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| The model in context | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| The urban water model is formulated as a stochastic dynamic programming problem. Dynamic programming techniques have been applied to a range of water related problems including estimating optimal extractions from ground water aquifers (Hafi 2002) and estimating optimal release rules from irrigation storages (Brennan 2007). Previous dynamic programming models of urban water pricing and investment include the work of Hirshleifer et al. (1960) and Riordan (1971a, 1971b), although these earlier models focused on capacity constraints rather than water scarcity as the driver of supply augmentation. Riordan (1971a) developed a model of optimal water pricing and investment by a regulated monopoly whereby price is set equal to short run cost plus a capacity charge which keeps demand within existing capacity constraints. The model makes use of deterministic dynamic programming techniques to derive the optimal timing of capacity expansion. Stochastic dynamic programming techniques are often used in engineering literature to estimate the optimal release rules for reservoirs given uncertain inflows. The engineering literature typically has a stronger focus on supply side issues than on issues of demand and pricing. For example Perera and Codner (1996) use stochastic dynamic programming to estimate the optimal distribution of stored water across individual dams so as to maximise supply reliability and minimise dam spills in a multiple reservoir system. |
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| The demand for urban water | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| For the purposes of this model a simple aggregate demand function is constructed, which accounts for long-term growth in urban water consumption, seasonal variation, response to weather variability (equation 1) and response to price changes (equation 2). The seasonal variation of demand and the response of demand to weather variability are estimated econometrically. A simple explanatory equation for urban water consumption was estimated based on quarterly ACT data over the period 1960 to 2000 with explanatory variables, including seasonal indicators, a time trend, total inflows, and two intervention terms. The intervention terms account for two structural breaks in the data: lower levels of consumption in the ACT before 1973 and after 1992. Inflows were used as a proxy for prevailing weather conditions (such as rainfall and temperature), as this allows the model to have a single source of risk. |
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| In the model demand is projected over an arbitrary 25-year time horizon, notionally the period 2000 to 2025. The long-term growth of demand over time adheres to a logistic functional form. The long-run growth projection should be considered an illustrative scenario rather than a forecast. The modelled growth rate is, however, broadly in line with the ‘high growth’ scenarios developed by Actew (2004). Equation 1 specifies the evolution of demand for water over time given a fixed short-run marginal cost price. |
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 |
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| Where: QtMC = the quantity demanded (ML) when price is equal to marginal cost INt = the level of inflows at time t (ML) bcon = the estimated demand equation constant bsseas = the estimated seasonality parameters bin = the estimated inflow coefficient g1 , g2 = long-term growth parameters t = time periods (quarters) s = seasons (summer, autumn, winter, spring) A constant elasticity relationship is assumed to exist between urban water demand and price, set to a value of –0.45 based on a search of the literature. For each consumer it is assumed there exists a certain proportion of urban water consumption which is essential and hence unresponsive to price. The level of essential water demand is assumed to be non-seasonal and is calculated in the model as a fixed proportion of winter equivalent aggregate water demand. The level of essential water demand is assumed to be non-seasonal and is calculated in the model as a fixed proportion of winter equivalent aggregate water demand. Equation 2 specifies the inverse demand function, which defines the relationship between price and aggregate water consumption. |
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| Where: Qt = the final quantity of water demanded (ML) mc = the short-run marginal cost ($/ML)  = the assumed demand elasticity |
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| The supply of urban water | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| For the supply side of the model it is assumed there is a single storage with stochastic inflows. No attempt has been made to incorporate multiple connected storages, distribution networks or water treatment services. In reality the ACT maintains three (soon to be four) major water storages. However a single storage model can be interpreted as an approximation of a multiple reservoir system. Perera and Codner (1988) demonstrate the use of a single storage model as a way of limiting the complexity of multiple storages and minimising the curse of dimensionality problem inherent in stochastic dynamic programming models. Seasonal inflow probability distributions were estimated using historical inflow data for the ACT. When estimating inflow distributions the sample was limited to the period 1980 to 2006 in an attempt to represent the lower mean inflows occurring in recent decades. Log normal distributions were fitted using maximum likelihood estimators. The estimated parameters are shown in table 5. |
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| The supply side of the model also accounts for evaporation from storages and environmental flows. In the interests of simplicity it is assumed that evaporation from storages adheres to a simple seasonal average and that environmental flows are a fixed proportion of total inflows, both based on historical data. The water storage level evolves over time according to equation 3. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
 |
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| Where: St = the storage level TotInt = total inflows (dependent on supply augmentation) Et = total storage losses (dependent on supply augmentation) Kt = total storage capacity (dependent on supply augmentation) SMIN = storage minimum feasible level The model allows for two forms of supply augmentation investment, rain-dependent and rain-independent. Rain-independent augmentation is assumed to generate a fixed increase in inflows and no increase in storage capacity. Rain-independent augmentation is meant to broadly represent either desalination or water recycling. Rain-dependent augmentation represents construction of dams and is assumed to generate additional storage capacity and additional stochastic inflows. In the model rain-dependent augmentation implicitly assumes that a new dam generates additional inflows by capturing water from a stream/catchment that was previously not captured. This is again a simplifying assumption: in an interregional context new dams (and even extensions to existing dams) may involve capturing water that would otherwise be captured (at least partially) by other dams or water users downstream. For simplicity the model assumes a single storage, to maintain this assumption rain dependent augmentation (i.e. a new dam) is modelled as an increase in storage capacity and inflows into the representative single storage. As noted above a single storage model can be interpreted as an approximate representation of a multiple storage system. For simplicity it is also assumed that new dam inflows are perfectly correlated with existing inflows. Equations 4 and 5 specify the relationship between inflows, storage capacity and supply augmentation investment. |
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  |
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| Where: It,i = 1 for all t > T where T is the time of execution for investment I (and 0 otherwise) ef = the proportion of inflows released to the environment nrini = fixed non rain-dependent inflows of investment i (ML)  = proportional increase in rain-dependent inflow for investment i iki = additional storage capacity of investment i (ML) LTi = lead time (construction period) of investment i (quarters) Each option also has a capital expenditure cost, an annual running cost and a lead time or construction period. The capital cost is spread equally over the construction period while the ongoing running costs are incurred every season following the construction period. It is assumed that each augmentation option is operated permanently at full capacity such that inflows are always generated and costs always incurred. It is assumed that the SRMC of water supply is constant over time and is independent of the level of water consumption and the level of supply augmentation. An arbitrary value of $1 per kilolitre (kL) is assumed for the SRMC. |
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| Solving the model | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| The objective function of the model is the expected discounted sum of market (consumer and producer) surplus less the cost of any new investment and less a penalty imposed for an inability to meet essential water demand, see equations 6 and 7. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
  |
|||||||||||||||||||||||||||||||||||||||||||||||||||||
| Where: MSt = the market surplus (consumer and producer) Qt = the level of essential water demand at time t TCt = the total cost (capital and/or ongoing costs) of augmentation investment at time t PENALTYt = penalty imposed in the event essential water is unable to be met  = is the discount factor The costs associated with a failure to supply essential water are difficult to estimate in practice and may include costs imposed on consumers as well as the costs of implementing contingency water supply options. The approach taken in this study is to set a penalty term in the objective function to a sufficiently high level to ensure the probability of a shortage of essential water is near zero. This would be consistent with the behaviour of water utilities that appear unwilling to accept any significant probability of such an event. The model is formulated as a discrete time, finite time horizon, stochastic dynamic programming problem. The problem has two state variables — the storage level and the level of supply infrastructure (owing to previous supply augmentation) — and two policy or control variables, price and supply augmentation investment. The problem is specified in the standard Bellman (1957) equation form with the continuos variables evaluated over a discrete grid. Given the discrete state and policy space and the finite time horizon the model can be solved using backwards induction. The model is evaluated over a length of time greater than the 25-year simulation period to estimate a terminal value: the value function at the end of the last time period. |
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This appendix displays results of a sensitivity analysis of key model parameters. Sensitivity analyses were performed for four parameters: augmentation capital cost c, demand price responsiveness ( ), net inflows (environmental flows — ef) and the penalty for failure to meet essential demand (PENALTY). Each sensitivity analysis involves solving the model and running Monte Carlo simulations for each of a range of parameter values, bracketing the base case value. The results of these simulations are summarised by calculating the mean (over the entire simulation period) of key variables such as: the price, the timing of investment and the volume of water consumption. The mean discounted value of the objective function is also reported. |
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| Capital cost | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| Results of a sensitivity analysis on augmentation capital cost are shown in table 6 and table 7. Increasing the capital cost of augmentation results in an increase in the mean investment time for both rain-dependent and rain-independent scenarios. In the rain-independent scenario, increasing the cost of investment also reduces the probability of investment occurring within the 25-year simulation period. As would be expected with a delaying of supply augmentation, an increase in mean price and a reduction in mean consumption are observed in both scenarios. | |||||||||||||||||||||||||||||||||||||||||||||||||||||
|
||||||||||||
| Investment cost ($ million) | 120 |
(–40.0) |
160 |
(–20.0) |
200 |
(0) |
240 |
(20) |
280 |
(40) |
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| Probability of investment | ||||||||||||
| occurring in simulation | 1 |
(3) |
0.99 |
(3) |
0.97 |
(0) |
0.9 |
(–7.6) |
0.72 |
(–26.0) |
||
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||||||||||||
| Mean investment time t | 15.3 |
(–13.7) |
16.5 |
(–6.4) |
17.7 |
(0) |
18.6 |
(5) |
19.6 |
(11) |
||
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||||||||||||
| Mean discounted objective | ||||||||||||
| value ($ million) | 1 316 |
(2) |
1,303 |
(1) |
1,290 |
(0) |
1,283 |
(–0.7) |
1,283 |
(–0.7) |
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| Mean price ($ per kL) | 1.54 |
(–5.7) |
1.59 |
(–2.6) |
1.64 |
(0) |
1.69 |
(4) |
1.78 |
(9) |
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| Mean consumption | ||||||||||||
| (ML per season) | 24.2 |
(2) |
23.9 |
(1) |
23.7 |
(0) |
23.4 |
(-1.3) |
23 |
(-3.0) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||||
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| Investment cost ($ million) c | 60 |
(–40.0) |
80 |
(–20.0) |
100 |
(0) |
120 |
(20) |
140 |
(40) |
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| Probability of investment | 1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
||
| occurring in simulation | ||||||||||||
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||||||||||||
| Mean investment time (years) | 6.1 |
(–23.7) |
7.1 |
(–12.0) |
8 |
(0) |
9 |
(12) |
9.8 |
(22) |
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||||||||||||
| Mean discounted objective value ($ million) |
1 360 |
(2) |
1 350 |
(1) |
1 333 |
(0) |
1 320 |
(–1.0) |
1 310 |
(–1.7) |
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|
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| Mean price ($ per kL) | 1.43 |
(–2.1) |
1.44 |
(–1.0) |
1.46 |
(0) |
1.49 |
(2) |
1.51 |
(4) |
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| Mean consumption | ||||||||||||
| (GL per season) | 24.9 |
(1) |
24.8 |
(0) |
24.7 |
(0) |
24.5 |
(–0.8) |
24.4 |
(–1.3) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||||
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| Elasticity parameter, |
1.33 |
(–40.0) |
1.78 |
(–20.0) |
2.22 |
(0) |
2.67 |
(20) |
3.11 |
(40) |
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| Elasticity (1/) |
0.75 |
0.56 |
0.45 |
0.37 |
0.32 |
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| Probability of investment | 0.03 |
(–96.8) |
0.31 |
(–68.2) |
0.97 |
(0) |
1 |
(4) |
1 |
(4) |
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| occurring in simulation | ||||||||||||
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||||||||||||
| Mean investment time t | 21.7 |
(23) |
20.9 |
(18) |
17.7 |
(0) |
15.1 |
(–14.6) |
13 |
(–26.5) |
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| Mean discounted objective | ||||||||||||
| value ($ million) | 525.4 |
(–59.2) |
844 |
(–34.5) |
1 290.0 |
(0) |
1 968.7 |
(53) |
2 951.7 |
(129) |
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||||||||||||
| Mean price ($ per kL) | 1.48 |
(–9.6) |
1.65 |
(1) |
1.64 |
(0) |
1.7 |
(4) |
1.77 |
(8) |
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| Mean consumption | ||||||||||||
| (GL per season) | 22.2 |
(–6.3) |
22.5 |
(–5.1) |
23.7 |
(0) |
24.1 |
(2) |
24.4 |
(3) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||||
|
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| Elasticity parameter, |
1.33 |
(–40.0) |
1.78 |
(–20.0) |
2.22 |
(0) |
2.67 |
(0) |
3.11 |
(40) |
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|
||||||||||||
| Elasticity (1/) |
0.75 |
0.56 |
0.45 |
0.37 |
0.32 |
|||||||
|
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| Probability of investment | ||||||||||||
| occurring in simulation | 1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
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| Mean investment time t | 13 |
(61) |
9.9 |
(23) |
8 |
(0) |
6.6 |
(–17.3) |
5.2 |
(–35.0) |
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| Mean discounted objective | ||||||||||||
| value ($ million) | 512 |
(–61.7) |
853 |
(–36.2) |
1 333 |
(0) |
2 017 |
(51) |
3 002 |
(125) |
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| Mean price ($ per kL) | 1.3 |
(–10.3) |
1.38 |
(–5.0) |
1.46 |
(0) |
1.57 |
(8) |
1.71 |
(18) |
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| Mean consumption | ||||||||||||
| (GL per season) | 24 |
(–2.9) |
24.5 |
(–1.1) |
24.7 |
(0) |
24.8 |
(0) |
24.7 |
(–0.1) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||||
| Price elasticity |
| Results of a sensitivity analysis on the demand elasticity parameter (a) are shown in table 8 and table 9. When the demand for water is less (more) elastic, the associated surplus per unit of water consumed is higher (lower) and as a result investment occurs earlier (later). Rain-independent investment is particularly sensitive to a change in price elasticity. All else held constant, a smaller price elasticity would be expected to result in substantially higher mean prices. However, increased (earlier) investment limits the increase in price by providing an increase in supply, which results in increased mean consumption. In the rain-dependent case the price is more sensitive to changes in demand elasticity. |
| Net inflows |
| Results of a sensitivity analysis on the environmental flow parameter (ef) are shown in tables 10 and 11. The environmental flow parameter specifies a proportion of rain dependent inflows which are assumed to be released downstream. A sensitivity analysis over this parameter is effectively a sensitivity analysis over mean net inflows. A reduction in mean net inflows results in a bringing forward of investment. A decrease (increase) in net inflows in general results in an increase (decrease) in mean price and a fall (rise) in mean consumption. In the rain-independent scenario the changes in price and consumption are relatively small, while in the rain-dependent case price and consumption are much more sensitive to variations in mean net inflows. |
| Penalty value |
| Results of a sensitivity analysis on the penalty for failure to meet essential water (PENALTY) are shown in table 12 and table 13. When storage levels reach low levels, the probability of a system failure (an empty dam) becomes significantly different from zero and the penalty value begins to influence the estimated price and investment policy functions. As shown in the sensitivity analysis, a higher penalty value results in higher prices on average and earlier investment on average in both scenarios. The impact of the penalty value on mean prices is relatively small, as it will tend to increase prices only during substantial droughts. The penalty value does, however, have a significant impact on investment timing in the rain-independent scenario. In the absence of a penalty for system failure, the rain-independent supply option is likely to be executed in only one in four simulations. A higher penalty value also decreases the probability of a system failure. However, given that this probability is very low, it is difficult to accurately measure it without increasing the number of simulations substantially. |
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| Environmental flow parameter, ef | 0.15 |
(–40.0) |
0.2 |
(–20.0) |
0.25 |
(0) |
0.3 |
(20) |
0.35 |
(40) |
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| Effective change in mean inflow | (13) |
(7) |
(0) |
(–6.7) |
(–13.3) |
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| Probability of investment | ||||||||||||
| occurring in simulation | 0.03 |
(–97.4) |
0.59 |
(–39.3) |
0.97 |
(0) |
1 |
(2) |
1 |
(2) |
||
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||||||||||||
| Mean investment time t | 22.4 |
(26) |
20.4 |
(15) |
17.7 |
(0) |
14.9 |
(–16.4) |
12.4 |
(–30.1) |
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| Mean discounted objective | ||||||||||||
| value ($ million) | 1 375 |
(7) |
1 333 |
(3) |
1 290 |
(0) |
1 258 |
(–2.5) |
1 220 |
(–5.4) |
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|
||||||||||||
| Mean price ($ per kL) | 1.6 |
(–2.15) |
1.64 |
(0) |
1.64 |
(0) |
1.68 |
(3) |
1.78 |
(9) |
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| Mean consumption | ||||||||||||
| (GL per season) | 24 |
(1) |
23.7 |
(0) |
23.7 |
(0) |
23.3 |
(–1.36) |
22.8 |
(–3.73) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||||
|
||||||||||||
| Environmental flow parameter, ef | 0.15 |
(–40.0) |
0.2 |
(–20.0) |
0.25 |
(0) |
0.3 |
(20) |
0.35 |
(40) |
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| Effective change in mean inflow | (13) |
(7) |
(0) |
(–6.7) |
(–13.3) |
|||||||
|
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| Probability of investment | ||||||||||||
| occurring in simulation | 1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
||
|
||||||||||||
| Mean investment time t | 11.8 |
(47) |
9.8 |
(23) |
8 |
(0) |
6.3 |
(–21.8) |
4.6 |
(–42.1) |
||
|
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| Mean discounted objective | ||||||||||||
| value ($ million) | 1 370 |
(3) |
1 360 |
(2) |
1 333 |
(0) |
1 300 |
(–2.3) |
1 260 |
(–5.3) |
||
|
||||||||||||
| Mean price ($ per kL) | 1.29 |
(–12.11) |
1.36 |
(–7.43) |
1.46 |
(0) |
1.6 |
(9) |
1.81 |
(24) |
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| Mean consumption | ||||||||||||
| (GL per season) | 26 |
(5) |
25.4 |
(3) |
24.7 |
(0) |
23.8 |
(–3.59) |
22.7 |
(–8.06) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||||
|
||||||||||
| Penalty value PENALTY | 0 |
(–100) |
5 × 109 |
(–50) |
1 × 1010 |
(0) |
1.5 × 1010 |
(50) |
||
|
||||||||||
| Probability of investment | ||||||||||
| occurring in simulation | 0.24 |
(–75.2) |
0.96 |
(–1.6) |
0.97 |
(0) |
0.98 |
(0) |
||
|
||||||||||
| Mean investment time t | 21 |
(18) |
18 |
(2) |
17.7 |
(0) |
17.4 |
(–2.1) |
||
|
||||||||||
| Mean discounted objective | ||||||||||
| value ($ million) | 1 356 |
(5) |
1 294 |
(0) |
1 290 |
(0) |
1 289 |
(–0.1) |
||
|
||||||||||
| Mean price ($ per kL) | 1.66 |
(1) |
1.65 |
(1) |
1.64 |
(0) |
1.63 |
(–0.33) |
||
|
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| Mean consumption | ||||||||||
| (GL per season) | 23.3 |
(–1.62) |
23.6 |
(–0.30) |
23.7 |
(0) |
23.7 |
(0) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||
|
||||||||||
| Penalty value PENALTY | 0 |
(–100) |
5 × 109 |
(–50) |
1 × 1010 |
(0) |
1.5 × 1010 |
(50) |
||
|
||||||||||
| Probability of investment | ||||||||||
| occurring in simulation | 1 |
(0) |
1 |
(0) |
1 |
(0) |
1 |
(0) |
||
|
||||||||||
| Mean investment time t | 10.1 |
(26) |
8.3 |
(3) |
8 |
(0) |
7.9 |
(–1.2) |
||
|
||||||||||
| Mean discounted objective | ||||||||||
| value ($ million) | 1 354 |
(2) |
1 337 |
(0) |
1 333 |
(0) |
1 333 |
(0) |
||
|
||||||||||
| Mean price ($ per kL) | 1.39 |
(–4.53) |
1.46 |
(–0.12) |
1.46 |
(0) |
1.47 |
(1) |
||
|
||||||||||
| Mean consumption | ||||||||||
| (GL per season) | 25 |
(1) |
24.7 |
(0) |
24.7 |
(0) |
24.7 |
(–0.20) |
||
| Note: Percentage change shown in parenthesis. | ||||||||||
