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date: 27 November 2020

# Dynamic Water Pricing

• R. Quentin Grafton, R. Quentin GraftonCrawford School of Public Policy, Australian National University
• Long ChuLong ChuCollege of Asia and the Pacific, Australian National University
•  and Paul WyrwollPaul WyrwollCrawford School of Public Policy, Australian National University

### Summary

Water insecurity poses threats to both human welfare and ecological systems. Global water abstractions (extractions) have increased threefold over the period 1960–2010, and an increasing trend in abstractions is expected to continue. Rising water use is placing significant pressure on water resources, leading to depletion of surface and underground water systems, and exposing up to 4 billion people to high levels of seasonal or persistent water insecurity. Climate change is deepening the risks of water scarcity by increasing rainfall variability. By the 2050s, the water–climate change challenge could cause an additional 620 million people to live with chronic water shortage and increase by 75% the proportion of cropland exposed to drought. While there is no single solution to water scarcity or water justice, increasing the benefits of water use through better planning and incentives can help.

Pricing is an effective tool to regulate water consumption for irrigation, for residential uses, and especially in response to droughts. For a water allocation to be efficient, the water price paid by users should be equal to the marginal economic cost of water supply. Accounting for all costs of supply is important even though, in practice, water prices are typically set to meet a range of social and political objectives.

Dynamic water pricing provides a tool for increasing allocative efficiency in short-term water allocation and the long-term planning of water resources. A dynamic relationship exists between water consumption at a point in time and water scarcity in the future. Thus, dynamic water pricing schemes may take into account the benefit of consuming water at that time and also the water availability that could be used should a drought occur in the future. Dynamic water pricing can be applied with the risk-adjusted user cost (RAUC), which measures the risk impact of current water consumption on the welfare of future water users.

### Introduction

Water insecurity poses threats to both human welfare and ecological systems. Global water abstractions (extractions) have increased threefold over the period 1960–2010 (Wada & Bierkens, 2014), and an increasing trend in abstractions is expected to continue (Haddeland et al., 2014; Shen,Oki, Utsumi,Kanae, & Hanasaki,2008; Wada & Bierkens, 2014). Rising water use is placing significant pressure on water resources (Haddeland et al., 2014), leading to depletion of surface and underground water systems (Famiglietti, 2014; Gleeson,Wada, Bierkens, & van Beek, 2012; Grafton et al., 2012; Vörösmarty et al., 2010). This has already contributed to up to 4 billion people experiencing high levels of seasonal or persistent water insecurity (Mekonnen & Hoekstra, 2016; Vörösmarty et al., 2010).

Climate change is deepening the risks of water scarcity by increasing rainfall variability (Pendergrass,Knutti,Lehner, Deser, & Sanderson, 2017). A warmer climate increases evaporation and frequency of storms, making storm-affected areas more exposed to risks of flooding. Areas located far away from storm tracks may also experience less precipitation and increased risk of drought (Climate Reality Project, 2017). By the 2050s, the water–climate change challenge could cause an additional 620 million people to live with chronic water shortage and increase by 75% the proportion of cropland exposed to drought (King,Schrag, Dadi,Ye, & Ghosh, 2016). While there is no single solution to water scarcity (Meinzen-Dick, 2007), or to water justice (Taylor et al., 2019), increasing the benefits of water use through better planning and incentives can help (World Bank, 2016).

Pricing is an effective tool to regulate water consumption for irrigation (Dinar,Knapp, & Letey, 1989; Shomar,Darwish, & Rowell, 2014; Ward & Pulido-Velazquez, 2009), for residential uses (Arbués,Barberán, & Villanúa, 2004; Ward & Pulido-Velazquez, 2008; Zhao et al., 2016), and especially in response to droughts (Karamouz,Nazif, & Ahmadi, 2013). For water allocation to be efficient, the water price paid by users should be equal to the marginal economic cost of water supply (James & Pollock, 1988), including explicit components (e.g., pumping and treatment) and implicit components (e.g., forgone environmental or downstream economic benefits); this condition provides incentives for users to extract water to the point where further extraction is no longer desirable. Accounting for all the costs of supply is important even though, in practice, water prices are typically set to meet a range of social and political objectives (Whittington, 2010).

An important feature of water, as an economic resource, is that water supply and water demand are typically out-of-phase (Riley & Scherer, 1979); in other words, there is a negative correlation between rainfall and the demand for both irrigation water (Isselhorst,Berking, & Schütt, 2018) and residential water (Ghimire,Boyer, Chung, & Moss, 2016). This out-of-phase relationship between water supply and demand has important implications for water pricing. To ensure long-term water availability and sustainability, water prices should explicitly account for: (a) water scarcity, because inter-seasonal variations in price can incentivize more sustainable water consumption (Pesic,Jovanovic, & Jovanovic, 2012); and (b) the welfare losses, such as decline in ecosystem services, associated with extracting water from lakes, streams, and aquifers.

An important implication of the out-of-phase relationship between water supply and demand is the dynamic connection between present and future consumption. To reduce the impact of seasonality, water storages are built to distribute water from wet periods (when supply is abundant, and demand is low) to dry periods (when supply is limited, and demand is high). The presence of storage in water supply systems also means that reducing water consumption today can increase future water availability and vice versa. Thus, water prices that vary with water availability (or scarcity), such as in water storages, can play an important role in the intertemporal substitution of water use and also in long-run planning of water resources management (Gys, 1971).

Dynamic water pricing is an approach that includes seasonal water scarcity and dynamic connections between present and future consumption. It provides an effective tool for conserving water resources and to ensure that water supply is able to satisfy peak-season demand (Molinos-Senante, 2014; Pesic et al., 2012; Saʇlam, 2015). Dynamic water pricing encourages investment in water conservation technology in response to increased climate uncertainty (Bhaduri & Manna, 2014) and can be implemented with risk-adjusted user cost (RAUC). The RAUC is the component in the cost of water supply that represents the scarcity of water while taking into account possible realizations of uncertain future outcomes. In other words, it is the implicit and intertemporal cost of supplying water when water consumption in the present poses a risk of causing water scarcity in the future.

Here, we: describe key water pricing concepts with illustrative examples; introduce dynamic water pricing; formally derive the RAUC; present a case study of the RAUC in Canberra, Australia; and offer our conclusions.

### Water Pricing, Water Tariffs, and Water Services

The terms “water price” and “water tariff” are often viewed as the same but are different. A water price is a charge per volume of water delivered. A water tariff, on the other hand, is a collection of charges associated with water provision. Tariffs can include: (a) fixed charges that are independent of the amount of water delivered; and (b) volumetric pricing, which may depend on the amount of water delivered. Only when there are no fixed charges associated with water provision will a water price and water tariff be equivalent.

Typically, a water tariff does not include all welfare loss (such as deterioration in ecosystem services) from the abstraction or extraction of water from a natural source (lake, stream, or aquifer). Rather, the water tariff will only include the “added value” of extracting, storing, processing, and delivering water to users. Such actions comprise both fixed and variable costs and, hence, a water tariff can include both a fixed and variable component.

Water tariffs are almost always regulated in some way or other because the provision of water services is, typically, a “natural monopoly.” That is to say, large, fixed costs of water storage and transmission infrastructure result in a declining average cost of water supply, and this can hinder competition from alternative water suppliers (Sibly, 2006). Typically, the water tariff regulations set by government agencies or pricing tribunals are intended to ensure that consumer outcomes are consistent with social norms regarding accessibility and affordability.

#### Objectives of Water Tariff Design

There are multiple ways in which water tariffs and pricing are used to allocate water in urban, agricultural, and industrial water systems across the world (see Dinar,Pochat, & Albiac-Murillo, 2015; OECD, 2010 for reviews). This is because there are, typically, multiple goals in relation to water pricing. Common pricing objectives, as drawn from Griffin (2016), Boland (1993), and Hanemann (1997), include:

1.

Revenue sufficiency or cost recovery—The revenue collected from consumers offsets the total costs of water services provision on an annual basis and over time.

2.

Allocative efficiency—The net benefits of water allocation are maximized across all consumers in a given time period.

3.

Dynamic efficiency—The net benefits of water allocation are maximized across different time periods.

4.

Equity and fairness—Consumers with equivalent characteristics face equivalent tariffs, and social outcomes, such as affordability of and access to water services, are considered fair and just.

5.

Transparency and simplicity—Tariff structures should be transparent to all water users.

6.

Accuracy—The representation of supply costs in tariffs are based on actual, updated costs rather than arbitrary assumptions or outdated estimates.

There can be, and often are, trade-offs between water pricing objectives such as cost recovery and allocative efficiency. Due to the high up-front costs of dams and other water infrastructure, the average cost of providing a unit of water is frequently greater than the marginal costs from using that infrastructure to deliver water. Consequently, allocative efficiency is achieved when the marginal cost of supplying the final unit of water equals the marginal benefit of consuming that water. The challenge is that marginal cost pricing alone does not cover the fixed costs of water provision if average costs are declining. As a result, a water tariff with only a marginal cost volumetric price would mean that the water supplier is unable to fully recover all of its (fixed and variable) supply costs. When this arises, government subsidies or transfers are required to defray the fixed costs of the water supplier that, in turn, impose a tax burden on other sectors of the economy. By comparison, a volumetric price set at the average cost of water provision would mean that the marginal benefit of consuming the final unit of water is higher than the marginal cost. While average cost water pricing means that all the supplier’s costs (fixed and variable) are recovered, it also means that water is not allocated efficiently as water could be supplied at a lower (marginal) cost and at a volumetric price that some water consumers would be willing to pay.

#### Water Tariff Design Options

The “textbook approach” to cost recovery for water supply that is a natural monopoly is the two-part tariff, which is commonly used in urban water systems (see Coase, 1946; Feldstein, 1972; Lewis, 1941 for the origins of this approach). A two-part tariff includes a fixed charge independent of water consumption and a per unit volumetric charge based on the water used. Possible options in relation to a two-part tariff design include:

A fixed charge for water supply (or flat charge) that does not vary with the water quantity; and

A volumetric pricing scheme, which could either be uniform volumetric price or block-rate pricing where the volumetric price increases or decreases by a given amount as specified thresholds of delivery or withdrawals are exceeded.

The fixed charge is paid by each water user and may change from year to year. It can help cover all or part of the fixed costs of providing water each year before any water is supplied. These fixed costs may include permanent staff and administration, annual operations and maintenance (O&M) (e.g., scheduled annual pump maintenance), additional infrastructure investment (e.g., water meter installation), loan repayments, and depreciation.

The volumetric price should cover the marginal cost per unit of water supplied to users to meet the objective of revenue sufficiency. It includes the costs that vary with the amount of water that is supplied, for example, pumping and water conveyance costs (e.g., electricity), variable operations, and maintenance that depend on the volume of water delivered in the water supply system. The volumetric price can also change on an annual basis.

#### Table 1. Example of Water Supply Cost Structure

Fixed Costs ($) Variable Costs ($/L)

Staff (S)

9000

Operations & Maintenance (O&M)

5000

Annual loan repayments (L)

22,000

Total fixed costs

36,000

Pumping & water conveyance (P)

0.03

Table 1 demonstrates an example of the water supply cost structure. In this example, the cost would be $36,000 even when no water is supplied, and it would increase by$0.03 for each additional liter of water extracted from the water supply system. Table 2 shows an example of applying the two-part tariff with the supply cost structure in Table 1. The example in Table 2 assumes 300 customers in three categories, namely category A has 100 customers each consuming 5,000 L, category B also has 100 customers each consuming 10,000 L, and category C has 100 customers each consuming 2,000 L.

The fixed charge per customer, in this example, is $120, that is, the total fixed cost divided by the number of customers in all categories. The volumetric price would be$0.03/L. The total water tariff would be $270,$420, and $180 per customer of category A, B, and C respectively. The total water revenue would be$87,000. On the supply side, the total cost is the sum of the fixed cost ($36,000) plus the additional cost of supplying 1,700,000 L of water at$0.03/L, which amounts to $87,000. The revenue is equal to the cost, achieving the cost-recovery objective. #### Table 2. Example of Two-Part Water Tariff Customer Categories A (100 customers) B (100 customers) C (100 customers) Fixed charge per customer ($)

120

120

120

Water demand per customer (L)

5000

10,000

2000

Volumetric water charges ($) 150 300 60 Total water tariff ($)

270

420

180

In addition to these general approaches to water pricing, there may be specific arrangements for particular applications. These pricing schemes could be considered when water quantity cannot be measured. For example, in agriculture, farmers may be charged a non-volumetric price that may be based on farmers’ production, such as output, use of an input, farm area, crop type, or irrigation method.

A key challenge in water tariff design is to manage trade-offs between equity and allocative efficiency. For instance, lower prices for an essential service may be politically popular in the short run, but this may come at a high social cost if water utilities cannot fully recover their fixed or even some of their operating costs. This is because a water tariff that does not cover all costs means the water supplier does not have the capacity nor the incentives, in the absence of other transfers or subsidies, to maintain or to expand its water infrastructure.

One way in which regulators and suppliers seek to promote equity in urban water tariffs is the use of Increasing Block Tariff (IBT) structures. This approach involves low or zero volumetric prices on initial volumes of consumption with higher prices for higher blocks of water uses. The intention of IBT is to impose more of the cost recovery burden on higher income households who, it is assumed, use more water. IBT water pricing, however, may impose higher burdens on poorer households if poorer households also have larger families or if poorer households have a reduced ability to purchase water-saving appliances (Wichelns, 2013). Further, multiple poor households in densely populated cities may share a single connection, which results in all the families on the one connection paying a higher volumetric tariff than a high-income household on a single water connection. The unintended consequence is that IBTs may result in poorer and larger households subsidizing the water consumption of smaller, higher-income households (Dahan & Nisan, 2007; Whittington, 1992; Young & Whittington, 2016).

From an allocative efficiency perspective, the most important attribute of a water tariff is that the volumetric price for the final unit of water consumption is the same for all customers. It may be possible to meet this criterion, achieve full cost recovery, and ensure that poorer households receive affordable water services. Table 3 and Table 4 present two examples, drawn on the earlier two-part tariff examples, to demonstrate two approaches to respond to equity considerations, cost recovery, and allocative efficiency criteria simultaneously.

#### Table 3. Water Tariff with Cross-Subsidy

Customer Type

A

B

C

Fixed charge ($) 150 150 60 Water demand (L) 5000 10,000 2000 Volumetric water charges ($)

150

300

60

Total water tariff ($) 300 450 120 The example in Table 3 shows a cross-subsidy. In this example, the fixed charge for category C customers can be funded through imposing a higher fixed charge for customers in categories A and B. The volumetric charge remains constant for all customers. The cross-subsidy results in a lower tariff of category C and higher tariffs for categories A and B. The total revenue would be the same, and the cost-recovery objective is maintained. Implementation of such a cross-subsidy scheme requires that the water supplier can identify customer categories. Table 4 demonstrates an alternative subsidy scheme to category C. In this situation, the subsidy of the fixed charge is financed through government funding. The water tariff would be lower for category C while remaining unchanged for categories A and B. This approach may be an attractive option to support equity outcomes in jurisdictions where there are limited options for governments to directly transfer financial support to households via the tax and social welfare systems. #### Table 4. Water Tariff with Government Subsidy Customer Type A B C Fixed charge ($)

120

120

60

Water demand (L)

5000

10,000

2000

Volumetric water charges ($) 150 300 60 Total Water Tariff ($)

270

420

120

#### Institutions and Policies

There is no one-size-fits-all approach to water tariff design (see Grafton,Chu, & Wyrwoll, 2020 for a discussion of this condition and alternative approaches if it does not hold). The Tinbergen principle of ‘one policy instrument for one policy objective’ is also a useful guide for water pricing, and particularly in the case of IBTs (Young & Whittington, 2016).

Volumetric pricing requires water meters to be installed, monitored, and maintained; these factors represent transaction costs that raise the fixed costs of water delivery regardless of whether the system is a small irrigation scheme or a large city. Importantly, institutional capacity must exist for any water tariff structure to achieve a given set of objectives and sustain the water delivery system. For instance, water suppliers must be able to deliver the volume and quality of water services for which customers are willing to pay. Further, non-payment of water charges must result in some penalties. Participation of customers in planning and management is also important. The absence of these and other institutional requirements for effective water pricing are key reasons why cost-recovery rates are low in poorer countries (Easter & Liu, 2005).

### Dynamic Water Pricing

An important extension to water tariff design is that it may be in the long-term interest of water consumers to increase the volumetric water price to (a) account for the scarcity of water in a given period and provide signals for efficient capacity expansion, or (b) incorporate the future value of in situ water (water left in storage or the environment) into decisions regarding current consumption. These are the primary motivations for using “dynamic water pricing” to manage the fundamental uncertainty regarding future water supply and demand.

#### Marginal Capacity Cost

Under droughts or other supply constraints, water regulators and utilities frequently pursue non-price approaches, such as rationing or supply-side investments, to manage excess demand for water (Olmstead & Stavins, 2009). Rationing of water for outdoor use generates welfare losses (see Grafton & Ward, 2008) because it prevents consumers from paying for water for uses that they may value (such as gardening). Further, premature investments in additional supply infrastructure, such as desalination plants, unduly burdens consumers with higher water charges to recover construction and operating costs (e.g., Grafton & Kompas, 2007). Instead of these quantity-based approaches to managing short-term water supply constraints, regulators and suppliers could use scarcity- and seasonal-based water pricing to match supply and demand within a given season (Grafton & Kompas, 2007; Grafton & Ward, 2008; Lopez-Nicolas,Pulido-Velazquez, Rougé, Harou, & Escriva-Bou, 2018; Renzetti, 1992; Saʇlam, 2015).

A key benefit of scarcity-based water pricing is that water supply expansions tend to be “lumpy” in relation to the construction of large water storages, diversions, and other supply projects. This means that many water supply systems alternate between states of under-capacity and excess capacity (Turvey, 1976). Given the high costs of supply augmentation, the timing of new investments should explicitly account for the forgone benefits from under-capacity (i.e., not being able to supply total water demand at a volumetric price equal to short-run marginal cost) and the net costs of excess capacity (i.e., the recovery of higher fixed and variable costs from consumers for a level of available water supply that exceeds total demand) (Grafton,Chu, & Kompas, 2015). One approach to optimally invest in water supply augmentation is for the volumetric price to increase by an amount that would reduce current water demand to the present (and restricted) water supply. When this price premium equals the marginal cost of supply augmentation, or “marginal capacity cost,” it is optimal for the next supply-side investment to occur (for applications, see Dandy,McBean, & Hutchinson, 1984; Grafton,Chu, Kompas, & Ward, 2015; Gysi & Loucks, 1971; Hanke & Davis, 1971). In practice, a potential barrier to deploying this dynamic water pricing approach is that water utilities will receive higher revenues from scarcity pricing and, thereby, violate the regulated return that they are allowed to receive from operating their assets. This can be mitigated, however, by allocating the additional revenues from higher short-run water prices to reduce the fixed access charge in the tariffs of all customers or for only low-income groups.

Table 5 provides an example of a water pricing scheme that responds to water scarcity. When the storage is 85% of the dam capacity or above, the volumetric price is $0.03/L and the water quantity demanded is 1,700,000 L. When the storage reduces to 80% of the dam capacity, the volumetric price would be increased to reduce the water demanded. Assuming the price elasticity is 0.5, that is, water consumers would reduce consumption by 0.5% in response to a 1% increase in the water price, a premium of$0.004/L could be added to the volumetric price to reduce the water quantity extracted from 1,700,000 L to 1,600,000 L. Lower storage levels would require higher premiums to manage water demand, as illustrated in Table 5.

#### Table 5. Volumetric Charge and Storage

Storage

Water quantity (L)

Total volumetric price

85%

1,700,000

$0.00$0.03

80%

1,600,000

$0.004$0.034

75%

1,500,000

$0.007$0.037

70%

1,400,000

$0.011$0.041

65%

1,300,000

$0.014$0.044

60%

1,200,000

$0.018$0.048

55%

1,100,000

$0.021$0.051

50%

1,000,000

$0.025$0.055

#### Marginal User Cost

The consumption of a given volume of water today generates an opportunity cost: that same water is not available for consumption in the future. For groundwater, higher extractions in the current period will increase pumping costs in the next period if extractions exceed aquifer recharge. These trade-offs are embodied in “marginal user cost”: the future value of abstracted water discounted to today (Griffin, 2016). Incorporating marginal user cost into the volumetric water price ensures that consumers account for the intertemporal opportunity costs of their current consumption, such as the possibility of water rationing being imposed in a future period (Chu & Grafton, 2019).

Typically, the marginal user cost of water extractions is estimated through mathematical optimization techniques (e.g., Khadem et al., 2018; Macian-Sorribes,Pulido-Velazquez, & Tilmant, 2015). From the perspective of water tariff design, initial estimates of marginal user cost can be calculated for alternative states of future water supply and demand, and then converted into pricing schedules, such as a block-rate tariff where the volumetric price increases as storage falls below corresponding thresholds (Lopez-Nicolas et al., 2018). A key practical barrier to incorporating marginal user cost into water tariffs is that the calculations are determined on the basis of probabilities and expectations of future water availability (such as inflows to water storages), which in turn depend on multiple and uncertain factors such as precipitation.

The RAUC is an important approach to dynamic water pricing and applies when water pricing decisions incorporate intertemporal considerations, not only the welfare of water users today. As water storage connects present and future water uses, water consumption today will shift the welfare burden from the present to the future, especially when the future water supply is subject to uncertainty (Dery & Salomon, 1997). In other words, to maximize welfare across time, the volumetric water price should be set to incorporate the cost of water supply at present and the impact of present water use on the availability of water for future use.

This section formalizes a model to demonstrate the RAUC. The model incorporates dynamic features (e.g., Grafton,Chu, Stewardson, & Kompas, 2011) whereby decisions of water users today are made contingent on current information and take into account possible realizations of uncertainty in future water supply. In this setting, water prices maximize social welfare over a given time horizon (e.g., $T$ periods).

In the model, the subscript $t$ is for time, $St$ is for the storage level, and $Scap$ is for the storage capacity. Denote $It$ for the expected inflow of (usable) water into storage, $ϵt$ as the uncertainty in future water inflows that is unknown at the time of decision-making, $ht$ as the volume of water extracted by water users, and $r$ as the proportion of the abstracted water that is returned to the environment after use. These notations are summarized in Figure 1, which depicts the water supply cycle.

The dynamics of the storage level are presented in equation (1). This equation specifies that the change in dam storage is the difference between inflow and abstraction, factoring in the water losses denoted as $Lt$ (e.g., evaporation) and spillover if dams are overfilled.

$Display mathematics$(1)

When setting water prices, decision-makers should consider the dam level ($St$) and the explicit cost of water supply, such as the costs of catchment management, the explicit cost to supply water from storage to water users, and the cost of sewage treatment and pumping in the case of residential water users. In addition, the implicit cost (or opportunity cost) of water abstractions from where water is sourced (lake, streams, or aquifer), including the external costs, such as reductions in water availability or water quality that affect downstream water users and ecosystem services, should be included in the water price charged.

Denoting the water price as $pt$ and the demand for water as$ht(pt)$, the social welfare generated by the water resource is presented in equation (2). The right-hand side (RHS) of equation (2) has three components. The first component is the consumer welfare where $pbackstop$ is the water price that water users would pay if the water resource were not available (e.g., the price of imported water from other regions). The second component is the profit of the water supplier, which is the product of quantity and the difference between price and average explicit cost ($ECt$). The last component is the opportunity cost of the amount of water that is not returned to the environment and equals the quantity times the average implicit cost ($ICt$).

$Display mathematics$(2)

The dynamic water pricing problem can be specified in equation (3) where $ρ$ is the discount rate, and $E$ is the expectation (mean) operator given information at the time of decision-making

$Display mathematics$(3)

subject to the storage dynamics in equation (1).

The dynamic water pricing problem can be formulated using the Principle of Optimality (Bellman, 1957) and rewritten as per equation (4). This is the Bellman equation. $Vt(St)$ is the value function that represents the social benefit of the water resource.

$Display mathematics$(4)

The value function must satisfy two important properties. The first, derived from the first-order conditions of optimization, is the principle for water pricing given in equation (5).

$Display mathematics$(5)

Equation (5) highlights the pricing principle for efficiency, that is, the water price in the left-hand side (LHS) should include the three components on the RHS. The first component is the explicit user cost of water supply (Renzetti & Kushner, 2004). The second component is the implicit user cost, which is the benefit that would be obtained if water were used for another purpose. The last component connects the current water price to the expected future state of the world. In other words, the current water price should consider the possible risks that future water users may have to face as a result of the uncertainty in water supply.

The second property of the value function is its concavity, as proved by Benveniste and Scheinkman (1979). Concavity implies that the net benefits from increasing present consumption ($ht$) diminish the more water is used in the current period. In other words, the RAUC component in equation (5)—which is the first derivative of the value function—increases as the water in storage declines or as current water use increases. An important implication of concavity of the value function is the risk-aversion property that larger water users should pay a higher volumetric water price than smaller water users because they impose a greater future cost in terms of their marginal (and also average) water use, as represented by a higher risk of reduced water availability in the future.

The RAUC will, in general, depend on the water storage level. All else equal, the lower the dam levels the higher will be the RAUC for the same level of water use. Thus, when there is less water available in storage today, an incremental increase in water use imposes a larger expected cost on water users into the future.

As a risk premium, the RAUC depends on the level of risk-aversion of water users, that is, how much water users are willing to pay to avoid risks to their water supply. Their risk-aversion can be evaluated via the price elasticity of water demand, which measures how much water use reduces when water price increases. Lower price elasticity implies higher risk-aversion because water users have less flexibility to reduce their water use when the price increases. On the other hand, high price elasticity entails low risk-aversion because water users can be more flexible in responding to changes in price. Empirical estimates of water demand indicate that, in general, water use is price inelastic (Marzano et al., 2018; Nauges & Whittington, 2009; Olmstead, 2010).

Another determinant of the RAUC is the magnitude and frequency of possible risk events (i.e., the distribution of $ϵt$). More frequent droughts, for instance, will increase the risk. Consequently, the RAUC could be substantial if the weather is highly variable even if the long-term average rainfall does not change over time. This is because the greater the variation in inflows to water storages, then the more valuable is a known and certain volume of water held in storage relative to uncertain future inflows. Further, higher rainfall variability poses an additional difficulty because dry and hot weather will increase the likelihood of catastrophic events, such as bush fires, that can be disruptive to water supplies and water quality. Highly variable rainfall also poses challenges for infrastructure as it can result in dam spills when storages reach maximum capacity and, as a result of such spills, a share of the inflows not being available to water consumers.

### Case Study of the RAUC in Canberra, Australia

We illustrate the RAUC in a particular location, the Australian Capital Territory (ACT), which includes the capital city of Australia, Canberra. The ACT is located in a semi-arid climate with an annual average rainfall of about 600 mm. During a major drought over the period 2000–2010, welfare reducing water restrictions of various levels of severity were imposed on water consumers to ensure there were sufficient water supplies to meet water demand at the existing water price.

The water tariff in the ACT is unique in Australia because it contains a water abstraction charge (WAC) that incorporates a catchment management charge. The WAC is included in the water bill of consumers and the WAC revenue is collected by the ACT government and comprises (a) the (private) opportunity cost and (b) an environmental charge. The opportunity cost of water abstracted by the ACT is calculated using the price of irrigation water downstream of the ACT. The environment charge is estimated using the price of water entitlements for water downstream of the ACT and is intended to proxy the marginal value society places on increased environmental flows or stream flows in perpetuity (ICRC, 2003).

There are at least two types of climate risks evident in ACT. The first is the probable increase in seasonality. This is reflected by the increases in the maximum temperature (ICRC, 2015, f3.10) and the variation between monthly rainfall within a year (Grafton & Chu, 2017, f5). In other words, the increase in seasonality implies that future summers will probably be warmer and drier, putting more pressure on the water supply system during the peak time of the year. The second risk involves a crisis in water quality due to bush fires that can lead to poor water quality in the catchment area and, in 2003, forced the ACT to rely on a more costly standby water supply system. These two types of risks can result in water restriction measures that reduce consumer welfare in the ACT, for example, water must not be used for certain purposes, such as hosing or gardening, even when consumers are prepared to pay for the water they use.

Figure 2 illustrates how water consumption quantity can increase the risk of water restrictions. The left panel of Figure 2 shows two possible projections of the ACT population, while the right panel shows the corresponding probability of water restriction under the same climate scenario until 2062. The figures show that the ACT population is increasing arithmetically while the probability of water restrictions is increasing exponentially over time. In other words, an increase in population (and in water use) increases the risks of water restriction.

Information from Figure 2 can be used to estimate the change in the risk of welfare losses for water consumers depending on assumptions about the possible degree of water restrictions. Depending on the scenario, the RAUC for 2020 is estimated to be some \$0.17/kL when the risk of water restrictions becomes material and is equivalent to approximately 7% of the water price. Our results also find that the RAUC increases over time as the ACT population grows.

The RAUC is a core component of dynamic water pricing. It is a risk premium above the marginal cost when water supply is subject to variations (Riley & Scherer, 1979). The RAUC applies when water pricing takes into account climate variability, especially the risks of extreme weather. Under climate variability, the full cost of supplying water is not constant, but should vary over time depending on what weather eventuates. Consequently, the water price charged to water users should also vary depending on water availability and expectations of future water availability (Grafton et al., 2012).

Dynamic water pricing is characterized by two features: (a) the water price varies over time to reflect the full cost of supplying water, for example, seasonal or peak pricing during temporary water shortage (Hanemann, 1997; Pesic et al., 2012); and (b) water pricing considers the social benefit not only at a point in time but over multiple years ahead in a forward-looking manner (Fridman, 2015; Grafton et al., 2011; Macian-Sorribes et al., 2015). This dynamic cost component exists regardless of whether the water is used for residential or irrigation purposes, and whether it is surface or underground water.

An important implication of the RAUC is that an efficient water price should not be uniformly charged to all water consumers. In other words, larger water users should pay a higher volumetric price even if the volumetric treatment cost of water is constant because they impose a greater cost in terms of future water availability. While the RAUC may appear to be similar to IBT, whereby larger water users pay a higher volumetric price for water use beyond a defined block or threshold, they are calculated differently and seek different objectives. Unlike an IBT, the RAUC is an efficient volumetric water price premium that represents a user cost incorporating the risk of future water scarcity and the explicit costs of current water supply. The RAUC can, thus, be returned to poorer households without leading to dynamically inefficient water use provided the rebate or refund is independent of water use, such as in the form of a reduced fixed charge.

A policy implication of applying RAUC is that the water price may vary with the water storage level (Lopez-Nicolas et al., 2018). Thus, a key to successful implementation of dynamic scarcity-based water pricing is ensuring that water users are well informed about the volumetric price they are paying and why it varies with water storage levels. This involves a redesign of water bills that raises the prominence of the volumetric water price paid that quarter, and also gives information on the level of water storages.

A necessary condition for implementing dynamic water pricing is that water users are individually metered in terms of their consumption (Narayanan,Beladi, Hansen, & Bishop, 1987). Thus, a lack of prior investment (e.g., water meter installation) could be an obstacle (Heumesser,Fuss, Szolgayová, Strauss, & Schmid, 2012). When this investment is made and water users are charged volumetrically for their water use, dynamic scarcity pricing should prove no more difficult than the current system of regulatory pricing.

### Conclusions

Water scarcity is an ongoing challenge in many countries. While multiple responses are possible, dynamic water pricing offers an important option for governments or price regulators to ensure that water use is both allocatively efficient and sustainable. Dynamic water pricing can help resolve increasing water scarcity under climate change, population growth, and rising incomes, and thus should be part of the “toolbox” to mitigate declining per capita water availability. Its key benefit is that it ensures all water costs are accounted for, including: (a) fixed and variable costs of storing and supplying water; (b) the external costs and opportunity costs associated with extracting water; and (c) the risk of reduced water availability in the future, such as from droughts or an increasing population.

The implementation of dynamic water pricing requires (a) the use of volumetric water pricing and, thus, the existence of water metering and monitoring; and (b) a risk-adjusted user cost that would be a premium on the volumetric water price that increases with a decline in expected future water availability. Using data from the Australian Capital Territory, our case study illustrates how a RAUC arises in practice. Provided there is an effective system of water metering and monitoring, dynamic water pricing mitigates water scarcity and increases the intertemporal welfare of water users, including poorer households who can be compensated for a higher volumetric price from a RAUC through a reduced fixed charge as part of their water tariff.