Educational Systems and the Intergenerational Transmission of Inequality: A Complex Dynamical Systems Perspective

Abstract

Human beings have many observable and measurable characteristics that distinguish and segregate them in many ways. Some segregations result from the structure of organizations through which an individual transit along—like the educational system—that interact with other modes of inherited segregations like race, income, sex, geographical area, etc. This chapter is about the intergenerational transmission of income inequalities induced by educational systems that generate persistent effects on societies. One may naturally think of the influence that aggregate characteristics of a community have on schooling decisions. Youngsters in low-income families, for example, are more likely to fail and dropout the school, starting out at a disadvantage in the labor market and restricting their future earnings. These individual decisions, in turn, affect aggregate outcomes that could potentially explain why societies show self-perpetuating inequalities in education and income distribution that reinforce each other over time. This issue is attracting significant attention recently, and new interdisciplinary knowledge is needed to inform and guide the current and future debates. A Complex Dynamical Systems (CDS ) approach is essential to understand the persistence of income inequalities that educational systems generate in the context of global trends. A CDS simulation model will be applied to track down the complex interaction of such inequalities and draw alternative causal inferences to those documented by studies using correlation methods. Baseline simulations are contrasted to data compiled from Nicaragua (2000–2010) in order to assess the model ability to replicate historical observations. The method can be extended for the purpose of designing and evaluating feasible, cost-effective policies to attenuate social and economic disparities in society. We conclude that inequality is detrimental for the operational efficiency of the educational system itself.

Appendices

Appendix

The following description of the simulation model is an excerpt from Guevara et al. (2014) reprinted with permission from the journal Nonlinear Dynamics, Psychology, and Life Sciences.

The Simulation Model

The educational model has 3 state variables: Population (P), Population in Primary School (G), and Primary School Graduates (H). These are represented by stocks (rectangles) in Fig. 15.3. P stands for the country’s total population, disaggregated into age cohorts and it is the main input to the education system (Eq. 15.1). The arrows in Fig. 15.3 are differential equations that modify the stocks; hence, population increases with births and decreases with deaths. Equation 15.1 shows that the birth rate B, is the product of a constant fractional vector β multiplied by the country’s population (i.e., the sum of all age-cohorts). Similarly, death rate D, is the result of a constant ϕ multiplied by the stock of population. In the model, aging [A(t)] represents the transition of the population from one age cohort to the next, after it has remained an average length of time (υ) in that cohort. P _a (0) is the initial population.

$$ {P}_a(t)={\displaystyle \underset{t={t}_0}{\overset{T}{\int }}\left[B(t)+{A}_{a-1}(t)-{A}_a(t)-{D}_a(t)\right]\;\mathrm{d}t}+{P}_a(0) $$

(15.1)

$$ \mathrm{where}\kern1.12em B(t)=\beta {\displaystyle \sum_a{P}_a}\kern1em 0<\beta <1 $$

$$ {D}_a(t)=\phi {P}_a\kern1em 0<\phi <1 $$

$$ {A}_a(t)=\raisebox{1ex}{${P}_a$}\!\left/ \!\raisebox{-1ex}{$\upsilon $}\right.\kern1em \upsilon \kern0.5em =\kern0.5em 1, $$

$$ {P}_a(t)=\mathrm{stock}\ \mathrm{of}\ \mathrm{age}\hbox{-} \mathrm{a}\ \mathrm{population},a=0,\ 1,\ 2, \dots,\ 15,\ \mathrm{and}\ \mathrm{Adults}\ \left(16\ \mathrm{or}\ \mathrm{more}\right). $$

$$ {A}_a(t)=\mathrm{aging}\ \mathrm{rate},\kern0.37em B(t)=\mathrm{birth}\ \mathrm{rate},\;{D}_a(t)=\mathrm{death}\ \mathrm{rate}, $$

The second state variable, G, is a matrix broken down by grade and age, encompassing children currently enrolled in school. Equation 15.2 shows that it consists of 6 grades according to the official cycle length in the country. In words, G _1,a (t) represents the population of age-a students attending the first grade. Once children enter the school system they may follow three mutually exclusive directions: (1) passing to the next level through promotion (p _i,a (t)) from grade i to i + 1 and growing older by 1 year (from a to a + 1); (2) repeating the year (r _i,a (t)) just passing to the next age cohort (from a to a + 1) but remaining in the same grade (i); or (3), dropping-out of the grade i at age a (d _i,a (t)). Note that in Eq. 15.2 intake [e _1,a (t)] only occurs in the first grade, denominated by ${p}_{0,a-1}(t),$ and promotion replaces it as an inflow after the second grade. Thus,

$$ {G}_{i,a}={\displaystyle \underset{t={t}_0}{\overset{T}{\int }}\left[{p}_{i-1,a-1}(t)+{r}_{i,a-1}(t)-{p}_{i,a}(t)-{d}_{i,a}(t)-{r}_{i,a}(t)\right]\mathrm{d}t}+{G}_{i,a}(0) $$

(15.2)

$$ \mathrm{where}\ {G}_{i,a}(t)=\mathrm{population}\ \mathrm{in}\ \mathrm{grade}\ i=1,\;2, \dots,\;6;\;\mathrm{age}a=0,\ 1,\;2, \dots,\;15,\;16\ \left(\mathrm{age}\;16\;\mathrm{and}\ \mathrm{above}\right). $$

$$ {p}_{i,a}(t)=\mathrm{promotion}\ \mathrm{grade}\ i\ \mathrm{at}\ \mathrm{age} $$

$$ {e}_{1,a}(t)=\mathrm{intake}\ \mathrm{rate}\ \mathrm{grade}\ 1\ \mathrm{a}\mathrm{t}\ \mathrm{a}\mathrm{ge}\ \mathrm{a};\;{p}_{0,a-1}(t)\equiv {e}_a(t) $$

$$ {r}_{i,a}(t) = \mathrm{repetition}\ \mathrm{grade}\ i\ \mathrm{at}\ \mathrm{age}\ a $$

$$ {d}_{i,a}(t) = \mathrm{dropout}\ \mathrm{grade}\ i\ \mathrm{at}\ \mathrm{age}\ a $$

All transition rates are specified as the product of a vector of fractions such as intake (α _1,a ), repetition (ρ _i,a ), dropout (δ _i,a ), and promotion $\left({\uppi}_{i,a}\equiv \left(1-{\delta}_{i,a}-{\rho}_{i,a}\right)\right)$ multiplied by the stock of people in the respective grade (in the case of intake, by the population stock, P). In addition, these fractional values change across grades but remain constant within grades $\left({\delta}_{i,a},\;{\rho}_{i,a},\;{\uppi}_{i,a}={\delta}_i,\;{\rho}_i,\;{\uppi}_i\right)$. The corresponding formulations are Eqs. 15.3–15.6.

$$ {e}_a(t)=e\left({P}_a(t),{\alpha}_{1,a}\right)={\alpha}_{1,a}{P}_a(t) $$

(15.3)

$$ {d}_{i,a}(t)=\mathrm{d}\left({G}_{i,a}(t),{\delta}_{i,a}\right)={\delta}_i{G}_{i,a}(t) $$

(15.4)

$$ {r}_{i,a}(t)=r\left({G}_{i,a}(t),{\rho}_{i,a}\right)={\rho}_i{G}_{i,a}(t) $$

(15.5)

$$ {p}_{i,a}(t)=p\left({G}_{i,a}(t),{\uppi}_{i,a}\right)={\uppi}_i{G}_{i,a}(t) $$

(15.6)

$$ {\alpha}_{1,a},\;{\delta}_{i,a},\;{\rho}_{i,a},\;{\uppi}_{i,a}\in \left(0,\kern0.1em 1\right)\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{every}\ a $$

The third stock in Fig. 15.3, H, accumulates graduates from primary education as shown in Eq. 15.7. Equation 15.8 describes the construction of an index h of per-capita human capital which is the number of living people who have completed primary school compared to the country’s population. This index ranges from 0 to 1 where 0 implies that no adult (i.e., no person aged 16 and above) has completed primary education and 1 means that all adults have at least finished it. Therefore

$$ {H}_{a=16}={\displaystyle \underset{t={t}_0}{\overset{T}{\int }}\kern0.5em \left[{\displaystyle \sum_a{p}_{6,a}(t)}-{D}_{a=16}(t)\kern0.6em \mathrm{d}t\right]}+{H}_{a=16}(0) $$

(15.7)

$$ {h}_{a=16}=\frac{H_{a=16}}{P_{a=16}},\kern1.5em \mathrm{where}\ 0\le h\le 1 $$

(15.8)

Equations 15.1 to 15.8 allow the construction of the two performance indicators: the gross enrollment rate (from Eqs. 15.1 and 15.2) and the primary completion rate (from Eqs. 15.1 and 15.6):

$$ \mathrm{G}\mathrm{E}\mathrm{R}={\displaystyle \sum_{i,a}\frac{G_{i,a}(t)}{P_{7-12}(t)}} $$

(15.9)

$$ \mathrm{P}\mathrm{C}\mathrm{R}={\displaystyle \sum_a\frac{p_{i,a}(t)}{P_{12}(t)}} $$

(15.10)

Model Calibration

To calibrate the model it is necessary having a complete dataset for at least one point in time in which all stock variables are disaggregated by age, income group, and level of education attained. In this model that data point corresponds to year 2000 (LSMS 2001, 2005 and World Bank, 2015) and Table 15.3 shows this data point for the population variable used in the model, disaggregated by age and income. Likewise, Table 15.4 presents average parameter values for repetition and dropout rates across all grades for year 2000.