Designing service growth during the life-cycle of the water supply project

To value the opportunity cost of spare capacity, we compute the AIC by assuming a full use of the design capacity throughout the project design lifetime, which we call the average incremental cost at the full use of design capacity (AICF). Then we compute the difference between the average incremental cost at the expected use of design capacity (AICE) and AICF, namely AICE–AICF. This value assesses an opportunity cost of spare capacity expressing the opportunity cost of spare capacity per unit of service provided by the project during its design lifetime. For a project designed to provide an increasing annual level of services, the AICF is expected to understate the AICE by an amount determined by the time path of the unused production capacity. Exceptions may occur when recurrent costs are prominent with respect to investment costs, and strongly related to the level of service provided or to the evolution over time of the price of resources used in recurrent activities. In such exceptional situations the AICF may overstate the AICE, implying a negative unit opportunity cost of spare capacity.

5.5.5 designing service growth during the life-cycle of the water supply project

To quantify the services provided by a water supply system in each year t of its life-cycle, we consider three alternative indicators:

• the size of the population served, denoted by Pt;

• the number of household water connections, denoted by Ht;

• the quantity of water supplied, denoted by Qt.

To design consistent life-cycle scenarios for these production indicators, we start by specifying independent scenarios for the population served and for the following two other variables:

• the average size of the household served, denoted by Nt;

• the average per capita consumption of water of the population served, denoted by qt.

These last two variables allow us to derive (from the population-served scenario) consistent life-cycle scenarios for the number of household water connections and for the quantity of water supplied, simply by dividing Pt by Nt and by multiplying Pt by qt, respectively. Thus we compute: Ht = Pt/Nt and Qt = Ptqt.

The life-cycle scenarios of variables Pt, Nt and qt can be entirely designed by the user, by setting the value of these quantitative indicators for each year of the project life-cycle. The scenarios may also be modelled in a more parsimonious way by means of the following formula:

Xt = X1+(X_θ+1−X F t1) ( −1; , , ),α β θ

where X_t denotes the variable to be modelled, X₁ its initial value when the water supply system starts to be used (beginning of year t = 1), X_θ+1 its final value (beginning of year t =θ + 1) corresponding to full capacity use of the system reached after θ≤ T full years of the project life-cycle (T full years), and F(τ;α,β,θ) a beta cumulative distribution function of the continuous time variable τ defined on the interval [0;θ].

This function expresses the shape of the time trend followed by variable X_t to reach, after θ full years, its final value X_θ+1 from its initial value X₁. Therefore, the function depicts a growth scenario if X₁< X_θ+1 a decline scenario if X₁> X_θ+1 and a steady state scenario if X₁= X_θ+1.

The profile of this time trend is determined by the value of parameters α and β, which determine the shape of a beta cumulative distribution function and of its underlying density function, expressing the instantaneous rate of change (speed) of this time trend. By choosing appropriate values of these positive parameters α and β, a wide range of time trend profiles can be generated. A qualitative analysis of the shape of these profiles as a function of these parameters is presented in Annex V.

To summarize the detailed analysis of Annex V, five qualitatively different growth profiles of the cumulative distribution function F(τ;α,β,θ) can be generated, depending on the choice of one of the seven different time profiles that can be used to model the evolution of the instantaneous speed of growth during the water supply system’s life-cycle of θ= T = 30 years.

• As illustrated in Figure 5.5, by assuming a bell-shaped speed time profile (α> 1, β> 1), namely a speed that first increases from 0 to a maximum value then decreases to zero, we generate an S-shaped growth profile, namely accelerated growth at the beginning of the life-cycle followed by a decelerated growth at the end of the life-cycle. These profiles are symmetrical about τ=θ/2 if α=β.

figure 5.5 S-shaped growth profile.

• As illustrated in Figure 5.6, by assuming a U-shaped speed time profile (0 < α < 1 and 0 < β < 1), namely a speed that first drops from infinity to a minimum value then rises up to infinity, we generate a rotated S-shaped growth profile (about the segment joining the initial to the final growth trend value), namely decelerated growth at the beginning of the life-cycle followed by accelerated growth at the end of the life-cycle. These profiles are symmetrical about τ = θ/2 if α = β.

figure 5.6 Rotated S-shaped growth profile.

• As illustrated in Figure 5.7, by assuming a uniform-shaped speed time profile (α=β = 1), we generate a linear-shaped growth profile.

figure 5.7 Linear-shaped growth profile.

• As illustrated in Figure 5.8, a J-shaped growth profile is generated by assuming an increasing speed time profile (α ≥ 1 and 0 < β ≤ 1), either linear-shaped (α = 2 and β = 1), as in Figure 5.7, or J-shaped (α ≥ 1 and 0 < β < 1 or α > 2 and β = 1) with an increasing convex profile, or rotated J-shaped (1 < α < 2 and β = 1) with an increasing concave profile, obtained by rotating the J-shaped profile about the segment joining the initial to the final profile value.

figure 5.8 J-shaped growth profile.

• As illustrated in Figure 5.9, a rotated J-shaped growth profile is generated by assuming a decreasing speed time profile (0 < α ≤ 1 and β ≥ 1), either linear-shaped (α = 1 and β = 2) as in Figure 5.7, or reverse J-shaped (0 < α < 1 and β ≥ 1 or α = 1 and β > 2) with a decreasingly convex profile, or rotated reverse J-shaped (α = 1 and 1 < β < 2) with a decreasing concave profile obtained by rotating the reverse J-shaped profile about the segment joining the initial to the final profile value.

The choice of one of these growth profiles to model a scenario of the services provided by a water supply system during its life-cycle depends on the kind of indicator used to quantify the system’s production.

When the production indicator is represented by the services provided to the population served P_t measured in terms of inhabitant-year (services provided by the water supply system to an inhabitant during a full year), the growth profile of the cumulative distribution function F(τ;α,β,θ) should represent the growth path of the population served by the water supply system during its life-cycle. Demography shows that, in the very long run, the growth potential of a human population is bounded by the carrying capacity of the territory on which the population is settled. Therefore, in such a situation, the typical shape of a human population growth curve is that of an S-shaped profile. However, for shorter periods corresponding to the life-cycle of a water supply system, population growth corresponds to only a portion of such a long-run time path. In such a case, the use of another growth profile may be more appropriate. For example, a J-shaped profile will depict the initial accelerated growth of a population, a rotated J-shaped profile will depict the final decelerated growth of a population, and a linear-shaped profile will depict the population growth about the inflection point of an S-shaped profile, where the transition from the initial accelerated growth phase towards the final decelerated growth phase takes place.

figure 5.9 Rotated J-shaped growth profile.

When the production indicator is represented by the services provided to household water connections H_t measured in terms of household-year (services provided by the water supply system to a household during a full year), the previous population growth scenario should be supplemented by a scenario specifying the time path of the average size N_t of a household served by the system during its life-cycle, as indicator H_t is computed by dividing P_t by N_t. In the short run, this average size may be almost constant, but in the long run it will vary according to the demographic pattern of the households, by displaying either an increase, if the system is set up for a population of young households, or a decrease, if the system is set up for a population of mature households, for example in a rural settlement experiencing youth migration.

Similarly, if the production indicator is represented by the quantity of water supplied Q_t measured in terms of litre/day-year (litres of water per day supplied by the water supply system during a full year), the previous population growth scenario should be supplemented by a scenario specifying the time path of the average per capita daily consumption of water q_t of the population served by the system during its life-cycle, as indicator Q_t is computed by multiplying P_t by q_t. In the medium and long run, the per capita daily consumption of water may change as a consequence of the demographic composition of the population and especially of its socioeconomic development level. Therefore, before designing a life-cycle scenario for water consumption it is important to assess the effective demand of water by the community benefiting from the water supply project, as explained in section 4.3.