*In the following section we would first introduce the model of de Bartolome *
(1990), and then start with the revisions we made to show our work and results. The
model used by this thesis is mainly extended from the settings and assumptions made
*by de Bartolome (1990). Opposed to Tiebout (1956), which assumed that the *
production of public service is determined solely by physical inputs and community
*size, de Bartolome (1990) proposed that community composition, peer effect in the *
context of education, is also a major determinants of the quality of public service, and
thus developed a two-community model to address his viewpoint. In the following,
*we will provide a brief introduction to the model of de Bartolome (1990). Throughout *
the introduction we will also notify the differences between the model of this thesis,
which included new assumptions and revisions to the assumptions of the original
*paper, and the model of de Bartolome (1990). *

*The model of de Bartolome (1990) assumes that there are two communities, one *
is the suburban area and the other is the urban area. Each community can
accommodate a fixed number of families and each family has one child and is
distinguished by the ability of their child. The types of families can be divided to

“high-ability” and “low-ability” families. The total numbers of families in each community are fixed, and so are the numbers of families with child of different ability.

In addition, the total number of families of the two communities is equal to the total
number of families of different type. That is to say, regardless of the ability of the
child in each family, they all have a place to live. Either they live in the city or live in
the suburbs. The families attend public school pertaining to the community they live,
which means if they live in the suburbs, they attend public school there, and if they
*live in the urban area, they attend public school located in the city. Furthermore, de *

*Bartolome (1990) assumed that the production of human capital exhibits constant *
returns to scale in community size, implying that the efficiency of school does not
depend on the size of the community so that one can simply consider one public
school in each community. Families aim to maximize their household utility, which is
derived from consumption and educational output. The educational output has three
major components. The first component is the educational input of the school that
children attend in each community, which is financed by the tuition they pay to public
schools. The second component is the average quality of their schoolmates. In other
words, they care about how many “good” students are attending their school, which is
the so called peer effect. The last component of the educational outcome is the innate
ability of the child of each family. The families are able to migrate between the
suburbs and the city in order to fit their preference. As a result, they have to choose
where to reside through comparing the levels of input, which are decided by majority
voting by each community, peer effect, and also the housing rent in each of the two
communities. The community composition and rent premium in equilibrium is what
*de Bartolome (1990) was most interested in. *

In our model, we will introduce private school into the model, and discuss the results. We will also include the implementation of voucher programs to see how the program affects the allocation of peer effects as well as to find out its effect of promoting private school. Since the issue mentioned above will not be very interesting if there is no difference in income, in our model we need to allow the families to have different income endowments. That is to say, there are four types of families rather than two, which differ in the ability of their child and the income they are endowed with. The objectives of the family is still the same, except that aside from choosing where to live, they also need to decide whether they are to attend private school or not, given the terms private school is willing to offer. The private

school in our model attempts to maximize aggregate education output it created.

Intuitively, the students that attend private school have to be at least as well of as they were when attending public school. Therefore, the restriction for the private school is the fact that students have to gain no less utility from private school than what they originally get when attending public schools.

*In the model of de Bartolome (1990), there are N*1 families each with a child of
*low ability, denoted a*_{1}*, and N*_{2} families each with a child of high ability, denoted
*a*2*.The education achievement e is a function of input level I, the proportion of *
*families with more-able child θ, and the innate ability of their child a, which we *
*denote e(I,θ,a). It is assumed that more able children gain more marginal utility from *
educational input than less able children, or

1 2

On the other hand, less able children are assumed to gain no less from a peer group improvement, or

The utility of each family depends on consumption and educational achievement. For families with less able child, the utility function is

( , ( , , ))1 ( ) ( ) ( ).

*U C e I* θ *a* =*F C* +*G I* +*H* θ

And for families with more able child, the utility function is

( , ( , , 2)) ( ) ( ) ( ).

And the assumptions made on peer effect imply
communities, he tried to find out what the proportion of families with more able child
in each community will be in equilibrium. The proportion of families with more able
*child in each community is denoted θ**u **and θ**s**, where u and s stands for urban and *
suburban area, respectively. The total number of residents in each community is thus
*denoted n**u** and n**s*. He assumed that the educational inputs are financed by local taxes,
and the production of educational output is assumed to show constant return to
community size and are chosen so that one unit cost equals to one unit consumption.

*Each urban family therefore pays I*_{u}* and obtains I** _{u}* from school, and each suburban

*family pays I*s

*and obtains I*

*s*from school. The housing rent for residents in the urban

*area is r*

_{u}*, and is r*

*for the suburban area. The government collects the rent and gives*

_{s}*it back to the residents as a kind of lump-sum transfer, T. According to the*

*By substitution, the net suburban rent r and net urban rent becomes *

( )

*We use Figure 1 to illustrate the community model of de Bartolome (1990) *

Figure 1: Two Communities

In equilibrium, the role of the net rent can be regarded as the willingness to pay
*to move from the low θ community to the high θ community. Notice that the net urban *
rent is proportional to the net suburban rent, where the proportion is decided by the
number of residents in each community. This shows that the number of residents in
each community affects the rent premium of both the urban area and suburban area.

Given the net rent for both communities, the consumptions of each urban family and suburban family equals to their income less education input and net rent

( ) , community is decided by majority voting. However, in the original paper the author dealt with the problem that he is interested in. He assumed that the majorities in the

Suburban area Urban area

urban area are the families with child of low ability, while the majorities in the suburban area are those with more able children. In other words,

{ } migration condition” is fulfilled. The no migration condition requires that there is no way for the family to gain more utility by means of migrating from the current residence to another community. In other words, it means that the residents are indifferent of living in the suburban area and living in the urban area.

For families with less able children, the condition is

2

As one can see, on the left hand side of the equation is the utility they gain as if
they live in the suburban area, while on the right hand side of the equation is the
utility they gain when they live in the urban area. Given the proportion of families
with more able child in the suburban area, we can derive the proportion of families
with more able child in the urban area shown on the right hand side of the equation in
*function H. *

And for families with more able children, the condition is

2

Similarly, the left hand side of the equation is the utility they gain as if they live
*in the urban area, and on the right hand side is that of the suburban area. In de *

*Barolome (1990), the equilibrium values * ^{( ,}*I I**u*^{*} *s*^{*}^{,}θ*s*^{*}^{,}*r*^{*}^{)} are derived from the four
equations mentioned above. The former two are derived through majority voting, and
the latter two are derived from the no migration condition. Figure 2 illustrates this
equilibrium, which we will explain below.

Figure 2: Equilibrium ensuring no migration condition

From the graph we can see that the on the vertical axis is the rent premium, and
the horizontal axis is the allocation of peer effects. In equilibrium we can plot two
curves AA′ and BB′, where each point on AA′ stands for the rent premium that low
*ability students are willing to pay for a given level of θ** _{s}*. Similarly, each point on BB′

stands for the rent premium that high ability students are willing to sell for a given
*level of θ** _{s}*. The reason that the slope of AA′ is larger is because low ability students
gain more marginal utility from peer effects compared to what high ability students
get. From the two curves we can find the solution of the equilibrium. In de Bartolome
(1990) five kinds solutions are able to be derived. The first kind of solution is when

*r *

*r* *

θ*s*

*

θ*s*

*AA*′′′′

*BB′ *

the majority of the urban area is less able students and the majority of suburban area is more able students. The second kind of solution is when the majority of the urban area is less able students, and in the suburban area there are only families with more able child. The third solution is when there is only less able students in the urban area and the majority of the suburban are is families with more able child. The fourth solution is when there are only less able students in the urban area and only more able students in the suburban area. The last solution is that the two communities have same composition. However, only the solution that implies that the majority in the suburban area is families with more able child and the majority in the urban area is families with less able child is discussed. This is because that it is the solution that is more interesting and plausible solution that would be correspondent to reality. The loss of leaving other solutions out of the discussion is that we are being too specific, and might not be able to cover all the possible circumstances.

*Now that we have made a brief introduction of the model of de Bartolome (1990), *
we shall start introducing the revisions we have made based on the original model. In
*our model there are still N*1* families with children of ability a*1*, and N*2 families with
*children of ability a*_{2}. However, instead of two types of families we assumed that there
are four types in total, “low-ability low-income”, “low-ability high-income”,

*“high-ability low-income”, and “high-ability high-income”. Therefore, we denote N*_{11u}*, *
*N**12u** and N**21u** the total number of each type of families in the urban area, where the *
former number stands for ability and the latter stands for endowed income. “1” is
*assumed to be low, while “2” is assumed to be high. For example, N**11u* is the notation
for the number of “low-ability low-income” families who live in the urban area.

Similarly, the total numbers of each type of families in the suburban area are denoted
*N*_{22s}*, N*_{12s}* and N*_{21s}*, respectively. Endowed income is divided to y*_{1}* and y*_{2}, where the
former stands for lower endowed income, and the latter stands for higher endowed

income. The utility functions of the family with child of low ability is still ( , ( , , ))1 ( ) ( ) ( ).

*U C e I* θ *a* =*F C* +*G I* +*H* θ

And for the families with more able children ( , ( , , 2)) ( ) ( ) ( ).

*U C e I*θ *a* =*F C* +*R I* +*S* θ

The consumption level for the families who live in the urban area with high income endowment, in the urban area with low income endowment, in the suburbs with low income endowment, and in the suburbs with high income endowment are possibilities for who actually are the majority and makes decision through voting. In this thesis we assume that the input level of the urban community is decided by Below we show a graph that illustrate our model.

Figure 3: Two Communities with Private School

Now that we have the consumption level and the input level, we shall introduce
the last condition required to derive the community composition and rent premium in
*equilibrium. Following de Bartolome (1990), the equilibrium occurs when the “no *
migration condition” is fulfilled. However, the set of equilibrium that can make all
four types of families to mix over the two communities is impossible to exist, since
two out of four curves that show the indifference of migration are in parallel. Under
this circumstance, four possible combinations outcomes can be derived, which we
will show in the graph below.

Figure 4: Equilibrium with Different Ability and Income

**As we can see in the graph there are two AA′ curves and two BB′ curves. The H ***and L stands for high income and low income. Therefore the graph shows that *
different combinations of rent premium and composition of suburban area varies
according to the majority of each community. The four conditions that have to be
satisfied in order to solve for equilibrium are as follows.

For “low-ability high-income” families, the condition is

2

For “high-ability low-income” families, the condition is

1

condition, we can solve for( ,*I I*_{u}^{*} _{s}^{*},θ_{s}^{*},*r*^{*}). If these values also satisfy condition (5)
and (6), then we obtain our solution we are looking for.

For “low-ability low-income” families, the condition is

* * *

For “high-ability high-income” families, the condition is

*

*Additional to the assumption of de Bartolome (1990) that both communities have *
one public school each, we introduce private school into the model. The private
school itself recruits given number of students, where the source of students is from
the two communities. Private school itself can charge different tuition depending on
the ability and the income level of the student that attends, but each student receives
*identical level of input. In other words, the level of I** _{p}* they get each is the same
regardless of their differentiated ability, endowed income, and tuition paid. The
objective function of the private school is to maximize the educational output it

*creates by choosing the level of I and θ it offers to students, subject to its budget*constraint, which balances the total amount of tuition it charges and the total spending on the educational input it provided. This assumption implies that as long as there are students that are willing to attend private school, private school would raise the level

*of I it charges to increase the educational output it creates, thus in equilibrium, θ and r*should not only be able to make residents feel indifferent about migrating among the two communities, but also have to make them feel indifferent about attending public school and private school.

We would then like to see how the equilibrium will be when we include private school in the original community model. The optimization problem of private school is shown as follows.

The reason that there are only two constraints other than the budget constraint is that
the most appealing feature of private school is the high peer effect it offers. We think
this attracts students from urban area most because the majority in the urban area is
families with less able child, and that is the source of private school. In equation (7),
*N* is the total number of students that attend private school and *θ** _{p}* is the proportion

*of students attending private school. Moreover, in the constraint, I*1

*and I*2 stands for

*the tuition charged to students of different ability, and I*

*is the input expenditure that private school has chosen. The new no migration condition is as follows, since some students were taken away from public school.*

_{p}2

2 then we obtain our solution we are looking for.

* * * *
each family by giving a discount on the tuition they have to pay. Practically a certain
proportion to the total amount of public school tuition is given back to the families
who choose to attend private school. Suppose the original tuition they have to pay is
*I*_{u}* , then under a 50% voucher the tuition will become 0.5×I** _{u}* . Again we can derive the
optimal level of input from equations (1) and (2), and from equations (7), (8) and (9)
we solve for(θˆ

_{s}^{*},

*I I r I*1

^{*},

^{*}2, ,ˆ

^{*}

^{*}

*,θ*

_{p}^{*}

*) (θˆ*

_{p}

_{s}^{*},

*I I r I*1

^{*}, 2

^{*}, ,ˆ

^{*}

^{*}

*,θ*

_{p}^{*}

*) , if the values also satisfy the conditions (10) and (11), then we obtain our solution we are looking for. The difference is that the second constraint in equation (7) is different when we consider the effect of vouchers. The constraints become*

_{p}

We assumed in our model that the majority of the suburban area is high-ability high-income, and the majority of the urban area is low-ability low-income. However, there are also other possibilities as we have shown in figure 4. Suppose the majority changes, a new input level for both communities are derived, which further affects the level of consumption. The most significant influence is that the no migration condition is affected which implies that the equilibrium might not exist, or the conditions for equilibrium would be different than what we derived with our assumptions.

The general-form functions can only carry us this far. To further investigate the features of the equilibrium, we will apply the numerical method to explore the model in the following chapter.