A system dynamics model for long-term planning of the undergraduate education in Brazil

Abstract

Higher education in Brazil has experienced a rapid expansion since the 1990s as a consequence of the government’s pliability in launching new programs and educational institutions. This expansion was mainly driven by the private sector. Despite this expansion, Brazil has not yet achieved the enrollment goal expected in the National Education Plan launched in 2010. Moreover, the demand for undergraduate programs, is presenting signs of reduction, characterizing a system with fast initial growth followed by stagnation. This paper presents the construction and application of a system dynamics model for analyzing long-term policies concerning undergraduate programs in Brazil at an aggregate level. The main objective of the model is to conduct scenario analysis given by the different behavior of several aspects related to the system, such as government regulation, demand, places, and the balance between public and private sectors. A scenario analysis was conducted, considering different policies regarding the nature of education and economic development. The results are highly promising, demonstrating the potential of this approach for both understanding the dynamic behavior of higher education, improving policies, and developing effective strategies.

Appendix: Mathematical representation of stock and flow diagrams

There is more than one way to represent stock and flow diagrams. We adopted the notation used by Sterman (2000) in his classic book. Figure 17 presents a very simple stock and flow diagram to illustrate the related concepts. Stocks can change their state only through their flows, which, in turn, could have influences from other stocks or auxiliary variables. Some variables that represent information or status, or are external influences, are better modeled using auxiliary variables.

Fig. 17

Stock and flow diagram using iThink© notation. Source: Sterman (2000)

The simulation of the system behavior is made possible by the equations that characterize the behavior of flows and stocks in the system over time. Stocks are modeled with integral equations, as in Eq. (1), representing the accumulation from time t ₀ to moment t. Inflows(t) and Outflows(t) represent the values of an input stream and an output stream at any time t.

$$Stock(t) = \int\limits_{{t_{0} }}^{t} {[Inflows(t) - Outflows(t)]ds + Stock(t_{0} )}$$

(1)

Equivalently, the rate of change of any stock is the Inflow minus the Outflow defined by the differential Eq. (2) as follows:

$$d(Stock)/dt = Inflow(t) - Outflow(t)$$

(2)

In system dynamics, we transform the differential equations represented in (2) into the following difference equation:

$$Stock(t + 1) = Stock(t) + \varDelta t* \, [Outflow(t){-}Inflow(t)]$$

Using SD nomenclature, we can describe the system presented in Fig. 4 by the four level equations and the six rate equations presented below. Since the preliminary CLD in Fig. 2 was not implemented (rather, we implemented the CLD in Fig. 3), we use f(.) to represent how involved variables are related. In several cases, this function is a probabilistic one defined by using historic data.

Level equation 1:

$$\begin{gathered} {\text{Number\_of\_Students(t)}} = {\text{Number\_of\_Students(t}} - {\text{dt)}} + ( {\text{Demand\_Rate}} - {\text{Graduates)}}*{\text{dt}} \hfill \\ {\text{INIT Number\_of\_Students}} = {\text{Initial value of number of Students}} \hfill \\ \end{gathered}$$

INFLOWS:

$${\text{Demand}}\_{\text{Rate}} = f({\text{Places\_Capacity,}}\;{\text{Tuition\_Fee\_perception,}}\;{\text{Socioeconomics\_Status}})$$

OUTFLOWS:

$${\text{Graduates}} = {\text{f}}({\text{Graduation rate}})$$

Level equation 2:

$$\begin{gathered} {\text{Places\_Capacity(t)}} = {\text{Places\_Capacity}}({\text{t}} - {\text{dt}}) + ({\text{Opening\_Places}} - {\text{Closing\_Places}})*{\text{dt}} \hfill \\ {\text{INIT Places\_Capacity}} = {\text{Initial value of places capacity}} \hfill \\ \end{gathered}$$

INFLOWS:

$${\text{Opening}}\_{\text{Places}} = {\text{f}}({\text{Number}}\_{\text{of}}\_{\text{Students, Places}}\_{\text{Capacity}})$$

OUTFLOWS:

$${\text{Closing}}\_{\text{Places}} = {\text{f}}({\text{Number\_of Students, Places\_Capacity}})$$

Level equation 3:

$$\begin{gathered} {\text{Tuition}}\_{\text{Fee}}\_{\text{Perception(t)}} = {\text{Tuition\_Fee\_Perception(t}} - {\text{dt)}} + ({\text{Change\_in\_Tuition\_Fee\_Perception}})*{\text{dt}} \hfill \\ {\text{INIT Tuition\_Fee\_perception}} = {\text{Initial value of Tuition\_Fee\_Perception}} \hfill \\ \end{gathered}$$

INFLOWS:

$${\text{Change\_in\_Tuition\_Fee\_Perception}} = {\text{f}}({\text{Number\_of\_Students,}}\;{\text{Delay\_of\_Perception}})$$

Level equation 4:

$${\text{Young\_Population(t)}} = {\text{Young}}\_{\text{Population (t}} - {\text{dt)}} + ({\text{Birth}}\_{\text{Rate}} - {\text{Demand}}\_{\text{Rate}})*{\text{dt}}$$

$${\text{INIT}}\;{\text{Young\_Population = Initial value of young population}}$$

INFLOWS:

$${\text{Birth}}\_{\text{Rate}} = {\text{f}}({\text{birth}}\;{\text{rate}})$$

OUTFLOWS:

$$\begin{gathered} {\text{Demand}}\_{\text{Rate}} = f({\text{Places}}\_{\text{Capacity, Tuition}}\_{\text{Fee}}\_{\text{Perception}},{\text{ Socioeconomics}}\_{\text{Status}}) \hfill \\ {\text{Socioeconomics}}\_{\text{Status}} = {\text{f}}({\text{Socioeconomics}}\_{\text{Status}}) \hfill \\ \end{gathered}$$