Aggregate and Distributional Effects Appendix Figures and Tables

Appendix (for Online Publication) A. Appendix Figures and Tables Figure A.1: SHS Completion by Quartile of Test Score: Data vs Model 1 .9 .8 Control (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 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 100 60 80 60 40 40 20 20 0 0 0 20 40 60 80 100 100 Unconstrained Cutoff 80 100 Parent Ability (Percentile) Unconstrained Cutoff Parent Ability (Percentile) 100 80 60 80 60 40 40 20 20 0 0 0 20 40 60 80 Child Ability (Percentile) Child Ability (Percentile) (a) Attendance Probability (b) Increase in Attendance 100 Figure A.3: Child SHS Attendance Probability (Counterfactual Parameters) Table A.1: Free Secondary Schooling Policies in Developing Countries Country Year Requirement Benin 2007 Pass Brevet d’Etudes du Premier Cycle Gambia 2015 Pass Basic Education Certificate Exam Ghana 2017 Pass Basic Education Certificate Exam Kenya 2008 Pass Certificate of Primary Education Exam Malawi 2019 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 Rwanda 2012 Score ≥ ‘High’ on O-level Test Sierra Leone 2018 Score ≥ 6 on Basic Education Certificate Exam Tanzania 2015 Pass Standard 7 Exam Uganda 2007 Score ≥ 28 in Primary School Leaving Exam Zambia 2022 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. 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 Factor b 2.5 Std. Dev. of Exam Score Noise σε 1 Gumbel Scale Parameter of Taste Shock θ 0.5 Persistence of Ability Process ρ 0.8 Std. Dev. of Ability Process σ 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 χ 0.1 0.2 Savings Wedge 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 2 for both models. Table A.3: Labor Income Tax Schedule in Ghana Income Tax 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. Table A.4: Elasticities of Moments to Parameters νJ νS σ ρ Aggregate population growth 0.8 0.0 0.0 0.6 ηS b ΨS σε σζ χ -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. Table A.5: Sensitivity of Parameters to Moments νJ νS (i) TE on human capital x 1.5 3.6 (ii) TE on fertility x 1.5 σ ηS b ΨS σε σζ χ 2.1 -1.0 0.0 -0.7 4.2 -1.7 2.8 1.3 3.0 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 ρ 0.2 8.3 Note: This matrix displays the percent change in each parameter when the mode is re-estimated to match different target moments. Row (i) is when we target a fifty percent higher target for the treatment 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 (v) is when we target a difference in treatment 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.. Table A.6: Support for Free Secondary School Percent of respondents Difference P-value (JHS - SHS) Obs. All JHS SHS Urban respondents 1,829 69.9 74.0 65.2 8.8*** 0.00 Rural respondents 1,086 70.1 72.0 63.5 8.4** 0.01 Male respondents 1,427 67.6 70.2 64.5 5.7** 0.02 Female respondents 1,488 72.2 75.2 65.2 9.9*** 0.00 Respondents from the Volta region 265 42.3 45.0 37.2 7.8 0.22 Respondents from the Ashanti region 854 78.2 81.7 69.8 11.9*** 0.00 Sample restricted to only ... Notes: This table reports the share of respondents who support the free secondary school program. More specifically, respondents were asked whether the “expenditures on the free secondary school program should either be abolished, cut substantially, cut somewhat, maintained, expanded, or delayed/postponed.” The table shows the percent of respondents who agreed that the expenditures on free secondary school should either be maintained or expanded. To check for the robustness of the responses, we limited the sample to only respondents in urban and rural localities, male and female heads of households, and respondents from the Volta and Ashanti regions of Ghana. The JHS column shows the percent of respondents who supports the free secondary school program and had completed at most basic education (Junior High School or below). On the other hand, the SHS column reports the percent of respondents who support the free secondary and had completed at least secondary education (Senior High School or above). ∗∗∗ , ∗∗ , and ∗ represent significance at the 1%, 5%, and 10% levels respectively. B. Model Appendix In this appendix we define the concepts of recursive competitive equilibrium and balanced growth path for our model. Letting X denote the vector of individual state variables (τ, a, zp , sp , ζp , δJ , δS , zc , sc , z̃c , ζc ), a recursive competitive equilibrium is 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 a0 (X, f, P ), c(X, f, P ), s0c (X, f, P ) 3. Perceived laws of motion f 0 = F (f, P ), P 0 = H(f, P ) such that a) V, a0 , c, s0c solve the household’s optimization problem given wS , wU , F, G. 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 = 6≤τ ≤12,sp =J NS = h Z 9≤τ ≤10,s0c (X,f,P )=J Z 6≤τ ≤12,sp =S i ζc h(zc , s0c )f (X)dX P, ζp h(zp , sp )f (X)dX + Z ζp h(zp , sp )f (X)dX + ζc h(zc , s0c )f (X)dX i P. τ =10,sc =S d) Perceived laws of motion for f and P coincide with those induced from household policy functions a0 , c, s0c . The balanced growth path is a particular type of recursive competitive equilibrium defined below. Definition: A balanced growth path is a recursive competitive equilibrium that satisfies the following properties: 1) Aggregate population grows at a constant rate: P0 P = ν for some constant ν > 0. 2) The distribution of X is stationary: f 0 = 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 and discuss how model parameters translate to outcomes that are measured in data such as 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 case of the balanced growth path to provide more explicit formulas. By definition, the aggregate population growth rate is given by the formula Agg. Pop. Growth Rate = # births − # deaths P (11) Given the aggregate state variables of the economy f, P , we have the following accounting equations for births and deaths Z  # births = νJ Z  # deaths = Z f (X)dX + νS sp =J,τ =5  f (X)dX P (12) sp =S,τ =5  f (X)dX P (13) τ =14 In any given period, the aggregate population growth rate can be computed from state variables as Z Z Z ν − 1 = νJ f (X)dX + νS f (X)dX − f (X)dX (14) sp =J,τ =5 sp =S,τ =5 τ =14 Note that as written, ν > 1 is the aggregate population growth rate such that P 0 = νP . To compare to data, it must be converted to an annual percentage growth rate. Recall that the aggregate population growth rate ν 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 balanced growth path analytically. This calculation provides insight into the changes in population dynamics that can be expected due to changes in education. Such changes are important for our general equilibrium analysis. With the aggregate population growth rate fixed at ν, we know that the ratio of the population of households of age x and households of age y must be given by: R f (X)dX Rτ =x = ν y−x f (X)dX τ =y (15) R From that fact that τ ∈ {1, · · · , 14} and f (X)dX = 1 because f is a pdf, we can derive that along the balanced growth path with aggregate population growth rate ν 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 = (16) (17) 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 be ˆ Ŝ. replaced by the aggregate education shares J, R R f (X)dX f (X)dX s =J,τ ≥5 s =J,τ =5 Jˆ ≡ pR = pR f (X)dX f (X)dX R τ ≥5 R τ =5 f (X)dX f (X)dX s =S,τ ≥5 s =S,τ =5 Ŝ ≡ pR = pR f (X)dX f (X)dX τ ≥5 τ =5 (18) (19) Combining equations (16) to (19) with equation (14) 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 (20) which can be reduced to ν 5 = νJ Jˆ + νS Ŝ. (21) 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 BGP. To see why, consider a simplified model with two generations, each of whom do nothing other than live through their first 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 , (ν−1)N1 2 +1 and N +1 the aggregate population growth rate will oscillate between (ν−1)N N1 +N2 1 +νN2 indefinitely, never converging to a single constant rate, as there 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 neglibly 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 slight randomization in timing effectively mixes away any differences in the initial stocks of agents for each generation, ensuring that the model converges to the BGP 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 enough to ensure that this outcome has minimal impact on parents’ decisions. C. Household Survey Appendix We surveyed a nationally representative sample of Ghanaian households to explore residents’ support for the free secondary school program. To do so, we first split the country into three zones: northern, middle, and coastal. Then we chose two regions from the northern zone (Northern and Savannah), two from the middle zone (Ashanti and Bono), and three from the coastal zone (Greater Accra, Volta, and the Western regions). According to the 2021 Population and Housing Census (PHC), the selected seven areas account for almost 61 percent of the entire population. Figure C.1 depicts a map of the studied regions. The dark grey areas are the regions we surveyed, whereas the light grey parts are the regions we did not survey. In addition, the values in parenthesis in the dark grey zones – the regions we surveyed – show the number of households sampled in each region. Second, we obtained an exhaustive list of enumeration areas (EA) across the seven regions from the 2021 Population and Housing Census conducted by the Ghana Statistical Services. The list of EAs included details such as location, type of residence (urban or rural), and projected size (number of households). We divided the EAs into 14 strata based on geography (7) and kind of habitation (2). Following that, we selected a nationally representative sample of 3,500 homes from the list of EAs using a two-stage cluster sampling approach. In the first stage, we selected 151 EAs at random and independently using a probability proportional to size to allocate the total number of EAs per stratum. In the second stage, an average of 23 homes were chosen at random from each EA sampled in the first stage. Following that, twenty-four (24) enumerators were hired, trained, and tasked with interviewing an adult individual from each of the 3,500 chosen houses, preferably the head of the household. Figure C.1: Sampled Households in each Region Surveyed Upper East North East Upper West Northern (350) Savannah (210) Bono East Oti Bono (210) Ashanti Ahafo (1000) Eastern Western Volta (303) North Western Central Greater Accra (1030) (397) Not surveyed Surveyed (Sampled households) Notes: This figure plots the regions that were surveyed and the total number of households surveyed in each region. The numbers in the parenthesis represent the number of households sampled in each region.