Public Versus Private Investment in Education in a Two Tiers System: The Role of Income Inequality and Intergenerational Persistence in Education

Abstract

This paper provides a simple political economy model of hierarchical education to study the endogenous determination of the public education budget and its allocation between different tiers of education (basic and advanced). The model integrates private education decisions by allowing parents, who are differentiated according to income and human capital, to opt out of the public system and enrol their offspring at private universities. Majority voting decides the size of the budget allocated to education and the expenditure composition. The model exhibits a potential for multiple equilibria and ‘low education’ traps. Income inequality and the intergenerational persistence of educational attainments play a fundamental role in deciding the equilibrium. The main predictions of the theory are broadly consistent with descriptive cross-country evidence collected for 43 high-middle income countries.

Appendices

Appendix 1

Lemma 1.

There exists a threshold:

$$ \hat{x}\left( {G_{T}^{e} ,h_{p} ,\pi } \right) = \frac{{1 + \pi \gamma^{{\frac{1 + \pi \gamma }{{\pi \gamma }}}} }}{{h_{p} \pi \gamma }}max\left( {1,G_{T}^{e} } \right) $$

such that households, whose human capital is ${h}_{p},$ strictly prefer private education if and only if $x>\widehat{x}\left({G}_{T}^{e},{h}_{p},\pi \right)$.

Proof.

If the household is planning to choose public tertiary education, the expected utility is given by Eq. (6) substituting ${h}_{T}={G}_{T}^{e}$

$$ EU\left( {c, h_{B} ,G_{T}^{e} } \right) = ln\left( {\left( {1 - \tau } \right)h_{p} x} \right) + \gamma \ln \left( {h_{B} } \right) + {\varvec{I}}\gamma {\uppi }\ln \left( {G_{T}^{e} } \right) $$

with I = 0 for ${G}_{T}^{e}<1$ and I = 1 for ${G}_{T}^{e}\ge 1 .$

The expected utility of the opting out decision is instead given by Eq. (6) setting ${h}_{T}={e}^{*}$

$$ EU\left( {c,h_{B} , e^{*} } \right) = \ln \left( {\left( {1 - { }\tau } \right)\left( {h_{p} x - e^{*} } \right)} \right) + { }\gamma { }\left[ {\ln \left( {h_{B} } \right) + {\uppi }\ln \left( {e^{*} } \right)} \right] = ln\left( {1 - { }\tau } \right) + ln\left( {\frac{{{ }1}}{\gamma + 1}h_{p} x} \right) + { }\gamma ln \left( {h_{B} } \right) + \gamma {\uppi }ln\left( {\frac{{{ }\gamma }}{\gamma + 1}h_{p} x} \right) $$

Imposing$EU\left({c,{h}_{B} , G}_{T}^{e}\right)=EU\left(c,{h}_{B},{e}^{*}\right)$, we obtain the threshold in Eq. (10).

Lemma 2.

The public budget ${\text{F}}\left(\uptau ,{{\text{G}}}_{{\text{T}}}^{{\text{e}}};\updelta ,{\text{K}}\right)$ satisfies the following:

• (i) $\frac{\partial F}{{\partial \tau }} > 0$

• (ii) $\frac{\partial F}{{\partial G_{T}^{e} }} \ge 0$.

• (iii) $\frac{\partial F}{{\partial \delta }} \le 0\; if \; \hat{x}\left( {G_{T}^{e} } \right)^{2} > m^{2} - \delta ^{2}$.

• (iv) $\frac{\partial F}{{\partial K}} > 0\; if h < \frac{1}{{\left( {1 + \gamma } \right)}}$.

Proof.

Substituting Eq. (11) into Eq. (13), we obtain the following expression for the budget for ${G}_{T}^{e}\ge 0$:

$$ F\left( {\tau ,G_{T}^{e} ; \delta ,K} \right) = { }\left\{ {\begin{array}{ll} \tau \left[ {M - K\frac{\gamma }{{\left( {1 + \gamma } \right)}}m} \right] &\quad \hat{x}\left( {G_{T}^{e} } \right) \le m - \delta \\ \tau \left[ {M - K\frac{\gamma }{{4\delta \left( {1 + \gamma } \right)}}\left( {\left( {m + \delta } \right)^{2} - \left( {\hat{x}\left( {G_{T}^{e} } \right)} \right)^{2} } \right)} \right] &\quad m - \delta < \hat{x}\left( {G_{T}^{e} } \right) < m + \delta \\ \tau M &\quad\hat{x}\left( {G_{T}^{e} } \right) \ge m + \delta \end{array} } \right. $$

(A.1)

(i) It is straightforward. (ii) The rate of opting out decreases with ${G}_{T}^{e}$ ($\frac{\partial \widehat{x}}{\partial {G}_{T}^{e}}\ge 0)$. This implies that the level of private expenditures in tertiary education (which are tax deductible) decreases with ${G}_{T}^{e}$. (iii) Taking the first derivative of the budget in eq. (A.1) with respect to $\updelta$, when $m-\delta <\widehat{x}\left({G}_{T}^{e}\right)<m+\delta ,$ it is easy to show that it if ${\widehat{x}\left({G}_{T}^{e}\right)}^{2}>{m}^{2}-{\delta }^{2}$ the derivative is negative. In the other cases, $\frac{\partial F}{\partial\updelta }=0.$ (iv) The average income $M=\left(1-K\right)hm+Km$ is positively related to K, thus, from A.1 we obtain:

$$\frac{\partial F}{\partial K}=\left\{\begin{array}{ll}\tau m\left[1-h-\frac{\gamma }{(1+\gamma )}\right] &\quad\widehat{x}\left({G}_{T}^{e}\right)\le m-\delta \\ \tau \left[m\left(1-h\right)-\frac{\gamma }{4\delta (1+\gamma )}\left({\left(m+\delta \right)}^{2}-{\left(\widehat{x}\left({G}_{T}^{e}\right)\right)}^{2}\right)\right]&\quad m-\delta <\widehat{x}\left({G}_{T}^{e}\right)<m+\delta \\ \tau m\left(1-h\right)&\quad \widehat{x}\left({G}_{T}^{e}\right)\ge m+\delta \end{array}\right.$$

Noting that $\tau \left[m\left(1-h\right)-\frac{\gamma }{4\delta (1+\gamma )}\left({\left(m+\delta \right)}^{2}-{\left(\widehat{x}\left({G}_{T}^{e}\right)\right)}^{2}\right)\right]<\tau m\left[1-h-\frac{\gamma }{(1+\gamma )}\right]$, $\forall {G}_{T }^{e}\ge 0$, a sufficient condition for $\frac{\partial F}{\partial K}>0$ is $h<\frac{1}{(1+\gamma )}$<1.

Lemma 3.

For ${{P}}\in \left\{{{A}},{{B}}\right\}$ and $\forall\; {{\text{G}}}_{{\text{T}}}^{{\text{e}}}\ge 0$, the fiscal room is enough to guarantee that ${{\text{G}}}_{{\text{T}}}^{{\text{P}}}>1$ if

$$M-\frac{1+\gamma \left(\alpha +\eta \right)}{\eta \gamma }>K\frac{\gamma }{4\delta (1+\gamma )}\left({\left(m+\delta \right)}^{2}-max{\left(m-\delta ; \frac{{\left(1+ \gamma \right)}^{\frac{1+ \gamma }{\gamma }}}{ \gamma }\right)}^{2}\right)$$

Proof.

The above condition is obtained from A.1 by considering its minimum value and $\tau$ preferred by group A.⁴⁷ Consequently, it holds a fortiori when the budget increases, and, given the ranking of the preferred policies in expression (14), it is satisfied a fortiori for $P=B$.

Proof of Proposition 1.

For $P\in \left\{A, B\right\}$, given parameters’ restriction in Lemma 3, ${\Delta }^{P}\left({G}_{T}^{e}\right)>1$ and it is continuous monotone nondecreasing in ${G}_{T}^{e}$ (see Lemma 2), with ${G}_{T}^{e}\in {\mathbb{R}}_{+}$.⁴⁸ Its minimum value is obtained for $0\le {G}_{T}^{e}\le 1$

$$ \min {\Delta }^{P} \left( {G_{T}^{e} } \right) = z^{P} \left[ {M - K\frac{\gamma }{{4\delta \left( {1 + \gamma } \right)}}\left( {\left( {m + \delta } \right)^{2} - \max \left[ {m - \delta ,\left( {\frac{{\left( {1 + \gamma } \right)^{{\frac{1 + \gamma }{\gamma }}} }}{ \gamma }} \right) } \right]^{2} } \right)} \right] $$

The maximum value is obtained when $\widehat{x}\left({G}_{T}^{e}\right)\ge m+\delta$. In this case, public tertiary education spending is equal to its maximum

$${\text{max}}{\Delta }^{P}\left({G}_{T}^{e}\right)={z}^{P}M$$

There are two possibilities:

• (i) ${z}^{P}M<\frac{\gamma \left(m+\delta \right)}{{\left(1+ \gamma \right)}^{\frac{1+ \gamma }{\gamma }}}$

• (ii) ${z}^{P}M\ge \frac{\gamma \left(m+\delta \right)}{{\left(1+ \gamma \right)}^{\frac{1+ \gamma }{\gamma }}}$

First, we must show that the function ${\Delta }^{P}({G}_{T }^{e})$ crosses the 45-degree line exactly once when ${z}^{P}M<\frac{\gamma \left(m+\delta \right)}{{\left(1+ \gamma \right)}^{\frac{1+ \gamma }{\gamma }}}$. In this case, ${{\Delta }^{P}(G}_{T}^{e})< {G}_{T}^{e}$ when${G}_{T}^{e}=\frac{\gamma \left(m+\delta \right)}{{\left(1+ \gamma \right)}^{\frac{1+ \gamma }{\gamma }}}$, while ${\Delta }^{P}$ (${G}_{T}^{e})> {G}_{T}^{e}$ when${0\le G}_{T}^{e}\le 1$. We can exclude that ${\Delta }^{P}({G}_{T }^{e})$ crosses more than once, as the relationship between the budget and ${G}_{T}^{e}$ is quadratic, and it is easy to verify that the first derivative of the function ${{\Delta }^{P}(G}_{T}^{e})$ with respect to ${G}_{T}^{e}$ is monotone nondecreasing. Therefore, ${{\Delta }^{P}(G}_{T}^{e})$ crosses the 45-degree line only once and the fixed point ${G}_{T}^{P*}$ lies in the interval $1<{G}_{T}^{P*}<{z}^{P}M.$

In case (ii), there is instead the possibility of triple crossing.⁴⁹${{\Delta }^{P}(G}_{T}^{e})$ certainly crosses the 45-degree line at ${G}_{T}^{e}=$ ${z}^{P}M$ and therefore the fixed point ${G}_{T}^{P*}={z}^{P}M$ always exists. The function might double cross below ${z}^{P}M$ and in this case, we might have two additional interior fixed points with 1 $<{G}_{T}^{P*}<{z}^{P}M$.

For $P=C,$ as ${\Delta }^{C}\left({G}_{T}^{e}\right)=0 \; \forall \; {G}_{T}^{e},$ there exists a unique fixed point ${G}_{T}^{C*}=0$ for ${G}_{T}^{e}=0$.

Proof of Proposition 2.

2.1 To prove that ${G}_{T}^{B*}>{G}_{T}^{A*}$ recall that ${z}^{B}>{z}^{A}$. When ${G}_{T}^{P*}={z}^{P}M$ the result is obvious. Suppose that ${G}_{T}^{P*}$ is an interior fixed point and consider the following implicit function Z:

$$ Z\left( {G_{T}^{P} ,{\Delta }^{P} \left( {G_{T}^{P} } \right)} \right) = G_{T}^{P} - z^{P} \left[ {\left[ {M - K\frac{\gamma }{1 + \gamma }{\Omega }\left( {G_{T}^{P} } \right)\left( {m + {\updelta }\left( {1 - {\Omega }\left( {G_{T}^{P} } \right)} \right)} \right)} \right]{ }} \right] = 0 $$

(A.2)

As ${G}_{T}^{P*}$ is a solution of (A.2), by the implicit function theorem, we can write

$$\frac{d{G}_{T}^{P*}}{d{z}^{P}}=-\frac{\frac{\partial Z}{\partial {z}^{P}}}{\frac{\partial Z}{\partial {G}_{T}^{P*}}}<0$$

Indeed, the numerator ($\frac{\partial Z}{\partial {z}^{P}})$ has negative sign. The denominator ($\frac{\partial Z}{\partial {G}_{T}^{P*}})$ has positive sign because, when ${G}_{T}^{P*}$ is an interior solution and a focal point, the function ${{\Delta }^{P}(G}_{T}^{e})$ has a positive slope smaller than 1 in ${G}_{T}^{P*}.$

2.2 To prove that for $P=A$ $\frac{d{G}_{T}^{A*}}{d\eta }>0$, we first note that $\frac{d{z}^{A}}{d\eta }$>0. Hence, the result is obvious when ${G}_{T}^{A*}={z}^{A}M.$ Suppose that ${G}_{T}^{A*}$ is an interior solution and consider the function (A.2). By the implicit function theorem, we can write:

$$\frac{d{G}_{T}^{A*}}{d\eta }=-\frac{\frac{\partial Z}{\partial\upeta }}{\frac{\partial Z}{\partial {G}_{T}^{P*}}}$$

The numerator ($\frac{\partial Z}{\partial\upeta })$ has negative sign for P = A. The denominator ($\frac{\partial Z}{\partial {G}_{T}^{P*}})$ has positive sign because, when ${G}_{T}^{P*}$ is an interior solution and a focal point, the function ${{\Delta }^{P}(G}_{T}^{e})$ has a positive slope smaller than 1 in ${G}_{T}^{P*}$; thus, $\frac{d{G}_{T}^{A*}}{d\eta }>0$.

2.3 Firstly, note that when ${G}_{T}^{P*}={z}^{P}M,$ $\frac{d{G}_{T}^{P*}}{d\delta }=0.$ Focusing on interior solutions, to prove that $\frac{d{G}_{T}^{P*}}{d\delta }\le 0$, iff ${\widehat{x}\left({G}_{T}^{P*}\right)}^{2}\ge {m}^{2}-{\delta }^{2}$, consider the function (A.2). By the implicit function theorem, we can write

$$\frac{d{G}_{T}^{P*}}{d\delta }=-\frac{\frac{\partial Z}{\partial \delta }}{\frac{\partial Z}{\partial {G}_{T}^{P*}}}$$

The numerator sign depends on the sign of the public budget derivative with respect to inequality. Therefore, from Lemma 2$\frac{\partial Z}{\partial \delta }>0$ iff ${\widehat{x}\left({G}_{T}^{P*}\right)}^{2}>{m}^{2}-{\delta }^{2}$. The denominator ($\frac{\partial Z}{\partial {G}_{T}^{P*}})$ has positive sign because, when ${G}_{T}^{P*}$ is an interior solution and a focal point the function ${{\Delta }^{P}(G}_{T}^{e})$ has a positive slope smaller than 1; thus, $\frac{d{G}_{T}^{P*}}{d\delta }<0$ iff ${\widehat{x}\left({G}_{T}^{P*}\right)}^{2}>{m}^{2}-{\delta }^{2}$

2.4 Firstly, note that when ${G}_{T}^{P*}={z}^{P}M,$ $\frac{d{G}_{T}^{P*}}{dK}=0$. Focusing on interior solutions, to prove that $\frac{d{G}_{T}^{P*}}{dK}>0$ if $h<\frac{1}{(1+\gamma )}$ we have to consider the implicit function in A.2, the conditions stated in Lemma 2, and follow the same logical steps of the previous proofs.

Proof of Proposition 3.

Consider only focal fixed points.

3.1 For $\left[{\tau }^{*}= {\tau }^{A},{ \phi }^{*}= {\phi }^{A}\right]$ to be a political equilibrium of the voting game, with ${{G}_{T}^{e}=G}_{T}^{A*}$, it must be $\Omega \left({G}_{T}^{A*}\right)\le \frac{1}{2K}$ (group C is not majoritarian), and $1-\Omega \left({G}_{T}^{A*}\right)\le \frac{1}{2K}$ (group B is not majoritarian). Recalling that the measure of group A does not depend on the opting out rate, if A is majoritarian ($1-K>\frac{1}{2}$), these two conditions are always satisfied for any $1< {G}_{T}^{A*}\le {z}^{A}M$; thus, $\left[{\tau }^{*}= {\tau }^{A},{ \phi }^{*}= {\phi }^{A}\right]$ is the unique political equilibrium.

3.2 From Proposition 1 and Proposition 2, ${G}_{T}^{C*}<{G}_{T}^{A*}<{G}_{T}^{B*}$ and $\Omega \left({G}_{T}^{C*}\right)>\Omega \left({G}_{T}^{A*}\right)>\Omega \left({G}_{T}^{B*}\right)$. To show that at least one political equilibrium exists, note that if neither $\left[{\tau }^{*}= {\tau }^{C}, {\phi }^{*}= {\phi }^{C}\right]$ nor $\left[{\tau }^{*}= {\tau }^{B},{ \phi }^{*}= {\phi }^{B}\right]$ are political equilibria, that is if $1-\Omega \left({G}_{T}^{B*}\right)\le \frac{1}{2K}\; {\text{and}}\; \Omega \left({G}_{T}^{C*}\right)\le \frac{1}{2K},$ then $\left[{\tau }^{*}= {\tau }^{A},{ \phi }^{*}= {\phi }^{A}\right]$ is certainly a political equilibrium with ${{G}_{T}^{e}=G}_{T}^{A*}$. In fact,$1-\Omega \left({G}_{T}^{B*} \right)\le \frac{1}{2K}$ implies $1-\Omega \left({G}_{T}^{A*}\right)<\frac{1}{2K}$ (group B is not majoritarian); whereas $\Omega \left({G}_{T}^{C*}\right)\le \frac{1}{2K}$ implies $\Omega \left({G}_{T}^{A*}\right)<\frac{1}{2K}$ (group C is not majoritarian). In this political equilibrium, group A is pivotal, although not majoritarian (see preferred policies ranking in expression (14) in the text).

To show that a multiplicity of political equilibria might arise, note that for the fixed points ${{G}_{T}^{A*},{G}_{T}^{B*},G}_{T}^{C*}$ to be associated with political equilibria the following conditions must be satisfied:

$\left(i\right) 1-\Omega \left({G}_{T}^{A*}\right)\le \frac{1}{2K}\; {\text{and}} \; \Omega \left({G}_{T}^{A*}\right)\le \frac{1}{2K}$ for ${G}_{T}^{A*};$

$\left(ii\right) 1-\Omega \left({G}_{T}^{B*} \right)>\frac{1}{2K}$ for ${G}_{T}^{B*}$.

(iii) $\Omega \left({G}_{T}^{C*}\right)>\frac{1}{2K}$ for ${G}_{T}^{C*}$.

These conditions might be simultaneously satisfied.

Appendix 2: Descriptive evidence and multivariate correlation analysis

Figure 3 presents education expenditure as a share of GDP in 2019 and its private and public funding composition for a sample of 43 high-middle income countries. It demonstrates the substantial variability in education expenditure and the significant differences in the source of funding among countries.⁵⁰ On average, the share of GDP devoted to education was about 5%, ranging from the lowest value in Romania (2.5%) up to the highest in Cyprus (6.9%). In terms of composition, on average 18% of education expenditure was from private funding, with the highest value in Peru (43%) and the lowest in Romania (0.1%).

Fig. 3

Public and private education expenditure, % of GDP, year 2019 or latest available

Figure 4 presents the allocation of public expenditure on basic and tertiary education as shares of GDP. It shows a positive relationship between the two education tiers, but also a great variability: countries with the same share of public basic education might have very different expenditures on tertiary education.

Fig. 4

Public spending on basic vs tertiary education, %GDP, year 2019 or latest available

Building on our theoretical results, we endeavour to analyse countries’ differences in education expenditures by relating these differences to the variations in income inequality and intergenerational persistence in education. To characterise these relationships, we perform multiple regression correlations. This analysis has only a descriptive purpose as the limited number of observations and endogeneity issues, due to reverse causality and omitted variables, pose major challenges for identifying causal effects. We have collected data on three education-spending variables (public basic, public tertiary and total private) and the proportion of public basic education expenditure on total public education expenditure (public basic/total public) for 43 high and middle income countries covering two time periods.⁵¹ In the first period, expenditures are computed as averages over the years 2000 to 2009; in the second period, averages are computed over the years 2010 to 2019.⁵² To mitigate the issue of a potential reverse causality link, whereby low public expenditure in education leads to more inequality and higher education persistence, we have considered the values of income inequality and intergenerational education persistence that precede the observed values of education expenditure.⁵³ Income inequality is measured by the variable GINI, which is the Gini index of disposable income for the years 2000 and 2010.⁵⁴ To assess intergenerational education persistence, we use the variable COR, which measures the correlation between the years spent on education by parent and child. We obtain data from the 2023 Global Database on Intergenerational Mobility from the World Bank (WB 2023) for the 1970s and 1980s cohort.⁵⁵ Moreover, as different standard of living may affect government spending on education, we include the real GDP per capita as a control variable.⁵⁶ Table 1 contains the descriptive statistics.

Table 1 Summary statistics

Table 2 presents the outcome of a pooled linear regression between the four education spending variables (public basic, public tertiary, total private and public basic/total public) and our two main variables of interest (COR and GINI). To control for time effects, we add a dummy that takes value 0 in the first period and value 1 in the second.

Table 2 Regression results

In line with the model, public expenditure on education (as percentage of GDP) is negatively correlated with COR and GINI.⁵⁷ COR is significant in the relationship with public basic education (Table 2, column 1), while GINI is relevant in the relationship with public tertiary education (Table 2, column 2). In column 3, we again consider public tertiary education and add an interaction term between COR and GINI. Consistent with the model’s results, the coefficients of COR and GINI are both significant and negative. The coefficient of the interaction term is positive and significant, implying that the negative relationship between intergenerational persistence in education and public tertiary spending declines as inequality increases. To clarify this relationship, we compute marginal effects (Fig. 5).

Fig. 5

Average marginal effects of COR on Public Tertiary for increasing values of GINI

The sign of the marginal effect of COR on public tertiary spending is negative for low values of GINI and becomes positive for high values of income inequality. To explain why the negative correlation between COR and tertiary public expenditure becomes weaker as income inequality increases, note that private expenditures, which are mainly concentrated on tertiary education, are strongly and negatively correlated to GINI (see Table 2, column 4). Indeed, if providers of tertiary education are mainly private (e.g. in Anglo-Saxon countries), we observe a strong negative relationship between COR and tertiary education spending when considering private (tertiary) expenditure (see Table 2, column 4). As income inequality increases further, the marginal effect of COR on tertiary public education spending becomes positive. Indeed, at odds with our theoretical conjectures, in some countries where both income inequality and intergenerational persistence in education are remarkably high (e.g., Mexico, Turkey, Argentina and Chile), we observe that the allocation of public education expenditure is biased towards tertiary public education. This scenario might describe a situation where the private sector is not sufficiently developed and the political power is in the hands of well-educated minorities, who would benefit most from an increase in public tertiary education spending (see Su (2006)).⁵⁸ This circumstance is not captured by our model, whose assumptions about the political process are more apt to describe an environment where power is equally distributed.

In the last column of Table 2, we examine the share of public education spending devoted to basic education. We do not find any significant correlation between COR and the share of public basic education spending on total public expenditure. However, this share is significantly and positively correlated with GINI. This might be consistent with our model; as income inequality increases the weight of the middle class decreases and it is more likely the emergence of an equilibrium ends against the middle, as described by Epple and Romano (1996), where the interests of the well-educated elite who opts out of the public system—mainly at the tertiary level- become closer to the interests of poor, less educated agents who are not interested in tertiary public education. Finally, as expected, log real GDP per capita is positively and significantly correlated with public expenditure in both education tiers.