J Econ (2017) 120:65–78 DOI 10.1007/s00712-016-0513-5

Multistage public education, voting, and income distribution Katsuyuki Naito1

· Keigo Nishida2

Received: 2 September 2015 / Accepted: 31 August 2016 / Published online: 12 September 2016 © Springer-Verlag Wien 2016

Abstract This paper proposes a theory to study the formulation of education policies and human capital accumulation. The government collects income taxes and allocates tax revenue to primary and higher education. The tax rate and the allocation rule are both endogenously determined through majority voting. The tax rate is kept at a low level, and public funding for higher education is not supported unless the majority of individuals have human capital above some threshold. Although public support for higher education promotes aggregate human capital accumulation, it may create long-run income inequality because the poor are excluded from higher education. Keywords Majority voting · Public education · Multistage education · Human capital accumulation · Income distribution JEL Classification D72 · H52 · I24 · I25 · O43

1 Introduction Human capital accumulation can be an engine of economic growth, and the government plays significant roles in providing formal education. In many existing studies, human capital is produced in a single education sector, where an amount of government

B

Katsuyuki Naito [email protected] Keigo Nishida [email protected]

1

Faculty of Economics, Asia University, 5-24-10, Sakai, Musashino, 180-8629 Tokyo, Japan

2

Faculty of Economics, Fukuoka University, 8-19-1, Nanakuma, Jonan-ku, 814-0180 Fukuoka, Japan

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expenditure is a direct input. In reality, however, an education system is divided into multiple stages, such as primary, secondary, and tertiary education, and the government supports each of them. Primary and secondary (K-12) education has a particularly big difference from tertiary education in the sense that it is mandatory but serves everyone for free although college education is optional and requires private spending. It is relatively recently, however, that several studies have begun to analyze how policy changes for different education sectors affect economic growth and income distribution (Abington and Blankenau 2013; Arcalean and Schiopu 2010; Blankenau 2005; Blankenau et al. 2007a; Restuccia and Urrutia 2004; Su 2004). In particular, with heterogeneity in income or innate ability, changes in the size of public education funds or budget allocations across different education sectors may increase the welfare of some individuals at the expense of others as shown by Su (2004), Blankenau et al. (2007a), Abington and Blankenau (2013), and Hidalgo-Hidalgo and Iturbe-Ormaetxe (2012). This provides fertile ground for politico-economic analyses.1 For example, enhancing college education may most benefit the rich but may not benefit the poor who do not attend college. This paper proposes an overlapping generations model in which both a tax rate to finance overall government education expenditures and a budget allocation for multiple education sectors are determined via majority voting, and analyzes the effects on human capital dynamics. Since individuals vote for both a tax rate and an allocation rule, the policy is multidimensional. In general, it is extremely, and often impossibly, difficult to find political equilibrium when a policy is multidimensional (Persson and Tabellini 2000). In this paper, we provide a model of a special case to avoid this difficulty and prove that the individual with median income (median human capital) is the decisive voter. Although our model is simple, it enables us to analyze the interaction between the political determination of education policies and the dynamics of human capital. The outline of the model is as follows. Individuals live for two periods, namely childhood and adulthood, and they have children when they are adults. Individuals in adulthood make all the economic and political decisions, but they care about the human capital of their children. The government operates two education sectors, which we call primary and higher education sectors. Primary education is compulsory but serves all childhood individuals free of charge. On the other hand, higher education is optional and requires private expenditures. In each education sector, the private return on education depends on resource allocations from the government and parental human capital. Parents with low human capital are unwilling to have their children receive higher education because the private return is low. They prefer a low income tax rate and to allocate all public funds to primary education. Parents whose human capital is above some threshold are willing to have their children obtain higher education. They prefer a higher tax rate and to 1 Hidalgo-Hidalgo and Iturbe-Ormaetxe (2012) show that it is almost impossible to find a feasible policy reform that achieves a Pareto improvement. They develop a static model inhabited by individuals with heterogeneities in both innate ability and parental income, in which the government splits its resources between compulsory primary education and optional higher education. Hidalgo-Hidalgo and Iturbe-Ormaetxe (2012) introduce alternative measures of efficiency, and analyze the effects of policy reforms on the measures of efficiency and equity. As will be clarified soon, we consider the determination of an education policy through majority voting in a dynamic model.

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allocate some resources to higher education. The policy implemented in the political equilibrium depends on the median level of human capital. Our model generates various dynamics of human capital accumulation and steady state income inequalities. As long as the human capital of the median voter is below some threshold, the income tax rate, which determines budget size of education, is low and all resources are provided for primary education. In some cases, the median voter accumulates sufficient human capital through primary education, and, as the economy develops, a higher tax rate (i.e., a larger budget for education) and a positive allocation for higher education are realized. In other cases, however, only the initial income distribution matters for the policy implementation. Although the emergence of higher education unambiguously promotes human capital accumulation and increases the aggregate income level, it may lead to income inequality because the poor are excluded from higher education. This paper is related to the literature on the political economy of education and its macroeconomic implications.2 In our model, the tax rate is low and higher education is not realized if the median human capital (income) is below some threshold. There are several reasons why the median income may be low, but one of the main possibilities is large income inequality. If this is the case, our paper is also related to studies on income inequality and growth. Saint-Paul and Verdier (1993) develop a model in which income redistribution is in the form of public education. It is well known that, in majority voting models, large income inequality causes a large income redistribution. On the basis of the redistributive character of public education, SaintPaul and Verdier (1993) show that in majority voting, large income inequality creates strong support for public education, thereby promoting human capital accumulation and economic growth. However, the model prediction is not supported by empirical studies (e.g., Easterly 2001, Easterly 2007).3 We regard public education differently from Saint-Paul and Verdier (1993). The private return from public education is not equal among individuals but dependent on their inherited human capital. Moreover, higher education requires private spending. Thus, public education does not necessarily have a redistributive effect. In our model, if large income inequality makes the majority of individuals poor, it may have adverse effects on growth. Fernandez and Rogerson (1995) and Naito and Nishida (2012) also recognize that public support for education involves redistribution from the poor to the rich. In a static setting, Fernandez and Rogerson (1995) investigate majority voting on education subsidies. In their model, there is a single education opportunity, and individuals can obtain education if they pay a fixed cost. The government uses a proportional income tax scheme to provide subsidies that partially cover the fixed cost to individuals who take education. However, even with the subsidies, poor individuals may not be able to afford education because the credit markets are imperfect. Fernandez and Rogerson (1995) prove that there exists a majority voting equilibrium in which only rich and middle income individuals receive subsidies and obtain education although the poor 2 Notable examples include Glomm and Ravikumar (1992), Saint-Paul and Verdier (1993), Su (2006), and Galor et al. (2009). 3 Galor et al. (2009) present evidence that inequality in land, which is an early and primary form of wealth,

has an adverse impact on education expenditure per child in the United States.

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also pay proportional income taxes. Naito and Nishida (2012) show that taxation to improve the quality of public education is not implemented under majority voting if the majority of individuals are so poor that they cannot afford education and/or the returns from education are low. Although Naito and Nishida (2012) use a model with a similar structure to this paper’s, human capital is accumulated only through a single public education sector and individuals vote for a proportional tax rate that determines the quality of education. Some studies use political mechanisms other than majority voting. Galor et al. (2009) and Su (2006) consider environments in which rich individuals have strong political power. Galor et al. (2009) construct a model in which landowners, who are better endowed with a production factor, prevent public education that promotes human capital formation and growth. The study most closely related to ours is Su (2006). She studies how a public budget allocation between primary and higher education is determined when the top class has dominant political power as in many less-developed countries. In economies where the middle class is poor, the top class implements a policy that allocates more resources to higher education at the expense of primary education. The results in Su (2006) accord with observed cross-country differences of education policies. Because the purpose of this paper is to provide a theory, we follow most of the literature and use majority voting as the political mechanism. However, our model can be immediately extended to study an environment in which only the rich have political power. In a typical case of less-developed economies, where the majority of the population is very poor and only a small portion at the top has high human capital, our model predicts that public funds are allocated to higher education despite the fact that most individuals are excluded from higher education. This result matches the characteristics of the less-developed economies documented by Su (2004, 2006). Thus, our paper is not without empirical value. Nonetheless, our main interest and goal are theoretical. Here, we aim to provide one step to analyze how politically determined public budget size and its allocation rule interact with the dynamic behavior of human capital accumulation. The rest of this paper is organized as follows. Section 2 describes the basic environments of the model. Section 3 characterizes the static equilibrium, and Sect. 4 analyzes the dynamics. Lastly, Sect. 5 concludes the paper.

2 Basic environment We consider an overlapping generations economy in which individuals live for two periods, namely childhood and adulthood. The size of each generation is normalized to one. Every individual belongs to a lineage and has one child in adulthood. Hereafter, an individual who belongs to a lineage indexed by i is simply referred to as individual i, unless it causes confusion. Individual i born in period t derives utility from consumption in her adulthood, cit+1 , and the human capital of her child, h it+1 :4 4 In our setup, the human capital of individuals is equal to their pre-tax income, and we assume that parents

care about their children’s pre-tax income rather than their children’s disposable income. The children’s

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U (cit+1 , h it+1 ) = cit+1 + h it+1 .

(1)

The economy is small open. Individuals can freely lend and borrow at the gross interest rate determined at international financial markets. The gross interest rate is normalized to one. In childhood, individuals make no decisions, but they must receive compulsory primary education supplied by the government. After the completion of primary education, parents decide whether their children should receive higher education. Higher education is also supplied by the government, but in contrast to primary education, requires a private cost. For example, the cost includes the expense of migration to attend college as well as tuition fees. We assume that the size of the cost is fixed and normalized to one.5 In adulthood, individuals make all the economic and political decisions. They either become workers at a firm or school teachers. The firm transforms one unit of human capital into one unit of a final good. The final good market is perfectly competitive and therefore the wage per unit of human capital is one.6 Labor mobility across sectors is perfect. The income earned by workers and teachers is identical, and individuals are indifferent about their occupation, which they choose randomly. This makes the average quantity of human capital equal in each sector. In addition to supplying labor, individuals simultaneously vote for (i) an income tax rate, which determines the overall size of government expenditure, and (ii) an allocation of tax revenue between primary and higher education, which determines the quality of each. After observing the realized tax rate and the allocation rule, they decide whether their children should receive higher education, and then consume all their remaining wealth. Individuals accumulate human capital solely through public education funded by government tax revenue, as in Su (2004, 2006) and others.7 With regard to higher education, considering not only government expenditure to enhance the quality but also subsidies to cover its private costs captures an important aspect of education policies.8 Nonetheless, according to the OECD (2013), the majority of public education expenditure is used to directly improve the quality of higher education. In 2010, direct public expenditure for institutions accounted for 78.3 % of the total public education

Footnote 4 continued disposable income depends on the tax rate realized when the parents leave the economy and their children become adults. This assumption implies that parents do not derive utility from the state after their death. Parents can directly observe the human capital of their children, but they cannot observe the future tax rate and disposable income of their children. 5 Such a fixed cost is traditional in growth and development literature, at least since Galor and Zeira (1993). 6 The wage rate per unit of human capital may depend on the aggregate and average human capital. For

example, if there is a positive externality in production, the wage rate should increase with aggregate and average human capital. In such cases, individuals would prefer a lower tax rate because a higher tax rate would decrease their consumption by more than it would in cases where there is no externality. 7 For example, see Blankenau et al. (2007b), Galor et al. (2009), and Naito (2012). 8 In the Hidalgo-Hidalgo and Iturbe-Ormaetxe (2012) model, the government provides subsidies that par-

tially cover the private costs of higher education as well as improving the quality of primary and higher education.

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expenditure for tertiary education, in the overall average of OECD countries.9 Thus, subsidies for higher education are not as high as direct expenditures on institutions. We consider an environment in which government expenditure influences the quality of higher education rather than the private costs of higher education. Income earned at period t + 1 is taxed at a rate of τt+1 , and the tax revenue is τt+1 h¯ t , where h¯ t is the average (and aggregate) human capital supplied at period t + 1. Keeping a balanced budget, the government allocates the tax revenue to primary and higher education. Denoting the share of government expenditure on primary education by xt+1 , the government budget constraints are given by P G t+1 = xt+1 τt+1 h¯ t ,

(2)

A = (1 − xt+1 )τt+1 h¯ t , G t+1

(3)

P and G A are the government expenditures on primary and higher educawhere G t+1 t+1 tion, respectively. A child’s human capital depends on the quality of education and the level of parental human capital. Specifically, if individual i born in period t + 1 receives only primary education, she accumulates human capital according to P ˆ β ∈ (0, 1), hˆ ≥ 0, )β h it + h, h it+1 = (n t+1

(4)

P is the number of teachers in primary where h it+1 is the human capital output, n t+1 education, and hˆ is the equally endowed innate human capital with which each individual is born.10 The return from education is increasing in her parental human capital and the number of teachers.11 In contrast to primary education, not all individuals take higher education. Consequently, the student-teacher ratio changes endogenously, and there may be a congestion effect in higher education. That is, the greater the number of individuals who receive higher education, the higher the number of students per teacher will be. This congestion effect might adversely affect human capital acquisition in higher education. Although this point is theoretically important, there are some empirical studies that show that the class size has little effect on students’ achievements. Siegfried and Kennedy (1995) find that pedagogical methods of instructors do not change with class 9 Scholarships and other grants to households are 11.4 %. 10 Innate (preschool) human capital may be positively correlated with parental human capital because of

parental care at the preschool age. Suppose, instead, that the innate human capital of individual i is given ˆ where δ > 0. Using this specification of innate ability would not alter our qualitative by hˆ it = δh it + h, results. 11 We assume that human capital production exhibits the constant marginal product of parental human capital (see (4) and (5)). In the case where human capital production exhibits the decreasing marginal product of parental human capital, both primary and higher education have stronger redistributive effects from richer to poorer individuals. In such a case, the most preferred tax rates are represented as functions of individual human capital, and poorer individuals may have stronger incentives to raise the tax rate and obtain higher education. We question the redistributive effects of education, as described in Introduction, and our interest is the analysis within an environment in which the redistributive effects of education are sufficiently weak.

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size, and Kennedy and Siegfried (1997) show that class size does not affect academic achievement in undergraduate economics classes. The class size in Kennedy and Siegfried (1997) ranges from 14 to 109, and they report that their results are robust to various specifications on the way the class size is entered. In addition, Hill (1998) and Machado and Vera-Hernandez (2008) do not find evidence that class size has significant effects on educational outcomes in undergraduate classes. Building on these observations, we assume that class size does not matter in higher education. In other words, there is no congestion effect.12 Accordingly, we assume that if individual i receives higher education, she accumulates human capital according to P A β ˆ )β h it + (n t+1 ) h it + h, h it+1 = (n t+1

(5)

A is the number of teachers in higher education.13 where n t+1 Since the average quantity of human capital supplied at period t + 1 is h¯ t in every P = G P /h¯ sector, the average wage payment to teachers is also h¯ t . This leads to n t+1 t+1 t A A ¯ and n t+1 = G t+1 /h t , the same result as in Naito (2012). From (2), (3), (4), and (5), the human capital of individual i is given by β

β

h it+1 = xt+1 τt+1 h it + hˆ ≡ F P (xt+1 , τt+1 , h it )

(6)

if she receives only primary education, and it is given by β β h it+1 = [xt+1 + (1 − xt+1 )β ]τt+1 h it + hˆ ≡ F A (xt+1 , τt+1 , h it )

(7)

if she receives higher education.

3 Static equilibrium analysis Before finding the combination of an income tax rate and an allocation rule that each individual votes for, we discuss how an individual decides whether to invest in higher education under a given tax rate and an allocation rule. If individual i has her child obtain higher education, (1) and (7) give her utility, V A (xt+1 , τt+1 , h it ), as 12 In some disciplines in which laboratory teaching in small classes is necessary, a congestion effect might emerge (e.g., in some engineering and medical classes). We abstract the congestion effect in such disaggregated disciplines, and assume that class size does not matter in considering aggregate human capital production. Although it is not necessary to consider congestion effects in primary education in our model, there is evidence that class size does not matter in primary education either. See Hoxby (2000) and Leuven et al. (2008). 13 In Glomm and Ravikumar (1992), Su (2004, 2006), and Naito (2012), inherited human capital is the

only source of heterogeneity among individuals and has a positive effect on their educational outcome, as in our model specification. Blankenau (2005) and Blankenau et al. (2007b) also assume that parental human capital stock has a positive impact on the human capital acquisition of their children in school. On the empirical side, Hanushek (1986) points out that children of parents with higher human capital benefit more from school. Solon (1992), Zimmerman (1992), and Dearden et al. (1997) show empirically that the earnings of parents and their children are positively correlated, and that the degree of intergenerational mobility is limited.

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V A (xt+1 , τt+1 , h it ) = [(1 − τt+1 )h it − 1] + F A (xt+1 , τt+1 , h it ).

(8)

The first term is her consumption in adulthood, and the second term is the human capital of her child. If individual i decides not to have her child receive higher education, (1) and (6) give her utility, V P (xt+1 , τt+1 , h it ), as V P (xt+1 , τt+1 , h it ) = (1 − τt+1 )h it + F P (xt+1 , τt+1 , h it ).

(9)

Individual i is willing to have her child receive higher education if and only if V A (xt+1 , τt+1 , h it ) ≥ V P (xt+1 , τt+1 , h it ), which is equivalent to h it ≥

1 β

(1 − xt+1 )β τt+1

≡ H (xt+1 , τt+1 ).

(10)

Here, H (xt+1 , τt+1 ) is increasing in xt+1 and decreasing in τt+1 because a larger share of government expenditure on higher education (i.e., a lower value of xt+1 ) and a higher income tax rate raise the productivity of higher education and make more individuals willing to invest in higher education for their children. To understand (10) in terms of xt+1 , it is useful to define x(τt+1 , h it ) by H (x(τt+1 , h it ), τt+1 ) = h it . Individual i is willing to have her child receive higher education if and only if 0 ≤ xt+1 ≤ x(τt+1 , h it ). The level of the income tax rate, τt+1 , and the share of government expenditure on primary education, xt+1 , are both determined under majority voting. We begin by identifying the welfare-maximizing share of each individual for a given income tax rate. The value of V P (xt+1 , τt+1 , h it ) is maximized at xt+1 = 1 because it is monotonically increasing in xt+1 . To describe the maximum of V A (xt+1 , τt+1 , h it ), we define x ∗ by (11) x ∗ ≡ arg max V A (xt+1 , τt+1 , h it ) = 1/2. xt+1 ∈[0,1]

The maximizers of V P (xt+1 , τt+1 , h it ) and V A (xt+1 , τt+1 , h it ) are both independent of τt+1 . Individual i prefers xt+1 = x ∗ to xt+1 = 1 if V A (x ∗ , τt+1 , h it ) ≥ V P (1, τt+1 , h it ). Simple calculations show that this is equivalent to h it ≥ H˜ (τt+1 ) ≡

1 (21−β

β

− 1)τt+1

.

(12)

This equivalence relation means that the welfare of individual i is maximized at xt+1 = x ∗ if h it ≥ H˜ (τt+1 ), while it is maximized at xt+1 = 1 if h it < H˜ (τt+1 ). This result is similar to that of Su (2006): individuals with low human capital prefer to allocate all the tax revenue to primary education, while individuals with high human capital prefer a balanced budget allocation. There are two types of individuals who prefer xt+1 = 1 to xt+1 = x ∗ . If x ∗ > x(τt+1 , h it ), or equivalently h it < H (x ∗ , τt+1 ), individual i obviously prefers xt+1 = 1 because she decides not to have her child obtain higher education. However, even if x ∗ ≤ x(τt+1 , h it ), V A (x ∗ , τt+1 , h it ) < V P (1, τt+1 , h it ) may still hold. In this case,

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although individual i is willing to have her child receive higher education if xt+1 = x ∗ is realized, she prefers to make xt+1 = 1 rather than make xt+1 = x ∗ and invest in higher education for her child. The following lemma summarizes the obtained results. Lemma 1 For a given income tax rate τt+1 ∈ [0, 1], the welfare of individual i is maximized at xt+1 = x ∗ if h it ≥ H˜ (τt+1 ), while it is maximized at xt+1 = 1 if h it < H˜ (τt+1 ). Lemma 1 states that the budget share, xt+1 , preferred by individual i is either 1 or x ∗ , and there are no other possibilities. For any τt+1 , since V P (xt+1 , τt+1 , h it ) is maximized at xt+1 = 1 and V A (xt+1 , τt+1 , h it ) is maximized at xt+1 = x ∗ , all we need to do is to compare the maxima of V P (1, τt+1 , h it ) and V A (x ∗ , τt+1 , h it ) with respect to τt+1 in order to identify the most preferred set of xt+1 and τt+1 . Let us define τ P and τ A by 1

τ P ≡ arg max V P (1, τt+1 , h it ) = β 1−β ,

(13)

0≤τt+1 ≤1

1

τ A ≡ arg max V A (x ∗ , τt+1 , h it ) = 2β 1−β < 1.

(14)

0≤τt+1 ≤1

Comparing V A (x ∗ , τ A , h it ) with V P (1, τ P , h it ) gives the following equivalence relation: V A (x ∗ , τ A , h it ) ≥ V P (1, τ P , h it )

⇔

h it ≥

1 β

β 1−β

(1 − β)

≡ Hˆ .

(15)

Furthermore, simple calculations yield that H˜ (τ A ) < Hˆ < H˜ (τ P ). If h it < Hˆ , individual i prefers (xt+1 , τt+1 ) = (1, τ P ). If h it ≥ Hˆ , individual i prefers (xt+1 , τt+1 ) = (x ∗ , τ A ). The following lemma summarizes the obtained results. Lemma 2 Individual i prefers (xt+1 , τt+1 ) = (1, τ P ) if h it < Hˆ , while she prefers (xt+1 , τt+1 ) = (x ∗ , τ A ) if h it ≥ Hˆ . It is easy to show that the individual with median income is the decisive voter. This result follows immediately from Lemma 2. Let h mt denote the human capital of the individual with median income. If h mt < Hˆ , then the individual with h mt prefers (xt+1 , τt+1 ) = (1, τ P ). Since individuals whose human capital is less than h mt comprise 50 % of the total population, and they all prefer (xt+1 , τt+1 ) = (1, τ P ), this policy is chosen under majority voting. Similarly, if h mt ≥ Hˆ , then the individual with h mt prefers (xt+1 , τt+1 ) = (x ∗ , τ A ). Since individuals whose human capital is greater than h mt have the same preference and constitute 50 % of the total population, (xt+1 , τt+1 ) = (x ∗ , τ A ) is chosen under majority voting. Proposition 1 Under majority voting, (xt+1 , τt+1 ) = (1, τ P ) is implemented if h mt < Hˆ , while (xt+1 , τt+1 ) = (x ∗ , τ A ) is implemented if h mt ≥ Hˆ .

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4 Dynamic analysis This section analyzes the dynamics of human capital in the political equilibrium described in Proposition 1. Recall that the realization of higher education relies only on whether h mt exceeds Hˆ . In the case where (xt+1 , τt+1 ) = (1, τ P ) is realized, the human capital of all lineages is accumulated according to β

ˆ h it+1 = F P (1, τ P , h it ) = β 1−β h it + h.

(16)

In the case where (xt+1 , τt+1 ) = (x ∗ , τ A ) is realized, the human capital of lineage i with h it < H (x ∗ , τ A ) is augmented only through primary education. It is accumulated according to β ˆ (17) h it+1 = F P (x ∗ , τ A , h it ) = β 1−β h it + h. The human capital of lineage i with h it ≥ H (x ∗ , τ A ) is augmented by both primary and higher education: β

ˆ h it+1 = F A (x ∗ , τ A , h it ) = 2β 1−β h it + h.

(18)

Note that regardless of whether (xt+1 , τt+1 ) = (1, τ P ) or (xt+1 , τt+1 ) = (x ∗ , τ A ), the human capital of lineage i with h it < H (x ∗ , τ A ) accumulates in the same fashion because F P (1, τ P , h it ) = F P (x ∗ , τ A , h it ). Taking the income tax rate as given, a decrease in xt+1 lowers the quality of primary education. However, in the political equilibrium, the level of the income tax rate increases from τ P to τ A in parallel with the emergence of higher education. Consequently the quality of primary education is maintained at a constant level. Simple calculations show that the slope of F A (x ∗ , τ A , h it ) is less than one if and only if β > 1/2, and we focus on the human capital dynamics in this case. Let h P and h A denote the intersection of F P (1, τ P , h it ) and the 45-degree line and that of F A (x ∗ , τ A , h it ) and the 45-degree line, respectively: hP ≡

hˆ 1−β

β 1−β

,

hA ≡

hˆ β

1 − 2β 1−β

.

There are two cases that need to be considered: (i) Hˆ ≤ h P and (ii) Hˆ > h P . First, we consider the case where Hˆ ≤ h P . Simple calculations give the following equivalence relation: β

Hˆ ≤ h

P

⇔

hˆ ≥

1 − β 1−β β

β 1−β (1 − β)

≡ ψ1 (β).

(19)

Here, ψ1 is increasing in β for all β ∈ (1/2, 1), and, thus, economies with a high value of hˆ and/or a low value of β satisfy the condition Hˆ ≤ h P . Figure 1a sketches the dynamics of h mt . If h m0 < Hˆ , higher education is not realized at first and, consequently, all individuals accumulate their human capital according to F P (1, τ P , h it ).

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hmt+1

45◦ F A (x∗ , τ A , hmt )

(a)

hit+1

45◦ F A (x∗ , τ A , hit )

(b)

F P (1, τ P , hit ) = F P (x∗ , τ A , hit )

F P (1, τ P , hmt )

0

ˆ H

hP

hmt+1

hA

hmt

45◦

(c)

F A (x∗ , τ A , hmt )

0 h ˆ H(x∗ , τ A ) P it h hit+1

45◦

(d)

F A (x∗ , τ A , hit )

F P (1, τ P , hmt )

0

ˆ hP H

hA

hmt

hit

hA

F P (1, τ P , hit )

0

hi0

hP

hA H(x∗ , τ A )

hit

Fig. 1 Dynamics of human capital

However, the human capital of lineage m eventually exceeds Hˆ at some time period, say tˆ, following which higher education is always realized and h mt converges to h A . The human capital dynamics of the other lineages can also be easily analyzed. As shown in Fig. 1b, after tˆ, the human capital of lineage i with h i tˆ < H (x ∗ , τ A ) accumulates only through primary education for some periods. However, because Hˆ < h P , the human capital certainly exceeds H (x ∗ , τ A ) at some period, and thereafter, it is augmented by both primary and higher education.14 Thus, human capital of the lineage converges to h A . Individuals in lineage i with h i tˆ ≥ H (x ∗ , τ A ) always have their children receive higher education after period tˆ, and the human capital in the lineage also converges to h A . In this case, the human capital in all lineages converges to h A . Note that if h m0 ≥ Hˆ , then tˆ = 0, and the above analysis remains intact. Second, we consider the case where Hˆ > h P , that is, hˆ < ψ1 (β). In this case, the realization of higher education depends on the value of h m0 . Figure 1c depicts the dynamics of h mt . If h m0 < Hˆ , the human capital of lineage m converges to h P . Higher education is never realized, and the human capital of all lineages converges to h P . When a low value of h m0 is caused by large initial inequality, an implication of this result is that the large inequality prevents the implementation of institutions promoting human capital and harms growth. When low h m0 is caused by economywide underdevelopment, underdevelopment reproduces itself. If, on the other hand, h m0 ≥ Hˆ , higher education is realized at the beginning, and h mt converges to h A . Individuals in lineage i with h i0 ≥ H (x ∗ , τ A ) always receive higher education, and 14 Since H (x ∗ , τ A ) < Hˆ , Hˆ < h P implies H (x ∗ , τ A ) < h P .

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the human capital of the lineage converges to h A . For individuals in lineage i with h i0 < H (x ∗ , τ A ), the dynamics of human capital depend on the values of H (x ∗ , τ A ) and h P . Simple calculations give the following equivalence relation: β

∗

H (x , τ ) ≤ h A

P

⇔

1 − β 1−β hˆ ≥ ≡ ψ2 (β) < ψ1 (β). β 21−β β 1−β

(20)

Here, ψ2 is increasing in β for all β ∈ (1/2, 1), and, thus, economies with a high value of hˆ and/or a low value of β satisfy the condition H (x ∗ , τ A ) ≤ h P . If H (x ∗ , τ A ) ≤ h P , all individuals in the lineage eventually receive higher education, and the human capital of the lineage converges to h A . In this case, there is no income inequality in the longrun. In contrast, if h P < H (x ∗ , τ A ) (i.e., hˆ < ψ2 (β)), individuals in lineage i with h i0 < H (x ∗ , τ A ) never receive higher education, and the human capital of the lineage converges to h P as shown in Fig. 1d. Income inequality expands in the process of economic growth and there is substantial income inequality in the long-run.15,16 Although we have used majority voting as the political mechanism, the analysis can be immediately extended to the case where only the rich have political power, as in Su (2006). In the case where Hˆ > h P and h P < H (x ∗ , τ A ), let us consider the following characteristic of less-developed economies: most individuals are very poor, but only a small portion of individuals are so rich that their human capital exceeds Hˆ . Then, (xt+1 , τt+1 ) = (x ∗ , τ A ) is politically implemented. Public resources are allocated to finance higher education although most individuals are poor and never obtain higher education. To some extent, this result is consistent with the characteristics of the public education policies in less-developed economies provided by Su (2004, 2006). However, the insight from this result is different from that of Su (2006). Su (2006) assumes congestion effects in higher education, and that only individuals who acquire some threshold level of human capital in primary education can benefit from higher education. Rich individuals with political power may prefer decreasing public expenditure on primary education in order to lower its quality, disqualify poorer individuals from receiving higher education, and receive exclusive benefit from higher education. In contrast to Su (2006), we do not assume congestion effects, and rich individuals do not want to exclude poor individuals from higher education. The focus of our analysis is the effect of private costs of higher education rather than congestion effects. 15 The net income of lineage i with h ≥ H (x ∗ , τ A ) converges to (1 − τ A )h A , and the consumption i0 level converges to c A ≡ (1 − τ A )h A − 1. In contrast, the net income of lineage i with h i0 < H (x ∗ , τ A ) A P converges to (1 − τ )h , and the consumption level converges to c P ≡ (1 − τ A )h P . The pre-tax income inequality, h A −h P , is reduced by taxation, and the inequality measured by consumption, c A −c P , becomes even smaller because of the private cost of higher education. In particular, c A can be smaller than c P if hˆ is sufficiently small and/or β is sufficiently large. In this case, h A − h P is relatively small, and the taxation and private cost of higher education cause c P > c A . 16 This theoretical consequence may fit the cases of countries such as Mexico and Peru. According to World Development Indicators, since 1994, the Gini coefficients of Mexico have been around 50, and those of Peru have ranged between 45 and 55. Possibly as a result of this high income inequality, there are many individuals who do not receive higher education: the gross enrolment ratios in tertiary education in Mexico and Peru are about 30 and 40 %, respectively. This may be the cause of persistent income inequality in these two countries.

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The clear-cut results of the model analysis depend on the specifications of the human capital production functions (4) and (5). In particular, the congestion effect in higher education is abstracted. Although we discussed several empirical studies that can justify this simplification, the congestion effect may be important in some disciplines such as engineering in which teaching in small laboratory classes is indispensable. Moreover, as more people attend college in developing countries, politico-economic studies, such as Su (2006), that consider the congestion effect in higher education might become increasingly important.

5 Conclusion The government plays an important role in the funding of both primary and higher education. This paper proposes a model in which both the size of the public education budget and its allocation across the two education sectors are determined through majority voting, and analyzes the interaction between politically implemented education policies and human capital accumulation. In our model, the return from each education sector is positively correlated with the human capital level inherited from parents. A tax increase to finance higher education is not politically supported until the majority of individuals accumulate sufficient human capital. In some cases, higher education starts to be realized as the majority of individuals accumulate sufficient human capital through primary education. However, in other cases, only the initial distribution of human capital matters. Although the implementation of higher education accelerates aggregate human capital accumulation, it may generate large and persistent income inequality in the long-run because the poor do not receive higher education. We have assumed that individuals must cover private costs to obtain higher education supplied by the government. The logic of our model can be applied to other public policies by considering a situation in which access to publicly provided services requires private spending. For example, the government may consider two policies: (i) public support to enhance productivity in high-tech industries that employ skilled workers, and (ii) a lump-sum transfer of tax revenue to all individuals. Rich individuals who can cover education costs to become skilled workers would prefer public support for high-tech industries, while poorer individuals would prefer the lump-sum transfer. The implication of such a situation for economic growth would be a topic for future research. Acknowledgements We are grateful to the editor, Giacomo Corneo, and two anonymous referees for their helpful comments and suggestions. We also thank Akihisa Shibata, Shiro Kuwahara, Carlos Bethencourt, and conference participants of PET 12. Any remaining errors are ours. This research is financially supported by the Global COE program of Osaka University, the Central Research Institute of Fukuoka University (No. 144002), and the Joint Research Program of KIER.

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