In this section, we use the steady-state balanced growth rate to com-pare the economic performance between private and public education regimes. Along the balanced growth path, the growth rate of physical capital per worker and the growth rate of individual human capital ac-cumulation will be the same as the growth rate of output per capita. The following equations express the above concepts:
1+ gk=
where St= 1−ϕnstiwthtis the ratio of saving to wage income, Et=1−ϕnettwthtis the ratio of education expenditure to wage income, ntis fertility rate,and lt = 1 − ϕnt is an individual labor supply. Eq.32 devides into Eq.33
yields the law of motion of mt.
mt+1= A1 − θst
λA1 − θetltδnt
mθt1−δ (35)
At steady state, the fertility rate ntis a constant and we can neglect the time subscript. From Eqs.9 and 20, we know that the tax rates τtand ηt depends on n. Therefore, the tax rates τt and ηt are time-invariant.
In addition, the physical-human capital ratio is also constant over time, mt+1= mt= m. We substitute mtin either Eq.32 or Eq.33 and obtain the balanced growth rate of output per capita 1+ g:
1+ gi = HSi
ni
θδEiδ1−θliδ1−θ1−θ1−δ1 , i= r, u (36) where i = r, u indicates the private and public education regimes respec-tively, H= λ1−θA1 − θδis a constant. Eq.36 shows that Si, Eiand ni
are three determinants of growth. 4 The ratio of saving Si and education investment Eito labor income both cause positive effect on the balanced growth rate obviously. However, fertility rate ni has both positive and negative impacts on the balanced growth rate. Negative one is so-called
”resource-dilution effect”, that is, bearing more children dilutes educa-tional resources at present time and output per capita in the future. Pos-itive one is ”tax-sharing effect”; a higher number of children also implies that there are more labor force for sharing PAYG social security burden in the future.
Proposition 2 Without PAYG social security program (B = 0), that is, an economy with fully-funded social security system or without offering any social security program, given any life expenctancy p an economy with pri-vate education system has higher balanced-growth rate than the one with public education system.
4Since labor supply liis a function of fertility niwe do not treat lias another growth determinant.
Proof. See Appendix
Proposition 2 indicates that when longevity increases but government does not implement PAYG social security program, private education sys-tem can stimulate higher economic growth than public education syssys-tem does. The reason is that the education expenditure in public schooling regime is financed by tax revenue not households themselves, it gives par-ents incentive to have more children and free ride educational resource.
High fertility rate leads ”resource-dilution effect” and thereby has nega-tive impact on balanced growth rate. This result is consistent with the ho-mogenous agent case in Glomm and Ravikumar (1992) and low income inequality case in de la Croix and Doepke (2004). However, as Zhang and Zhang (2001) points out that an increase in longevity also has indi-rect effect on growth through the higher burden of PAYG social security system. Therefore, it is necessary and interesting to see whether the re-sult will be changed when considering PAYG social security system in our model economy.
Proposition 3 When implementing PAYG social security system (B 0), an economy with private education system has higher level of PAYG social security tax than the one with public education system. Moreover, a higher social security tax rate reduces the steady-state capital accumulation and balanced growth rate.
Proof. See Appendix
If government implements PAYG social security program,raising more children will have ”tax-sharing effect” by broadening future tax base of PAYG social security program and give posititve impact on economic growth. On balance, implementing PAYG social security system makes children involve a positive externality (Groezen, Leers and Meijdam, 2003).
Hence, government can use several policy tools to ”correct” the external-ity resulted from public pension policy, such as child allowance, which is
discussed a lot by previous studies (Groezen, Leers and Maijdam, 2003 ; Oshio, 2005). Proposition 3 suggests that providing public education may be another policy instrument to encourage parents to bear children and then mitigate the heavier and heavier PAYG social security burden in an aging economy. Furthermore, it also reveals that high level of PAYG social security tax has a negative impact on capital accumulation and balanced growth rate, for this reason, public education system may have possibil-ity to stimulate higher economic growth than private education system if ”tax-sharing effect” dominates ”resource-dilution effect”. The following section, we present the calibrated version of our model to obtain clearer picture of the above two opposite effects when comparing the economic performance between public and private education systems.
3 Computational Experiments
3.1 Calibration
In order to obtain credible quantitative results of our theory, we calibrate our model to match the growth features of the US or other OECD coun-tries. There are five features that we want to match: life expectancy, annual growth rate of output per capita, total fertility rate per woman (TFR=2nt 5), the share of education expenditure on output and the tax rate of PAYG social security. Because public school enrollment rate is higher than pri-vate school enrollment rate in most countries (Chen, 2005 ; de la Croix and Doepke, 2007), our baseline model, which is calibrated to fit the real world data, is an economy with public education system.
One preiod (generation) in our model is assumed 30 years and agents can survive safely for two preiods, that is, life expectancy in our model economy is at least 60 years old. We set p = 0.5 to match life expectancy in the United States at 2000 (about 76 years old). According to standard real business cycle literatures (Docquier and Paddison,2003), we set discount
5Since at least two people (a male and a female) can give a birth in the real world, but our model economy is ”asexual reproduction” (an agent can have his/her offspring individually.) Therefore, to match the data of total fertility rate per woman, we need to let n multiply 2.
factor (the weight of old age consumption) σ= 0.9930.
The parameter A= 5 in production function and λ = 3.5 in human cap-ital accumulation function, which does not influence qualitative results of our model, is used to match long-run growth rate of per capita out-put 2.5% (i.e. in the US 2.11% and 2.53% in Germany, Zhang and Zhang, 2003).
To calibrate total fertility rate, we need to adjust ϕ the fraction of time devoted to raise children and γ the weight of offspring quantity in the utility function. The studies of Robert Haveman and Barabra Wolfe(1995) and John Knowles(1999) show that parents spend about 15% of their time raising children.Accordingly, we choose ϕ = 0.15. The parameter γ is assigned to 0.26 (de la Croix and Doepke, 2003) to achieve average total fertility rate per woman 2.11 in United States during 2000− 2005.
Next, we use the elasticity δ of future human capital (wage income) with respect to public education expenditure and the weight of offspring quality in the utility function β to determine the ratio of public educa-tion expenditure to output. Johnson and Stafford(1973) estimated income elasticity for education expenditure was 0.198, another estimation of this figure provided by Card and Kreuger(1992) is 0.2. Since these estimations are similar, we set δ= 0.2.We choose β = 0.72 such that public education expenditure as a fraction of output fits the corresponding figure (public education expenditure for all level) in high income OECD countries at 2000, which is 4.8%.
The income replacement ratio of PAYG social security B is set to 0.43, which follows Pecchenino and Pollard(2002), for matching the social se-curity contribution rate 19%; this value is between the rates in France and US (Zhang and Zhang, 2003).
The remaining parameter θ is the share of income that goes to physical capital, following the previous literatures (see, for example, Boldrin,2005), we set θ = 0.3 as the calibrated value.The parameters of baseline model is summerized in Table 1.
Table 1: Calibrated values of baseline model
p= 0.5 λ = 3.5 A= 5 B = 0.43 θ = 0.3 δ= 0.2 ϕ = 0.15 γ = 0.26 β = 0.72 σ = 0.8 3.2 Comparing private and public education systems
In this section, we compare economic performance between public and private education regimes on the balanced growth path. From propsi-tion 2, we know that a private educapropsi-tion regime at steady state has higher growth rate than a public education regime for any degree of longevity in an economy without PAYG social security program.
However, many developed countries execute PAYG social security sys-tem, whose tax rate is positive related to life expectancy and pension re-placement rate nowadays but negative related to labor force at present (the number of children in last generation). Proposition 3 indicates that an economy with a private education system has to suffer more PAYG so-cial security tax burden than one with a public education system. When the tax burden expands, it will bring about larger distortion of economic activity and slow down the growth rate of GDP per capita. This opens the possibility for a public education system boosting higher economic growth even if there is no income inequality in our simple model.
Next, we want to show that the institution of PAYG social security mat-ters when analyzing the economic effects of two educational systems. The way we use here is by changing two key parameters of PAYG social secu-rity tax rate, life expectancy and pension replacement ratio, to emphasize the importance of joint consideration of these two policies.
3.2.1 The effect of longevity
To investigate the effect of longevity under different education regimes, we take balanced growth rate comparison by varying the life expectancy over the interval from 63 years to 90 years (p= 0.1 to p = 1.0).
Fig. 2a shows given the ratio of earning replacement B = 0.43, the tax burden of PAYG social security increase with the extension of life
ex-pectancy. Due to low fertility rate, a private education regime (green line) has higher level of the PAYG social security tax rate than a public educa-tion regime (blue line) and the gap of tax rate between two regimes en-larges as life expectancy raise. Heavy tax burden of PAYG social security program has a very strong negative impact on investment in human and physical capital accumulation.
Figure 2: Life expectancies and education systems (a)
PAYG Social Security Tax Rate, %
Public Education
Hence, we can find that a private schooling regime has less physcial capital investment than a public schooling regime at any extent of longevity (see Fig. 2b) and has less human capital investment than a public school-ing regime at sufficiently high level of life expectancy (see Fig. 2c). Be-cause of slow capital accumulation at the stage of high life expectancy (about 87 years), a private education system results in lower economic growth than a public education system (see Fig. 2d).
Table 2 gives two numerical examples to summerize the above find-ings. As shown in the first row of table 2, parents in a public
school-ing regime bear almost twice more number of children than in a pri-vate schooling regime. High fertility rate causes two opposite impacts on economic growth, the ”resource-dilution effect” and the ”tax-sharing effect”. However, the relative size of two effects depends on what degree of longevity an economy stays at. In a ”young” economy (life expectancy is 63 years), the ”tax-sharing effect” is smaller than the ”resource-dilution effect”, a public education system reuslts in less educational investment and then lower economic growth than a pirvate education system. On the contrary, in an ”old” economy (life expectancy is 87 years), the ”tax-sharing effect” dominates the ”resource-dilution effect”. A public educa-tion system leads faster capital accumulaeduca-tion and higher growth rate than a private education system. In sum, which education systems is better for long-run growth should hinge on how ”old” an economy is (the life ex-pectancy of an economy).
Table 2: Longevity and educational systems
Variables low longevity (p=0.1) high longevity (p=0.9) Private Public Private Public
FertilityTFR 1.16 2.59 1.01 2.10
Social Securityτ 7.38 3.31 76.04 36.78
SavingS 5.21 5.89 4.43 16.29
Education InvestmentE 18.89 8.75 4.82 4.64
Balanced Growthg 3.28 2.53 2.42 2.49
1 Except longevity, all parameters are the same as the setting in baseline model.
3.2.2 The effect of replacement ratio
In this section, we allow government can change her pension policy through varying pension replacement ratio from 10% of average earnings to 100%
of average earnings (B = 0.1 to B = 1.0).
If government raises pension replacement ratio (pension benefit for the aged), the level of PAYG social security tax will become higher and then have adverse impacts on physical capital accumulation (see Fig. 3b),
human capital accumulation (see Fig. 3c) and balanced growth rate (see Fig. 3d). Comparing two education systems, we find that a private ed-ucation system is more sensititve to the change of pension replacement ratio than a public education system, and furthermore a public education system can boost higher economic growth than a private education when government decides to provide sufficiently ”rich” pension benefit to the old.
Figure 3: Replacement ratios and education systems (a)
PAYG Social Security Tax Rate, %
Public Education
Table 3 gives two specific cases to illustrate that for maximizing eco-nomic growth what is the favored choice of education systems under dif-ferent policies of pension replacement rate. If govenment chooses the policy of low pension replacement ratio (B=0.2), growth rate in a pri-vate schooling regime is 3.52% higher than 2.86% in a public schooling regime. Contrariously, if govenment carries out the policy of high
pen-sion replacement rate (B=0.9), it is preferred to adopt a public education system (2.34%) rather than a private education system (2.02%). To sum up, in order to promote economic growth, government should coordinate educational policy and the benefit scheme of PAYG social security.
Table 3: Replacement ratio and educational systems
Variables low replacement (B=0.2) high replacement (B=0.9)
Private Public Private Public
FertilityTFR 1.01 2.18 1.11 2.35
Social Securityτ 19.82 9.17 81.35 38.20
SavingS 15.46 20.41 2.01 9.78
Education InvestmentE 16.15 7.76 3.78 4.89
Balanced Growthg 3.52 2.86 2.02 2.34
1 Except replacement ratio, all parameters are the same as the setting in baseline model.
3.3 Policy Implication: Subsidizing Private education
In previous section, we find that, because of low birth rate, a private ed-ucation system is more sensitive to the variation in life expectancy and pension replacement ratio than a public education system and then leads lower balanced growth rate when an economy with sufficiently high life expectancy and pension replacement rate. For this reason, it seems in-teresting to see whether growth can be promoted by implementing some policies, such as voucher program (subsidy of private education), which eliminates the educational expenditure per child and then encourage par-ents give more birth for sharing pension burden in an economy with
”great” PAYG social security program.
The intertemporal budget contraint for the households in a private schooling regime with voucher program can be revised as follows:
crtm+ 1 − vrtertnrt+ cort
rt+1 = 1 − τrt− qrt1 − ϕnrtwrthrt+ Vrt+1
rt+1 (37)
where vrt is subsidy rate of private education expenditure and qrt is a propotional tax for financing voucher program. Government also runs balanced budget to subsidize private education and the budget constraint of voucher program is:
vrtertnrt= qrt1 − ϕnrtwrthrt (38) where assume that the scale of voucher program is determined exoge-nously by government not by voting process.
Table 4 and 5 indicates that compared to private schooling regime (no subsidy, v=0), subsidizing educational fee per child can raise about 0.05–
0.07 (50% of subsidy, v=0.5) and 0.09–0.12 (90% of subsidy, v=0.9) total fertility rate and then reduce the tax burden of PAYG social security. The results also reveals that the relationship between the level of subsidy for private education and economic growth depends on the life expectancy of an economy and the policy of pension benefit.
In the case of low longevity or small pension replacement ratio, where
”resource-dilution” effect dominates ”tax-sharing” effect, more subsidy of educational investment results in lower economic growth. On the con-trary, when an economy has high life expectancy or implements the pol-icy of providing large pension benefit for old, ”tax-sharing” effect is stronger than ”resource-dilution” effect, government should provide more subsidy of educational investment to raise growth rate.
Table 4: Longevity and subsidy of education
Variables low longevity (p=0.1) high longevity (p=0.9) v=0 v=0.5 v=0.9 v=0 v=0.5 v=0.9
FertilityTFR 1.16 1.23 1.28 1.01 1.06 1.10
Social Securityτ 7.38 6.98 6.67 76.04 72.80 70.22
SavingS 5.21 5.27 5.32 4.43 5.14 5.73
Education InvestmentE 18.89 17.94 17.18 4.82 5.22 5.49 Balanced Growthg 3.28 3.23 3.19 2.42 2.49 2.54 1 Except longevity, all parameters are the same as the setting in baseline model.
Table 5: Replacement ratio and subsidy of education
Variables low replacement (B=0.2) high replacement (B=0.9)
v=0 v=0.5 v=0.9 v=0 v=0.5 v=0.9
FertilityTFR 1.01 1.06 1.10 1.11 1.16 1.21
Social Securityτ 19.82 18.94 18.24 81.35 77.42 74.29
SavingS 15.46 15.81 16.10 2.01 2.50 2.91
Education InvestmentE 16.15 15.55 15.08 3.78 4.34 4.73
Balanced Growthg 3.52 3.49 3.46 2.02 2.15 2.24
1 Except replacement ratio, all parameters are the same as the setting in baseline model.
4 Empirical Implications
The prediction of our model implies that comparing to a private school-ing system, a public schoolschool-ing system can encourage parents to have more births and then leads to a lower tax rate of PAYG social security in the fu-ture. Due to the lack of data, there are few empirical studies examining the impact of educational systems and policies on macroeconomic variables or demographic structure across countries. In this section, we use the in-ternationally comparable data provided by OECD and WDI (World De-velopment Indicators) to investigate preliminary relationships between educational systems, birth rate and social security burden.
Table 6 lists the whole 17 countires in our sample. We especially choose these high income OECD countries for two reasons. First, the life ex-pectancy at birth in these countries are sufficiently high and similar to each other, which matches the demographic feature of our model and also controls the effect of longevity on fertility rate and social security tax rate. Second, some countries, for example Italy and Germany, also satify our standard but their data is not reliable6.
6we also use secondary private school enrollment rate in 1985 from UNSCO to check the reliability of our classification for education systems
Table 6: Education systems, fertility rate and social security
Country Percentage Share of Total Fertility Rate Chnage in Social security
Private Funding,% tax rate 1990-2003, %
Australia 15.93 1.87 1.62
New Zeland 1.01 2.05 -2.25
Switzerland 6.56 1.51 3.39
To classify education systems in our sample, we follow de la Croix and Doepke (2007) and choose 90% of the public share in all level educaton as a criterion. If an economy has ”more” than 90% of public funding for education in 1993, we assort this country to a group of public educaton system. If an economy has ”less” than 90% of public funding for education in 1993, we assort this country to a group of private educaton system.
Table 7: Fertility rate and private educational funding
Size of N.obs Average Share of Total
Private Edu. Private Funding,% Fertiltiy Rate
Large( 10%) 8 30.19 1.66
Samll( 10%) 9 4.07 1.84
Mean difference test -0.18 (t-stat=-1.67)
Fig. 4 and table 6 reveal that the countries with larger share of private funding for education ”seem” to have lower birth rates than those with larger share of public funding for education. Computing the correlation between the propotion of private educational spending in 1993 and total fertility rate in 1993, we find that the correlation coeffcient is −0.4070, which is moderately negative. Table 7 provides the mean difference test and shows the difference of total fertility rate between the countries with larger private sector and those with larger public sector is−0.1845, whose
t-statistic is−1.67 and p-value is close to 10% significance.
Figure 4: Fertility rate and private funding on education
1.21.41.61.822.2
Total Fertility Rate per Woman in 1993
0 20 40 60 80
Friction of Private Educational Spending in 1993, %
Figure 5: The increase in social security program and private funding on education
−4−20246
Change in Social Security Tax rate 1990−−2003, %
0 20 40 60 80
Friction of Private Educational Spending in 1993, %
The predicted difference of fertility rate is roughly resemble the above empirical evidences. The another implication of our theory needed to
be examined is that countries with public education regime tend to have lower social security tax burden in the future years. Fig. 5 and table 6 show that the countries with larger percentage of private funding on education in 1993 have larger growth in a tax rate of social security during 1990 to
be examined is that countries with public education regime tend to have lower social security tax burden in the future years. Fig. 5 and table 6 show that the countries with larger percentage of private funding on education in 1993 have larger growth in a tax rate of social security during 1990 to