_<u>pL</u>_</sup> . We need to consider the following four sub-cases:

**Case 1.1:** Consider the parameters satisfying

so that the best response function for the low quality school is _PL_ ( _pH_ ) =<sup>_<u>p</u>_</sup> 2<sup>_<u>H</u>_</sup> _qH_<sup>_<u>qL</u>_.Solving the best</sup> response functions simultaneously yields Therefore, the inequalities (5) and (6) yield _xH ≥_ 4 _q_ 2 _Hq−HqL_<sup>andequation(7)yields</sup><sup>_xL≥_</sup> 4 _qHqH−qL_<sup>.</sup>

**Case 1.2:** Consider the parameters satisfying

12
so that the best response function for the low quality school is _PL_ ( _pH_ ) =<sup>_<u>pHqL−xL</u>_</sup> _q_<sup>_<u>q</u>_</sup> _H_<sup>_<u>L</u>_</sup><sup><u>(</u></sup><sup>_<u>qH</u>−qL_</sup><sup><u>)</u></sup> . Solving them simultaneously yields Therefore, the inequalities (5) and (8) yield _qH  qLxL_ + (2 _qH − qL_ ) _xH_ and equation (9) yields _<u>qH</u> xL <_ 4 _qH −qL_<sup>.</sup>

**Case 1.3:** Consider the parameters satisfying

so that the best response function for the low quality school is _PL_ ( _pH_ ) =<sup><u>(1</u></sup><sup>_−x_</sup> 2<sup>_<u>H</u>_</sup><sup><u>)</u></sup><sup>_<u>qL</u>_</sup> . Solving them simultaneously yields The inequality (10) yields _xH <_ 4<sup>2</sup> _q_<sup>_<u>q</u>_</sup> _H_<sup>_<u>H</u>_</sup> _−_<sup>_−_</sup> 3<sup>_<u>q</u>_</sup> _q_<sup>_<u>L</u>_</sup> _L_<sup>and the inequality (5) yields</sup><sup>_xH≥_</sup> 4<sup>2</sup> _q_<sup>_<u>q</u>_</sup> _H_<sup>_<u>H</u>_</sup> _−_<sup>_−_</sup> 3<sup>_<u>q</u>_</sup> _q_<sup>_<u>L</u>_</sup> _L_<sup>, which can-</sup> 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

so that the best response function for the low quality school is _PL_ ( _pH_ ) = _qL_ (1 _− xH − xL_ ). Solving them simultaneously yields The inequality (12) yields _xH <_<sup>_<u>q</u>_</sup> 2<sup>_<u>H</u>_</sup> _q_<sup>_−_</sup> _H_<sup>_<u>q</u>_</sup> _−_<sup>_<u>L</u>_</sup> _q_<sup>_x_</sup> _L_<sup>_<u>L</u>_and the inequality (5) yields</sup><sup>_xH≥_</sup><sup>_<u>q</u>_</sup> 2<sup>_<u>H</u>_</sup> _q_<sup>_−_</sup> _H_<sup>_<u>q</u>_</sup> _−_<sup>_<u>L</u>_</sup> _q_<sup>_x_</sup> _L_<sup>_<u>L</u>_, which can-</sup> 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

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

so that the best response function for the low quality school is _PL_ ( _pH_ ) =<sup>_<u>p</u>_</sup> 2<sup>_<u>H</u>_</sup> _qH_<sup>_<u>qL</u>_.Solving the best</sup> response functions simultaneously yields Therefore, the inequalities (14), (15), and (16) yield _xH <_ 4 _q_ 2 _Hq−HqL_<sup>,</sup><sup>_xH≥xH_, and</sup><sup>_qHxH_+(2</sup><sup>_qH −_</sup> _qL_ ) _xL ≥ qH_ respectively.

**Case 2.2:** Consider the parameters satisfying

so that the best response function for the low quality school is _PL_ ( _pH_ ) =<sup>_<u>pHqL−xL</u>_</sup> _q_<sup>_<u>q</u>_</sup> _H_<sup>_<u>L</u>_</sup><sup><u>(</u></sup><sup>_<u>qH</u>−qL_</sup><sup><u>)</u></sup> . Solving them simultaneously yields Therefore, the inequalities (14), (17), and (18) yield _qH > xLqL_ + _xH_ (2 _qH − qL_ ), _xH ≥ xH_ , and _qH xH_ + (2 _qH − qL_ ) _xL < qH_ respectively.

**Case 2.3:** Consider the parameters satisfying

so that the best response function for the low quality school is _PL_ ( _pH_ ) =<sup><u>(1</u></sup><sup>_−x_</sup> 2<sup>_<u>H</u>_</sup><sup><u>)</u></sup><sup>_<u>qL</u>_</sup> . Solving them simultaneously yields The inequality (19) yields _xH < xH_ implying that there cannot exist an equilibrium for the parameter values satisfying inequalities (14), (19), and (20).

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**Case 2.4:** Consider the parameters satisfying

so that the best response function for the low quality school is _PL_ ( _pH_ ) = _qL_ (1 _− xH − xL_ ). Solving them simultaneously yields 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

**Region 1:** Parameters satisfy _xH ≥_ 4 _q_ 2 _Hq−HqL_<sup>and</sup><sup>_xL≥_</sup> 4 _qHqH−qL_<sup>.Equilibriumpricesare</sup> _pL_ =<sup>_<u>qL</u>_</sup> 4<sup><u>(</u></sup> _q_<sup>_<u>q</u>_</sup> _H_<sup>_<u>H</u>_</sup> _−_<sup>_−_</sup> _q_<sup>_<u>q</u>_</sup> _L_<sup>_<u>L</u>_</sup><sup><u>)</u></sup> and _pH_ =<sup>2</sup><sup>_<u>qH</u>_</sup> 4 _q_<sup><u>(</u></sup> _H_<sup>_<u>qH</u>_</sup> _−_<sup>_−_</sup> _qL_<sup>_<u>qL</u>_</sup><sup><u>)</u></sup> . Therefore, enrollment and revenue (per student) of the high quality school are _eH_ = 4 _q_ 2 _Hq−HqL_<sup>and⇧</sup><sup>_H_=4</sup> (4<sup>_<u>q</u>_</sup> _<u>H</u>_<sup>2</sup> _qH_<sup><u>(</u></sup><sup>_<u>q</u>_</sup> _−_<sup>_H_</sup> _q_<sup>_−_</sup> _L_<sup>_<u>q</u>_</sup> )<sup>_L_2).Notethatthisisnot</sup> the profit function of the high quality school, and so the cost of choosing capacity _xH_ and high quality are excluded.

**Region 2:** Parameters satisfy _xL <_ 4 _qHqH−qL_<sup>and</sup><sup>_qLxL_+ (2</sup><sup>_qH −qL_)</sup><sup>_xH≥qH_.Equilibrium</sup> prices are _pL_ =<sup>_<u>qL</u>_</sup><sup><u>(</u></sup><sup>_<u>qH</u>_</sup> 2<sup>_−_</sup> _qH_<sup>_<u>qL</u>_</sup> _−_<sup><u>)(</u></sup> _q_<sup>1</sup> _L_<sup>_−_2</sup><sup>_xL_</sup><sup><u>)</u></sup> and _pH_ =<sup><u>(</u></sup><sup>_<u>qH</u>−q_</sup> 2<sup>_<u>L</u>_</sup> _q_<sup><u>)(</u></sup> _H_<sup>_<u>q</u>_</sup> _−_<sup>_<u>H</u>_</sup> _q_<sup>_−_</sup> _L_<sup>_<u>qLxL</u>_</sup><sup><u>)</u></sup> . Therefore, enrollment and revenue (per student) of the high quality school are _eH_ = _<u>q</u>_ 2 _<u>Hq−Hq−LqxLL</u>_ and ⇧ _H_ = ( _qH − qL_ )<sup><u>(</u></sup> (2<sup>_<u>qH</u>_</sup> _q_<sup>_−_</sup> _H_<sup>_<u>q</u>_</sup> _−_<sup>_<u>L</u>_</sup> _q_<sup>_x_</sup> _L_<sup>_<u>L</u>_</sup> )<sup><u>)22</u>.</sup>

**Region 3:** Parameters satisfy _xH <_ 4 _q_ 2 _Hq−HqL_<sup>and</sup><sup>_qHxH_+(2</sup><sup>_qH −qL_)</sup><sup>_xL≥qH_.Equilibrium</sup> prices are _pL_ =<sup>_<u>qL</u>_</sup><sup><u>(</u></sup><sup>_<u>qH</u>_</sup> 2<sup>_−_</sup> _qH_<sup>_<u>qL</u>_</sup> _−_<sup><u>)(</u></sup> _qL_<sup>1</sup><sup>_−xH_</sup><sup><u>)</u></sup> and _pH_ =<sup>2</sup><sup>_<u>qH</u>_</sup><sup><u>(</u></sup><sup>_<u>qH</u>_</sup> 2 _q_<sup>_−_</sup> _H_<sup>_<u>q</u>_</sup> _−_<sup>_<u>L</u>_</sup> _q_<sup><u>)(</u></sup> _L_<sup>1</sup><sup>_−xH_</sup><sup><u>)</u></sup> . Therefore, enrollment and revenue of the high quality school are _eH_ = _xH_ and ⇧ _H_ =<sup>2</sup><sup>_<u>qH</u>_</sup><sup><u>(</u></sup><sup>_<u>qH</u>−_</sup> 2 _q_<sup>_<u>q</u>_</sup> _H_<sup>_<u>L</u>_</sup> _−_<sup><u>)(</u></sup> _q_<sup>1</sup> _L_<sup>_−xH_</sup><sup><u>)</u></sup><sup>_xH_</sup> . Moreover, the profit of the low quality school is ⇧ _L_ = _pL_ ⇣ _<u>pqHH</u> −−pqLL_<sup>_−_</sup><sup>_<u>p</u>_</sup> _qL_<sup>_<u>L</u>_</sup> ⌘ =<sup>_<u>qHqL</u>_</sup><sup><u>(</u></sup> (2<sup>_<u>qH</u>_</sup> _qH_<sup>_−q_</sup> _−_<sup>_<u>L</u>_</sup> _q_<sup><u>)(</u></sup> _L_<sup>1</sup> )<sup>2</sup><sup>_−xH_</sup><sup><u>)2</u></sup> .

**Region 4:** Parameters satisfy _qH xH_ +(2 _qH − qL_ ) _xL < qH_ and _qLxL_ +(2 _qH − qL_ ) _xH < qH_ . Equilibrium prices are _pL_ = _qL_ (1 _− xH − xL_ ) and _pH_ = (1 _− xH_ ) _qH − xLqL_ . Enrollment and revenue of the high quality school are _eH_ = _xH_ and ⇧ _H_ = _xH_ [(1 _− xH_ ) _qH − xLqL_ ]. Enrollment and revenue of the low quality school are _eL_ = _xL_ and ⇧ _L_ = _pLxL_ = _qL_ (1 _− xH − xL_ ) _xL_ .

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<!-- Start of picture text --> xH<br>1<br>Region 2 Region 1<br>qH<br>2 qH −qL<br>2 qH<br>4 qH −qL<br>12 Region 3<br>Region 4<br>qH 1 qH 1 qH xL<br>4 qH −qL 2 2 qH −qL qL<br><!-- End of picture text -->

# **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 quality. Moreover, the schools’ initial capacity is _x_ 1 = _x_ 2 =<sup>_<u>M</u>_</sup> 2<sup>.Therefore,thebaselinemarket</sup> price is _P_ ( _M_ ) = _qL_ (1 _− M_ ). We make the following two assumptions regarding the size of the covered market, _M_ :

**Assumption 1:** 2 _ TM_ . That is, total private school enrollment is at least 2.

**Assumption 3:** _TrK_<sup>+</sup><sup>_<u>M</u>_</sup> 2<sup>__</sup> 4 _q_ 2 _Hq−HqL_<sup>.</sup>

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:** _x_ 1 = _x_ 2 =<sup>_<u>M</u>_</sup> 2<sup>satisfy the constraint</sup><sup>_qHx_1+</sup><sup>_x_2(2</sup><sup>_qH−qL_)</sup><sup>_< qH_if assumption</sup> 2 holds.

**Observation 2:** 4 _q_ 2 _Hq−HqL_<sup>_>_</sup><sup><u>1</u></sup> 2<sup>,andso</sup><sup>_<u>M</u>_</sup> 2<sup>_<_</sup> 4 _q_ 2 _Hq−HqL_<sup>.</sup>

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_ (2 _qH − qL_ ) _< qH and qLxL_ + _xH_ (2 _qH − 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 assumption 3 that school 1’s final capacity will never be above 2 _qH /_ (4 _qH − 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.

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School 2’s profit, if it picks _x_ such that _x_ +<sup>_<u>M</u>_</sup> 2<sup>and</sup><sup>_xH_remainsinRegion4,is</sup> The first order conditions imply that the optimal (additional) capacity is<sup>1</sup><sup>_−xH_</sup> 2<sup>_−r/qL_</sup> _−_<sup>_<u>M</u>_</sup> 2<sup>or</sup> less if the grant is not large enough to cover this additional capacity. On the other hand, the capacity school 2 needs to move to Region 3, _xL_ , must satisfy _xL ≥_<sup>_<u>qH</u>_</sup> 2 _q_<sup><u>(</u></sup> _H_<sup>1</sup><sup>_−_</sup> _−_<sup>_x_</sup> _q_<sup>_<u>H</u>_</sup> _L_<sup><u>)</u>,whichisstrictly</sup> higher _x_ +<sup>_<u>M</u>_</sup> 2<sup>.Therefore,givenschool1’schoice,school2’soptimalcapacitywillbesuchthat</sup> 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<sup>_<u>qH</u>_</sup> 2 _q_<sup><u>(</u></sup> _H_<sup>1</sup><sup>_−_</sup> _−_<sup>_x_</sup> _q_<sup>_<u>H</u>_</sup> _L_<sup><u>)</u></sup> because school 2’s profit does not depend on its capacity beyond this level. Therefore, the profit under this capacity level would be However, if school 2 picks _x_ and remains in Region 4, then its would be The yields 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_ .

<u>Low saturation treatment:</u> If school 1 invests in quality its is High saturation treatment with <u>(</u> _H, L_ <u>)</u> Equilibrium: We are trying to create an equilibrium where at least one school invests in high quality. In an equilibrium where only one school invests in quality, the low quality school’s optimal capacity choice is _x_<sup>_l_</sup> =<sup><u>1</u></sup> 2 ⇣1 _−_<sup><u>3</u></sup><sup>_<u>M</u>_</sup> 2<sup>_−_</sup> _q_<sup>_<u>r</u>_</sup> _L_ ⌘ and profit 17
is On the other hand, the high quality school’s equilibrium is where _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 However, if the high quality school deviates to low quality, then we are again in KS world. b Thus, given that the other school’s capacity is _xL_ , deviating school’s optimal capacity is _x_ = <u>12</u> ⇣1 _− M − xL − q_<sup>_<u>r</u>_</sup> _L_ ⌘ and optimal profit is High saturation treatment with <u>(</u> _H, H_ <u>)</u> Equilibrium: Because _w_ = _K_ , schools cannot increase their capacities. Moreover, we are in KS world, and so the equilibrium payoff is Note the following:

**Claim 1.** _If x_<sup>_l_</sup> _<_ min � _TrK_<sup>_, B_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup> � _, then x <_ b min � _TrK_<sup>_, B_(</sup><sup>_xL_)</sup> � _. Proof._ Assume that _x_<sup>_l_</sup> satisfies the above inequality. Then _xL_ =<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_xl_,</sup><sup>_B_(</sup><sup>_xL_) =</sup><sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_−_</sup><sup>_<u>x</u>_</sup> 2<sup>_l_,</sup> and _x_ b =<sup>_<u>xl</u>_whichislessthan</sup> _<u>K</u>_<sup>Moreover,</sup><sup>_x_b</sup><sup>_<B_(</sup><sup>_xL_)because</sup><sup>_xl<B_(</sup><sup>_<u>M</u>_andthusthe</sup> 2<sup>,</sup> _Tr_<sup>.</sup> 2<sup>),</sup> desired result.

**Claim 2.** _If Tr_<sup>_<u>K</u>< xlB_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_,theneitherx <_bmin</sup> � _TrK_<sup>_, B_(</sup><sup>_xL_)</sup> � _or Tr_<sup>_<u>K</u><x_b</sup><sup>_B_(</sup><sup>_xL_)</sup><sup>_._</sup>

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_Proof._ In this case _xL_ =<sup>_<u>M</u>_</sup> 2<sup>+</sup> _Tr_<sup>_<u>K</u>_,</sup><sup>_B_(</sup><sup>_xL_) =</sup><sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_−_</sup> 2 _<u>KTr</u>_<sup>,and</sup><sup>_x_b =</sup><sup>_xl −_</sup> 2 _<u>KTr</u>_<sup>.Therefore,wehave</sup> _x_ b _ B_ ( _xL_ ) because _x_<sup>_l_</sup> _< B_ (<sup>_<u>M</u>_</sup> 2<sup>).However,</sup><sup>_x_bmaybegreaterorlessthan</sup> _Tr_<sup>_<u>K</u>_,hencethedesired</sup> result.

_<u>K K</u>_ **Claim 3.** _If B_ (<sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_<_min</sup> � _Tr_<sup>_, xl_�</sup> _, then B_ ( _xL_ ) _<_ min � _Tr_<sup>_,_b</sup><sup>_x_</sup> � _. Proof._ In this case _xL_ =<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>),</sup><sup>_B_(</sup><sup>_xL_)=</sup><sup><u>1</u></sup> 2<sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>),and</sup><sup>_x_b=</sup><sup>_xl−_</sup><sup><u>1</u></sup> 2<sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>),Therefore,we</sup> have _x > B_ b ( _xL_ ) and _B_ ( _xL_ ) _< B_ (<sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_<_</sup> _Tr_<sup>_<u>K</u>_,andthusthedesiredresult.</sup>

> _<u>K</u>_ b _<u>K</u>_ **Lemma 1.** _Suppose that x_<sup>_l_</sup> __ min � _Tr_<sup>_, B_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup> � _and x _ min � _Tr_<sup>_, B_(</sup><sup>_xL_)</sup> � _. 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 school in low saturation treatment is _x_<sup>_l_</sup> , and thus _xL_ =<sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_xl_and</sup><sup>_x_b=</sup><sup>_<u>x</u>_</sup> 2<sup>_l_.Assumethatthe</sup> treated school in the low saturation treatment invests in quality. Then we must have or equivalently,<sup>_<u>TM</u>_</sup> 2 ⇥�1 _−_<sup>_<u>M</u>_</sup> 2 � _qH −_<sup>_<u>M</u>_</sup> 2<sup>_qL_</sup> ⇤ _≥ K_ + _T_ h <u>(2</u> _−_ 16 _M_ <u>)</u><sup>2</sup> _qL −_<sup><u>(2</u></sup><sup>_−_</sup> 4<sup>3</sup><sup>_M_</sup><sup><u>)</u></sup> _r_ + 4<sup>_<u>r</u>_</sup> _q_<sup>2</sup> _L_ i. 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.,

- (2) Alternatively, the schools do not deviate from ( _H, H_ ), that is

Note that if ⇧<sup>_L_</sup> ( _H,L_ )<sup>_<_⇧b</sup><sup>_L_</sup> ( _H,L_ )<sup>, then the inequality in (2) holds, and so we have an equilibrium</sup> where both schools pick high quality. Inversely, if the inequality in (2) does not hold, then ⇧<sup>_L_</sup> ( _H,L_ )<sup>_≥_⇧b</sup><sup>_L_</sup> ( _H,L_ )<sup>,i.e.,thelowqualityschooldoesnotdeviatefrom(</sup><sup>_H, L_).Ifweshowthatthe</sup> high quality school also doesn’t deviate from ( _H, L_ ), then we complete our proof. Because ⇧<sup>_H_</sup> _Low_<sup>_≥_</sup> ⇧<sup>_L_</sup> _Low_<sup>, showing ⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup><sup>_H_</sup> _Low_<sup>_≥_⇧b</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup><sup>_L_</sup> _Low_<sup>would prove that the second inequality in (1) holds</sup> as well. Therefore, we will prove that ⇧<sup>_H_</sup> _Low_<sup>_−_⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>+ b⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup> _Low_<sup>_L_=</sup><sup>_TM_</sup> 2<sup>_<u>qL</u>_</sup> _x_<sup>_l_</sup> +⇧<sup>b</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup> _Low_<sup>_L_0.</sup> Thus, either ( _H, L_ ) or ( _H, H_ ) is an equilibrium outcome.

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**Lemma 2.** _Suppose that TrK_<sup>_<xlB_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_andx_b</sup><sup>__min</sup> � _TrK_<sup>_, B_(</sup><sup>_xL_)</sup> � _. 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 _x_<sup>_l_</sup> is greater than _TRK_<sup>,and thus</sup><sup>_xL_=</sup><sup>_<u>M</u>_</sup> 2<sup>+</sup> _Tr_<sup>_<u>K</u>_.Moreover,because</sup><sup>_x <_bmin</sup> � _TrK_<sup>_, B_(</sup><sup>_xL_)</sup> � holds, we have _x_<sup>_l_</sup> _<_ 2<sup><u>3</u></sup> _Tr_<sup>_<u>K</u>_.Assume that the treated school in the low saturation treatment invests</sup> in quality. Then we must have or equivalently, _<u>TM</u>_ 2 ⇥�1 _−_<sup>_<u>M</u>_</sup> 2 � _qH −_<sup>_<u>M</u>_</sup> 2<sup>_qL_</sup> ⇤ _≥ TqL_ � _TrK_<sup>+</sup><sup>_<u>M</u>_</sup> 2 ��1 _− M − Tr_<sup>_<u>K</u>_</sup> �. 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.,

- (2) Alternatively, the schools do not deviate from ( _H, H_ ), that is

Note that if ⇧<sup>_L_</sup> ( _H,L_ )<sup>_<_⇧b</sup><sup>_L_</sup> ( _H,L_ )<sup>, then the inequality in (2) holds, and so we have an equilibrium</sup> where both schools pick high quality. Inversely, if the inequality in (2) does not hold, then ⇧<sup>_L_</sup> ( _H,L_ )<sup>_≥_⇧b</sup><sup>_L_</sup> ( _H,L_ )<sup>,i.e.,thelowqualityschooldoesnotdeviatefrom(</sup><sup>_H, L_).Ifweshowthatthe</sup> high quality school also doesn’t deviate from ( _H, L_ ), then we complete our proof. Because ⇧<sup>_H_</sup> _Low_<sup>_≥_</sup> ⇧<sup>_L_</sup> _Low_<sup>, showing ⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup><sup>_H_</sup> _Low_<sup>_≥_⇧b</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup><sup>_L_</sup> _Low_<sup>would prove that the second inequality in (1) holds</sup> as well. Therefore, we will prove that ⇧<sup>_H_</sup> _Low_<sup>_−_⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>+ b⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup><sup>_L_</sup> _Low_<sup>=</sup><sup>_KM_</sup> 2 _r_<sup>_<u>qL</u>_</sup> + ⇧<sup>b</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup><sup>_L_</sup> _Low_<sup>__0.</sup> Thus, either ( _H, L_ ) or ( _H, H_ ) is an equilibrium outcome.

_<u>K K</u>_ **Lemma 3.** _Suppose that Tr_<sup>_<xlB_(</sup><sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_and_</sup> _Tr_<sup>_<x_b</sup><sup>_B_(</sup><sup>_xL_)</sup><sup>_.Ifthetreatedschoolinthe_</sup> _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 ⇧<sup>_H_</sup> _Low_<sup>_≥_⇧</sup><sup>_L_</sup> _Low_ or equivalently, _<u>TM</u>_ 2 ⇥�1 _−_<sup>_<u>M</u>_</sup> 2 � _qH −_<sup>_<u>M</u>_</sup> 2<sup>_qL_</sup> ⇤ _≥ TqL_ � _TrK_<sup>+</sup><sup>_<u>M</u>_</sup> 2 ��1 _− M − Tr_<sup>_<u>K</u>_</sup> �. 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.,

- (2) Alternatively, the schools do not deviate from ( _H, H_ ), that is

Thus, either ( _H, L_ ) or ( _H, H_ ) is an equilibrium outcome.

**Lemma 4.** _Suppose that B_ (<sup>_<u>M</u>_</sup> 2<sup>)</sup><sup>_<_min</sup> � _TrK_<sup>_, xl_</sup> _and B_ ( _xL_ ) _<_ min � _TrK_<sup>_,_b</sup><sup>_x_</sup> _. 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 _B_ (<sup>_<u>M</u>_</sup> 2<sup>) =</sup><sup><u>1</u></sup> 2<sup>_−_</sup><sup>_<u>M</u>_</sup> 4<sup>,</sup><sup>_xL_=</sup><sup>_<u>M</u>_</sup> 2<sup>+</sup><sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>),and</sup><sup>_B_(</sup><sup>_xL_) =</sup><sup><u>1</u></sup> 2<sup>_B_(</sup><sup>_<u>M</u>_</sup> 2<sup>).</sup> Assume that the treated school in the low saturation treatment invests in quality. Then we must have ⇧<sup>_H_</sup> _Low_<sup>_≥_⇧</sup><sup>_L_</sup> _Low_<sup>or equivalently,</sup><sup>_<u>TM</u>_</sup> 2 ⇥�1 _−_<sup>_<u>M</u>_</sup> 2 � _qH −_<sup>_<u>M</u>_</sup> 2<sup>_qL_</sup> ⇤ _≥ TqL_ � _B_ (<sup>_<u>M</u>_</sup> 2<sup>) +</sup><sup>_<u>M</u>_</sup> 2 ��1 _− M − B_ (<sup>_<u>M</u>_</sup> 2<sup>)</sup> �+ _K−TrB_ (<sup>_<u>M</u>_</sup> 2<sup>).Then we need to show that either (</sup><sup>_H, L_) or (</sup><sup>_H, H_) is an equilibrium.Equivalently,</sup> we need to show that either the inequalities in (1) or (2) below hold:

Same as before if we show that the high quality school doesn’t deviate from ( _H, L_ ), i.e., 21
Thus, either ( _H, L_ ) or ( _H, H_ ) is an equilibrium outcome.

Finally, the converse of the claim is not necessarily true because ⇧<sup>_H_</sup> _Low_<sup>_−_⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>+ b⇧</sup><sup>_H_</sup> ( _H,L_ )<sup>_−_⇧</sup> _Low_<sup>_L_</sup> 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, <u>37</u><sup>ofthevillagesareassignedtothe</sup><sup>_L_arm,and</sup><sup><u>2</u></sup> 7<sup>eachtothe</sup><sup>_H_armandthecontrolgroup.</sup> 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

- _slowuntreated_ = _BB−_ 1

To see why weighting is necessary, consider this example. Assume we are interested in the following unweighted simple difference regression: _Yij_ = _↵_ + _βTij_ + _✏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 _β_ = [ _E_ ( _TT_<sup>_0_</sup> )]<sup>_−_1</sup> _E_ ( _TY_ ).

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: _Yij_ = _↵_ + _β_ 0 _T_<sup> ̃</sup> _ij_ + _✏ij_ , and our _β_ 0 = [ _E_ ( _T_<sup> ̃</sup> _T_<sup> ̃</sup><sup>_0_</sup> )]<sup>_−_1</sup> _E_ ( _T_<sup> ̃</sup> _Y_<sup> ̃</sup> ), where _T_<sup> ̃</sup> and _Y_<sup> ̃</sup> are obtained by multiplying through by ~~p~~ _Bj_ where _Bj_ is 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:

- _shigh_ = _scontrol_ =<sup><u>7</u></sup> 2

- _slowtreated_ =<sup><u>7</u></sup> 3<sup>_B_</sup>

> _• slowuntreated_ =<sup><u>7</u></sup> 3 _BB−_ 1

23
# **B.2 Tracking Weights**

In addition to the saturation weights, tracking weights are required to account for the randomized 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 concerns, 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 under 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 _L_<sup>_t_</sup> 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 numbers 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 information from school owners. The financial 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.<sup>6</sup> 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.<sup>7</sup> 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. 6The 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.

> 797 percent of schools (394/408) had positive enrollment in grade 4.

26
# **Data**

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**<br>**Source**|
|---|---|---|
|_Group A: Village L_|_evel_||
|Grant per capita|Grant amount per private school going child in<br>treatment villages.<br>For _L_ villages, this is Rs<br>50,000/total private enrollment, and for_H_ villages,<br>this equals (50,000*# of private schools in vil-<br>lage)/total private enrollment. Control schools are<br>assigned a value of 0.|School|
|_Group B: School L_|_evel_||
|Closure|An indicator variable taking the value ‘1’ if a school<br>closed during the study period|School|
|Refusal|An indicator variable taking the value ‘1’ if a school<br>refused a given survey||
|Enrollment|School enrollment in all grades, verifed through<br>school registers. Coded as 0 after school closure.|School|
|Fees|Monthly tuition fees charged by the school averaged<br>across all grades.|School|
|Posted Revenues|Sum of revenues across all grades obtained by mul-<br>tiplying enrollment in each grade by the monthly<br>fee charged for that grade. Coded as 0 after school<br>closure.|School|
|Collected<br>Rev-<br>enues|Self-reported measure on total monthly fee collec-<br>tions from all enrolled students. Coded as 0 after<br>school closure unless otherwise specifed.|School|
|Test Scores|Child test scores in English, Math and Urdu, are av-<br>eraged across enrolled children to generate school-<br>level test scores in these subjects. Tests are graded<br>using item response theory (IRT), which appropri-<br>ately adjusts for the difculty of each question and<br>allows for comparison across years in standard de-<br>viation units.|Child<br>tests|
|Stayer|A stayer is a child who self-reports being at the<br>same school for at least 18 months in round 3.|Child<br>survey|
|Fixed Costs|Sum<br>of<br>spending<br>on<br>infrastructure<br>(construc-<br>tion/rental of a new building, additional classroom,<br>furniture and fxtures), educational materials, and<br>other miscellaneous items in a given year. Data is<br>collected at the item level, e.g.<br>furniture, equip-<br>ment, textbooks etc. Coded as 0 after school clo-<br>sure.|School|

_Continued on next page_ 27
_Continued_ _<u>from</u> previous page_

|**Variable**|**Description**|**Survey**<br>**Source**|
|---|---|---|
|Variable Costs|Sum of spending on teacher salaries, non-teaching<br>staf salaries, rent and utilities for a given month.<br>Coded as 0 after school closure.|School|
|Sources of school<br>funding (Y/N)|Indicator variables for whether school items were<br>purchased through (i) self-fnancing- school fees or<br>owner’s own household income, or (ii) credit- loans<br>from a bank or MFI|School|
|Household<br>bor-<br>rowing (Y/N)|Indicator variables for borrowing behavior of the<br>school owner’s household: whether household ever<br>borrowed from any sources, formal sources (e.g.<br>bank, MFI) and informal (e.g. family, friend, pawn-<br>shop, moneylender) sources.|School<br>owner|
|Household<br>bor-<br>rowing:<br>Loan<br>value|Value of total borrowing in PKR by the owner<br>household from any source for any purpose.|School<br>owner|
|_Group C: Teacher _|_Level_||
|Teacher salaries|Monthly salary collected for each teacher present<br>during survey.|Teacher<br>roster|
|Teacher start date|YYYY-MM at which the teacher started her tenure<br>at the school. This allows us to tag a teacher as<br>a newly arrived or an existing teacher relative to<br>treatment date.|Teacher<br>roster|

28
# **Appendix(Figure(C1:(Sample(Details**

<!-- Start of picture text --> Sample Control( HighDsaturaEon LowDsaturaEon<br>776villages,666 756villages,666666 1146villages,666666<br>2496schools6 2286schools6 3786schools6<br>Treatment+Offers<br>06schools6 2286schools6 1146schools6<br>Take0up 06schools6 2136schools6 1096schools6<br><!-- End of picture text -->

# **Appendix(Figure(C2:(Project(Timeline**

<!-- Start of picture text --> 2012 2013 2014<br>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<br>Baseline6Survey<br>Baseline6Child6Testing6<br>Randomization6Ballot<br>Disbursements<br>Round61<br>Round62<br>Round636<br>Round64<br>Round656<br><!-- End of picture text --> Notes:6Rounds61T36correspond6to6the6first6school6year6and6rounds646and656refer6to6the6second6school6year6after6treatment.6A6school6 year6in6this6region6is6typically6from6AprilTMarch,6with6a6three6month6break6for6summer6between6JuneTAugust.6

# **Appendix(Figure(C3:(Data(Availability(by(Survey(Rounds**

|Outcome|Baseline|Round1|Round2|Round3|Round4|Round5|
|---|---|---|---|---|---|---|
|Enrollment|✓|✓|✓|✓|✓|✓|
|Fees|✓|✓|✓|✓|✓|✓|
|PostedRevenues|✓|✓|✓||✓||
|CollectedRevenues|||✓|✓|✓|✓|
|Expenditures|✓|✓||||✓|
|TestScores*|✓|||✓|||
|Teachervariables*|✓|✓||||✓|

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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> schools in comparison with 228 _H_ and 249 control schools. The smaller sample size for the _L_<sup>_t_</sup> group increases the likelihood that the distributional overlap for a given covariate between the _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> and control, or _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> versus control; we observe some imbalance at the 10% level for _H_ vs _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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.<sup>8</sup> The weight is the inverse probability of response (1 _− prob_ ( _attrition_ ))<sup>_−_1</sup> , 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.

> 8The 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)|
|---|---|---|---|---|---|---|---|---|---|
|_PanelA:Villagelevelvariables_||Control<br>|Test|sofdiffere|nce|K-|STestp-valu<br>|es||
||N|Mean|H-C=0|L-C=0|H-L=0|H=C|L=C|H=L||
|Numberofpublicschools|266|2.5|0.011|0.010|0.001|0.95|1.00|1.00||
||||[0.95]|[0.95]|[0.99]|||||
|Numberofprivateschools|266|3.3|0.021|0.162|-0.141|1.00|1.00|0.99||
||||[0.85]|[0.16]|[0.18]|||||
|Privateenrollment|266|523.5|-23.549|11.202|-34.750|0.28|0.86|0.30||
||||[0.51]|[0.71]|[0.29]|||||
|Averagemonthlyfee(PKR)|266|232.1|12.668<br>[0.41]|-12.855<br>[0.20]|25.523<br>[0.07]|0.46|0.85|0.57||
|Averagetestscore|133|-0.222|-0.013<br>[0.88]|0.031<br>[0.75]|-0.044<br>[0.57]|0.27|0.51|0.35||
|OverallEffect:p-value|||0.95|0.96|0.99|||||

_Panel&B:&Private&school&level&variables_

|||Control||Testsofd|ifference||K-|STestp-val|ues|
|---|---|---|---|---|---|---|---|---|---|
||N|Mean|H-C=0|L<sup>t</sup>-C=0|L<sup>u</sup>-C=0|H-L<sup>t</sup>=0|H=C|L<sup>t</sup>=C|H=L<sup>t</sup>|
|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]||||
|Monthlyfee(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]||||
|Annualexpenses(PKR)|837|78860.9|21,559.2<br>[0.13]|-16,659.5<br>[0.15]|-5,747.2<br>[0.60]|38,218.7<br>[0.01]|0.58|0.88|0.57|
|Monthlyexpenses(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]||||
|Infrastuctureindex(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]||||
|Schoolage(inyears)|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]||||
|Numberofteachers|851|8.2|0.015<br>[0.97]|-0.408<br>[0.39]|0.242<br>[0.48]|0.423<br>[0.37]|1.00|0.95|0.81|
|Averagetestscore|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]||||
|OverallEffect:p-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.
||TableD2:|Randomization|Balance,Trimm|edSample|||
|---|---|---|---|---|---|---|
||(1)|(2)|(3)|(4)|(5)|(6)|
|||Control||<sup>t</sup><br>Testsofd|ifference|<sup>t</sup>|
|_Privateschoollevelvariables_|N|Mean|H*C=0|L*C=0|L<sup>u</sup>*C=0|H*L=0|
|Enrollment|836|154.1|*5.7<br>[0.39]|*13.8<br>[0.14]|*2.0<br>[0.77]|8.1<br>[0.35]|
|Monthlyfee(PKR)|834|221.6|2.5|*20.3|*8.4|22.8|
||||[0.81]|[0.13]|[0.38]|[0.07]|
|Annualexpenses(PKR)|821|65441.7|5,875.8|*5,477.6|*4,902.8|11,353.3|
||||[0.53]|[0.60]|[0.57]|[0.32]|
|Monthlyexpenses(PKR)|832|22293.5|1,061.4|*2,774.9|2,720.2|3,836.4|
||||[0.49]|[0.14]|[0.10]|[0.05]|
|Infrastuctureindex(PCA)|819|*0.141|0.077|0.133|*0.012|*0.056|
||||[0.41]|[0.31]|[0.88]|[0.69]|
|Schoolage(Noofyears)|836|7.9|*0.191|0.615|0.171|*0.806|
||.|.|[0.69]|[0.40]|[0.74]|[0.25]|
|Numberofteachers|834|7.7|*0.045|*0.290|0.316|0.245|
||||[0.88]|[0.44]|[0.31]|[0.47]|
|Averagetestscore|393|*0.242|*0.020|0.074|*0.029|*0.095|
||||[0.81]|[0.48]|[0.75]|[0.34]|
|OverallEffect: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)<br>Enrollment|(2)<br>Fees|(3)<br>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 (_L_<sup>_t_ </sup>= 0)|0.00|0.84|0.96|
|Test pval (_L_<sup>_t_ </sup>= 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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

34
Table4D4:4Differential4Attrition

||(1)<br>Control|(2)<br>High|(3)<br>LowTreated|(4)<br>LowUntreated|(5)<br>N|
|---|---|---|---|---|---|
|**_PanelA:DifferentialSurveyAttrition_**||||||
|Round1|0.059|@0.032|@0.044**|@0.035*|824|
|||(0.02)|(0.02)|(0.02)||
|Round2|0.052|@0.028|@0.045**|@0.031|806|
|||(0.02)|(0.02)|(0.02)||
|Round3|0.087|@0.063***|@0.079***|@0.038|798|
|||(0.02)|(0.02)|(0.02)||
|Round4|0.054|@0.030|@0.054***|@0.029|781|
|||(0.02)|(0.02)|(0.02)||
|Round5|0.126|@0.084***|@0.106***|@0.030|758|
|||(0.02)|(0.02)|(0.03)||
|Alwaysrefused|0.033|@0.007|@0.033**|@0.025*|758|
|||(0.02)|(0.01)|(0.01)||
|**_PanelB:DifferentialBaselineCharacteri_**|**_sticsforAttriters_**|**_(Atleastonceref_**|**_used)byTreatment_**|**_Status_**||
|Enrollment|191.4|8.4|6.4|@33.0*|79|
|||(44.68)|(28.77)|(18.74)||
|Monthlyfee(PKR)|257.5|@28.5|@47.5|37.2|79|
|||(60.78)|(42.46)|(50.90)||
|Annualfixedexpenses(PKR)|103745.0|55017.7|20106.0|@49684.0|77|
|||(90071.94)|(26347.19)|(39480.86)||
|Monthlyvariablecosts(PKR)|31768.8|7830.1<br>(19060.95)|44448.2<br>(31225.62)|@4501.2<br>(9184.26)|79|
|Infrastructureindex|0.062|0.536|1.140|@0.192|78|
|||(0.39)|(0.74)|(0.36)||
|Schoolage(Noofyears)|8.8|6.3*|@3.47|0.59|79|
|||(3.64)|(2.79)|(2.62)||
|Numberofteachers|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)<br>Enrollment|(2)<br>Fees|(3)<br>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 (_L_<sup>_t_ </sup>= 0)|0.02|0.49|0.72|
|Test pval (_L_<sup>_t_ </sup>= 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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

36
# **E Additional Results**

This section includes additional tables referenced in the main text.

37
Table E1: Credit Behavior <u>(Year</u> 1)

||School funding|sources (Y/N)|HH|borrowing (|Y/N)|HH loan value|
|---|---|---|---|---|---|---|
||(1)<br>Self-fnanced|(2)<br>Credit|(3)<br>Any|(4)<br>Formal|(5)<br>Informal|(6)<br>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 (_L_<sup>_t_</sup>= 0)|0.97|0.68|0.45|0.64|0.27|0.56|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

38
Table E2: Enrollment by Grades

||(1)<br>Lower than 1|(2)<br>1 to 3|(3)<br>4 to 5|(4)<br>6 to 8|(5)<br>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 (_L_<sup>_t_</sup>= 0)|0.01|0.00|0.03|0.03|0.13|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

39
Table E3: Enrollment Decomposition Using Year 1 Child Data

||(1)<br>Enrollment|(2)<br>% New|
|---|---|---|
|High|0.348|0.025*|
||(0.702)|(0.015)|
|Low Treated|0.776|0.056**|
||(0.740)|(0.024)|
|Low Untreated|-0.382<br>(0.706)|0.024<br>(0.017)|
|Baseline|0.641***<br>(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 (_L_<sup>_t_</sup>= 0)|0.30|0.02|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =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 (_L_<sup>_t_</sup>= 0)|0.37|0.61|0.32|0.14|0.26|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

41
Table E5: School Test Scores, Controls

|||No co|ntrols|||Additiona|l controls||
|---|---|---|---|---|---|---|---|---|
||(1)<br>Math|(2)<br>Eng|(3)<br>Urdu|(4)<br>Avg|(5)<br>Math|(6)<br>Eng|(7)<br>Urdu|(8)<br>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 (_L_<sup>_t_</sup>= 0)|0.59|0.34|0.60|0.96|0.43|0.51|0.40|0.70|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

42
Table E6: Test Scores, Stayers Only

|||Schoo|l level||Child level|
|---|---|---|---|---|---|
||(1)<br>Math|(2)<br>Eng|(3)<br>Urdu|(4)<br>Avg|(5)<br>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 (_L_<sup>_t_</sup>= 0)|0.32|0.62|0.42|0.71|0.38|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

43
Table E7: Main Outcomes, Interacted with Baseline Availability of Bank Account

||(1)<br>Enrollment|(2)<br>Fees|(3)<br>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 <u>per</u> capita

||(1)<br>Enrollment|(2)<br>Fees|(3)<br>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 (_L_<sup>_t_</sup>= 0)|0.03|0.80|0.97|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

45
Table E9: School Infrastructure <u>(Year</u> 2)

||Spending|Number|purchased|Facility|present (Y|/N)|Other|
|---|---|---|---|---|---|---|---|
||(1)<br>Amount (PKR)|(2)<br>Desks|(3)<br>Chairs|(4)<br>Computers|(5)<br>Library|(6)<br>Sports|(7)<br># 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 (_L_<sup>_t_</sup>= 0)|0.96|0.53|0.12|0.03|0.95|0.46|0.39|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =H) schools.

46
Table E10: Revenues, excluding Closed schools

||Ov|erall Posted (mo|nthly)|Ov|erall Collected (m|onthly)|
|---|---|---|---|---|---|---|
||(1)<br>Full|(2)<br>Top Coded 1%|(3)<br>Trim Top 1%|(4)<br>Full|(5)<br>Top Coded 1%|(6)<br>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 (_L_<sup>_t_</sup>= 0)|0.09|0.07|0.10|0.24|0.16|0.28|
|Test pval (_L_<sup>_t_</sup>=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 ( _L_<sup>_t_</sup> =0) schools, or whether we can reject equality of coefficients between high and low treated ( _L_<sup>_t_</sup> =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|
|_L_<sup>_t_</sup>|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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> and Rs.3,010 for _H_ schools. Thus, we have a monthly profit of Rs.3,639 for _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> over three villages decrease their overall income by Rs.2,514.

**Parents:** Calculating consumer surplus requires some strong assumptions on the demand function. 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 ( _P_ 0 _, Q_ 0) in the figure, and the _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> PQ-combination is _PL_ =230Rs and _QL_ =176. Hence, our linear demand curve is _Q_ = 521 _−_ 1 _._ 5 _P_ .

From Appendix Figure F1, Panel A, we can calculate the baseline consumer surplus, the triangle _CS_ 0, and the additional surplus gain in _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _._ 3 _P_ . 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 _L_<sup>_t_</sup> 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.<sup>9</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> .

> 9Our 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(**

<!-- Start of picture text --> P<br>CS0<br> P0<br>PL<br>CSL<br>Q0 QL Q<br><!-- End of picture text --> _Panel&A:_ !Consumer!surplus!at!baseline,!CS0,!and!in!L<sup>t</sup> !from!movement along!demand!curve <!-- Start of picture text --> P<br>CSH<br>PH<br>P0<br>CS0<br>Q0 QH<br>Q<br><!-- End of picture text --> _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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> and _H_ groups by 7pp (Table 3, col 4). The closure rate in _L_<sup>_t_</sup> 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 _L_<sup>_t_</sup> 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.7,560Rs.4,080]*12) in the _H_ arm as compared to the _L_ arm.

51