Online Appendix: A Model in which the Religious Composition of Schools, Rather than of the Population, Affects Religious Identity

Transcription

1 Online Appendix: A Model in which the Religious Composition of Schools, Rather than of the Population, Affects Religious Identity Consider a model in which household preferences are identical to those given in Section 2, but in which the probability that a child from denomination ' belongs to denomination as an adult is a function of the share of students in the child s school who belong to group, s, rather than the corresponding share in the local population, r. In this case, the probability of belonging to each denomination as an adult is given by the following transition matrix (B1) T ', ( s ω + (1 ω) s ) = (1 ω) s,, = ' '. If a child attends religious school of type s =, in which all students belong to group, s equals one (s equals zero for all religious schools of type s ). If a child instead attends a public school, s equals the share of students in that school who belong to group, g. Formally, the effect of the school environment on the probability of belonging to each group equals 1 if s = (B2) s = g if s = g. otherwise n Because s = 1, the sum of the group-membership probabilities across all possible groups n = is T ( s, g ) = 1. = ', From (B1) and (B2), the probability z that a child in group {,, n} who attends a public school will remain in that group equals ω + ( 1 ω) g. If the child instead attends a private school of the household s denomination s =, then z = 1. 1

2 As expression (B2) shows, the share of students in the public schools who belong to group, g, is positively related to the probability that a publicly-educated child from that group remains in the group as an adult. Thus, parents have a weaker motivation to incur the added expense of sending their children to private religious schools when the fraction of students that belong to their group in the public schools is higher. Unlike expression (12) in the main text, the relevant threshold income level in this model is now a function of g rather than of r : (1 α) x (B3) y ( x, g,ω) = δ + γ δ [( ω + (1 ω) g ) 1] d α Thus, the share of households from group whose children attend public schooling is F(y ( x,g,ω), and the share of households from group whose children attend group private schools is 1 δ. (B4) θ = 1 F(y ( x,g,ω). Lemma 1. The share of households from group who send their children to private schools is a decreasing function of the anticipated fraction of children in the public schools who belong to group. Proof. Partial differentiation of (B4) with respect to g yields θ y γ 1 δ δ 1 (B5) = f ( y ) = f ( y ) y [ ω + ( 1 ω) g ] ( 1 ω). g g d α Note that in the case that γ =, in which the only motive of parents to send their children to private schooling is to increase the quality of their education, the threshold income y 2

3 simplifies to y (1 α) x = d α 1 δ δ. In this case, α α x d = (1 ), which is positive y δ because x and y are positive and because < 1 δ 1 <α. In addition, [ + (1 ) g ] (1 ) ω ω ω is always positive because <ω < 1 and g < 1. Because γ and y are also positive, θ ( g g ) <. < Q.E.D. In equilibrium, the share of each group that attends religious schooling, θ, and the share of that religion in the public schools, g, are determined simultaneously: (B6) θ = 1 F(y ( x,g, γ, ω)). (B7) g ( r, r,..., r ) 2 n = n [ y ( x, γ, g )] { rk F[ yk ( x, γ k, gk )]} k= r F Unlike in the simple model presented in the main text, in which the share of each group that goes to religious schooling, θ, is a function of the market share of only group (and does not depend on how the shares of all other religions are distributed), in the current model it is a function of the entire distribution of the market shares of all religions in the population. B2. Calibration The calibration of this model is very similar to that of the simpler model given in the main text (as described in Appendix A). All of the parameters of the model except for γ are determined similarly, except here we use national-level data rather than county-level data. 1 We must also calibrate the share of each group in the public schools, g. We derive this 1 We assume that the choice between public schooling and non-sectarian private schooling does not incorporate a religious motive. Therefore, we assume that γ =. This assumption implies that the calibrated value of δ (which sets the non-sectarian enrollment rate equal to 1.56%) is identical to that found in the simpler model. 3

4 vector from the actual enrollment rates in each type of private schooling in the U.S. in 2. For example, the Catholic enrollment rate, r cath θ cath, equals 4.81%, and the Catholic market share, r cath, equals 22.4%. These numbers imply that the share of Catholics whose children attend Catholic schooling, θ cath, is %, so that the share of Catholics whose children attend public schooling is %. It then follows that the share of all students who are both Catholic and attend public schools, r cath F[y cath ], equals 17.23%. As public school pupils comprise 9.1% of all students, the share of Catholic students in the public schools is 19.12% (= / 9.1). Similarly, we derive the share of each of the other religions in the public schools (see Table OA1). Thus, we solve four equations in four unknowns: δ, γ cath,γ mainline and γ evangelical : 1. r θ ( x = 394.5, g =.5351, γ =, δ, ω=.5) = r cath θ cath ( x = 394.5, g cath =.1912, γ cath, ω=.5) = r evangelical θ evangelical ( x = 394.5, g evangelical =.1218, γ evangelical, ω=.5) = r mainline θ mainline ( x = 394.5, g mainline =.119, γ mainline, ω=.5) =.47 Solving these equations yields δ =.269, γ cath =.39, γ evangelical =.31 and γ mainline =.6, which are very similar to the corresponding values in the model in the text. The reason for this similarity is that, as a comparison of columns (2) and (7) in Table OA1 indicates, g is very similar to r in the U.S. as a whole. B3. Simulations We turn next to simulating the relationship between θ and r. As mentioned above, θ is a function of the market shares of all religions. In the simulations, we will assume that for each value of r cath, the remaining religions are distributed as in the U.S. in the year 2. That is, as the Market share of Evangelicals, Mainlines, non-adherents together is 73.61% (= 4

5 14.19% % %) and the Market share of Evangelicals is 14.19%, the proportion of Evangelicals in the non-catholic population is 19.28% (= 14.19% / 73.61%). Similarly, the shares of Mainlines and non-adherents in the non-catholic population are 13.11% and 67.61%, respectively. Therefore, in the simulations, when r cath increases from zero to one we rescale the market shares of all other groups, e.g., we rescale the Evangelical enrollment rate to be r ( 1 r ) evangelical = cath. Then, for each value of r cath we solve six equations in six unknowns (θ cath, θ mainline, θ evangelical, g cath, g mainline,, g evangelical ), corresponding to expressions (B6) and (B7) above. Table OA2 presents the results of the simulations of the model described in this Appendix (labeled Endogenous model in the table). Columns (2) and (3) show how g cath, θ cath, and (r cath θ cath ) varies by the Catholic share in the population, r cath. Note that g cath and r cath are quite similar across all values of r cath, which is compelling evidence that there is little practical difference in modeling enrollment decisions as a function of within-public school shares g rather than a function of population shares r. Column (3) shows that the share of Catholics in Catholic schools is decreasing function the Catholic share, and column (4) shows that the overall Catholic enrollment rate is an inverse u-shaped function of the Catholic share. Columns (5) and (6) show the analogous values of r cath θ cath and r cath implied by the model given in Section 2 (labeled the exogenous model in the table, as Catholic population shares are exogenous determinants of Catholic school enrollment rates). The similarity of these two columns to columns (3) and (4) is striking; in all cases, the parameters implied by the two models differ by less than 1 percentage point. Again, this similarity implies that there is little practical difference in modeling enrollment decisions as a function of within-public school shares g rather than a function of population shares r. In equilibrium, the relationships between population shares and both within-denomination 5

6 enrollment rates and overall religious school enrollment rates are quite similar across the two models. 6

7 Religion Table OA1. The Share of Each Religion in Public Schools Relative to its Share in the Population Market share Enrollment rate in school type Share of group in school type Share of group in public schools Share of all students who belong to group and attend a public school Share of group in public schooling (1) (2) (3) (4) (5) (6) (7) r r θ θ = 1 F(y ) 1 θ = F(y ) r (1 θ ) r (1 θ ) /.91 Catholic Evangelical Mainline other religions Non- Adherents Total Note: All numbers listed in the table are percentages. 7

8 Table OA2. Comparison of the Exogenous and Endogenous models Endogenous model Exogenous model (1) (2) (3) (4) (5) (6) Share of Share of Catholics in Catholics in Catholic Enrollment Share of Catholics in public Catholic rate Catholic schools schooling schools Catholic share in the population Catholic Enrollment rate r g θ = 1 F(y ) r θ θ = 1 F(y ) r θ Note: All numbers listed in the table are percentages. 8