Macroeconomic Effects of ‘Free’ Secondary Schooling in
the Developing World
Junichi Fujimoto (GRIPS)David Lagakos (BU and NBER)
Mitchell VanVuren (Yale)
March 2024

Abstract
This paper studies the macroeconomic effects of publicly funded (’free’) secondary schooling in the developing world. Our analysis is based on an overlapping generations model of human capital accumulation that we estimate tomatch experimental evidence on the effects of scholarships for poor but talented
students in Ghana. The model predicts that nationwide free secondary schooling increases average education levels but reduces GDP per capita in the longrun. The human capital gains from free schooling in the model are offset by lost
income during schooling years, negative selection of new students, and reductions in fertility by high-ability households.

Email: j-fujimoto@grips.ac.jp, lagakos@bu.edu, mitchell.vanvuren@yale.edu. For helpful comments we thank Ufuk Akcigit, Marco Bassetto, Natalie Bau, Kevin Donovan, Raquel Fernández,
Chang-Tai Hsieh, Aubhik Khan, Jeremy Pearce, Karthik Muralidharan, Michael Peters, Tommaso
Porzio, Pascual Restrepo, Todd Schoellman, Julia Thomas and seminar/conference participants at
Berkeley, Chicago, City University of Hong Kong, the Development Bank of Japan, the Federal Reserve Banks of Minneapolis and St. Louis, Georgetown, Hitotsubashi, Kyoto, NYU, Ohio State, Penn
State, Riverside, Tohoku, the Cowles Foundation Conference on Structural Change, the UCLA Human Capital and Development Conference and the NBER Summer Institute. We thank the IGC, JSPS
(Kakenhi No. 18K01505 and No. 22K01399), and GRIPS Policy Research Center for financial supportand Ariel Akerman, Soala Ekine, and Isaac Otoo for outstanding research assistance. All potential
errors are our own.

1

1. Introduction
It has been said that talent is universal but opportunity is not. This saying could
hardly apply better than to the millions of school-aged children throughout the developing world who are not enrolled in school. A common explanation of the low
secondary schooling levels is credit constraints that prevent poorer households fromborrowing for education. Keeping bright young people out of secondary school
may lead to a significant misallocation of talent in the education system, which
can reduce aggregate productivity levels (Hsieh, Hurst, Jones, and Klenow, 2019).
More generally, low levels of human capital in developing countries are thought
to be one of the most important proximate causes of their low income levels (e.g.
Hall and Jones, 1999; Bils and Klenow, 2000; Erosa, Koreshkova, and Restuccia,
2010; Hanushek and Woessmann, 2012; Manuelli and Seshadri, 2014; Hendricks andSchoellman, 2018) as well as a barrier to structural change (Porzio, Rossi, and Santangelo, 2022; Buera, Kaboski, Rogerson, and Vizcaino, 2022).
In this paper, we analyze the macroeconomic effects of an increasingly common
policy aimed at raising secondary enrollments: publicly funded (‘free’) secondary
schooling. A number of low-income countries have recently adopted free schoolingin the hopes of raising human capital levels, and others are seriously considering it(see Center for Global Development, 2022). To our knowledge, however, ours is thefirst macroeconomic study of free secondary schooling in the developing world.
Our analysis is based on an overlapping generations (OLG) model of human capital accumulation with heterogeneous households and credit constraints in the spiritof Galor and Zeira (1993) and Bénabou (2002). Our model incorporates three key
features necessary for understanding the long-run general-equilibrium impacts ofeducation policy in the developing world. The first is opportunity cost of schoolingin the form of lost labor income for those attending secondary schooling rather thanworking. It is hard to ignore this opportunity cost since people of secondary-school
age are close to their prime working years in developing countries. The second
feature is a secondary-school entrance exam. This merit requirement is relevant
since secondary school currently selects students with relatively high ability, meaning that free schooling for all could lower average student ability levels. The thirdfeature is fertility that depend on education, and in particular lower fertility for
those with more education. Evidence from a large set of developing economies has
linked education expansions and economic growth to significant declines in fertility(Chatterjee and Vogl, 2018).
2

We estimate the model using experimental evidence from a randomized controlled trial (RCT) that offered secondary school scholarships to a randomly selected
set of poor but high-ability students in Ghana (Duflo, Dupas, and Kremer, 2021).
Those offered the scholarships were about 25 percent more likely to finish secondary
school than a control group four years hence. Scholarship winners performed about
0.2 standard deviations higher on tests of literacy and mathematics, which is comparable to the effects found in other successful education interventions. Moreover,scholarship winners had significantly fewer children than the control group.
We use our estimated model to simulate the aggregate effects of free secondary
schooling in general equilibrium. The model predicts a one-third increase in thenumber of secondary school graduates, which is broadly consistent with the substantial increases in attendance described by the Center for Global Development(2022). Yet the model predicts a negative effect on GDP per capita, which falls byaround 1 percent in the long run. Our model’s confidence interval for this prediction is entirely negative, suggesting that even modest gains in GDP per capita fromfree secondary schooling are unlikely. Taxes increase by about 1 percent per capita
in the long-run, meaning that the policy pays for only a fraction of its costs.
To better understand why the model makes such pessimistic quantitative predictions we conduct a series of counterfactual experiments. In the first, we eliminate theopportunity cost of secondary schooling by exogenously disallowing work by thoseof secondary school age. We then simulate the impacts of free secondary schooling on this economy. We find that it would have raised GDP per capita by about 3
percent, signaling that the earnings gains from free school are largely offset by lostearnings during secondary school years in our main analysis.
In a second counterfactual, we simulate a hypothetical free schooling policy thatmaintains a similar skill distribution as the poor but talented students studied in the
RCT. Free schooling causes no change in GDP per capita in this economy, pointing to
the less favorable selection of students once everyone is allowed to attend secondary
school (like Hendricks and Schoellman (2014) find for college in the United States).
In a third counterfactual we eliminate the differential fertility rate by educationlevel, meaning there is no way the population share of the educated can fall after
education levels rise. In this counterfactual, GDP per capita rises by about 1 percent
in the long run due to free secondary school, highlighting how long-run changes in
fertility in the main analysis play a quantitatively important role in lowering GDP
per capita once secondary schooling is made free.

3When we simulate the counterfactual impacts of shutting down all three channels we find that free secondary schooling would lead to a GDP per capita gain ofaround 7 percent in the long run. Taxes per capita would change little in this caseas free schooling pays for itself in the long run. The sizable GDP per capita gainin this counterfactual illustrates how the model’s pessimistic conclusions about freeschooling are not hard wired into the findings of the scholarships RCT, but are rather
due to the economic forces of opportunity cost, negative selection of new graduates,
and reductions in fertility by high-ability households.
An alternative class of education policies studied in the development literature
have focused on raising the quality of schooling, rather than making schools moreaccessible. We use our model to simulate an economy-wide improvement in schooling quality, which could represent pay-for-performance incentives for teachers (Muralidharan and Sundararaman, 2011; Duflo, Hanna, and Ryan, 2012; Mbiti, Muralidharan, Romero, Schipper, Manda, and Rajani, 2019), additional teachers in theclassroom (Banerjee, Cole, Duflo, and Linden, 2007) or other interventions shown tobolster student academic performance. We find that school quality improvements
are significantly more effective at raising average income levels than free secondary
schooling. A nationwide school quality improvement raising test scores by 0.1 standard deviations – consistent with the experiments above – leads to a long-run GDP
increase of 3 percent per capita, and pays for itself in equilibrium. We conclude that
improving the quality of secondary schools in Africa is a more effective way to raise
living standards than making the current schools free.
A welfare analysis of our model predicts that the richest quartile of households
suffer welfare losses from free schooling, largely since they pay higher taxes, whereas
the poorest quartile gain relatively most. We conclude that free secondary schooling is predominantly redistributionary in nature, and that low secondary schoolinglevels are largely an efficient response to low quality school options. This implication is broadly in line with the conclusions of the macro development literatureemphasizing low schooling quality, rather than low average years of schooling, as
the proximate cause of low human capital levels in poor countries (Hanushek andWoessmann, 2007; Schoellman, 2012).
Related Literature. This paper belongs to the vast literature on macroeconomic
models of human capital investment (e.g. Becker and Tomes, 1979; Loury, 1981; Galor and Zeira, 1993; Restuccia and Urrutia, 2004; Erosa et al., 2010; Manuelli and Seshadri, 2014; Lee and Seshadri, 2019). In particular, our quantitative exercises build4

on the literature on the macroeconomic effects of credit constraints in educationin advanced countries, such as the seminal work of Lochner and Monge-Naranjo(2011). Our paper is most closely related to the studies by Abbott, Gallipoli, Meghir,and Violante (2019) and Daruich (2020), both of whom study expansions in publicly
funded education in the United States. Both studies reach fairly positive conclusions
about the effects of expanding public education, unlike our study, which arguablyreflects disparities in school quality between rich and poor countries. As in Daruich(2020), we discipline our model using experimental evidence from an RCT.
Similar to our paper, Khanna (2023) studies the impact of expanding education in
general equilibrium and finds a large role for relative wage effects. Comparatively,
we find a smaller role for wage impacts and instead emphasize the importance ofother channels such as fertility and opportunity cost.
Our paper also builds on the recent literature attempting to quantify the extent
to which credit market imperfections drive misallocation in developing countries.
Bassi, Muoio, Porzio, Sen, and Tugume (2022) and Caunedo and Kala (2022) show
how rental markets for large indivisible capital goods can reduce capital misallocation, and Moll (2014) and Midrigan and Xu (2014) find a significant ability for firms
to save their way around credit constraints.
In estimating our model to a field experiment, we build on a growing body ofmacroeconomic research on development that uses randomized experiments in order to guide general-equilibrium counterfactuals (see Buera, Kaboski, and Townsend,2023). Our paper is the first to take this approach when studying the macroeconomic effects of education policy in the developing world. Other studies using
this methodology have studied small business investment (Kaboski and Townsend,2011), occupational choice (Buera, Kaboski, and Shin, 2021), infrastructure investments (Brooks and Donovan, 2020), rural-urban migration (Lagakos, Mobarak, and
Waugh, 2023), and firm training programs (Akcigit, Alp, and Peters, 2021). Our
methodology relates to the papers by Todd and Wolpin (2006) and Attanasio, Meghir,and Santiago (2012) that use structural labor models to interpret experimental evidence; see Todd and Wolpin (2023) for an engaging recent review.

2. Secondary Schooling in the Developing WorldIn this section we summarize the macro and micro evidence on secondary schooling outcomes and free secondary schooling policies in the developing world. We
also discuss how this evidence informs our modeling choices and counterfactual
simulations in the sections that follow.
5120

Figure 1: Primary
Secondary
SchoolEnrollment Rates
Figure 2: andNet Enrolment
Rate AcrossCountries

80
60
40
0

20

Net Enrolment rate (percent)100

Primary School
Secondary School

.5

1

2

4

8

16

32

64

128

GDP per capita in constant PPP x ($1,000)
Notes: Net enrolment rate is the ratio of total enrolment students of official age-going school to the
population of the age group that officially corresponds to that level of education shown. Data wassourced from World Banks (World Development Indicators).

Aggregate data on schooling enrollment show plainly that developing countries
mainly lag behind richer ones when it comes to secondary schooling (as opposed toprimary schooling). Figure 1 plots net enrollment rates in primary school (blue dots)
and secondary school (red x’s) in 2019 against GDP per capita using data from theWorld Bank. Net enrollment rates are defined as the number of people enrolled inschool relative to the population of school-aged individuals. In the world’s poorest
countries, roughly four out of five children of primary-school age are enrolled inschool, compared to nearly every child in the richest ones. For secondary schooling,
the differences are much starker. At the bottom of the world income distribution,
only around one-third of those of secondary-school age are enrolled in secondary
school, whereas at the top, enrollment rates
are again near one hundred percent.
2
One salient difference between rich and poor countries in terms of education policy is that richer countries are much more likely to publicly finance secondary education. It is not surprising then that many developing countries have recently considered implementing ‘free’ schooling policies, in which the government finances
school fees for at least some secondary-age students (see Center for Global Development (2022) and Appendix Table A.1, which lists the developing countries, like
Ghana, that already have some form of free secondary schooling in place). One
main rationale for publicly funded schooling is to help raise average schooling lev6

els and hence GDP per capita. A second rationale is to make secondary education
more accessible to poorer households, consistent with redistributionary motives.
These two objectives are not necessarily in contrast with one another, since raising
average years of schooling is likely to require expanding schooling access to poorer
households that were previously unable to pay for secondary school fees.
Recently, a number of micro studies have estimated the impacts of merit-based
scholarship programs in developing countries, though with mixed results. BrudevoldNewman (2021) found, using a difference-in-difference approach, that free secondary
schooling in Kenya increased educational attainment, reduced fertility, and increased
the likelihood of skilled work. Using a regression discontinuity design in Cambodia,
Filmer and Schady (2014) found that scholarships increased educational attainment
but did not increase earnings, fertility, or test scores. Both studies highlight credit
constraints as a reason more students were not already enrolled.
Duflo, Dupas, and Kremer (2021) conducted the first long-run RCT evaluationof a merit-based scholarship program for secondary school. Their study is set in
Ghana, where the education system consists of primary school and junior highschool (JHS) until age 14, at which point students are required to pass the Basic Education Certificate Examination (BECE) in order to attend senior high school (SHS).
The authors identified approximately two thousand students who had passed the
BECE in 2008 but had not enrolled in SHS by the deadline for the next school year.
Among these students, one-third were randomly selected to receive a four-year
scholarship covering one hundred percent of tuition and fees.
Students who received a scholarship were substantially more likely to complete
SHS relative to those in the control group, and on average completed 1.2 more yearsof school than the control group. Scholarship winners similarly exhibited higher
human capital, as measured by math and reading tests. The magnitudes were substantial, and translate into 7.6 percent higher human capital per year of school completed. There was also a significant negative impact on fertility, with scholarshipwinners having fertility declines of around 10 percent after 12 years. We view thisexperiment as the most comprehensive and credible evaluation of free secondary
schooling to date. Consequently, we use these experimental moments to parameterize our model, which we develop in the next section.
Taking these results at face value, what might one infer about the aggregate gains
from a national free schooling policy? Suppose, like in the RCT, that the policy could
raise average schooling by 1.2 years among the 70 percent of Ghanaians of SHS age

7

who do not attend school. With a human capital increase of 7.6 percent per year,
consistent with impacts on test scores in the experiment described above, this would
result in a roughly 6 percent increase in GDP in the long run (=0.7 x (1.0761.2 − 1)).
This paints a promising picture for free secondary school, and suggests that theaggregate effects of the policy could in principle be significant.

3. Overlapping Generations Model
We now describe the model, which, at its heart, is an OLG model of human capital accumulation with credit constraints a la Galor and Zeira (1993) and Bénabou(2002). The model is tailored to capture key features of developing countries relevant for how free secondary schooling policies impact average income per capita.
The model features opportunity cost of schooling, in the form of lost labor income;merit requirements to attend secondary school, modeled as a test score cutoff; andfertility levels that depend on education levels. As in most models in this literature,ours allows for misallocation of talent through borrowing constraints that keep highability children of poor parents out of school.
We also include several other features that are relevant for estimating and quantifying the effects of free secondary schooling. Consistent with the evidence of
Khanna (2023), we model the labor of different education types as imperfect substitutes, so that an increase in the supply of educated workers depresses their relative wages. As in a growing literature in macro development we allow for saving
constraints, which help match the low average levels of liquid asset holdings, andimpede households from simply saving themselves out of borrowing constraints(see e.g. Donovan, 2021). Following the literature on public finance and development we posit a tax system in which a narrow base of high earners pay the majorityof the taxes used to finance public expenditures (see e.g. Jensen, 2022).
3.1. Environment
Time is discrete and goes from 0 to infinity. There is a single good which canbe used for consumption, savings, and investment in education. The economy ispopulated by overlapping generations of households that are heterogeneous in theirparental human capital, child ability, taste for schooling, and savings. The timelineof events for these households is shown in the graphic below.
Individuals live for 14 periods, where each period corresponds to 5 years. For
their first five periods of life (ages 0–24) children live with their parents. In the third
period (ages 10–14), all children attend JHS. We abstract from the choice of attend8

ing school at this age based on the evidence of the previous section that virtually
all children already attend JHS. In the fourth period, (ages 15–19), children either
attend SHS (i.e. secondary school) or work. This is the key schooling choice in themodel. The fact that a household must give up a period of a child’s work, and thus
income, in order to attend secondary school captures the notion that there remains
an opportunity cost even in the case where schooling is made free. In the fifth period, all children work using their respective education level, which is fixed for theremainder of their life.
SHS
or
JHS work

Parent0work

10 15 20 25

Child

SHS
or
JHS work0

retirement

60

70

work

10 15 20At the beginning of period six, when turning age 25, children leave their parents,
have children, and become parents themselves. We abstract from household formation decisions since they do not seem crucial for our task at hand. Instead, we
model households as continuous dynasties that do not mix. These new parents thenwork from age 25 to 60, at which point they retire, and die at age 70. This is roughly
the average life expectancy in Ghana, for example, whose features we will use to
parameterize our model in the following section.
Each new household consists of a parent aged 25 and newborn children. The
model features population growth, and the number of children, denoted as 1 + νsp ,decreases with the parent’s schooling level sp . As a result, policies that increase anindividual’s level of schooling will decrease their fertility, consistent with a varietyof evidence for developing countries.

Individuals are heterogeneous in learning ability z ∈ Z = z 1 , z 2 , . . . , z N . The
ability within a household follows a first-order Markov chain which, by Tauchen’s(1986) method, approximates the AR(1) process:log zc = ρ log zp + υ, ρ ∈ (0, 1) .

(1)Here, zp and zc denote the parent and children’s ability. Throughout, variables with9

subscripts p and c pertain to parents and children, respectively. The random shockυ is independently and identically distributed (i.i.d.) and follows N (0, συ2 ). Thus,
ability is transmitted within each household but only imperfectly, and is identicalacross siblings. Following the evidence in e.g. Cunha and Heckman (2007), weinterpret ability to be a function of inherited capabilities and parental inputs.
All household decisions are made by parents, who derive flow utility U (c) =log (c) from household consumption c ≥ 0 and discount the future at rate β ∈ (0, 1).
The assumption that parents make educational decisions follows the tradition of
Becker and Tomes (1979) and is consistent with evidence that parents in low-income
countries predominantly take an authoritarian approach to parenting, dictating decisions directly rather than trying to reach an agreement with children (Doepke andZilibotti, 2017). Parents and children (from ages 15 to 25) have a single unit of timeeach period which they supply inelastically to wage work or education. Parents
are imperfectly altruistic toward children and therefore derive utility also from children’s well-being (as in e.g. Laitner, 1997).
Parents make schooling decisions for their children when the children turn age
15, after observing the children’s ability and test scores as well as the children’s
realization of a(schooling taste shock. More precisely, children enjoy random utility(internalized by the parent through imperfect altruism) δs from schooling level s ∈S = {J, S} (JHS, SHS), where δs follows a standard Gumbel distribution with scaleparameter θ. Parents must forgo a period of children’s income to send their children
to an additional period of school, and further, providing children final schoolinglevel s ∈ S requires goods costs Ψs . These goods costs represent school fees andsatisfy ΨS > ΨJ = 0, where the equality reflects the free primary education thatprevails in most developing countries. Thus when deciding whether to send theirchildren to school, parents must consider the opportunity cost of their children’s
work as well as the explicit goods cost.
Households face incomplete markets as in Aiyagari (1994), Bewley (1977), andHuggett (1993) and cannot borrow but can save at an exogenous rate r. Whilehouseholds do face idiosyncratic income risk, the most important feature of thisborrowing constraint is that it prevents parents from borrowing against their child’s
future income in order to fund school attendance. This allows for the possibility thata high ability child, whose return to additional schooling far exceeds the cost, may
not attend if born to a poor parent, resulting in misallocation.
To capture the fact that one must pass an entrance test to enter secondary school-

10ing in most developing countries, we set a threshold test score for entering SHS.
One’s test score z̃ is related to ability asz̃ = z + ε,(2)where the noise ε follows N (0, σε2 ).
The human capital of an individual with ability z and schooling s is given by

h (z, s) =(1

if s = J,z · ηS if s = S,

(3)

where ηS > 0 and represents the efficiency, or quality, of schooling. Thus, ability
affects human capital only for those with SHS education, and the resulting humancapital of a secondary education depends on the product of the student’s ability andthe schooling quality.
Markets are competitive and the aggregate production function, operated by arepresentative profit-maximizing firm, is given by:
i 1−α
h
λ
Y = AK α (NJ )λ + (NS )λ, α, λ ∈ (0, 1) .

(4)Here, A is aggregate productivity, K is physical capital, and Ns is aggregate efficiency units of labor of individuals with schooling level s. The firm rents physicalcapital from households or foreign investors at an exogenous international market
rate r∗ . Due to savings frictions, however, the return to physical capital for households is lower, at r = r∗ −χ < r∗ . This lower return to capital helps us match the low
savings rates among households in low-income economies (as in Donovan, 2021).
The labor income y of an individual equals the product of three terms. The firstterm is the wage rate per efficiency units of unskilled (s = J) or skilled (s ∈ S) labor, denoted as wU or wS , respectively. The second term, ζ, represents idiosyncratic

shocks to labor productivity, where log ζ is drawn each period from N 0, σζ2 . The
third term is human capital h (z, s), given by (3). For example, the labor income of
an individual with education level S is given byy(z, S, ζ) = wS ζh (z, S) = wS ζzηS .

11

(5)

3.2. Parents’ Problems
Parents make consumption and saving decisions in each period, and additionally,
schooling decisions when their children reach the age for secondary school. We
discuss below the parents’ problems in the key periods in the life-cycle; we omit thedescription of their problems in other periods, which are standard consumptionsavings problems. In addition to individual state variables described below, theparent’s problems depend on the p.d.f. f describing the distribution of households
across individual states and the aggregate population level P .
When τ = 9, and children turn 15, parents observe the realizations of the schooling taste shocks (δJ , δS ), children’s ability and test score (zc , z̃c ), and their own andchildren’s labor productivity (ζp , ζc ). Then, if z̃c weakly exceeds the threshold testscore z̄, parents have the option of sending children to an additional period ofschooling (s = S). The value function of such parents with ability zp , schoolinglevel sp , and assets a is given by
V9 (a, zp , sp , ζp , δJ , δS , zc , ζc ; f, P |z̃c ≥ z̄) =maxlog(c)
+ δJ I (s′c = J) + δS I (s′c = S) + βE V10 (a′ , zp , sp , ζp′ , zc , s′c , ζc′ ; f ′ , P ′ )c≥0,a′ ≥0,s′c ∈{J,S}

where the maximization is subject to the flow budget constraint
a′ + c + 1 + νsp I s′c = S ΨS
(6)=yp (zp , sp , ζp ) + (1 + r)a + 1 + νsp (1 − I s′c = S )yc (zc , J, ζc ) − T (zp , sp , ζp , zc , J, s′c , ζc )and the perceived laws of motion for the aggregate state variables f and P , given
by f ′ = F (f, P ) and P ′ = H(f, P ), respectively. Here, the prime denotes values ofvariables in the next period and T is total amount of taxes paid by the household,which depends on the parent and children’s labor income, and is therefore a function of (zp , sp , ζp , zc , J, s′c , ζc ).1 We suppress the dependence of yp , yc , and T on f andP except where it is necessary to make that dependence explicit.
When τ = 10, children live one final period with their parents and work withthe human capital given by their education decision the previous period. The value
1

Note that T depends on both the children’s current schooling level sc (= J) and next period’s
schooling level s′c . This is because the labor income depends on educational attainment, and only thechildren who do not go to school (s′c = sc ) earns the labor income in the current period.

12

function of such parents is expressed aslog(c) + βE V11 (a′ , zp , sp , ζp′ ; f ′ , P ′ )c≥0,a ≥0+ βb 1 + νsp E [V6 (0, zc , s′c , ζc′ ; f ′ , P ′ )]V10 (a, zp , sp , ζp , zc , sc , ζc ; f, P ) = max′(7)

subject toa′ + c = yp (zp , sp , ζp ) + (1 + r)a+ 1 + νsp yc (zc , sc , ζc ) − T (zp , sp , ζp , zc , sc , s′c , ζc ),(8)

f ′ = F (f, P ), P ′ = H(f, P ), and s′c = sc . On the right-hand side of (7), V11 denotes
the parent’s value function in the following period, which no longer depends on theability and schooling of children who become independent from parents. The lastterm on the right-hand side of (7) denotes utility that imperfectly altruistic parents
derive from their children’s well-being, where b > 0 is the altruism parameter andV6 (0, zc , s′c , ζc′ ; f ′ , P ′ ) is the value function of children who form new households withzero assets.
3.3. Government, Taxes and Equilibrium
The government collects tax revenue from households which it then spends on“public goods,” and, in the policy counterfactuals, free secondary schooling. The
government budget constraint in per capita terms is given by:Z
G+ξ

I (s′c = S ∧ τ = 9) df =Z

T (zp , sp , ζp , zc , sc , s′c , ζc )df(9)where G is spending on public goods per capita and ξ is expenditure on free secondary education per capita. Since the paper is about public financing of secondary
education, and not other public expenditures, we abstract from how G affects households or producers in the economy. When we simulate the effects of free publicschooling, we assume that G remains unchanged, so that any schooling subsidy
must be funded through per-period adjustments in the tax function T .
We focus our quantitative analysis on the balanced growth path of the economy.
We relegate the full definition of recursive competitive equilibrium and the balanced
growth path to Appendix B. In essence, the balanced growth path is the equilibrium
in which the aggregate population level grows at a constant rate, but the relative
distribution of households across individual states is constant. In this situation,
13

household behavior does not depend on the aggregate population level. In all of
our analyses, we assume the economy starts on a balanced growth path. To examine
the effects of a policy change, we introduce the policy into the balanced growth
path of the economy and compute transition dynamics by calculating sequences of
population growth rates and prices that converge to the new balanced growth path.
3.4. Illustration: Misallocation of Talent and Impacts of Free Schooling
The key decision for a household is whether or not their children attend secondary school. The benefit of attending is higher future wages, increasing in proportion to the ability level of the child. The costs of schooling are the goods cost, ΨS ,and the opportunity cost, represented by foregone earnings. Neither of these costs
depends directly on household characteristics, but, due to borrowing constraints,
the utility cost ends up being higher for households with low income and assets,
who have higher marginal utility of consumption. Intuitively, misallocation arises
whenever a poor family chooses not to send a high ability child to secondary school
even though doing so would increase lifetime household income.
Consider two example cases of our model that vary in the goods cost of schooling,ΨS , the quality of secondary education, ηS , and the savings wedge, χ. The firsteconomy, which we call the low misallocation economy, features a relatively low costof schooling, relatively low schooling quality, and a small savings wedge. As weshow below, borrowing constraints bind for few households in this economy, andmany households choose not to send their children to secondary schooling because
the returns are low. The second economy, namely the high misallocation economy, hasa higher cost of school, higher schooling quality, and a larger savings wedge. We
take other parameters to be the same across economies.2
Figure 2 panels (a) and (c) plot the probability that a child attends secondary
school – conditional on passing the entrance exam – as a function of the child’s ability and their parent’s ability (a proxy for parental income and wealth). Panel (a)represents the low misallocation case, and panel (c) represents the high misallocation case. The dotted gray line labeled unconstrained cutoff, marks the child ability
level at which the net-present-value of the additional period of schooling is exactly
equal to the total cost of attendance. If households faced no borrowing constraint,
this is the ability level above which all children would attend SHS, and below which2

In particular, the low misallocation economy features values (ΨS , ηS , χ) of (5.5, 1.5, 0.1) while
the high misallocation economy has values (14.0, 5.0, 0.2). The rest of the parameter values, which
are not crucial for the conclusions in this section, can be found in Appendix Table A.2.

14

Figure 2: Child SHS Attendance Probability
Low Misallocation
100

60

80

60

40

40

20

20

0
20

40

60

80

80
60

80

60

40

40

20

20

0

0

0

100

Unconstrained Cutoff

80

100

Parent Ability (Percentile)

Unconstrained Cutoff

Parent Ability (Percentile)

100

100

0

0

20

40

60

80

100

Child Ability (Percentile)

Child Ability (Percentile)(a) SHS Attendance Probability(b) Increase in Attendance from Free SHSHigh Misallocation
100

60

80

60

40

40

20

20

0
20

40

60

80

80
60

80

60

40

40

20

20

0

0

0

100

Unconstrained Cutoff

80

100

Parent Ability (Percentile)

Unconstrained Cutoff

Parent Ability (Percentile)

100

100

0

0

20

40

60

80

100

Child Ability (Percentile)

Child Ability (Percentile)(c) SHS Attendance Probability(d) Increase in Attendance from Free SHS

none would attend (assuming the average taste shock). In Panel (a), children’s attendance probabilities are roughly in line with this cutoff: those above the cutoff
largely attend regardless of their parent’s ability, and those below largely do not.
In Panel (c), in contrast, children born to sufficiently high ability parents attend
SHS roughly according to the unconstrained cutoff, and children born to low-ability
parents are unlikely to attend SHS regardless of their ability level. In this economy,
there is substantial misallocation in the sense that many children for whom the netpresent-value of education outweighs the costs of education do not attend. As aresult, one can imagine substantial scope for gains in output from reducing the costof schooling.

15

Panels (b) and (d) illustrate these potential gains by displaying the increase in
the probability of attendance when SHS is offered for free. In the low misallocation
economy, the increase in attendance is small in magnitude and concentrated among
marginal children; many of those induced to attend are actually below the unconstrained cutoff and gain little to no lifetime income from schooling. In contrast, thehigh misallocation economy exhibits large increases in attendance probabilities concentrated at the top of the ability distribution. These children exhibit substantial
income gains from schooling, and their attendance results in higher output.
It is important to note that aggregate data alone are not necessarily informative
about the reasons why secondary attendance rates are so low. Although the two
economies in Figure 2 are very different, they both have aggregate attendance rates
of around 30 percent. In the Panel (a) economy, attendance is low simply because
the return to schooling is low on average relative to the cost, as shown by the highunconstrained cutoff. In Panel (c), however, attendance is low in large part because
many high-ability children face borrowing constraints and hence forgo secondary
school. This motivates our use of experimental moments, in addition to aggregate
moments, in estimating the model.

4. Model Estimation
While our estimation is largely focused on experimental moments, we first choosea handful of parameters directly, either as normalizations or to match standard values from the literature. We then estimate the rest using the Simulated Method ofMoments (SMM).
4.1. Directly Chosen Moments and Aggregate Moments
Table 1 presents parameters chosen directly. We start by normalizing aggregate
productivity, A, to be one. We set capital’s share in production, α, to be 0.33, and thediscount factor, β, to be 0.965 , which are standard values. The international market
interest rate r∗ is chosen to generate a (depreciation-inclusive) user cost of capitalequal to 10 percent per year.
We choose the parameter governing the substitutability of skills, λ, to be 0.75,
which generates an elasticity of substitution of 4. This is consistent with the longrun estimates of Bils, Kaymak, and Wu (2022) based on cross-country school attainment and wage data by attainment level. We are primarily interested in the long-run
effects of schooling expansions, making a long-run elasticity of substitution appropriate for our study. We have experimented with lower values of this elasticity,16

down to a value of 1.4, but these do not substantively affect our main conclusions.
We pick the standard deviation of the idiosyncratic income shock, σζ , to be 0.32,
which matches the variance of the transitory component of log wages estimated by
Lagakos and Waugh (2013). In the model, this transitory component is calculated
by subtracting the variance of the permanent component of log income (explained
further below) from the total variance of log income.
Finally, we set the income tax function to match Ghana’s statutory income tax
rates at the time of the experiment, summarized in Appendix Table A.3, which focuses on a narrow tax base consisting of only the highest income earners. This specification is consistent with the overall view that taxation in the developing world ishighly progressive and absent for the poorest households (see e.g. Jensen, 2022). We
then set per capita government spending on public goods G such that the government budget is exactly balanced each period along the balanced growth path.
Table 1: Directly Chosen Parameters
Description

Parameter Value

Agg. Productivity

A

1

Share of Capital

α

0.33

Discount Factor

β

0.965

International Market Rater∗

1.15 − 1

Skill Substitutability

λ

0.75

Std. Deviation of Income Shock σζ

0.32

Note: The table reports the values of directly chosen parametersin the model.

4.2. Simulated Method of Moments
We estimate the remaining parameters of the model using the SMM. There areten such parameters, which we estimate using ten moments. Formally, we solve forthe parameter vector:
Θ = {νJ , νS , ηS , ΨS , b, σε , θ, χ, ρ, συ }(10)that minimizes the sum of squared differences between the moments in Table 2
and their model counterparts. We also compute 95-percent confidence intervals for17

our parameters through bootstrapping, treating non-experimental moments (thoseabove the line in Table 2) as fixed values, and re-sampling the experimental moments.
The first five moments we target do not use experimental variation; these arelisted in the top portion of Table 2. The first is a population growth rate of 2.2 percent
per year, which is the value estimated by the World Bank for Ghana. The next threetargets are the secondary school completion rate in the aggregate – meaning for all
individuals of school age – and the secondary school completion rates in the top
and bottom test score quartiles of the control group. The final non-experimental
moment we target is the variance of the permanent component of log wages.3 Wetarget a value of 0.22 from the estimate of Lagakos and Waugh (2013), which is inline with other estimates found in the literature.
The remaining five moments come primarily from the experiment of Duflo, Dupas, and Kremer (2021), described in Section 2. To match these moments, we need
to be able to replicate their experiment within our model. We describe how we dothis in the following subsection. The last moment we target is the intergenerationalcorrelation of schooling in Ghana, taken from Azomahou and Yitbarek (2021), andcomputed from regressions of children’s educational attainment on parents’ educational attainment. We target these regression coefficients by running these same
regressions in our model.
4.3. Running the Experiment in the Model
We replicate the experiment in partial equilibrium. Since the experiment affectedjust 2,064 students, we find it implausible that the experiment had any significant
general equilibrium effects. We also abstract away from the difference between day
schools, which are the subject of the experiment, and boarding schools, which may
be of higher quality, since day schools are more likely to be the focus of secondary
schooling expansions in the future.
Importantly, we mimic the sample selection in the experiment, which consisted ofpicking “smart kids from poor families.” To match the requirement that students inthe sample have passed the BECE, we choose a test score cutoff so that only the top
42 percent of students in the model pass, consistent with the actual BECE passing
3

Along the balanced growth path of the model, this object corresponds to Cov (log yi,t , log yi,t−1 ),where the subscript i denotes agent and t denotes time. To see this, note that (5) implies log yi,t =log ζi,t + log (wi,t hi ). Since log ζi,t , the transitory component of log yi,t , is i.i.d. and is uncorrelatedwith the permanent component, log (wi,t hi ), we have Cov (log yi,t , log yi,t−1 ) = V ar (log (wi,t hi )) .

18

rate. Selection into the experiment also required that students had not registered forsecondary school in the fall semester following their exam, which is harder to match
literally within the model (particularly since one period in the model represents five
years). Our strategy is to choose a parental income cutoff such that, among thestudents passing the BECE in the model, the eventual secondary school completion
rate for those below the cutoff is 47.5 percent, just as in the control group of theexperiment. We then choose the experimental sample in our model to be a subset of
those with test scores above the test score cutoff and income below the income cutoff.
We treat the experiment as unanticipated and assume that model households
know that the experiment ends after a single generation. Households selected into
the control group solve their optimization problem as usual. Households selected
into treatment experience an exogenous reduction in the goods cost of secondary
school ΨS to 0 for the current period and then re-optimize. We construct simulated
equivalents of the experimental moments by taking simple differences of averageoutcomes between the treated and control households in the model, which correspond to the intent-to-treat estimates in the experiment.
We target the negative treatment effects on fertility and positive treatment effects
on human capital in the experiment, which we view as the most important – andprecisely estimated – findings of the experiment. The treatment effect on fertility
is large, and consistent with a 10.6 percent reduction in fertility after 12 years. The
experiment found substantial positive impacts on test scores in reading and mathof 0.16 standard deviations, which are consistent with the impacts of other successful interventions found in this literature (e.g. Duflo, Hanna, and Ryan, 2012; Mbiti,
Muralidharan, Romero, Schipper, Manda, and Rajani, 2019).
Regarding their estimates of the labor market impact of the scholarship, Duflo
et al. (2021) say “the total impact on earnings is 37 shilling (3% of the control groupmean), a very imprecise estimate (95% CI [-10%,+15% of the control group mean],p-value 0.65). We cannot reject that returns are either zero or high compared tostandard estimates of Mincerian returns.” Despite heroic efforts to follow the RCTparticipants for around a decade, the estimated monetary returns to schooling are
imprecise and cannot rule out anything ranging from substantial reductions in earnings to Micerian returns larger than those typically estimated in advanced countries
like Denmark (see e.g. Schoellman, 2012).
For this reason, we instead choose to target the treatment effect on test scores as
the experimental measure of the impact of receiving the scholarship on human cap-

19

ital. The estimate impact is 0.16 standard deviations and, in the model, we convert
this increase in test scores to an increase in wages by assuming that a 0.16 standard
deviation increase in test scores for the treatment group relative to the control groupcorresponds to a 0.16 standard deviation increase in wages for the treatment group,
equivalent to wage gains of 7.6 percent in the quantitative model. This is higher
than the point estimate for earnings of 3 percent from the experiment but is wellwithin the confidence interval. An additional advantage of this approach is thattargeting a direct measure of human capital gains such as test scores circumvents
the concern that increases in earnings may (at least partially) reflect increased accessto rent-earning public sector jobs and thus overstate the extent to which education
improves labor productivity.
The treatment effects on school attendance are also informative about the extent
of misallocation in education. We hence target the experiment’s treatment effecton school completion, which was 27 percent. Additionally, we target the treatment
effect on secondary school completion in the top quartile of test scores relative to thebottom one. This difference is small, at 4 percent, meaning that the overall treatment
effect on secondary school completion was not particularly skewed toward thosewith high test scores relative to those with low scores.
4.4. Model Fit and Validation
Table 2 reports the targeted moments and their values in the estimated model. We
also report the 95-percent confidence intervals for the moments that we resample inthe bootstrap procedure. The fit is good for most moments, but a bit off for severalof them, it must be said. On the plus side, the model does well in matching thetreatment effects on human capital (6.7 percent versus 7.6 percent in the data) andfertility (-11.6 percent versus -10.6 percent in the data). The population growth rate
and the variance of the permanent component of income are matched more or lessexactly, and the model’s treatment effect on secondary school completion is only 3
percentage points higher for the top quartile of the test score distribution than thebottom quartile, which is close to the 4 percent in the data.
The model is less successful in matching the average secondary schooling completion rates (30 percent versus 34 percent in the data), and the model’s completion
rates are a bit too high in the top test quartile and a bit too low in the bottom test
quartile. Overall, though, the model captures the slight increase in completion rates
by test score quartiles in both the control and treatment groups (see Appendix Figure A.1). The treatment effect on secondary school completion is too low in the20Table 2: Targeted Moments and Model Predictions
Moments

Data

Model

Aggregate Population Growth

2.2

2.2

Aggregate SHS Completion Rate

34

30

SHS Completion, Q4 of Test (Control Group)

53

65

SHS Completion, Q1 of Test (Control Group)

41

35

Var(Permanent Component of Income)

0.22

0.22

7.6
(3.2, 12)
-10.6(−20.8, −0.4)
27.0(22.7, 31.3)
4(0, 8)
0.45(0.43, 0.47)

6.7

Treatment Effect on Human Capital
Treatment Effect on Fertility
Treatment Effect on SHS Completion
Treatment Effect on SHS Completion, Q4 - Q1
Intergenerational Schooling Correlation

-11.6
21.3
3
0.32

Note: This table reports the moments targeted in the estimation and their values
in the data and in the model. The range reported below each moment in thebottom half of the table (below the line) is its 95 percent confidence interval.

model, and the same is true of the intergenerational schooling correlation. In Appendix C, we perform some robustness exercises and show that missing on these
two moments is not important for our main conclusions.
The estimated parameter values, and their bootstrapped confidence intervals, arepresented in Table 3. While there is certainly some uncertainty in the estimatedvalues, the confidence intervals for each parameter are fairly reasonable, suggestingthat the model is precisely estimated in a statistical sense.
The estimated parameters seem reasonable from an economic sense as well. The
estimated fertility parameter νJ implies that each less-educated family has 1+1.07 =2.07 offspring, implying an average of 4.1 children in a less-educated family with
two adults. Similarly, the estimate of νS implies that each more-educated family
has 1+0.19 = 1.19 offspring, which corresponds to around 2.4 children per familywith two adults. These predictions are quite similar to (non-targeted) averages fromthe Demographic and Health Surveys for Ghana, which show average fertility of21

Table 3: Parameter Estimates and Confidence Intervals
Parameter

Description

Estimate(Confidence Interval)

νJ

Fertility of Primary School Graduates

1.07(1.03, 1.17)

νS

Fertility of Primary School Graduates

0.19(0.17, 0.21)

ηS

Efficiency of Secondary School

5.66(4.39, 6.14)

ΨS

Goods Cost of Secondary School

1.52(1.48, 1.71)

bIntergenerational Altruism Factor

2.26(2.1, 2.45)

σεStd. Deviation of Exam Score Noise

0.92(0.89, 1.04)

θGumbel Scale Parameter of Taste Shock

0.42(0.39, 0.46)

χSavings Wedge

0.09(0.09, 0.10)ρPersistence of Ability Process

0.79(0.77, 0.92)

συ

Std. Deviation of Ability Shock

0.36(0.34, 0.39)

Note: This table reports the estimated parameters. The confidence interval is the 2.5th
and 97.5th percentiles of 100 bootstrapped parameter estimates.

4.1 children for women with JHS only, and 2.6 children for those with secondary
education or more.
The estimated efficiency of schooling, ηS , is hard to interpret directly but implies(with all the other parameters) an annual return to education of 7.9 percent per year
for this experimental sample. This is generally in line with other estimates of returns
to education in developing countries, and if anything is on the high side. Schoellman (2012), for example, estimates returns of around 4 percent in Ghana and values
generally under 5 percent for Sub-Saharan Africa (with large confidence intervals).
The cost of schooling, ΨS , amounts to 25 percent of GDP per capita, which is close
to the 21 percent reported by Duflo et al. (2021).
To better understand how plausible the estimate of b is, we compute the compensating variation of secondary schooling for all children in the model, at age 15
when their schooling decision is being made. We find that the average compen22

sating variation is similar to the average cost of schooling (including opportunitycost), modestly lower for children who receive only a JHS education, and substantially larger for those whose parents choose a secondary education (see Appendix
Figure A.2). These calculations imply that the children’s valuation of schooling ismostly in line with that of their parents, suggesting that the value of b is reasonable.
The savings wedge, χ, has a value of 0.09, which implies that households save ataround 11 percent per period, or 2 percent per year. This is a low return to savings
but not as low as the negative returns posited by other similar incomplete-markets
models estimated to data from developing countries (e.g. Lagakos, Mobarak, and
Waugh, 2023; Donovan, 2021).4
The estimated value for the intergenerational persistence of ability, ρ, is 0.79, implying a strong correlation between parents’ and children’s ability. This is broadly
consistent with the recent conclusions of Lee and Seshadri (2019) that parental traits,summarized by ability in our model, explain a substantial amount of the variationin children’s income levels. The estimated standard deviation of the ability shock,συ , is 0.36. While not directly interpretable, this value (along with the other parameters) generates a Gini coefficient within the model of 0.31. This is somewhat lower
than the Ghanaian value of 0.43 but within the range of 0.3 to 0.6 reported in theWorld Development Indicators for other Sub-Saharan African countries.5

5. Simulating the Effects of Free Secondary School
Using the estimated model, we simulate the effects of a national free secondary
schooling policy. To do so we make two specific changes to our model. First, we
4

The average household asset-to-income ratio in the model is 0.5. This is broadly in line withother estimates from low-income countries, such as Samphantharak and Townsend (2018), who finda ratio of around 0.6 in Thai villages. Unfortunately, we know of no reliable household asset data in
Ghana to which we can make a direct comparison.
5
In Appendix C we discuss how we view each parameter as being identified from the targeted
moments. We do so by computing the elasticities of each model moment to each parameter, following
Kaboski and Townsend (2011), and by computing the sensitivity of each parameter to each targeted
moment, following Andrews, Gentzkow, and Shapiro (2017). In short, the population growth parameters, νJ and νS , largely shape the model’s aggregate population growth rate and treatment effects
on fertility. The variance and persistence parameters of the ability process, συ and ρ, relate mostclosely to the permanent component of income variance and the intergenerational schooling correlation, though they affect other moments significantly as well. The efficiency of schooling, ηS , andthe intergenerational altruism parameter, b, drive the benefits of schooling, and play a big role in themodel’s aggregate secondary attendance; ηS affects the treatment effect on human capital more directly. The cost of schooling, ΨS , reduces school attendance, raises the treatment effect on schoolingand lowers the treatment effect on fertility. The savings wedge, χ, and the standard deviations of thetest score noise and taste shocks all impact secondary attendance in the top versus bottom quartilesof the test score distribution.

23eliminate the goods cost of schooling for parents, and force the government to pay
for all the goods costs through tax revenues. Second, we eliminate the test score cutoff required for SHS admission. This is consistent with Ghana’s actual free schoolingpolicy. We assume that households do not anticipate the policy and that the economy is on the balanced growth path at the time of implementation.
Public funding of SHS in the model works as follows. We require that the government raise taxes in proportion to the existing tax rates. Before the policy, eachhousehold paid taxes according to the tax function T that is a function of parent’s
and child’s income. The post-policy tax function takes the form (1 + τt )T where τt
is the proportional increase in taxes each period. Taking this approach maintains
the current structure of the labor tax schedule, and in particular, the feature that thepoorest half of households pay no taxes (see Appendix Table A.3).
We choose τt so that per period tax revenue along the post-policy balanced growth
path is equal to per period tax revenue along the pre-policy balanced growth pathplus the additional education expenditure. In other words, we assume that the policy does not change per capita spending on public goods G.6
5.1. Quantitative Results
The general equilibrium effects of the policy are summarized in Table 4. We also
report confidence intervals for each aggregate outcome using the bootstrapped parameter estimates summarized above. While this is a natural use of bootstrapped
parameter estimates, it is not commonly done in G.E. counterfactual simulations
in macroeconomics. The goal is simply to quantify the uncertainty in the model’s
predictions arising from sampling uncertainty in the targeted moments — the experimental moments in particular as these are estimated with large standard errors.
The number of secondary schooling graduates increases by about 12 percentage
points, under free schooling, from 30 percent of the population to 42 percent. The
increase is small relative to the changes in secondary school completion in the experiment, in large part because the experimental sample is highly selected relative tothe general population. Fertility falls due to the schooling expansion, and the population growth rates fall by 0.2 percentage points. Adult earnings increase by about
1.2 percent from the policy, stemming largely from the higher wages for the 12 percent of the population now receiving secondary education. On the other hand, child6

We have experimented with alternative public finance arrangements but find that they make no
substantive difference in our conclusions. For this reason we stick with the simpler assumption of
period-by-period budget balance.

24

earnings decrease by 7.5 percent, representing the opportunity cost of the newly
educated workers.
Table 4: G.E. Effects of Free Secondary Schooling
Change Under
Free Schooling

Statistic
GDP per Capita (%)
Secondary School Completion (p.p.)
Population Growth Rate (p.p.)
Adult Earnings (%)
Child Earnings (%)
Taxes per Capita (%)
Skilled Wage/Unskilled Wage (%)
Gini Coefficient-1.0(-4.2, -0.7)
11.8(4.2, 13.1)-0.2(-0.3, -0.1 )
1.2(-4.4, 1.8)
-7.5(-9.2, -1.5)
1.4(1.1, 1.7)
-9.8(-11.8, -6.5)-0.04(-0.05, -0.04)

Note: This table reports the estimated aggregate effects of free secondary schooling. The changes insecondary school completion rates and population growth rates are expressed in percentage points.
The changes in the Gini coefficient is measured in levels. The changes in all other statistics are expressed in percentage changes. The range reported below each estimated value is its bootstrapped
95 percent confidence interval for the change.

Rather than increasing, GDP per capita falls by about 1 percent in the long-run.
Our confidence interval excludes decreases smaller than 0.7 percent in magnitude.
Thus, from the perspective of sampling uncertainty in the targeted moments, we
can reject any positive impact on GDP. The long-run cost of the policy is 1.4 percent
of GDP, implying a total cost of the program of about 2.4 percent of GDP (the direct
cost plus lost GDP). Relative wages of the skilled fall by about 10 percent, pointing
to clear distributional impacts of free schooling policies, even for those who remain
unskilled after the policy change. Our predictions here are similar at least qualitatively to those of Khanna (2023), who finds substantial declines in the relative wagesof skilled workers after an education expansion in India. His wage effects are larger
than ours quantitatively, though his study focuses on the short run where elasticities
25

of substitution between low and high skilled workers are likely smaller.
Figure 3: Child SHS Attendance Probability in Estimated Model
100

60

80

60

40

40

20

20

0
20

40

60

80

80
60

80

60

40

40

20

20

0

0

0

100

Unconstrained Cutoff

80

100

Parent Ability (Percentile)

Unconstrained Cutoff

Parent Ability (Percentile)

100

100

0

0

20

40

60

80

100

Child Ability (Percentile)

Child Ability (Percentile)(a) Attendance Probability(b) Increase in Attendance

The modest declines in GDP suggest that the estimated model does not featurehigh levels of misallocation in education. Recall that Figure 2 provided examplesof economies with high and low misallocation by displaying the probability of SHS
attendance as a function of child and parent ability. Figure 3, Panel (a), displays an
identical plot using the fully estimated model. While it falls between the two extremes shown in Figure 2, it is clear that the estimated model corresponds moreclosely to the case with low misallocation; the highest ability children are fairly
likely to attend SHS even if they are born to low ability parents. Attendance does
have some dependence on parent ability; however, this effect is fairly minimal andthe probability of attendance looks much closer to that of Panel (a) in Figure 2. Increases in attendance from free secondary schooling (Panel b) also more closely resemble those of the low-misallocation economy.
5.2. Opportunity Cost, Selection, and Differential Fertility
Given that the back-of-the-envelope calculation at the end of Section 2 suggested
a sizable large GDP gain of around 6 percent based on the experimental estimates, it
is natural to wonder what mechanisms lead the model to instead predict a modest
reduction in GDP as a result of expanding secondary schooling. Here we perform
counterfactual experiments using the model to highlight the role of three channels
that each explain some portion of the difference.
We present the results of counterfactuals, which we describe below, in Figure 4.
In each panel, the left-most brown bars reproduce the predictions for free secondary26

schooling coming from the baseline economy. The other four sets of bars, to theright of the baseline, represent the results of the four different counterfactual exercises. The panels represent the increases in GDP gains (top left), secondary completion rates (top right), adult earnings (bottom left), and child earnings (bottom right)
coming from free secondary schooling.
First, we highlight the role of opportunity cost. To do this, we consider a counterfactual world in which the opportunity cost of secondary school is zero and simulate
the effects of the free schooling policy in this world. We implement this in the modelby simply eliminating the option for children to spend ages 15 through 20 working,so that parents’ choice is between education and idleness (which neither the childnor the parent value) rather than education and income. We then recompute thepre-policy steady state of the model and simulate the effects of the free SHS in thisnew world.
The results of this exercise can be seen in the beige columns of Figure 4, marked‘no opportunity cost.’ In this counterfactual, GDP per capita increases by 2.9 percent
compared to the -1.0 percent drop in the baseline. This larger income gain is driven
in part by substantially larger increases in secondary attendance of 23 percentage
points, compared to 11.8 percentage points in the baseline. This larger increase insecondary completion leads to a larger increase in adult earnings (4.8 percent), andthe lack of work during secondary years eliminates almost all of the decline in children’s earnings. Overall, this counterfactual suggests that opportunity cost reduces
the gains from free schooling by almost 4 percentage points (2.9 percent minus -1.0
percent).
The second counterfactual examines the role of selection stemming from the nonrepresentativeness of the experimental sample. We simulate the effects of free schooling while simultaneously boosting the abilities of new secondary attendees so thatthey draw from the ability distribution of the experimental sample. More precisely,
we start from the pre-expansion steady state and implement the free secondary
school policy as in the baseline; however, every child who would not have goneto school according to the pre-expansion policy functions redraws their ability z
randomly from the ability distribution of the experimental sample. After seeing theresult of this draw, the parent decides whether or not to send the child to school.
As a result, the set of new attendees looks identical to the experimental sample,
effectively eliminating the role of selection.
The results are displayed as the orange bars marked ’No selection on ability’27

Figure 4: Effects of Free SHS under Counterfactual Scenarios
8

24.5

25

23.4

6.9

Percentage point change

Percentage point change

6

4
2.9

2

1.2
0.0

0

20

15
12.7
11.8

10
7.6

-1.0

-2

5

Baseline

No opportunity No selection Same fertility
on ability (ii)cost (i)rate (iii)All(i+ii+iii)

Baseline No opportunity No selection Same fertility
on ability (ii)rate (iii)
cost (i)(a) GDP per Capita

All(i+ii+iii)(b) SHS Completion Rate
2.8

10

9.5

2

Percentage point change

Percentage point change

8

6
4.8

4
2.5

2

2.7

0

-0.1

-2
-2.6

-4

-6

1.2

0

-8

Baseline No opportunity No selection Same fertility
rate (iii)
cost (i)on ability (ii)All
(i+ii+iii)(c) Adult Earnings

-7.5

-7.5

Baseline No opportunity No selection Same fertility
rate (iii)
on ability (ii)cost (i)All(i+ii+iii)(d) Child Earnings

in Figure 4. Rather than the 1 percent decrease predicted by the baseline model,
GDP per capita is essentially unchanged in this counterfactual. The reason is thatearnings of adults now increase by 2.5 percent – around twice the increase observedin the baseline. In other words, the fact that the experimental sample exhibits higher
average ability than the general population knocks off around a percentage pointfrom the GDP gains from expanding secondary access.
We investigate the role of fertility changes in the third counterfactual. Like withthe selection counterfactual, we begin from the pre-expansion steady state of themodel, and then implement the free schooling policy. We then change the fertility
level of new secondary attendees to be 1 + νJ , the fertility level of those without
28

Figure 5: Changes in Ability Distribution Induced by Free Schooling
0.5

Parent Ability (Percentile)

100
80
60

0

40
20
0

-0.5

0

20

40

60

80

100

Child Ability (Percentile)

secondary education. This counterfactual therefore allows free schooling to operate
as before but now without affecting fertility for new attendees.
The dark orange bars labeled ’Same fertility rate’ report the results of this exercise. The GDP per capita gains are now modestly positive at 1.2 percent. Like inthe case of selection, this is largely driven by an increase in adult earnings, which
expand by 2.7 percent. Secondary attendance increases by somewhat less than thebaseline, and child earnings decrease by less. To see why adult earnings increase
once the fertility channel is shut off, Figure 5 plots the percentage point changes inthe (post-policy) distribution of ability in this case relative to the baseline. The blue
portions of the figure are where population declines (by around 0.5 percent of the
population) and the red portions indicate places where population increases (also byaround 0.5 percent). The magnitudes are not dramatic, but the overall patterns are.
There is a significant shift from higher ability parents and children to lower-ability
ones. This figure highlights how unfavorable long-run movements in the ability
distribution serve to reduce the gains from free schooling by about 2.2 percentage
points.
The red (rightmost) bars of Figure 4 display what happens when we combine
all three counterfactuals, shutting down the opportunity cost, selection and differential fertility channels. In this case the GDP gains from free schooling are large,29

at 6.9 percent, and close to 8 percentage points higher than in the baseline. Now
secondary schooling expands by a robust 24.5 percentage points and adult earnings
grow by close to 10 percent, while child earnings also grow modestly. Unlike in themain analysis, taxes per capita stay roughly the same in this counterfactual as freeschooling largely pays for itself. Note that the overall GDP gain in this counterfactual is close to the back-of-the-envelope gains we reported in Section 2 taking into
account just the returns to schooling proxied by test scores and the increase in yearsof schooling.
A key takeaway from these counterfactuals is that the model has no problem
delivering a sizable gain in GDP from free schooling once we shut down its opportunity cost, selection and differential fertility channels. Thus, we conclude that theseforces are the key ones that combine to eliminate nearly all the gains from higher human capital from free schooling implied by the significant test score and secondary
completion increases found by Duflo, Dupas, and Kremer (2021).7
5.3. Welfare and Distributional Impacts
We turn next to the welfare and distributional impacts of the free secondary
school. Welfare is inherently hard to discuss in this setting due to the presence ofendogenous fertility. Any policy that changes schooling attendance decisions alsochanges fertility, and the set of agents that exists in the post-policy balanced growth
path is not the same as the set that would have existed had the policy never been
implemented. In our case, free secondary school leads to higher attendance andlower fertility. Thus some individuals who would have existed had the policy not
been implemented are instead never born.
Rather than taking a stance on how to aggregate welfare across agents who may
or may not exist after a policy change (as in Golosov, Jones, and Tertilt, 2007; De la
Croix and Doepke, 2021), we simply report consumption-equivalent welfare separately for different groups of agents. In particular, we focus on parents with newborns at the time of policy implementation and their eventual children. Because
these parents have already given birth, their fertility decisions are determined, andall agents in these two groups exist both with and without the policy, making traditional welfare comparisons possible. Additionally, to get a sense for how welfare
and fertility changes interact, we examine welfare for two groups of the parents’7

The results of this section highlight that our model’s negative predictions about GDP per capita
are not driven by the fact that the experiment’s treatment effect on labor earnings is wide and includes zero. We do not target this moment in our estimation.

30

Table 5: Welfare Change of Select Groups under Free Schooling
Overall

Bottom 25 Pct Top 25 Pct

Parents

3.9

7.3

-4.2

Children

12.4

23.1

-5.3

Grandchildren (always born)

9.0

-

-

Grandchildren (born w/o policy)

16.9

-

-

Note: This table reports the average change in consumption-equivalent welfare for select groups ofindividuals under the free schooling policy.

grandchildren: those who would always exist regardless of schooling policy andthose who would exist only if schooling were not expanded.
For each of these groups, we compute lifetime utility from consumption, ignoring utility gained from altruism towards children (or grandchildren or greatgrandchildren, etc.). We compute this value under both the free secondary schooling policy and the case of no policy change, and our measure of consumptionequivalent welfare reports the percentage increase in the consumption of all individuals (within the relevant group) required to raise the average utility level underno policy change to that achieved by the policy.
We are also interested in the redistributive component of the policy; that is, howmuch of the welfare gains accrue to poor households relative to rich ones. As in
Fernández and Rogerson (1995), rich households in our model are more likely to
go to school. Thus a free SHS policy risks being regressive. Unlike Fernández andRogerson (1995), who model schooling as funded through proportional taxation,
this effect is mitigated by the fact that our tax schedule is strongly progressive. Thus
the redistributive nature of the policy is a quantitative question. In order to answer
this question, we also report the change in welfare for parents and children in the
top and bottom 25 percent of the (pre-policy) income distribution.
The first two rows of Table 5 report our welfare measure for parents (those withnewborn children at the time of policy implementation) and their children. Welfare
for parents increases on average by 3.9 percent of consumption. This is somewhat
surprising as, due to higher school attendance and the loss of children’s wages, thepolicy actually reduces income in this group. Instead, the welfare gains come entirely from redistribution. The poorest 25 percent of parents gain 7.3 percent of consumption while the richest lose 4.2 percent. Despite concerns that wealthy parents
31are more likely to send their children to secondary school and thus are more likely
to be beneficiaries of the policy, the tax schedule is sufficiently progressive that thepolicy acts redistributively overall.
The gains for children are large and positive at 12.4 percent. As was the case forparents, the policy is highly redistributive. The children who would have ended up
in the top 75 percent of the income distribution absent the policy lose 5.3 percent,roughly the same amount as the parents, due to higher taxes. The (pre-policy) poorest 25 percent of children, however, make substantial gains of 23 percent. While asmall portion of these gains occur broadly within this group due to higher unskilledwages, the majority are accrued by the relatively small number of misallocated children who make substantial income gains.
The welfare results discussed so far present a small puzzle. Despite a decline inlong-run output per person of 1 percent, the policy seems to lead to large gains in
welfare, particularly for children who gain more than can be plausibly attributed
to redistribution. This tension is resolved by examining the welfare impacts in thelonger run. The third and fourth rows of Table 5 list welfare gains for the grandchildren of the parents described in the first row, separated into those who will always
be born regardless of policy and those who would only be born in the pre-policy
steady state.8 While the welfare gains for the grandchildren who exist regardlessof policy are positive, they are smaller than those of the children (9 percent vs 12
percent). The central reason is that the grandchildren who do not get born under the free schooling policy experience disproportionately large benefits (almost
17 percent). Essentially, the decline in fertility serves to shift the composition of the
population towards those who experience smaller gains. This does not stop after
the grandchildren are born and continues to compound with each generation, eventually leading to the long-run decline in GDP per capita described in Table 4.
5.4. Robustness and Discussion
Overall, the results of this section suggest that the free secondary schooling policy
does not lead to a long-run increase in overall income per capita and instead serves
mostly as redistribution. In order to focus on the channels of opportunity cost, selection, and fertility, the model used to draw these conclusions simplifies away somepotentially important features. Here, we discuss some of these features and perform
8

Conceptually, the welfare for grandchildren who would not be born if the policy were implemented is computed by asking the question “if the policy were implemented and your dynasty alone
was exogenously assigned to experience no fertility change (so that you, but none of the others inyour category, exist), what would your change in welfare be?”32

some basic quantitative exercises to assess the robustness of these conclusions.
Familial Bargaining and Transfers: One restriction placed upon the model is an
abstraction away from household bargaining in favor of the assumption that parents
dictate the decisions of children and that there are no transfers (in either direction)
between parents and children. While this provides tractability, it raises the possibility that parents and children disagree on the value of schooling and lack a market inwhich to interact and resolve this disagreement. Further, the lack of non-schoolingtransfers means that secondary school is the only mechanism through which parents can make transfers towards children. These features raise the possibility thatthe conclusion that the gains from free schooling are small arises from parents overvaluing secondary school and “over-schooling” by enrolling children whose private
valuation of school would not have them attend.
To investigate whether or not this is a concern, we perform a compensating variation calculation to assess children’s valuation of schooling. In particular, we compute the level of assets at which a child becomes indifferent between being “born”(i.e. forming their own household) possessing a JHS education and the stock ofassets or being born possessing an SHS education and no assets. In essence, thiscalculation computes the child’s value of secondary education.
Appendix Figure A.2 displays the average compensating variation (as a percent
of GDP per capita) for SHS and JHS graduates in the pre-policy steady-state of themodel. Reassuringly, the average valuation of secondary schooling among those
who receive it is much larger than the total (i.e. opportunity-cost-inclusive) cost of
the schooling (217 percent of GDP per capita vs 83 percent). Similarly, the averagevaluation for those who do not attend secondary school is smaller than, but close
in magnitude to, the cost (78 percent of GDP per capita vs 83). At least on average,children who value secondary schooling above its cost are attending while those
who value it under the cost are not, assuaging concerns that the lack of familial
bargaining or transfers may be driving the headline results.
Human Capital Externalities: Externalities through which increases in the stockof skills improve overall productivity, leading the aggregate gains in output due toan increase in education to be larger than the sum of the private gains (e.g. Acemoglu and Angrist, 2000), are often referenced as reasons for education expansion
and are completely absent from the baseline model. The absence of such humancapital externalities would lead the model to underpredict the increase in GDP per
capita arising from an increase in educational attainment due to the free secondary33

school policy.
Fortunately, extending the model to include such an effect is fairly simple; addingβ̃a term Hext
to the production function yields
h
i 1−α
λ
β̃
Y = AHext
K α (NJ )λ + (NS )λ(11)

which can then be combined with the equilibrium condition Hext = NS /P to implement the human capital externality, ensuring the neither firms nor individuals
internalize the social benefit of increasing the average skill level in the population.
With this production function, the ratio of the marginal social benefit of moving
one individual from primary to secondary education (i.e. decreasing NJ by unity
and increasing NS by the individual’s z) and the marginal private benefit does notβ̃ NJλ +NSλ
depend on z and is given by 1 + 1−α. This expression yields a correspondenceNλS

between β̃ and the spillovers generated by the externality — if an individual’s return to a year of education is 10 percent, but increasing the economy-wide level of
education by one year increases output by 12 percent (suggesting a spillover of 20
NSλpercent), the corresponding value of β̃ is given by (1 − α)(1.2 − 1) N λ +N
λ.
JS

We choose β̃ to yield a generous spillover of 50 percent (i.e. the social gains ofeducation are 1.5 times larger than the private gains), corresponding to a value ofabout 0.16. This is substantially larger than many estimates in the literature, which
often find a small role for such externalities (Acemoglu and Angrist, 2000; Ciccone
and Peri, 2006), particularly in the case of secondary schooling (Iranzo and Peri,2009), and even exceeds the larger estimates (e.g. Gennaioli, La Porta, Lopez-de
Silanes, and Shleifer, 2013, who find a spillover of about 40 percent).9
Even after incorporating this large externality, implementing the secondary school
expansion policy continues to yield results similar to those in the baseline model.
Rather than decreasing by 1 percent, GDP now increases by 0.2 percent. Although it
is now positive, the impact of the policy on GDP remains anemic and substantiallysmaller than the back-of-the-envelope calculation in Section 2 would suggest. The
aggregate return human capital is larger than in the baseline model but, as discussed
above, the estimated model does not feature particularly high levels of misallocation
and, consequently, the schooling expansion policy only modestly increases aggregate human capital, leading to small results even when an externality is included.
9

In particular, Gennaioli et al. (2013) estimate a Mincer return to schooling of 20 percent and aregional TFP increase from an additional year of education of 7.5 percent, suggest a spillover of 37.5
(7.5/20) percent

34

Rationed Public Sector Jobs: In many developing countries, the ability to obtain
public sector jobs, which command a large wage premium relative to the private
sector, is a large part of the reason that many choose to complete secondary education (or beyond). While these jobs lead to a substantial increase in earnings forthe individuals who obtain them, the common view among economists is that thispremium reflects government rents rather than a higher productivity of the public
sector. Because these rents are not subject to market pressure and effectively act asa wage floor, these jobs end up being rationed.
It is unclear how much of an issue this is for our results. To the extent that higher
earnings for secondary school graduates reflect these rents, rather than higher productivity, the model will overestimate the impact of free schooling. However, themodel is estimated to match the increase in human capital, measured via test scores,in the treatment group rather than the increase in wages which is less subject to thisconcern. This increase in human capital increases output, even if government rents
impact how that output is distributed across individuals.
Still, it is useful to extend the model to examine the potential magnitude of thisissue. Duflo et al. (2021) report that 7.7 percent of their control group (secondary
graduation rate of 42.3 percent) end up employed in a public sector job. Assuming that all public sector jobs are allocated to secondary graduates,10 this implies
roughly an 18 percent chance of obtaining public employment conditional on completing secondary school. Finan, Olken, and Pande (2017) estimate a public sector
earnings premium (in Ghana) of 76 percent. Attributing all of this earnings premium to rents (rather than higher productivity) and combining these two numbers((1 − 0.177) + 1.76(.177) − 1 = 0.13) suggests that, at most, public sector rents accountfor 13 percent of the average return to secondary schooling.
To account for this in the model, we augment the human capital function h sothat the human capital for someone with SHS education becomes zηS 1 − 0.13(1 −1ˆ is the percent deviation of the aggregate SHS share from the pre)where SHSˆSHS
policy steady-state. Essentially, if the number of secondary school graduates douˆ = 2), the return to secondary schooling falls by 6.5 percent, reflecting thatbles (SHS
the fact that the probability of any particular student obtaining a public sector job(which accounted for 13 percent of the pre-policy return) has been cut in half.
In this extension of the model, the impact of the free secondary schooling ex10

Violations of this assumption would imply that our estimate for the contribution of the public
sector earnings premium to the total SHS earnings premium is an upper bound.

35

pansion policy is almost identical to that of the baseline model. Unsurprisingly, thedecline in GDP is slightly larger than the baseline model at -1.8 percent, a result of
the fact that this extension to the model unambiguously shrinks the potential output
gains from secondary school. The fact that the decline is only slightly larger than thebaseline model reflects the fact that the increase in secondary attendance is modest,
only about 35 percent (11 percentage point) and, consequently, the decline in the
return to schooling induced by the model extension is small (about 5 percent).

6. Aggregate Effects of Alternative Policies
If making secondary school free to parents does not raise living standards, arethere any alternative policy levers that governments in low income countries canpull to bolster their education systems and raise their average income levels? In
this section we address this question by using our estimated model to simulate theaggregate effects of some alternative education policies.
Table 6: Aggregate Effects of Alternative Policies

Free Secondary School (Main Analysis)
Free Secondary School + Relaxed Test
Free Secondary School + Keep Test As-Is
Only Remove Test
Tax + Transfer to Bottom 25%
Raise Schooling Quality

GDP
Gain
(%)

SHS
Increase(p.p.)

-1.0
-0.3
0.1
-0.4
0.1
2.7

11.8
6.3
3.0
8.4
-1.6
13.8

Note: This table reports the gains in GDP and the increase in the SHS graduation rate under freeschooling and several alternative policies (described in the text).

Table 6 summarizes the results of various alternative policy counterfactuals. The
first row reproduces the key aggregate statistics from the free schooling policy counterfactual from the previous section (i.e. the gains in GDP and the increase in secondary schooling completion). The second reports the same outcomes in an alternative simulation where free secondary schooling is offered but, unlike the main
analysis, the requirement that secondary students pass an admissions test is maintained. This policy mitigates GDP losses by maintaining higher standards for who
can enter secondary schooling, reducing the problems of negative selection found36

in the main analysis. GDP per capita now only falls by 0.3 percent and the change
in secondary attendance is smaller. The third row performs a similar exercise but
does not change the stringency of the test at all, leading to basically no change in
GDP per capita. On the other hand, the fourth row reports the effects of eliminating
the test requirement but maintaining a positive cost of schooling. As in the main
analysis, this leads to an increase in attendance but a decline in GDP.
The fifth row simulates a pure tax-and-transfer scheme to isolate the redistributive component of the free secondary schooling policy. We increase the tax rates by
the same magnitudes as in the baseline policy but now simply distribute the extra
revenues as a lump-sum transfer to the bottom 25 percent of earners. SHS decreases
slightly as the increase in taxes distorts parents’ educational decisions for their children; however, GDP is left essentially unchanged (as the students pushed out of
SHS have marginal opportunity-cost-inclusive return to schooling).
The last row of Table 6 summarizes the effects of improving schooling quality
in such a way that average test scores rise by 0.1 standard deviations and eliminating the admissions test. This effect is conservative relative to the average effect size
estimated in a number of different randomized interventions aimed at improving
schooling quality in the developing world, many of which find effects of around 0.2
standard deviations or higher. One such intervention is to offer financial incentives
to teachers based on the test scores of their students. Muralidharan and Sundararaman (2011) and Duflo et al. (2012) found that this raised test scores in India forexample, while Mbiti et al. (2019) found effects of a similar size for teacher incentives plus block grants for schools in Tanzania. Another successful schooling quality
intervention is to increase the number of teachers in the classroom, as in the studies
of Banerjee et al. (2007) and Muralidharan and Sundararaman (2013) in India. For
our simulated intervention, we use the policy cost from Mbiti et al. (2019) who report that the cost of increasing test scores by 0.1 standard deviations per student in
Tanzania was US $5.78.
Our model implies larger effects on GDP and welfare of improving schooling
quality than providing free schooling. GDP increases by 2.7 percent under such anintervention. Even though this policy has no provisions aimed at expanding secondary enrollment directly, improved schooling quality raises school enrollments
by 13.8 percentage points, slightly more than the free schooling policy. The implication is that many students were not attending secondary schooling to begin withbecause they felt the returns were not high enough to justify the costs (including

37

Table 7: Welfare Change of Select Groups under Improved Schooling Quality
Overall

Bottom 25 Pct Top 25 Pct

Parents

3.0

4.8

-0.8

Children

18.6

24.4

6.9

Grandchildren (always born)

14.5

-

-

Grandchildren (born w/o policy)

26.4

-

-

Note: This table reports the average change in consumption-equivalent welfare for select groups ofindividuals when schooling quality is improved.

opportunity cost).
These gains to GDP suggest that the channels emphasized in Section 5 are not solarge as to prevent any increases in output from education interventions. While opportunity cost is still present, schooling quality improvements amplify the benefitsof attendance directly and increase incomes even for inframarginal students, leading
to larger output gains per “unit” of opportunity cost paid. Quality improvements
also mitigate the impact of the selection channel; although new attendees may still
have lower average ability than the experimental sample or current attendees, thefact that higher schooling quality affects both new and current attendees means thathuman capital increases even for the highest-ability students .
Table 7 lists the changes in consumption-equivalent welfare under improved
schooling quality for the same groups of individuals as the discussion of welfare
under the baseline policy in Section 5.3. Unsurprisingly, the gains in welfare forparents are smaller under the policy of improved quality (3.0 percent vs 3.9 percent)as, unlike the free secondary schooling policy, it does not represent a direct transfer
of resources to parents. Instead, the gains in welfare are largely due to redistributionthat occurs due to the change in relative wages.
Children and grandchildren benefit substantially more under improved schooling quality (18.6 percent vs 12.4 percent for children). This difference is almost entirely attributable to improvements in the welfare gains for the richest 25 percent
of children who go from losing 5.3 percent under the baseline policy to gaining 6.9
percent when school quality is improved. The reason for this substantial increaseis twofold. First, unlike the baseline policy of free schooling which improves theincome of only marginal attenders, improvements in school quality also improveoutcomes for inframarginal attenders (who make up almost all the top 25 percent).
38

Second, the policy is highly cost effective and pays for itself in the long run, resulting in a reduction in taxes — a benefit that mostly accrues to the richest. Despite
this substantial improvement for the richest 25 percent, the poorest 25 percent still
do better than under the baseline policy (24.4 percent vs 23.1 percent), suggestingthat these improvements do not come at the cost of welfare for the very poorest.

7. Conclusions
One of the main reasons income per capita is so low in the developing world isthat human capital levels are so low (Hall and Jones, 1999; Bils and Klenow, 2000;
Erosa et al., 2010; Manuelli and Seshadri, 2014; Hendricks and Schoellman, 2018).
Since attendance rates in primary schools are high in most low income countries,
attention has turned to increasing secondary education. Making secondary schooling free for students, and funding the costs through higher taxes, is a natural policy
option to consider. Not surprisingly, many developing countries are currently considering or implementing free secondary schooling policies of some kind.
In this paper we analyze the macroeconomic effects of free secondary schoolingpolicies in the developing world, looking through the lens of an OLG model of human capital accumulation with credit constraints. We focus on the case of Ghana, forwhich we can draw on recent experimental evidence on the outcomes of students
randomly assigned to receive free secondary schooling, leading to higher secondary
school completion rates and higher average test scores (Duflo et al., 2021). Ghana
is also a country that has recently adopted free secondary schooling, and the policy
is viewed as a success there and in other developing countries (Center for Global
Development, 2022).
Our conclusions are less optimistic. When we simulate the general equilibrium
effects of free secondary school in our model, we find that it raises secondary enrollment substantially but has a negative impact on GDP per capita. Three forces in
the model serve to offset the human capital gains from the policy. The first is lostearnings of secondary aged individuals, who forgo work in order to attend school.
The second is the worse selection of students by ability level in school, which lowers
the average effect of secondary education. The third is relative declines in fertility
for the high ability, whose share in the overall population declines.
Of course, there are benefits of education that are beyond the scope of our model.
For example, education expansions have been shown to reduce crime (e.g. Lochnerand Moretti, 2004), create more informed voters, and improve parental ability (e.g.
Daruich, 2020). We abstract from these channels largely due to a lack of evidence for39

our setting (though Duflo et al. (2021) found no evidence that increased school attendance altered voting behavior) and because we view them as secondary relative
to the direct impact of human capital on productivity. Moreover, we conjecture thatgiven our modest estimated effects of free schooling policies on schooling completion, adding these effects would be unlikely to have much additional impact.
We conclude that free secondary education policies are mostly redistributive innature, rather than a path to economic growth, at least at current low levels ofschooling quality. Improving schooling quality would lead to substantially higher
GDP per capita and welfare across the income distribution for less fiscal cost. Improving schooling quality would also expand schooling enrollments by around thesame amount as free schooling policies. Our analysis suggests that the best way forpoor countries to raise human capital levels is to focus on improving the quality of
their current schooling systems rather than giving away a mediocre education tomore young people.

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46

Appendix (for Online Publication)
A. Appendix Figures and Tables
Figure A.1: SHS Completion by Quartile of Test Score: Data vs Model1.9.8Control (Model)
Control (Data)
Treatment (Data)

.6
Ratio

0.71

0.69

.7

0.67

0.67

0.43

0.70
0.65
0.56

0.53

.5
.4

0.86 0.84

Treatment (Model)

0.44

0.43

0.47

0.39
0.35

.3
.2
.1
0

Q1

Q2

Q3

47

Q4

217

Percent of GDP per capita

200

150
Cost of SHS
100
78

84

50

0
SHS Graduates

Non-graduates

All 25 year olds

Figure A.2: Compensating Variation of Secondary School to Children

48Table A.1: Free Secondary Schooling Policies in Developing Countries
Country

Year RequirementBenin2007 Pass Brevet d’Etudes du Premier Cycle

Gambia2015 Pass Basic Education Certificate Exam

Ghana

2017 Pass Basic Education Certificate Exam

Kenya

2008 Pass Certificate of Primary Education Exam

Malawi2019 Pass Primary School Leaving Certificate Exam

Mauritius

2016 Pass General Certificate of Education Exam

Nepal

2018 Pass final district-level exam

Philippines 1988 Do not fail in two consecutive years
Rwanda2012 Score ≥ ‘High’ on O-level Test

Sierra Leone 2018 Score ≥ 6 on Basic Education Certificate Exam
Tanzania

2015 Pass Standard 7 Exam

Uganda2007 Score ≥ 28 in Primary School Leaving Exam

Zambia2022 Pass Baccalaureate Exam

Note: This table reports the year that each country adopted a free secondary schooling policy and the merit requirement to attend secondary schooling.

49

Table A.2: Parameter Values Used in Discussion
Misallocation
Description

Parameter

Low

High

Fertility of Primary School Graduates

νJ

1

Fertility of Secondary School Graduates

νS

0.2

Intergenerational Altruism Factorb

2.5

Std. Dev. of Exam Score Noiseσε1Gumbel Scale Parameter of Taste Shockθ

0.5

Persistence of Ability Shockρ

0.8

Std. Dev. of Ability Shock

συ

0.5

Std. Dev. of Idiosyncratic Income Shock

σζ

0.3

Gains from Secondary School

ηS

5.5

14.0

Goods cost of Secondary School

ΨS

1.5

5.0

Savings Wedge

χ

0.1

0.2

Note: This table lists the parameter values used for creating the figures in subsection 3.4.
The two versions of the model share many parameters and differing only on three key
parameters. Parameters not listed here take the value given by Table 1 for both models.

Table A.3: Labor Income Tax Schedule in Ghana
IncomeTax Rates

First 1,008 GHC (=up to 42% of GDP p.c.)0%

Next 240 GHC (=up to 52% of GDP p.c.)5%

Next 720 GHC (=up to 82% of GDP p.c.)

10%

Next 14,232 GHC (=up to 675% of GDP p.c.)

17.5%

Exceeding 16,200 GHC ( ≥ 675% of GDP p.c.)25%

Note: The table reports the marginal labor tax schedule in Ghana in 2011.
It shows, by income in Ghanaian Cedis (GHC), the marginal tax rate assessed on labor income, and the corresponding ratio of GDP per capita in
Ghana in 2011.

50

B. Model Appendix
In this appendix we define the concepts of recursive competitive equilibrium andbalanced growth path for our model. Letting X denote the vector of individualstate variables (τ, a, zp , sp , ζp , δJ , δS , zc , sc , z̃c , ζc ), a recursive competitive equilibriumis defined as follows.
Definition: A recursive competitive equilibrium consists of
1. A price system wS (f, P ), wU (f, P )2. Household value functions V (X, f, P ) and policy functions a′ (X, f, P ), c(X, f, P ),s′c (X, f, P )3. Perceived laws of motion f ′ = F (f, P ), P ′ = H(f, P )such thata) V, a′ , c, s′c solve the household’s optimization problem given wS , wU , F, P .
b) For all f, P ,
h
i 1−α
−1
λ
,
wS (f, P ) = (1 − α) AK α (NJ )λ−1 (NJ )λ + (NS )λ
i 1−α
h
−1
λ
wU (f, P ) = (1 − α) AK α (NS )λ−1 (NJ )λ + (NS )λ
,
h
i 1−α
λ
λ
λ
∗
α−1
r = αAK.
(NJ ) + (NS )c) Markets clear:
Z
h
NJ =

Z
9≤τ ≤10,s′c (X,f,P )=J6≤τ ≤12,sp =J

NS =

h

i
ζc h(zc , s′c )f (X)dX P,

ζp h(zp , sp )f (X)dX +ZZζp h(zp , sp )f (X)dX +6≤τ ≤12,sp =Sζc h(zc , s′c )f (X)dXiP.

τ =10,sc =Sd) Perceived laws of motion for f and P coincide with those induced from household policy functions a′ , c, s′c .
The balanced growth path is a particular type of recursive competitive equilibrium defined below.
Definition: A balanced growth path is a recursive competitive equilibrium thatsatisfies the following properties:51′1) Aggregate population grows at a constant rate: PP = ν for some constant ν > 0.
2) The distribution of X is stationary: f ′ = f .
3) The household value and policy functions do not depend on P .
Along the balanced growth path, aggregate population grows but the distribution of households across individual states remains stationary. Further, the household value and policy functions are independent of aggregate population, and thus
household behavior remains the same over time conditional on the individual states.
Now we walk through the details of population growth within the model anddiscuss how model parameters translate to outcomes that are measured in data suchas the aggregate population growth rate and the number of children per household.
We start with the most general case that applies to any equilibrium whether it satisfies the properties of a balanced growth path or not. Later, we specialize to the caseof the balanced growth path to provide more explicit formulas. By definition, theaggregate population growth rate is given by the formulaAgg. Pop. Growth Rate =# births − # deaths
P(12)

Given the aggregate state variables of the economy, f and P , we have the following
accounting equations for births and deathsZ# births = νJZ# deaths =Z
f (X)dX + νS

sp =J,τ =5

f (X)dX P(13)

sp =S,τ =5

f (X)dX P

(14)

τ =14

In any given period, the aggregate population growth rate can be computed fromstate variables asZZZν − 1 = νJ
f (X)dX + νS
f (X)dX −f (X)dX(15)
sp =J,τ =5

sp =S,τ =5

τ =14Note that as written, ν − 1 > 0 is the aggregate population growth rate such thatP ′ = νP . To compare to data, it must be converted to an annual percentage growth
rate.
Recall that the aggregate population growth rate is constant along the balanced
growth path by definition. By leveraging this assumption we can calculate the aggregate population growth rate as a function of educational shares along the bal52

anced growth path analytically. This calculation provides insight into the changes
in population dynamics that can be expected due to changes in education. Suchchanges are important for our general equilibrium analysis.
With the aggregate population growth rate fixed at ν − 1, we know that the ratioof the population of households of age x and households of age y must be given by:R
f (X)dXRτ =x= ν y−x
f
(X)dX
τ =y

(16)

R
From that fact that τ ∈ {1, · · · , 14} and f (X)dX = 1 because f is a pdf, we canderive that along the balanced growth path with aggregate population growth rateν − 1 the following equations are true
ν−1
ν 14 − 1
Zτ =14(ν − 1)ν 9
f (X)dX = 14
ν −1
τ =5

Z

f (X)dX =

(17)
(18)

Finally, because household policy functions are invariant with respect to P and
f is stationary along the balanced growth path, we have that the share of the adult
population with a given level of education is the same for all ages. In particular,
this implies that the education shares of the parents giving birth this period can beˆ Ŝ.
replaced by the aggregate education shares J,
R
sp =J,τ ≥5

Jˆ ≡

R
τ ≥5

R

f (X)dX=f (X)dX

sp =J,τ =5

R
τ =5

R

f (X)dXs =S,τ ≥5Ŝ ≡ pR=f (X)dXτ ≥5f (X)dXR
sp =S,τ =5R
τ =5f (X)dX(19)

f (X)dXf (X)dX(20)

Combining equations (17) to (20) with equation (15) yields the following equation
which describes the aggregate population growth rate along the balanced growth
path as an implicit function of the education shares of the population:


 ν−1
ν − 1 = ν 9 νJ Jˆ + νS Ŝ − 1 14
ν −1

53

(21)

which can be reduced toν 5 = νJ Jˆ + νS Ŝ.

(22)

One wrinkle not yet addressed is the fact that, as written, the balanced growth
path of the model is not an attractor. That is, the model does not necessarily converge over time to the balanced growth path. To see why, consider a simplified
model with two generations, each of whom do nothing other than live through theirfirst period of life and, at the end of their second period of life, die and have ν children who become the new first generation. If the initial stocks of age 1 and age 2
agents are N1 and N2 , the aggregate population growth rate will oscillate between(ν−1)N1(ν−1)N2
and N
indefinitely, never converging to a single constant rate, as there
N1 +N2
1 +νN2
is no mechanism to close ”gaps” in size between the initial stocks.
To address the computational issues arising from this fact, we assume that a negligibly small fraction of children leave their parents and have their own children one
period earlier than the typical timing (that is, at age 20 rather than 25). This slightrandomization in timing effectively mixes away any differences in the initial stocksof agents for each generation, ensuring that the model converges to the balanced
growth path over time regardless of the initial state. In our computations, we assume the probability that any given child leaves early is 0.1 percent, small enoughto ensure that this outcome has minimal impact on parents’ decisions.

54

C. Intuition and Details on Model Identification
An important question is which of the targeted moments are most informative foreach of the estimated parameter values. To help answer this question, we follow Kaboski and Townsend (2011) and compute the percent change in each moment wheneach parameter is increased by one percent. We then re-estimate the model multiple times under alternative values of the targeted moments, to see how sensitive
each parameter is to each target, following Andrews et al. (2017). While in general
all moments jointly discipline all the parameters, some parameters correspond moreclosely to certain moments. The Jacobian matrix is presented in Appendix Table C.1,
and the sensitivity matrix is presented in Appendix Table C.2; here, we summarize
what we see as the main lessons from these matrices.
The population growth parameters νJ and νS are, perhaps unsurprisingly, significant determinants of the aggregate population growth rate and the treatment effects
on fertility. The variance and persistence parameters of the ability process, συ andρ, naturally increase the variance of the permanent component of income and theintergenerational schooling correlation, but also have sizable effects on many othermoments in equilibrium.
The effectiveness of schooling, ηS , and the intergenerational altruism parameter, b, govern the benefits of schooling and thus result in similar changes, notably asizable increase in aggregate secondary attendance. The key difference is that ηS increases the treatment effect on human capital while b has a minimal effect, as it only
impacts the parent’s valuation of better schooling. Intuitively, the cost of schooling,ΨS , decreases school attendance, increases the treatment effect on schooling, andconsequently increases (in absolute value) the treatment effect on fertility.
Finally, the savings wedge, χ, and the standard deviations of the test score noise
and taste shocks, σε and θ, all jointly impact secondary attendance in the top andbottom quartiles of the test score distribution as well as the difference in treatment
effect between the quartiles. In fact this was the purpose of introducing these shocks
into the model, and without them schooling completion and the treatment effecton schooling are always (counterfactually) much larger for those with higher testscores.

55

Table C.1: Elasticities of Moments to Parameters
νJ

νS

συ

ρ

ηS

b

ΨS

σε

θ

χ

Aggregate population growth 0.8

0.0

0.0

0.6

-0.2 -0.1 0.0

0.0

0.0

0.0

Aggregate SHS attendance-0.5 0.1 -0.1 -2.2

0.9

0.3 -0.1 -0.1 0.0

0.0

Intergenerational school corr.

0.0

0.0

0.5

1.7

-0.7 -0.2 0.2 -0.4 0.0

0.0

Var(permanent income)-0.4 0.1

0.8

-1.1

1.7

0.0

0.0

0.0

SHS in top quartile

0.0

0.2

-0.4

0.3

0.0

0.0 -0.2 0.0 -0.1

SHS in bot quartile

0.0 -0.2 -0.4 -1.0

0.6

0.3

0.0

0.2

0.0

0.0

TE on human capital

0.0

0.1

0.0

0.0

0.0

0.0

0.0

TE on fertility-0.5 0.3

TE on SHS completion-0.4 0.0 -0.8 -2.8

TE on SHS, Q4-Q1 difference

1.0 -0.3 -0.5 12.8 -8.8 2.7 -1.5 3.6 -1.6 1.1

0.0

0.0 -0.1 -0.2
0.8

3.0

0.0

-1.9 0.1 -0.4 0.5

0.0

0.0 -0.1

1.8 -0.2 0.4 -0.5 0.0

0.1

Note: This matrix represents the elasticities of each moment to each parameter. The entry in row
r and column c represents the percentage change in model moment r resulting from a one-percent
increase in model parameter c.

56

Table C.2: Sensitivity of Parameters to Moments
νJ

νS

συ

ρ

ηS

b

ΨS

σε

θ

χ

(i) TE on human capital x 1.5

3.6

2.1 -1.0 0.0

-0.7

4.2

-1.7

2.8

1.3

3.0

(ii) TE on fertility x 1.5

8.4-3.1 4.5 -0.4

6.9

-2.9

4.7

3.8

4.7 -2.2(iii) TE on SHS completion x 1.5

6.4

3.1 -3.0 -1.1

3.3

0.2

1.2

-0.7 1.4(iv) TE on SHS, Q4-Q1 difference x -1

2.3

1.8

-0.6 -1.9

7.1

2.2 -2.8 -7.6(i), (ii), (iii) and (iv) combined

10.3 -5.9 -2.8 1.5 17.9 -4.8 28.9 7.0 -2.8 -4.4

1.0

0.2

8.3

Note: This matrix displays the percent change in each parameter when the mode is re-estimatedto match different target moments. Row (i) is when we target a fifty percent higher target for thetreatment effect on human capital but keep all other targets the same. Row (ii) is when we target
a fifty percent higher treatment effect on fertility. Row (iii) is when we target a treatment effect on
secondary school completion that is fifty percent higher. Row (iv) is when we target a difference intreatment effects on schooling between Q4 and Q1 that is -4 instead of 4. The bottom row is when
we re-estimate the model to match all of the higher targets..

C.1. Robustness
In a similar vein as the above exercises, here we confirm that the fact the estimated model undershoots the intergenerational schooling correlation and the treatment effect on schooling attendance does not substantially alter the model’s conclusions.
First, we increase the cost of schooling parameter ΨS by 40 percent which allows the model to match the treatment effect on schooling, albeit at the expense of aslightly worse fit to other model moments. Under this alternative parameterization,
the free schooling policy results in a 0.8 percent decline in GDP (vs 1.0 percent in
the baseline) and a 14 percent increase in school attendance (vs 12 percent baseline).
Neither of these results are substantially different from the baseline.
We also perform a similar exercise by increasing the parameter governing the persistence of the ability ρ process by 15 percent to match the intergenerational schooling correlation coefficient (again with a slightly worse fit for other moments). In thiscase, the free schooling policy results in a 2.0 percent decline in GDP and only a 4
percent increase in attendance. The intutition for this result is that a higher persistence parameter implies less misallocation as fewer high ability children are born topoor families. Thus missing on this moment means, if anything, that our conclusions are too generous to the free schooling policy.

57