The conceptual model: system dynamics framework
Climate change presents a number of challenges for sustainability around the globe. South Africa is no exception, as shown by several disasters such as floods, droughts and fires, which have resulted in huge and frequent economic and social damages. What is needed is to improve the representation of climatic and biophysical systems and, in particular, the anthropogenic (human) activities that influence or control the impacts of climate change at national, provincial and local levels.
System dynamics can help decision-makers to understand the structure and characteristics of a complex system. It integrates system-thinking theory, cybernetics and information theory, and is useful when dealing with problems of high-level, non-linear and multiple feedbacks. It is a well-established approach for describing the behaviour of a system (Forrester 1961, 1968) and uses causal loop diagrams or stock and flow diagrams. A system dynamics model provides a flexible way of understanding the dynamic relationships between the variables and their interactions/linkages. The principles used to develop such a model can be found in a number of studies, including Forrester (1961), Randers (1980), Richardson & Pugh (1981) and Mohapatra (1994).
A causal loop diagram (see Fig. 4) was developed to illustrate the relationship between climate variability and change, municipal economic development and population, as well as factors associated with climate change and demand for electricity at municipal levels. South African municipalities need to put appropriate strategies in place to ensure sustainable provision of water services. Yet population and industrial growth increase demand for water significantly, and climate conditions such as rainfall and temperature can affect its supply. The interactions shown in Fig. 4 guided the selection of variables used to estimate the econometric models.
Figure 4. Causal-loop diagram.
Source: Authors analysis.
Note: A causal loop diagram explains the impact dynamics and feedback of the system being studied. The term ‘causal’ refers to a cause–effect relationship. ‘+’ on an arrow connecting two variables indicates that the variable at the tail of the arrow causes a change in the variable at the head of the arrow in the same direction; ‘−’ indicates a change in the opposite direction. GDP, gross domestic product.
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Figure 4 highlights a number of possible interactions between three systems: water (total water demand and supply), socio-economic (municipal population, investment in water-related infrastructure and GDP), and climatic (rainfall and evaporation, which is a function of temperature and humidity). GDP growth is expected to lead to reduced population, but GDP and population growth will also raise water demand, which in turn will increase the water demand–supply gap, hence the need for more investment in infrastructure to meet the needs for water provision.
The causal loop shown in Fig. 4 reasonably represents the real-world relationships (water demand/supply, electricity demand/supply and climatic conditions) that empirical models need to consider. For instance, in the real world, water demand for industry, agriculture and other municipal activities needs to indicate the feedback influences of rainfall, temperature and storage capacities.
The econometric models
The water and energy sectors are central to the welfare of citizens and functioning of municipalities in South Africa and beyond. Figure 4 shows how climate conditions (particularly changes in rainfall) affect water demand, which in turn affects the energy supply–demand gap. Using municipal expenditures on water and electricity as proxies for municipal water and electricity demand, the hypothesised link between these indicators and climate variability and/or change for municipality m at time t can be modelled as a system of equations as follows:
for m = 1,… … N municipalities over t = 1,… … T years.
The dependent variable Water is the municipal expenditures (in Rand) on water-related services and infrastructure, and Electricity is the municipal expenditures (in Rand) on electricity-related services and infrastructure. The climate change variables are represented by C, as the focus is on rainfall variability as a proxy for climate variability and change. In addition to these climate variables, characteristics of municipalities believed to affect municipal water, and electricity expenditures are controlled for. They are captured by the vectors W and X for water and electricity expenditures, respectively. The parameters to be estimated are β 0, β 1, β2, α 0, α 1 and α2. The error terms for each equation are denoted by ε and μ. These error terms are assumed to be such that (C,ε), (W,ε), (C,μ), and (X,μ) ∼ i.i.d and N(0,σ 2).
Time-series data on water and electricity expenditures by municipalities, as well as rainfall variability and other control variables, are combined to form a panel dataset. In estimating the systems of equations in Eq. (1), this structure of pooled data can be taken into account in three ways, each introducing the intercept into the model in a different way. Specifically, the estimated model may have a common, fixed or random intercept term. In other words, the three model specifications are the common effects regression, fixed effects and random effects models.
The common effects regression model involves pooling all the data for the municipalities, assuming that the parameters do not vary across sample observations, as done in Eq. (1). In this model, the level of demand for water and electricity – proxied by municipal expenditures on water- and electricity-related infrastructure and/or services, respectively – is assumed to be homogeneous across municipalities. These assumptions imply that the equations in Eq. (1) can be estimated using panel data ordinary least squares (OLS).
Alternatively, the intercept term could be allowed to vary across municipalities. This is done by introducing dummy variables to take into account the differences in the levels of water and electricity expenditure across the municipalities. Referred to as the fixed effects model, it can be represented as:
where βi and αi represent the intercept coefficient for the ith cross-sectional municipality in each respective equation, and Mit are dummy variables that take a value of one for observations belonging to the ith municipality and zero otherwise. This is appropriate when specifying a different intercept coefficient for each cross-sectional unit that adequately captures differences in municipalities.
A random effects model can be used as an alternative to the fixed effects model. It assumes that the coefficients are random variables drawn from a larger population such that:
where it is assumed that E[ui] = 0, E[ui 2] = σu 2, E[uiuj] = 0 for i ≠ j, E[uiεit] = 0; and E[ηi] = 0, E[ηi 2] = ση 2, E[ηiηj] = 0 for i ≠ j, E[ηiηit] = 0.
The structure of the model means that, for a given municipality, the correlation between any two disturbances in different time periods is the same. In other words, the correlation is constant over time and identical for all municipalities. Using the random effects estimator means that, in addition to controlling for unobserved effects, the intra-municipality correlation can also be controlled because of unobserved clusterxeffects (Wooldridge 2002). The principle is that the results from a random effects model can be generalised to the whole population from which the sample is taken. Thus, while the fixed effects specification assumes that municipality-specific effects are fixed parameters, the random effects specification assumes that municipalities are made up of a random sample, and the municipality-specific effects are assumed to be independently distributed with a mean of zero and a constant variance.
In principle, both the fixed and the random effects specifications provide some estimation efficiency gains over the simple common effects regression model when using panel data. However, in deciding which model specification to base the inferences on, several post-estimation statistical tests are performed. An F-test is used to test for the common coefficients specification against the fixed and random effects specifications. In case the specification is rejected, which implies that the common effects estimators are biased and spurious, a Hausman test is used for the fixed versus the random effects specification.
The municipality is the unit of analysis. A municipal-level panel data set is used, containing comprehensive information on the characteristics of the municipalities, as well as the average socio-economic characteristics of the households or individuals residing in those municipalities. The empirical panel data cover 283 municipalities across South Africa for the period 2005–2009.
The dependent variable, Water, is expenditure (in millions of Rand) on water-related services and infrastructure, and Electricity is expenditure (in millions of Rand) on electricity-related services and infrastructure. The level of expenditures is used as a proxy for municipal water and electricity demand. The assumption is that, in response to changes in water and electricity demand from their residents, municipalities change the level and composition of water- and electricity-related infrastructure and/or service provision, which in turn changes the level of water and electricity-related expenditures.
The main advantage of using the coefficient of variation is that, in contrast to the variance and standard deviation that need to be understood in the context of the mean of the variable, the coefficient of variation is a dimensionless number. The coefficient of rainfall variation thus illustrates how wide the spread of annual rainfall values is, recorded by municipalities. The larger the coefficient of variation, the more scattered the observed rainfall levels are on average. Rainfall variability indicators are computed at municipality level, implying that the values will be the same within municipalities but vary across municipalities.
The other control variables used in the empirical analysis include the value of unconditional grants allocated to municipalities by the national government (in thousands of Rand); gross-value added (in thousands of Rand), which is an equivalent of GDP at municipal level; the municipality's population; the land area covered by each municipality; population density; the share of population that is less than 19 years old; the share of population that is over 65 years old; and the number of councillors in a municipality, which indicates the size of the local government. The definition and descriptive statistics of these variables are presented in Table 1.
Table 1. Definition and descriptive statistics of the variables used in the econometric model
|Variable name||Variable definition||Mean||Standard deviation||Observations|
|Source: Authors estimates.|
|Water||Municipal expenditures on water-related infrastructure and provision (in million Rand)||39.92||118.80||678|
|Electricity||Municipal expenditures on electricity-related infrastructure and provision (in million Rand)||24.21||98.72||646|
|Rainfall||Annual rainfall in millimetres||417.83||1348.86||443|
|Variance of rainfall||Variance of annual rainfall in millimetres||1.30E+06||1.23E+07||470|
|Standard deviation of Rainfall||Standard deviation of annual rainfall in millimetres||228.95||1118.26||470|
|Rainfall variability||Coefficient of variation of annual rainfall||0.44||0.30||470|
|Population||Number of persons in municipality||167356.50||380108.50||1285|
|Municipal area||Area covered by the municipality, in square kilometres||4626.27||4423.01||1350|
|Population density||Population/municipal area, persons per square kilometre||80.12||208.36||1260|
|Grants||Unconditional grants allocated to the municipality by the national government (in thousand Rand)||4.87E+07||9.61E+07||1410|
|Gross value added||Gross value added (in thousand Rand)||5592.33||23047.13||1310|
|Population under 19||Proportion of population under the age of 19||0.44||0.04||1285|
|Population over 65||Proportion of population over the age of 65||0.05||0.01||1275|
|Councillors||Number of councillors||31.41||28.99||1406|
Table 1 shows that an average municipality spends close to R40 million per year on water-related infrastructure and/or services and about R24 million per year on electricity-related services and infrastructure. The average annual precipitation, for the municipalities where rainfall data were recorded, is 418 mm. Indicators reveal huge variability of rainfall across the years within municipalities, with an average coefficient variation of close to 44%. An average municipality covers around 4626 square kilometres and is home to about 167 357 residents, an average population density of close to 80 persons per square kilometre. The GDP equivalent for each municipality, on average, is around R6 million, while the average unconditional grant received by municipalities from the national government is approximately R49 billion. In an average municipality, 44% of the population is below the age of 19 years old, while 0.5% is over 65 years old.
The variables discussed previously are used to estimate Eq. (1). However, because of missing data for these variables, especially data on annual rainfall in most of municipalities (as revealed in Table 1, column 5), the effective sample used in the estimations covers 74 municipalities for which comprehensive rainfall data were available.