Social status, compulsory education, and growth

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Economic Modelling journal homepage: www.elsevier.com/locate/econmod

Social status, compulsory education, and growth Chia-Hui Lu Department of Economics, National Taipei University, 151, University Rd., San-Shia, 23741 New Taipei, Taiwan

A R T I C L E I N F O

A BS T RAC T

JEL classification: E24 I25 O41

In the long run, if agents pay more attention to social status, the time allocated to higher education and economic growth both increase. However, if the education provided by the government is less efficient than that provided by the private sector, a longer period of compulsory education not only decreases the time allocated to higher education, but also reduces the total time spent in education. Therefore, economic growth declines because of the greater amount of inefficient education provided by the government.

Keywords: Social status Compulsory education Economic growth

1. Introduction Undoubtedly, education (the accumulation of human capital) is very important for a country to enhance economic growth. Public education, especially at the primary and high school levels, enjoys wide political support in many countries. Some studies even highlight the potential benefits of government intervention in the form of delivering more human capital accumulation or increasing equality. However, these are essentially arguments for the public financing of education; that is, the government does not necessarily provide education directly. In most developed and developing countries, governments not only provide schooling, but also compel the citizenry to receive education as so-called compulsory education. In most Western countries, Japan, and China, governments oblige their citizens to receive at least 9 years of compulsory education. In Taiwan, citizens must receive 12 years of compulsory education.1 In recent decades, more and more countries have increased the years of compulsory education. For example, the duration of compulsory education in Taiwan was increased from 9 years to 12 years in 2014. Murtin and Viarengo (2011, Table 1) illustrated the extension of compulsory schooling in 15 Western European countries over the period 1950–2000. Those living in a country with compulsory education will be educated for at least the mandatory number of years stipulated by the government. However, the evidence shows that people usually pursue education that exceeds the length of compulsory education (see (OECD, 2014)). This means that compulsory education encourages people to develop social status norms related to their education level. Therefore, compulsory education contributes to growth, not only

through delivering human capital accumulation, but also by forming social status norms. The latter stimulate people to pursue higher education, which facilitates their further accumulation of human capital and so increases economic growth. However, these advantages do not necessarily imply that governments need to increase the duration of compulsory education continuously. In this paper, we set up a two-sector endogenous growth model in which households pursue social status that is based on the agent's relative education level. The purpose of this paper is to investigate the connection between social status, compulsory education, and economic growth. We are particularly interested in the impact of an excessively long period of compulsory education on economic growth especially when the education provided by the government is less efficient than that provided by the private sector. In addition, we also check the effects of the government's educational subsidies, the cost of education for compulsory or higher education, and the intensity of the agent's preference for social status. The structure of the paper is as follows. In Section 2, we discuss the related literature. In Section 3, we construct a benchmark model and analyze the individual's optimization. In Section 4, we analyze the comparative statics and provide numerical exercises. In Section 5, we consider some robustness checks concerning different social status, dissimilar educational efficiencies, and various educational costs. We also extend the model to include a labor–leisure trade-off in agents’ preferences and symmetrical technologies in both production sectors. Finally, we offer some concluding remarks in Section 6.

E-mail addresses: [email protected], [email protected]. It is worth noting that while the government provides most schools for compulsory education, not all institutions are public schools. In this paper, to simplify the analysis, we assume that all compulsory instruction schools are provided by the government. 1

http://dx.doi.org/10.1016/j.econmod.2017.08.013 Received 30 March 2017; Received in revised form 20 June 2017; Accepted 15 August 2017 0264-9993/ © 2017 Elsevier B.V. All rights reserved.

Please cite this article as: Lu, C.-H., Economic Modelling (2017), http://dx.doi.org/10.1016/j.econmod.2017.08.013

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growth. Different from their model in which public spending, as externalities, directly improves the productivity of schooling efforts, in this paper, we assume that government spending has no spillover effect; and our results can support their prediction. Moreover, Tournemaine and Tsoukis (2015) set up an endogenous growth model with heterogenous status-motivation individuals who decide whether to attend a publicly funded education regime or a privately funded one, and obtained an inverted-U shaped relationship between growth and the size of the public education sector. In a departure from Tournemaine and Tsoukis (2015) who focus on the same level of education but with different funding, we use the setting of the representative agent, and focus on the effects of compulsory education and its related social status on higher education (different level of education) and growth. In addition, the evidence we mentioned in Introduction shows that compulsory education encourages people to develop social status norms related to their education level, while existing studies have not considered that directly. For example, social status is based on the agent's relative human capital level in Fershtman et al. (1996), and is based on the agent's relative consumption level in Tournemaine and Tsoukis (2015). Fershtman et al. (1996) further mentioned that education appears to be the more important determinant of social status. Therefore, in this paper, we set that social status is directly based on the agent's relative education level. It is worth noting that achieving some kind of social status stands for one's relative position in society. When social status are prioritized, the agent's decisions are not made through market, as in Cole et al. (1992), who used higher income to represent higher status.3 Such situations are similar to those where externalities exist in agents' preferences. Akerlof (1980),Cole et al. (1992) and Fukumura (2017) revealed that models including social customs or nonmarket decisions may inherently have multiple equilibria. However, in our paper, we prove that a unique equilibrium exists in the long run.

2. Literature review Much of the literature investigates the effect of public education on growth. For example, Glomm and Ravikumar (1992) set up an overlapping generations (hereafter, OG) model with public and private education, and human capital investment through formal schooling is the engine of growth. They found that public education reduces income inequality more quickly than private education, whereas private education yields higher per capita income. Glomm and Ravikumar (1997) considered infrastructure and public education in an OG model and investigated the effects of those expenditures on long-run growth. Glomm and Ravikumar (1998) extended Lucas (1990) model to include government educational spending. They found that the growth effects of changes in capital income tax rates are still negligible as those in Lucas (1990), and simultaneous changes in taxes and spending on education have modest effects on growth. Besides, Blankenau and Simpson (2004) constructed an OG growth model with private and public investment in human capital accumulation and found that the response of growth to public education expenditures depends on the level of government spending, the tax structure and the parameters of production technologies. Blankenau et al. (2007) developed an OG growth model with public education and found no significant growth effects of public education expenditures when crowding-out effects are not properly taken into consideration. Blankenau and Camera (2009) used a two-period-lived model with heterogeneous skilled labor, and obtained that increased tuition subsidies may increase school enrolment but lower the incentives for student achievement, hence the skill level. In addition, Dissou et al. (2016) developed a multisector endogenous growth model and considered several fiscal instruments to finance the increase in government spending. They found that the non-distortionary financing method provides the highest increase in output. Moreover, some studies mentioned that public education contributes to growth not only by building human capital but also by instilling common norms that increase social cohesion. For example, Gradstein (2000) built a model whereby the household has a two-period life horizon and found that the system of public education is likely to generate a more rapid accumulation of human capital, to bolster economic growth, and to constitute a preferred choice for the majority of voters. Gradstein and Justman (2000) used a two-period economy with different social groups to compare the results under private and public schooling systems, and tried to explain why education is commonly publicly administered and financed. Fukumura (2017) set up a model utilizing the keeping up with the Joneses effect regarding schooling decisions and obtained multiple equilibria, which can explain the difference between the two groups of countries. However, the above-mentioned papers focused on the effect of public education, and did not consider the impact of enhancing the duration of compulsory education. Furthermore, in this paper, we find that if agents are concerned with the social status whereby their relative education level is a marker of social status, they will accumulate greater human capital, i.e., pursue higher education, and thus the growth rate of the economy rises. This result is consistent with the case where individuals are identical in Fershtman et al. (1996).2 However, longer compulsory education may have a negative effect on economic growth when the education provided by the government is less efficient than that provided by the private sector. Other related papers like Basu and Bhattarai (2012) who examined the effects of public educational spending on the long-run growth rate and the returns to schooling, predicted that a greater government involvement in education will lower schooling efforts and economic

3. The model This section builds the basic analytical framework. This framework draws on the Lucas (1988) two-sector endogenous growth model extended to include social status which is based on the agent's relative education level. People who spend more time on education will have a higher academic degree. When most people in the economy have a higher degree of education, firms will hire highly educated workers even if such high degree is not necessary to the position. In addition, people will pursue higher degree when high educational background is a common social phenomenon. As a result, the relative education level appears to represent social status. The representative agent is endowed with one unit of time. At an instant in time, a fraction e of the agent's time is spent in education and the remaining fraction 1 − e is devoted to working. The total time in education e includes the time allocated to compulsory education (ec), which is provided by the government and is mandatory, and the time allocated to higher or noncompulsory education (eh), which is decided by the agent. Thus, e = ec + eh . An agent's lifetime utility is as follows: ∞

U=

⎞

⎛

∫t=0 u⎜⎝c, ee ⎟⎠exp( − ρt )dt, e

where u(c , e ) = ln (c ) + χ time preference rate.

(1) 1− γ

(e / e ) 1−γ

, c is consumption, and ρ > 0 is the

3 The discussion about social status and economic performance can be found in Fershtman and Weiss (1993) and Weiss and Fershtman (1998). Applications concerning social status include those relating to endogenous fertility ((Palivos, 2001) and (Munshi and Myaux, 2006), money as a medium of exchange (Araujo, 2004), the spillover effects of human capital (de la Croix, 2001), and inequality ((Corneo and Jeanne, 1999) and (Kawamoto, 2009)), among others.

2 If individuals differ with respect to both their wealth and learning ability, Fershtman et al. (1996) found that the status motive may induce an inefficient allocation of talent and may have the opposite effect on economic growth.

2

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Furthermore, the government subsidizes a fraction of the agent's educational cost, C (eh ). Thus, S (eh ) = sη1y eh with s ∈ [0, 1].

In (1) we use a conventional additively separable utility function between consumption and social status with a unit intertemporal elasticity of substitution (hereafter, IES) for consumption. According to King and Rebelo (1999), the IES for consumption must be unity in order to be consistent with a balanced growth path (hereafter, BGP) in this separable utility. The variable e is the average education level in society, which is taken as given by the agent. The parameter χ > 0 is the intensity of the agent preference for social status relative to consumption. The agent relative education level positively influences his or her utility and induces status-seeking behavior. Thus, the parameter χ > 0 is also the strength of the agent preference for higher status. The average education level in society (the educational externality) may also affect the marginal utility of an individual agent's education level. When the average education level rises, the marginal utility of individual agents’ own education levels may increase (keeping up with the Joneses, or KUJ) or decrease (running away from the Joneses, or RAJ). The parameter γ denotes the elasticity of the marginal utility of relative education level. If the elasticity is larger (less) than unity, then the utility exhibits the KUJ (RAJ) pattern.4 The economy is composed of two production sectors: the goods sector y and the education sector x. Following Lucas (1988), we assume that the two sectors have general technologies in which the goods sector uses both physical and human capital as inputs and the education sector uses only human capital as an input. For simplicity, we employ the Cobb–Douglas form for each:

3.1. Optimization The agent's optimization problem is to maximize lifetime utility (1) by choosing between consumption, higher education, and investment in the goods and education sectors, all of which are subject to the constraints in (2a)–(2b) and the given initial stocks of physical and human capital, k(0) and h(0). Let λ and μ be the costate variables (i.e., the shadow prices) associated with k and h, respectively. The necessary conditions are:

1 = λ, c

(3a)

⎛e⎞ 1 (1 − α )y χ⎜ ⎟ −λ − λ(1 − s )η1y + μBh = 0, ⎝e ⎠ e 1 − ec − eh

(3b)

−γ

−λ

αy = λ˙ − ρλ , k

(3c)

−λ

(1 − α )y − μB(ec + eh ) = μ˙ − ρμ. h

(3d)

We also have the following transversality conditions:

lim e−ρt λ(t )k (t ) = 0,

(3e)

t →∞

lim e−ρt μ(t )h(t ) = 0.

y = Ak α [(1 − e)h]1− α , x = Beh ,

(3f)

t →∞

The conditions above are standard: (3a) equates the marginal utility of consumption to the shadow price of physical capital, while (3b) indicates the trade-offs between education and working. An increase in the time allocated to higher education increases its marginal utility and the accumulation of human capital, while it reduces the goods output and requires the agent to pay the additional cost of higher education. Finally, (3c) and (3d) are the Euler equations, and (3e) and (3f) are the usual transversality or “no-Ponzi game” conditions on physical and human capital, respectively.

in which k is physical capital and h is human capital, with given initial values k(0) and h(0). We assume both technologies exhibit constant returns to scale for consistency with perpetual growth. The parameter α ∈ (0, 1) is the income share of physical capital in the goods sector, and A and B are the technology coefficients, or factor productivities, in the goods and education sectors, respectively. While the goods output is used either for the formation of physical capital or for expenses, which include consumption and educational spending, the education output can only serve for the accumulation of human capital. Given that the government provides compulsory education, the agent only needs to pay the cost of any higher education pursued. In addition, in order to encourage more people to engage in accumulating human capital, we assume that the government subsidizes those who devote their time to higher education.5 For simplicity, we assume that there is no depreciation of physical and human capital. The laws of motion are as follows:

k˙ = Ak α [(1 − ec − eh )h]1− α − c − C (eh ) + S (eh ) − T ,

(2a)

h˙ = B(ec + eh )h ,

(2b)

3.2. The government To simplify the analysis, we assume that compulsory education is directly provided by the government. The government finances the cost of compulsory education and subsidizes agents pursuing higher education by levying lump-sum taxes. To simplify the model, we assume that the government has no other public expenditure and no distortionary tax, and that the cost of compulsory education, C (ec ), is similar to that of higher education. Thus, we set C (ec ) = η2yec with η2 > 0 . The government's flow budget constraint is as follows:

T = S (eh ) + C (ec ) = sη1yeh + η2yec .

(4)

6

where C (eh ) is the cost of higher education, S (eh ) is the government's subsidy for higher education, and Tt is lump-sum taxes. To simplify the analysis, we assume that the cost of higher education is linear in terms of eh. Moreover, the unit cost of higher education should be the same for all agents and not depend on the agent's personal income. In order to be consistent with a perpetual growth setup, the cost of higher education is as follows: C (eh ) = η1y eh , in which y is the economy-wide average goods output, and η1 > 0 .

3.3. Equilibrium The equilibrium, with e (t ) = e(t ) and y (t ) = y(t ), defines the time paths of {k (t ), h(t ), c(t ), eh(t ), λ(t ), and μ(t )}, which satisfy (2a), (2b) and (3a)–(3d), along with (4). We can rearrange the equilibrium conditions as follows. First, (3a) and (3c) imply that

y c˙ λ˙ = − = α − ρ, c λ k

4

Related studies include (Gali, 1994; Dupor and Liu, 2003; Long and Shimomura, 2004; Kawamoto, 2009), and so forth. 5 In reality, some higher education is provided directly by the government (public schools) and private higher education institutions will receive government subsidies. For example, in Taiwan, nearly half of the universities are publicly funded, and the Taiwan Ministry of Education also subsidizes private universities. 6 In some countries access to higher education is restricted by test scores etc. Those related costs are also included in C (eh ) .

y k

(5a) h 1− α eh )1− α ( k )

is a function of eh and in which = A(1 − ec − Next, (3b) yields the following relationship:

⎧k λ =⎨ μ ⎩c ⎪ ⎪

3

−1 ⎤ y⎡ 1 − α χ ⎫ kh ⎬ B + (1 − s )η1⎥ − . ⎢ k ⎣ 1 − ec − eh ec + eh ⎭ ck ⎦

h . k

⎪ ⎪

(5b)

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⎡ (1 − α )ρ B(ec + eh ) + ρ + (1 − s )η1ρ ⎢ (1 − η1eh − η2ec )[B(ec + eh ) + ρ] − αB(ec + eh ) ⎣ 1 − ec − eh ⎤ ρχ − (1 − α )B⎥ = . ec + eh ⎦ Proposition 1. Define ρχ ec

M≡

(7c) ⎡ (1 − α )ρ Bec + ρ ⎢ (1 − η2ec)(Bec + ρ) − αBec ⎣ 1 − ec

⎤ + (1 − s )η1ρ − (1 − α )B⎥ . ⎦

> M . Under the above condition, (7c) has a unique solution in which eh ∈ (0, 1 − ec ). Then there exists a unique BGP.

Assume that

Proof. We can rewrite (7c) as the following equation: LHS (eh ) = RHS (eh ) , where LHS (eh ) = and RHS (eh ) =

Thus, combining (3d) with (5b) yields −1 ⎧k y⎡ 1 − α ⎤ χ ⎫ μ˙ ky ⎬ (1 − α )B =ρ−⎨ + (1 − s )η1⎥ − ⎢ μ ec + eh ⎭ ck ⎦ ⎩ c k ⎣ 1 − ec − eh ⎪

⎪

⎪

− B(ec + eh ).

(5c)

4. Effects of social status and education on economic growth

Furthermore, substituting (4) into (2a) yields

y k˙ c = (1 − η1eh − η2ec ) − . k k k

This section analyzes the effects of social status and education on economic growth. We focus on the effects of changes in the agent's attitude toward social status (the intensity of the agent's preference for social status relative to consumption, χ) and the costs of education (η1 and η2). In addition, we are also interested in the effects of the government's policy on education (s and ec). It is worth noting that our results are unaffected by the elasticity of the marginal utility of the relative education level (γ). We check the other settings for social status seeking later.

(5d)

By using (2b) and (5a)–(5d), we can simplify the equilibrium conditions by transforming them into a three-dimensional dynamical system with vector (z, q, p) ≡ (c / k , h / k , λ / μ). Following Bond et al. (1996), the above equilibrium conditions then yield

y y z˙ c = α − ρ − (1 − η1eh − η2ec ) + , z k k k

(6a)

q˙ y c = B(eh + ec ) − (1 − η1eh − η2ec ) + , q k k

(6b)

4.1. Comparative static analysis

−1 ⎤ p˙ ⎧ k y ⎡ 1 − α χ ⎫ ky ⎬ (1 − α )B =⎨ + (1 − s )η1⎥ − + B(ec + eh ) ⎢ p ⎩ c k ⎣ 1 − ec − eh ec + eh ⎭ ck ⎦ y −α . (6c) k ⎪

⎪

⎪

⎪

By totally differentiating (7c), we obtain the effects of changes in the structural parameters on the time allocated to higher education as follows:

deh de de de de > 0, h < 0, h < 0, h > 0, h < 0. dχ dη1 dη2 ds dec

Thus, (6a)–(6c) describe a three-dimensional dynamical system, since (5b) implies that eh is a function of z, q, and p.

Along any BGP, we have z˙ = q˙ = p˙ = 0 , and thus the variables z, q, and p are constant. Along the path, the fraction eh is also constant, while c, k, and h grow at the same rate, denoted by θ. To determine the BGP,7 first we use (6a) and (6b) to obtain

θ = B(ec + eh ).

1

B(ec + eh ) + ρ − B(ec + eh ). α

(9)

Thus, except for compulsory education, all the effects of the changes in the structural parameters on economic growth are the same as those on eh. Intuitively, if the agents think highly of social status, they will

(7a)

Next, by substituting (7a) into (6b), we obtain

z = z(eh ) = (1 − η1eh − η2ec )

(8)

The above results can also be obtained from Fig. 1. An increase in η1, η2, or ec increases the LHS (eh ) locus, while an increase in s decreases it. An increase in χ increases the RHS (eh ) locus, while an increase in ec reduces it. Thus, a higher χ (s) increases eh, which is the point eh1 (eh3) in Fig. 1, while a higher η1 (η2, ec) decreases eh, which is the point eh2 (eh2, eh4) in Fig. 1. It is easy to derive the growth rate of the economy from (2b) as follows:9

3.4. The balanced growth path

⎡ B(e + eh ) + ρ ⎤ 1− α 1 q = q(eh ) = ⎢ c . ⎥ ⎣ ⎦ αA 1 − ec − eh

ρχ . ec + eh

⎤ + (1 − s )η1ρ − (1 − α )B⎥ , ⎦

It is obvious that RHS (eh ) is decreasing in eh. We also find that LHS (eh ) is increasing in eh when we differentiate it with respect to eh.8 Moreover, it is easy to show that the RHS (eh ) locus equals ρχ / ec when eh = 0, and equals ρχ < ∞ when eh = 1 − ec , while the LHS (eh ) locus equals M when eh = 0, and goes to infinity when eh = 1 − ec . Under the condition in Proposition 1, we have RHS (0) > LHS (0). Therefore, a positively sloping LHS (eh ) and a negatively sloping RHS (eh ) must intersect, and the intersection is unique. Thus, there exists a unique solution for the time allocated to higher education in which eh ∈ (0, 1 − ec ). See Fig. 1 (solid line, point eh0). □

Fig. 1. Balanced growth path and comparative statics.

⎪

⎡ (1 − α )ρ B(ec + eh ) + ρ ⎢ (1 − η1eh − η2ec)[B(ec + eh ) + ρ] − αB(ec + eh ) ⎣ 1 − ec − eh

(7b)

8-

⎡ ⎤ B(ec + eh ) + ρ d⎢ ⎥/de = ⎣ (1 − η1eh − η2ec)[B(ec + eh) + ρ] − αB(ec + eh) ⎦ h ⎡ (1 − α )ρ ⎤ (1 − α )ρ > 0. and d ⎢ ⎥/de = ⎣ 1 − ec − eh ⎦ h (1 − ec − eh)2

Finally, we substitute (7a) and (7b) into (6c) and obtain 7 As (6a)–(6c) along the BGP do not include the variable p, we can use these three equations to solve for the long-run values of z, q, and eh, and then the long-run value of p can be found by (5b) independently.

[B(ec + eh ) + ρ]2 η1 + αρB {(1 − η1eh − η2ec)[B(ec + eh ) + ρ] − αB(ec + eh )}2

> 0,

9 In the long run, the growth rates of c, k, h and y are the same according to Eqs. (5a)– y˙ c˙ k˙ h˙ (5d), i.e., = = = = B(ec + eh ) in the long run.

c

4

k

h

y

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Moreover, the total time spent in education and economic growth are not changed.10 All other comparative statics are the same as in our benchmark model.

spend more time in education to increase their social status. Thus, the time spent in higher education increases, as do the accumulation of human capital and economic growth. As for the costs of education, regardless of their type (η1 or η2), increased costs will reduce the time spent in higher education, human capital accumulation, and economic growth. An increase in the costs of higher education directly decreases eh by reason of its lower surplus (marginal benefit minus marginal cost), while an increase in the cost of compulsory education indirectly decreases eh. The agents do not need to pay tuition when they receive compulsory education, but its higher cost implies that lump-sum taxes are also higher, and these will lower agents’ disposable income. So the total time spent in education reduces. As the time allocated to compulsory education is mandatory and taken as given by the agents, they allocate less time to higher education. In the case of an increase in the government's subsidies for those pursuing higher education, the intuition is similar to that for a decrease in η1. Given that the cost of higher education falls, the agents increase the time spent in higher education, given the larger surplus. Although the agents eventually need to pay the same tuition, given that the higher subsidies imply higher lump-sum taxes, the agents take T as given and only consider the cost (they only need to pay the fraction 1 − s of the cost) when they make a decision. Regarding compulsory education, although longer compulsory education reduces the time allocated to higher education, if the total time spent in education (ec + eh ) increases, compulsory education may still have a positive effect on economic growth. (7c) suggests that if s=1 and η1 = η2 , there exists a unique total allocated time to education (ec + eh ). This implies that there is a trade-off between compulsory education and higher education. The agents allocate this unique total time to education optimally: if the government increases the duration of compulsory education, agents will reduce by the same amount the time they allocate to higher education. That is, except for eh, the other variables are unchanged, as is economic growth. Furthermore, even if s = 1, a change in ec still affects economic growth when η1 ≠ η2 . If η2 > η1, (7c) suggests that higher ec not only reduces eh, but also reduces agents’ total time in education. Therefore, the growth rate of the economy declines. Intuitively, when education provided by the government is less efficient than that provided by the private sector, i.e., the cost of education provided by the government is higher than that provided by the private sector, longer compulsory education not only has a crowding-out effect on agents’ time allocated to higher education, but also has a negative effect on agents’ total time in education given their lower disposable incomes. The above inefficient situation leads to lower economic growth. However, if the education provided by the government is more efficient than that provided by the private sector, longer compulsory education may have a positive effect on economic growth. Education provided by the competitive market (private sector) is the most efficient. Although education (the accumulation of human capital) is a key factor for economic growth, it does not need to be provided by the government. Compulsory education stimulates agents to pursue social status, and can increase their incentive to devote more time to education (e.g., higher education), while excessively long compulsory education may have the opposite effect. Our results suggest that the government should subsidize education. However, the target of the government's subsidies should be those individuals pursuing education, rather than the institutions providing education. The latter approach is similar to public schooling, which may be a source of inefficiency. The above conclusion is also consistent with the idea in Friedman (1962, ch. 6). It is worth noting that if the agents are unconcerned with social status, i.e., χ = 0 , the duration of compulsory education only has a crowding-out effect on the agents’ time allocated to higher education. The time allocated to higher education decreases by the same amount as the duration of compulsory education increases.

4.2. Calibration This subsection quantifies the effects of social status and education on economic growth. We calibrate the model along a BGP to reproduce key features representative of the US economy at annual frequencies. As pointed out by Prescott (2006), the fraction of time allocated to the market is around 25 percent. To simplify the analysis, we choose 1 − e = 0.25.11 In the real business cycle literature, it is more common to use a capital share of output between 0.30 and 0.40 (see the examples in (Cooley, 1995)). We set the capital share of output at α = 0.36 , following Kydland and Prescott (1982). There are no data for human capital, but human capital is at least as large as physical capital, as argued by Kendrick (1976). We normalize the initial ratio of human to physical capital at q = h / k = 1. Moreover, Kydland and Prescott (1991) used 4% as the annual rate of time preference, so ρ = 4%. To match the annual real economic growth rate in the long run at θ = 2.1%, we use (9) and (7a) to calibrate the technology coefficients in the goods and educational sectors at A = 0.4115 and B = 0.0280, respectively. The data from the Penn World Table suggest that the consumption–output ratio of the US economy is about 0.67, and thus we set the initial c / y = 0.67. Assume that the coefficients for the costs of compulsory education and higher education are initially the same, and thus η1 = η2 . By using (7b), we obtain η1 = η2 = 0.2748. To simplify the analysis, we first assume that the government subsidizes the total cost of education, i.e., s = 1. Furthermore, the intensity of agents’ preferences for social status relative to consumption can be calibrated by (7c) at χ = 2.3642 . Finally, we need to calculate the time allocated to compulsory education and higher education. According to data in OECD (2014), the share of the US population aged 25–64 years in 2013 who attained less than upper secondary education is 10.37%, compared with 45.71% who attained upper secondary or postsecondary nontertiary education and 43.91% who attained tertiary education. Assume that the time required to complete lower secondary education is about 9 years, that a further 3 years are required for upper secondary or postsecondary nontertiary education (so the total duration of education becomes 12 years), and that a further 4 years are required for tertiary education (so the total duration of education becomes 16 years). Compulsory education lasts 9 years in the US. Therefore, we can calibrate the ratio of time in compulsory education to the total time in education at ec / e = 0.6694 . That is, ec = 0.5021. So eh = e − ec = 0.2479.12 Now, we can investigate the quantitative effects of social status and compulsory education on economic growth. At every turn, we increase each parameter by 10%, except for s, which is changed from 1 to 0. The quantitative results are listed in Table 1. The numerical exercises support our theoretical inferences. In the long run, both the time allocated to higher education and economic growth are increasing in the intensity of agents’ preferences for social status relative to consumption and in the government's subsidy for higher education, while they are decreasing in the costs of compulsory and higher education. However, if the government subsidizes the total costs of education and if the unit costs of both kinds of education are equal, a longer duration of compulsory education reduces the time in higher education, but does not affect economic growth or the other economic variables. The comparative statics of the benchmark model are in Table 1, panel (A). 10

11

When χ = 0 , (7c) becomes ec + eh = 1 −

(1 − α )ρ . (1 − α )B − (1 − s )η1ρ

A change in the initial value of e does not alter our numerical results. Here we only use data from 2013 to calibrate ec and eh. If we select data from another year or use data over longer periods, the following results are unchanged. 12

5

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decision. In order to understand whether this externality will affect our results, we assume that the cost of higher education is C (eh ) = η1yeh . By using the same methods of calculation and calibration, we obtain results similar to those in our benchmark model. In addition, when we solve for eh, as ec and eh are not tied together under s ≠ 1 and η1 = η2 , an increase in the duration of compulsory education will not reduce the same amount the time allocated to higher education.13 Even if η1 = η2 , longer compulsory education not only has a crowding-out effect on higher education, but also has negative effects on the total time in education and economic growth. All other comparative statics are similar to those in the benchmark model. The results are in Table 1, panel (D). The above result supports our theoretical inferences. Although education is important for economic growth, it does not need to be provided by the government. Further, excessively long compulsory education may have negative effects on the total time in education and economic growth.

Table 1 Effects of social status and compulsory education. eh

e

(A) Comparative statics in the benchmark model Benchmark 0.2479 0.7500 0.2618 0.7639 χ = 2.6006 0.2464 0.7485 η1 = 0.3022

θ(%)

h /k

c /k

2.1000 2.1389 2.0958

1.0000 1.0695 0.9929

0.1135 0.1133 0.1124

0.2448

0.7469

2.0914

0.9856

0.1112

s=0 0.2277 ec = 0.5523 0.1977 (B) η1 < η2 in the benchmark model

0.7298 0.7500

2.0435 2.1000

0.9119 1.0000

0.1138 0.1135

η1 = 0.2191, η2 = 0.3022

0.2479

0.7500

2.1000

1.0000

0.1135

η1 = 0.2191, η2 = 0.3022 , ec =

0.1968

0.7491

2.0974

0.9956

0.1128

η1 = 0.3022 , η2 = 0.2612

0.2479

0.7500

2.1000

1.0000

0.1135

η1 = 0.3022 , η2 = 0.2612 , ec =

0.1982

0.7505

2.1013

1.0021

0.1139

0.5523 (D) No externality in the cost function s=0 0.2397 0.2560 χ = 2.6006 0.2370 η1 = 0.3022

0.7418 0.7581 0.7391

2.0769 2.1227 2.0695

1.0705 1.1650 1.0679

0.1231 0.1236 0.1229

5. Robustness checks

η2 = 0.3022

0.2361

0.7382

2.0669

1.0514

0.1205

ec = 0.5523

0.1868

0.7391

2.0694

1.0322

0.1209

5.1. Different social status

η2 = 0.3022

0.5523 (C) η1 > η2 in the benchmark model

In our benchmark model, the agents’ social status depend on their relative education level in society. Given that the economy-wide average education level is equal to the agent's optimal education level in equilibrium, i.e., e = e , the elasticity of the marginal utility of relative education level does not affect our results. To determine the effect of the agent's attitude to education, we now assume that agents’ preferences for social status may differ regarding relative and absolute education levels. ∞ e An agent's lifetime utility becomes U = ∫ u(c, e, e )exp( − ρt )dt ,

Note: The initial parameters in the benchmark case are: χ = 2.3642 , η1 = η2 = 0.2748, s = 1, and ec = 0.5021. In the case of panel (D), all exercises are under s = 0.

In order to discuss the effect of compulsory education on an agent's education choice and economic growth, we investigate the situation under different unit costs of compulsory and higher education. First, we consider the case where the education provided by the government is less efficient than that provided by the private sector, i.e., η1 < η2 . We increase η2 by 10% and reduce η1 in order to achieve the same levels as the whole variables in the benchmark case, and find that η1 = 0.2191 and η2 = 0.3022 . Then, we increase ec by 10%. We find that longer compulsory education not only decreases the time allocated to higher education, but also reduces the total time in education. Therefore, economic growth declines because of the relatively inefficient education provided by the government. The result is in Table 1, panel (B). Next, we examine the case where the education provided by the government is more efficient than that provided by the private sector, i.e., η1 > η2 . Similarly, we increase η1 by 10% and reduce the level of η2 in order to achieve the same levels as the whole variables in the benchmark case, and find that η1 = 0.3022 and η2 = 0.2612 . Then, we increase ec by 10%. We find that although longer compulsory education has a negative effect on the time allocated to higher education, it instead increases the total time in education. This is because the government provides more efficient education. Therefore, economic growth rises. The result is in Table 1, panel (C). In order to more clearly see the impact of each parameter on economic growth, we graphed the growth rate change under different values of χ, η1, η2, s and ec. The results which are consistent with our theoretical inferences are shown in Fig. 2. Note that when plotting the graph of changes in ec, we consider the case where the education provided by the government is less efficient than that provided by the private sector, i.e., η1 < η2 . This is because if the unit costs of both kinds of education are equal, a longer duration of compulsory education reduces the time in higher education at the same level, but does not affect the economic growth or the other economic variables. Finally, in our benchmark model, we assume that the unit cost of higher education does not depend on the agent's personal income. Thus, the cost of higher education only depends on the time allocated to higher education and has an externality, y , when the agents make a

t =0

e

e1− γ1

where u(c, e, e ) = ln(c ) + χ1 1 − γ + χ2 1

(e / e )1− γ2 . 1 − γ2

The parameters χ1 and χ2

are the intensities of the agent's preferences regarding absolute and relative education level, respectively, relative to consumption. The parameters γ1 and γ2 denote the elasticities of the marginal utility of the absolute and relative education level, respectively. By using the same method of calculation, we find that the right⎛ χ χ ⎞ hand side of (7c) becomes ρ⎜ (e + 1e )γ1 + e +2e ⎟, and that (7a), (7b), and c h⎠ ⎝ c h the left-hand side of (7c) are unchanged. Therefore, we obtain similar theoretical results as those in the benchmark model. Besides, an increase in the elasticity of the marginal utility of the absolute education level increases the agent's incentive to pursue more higher education as it causes agents to pay more attention to social status (their own education level). The above result can also be obtained from Fig. 1. An increase in γ1 increases the RHS (eh ) locus and increases eh, which is similar to point eh1 in Fig. 1. To quantify these results, we use the same calibration steps as those in our benchmark model. We assume that χ1 = χ2 and γ1 = γ2 = 1 at the beginning, and find that χ1 = χ2 = 1.1821 and that the other parameters and variables are the same as those in Section 4.2. Similarly, with each exercise, we increase each parameter by 10%, except for s, which is changed from 1 to 0, and γ1, which is changed from 1 to 2. The quantitative results are consistent with our theoretical inferences and are listed in Table 2, panel (A). 13

Now (7c) becomes

⎡ (1 − α )ρ [B(ec + eh ) + ρ][1 − (1 − s )η1eh] ⎢ (1 − η1eh − η2ec)[B(ec + eh ) + ρ] − α[1 − (1 − s )η1eh]B(ec + eh ) ⎣ 1 − ec − eh =

+

(1 − s )η1ρ 1 − (1 − s )η1eh

⎤ − (1 − α )B⎥ ⎦

ρχ . ec + eh

Given that ec and eh are still tied together when s = 1, all comparative static results, except s = 0, are the same as those in Table 1, panel (A). Therefore, we use the result under s = 0 as the new benchmark, and compare all the other changes (χ, η1, η2, and ec) with the case where s = 0 when we analyze the comparative statics.

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Fig. 2. The change in the growth rate under different values of χ, η1, η2, s and ec.

different kinds of education should be different. We assume that higher education delivers more human capital accumulation than does compulsory education, and thus the technologies in the education sector are as follows: x = B(ec + κeh )h , in which κ > 1. By using the same method of calculation, we obtain a similar theoretical outcome: differential results (8) and Fig. 1.14 The time allocated to higher education and economic growth increase if the agents place more emphasis on social status or if higher education receives much larger subsidies, but decrease if the costs of education (compulsory or higher education) increase or if the government increases the duration of compulsory education. A longer ec definitely reduces the time allocated to higher education and economic growth. This is because higher education delivers more human capital accumulation than does compulsory education, and thus a lower eh (because of a longer ec) certainly reduces economic growth, even if s = 1 and η1 = η2 . The remaining intuitions are the same as in the benchmark model. Although the effect cannot be directly derived from the equations along the BGP, improving the efficiency of higher education (increasing κ) encourages agents to pursue more higher education. Therefore, a larger κ should increase the time allocated to higher education and thus economic growth. The result should be similar to point eh3 in Fig. 1. To quantify the results, we use the same calibration steps as in our benchmark model. We assume that κ = 1.1 at the beginning, and find that B = 0.0271, χ = 2.3317, and the other parameters and variables are the same as those in Section 4.2. Similarly, we increase each parameter sequentially by 10%, except for s, which is changed from 1 to 0. The quantitative results are in accord with our theoretical inferences and are shown in Table 2, panel (B). Regarding a longer ec, even if it increases the total time in education, it still reduces economic growth because the impact of a lower eh on θ is greater than that of a longer ec on θ.

Table 2 Effects of social status and compulsory education: Robustness checks. eh

e

θ(%)

h /k

c /k

Benchmark (A) Different social status χ1 = 1.3003

0.2479

0.7500

2.1000

1.0000

0.1135

0.2551

0.7572

2.1201

1.0348

0.1134

χ2 = 1.3003

0.2551

0.7572

2.1201

1.0348

0.1134

η1 = 0.3022

0.2464

0.7485

2.0958

0.9929

0.1124

η2 = 0.3022

0.2448

0.7469

2.0914

0.9856

0.1112

γ1 = 2

0.2681

0.7702

2.1564

1.1034

0.1133

s=0 ec = 0.5523 η1 = 0.2191, η2 = 0.3022 , ec =

0.2277 0.1977 0.1968

0.7298 0.7500 0.7491

2.0435 2.1000 2.0974

0.9119 1.0000 0.9956

0.1138 0.1135 0.1128

0.5523 η1 = 0.3022 , η2 = 0.2612 , ec =

0.1982

0.7505

2.1013

1.0021

0.1139

0.5523 (B) Different educational efficiencies 0.2617 χ = 2.5648 0.2464 η1 = 0.3022

0.7638 0.7485

2.1410 2.0956

1.0694 0.9930

0.1134 0.1124

0.2449

0.7470

2.0909

0.9857

0.1112

0.2506 κ = 1.2100 s=0 0.2277 ec = 0.5523 0.1978 (C) Different cost functions of education 0.2618 χ = 2.6006 0.2465 η1 = 0.3312

0.7527 0.7298 0.7501

2.1828 2.0397 2.0867

1.0325 0.9109 0.9971

0.1144 0.1137 0.1134

0.7639 0.7486

2.1388 2.0960

1.0693 0.9933

0.1133 0.1125

η2 = 0.3312

0.2448

0.7469

2.0912

0.9853

0.1112

s=0 ξ1 = 1.2100 ξ2 = 1.2100 ec = 0.5523

0.2270 0.2499 0.2502 0.1974

0.7290 0.7520 0.7523 0.7497

2.0413 2.1057 2.1063 2.0992

0.9088 1.0096 1.0107 0.9987

0.1138 0.1151 0.1152 0.1133

η2 = 0.3022

Note: In all cases, s = 1 and ec = 0.5021 at the beginning. Other initial parameters in (A) are: χ1 = χ2 = 1.1821, η1 = η2 = 0.2748, and γ1 = 1. Those in (B) are: χ = 2.3317,

η1 = η2 = 0.2748, and κ = 1.1. Those in (C) are: χ = 2.3642 , η1 = η2 = 0.3011, and ξ1 = ξ2 = 1.1.

14

Now (7a)–(7c) become 1 ⎡ B(κec + eh) + ρ ⎤ 1− α 1 q = q(eh ) = ⎢ , ⎥ αA 1 − ec − eh ⎣ ⎦

5.2. Different educational efficiencies As regards human capital accumulation, our benchmark model implies that compulsory education and higher education have the same efficiency. However, the purpose of compulsory education is to acquire basic knowledge, whereas higher education usually aims to build professional capacity. Thus, the human capital accumulated from

z = z(eh ) = (1 − η1eh − η2ec ) and

− B(κec + eh ),

⎡ (1 − α )ρ B(κec + eh ) + ρ ⎢ (1 − η1eh − η2ec)[B(κec + eh ) + ρ] − αB(κec + eh ) ⎣ 1 − ec − eh

respectively.

7

B(κec + eh ) + ρ α

⎤ + (1 − s )η1ρ − (1 − α )κB⎥ = ⎦

ρχ , ec + eh

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Table 3 Effects of social status and compulsory education: With leisure. eh

e

el

θ(%)

h /k

c /k

l (1 − e)

Benchmark χl = 1.6024

0.2479 0.2468

0.7500 0.7489

0.7500 0.6830

2.1000 1.9124

1.0000 1.0395

0.1135 0.1113

0.2500 0.2290

χ = 2.6006 η1 = 0.3022

0.2620 0.2465

0.7641 0.7486

0.7749 0.7550

2.1698 2.1139

1.0635 0.9896

0.1137 0.1126

0.2393 0.2535

η2 = 0.3022

0.2451

0.7472

0.7602

2.1286

0.9789

0.1117

0.2573

s=0 ec = 0.5523 η1 = 0.2191,

0.2275 0.1977

0.7295 0.7500

0.7150 0.7500

2.0019 2.1000

0.9196 1.0000

0.1133 0.1135

0.2651 0.2500

0.1968

0.7491

0.7531

2.1086

0.9935

0.1130

0.2522

0.1981

0.7504

0.7485

2.0959

1.0032

0.1138

0.2489

η2 = 0.3022 ec = 0.5523 η1 = 0.3022 ,

η2 = 0.2612 ec = 0.5523

Note: The parameters in the benchmark case are: χl = 1.4567 , χ = 2.3642 , η1 = η2 = 0.2748, s = 1, and ec = 0.5021.

⎡ (1 − α )ρ Bl (ec + eh ) + ρ + (1 − s )η1ρ ⎢ (1 − η1eh − η2ec )[Bl (ec + eh ) + ρ] − αBl (ec + eh ) ⎣ 1 − ec − eh ⎤ ρχ − (1 − α )Bl ⎥ = , ec + eh ⎦ ⎡ 1−α ⎤ ρχ /(ec + eh ) + (1 − s )η1(ec + eh )⎥ ⎢ (1 − α )ρ 1 − e − e ⎣ ⎦ c h + (1 − s )η1ρ − (1 − α )Bl 1 − ec − eh = χl l ε +1 + χ .

Agents in the above-mentioned models only devoted their time to working or learning (accumulating human capital). However, people have other time allocation options. In this subsection, we discuss the effect of leisure. Assume that the agent is endowed with L units of time. At an instant in time, l units are devoted to learning and working with proportions e and 1 − e , respectively, and the remaining L − l units are devoted to leisure. Thus, the total time allocated to higher education is leh, that allocated to compulsory education is lec, and that allocated to working is l (1 − e). The utility function is increasing in consumption and social status, but decreasing in nonleisure time. Therefore, an agent's lifetime utility now B(ec + eh ) + ρ α

− B(ec + eh ) and

⎡ (1 − α )ρ ⎤ + (1 − s )η1ρξ1ehξ1−1 − (1 − α )B⎥ = ⎢ ξ ξ ⎦ (1 − η1e 1 − η2ec 2)[B(ec + eh ) + ρ] − αB(ec + eh ) ⎣ 1 − ec − eh B(ec + eh ) + ρ

h

(7d)

The above equations determine the long-run l and eh, and the growth rate of the economy is θ = Bl (ec + eh ). To analyze the effects of social status and compulsory education on economic growth in this case, we quantify the results. First, the IES for labor ranges from close to 0 (MaCurdy, 1981) to 3.8 (Imai and Keane, 2004). Following Hansen and Imrohoroğlu (2009), we select a middle value for the IES of labor at 2 as our benchmark case, which implies that ε = 0.5. Next, we use the same calibration steps as in our benchmark model and obtain the same parameters and variables as in Section 4.2. Finally, we use (7d) to calibrate the degree of disutility of nonleisure time relative to consumption and obtain χl = 1.4567. Now, we can investigate the quantitative effects. Similarly, we increase each parameter sequentially by 10%, except for s, which is changed from 1 to 0. The results for the comparative statics are in Table 3. A higher χl means that agents care more about leisure. In this situation, leisure is more important to agents than seeking social status. Thus, the time allocated to higher education falls. Agents also reduce learning and working time when χl rises, so the growth rate of the economy also falls. If agents pay more attention to social status, i.e., χ rises, they will increase the proportion of time devoted to higher education and reduce

5.4. Labor–leisure trade-offs

ξ Now (7a) and (7c) become z = z(eh ) = (1 − η1eh 1 − η2ecξ2 )

1− γ

becomes U = ∫t =0 u(c, l, ee )exp( − ρt )dt , where u(c, l, ee ) = ln (c ) − χl 1l + ε + χ (e 1/ e−) γ . The parameters χl measure the degree of disutility of working and learning relative to consumption, and the parameters ε denote the inverse of the IES for labor (nonleisure). In addition, the technologies in the goods and education sectors are y = Ak α [l (1 − ec − eh )h]1− α and x = Bl (ec + eh )h , respectively. By using the same method of calculation, along the BGP, we can simplify the dynamical system into the following two equations:16

In the benchmark model, we simplify the cost function of education as a linear function of the time in education. Now we check the effects of different cost functions. Assume that the cost functions of both kinds of education are convex with respect to the time in education. Thus, C (eh ) = η1y ehξ1 and C (ec ) = η2yecξ2 , where ξ1 > 1 and ξ2 > 1. By using the same method of calculation, we obtain similar theoretical results: (8) and Fig. 1.15 As for higher ξ1 or ξ2, because eh and ec are smaller than 1, the above change is just like a lower η1 or η2, which implies a lower cost of higher education or compulsory education. Therefore, both the time allocated to higher education and economic growth increase as ξ1 or ξ2 increases. The result is similar to point eh3 in Fig. 1. In addition, even if s = 1 and η1 = η2 , because ec is not always tied to eh when we solve for eh, longer compulsory education not only reduces the time allocated to higher education, but also decreases economic growth. Our other intuitions are the same as in the benchmark model. To quantify the results, we use the same calibration steps as in our benchmark model. We assume that ξ1 = ξ2 = 1.1 at the beginning, and find that χ = 2.3642 , η1 = η2 = 0.3011, and the other parameters and variables are the same as in Section 4.2. Similarly, with each turn, we increase each parameter by 10%, except for s, which is changed from 1 to 0. The quantitative results agree with our theoretical inferences and are reported in Table 2, panel (C).

15

1+ ε

∞

5.3. Different cost functions of education

1

16

ρχ , ec + eh

⎡ Bl(e + e ) + ρ ⎤ 1− α 1 Now (7a) and (7b) become q = q(eh , l ) = ⎢ c h and ⎥ αA l (1 − ec − eh ) ⎣ ⎦

z = z(eh , l ) = (1 − η1eh − η2ec )

respectively, while (7a) is not changed.

8

Bl (ec + eh ) + ρ α

− Bl (ec + eh ) , respectively.

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their leisure time. That is, the time allocated to higher education and the total time in education (el ) increase, as does the growth rate of the economy. Similar to the intuitions in the benchmark model, an increase in educational costs (η1 or η2) reduces the time allocated to higher education. However, because agents need to pay higher educational costs, they need to work harder. In order to increase their working time, they must increase their nonleisure time. Thus, the total time allocated to education (le ) is uncertain. According to our numerical exercise, le increases, and so does the growth rate of the economy. The effect of a decrease in the government's subsidy for higher education is the same as that in the benchmark case; i.e., it reduces the time allocated to higher education and economic growth. In addition, similar to the result in the benchmark exercise, an increase in the time allocated to compulsory education decreases by the same amount the time allocated to higher education, but does not affect the other variables. To describe the effect of compulsory education, we analyze the cases under different unit costs for both kinds of education and obtain similar results. When η1 < η2 , a longer e c indeed reduces the total ec + eh . While e c is an obligation, agents still need to pay the higher costs of compulsory education via taxes. This will force agents to reduce their leisure time; i.e., nonleisure time increases. Our numerical exercise suggests that the longer ec increases the total time in education, (ec + eh )l , and economic growth. The results under η1 > η2 are opposite to those under η1 < η2 . Here, the effects of η1, η2, and ec on economic growth are different from those in the benchmark model. This is because the growth rate of the economy depends not only on eh, but also on nonleisure time. Although an increase in η1, η2, or ec definitely reduces eh, which has a negative impact on economic growth, the agents need to increase their nonleisure time to pay the higher costs of education because of the higher η1, η2, or ec. The latter effect has a positive impact on economic growth. In our quantitative analysis, the positive impact dominates. This is why we obtain different results for economic growth in this model. However, the overall results remain consistent with those in the benchmark model in which eh is increasing in χ and s, but decreasing in η1, η2, and ec.

Table 4 Effects of social status and compulsory education: k in two sectors.

0.7500 0.7645 0.7486

2.1000 2.1076 2.0991

1.0000 1.0631 0.9942

0.0971 0.0966 0.0963

0.14452 0.14492 0.14447

η2 = 0.2715

0.2450

0.7471

2.0982

0.9883

0.0954

0.14442

s=0 ec = 0.5523 η1 = 0.1968, η2 = 0.2715

0.2302 0.1977

0.7323 0.7500

2.0877 2.1000

0.9316 1.0000

0.0977 0.0971

0.14386 0.14452

ec = 0.5523 η1 = 0.2715, η2 = 0.2346

0.1968

0.7491

2.0995

0.9964

0.0966

0.14449

ec = 0.5523

0.1981

0.7504

2.1003

1.0017

0.0974

0.14453

1− α

(1 − ec − eh )

⎡ β (1 − α − u )B ⎤ ⎢ ⎥ ⎣ ⎦ ραu

1− α β

(1− α )(1− β ) β

u1− α (ec + eh )

⎡ ⎤ αu = ρ ⎢1 + ⎥. ⎣ β (1 − α − u ) ⎦ To quantify the results, we use the same calibration steps as in our benchmark model. We assume that the goods sector is relatively more capital intensive than the human capital sector, and thus, α > β . We set β = 0.2 , and find that A = 0.3724, B = 0.0389, χ = 2.5026 , η1 = η2 = 0.2468, z = 0.0971, and u = 0.14452, and the other parameters and variables are the same as those in Section 4.2. Similarly, when analyzing the comparative statics, we increase each parameter sequentially by 10%, except for s, which is changed from 1 to 0. The comparative static results are in Table 4. The results in Table 4 are similar to those in Table 1, panels (A)– (C), as are the intuitions. The effects of the changes in the structural parameters on u are similar to those on eh, and therefore the impacts on the growth rate of the economy are similar. Note that in the long ραu run, θ = β(1 − α − u) . Thus, using symmetrical technologies in the two sectors does not change the main results in this analysis.

6. Concluding remarks This paper investigates the connection between social status, compulsory education, and economic growth. Along the BGP, if agents pay more attention to social status, or if the government increases subsidies to those pursuing higher education, the time allocated to higher education and the level of economic growth both increase. By contrast, if the costs of compulsory or higher education increase, the time allocated to higher education and the level of economic growth both decrease. However, if the government subsidizes the total costs of education, and if the unit costs of both kinds of education (compulsory and higher) are the same, a longer period of compulsory education only reduces the time allocated to higher education; it does not affect economic growth or the other economic variables. In addition, if the education provided by the government is less efficient than that provided by the private sector, i.e., the unit cost of compulsory education is higher than that of higher education, longer compulsory education not only decreases the time allocated to higher education, but also reduces the total time in education. Therefore, economic growth declines because of the greater amount of inefficient education provided by the government. We find that although education (the accumulation of human capital) is a key factor for economic growth, it does not need to be provided by the government. Compulsory education can increase agents’ incentive to

⎪

1

1− β ⎡ β (1 − α − u)B ⎤ β q = q(eh , u ) = ⎢ ⎥ u(ec + eh ) β ραu ⎣ ⎦ ⎤ ρ(1 − u ) ⎡ ραu αu z = z(eh , u ) = (1 − η1eh − η2ec ) ⎢1 + β (1 − α − u) ⎥ − β (1 − α − u) , respectively. α ⎣ ⎦

0.2479 0.2624 0.2465

αA(1 − u )

⎪

become

Benchmark χ = 2.7529 η1 = 0.2715

α −1

⎤⎫ ec + eh ⎡ 1 − α (1 − β )αu + (1 − s )η1 − ⎢ ⎥⎬ = 1, χ ⎣ 1 − ec − eh β (1 − u )(ec + eh ) ⎦⎭

(7b)

u

η2 = 0.2346 .

⎪

and

c /k

same levels as for the variables in the benchmark case, and find that η1 = 0.1968 and

⎪

(7a)

h /k

η2 = 0.2715. Similarly, when η1 > η2 , using the same method, we find that η1 = 0.2715 and

⎤ (1 − u )(1 − β ) ⎧ ⎡ (1 − α )β (1 − u ) ⎨1 − η1eh − η2ec + 1⎥ ⎢ ⎦ ⎣ α(1 − β )u α ⎩

Now

θ(%)

Note: The parameters in the benchmark case are: χ = 2.5026 , η1 = η2 = 0.2468 , s = 1, and

The above models follow the setting in Lucas (1988), in which the goods sector uses both physical and human capital as inputs, while the education sector only uses human capital as an input. We now augment our analysis by including the symmetrical technologies in Bond et al. (1996) and Mino (2001), in which both sectors use both physical and human capital as inputs. Therefore, the technologies in the goods and y = A[(1 − u )k ]α [(1 − ec − eh )h]1− α education sectors are and x = B(uk ) β [(ec + eh )h]1− β , respectively. The variable u denotes the share of physical capital allocated to the education sector. The parameter β is the share of physical capital in the education sector. By using the same method of calculation, along the BGP, we can simplify the dynamical system into the following two equations, which determine the long-run eh and u:17

17

e

ec = 0.5021. When η1 < η2 , we increase η2 by 10% and reduce the level of η1 to achieve the

5.5. Physical capital in two sectors

−

eh

and

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devote more time to education (e.g., higher education), while an excessively long time allocated to compulsory education may have the opposite effect. In addition, our results suggest that the government should subsidize education. However, the target of the government's subsidies should be those pursuing education, rather than the institutions providing education. The latter situation is similar to public schooling, which may be a source of inefficiency. Thus, the government should encourage people to pursue higher education through the use of subsidies or vouchers, as suggested by Friedman (1962, ch. 6), but should not oblige people to receive higher education, i.e., it should not stipulate an excessively long period of compulsory education. In this paper, we only focus on a representative agent model without any heterogeneity, as reflected by differently skilled workers or jobs. In addition, countries with different levels of development may experience different effects from compulsory education and social status. In addition, even if this paper can obtain a clear comparative statics result, in reality, the true impact of those changes in structural parameters needs to be measured using econometric methods. We leave such analyses to future research. References Akerlof, G., 1980. A theory of social custom, of which unemployment may be one consequence. Q. J. Econ. 94, 749–775. Araujo, L., 2004. Social norms and money. J. Monet. Econ. 51, 241–256. Basu, P., Bhattarai, K., 2012. Government bias in education, schooling attainment, and long-run growth. South. Econ. J. 79, 127–143. Blankenau, W., Camera, G., 2009. Public spending on education and the incentives for student achievement. Economica 76, 505–527. Blankenau, W., Simpson, N., 2004. Public education expenditures and growth. J. Dev. Econ. 73, 583–605. Blankenau, W., Simpson, N., Tomljanovich, M., 2007. Public expenditure on education, taxation and growth: linking data to theory. Am. Econ. Rev. 97, 393–397. Bond, E., Wang, P., Yip, C., 1996. A general two sector model of endogenous growth with human and physical capital: balanced growth and transitional dynamics. J. Econ. Theory 68, 149–173. Cole, H., Mailath, G., Postlewaite, A., 1992. Social norms, savings behavior, and growth. J. Polit. Econ. 100, 1092–1125. Cooley, T., 1995. Frontiers of Business Cycle Research. Princeton University Press, Princeton. Corneo, G., Jeanne, O., 1999. Pecuniary emulation, inequality and growth. Eur. Econ. Rev. 43, 1665–1678. de la Croix, D., 2001. Growth dynamics and education spending: the role of inherited tastes and abilities. Eur. Econ. Rev. 45, 1415–1438. Dissou, Y., Didic, S., Yakautsava, T., 2016. Government spending on education, human capital accumulation, and growth. Econ. Model. 58, 9–21. Dupor, B., Liu, W.-F., 2003. Jealousy and equilibrium overconsumption. Am. Econ. Rev. 93, 423–428. Fershtman, C., Murphy, K., Weiss, Y., 1996. Social status, education, and growth. J. Polit. Econ. 104, 108–132.

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