# 社區模型存在私立學校及教育券下的同儕效果分析

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(2) 社區模型存在 社區模型存在私立學校及教育券下的同儕效果分析 指導教授：宋皇叡 博士 國立高雄大學應用經濟學系 學生：周孚陽 國立高雄大學應用經濟學系碩士班 摘要. 本文主要探討的是在一個存在公立學校、私立學校及教育券的發放的社區 模型之下同儕效果的分配。本文主要是參考 de Bartolome (1990) 的社區模型，然 而其文中假設所有家庭的所得皆相同，也沒有將私立學校納入考量，這就目前我 們要探討的問題而言，這些假設還有延伸的空間。因此本文除了在其社區模型中 將所得差異的要素納入考慮之外，還在模型中加入了私立學校及教育券的發放。 我們首先概略地介紹了 de Bartolome (1990)的模型，探討其均衡存在的條件 及假設。接著我們將所得差異加入模型中探討其均衡存在的條件之變化，再來考 慮私立學校存在的空間，以及其存在對本來的社區之同儕效果分配的影響。除此 之外我們還探討了私立學校的學生組成如何影響到教育產出。最後文中還探討了 教育券的發放是如何影響到私立學校的存在空間以及教育券對教育產出的影 響。我們利用數值分析得到的模型的推論是要讓私立學校存在必須滿足以下幾個 條件：高能力學生的教育投入不能佔所得太高的比例、同儕效果的邊際效用不能 太小以及相對於低能力的學生高能力學生從同儕效果得到的好處不能小太多。除 此之外，不論有沒有發放教育券，私立學校最佳組成就是全部皆為高能力的學 生。最後，教育券在某些條件下是有可能促進私立學校的存在的。 關鍵字：同儕效果 同儕效果、社區模型、 社區模型、教育券 關鍵字 同儕效果、. I.

(3) Effect of Private School and Vouchers on the Equilibrium of a Community Model with Peer Effect Advisor: Hwang-Ruey Song Department of Applied Economics National University of Kaohsiung Student: Fu-Yang Chou Master program, Department of Applied Economics National University of Kaohsiung ABSTRACT In this thesis we will discuss the allocation of peer effect with the existence of public school, private school, and the issuing of school vouchers. This thesis mainly follows the community model of de Bartolome (1990). However, the model assumed that family income are identical, and also assumed that there are no private school in the model. However, when looking at the problem we would like to discuss, these assumptions should be further improved. Therefore, we not only include the possibility of different income endowment, but also add private school to the model. We also discussed the distribution of school vouchers. Through the use of a numerical method, we first give a brief introduction to the model of de Bartolome (1990), discussing the conditions for the equilibrium to exist. We then discuss how different income endowment affects the conditions for equilibrium to exist. Next we discuss the conditions for private school to exist, and how the composition of students affects the educational output it creates. Finally we discuss how school vouchers affect the existence of private school and educational output. The conclusion is that for private school to exist, the following conditions have to be satisfied. The proportion of income spent on educational input for high ability students cannot be too high, marginal utility of peer effect cannot be too small, and the marginal benefit students of high-ability gain from peer effect should not be too small relative to the gain from peer effect for students of low-ability. Moreover, no matter if there is school voucher or not, the best composition for private school is to recruit students of high-ability only. Finally, it is possible for school vouchers to promote the existence of private school. Keywords: Peer Effect, Community Model, School Voucher. II.

(4) Table of Contents 1. Introduction.............................................................................................1 1.1 1.2 1.3. Motivation and Objectives.........................................................................1 Contributions..............................................................................................4 Outline........................................................................................................4. 2. Literature Review ...................................................................................5 3. Model ......................................................................................................9 4. Numerical Analysis...............................................................................24 5. Conclusion ............................................................................................37 Appendix...................................................................................................38 References.................................................................................................41. III.

(5) List of Figures Figure 1: Two Communities…………………………………………….13 Figure 2: Equilibrium ensuring no migration condition………………...15 Figure 3: Two Communities with Private School……………………….18 Figure 4: Equilibrium with Different Ability and Income………………19. IV.

(6) 1. Introduction. 1.1 Motivation and Objectives. Schools in Taiwan often screen students by ability. This might be unfair and causes potential pressure on students. The pressure is derived from the competition among themselves to get in better classes or better schools. After all, one takes the behavior of one's company, so parents try to cram kids in the best class, and students receive a lot of pressure from both the teacher and their parents. In some sense, parents are trying to purchase “peer effects”. One might see better students separated from those who perform worse relatively to those who are known as elite, which is pretty common here in Taiwan. However, this causes both a gain and loss in utility. The gain comes from the elite students taking advantage of each other, while the loss part comes from the deprivation of good students who stimulates the relatively poor performing students. Furthermore, not only does the educational input such as facilities and faculties matters, but also the proportion of students of different ability alters. What motivates me most is the problem that why students spend more money on attending cram schools so often when the teachers in cram school actually teach the same thing as teachers in public school does. Therefore, I would like to find out the answer to this question in this thesis. Education has become a very important issue in the last few decades. Economists have attempted to analyze from many aspects of the issue and the most common way is to view education as one kind of industry or business, where the school is the firm, as mentioned in Rothschild and White (1995), which tries to maximize output, subject to several technological constraints. The main problem for the central planner to solve here of course is to choose the optimal amount of inputs subject to technological constraints in order to achieve the first best 1.

(7) solution. Conversely we can view the education industry as a decentralized market and compare the outcomes to that of the central planner’s problem, as mentioned in de Bartolome (1990). The comparison between the two enables us to know whether the central planner outcome can be achieved in a decentralized market, which means that we are trying to know if the solution is efficient, or generates more welfare in total. However, sometimes first best solutions are very hard to find or impossible to achieve due to some constraints, so finding for a second best alternative becomes the solution to the problem we are facing. When we talk about the production function of human capital, education is always an indispensable element that we have to focus on, but what kind of inputs and how should it be invested and how does the agents interact with the inputs has been the main problems of educational economics. Several issues have been discussed about, such as, voucher programs that allows students with low socioeconomic status to attend private schools or schools that are not near their residence, as mentioned in Epple and Romano (1998). There are also topics on subsidies, as mentioned in Winston (1999), in which the government tries to give aid to those who can not afford but are eager to study. Hierarchy has also been discussed before since it talks about the distribution of students are related to their income. One of the most popular topics that have been talked about is peer effect. The reason that it is important is that people cannot live alone. As long as they have interpersonal relationships they get influenced by their peers, more or less, even if they do not want to get affected. What is peer effect then, if it is so important an issue? Peer effect is when the behavior or background of your peer influences your utility, education outcome to be specific, whether the effect is good or bad. The reason there is peer effect is because there exists externality in the production of education due to the presence of peers. Some authors view education as a kind of club good, which means that the consumers 2.

(8) gain more than what they intended to obtain. For example in a ball game, fans get more than just the excitement of enjoying the game but also the experience as being part of the crowd. The price cannot be directly charged for the experience they share and the consumers simply have to pay only for the ticket fare. So, apparently peer effect causes a market failure if there is the effect of peers is not considered. Therefore, how to deal with this peer effect differs from model to model. Our model follows the model of de Bartolome (1990), which used a community model to find out the proportion of families with more able children and rent premium and also discussed the existence of equilibrium. Although in de Bartolome (1990) it was mentioned that Stiglitz (1974) did a research on private schools and conclude that private schools are a minority in education and thus assumed that private schools does not exist, this assumption can be further revised in order to focus on our model. Therefore, the objective of this thesis is to extend the community model of de Bartolome (1990) and find out how proportion of families with more-able children and the rent premium will be affected when adding private school to the model. We also attempt to find the conditions that enables private school to exist. Moreover, we want to see if voucher programs increase aggregate educational output under different policies of private school.. 3.

(9) 1.2 Contributions. The contribution of this thesis is based on the improvements we made on the theoretical model in the field of educational economics that deals with the problem of peer effect with public schools, private schools, vouchers. Epple and Romano (1998) discussed about the same issue with a similar structure and assumptions on income endowments, however in that paper public schools were treated in a much simplified method. In our model, public school is incorporated in a community model. We hope that this thesis is able to be a reference for the government before they make educational policies.. 1.3 Outline. In the following, the second chapter will be the literature review, providing an overlook on what researchers have done in the past. In chapter three we will introduce the model in this paper, and notify the difference between other models. The fourth chapter will be the analysis and results of the model. Chapter five will be the conclusion.. 4.

(10) 2. Literature Review. What we would like to discuss in this thesis is how the existence and composition of private school affects the educational output when the main issue is about peer effects. In order to do so we need to show what aspects educational economists have been focusing on according to the past research related to peer effects. Peer effects have been an indispensable element to take into consideration in most of the preceding works on the production of education. Some consider the issue in a framework with public schools only, and some even include both public and private schools in their model. When dealing with peer effects that affect the quality of educational output in classrooms, Arnott and Rowse (1987) discussed how educational expenditures should be allocated over classrooms when peer group effect is present with the use of a model where there is an education planner who wants to maximize social welfare subject to the budget constraint and the ability of students. Numerical tests were taken to see what the conditions are for streaming and mixing to occur as the best way to distribute students. The result is that the optimal allocation depends on the production function. Different production functions may provide different results. Similarly, Lazear (2001) analyzed peer effects by focusing on the problem of sorting of students and the quality of teachers by using a disruption model of educational production. The model takes the interruption of students’ asking questions into consideration, with the result that the optimal class size rises when the teacher’s age increases, when the probability of students behaving well increases, and when the value of a unit of education falls. Also, when the optimal class size is decided with certain assumptions, educational output is higher in classes that are larger in size with students that relatively behave well, vice versa. When dealing with peer effects that affect the 5.

(11) quality of educational output in “communities”, de Bartolome (1995) showed different composition of families by using a community model. First the model was used to see how families are allocated through migrating. Then the problem was dealt with as a central planner problem. The result shows that in order for a heterogeneous equilibrium to exist, the peer effect should neither be too strong nor too weak. Some research treats schools as firms that attempt to maximize profit subject to their budget constraint. Rothchild and White (1995) concerned the problem of pricing of higher education in which the customers are also inputs. In this paper, students themselves which are simultaneously customers and the input of producing human capital may bring externality since the existence of better students are considered as a component of educational output for those who are less able. This externality is claimed to be internalized by charging different tuition in the paper. The result is that competitive firms are able to find a zero-profit price that efficiently allocates peer effects. Education can be viewed as public goods. Tiebout (1956) intended to show that the Musgrave-Samuelson analysis need not apply to local expenditures. The paper showed that given several local governments with different economic structures and provision of public goods and assuming that the consumers are fully mobile, which means they choose where to live by their feet, then the optimal level of expenditure on public goods are actually decided by the consumers themselves. This is because the choice they make fits their preferences the best so that the governments do not have to play as a central planner and decide different expenditure levels on public goods according to different types of customers. That is to say, the customers distinguish themselves. Sandler and Tschrhart (1980) did a survey on the past research of club theory. Methods such as the Buchanan’s model, the general model and the game theory model were discussed. Miscellaneous issues were discussed in the theory of 6.

(12) club goods. For example, the membership heterogeneity, discrimination, and crowding effects, etc. The conclusion is that the theories of club goods can be applied to numerous issues, such as education. One of the extensions of club goods model they mentioned is the intergenerational clubs, which stands for clubs with multiple over-lapping generations of members. They claim it can be used to examine the principles of school district design. Epple and Romano (1998) discussed the effect of vouchers and peer effects on social welfare when using both a theoretical and computational model with tax-financed, tuition free public schools and competitive, tuition financed private schools. In the model, achievement depends on the students own ability and ability of their peers. Private schools can discriminate in their tuition policies. The paper also included the effects of vouchers. The result is that the equilibrium has a strict hierarchy of school qualities and two-dimensional student sorting with stratification by ability and income. Also it showed that voucher programs could have significant distributional consequences. Winston (1999) tried to find out how well our knowledge of microeconomic theories can help us on understanding the economics of higher education. A review of literature regarding non-profit enterprises subsides and peer effects were cited and discussed. The paper also mentioned that colleges can buy important inputs to their production only from the customers who buy their products, which is similar to Rothchild and White (1995). The paper also discussed the hierarchy of colleges and universities, which is different from Epple and Romano (1998) where the focus was on high-school education, also the issue was discussed empirically. The conclusion is that standard economic intuition and analogies based on the knowledge of profit making firms are likely to be a poor guide to understanding higher education.. 7.

(13) There has been controversy on how school voucher programs influence education. Ladd (2002) tried to provide evidence that large-scale vouchers will not derive significant gains in students’ achievement and might even be detrimental to disadvantaged students. By making a discussion on the past research, the conclusion is that widespread use of school vouchers is not likely to generate substantial gains in the productivity of the U.S. education system. Despite the fact that many claim that school vouchers play a small role in the education system, Neal (2002) argued that we cannot confidently predict the outcomes that would result from various voucher schemes. By describing both theoretical and empirical work that has been done on this topic, the conclusion is that vouchers cannot prove to be efficiently used, or might even create significant social costs. The peer effect in this thesis affects the allocation of families among communities. We will not only borrow the community model of de Bartolome (1990), but will also make some revisions on the model itself, where the revisions are close to the assumptions made on the standard by which we discriminate the types of schools, types of students in Epple and Romano (1998), except that in our thesis, public schools are not exempted from charging tuition. The level of educational input of public schools is chosen the same way as in Tiebout (1956), which is by means of majority voting, while the level of input of private school is chosen similar to Rothschild and White (1995), where private school is treated as a kind of firm trying to solve its maximization problem subject to some constraints. In the following section, we will introduce the work of de Bartolome (1990) briefly and display the model. To confirm the conclusions in Ladd (2002) and Neal (2002), we try to find out how the voucher program promotes private school and its influence on educational output.. 8.

(14) 3. Model. 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 9.

(15) 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 10.

(16) 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 N1 families each with a child of low ability, denoted a1, and N2 families each with a child of high ability, denoted a2.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. ∂e( I ,θ , a1 ) ∂e( I ,θ , a2 ) < . ∂I ∂I On the other hand, less able children are assumed to gain no less from a peer group improvement, or. ∂e( I ,θ , a1 ) ∂e( I ,θ , a2 ) ≥ . ∂θ ∂θ The utility of each family depends on consumption and educational achievement. For families with less able child, the utility function is U (C, e( I ,θ , a1 )) = F (C ) + G( I ) + H (θ ). And for families with more able child, the utility function is U (C , e( I ,θ , a2 )) = F (C ) + R( I ) + S (θ ).. where F(.), G(.), H(.), R(.), and S(.) are strictly concave functions. Greater input sensitivity of more able child implies ∂G ( I ) ∂R ( I ) . < ∂I ∂I. 11.

(17) And the assumptions made on peer effect imply ∂S (θ ) ∂H (θ ) ≤ . ∂θ ∂θ. Since de Bartolome (1990) is interested in how families with child of different abilities are affected by peer effects and allocate themselves between the two 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 θu and θs, where u and s stands for urban and suburban area, respectively. The total number of residents in each community is thus denoted nu and ns. 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 Iu and obtains Iu from school, and each suburban family pays Is and obtains Is from school. The housing rent for residents in the urban area is ru , and is rs for the suburban area. The government collects the rent and gives it back to the residents as a kind of lump-sum transfer, T. According to the government budget constraint,. T=. ns rs + nu ru ns + nu .. By substitution, the net suburban rent r and net urban rent becomes. r ≡ rs − T = ru − T =. nu ( rs − ru ) , and ns + nu n ns (ru − rs ) = − s r. ns + nu nu . We use Figure 1 to illustrate the community model of de Bartolome (1990). 12.

(18) nu. ns Suburban area. θs. Urban area. Is. Iu. rs. ru. θu. N1 ,N2. 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. Cu = y − I u + (. ns )r , and nu. Cs = y − I s − r . According to the assumptions of de Bartolome (1990), the input level of each 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 13.

(19) 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,. n I u = arg max F ( y − I u + s {I } nu. r ) + G ( I ), . and I s = argmax F ( y − I s − r) + R(I ). {I }. Following de Bartolome (1990), the equilibrium is reached when the “no 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 F ( y − I s − r ) + G ( I s ) + H (θ s ) n N − nsθ s = F ( y − I u + s r ) + G ( I u ) + H ( 2 ) nu nu . 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 F ( y − I s − r ) + R( I s ) + S (θ s ) n N − nsθ s = F ( y − I u + s r ) + R ( I u ) + S ( 2 ) nu nu . 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. 14.

(20) Barolome (1990), the equilibrium values ( I u* , I 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. r AA′. BB′. r*. θ s*. θs. 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 θs. Similarly, each point on BB′ stands for the rent premium that high ability students are willing to sell for a given level of θs. 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 15.

(21) 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 N1 families with children of ability a1, and N2 families with children of ability a2. 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 N11u, N12u and N21u 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, N11u 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 N22s, N12s and N21s, respectively. Endowed income is divided to y1 and y2, where the former stands for lower endowed income, and the latter stands for higher endowed 16.

(22) income. The utility functions of the family with child of low ability is still U (C, e( I ,θ , a1 )) = F (C) + G( I ) + H (θ ).. And for the families with more able children U (C , e( I ,θ , a2 )) = 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 Cu , high = y2 − I u + (. ns )r nu. ,. Cu , low = y1 − I u + (. ns )r nu. ,. Cs , high = y2 − I s − r. ,. and. Cs ,low = y1 − I s − r , respectively.. We still keep the assumption that input level is decided by majority voting. However, since there are four rather than two types of families, there are different 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 families of “low-ability low-income”, and for the suburban community, families of “high-ability high-income” makes the decision, we think this setting would be more interesting and believe it would capture the fact more closely. Therefore, the input level for the urban area is n I u = arg max F ( y1 − I u + s r ) + G ( I ), {I } nu . (1). And for the suburban area I s = arg max F ( y2 − I s − r ) + R( I ).. (2). {I }. Below we show a graph that illustrate our model.. 17.

(23) N. This is private school. θp Suburban Area. Urban Area. nu. Low-ability High-income. Low-ability. Is. High-income Low- ability Low- income. Iu. ns. High-ability High-ability. low-income. low-income. θu. θ. s. 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.. 18.

(24) r. AA′(H). AA′(L). BB′(H). BB′(L). θs 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 F ( y2 − I s − r ) + G ( I s ) + H (θ s ) n = F ( y2 − Iu + s nu. N 2 − nsθ s ). r ) + G ( I u ) + H ( nu . (3). For “high-ability low-income” families, the condition is F ( y1 − I s − r ) + R ( I s ) + S (θ s ) n N − nsθ s ). = F ( y1 − I u + s r ) + R ( I u ) + S ( 2 nu nu . (4). From equations (1) through (4), which are the input decision and no migration 19.

(25) * * * * condition, we can solve for ( I u , I 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 F ( y1 − I s* − r ) + G( I s* ) + H (θ s* ) n < F ( y1 − I u* + s nu. N 2 − nsθ s* * r + G I + H ) ( ) ( ). u nu . (5). For “high-ability high-income” families, the condition is n N − n θ* F ( y2 − I u* + s r ) + R( I u* ) + S ( 2 s s ) nu nu < F ( y2 − I s* − r ) + R( I s* ) + S (θ s* ).. (6). 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 Ip 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. 20.

(26) 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.. Max. (θ P , I p , I1 , I 2 ). (1 − θ P )[G ( I p ) + H (θ P )] + θ P × [ R( I p ) + S (θ P )]. s.t. (1 − θ P ) × I1 + θ P × I 2 = I p n N − (ns θs + Nθ P ) ) F ( y2 − Iu + ( s )rɵ ) + G ( Iu ) + H ( 2 nu nu − N n ≤ F ( y2 − Iu + ( s )rɵ − I1 ) + G ( I p ) + H (θ P ) nu N − (ns θs + Nθ P ) n ) F ( y1 − Iu + ( s )rɵ ) + R( Iu ) + S ( 2 nu nu − N. (7). n ≤ F ( y1 − Iu + ( s )rɵ − I 2 ) + R ( I p ) + S (θ P ) nu 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 θp is the proportion of students attending private school. Moreover, in the constraint, I1 and I2 stands for the tuition charged to students of different ability, and Ip 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. F ( y2 − I s − rˆ) + G ( I s ) + H (θˆs ) n N − ( nsθ s + Nθ P ) = F ( y2 − I u + s rˆ) + G ( I u ) + H ( 2 ) nu − N nu . 21. (8).

(27) n N − ( nsθ s + Nθ P ) F ( y1 − I u + s rˆ) + R ( I u ) + S ( 2 ) nu − N nu = F ( y − I − rˆ) + R ( I ) + S (θˆ ) 1. s. s. (9). s. * * From equations (1), (2) we solve for ( Iˆu , Iˆs ) , and from equations (7), (8) and (9), we. * * * * * * solve for (θˆs , I1 , I 2 , rˆ , I p ,θ p ) , if the values also satisfy the conditions (10) and (11),. then we obtain our solution we are looking for. F ( y1 − Iˆs* − rˆ* ) + G( Iˆs* ) + H (θˆ*s ) n < F ( y1 − Iˆu* + s nu. N 2 − (nsθ s + Nθ P ) * ) rˆ) + G( Iˆu ) + H ( nu − N . n N − ( nsθ s + Nθ P ) F ( y2 − Iˆu* + s rˆ* ) + R ( Iˆu* ) + S ( 2 ) nu − N nu < F ( y − Iˆ* − rˆ* ) + R ( Iˆ* ) + S (θˆ* ) 2. s. s. (10). (11). s. In addition to the existence of private school we are interested in how the voucher program affects the equilibrium. We assume that the voucher lessens the burden of 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 Iu , then under a 50% voucher the tuition will become 0.5×Iu . Again we can derive the optimal level of input from equations (1) and (2), and from equations (7), (8) and (9) * * * * * * * * * * * * we solve for (θˆs , I1 , I 2 , rˆ , I p ,θ p ) (θˆs , I1 , I 2 , rˆ , I p ,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. 22.

(28) n N − (ns θs + Nθ P ) F ( y2 − V × Iu + ( s ) rɵ ) + G ( Iu ) + H ( 2 ) nu nu − N n ≤ F ( y2 − V × Iu + ( s )rɵ − I1 ) + G ( I p ) + H (θ P ) nu n N − ( ns θs + Nθ P ) F ( y1 − V × Iu + ( s )rɵ ) + R ( Iu ) + S ( 2 ) nu nu − N n ≤ F ( y1 − V × Iu + ( s )rɵ − I 2 ) + R ( I p ) + S (θ P ) nu 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.. 23.

(29) 4. Numerical Analysis. In the last chapter, we discussed the equilibrium conditions with and without private school as well as taking voucher programs into consideration by means of a general model. By setting the utility functions in the form of logarithms, we are able to discuss the problems numerically to see how the results in each case are. Therefore, in this chapter we would first like to discuss how different income endowments affect the results. Then we will discuss when private school exists, and how does the existence affect the behavior of the residents of the two communities. Finally, we would like to see if the voucher program is feasible, and how it promotes the establishment of private school. We assume that there are one hundred people in each community, and the total number of low ability and high ability students are one hundred each, which means that N1=100, N2=100, nu=100, ns=100. The utility function is assumed to be in logarithm form. For families with child of low ability, the utility function is U (C, e( I ,θ , a1 )) = ln C + β1 ln I + β2 ln(1 + θ ). And for families with child of high ability the utility function is U (C , e( I ,θ , a2 )) = ln C + α1β1 ln I + α 2 β 2 ln(1 + θ ). The parameter β1 falls in the range (0,1). It affects the proportion spent on educational inputs to total consumption, which means the higher β1 is the more income is spent on I. Moreover, β2 indicates the importance of peer effect to families, which means that the higher β2 is, the marginal utility of peer effects is larger. The parameter α1 falls in the range [1,∞). It is to show that families with child of higher ability obtain higher marginal benefit from educational input than what the families with child of lower ability does. And finally, the parameter α2 falls in the range (0,1) and is to capture the extent that low-ability family obtains relatively higher marginal 24.

(30) benefit from peer effect. To explore the features of the equilibrium numerically, we substitute numbers into the equilibrium conditions to solve for comparable solutions. However, substituting different values to the parameters derive different outcomes, we would like to discuss whether the equilibrium exists or not, thus we have to find out the conditions for an equilibrium to occur. Also, the numerical results are shown in the appendix A through F at the end of this paper, while we will cover the interpretations in this chapter. Result 1: We derive the following results with our setting of the numerical model for heterogeneous equilibrium of de Bartolome (1990) to exist . 1.. α1 has to be a number greater than one, but cannot be too large.. 2.. When α2 is too large, the equilibrium does not exist.. 3.. The maximum value of α2 that enables the existence of equilibrium is lowered in the increase of α1.. 4.. The change in income does not change the allocation of peer effects, but the rent premium is different.. 5.. When β1 or β2 increases, there is less chance for a heterogeneous equilibrium to exist. The most important two parameters that matters when finding the conditions for. the equilibrium to exist are α1 and α2, which is because they measure the difference of the magnitude of marginal peer effects and marginal input effects between the families with child of low ability and families with child of high ability. Thus, given the value of y, β1, and β2, we try to find out how the variation of values substituted into these to parameters affect the existence of equilibrium. When α1 is a number greater than one, an increase in α2 increases the equilibrium values of θ and r. This is because when α2 increases, the amount of rent the families of high ability would 25.

(31) demand for them to give up a certain amount of peer effect increases, and for families of low ability to become willing to purchase peer effect at that price of rent, peer effect has to increase. To be more explicit, we discuss the problem with Figure 3 that plots the equilibrium which we have shown in the last chapter. As we can see in the graph, AA′ curve is all the combinations of θ and r for low ability students to be indifferent between living in the city and living in the suburbs, and BB′ curve is for that of high ability students. When α1 is a number greater than one, an increase in α2 pivots the BB′ counterclockwise leaving less room for transaction, since now the rent premium offered by families with child of high-ability to sell the peer effects is now higher for any given θ, while the willingness to pay for the peer effect remains unchanged. However, when α2 is large and close to one, a mixed equilibrium does not exist anymore, since it makes AA′ parallel to BB′. This is because when α2 is too large given the value of α1, the price that the residents in the suburbs are willing to sell peer effect is too high to induce low ability students to migrate, and thus a separated equilibrium is formed. Moreover, when α1 increases, the threshold for the value of α2 to derive a mixed equilibrium becomes lower. When α1 approaches to one, which means the two communities only differ in their preference on peer effect, then any value of α2 that is below one would cause a separated equilibrium, which implies that the rent to make residents feel indifferent of where to live does not exist. Now given the value of α2, we can see that when α1 increases, the equilibrium values increase, vice versa. This is because the increase in the value substituted to α1 increase the difference between the educational inputs Iu and Is in the two communities, which makes families of low ability prefer living in the urban area, and families of high ability prefer to live in the suburban area. When we look at the graph, an increase in α1 given the value of α2 makes AA′ curve pivot counterclockwise, and BB′ curve pivots clockwise, 26.

(32) intersecting at a point where both θ and r are larger in equilibrium. To offset the effect of changing the values substituted into α1, α2 has to be moved towards the opposite position. In other words, to sustain a mixed equilibrium, when α1 increases, α2 has to decrease. Now we take a look at the common parameters of the utility functions of both types of families. An increase in the values of β1 increases the importance of educational input and decreases the relative importance of peer effect in both communities, vice versa. However, this increase also reduces the room for trade since the educational input the residents have to pay increased in both communities, which means the willingness to pay rent for a given proportion of more able students has decreased, and the decrease is more for families with child of low ability since they prefer peer effect to input effect. Thus both θ and r will increase when β1 increases. When we look at the graph, this change in β1 pivots both AA′ and BB′ counterclockwise, with a larger pivot in BB′, and lessens the room for trade of peer effect, which is shown as the distance between the curves. Moreover, when the value of β2 increases, the room for trade increases, vice versa.. When β1 equals to β2, a simultaneous increase in the two parameters keeps the proportion of more able students in equilibrium almost unchanged but increases the rent premium. Now we keep parameters other than income fixed to see how the values of y1 and y2 affect the equilibrium values. We find out that the value of income does not affect the outcomes, although the increase in income affects the amount of consumption and input decision, this only affects the amount of rent premium, and will not change the preference of peer effects and input effects.. 27.

(33) Result 2: After we choose y1=10, y2=10, α1=2, α2=0.8, β1=0.04, β2=0.25 to be the benchmark case, the comparative statics of the numerical model is as following: Other things unchanged, 1. When y increases, only r increases, θ is unchanged. 2. When α1 increases, both θ and r increase. 3. When α2 increases, both θ and r increase. 4. When β1 increases, both θ and r increase 5. When β2 increases, only θ increases, r is unchanged. From the preceding paragraphs, we choose our benchmark values for substitution to be y1=10, y2=10, α1=2, α2=0.8, β1=0.04, β2=0.25. Since we have no explicit function of the equilibrium values, we now simulate a comparative statics by keeping other parameters fixed and see how the equilibrium is affected when each of the values of parameters increases. The results are shown in the appendix E. In the original equilibrium, approximately 60 out of 100 families with child of high ability live in the suburbs. The rent premium is relatively low compared to the amount of consumption, while the input expenditure is about one tenth of the total consumption. An increase in income affects everything but the equilibrium, which is because income has nothing to do with the preference of families. An increase in α1 increases the equilibrium θ and r, and also the level of educational input. However, the consumption level of residents in the suburbs has decreased because they have to pay more educational input. An increase in α2 has similar effects as α1, but the decrease in educational input and consumption for suburban families is caused by the increase in relative importance of peer effects. An increase in β1 reasonably causes consumption and educational input of both communities to increase since it stands for the proportion of income spent on those expenses. An increase in β2 caused θ to decrease since it makes peer effects more expensive. 28.

(34) Before going through the discussion on the conditions that enables private school to exist, we need to know how our model with different income endowments affects the conditions for the equilibrium to exist. Result 3: In our model, the conditions for the equilibrium to exist are 1 . The difference between y1 and y2 cannot be too big. 2. α1 has to be a number greater than one. 3. α2 cannot be too large. 4. β1 has to move oppositely according to the value of α1 5. β2 cannot be too small The difference between this model and the model of de Bartolome (1990) is that there is difference in income endowment. In addition to the difference, we assumed that the input level of urban area is decided by families of low-ability low-income, and that of the suburban are is decided by families of high-ability high-income, this cause the gap of the level of educational input between the two communities to increase in size, and lowers the chance for a mixed equilibrium to exist if the gap is too big, since none of the high-ability low-income students can afford to live in the suburbs. β1 has to move oppositely according to the value of α1, because if they move in the same direction, it will make the educational input level of the suburbs become too high, and the residents there would rather move to the urban area. If β2 is too small, then the peer effect would be trivial to influence the equilibrium, and causes separation of different abilities of families to emerge.. 29.

(35) Result 4: After we choose y1=10, y2=15, α1=2, α2=0.8, β1=0.04, β2=0.25 to be the benchmark case, the comparative statics of the numerical model is as following: Given other things unchanged, 1. When y1 increases, both θ and r increase. 2. When y2 increases, both θ and r decrease. 3. When α1 increases, both θ and r increase. 4. When α2 increases, both θ and r increase. 5. When β1 increases, both θ and r increase. 6. When β2 increases, only θ has decreases, r is unchanged. In equilibrium the values of θ and r are 0.577234 and 0.123756. We can know from the equilibrium values that approximately 57 out of 100 high ability students are in the suburbs. The rent premium is relatively low to the level of consumption and educational input. The level of consumption and educational input are not close to each other anymore, which is the result we want to see from adding the difference of income endowment to the model. The results are very similar to that of no difference in income, except for the fact that variation in the endowed income now affects the value of θ and r in equilibrium. The results for the comparative statics are shown in appendix F. Now that we know what the conditions for equilibrium in this model are, we can discuss the conditions for private school to exist. The definition of exist in this thesis means that given the no migration conditions are satisfied in equilibrium, students gain higher utility from attending private school than attending public schools, which implies that students are willing to attend private school. Otherwise, no student can be recruited.. 30.

(36) Result 5: Private school exists when 1. The product of α1 and β1 is not too large. 2. The value of α2 cannot be too small. 3. The value of β2 cannot be too small. Thus we choose y1=10, y2=15, α1=2, α2=0.8, β1=0.04, β2=0.25 to be the benchmark case for the existence of private school. The most appealing feature of private school is that it provides high peer effects, however, without the aid of vouchers, it is somewhat expensive for students to attend private school since they not only have to pay tuition for public school but also have to pay an additional fee for attending private school. Therefore, when peer effect is much important than input effect, the product of α1 and β1 cannot be too large, and if β2 is larger, then the original θ in equilibrium should be small relative to the peer effect that private school offers to appeal students from the urban area. When α1 increases, α2 has to decrease, vice versa. This is to keep the proportion of input expenditure fixed near a certain value; otherwise private school will not exist, since if the tuition they have to pay for public school is too high, they would not pay additional tuition to attend private school, no matter how high the peer effect is. The second point is that when α2 is too small, high ability students would not necessarily be better off attending private school. The thirds point is that when β2 is too small, there will be no mixed equilibrium in the community. The last point is that the larger the gap between low income groups and high income groups, the higher α1 has to be for private school to exist. This is because if α1 is too low, the equilibrium values would violate the assumption that the majority of suburban area is families with child of high ability. It is also because this change keeps the proportion of educational input fixed at a certain value. After we discuss the conditions of when private school exists, we would then 31.

(37) move on to explore the properties of the equilibrium with private school. Since the most appealing feature of private school is that it gathers students of high-ability so that it is most like that students of high-ability low-income who lives in the urban area would like to attend private school. Suppose private school recruits ten high-ability low-income students from the urban area. Then we re-calculate the equilibrium for the community model. The new equilibrium θ and r are 0.545376 and 0.123756, respectively. This means that 54 out of 100 high ability students remain in the suburbs. We then substitute the equilibrium values into the utility functions and confirmed that the residents are indifferent of migrating. The next step is what level of input expenditure private school should set to make the families indifferent of attending private school and public school. To do so we try to find the level of input that maximizes the educational output private school creates subject to the fact that students who attend private school are at least as well of as they are when they attend public school which is the optimization problem (11) mentioned in chapter three. In the following we are going to discuss the behavior of private school. In the preceding paragraph, we first consider the situation that there are ten students attending private school, and all of them are high-ability low-income students from the urban area. When discussing peer effect in educational economics, private school usually plays the role of a broker who facilitates the transaction of peer effects. Therefore, we would like to see if recruiting some students with low ability but high income who would like to pay higher tuition in order to attend private school would improve the educational output it creates. Suppose now in another scenario privates school recruits nine high-ability low-income students and one low-ability high-income student instead of all high-ability low-income students, and gives a discount to high-ability low-income students while low-ability high-income student burdens the rest of the payments. This is because the attending of low-ability 32.

(38) high-income student lowers the peer effect, thus he has to pay more tuition to compensate for the loss. We assume students of high-ability low-income receive a 4% discount, then after private school recruits students, the new equilibrium is 0.550895, and the rent premium is unchanged. Again we try to find the optimal level of tuition private school should charge. Finally, we compare the total educational output that private school created in each case. The following chart shows the results.. H-L/ L-H. θ. r. Total Educational Output. 10/0. 0.54376. 0.123756. 1.74242. 9/1. 0.550895. 0.123756. 1.63375. 8/2. 0.556414. 0.123756. 1.44838. To conclude, when private school exists, it is better to have all high-ability low-income students attending private school. This is probably because the value we substituted into β1 is too small or because we considered peer effects linearly so that it could not capture the significance of peer effect in the consideration of private school. Also we have found that when private school exists, the students that attend public school obtain lower utility. This is because when some of the students in the community attend private school, the number of students that attend public school decreases. However, the ones that remain in public school did not gain additional educational input. We assume that the government collected the redundant inputs, so that with lower peer effects and paying the same amount of tuition makes the students who still attend public school after private school recruited students are worse off. Another reason is that private school chooses the tuition that makes students indifferent of attending public and private school and maximizes the educational input it creates, which means that private school is actually better off after recruiting 33.

(39) students. The final part of this section we would like to look at the effects of school vouchers. We attempt to find out how vouchers affect the existence and the operation of private school. Also we would like to know the affect on educational output. First we would like to see how different levels of vouchers affect the existence of private school. The final part we would like to look at how different levels of vouchers affect the total educational output created by private school.. Result 6: School vouchers can promote the existence of private school, on condition that the community was willing to attend private school but could not due to some certain reasons such as insufficient income.. To see why, we first look at the chart below.. Level of Vouchers (The students only have to pay V×tuition) Values of Parameters V=0.25. V =0.5. V =0.75. V =0.95. Exist. Exist. Exist. Do not Exist. Do not Exist. Do not Exist. Do not Exist. Do not Exist. y1=10, y2=15, α1=2, α2=0.8, β1=0.1, β2=0.25 y1=10, y2=15, α1=2, α2=0.8, β1=0.2, β2=0.25. From the chart we can see that when private school does not exist, as long as the parameters are not too extreme in their values, it is still possible for private school to exist after the school voucher is issued. Take the case here for example, when the value of β1 is equal to 0.1, which means that twenty percent of consumption is due to educational input, private school will exist in most levels of school vouchers except 34.

(40) when the level is 0.95, which means that only a small proportion of the tuition that has to be paid to public school has been subsidized. On the other hand, when β1 is equal to 0.2, which means forty percent of consumption is due to educational input, private school will not exist no matter how high the level of school voucher is.. Result 7: The level of school vouchers does not change the result that private school creates the most educational output when recruiting only high-ability students.. When students only have to pay 25% of original tuition H-L/ L-H. θ. r. Total Educational Output. 10/0. 0.549865. 0.123756. 1.80945. 9/1. 0.550895. 0.123756. 1.69261. 8/2. 0.556414. 0.123756. 1.51407. When students only have to pay 50% of original tuition H-L/ L-H. θ. r. Total Educational Output. 10/0. 0.548453. 0.318688. 1.78774. 9/1. 0.553983. 0.318688. 1.6716. 8/2. 0.559514. 0.318688. 1.4929. When students only have to pay 75% of original tuition H-L/ L-H. θ. r. Total Educational Output. 10/0. 0.546959. 0.222717. 1.76541. 9/1. 0.552484. 0.222717. 1.64998. 8/2. 0.55808. 0.222717. 1.47102. From this chart we can see that the level of vouchers affects the total educational. 35.

(41) output. Moreover, no matter how much the level of discount is, recruiting no low-ability high-income students creates the highest educational output, which is the same result we get even without distributing vouchers. To conclude, when the government gives out school vouchers to promote private school, which is successful since students of high-ability low-income and low-ability high-income are more willing to attend private school, the total educational output created by private school slightly increases. However, the best policy for private school is to only recruit students of high-ability, since the educational output is higher this way rather than recruiting students of low-ability high-income.. 36.

(42) 5. Conclusion In this thesis we extended the model of de Bartolome (1990) by including different income endowments and private school as well as vouchers in the model. We discussed the conditions that would enable the equilibrium in the original model to exist, the conditions for private school to exist, how the private school should recruit students to obtain the maximized educational output, and how vouchers affect the existence of private school and the educational outcome. The results of our analysis show that private school exists when the proportion of expenditure on educational input for high ability students is not too high, marginal utility of peer effect is not too small, and the marginal benefit students of high-ability gain from peer effect is not much lower. Moreover, no matter if school vouchers are issued by the government, the best policy for private school is to recruit only high-ability students, rather than mixing low-ability students. Finally, it is possible for school vouchers to promote the existence of private school, on condition that the environment was not too far from enabling its existence.To make this thesis more complete and close to the real world, we believe more revisions can be made. Thus in the future we may add more private schools to the model to show competition among private school, or consider the problem that people outside the community might move in because of the reputation of private school.. 37.

(43) Appendix A. Input decisions (No Difference in Income Endowment) 1 β 1 β F '(Cu ) = G '( I u ) → = → (Y − I u + r ) = I u → I u* = (Y + r ) β Cu I u 1+ β F '(Cs ) = G '( I s ) →. 1 α1 β 1 α1 β = → (Y − I s − r ) = I s → I s* = (Y − r ) α1β Cs Is 1 + α1 β. B. No Migration Condition (No Difference in Income Endowment) α1 β αβ (Y − r ) − r ) + β ln( 1 (Y − r )) + ln(1 + θ ) 1 + α1β 1 + α1β β β = ln(Y − (Y + r ) + r ) + β ln( (Y + r )) + ln(2 − θ ) 1+ β 1+ β. ln(Y −. ln(Y −. β 1+ β. = ln(Y −. (Y + r ) + r ) + α1β ln(. β 1+ β. (Y + r )) +. 1. α2. ln(2 − θ ). 1 α1β αβ (Y − r ) − r ) + α1β ln( 1 (Y − r )) + ln(1 + θ ) 1 + α 1β 1 + α1β α2. C. Utility if attending public school: α 1β αβ (Y2 − r ) − r} + β ln{ 1 (Y2 − r )} + ln(1 + θ ) 1 + α 1β 1 + α 1β β β U 1,1, urb = ln{Y1 − (Y1 + r ) + r} + β ln{ (Y1 + r )} + ln(2 − θ ) 1+ β 1+ β β β U 1,2, urb = ln{Y2 − (Y1 + r ) + r} + β ln{ (Y1 + r )} + ln(2 − θ ) 1+ β 1+ β 1 β β (Y1 + r )} + ln(2 − θ ) U 2,1, urb = ln{Y1 − (Y1 + r ) + r} + α 1β ln{ 1+ β α2 1+ β α 1β αβ (Y2 − r ) − r} + β ln{ 1 (Y2 − r )} + ln(1 + θ ) U 1,2, sub = ln{Y2 − 1 + α 1β 1 + α 1β 1 α 1β αβ U 2,1, sub = ln{Y1 − (Y2 − r ) − r} + α 1β ln{ 1 (Y2 − r )} + ln(1 + θ ) 1 + α 1β 1 + α 1β α2 1 α 1β αβ U 2,2, sub = ln{Y2 − (Y2 − r ) − r} + α 1 β ln{ 1 (Y2 − r )} + ln(1 + θ ) 1 + α 1β 1 + α 1β α2 1 β β U 2,2, urb = ln{Y2 − (Y2 + r ) + r} + α 1β ln{ (Y2 + r )} + ln(2 − θ ) 1+ β 1+ β α2 U 1,1, sub = ln{Y1 −. 38.

(44) D. Utility when attending private school (Assuming that private school only recruits high ability students from urban area): U 1,1, urb , pri = ln(C1,1, urb − I 2,1, u ) + β ln( I 2 ,1, u ) + ln 2 U 1, 2, urb , pri = ln(C1, 2 , urb − I 2,1, u ) + β ln( I 2 ,1, u ) + ln 2 U 2,1, urb , pri = ln(C 2 ,1, urb − I 2,1, u ) + α 1 β ln( I 2 ,1, u ) +. 1. α2. ln 2. U 1, 2, sub , pri = ln( C1, 2, sub − I 2,1, u ) + β ln( I 2 ,1, u ) + ln 2 U 2,1, sub , pri = ln( C 2,1, sub − I 2,1, u ) + α 1 β ln( I 2 ,1, u ) + U 2, 2, sub , pri = ln( C 2, 2, sub − I 2,1, u ) + α 1 β ln( I 2 ,1, u ) +. 1. α2 1. α2. ln 2 ln 2. E. Comparative statics (No Difference in Income Endowment) θ. r. Cu. Cs. Iu. Is. 10.0786. 8.81323. 0.403143. 0.705058. 11.0864. 9.69455. 0.443457. 0.775564. 10.1517. 8.71059. 0.406066. 0.731689. 10.2153. 8.6816. 0.408611. 0.694528. 10.0634. 8.57584. 0.50317. 0.857584. 10.0786. 8.81323. 0.403143. 0.705058. Original Equilibrium 0.829914. 0.481713. Increase in y1, y2 (y1= y2=11) 0.829914. 0.529884. Increase in α1 (α1=2.1) 0.880165. 0.557726. Increase in α2 (α2 =0.85) 0.913404. 0.623874. Increase in β1 (β1=0.05) 0.890681. 0.566578. Increase in β2 (β2=0.3) 0.776305. 0.481713. 39.

(45) F. Comparative statics (With Different Income Endowment) θ. r. Cu. Cs. Iu. Is. 9.73438. 13.7743. 0.389375. 1.10194. 10.7754. 13.6977. 0.431018. 1.09582. 9.69318. 14.7399. 0.387727. 1.17919. 9.76716. 13.692. 0.390686. 1.15013. 9.7612. 13.7485. 0.390448. 1.09988. 9.66077. 13.5056. 0.483039. 1.35056. 9.73438. 13.7743. 0.389375. 1.10194. Original Equilibrium 0.577234. 0.123756. Increase in y1 (y1=11) 0.615012. 0.20646. Increase in y2 (y2=16) 0.554507. 0.08091. Increase in α1 (α1=2.1) 0.597073. 0.157843. Increase in α2 (α2 =0.9) 0.588839. 0.151646. Increase in β1 (β1=0.05) 0.591866. 0.143814. Increase in β2 (β2=0.26) 0.574269. 0.123756. 40.

(46) References Arnott, R. and J. Rowse, (1987), “Peer Group Effects and Educational Attainment”, Journal of Public Economics, Vol. 88, No.1, pp.287-205 De Bartolome, C. (1990), “Equilibrium and Inefficiency in a Community Model with Peer Group Effects”, Journal of Political Economy, Vol.98, No.1, pp.110-133 Epple, D. and R. Romano, (1998), “Competition between Private and Public Schools, Vouchers, and Peer-Group Effects,” American Economic Review, Vol.88, No.1, pp.33-62 Lazear, E. (2001), “Educational Production”, The Quarterly Journal of Economics, Vol. CXVI, Issue 3, pp.777-803 Ladd, H. (2002), “School Vouchers: A Critical View”, Journal of Economic Perspectives, Vol. 16, No. 4, pp.3-24. Neal, D., (2002), “How Vouchers Could Change the Market for Education”, Journal of Economic Perspectives, Vol.16, No.4, pp.25-44 Rothschild, M. and L. White, (1995), “The Analytics of the Pricing of Higher Education and Other Services in Which the Customers Are Inputs”, Journal of Political Economy, Vol. 103, No. 3, pp.573-586 Sandler, T. and J. Tschirhart, (1980), “The Economic Theory of Clubs: An Evaluative Survey”, Journal of Economic Literature, Vol. XVII, pp.1481-1521 Tiebout, C., (1956), “A Pure Theory of Local Expenditures”, Journal of Political Economics, Vol. 64, No.5, pp.416-24 Winston, G., (1999), “Subsidies, Hierarchy and Peers: The Awkward Economics of Higher Education”, Journal of Economic Perspectives, Vol. 13, No.1, pp.13-36. 41.

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