Price elasticity of water demand.

Flat rate, constant rate, and block rate are the three most commonly used water pricing structures. A single price for an unlimited amount of water is called a flat rate, while uniform rate for each unit of water consumed is constant price.¹⁶ In a block rate type of billing structure, the price per unit of water changes as the volume consumed increases.¹⁶ Generally, the price elasticity of the water demand is calculated using regression analysis with several independent variables, and water use as the dependent variable. The most common independent variables are median household income, average household size, precipitation, and average water price. Because of the limited availability of data, household size must often be estimated indirectly from the population and the number of dwellings.¹⁷ Either annual or seasonal precipitation values can be used in the regression equation for calculating the price elasticity of water demand. Seasonal precipitation is generally taken as the summer period, from May to September, because of the high fluctuation in demand, use, and availability. Foster and Beattie¹⁸ include precipitation as a variable during those months where the average monthly temperature is at least 45°F and 60°F in the northern and southern regions of the United States, respectively. Water demand is directly proportional to temperature and inversely related to precipitation.¹⁹ The common exponential form of a regression equation for calculating the price elasticity of water demand is shown in Eqn. (1):

After taking the log of both sides, Eqn. (1) can be written as Eqn. (2). This logarithmic form of regression equation is used to model price elasticity of water demand:

Study area and data.

The Palouse Basin spans eastern Washington and northern Idaho. The largest portion is located within Washington's Whitman County and Idaho's Latah County, with a very small area in Benewah County in Idaho. Figure 1 shows the Palouse Basin watershed with major cities and surface water tributaries. The Palouse region is a semi-arid area where precipitation ranges from approximately 59 to 85 cm per year. As elevation increases towards the east, so too does precipitation. The mean temperature of the Palouse Basin decreases from west to east. The precipitation of the Palouse Basin comes either in the form of rain or snow. According to the dominant geologic formations, there are two groundwater aquifers in the Palouse Basin, identified as the Wanapum and Grande Ronde aquifers. Both aquifers have satisfactory groundwater quality for domestic, agricultural, and industrial purposes. The Wanapum aquifer is the shallower of the two at approximately 110 m deep, while the Grande Ronde aquifer is approximately 290 m deep. The shallower Wanapum aquifer is the primary water supply for rural residents of Latah County within the basin limits and in some areas of Whitman County (McKenna);²¹ it also supplies approximately 32 percent of Moscow's drinking water (Ralston,²² Palouse Basin Aquifer Committee (PBAC)).²³ The rest of Moscow's and 100 percent of the City of Pullman's drinking water demands are fulfilled by the Grande Ronde aquifer.

Figure 1.

The City of Pullman overlaid with the Palouse basin watershed with major cities, and North Fork and South Fork Palouse River bordered with Idaho and Washington State.

Source: Palouse Basin Community Information System, 2007.

The Palouse Basin area includes three major cities: Moscow, Pullman, and Colfax, as well as other small cities like Viola, Potlatch, etc. (Fig. 1). The City of Pullman is the largest population center in the area, with approximately 31,000 residents in 2014. Half of the city's population is comprised of students attending Washington State University. The population within 7 miles of Moscow and Pullman is denser compared to the rest of the region (ie, Colfax, Viola, and Palouse). Because of the limited availability of data across the basin, the City of Pullman is taken as the representative of the basin, and is used to develop a single price elasticity relationship. This study thus discusses water pricing and demand scenarios of a representative college town where groundwater is the sole source of drinking water.

As discussed above, some of the commonly used demographic variables for calculating the price elasticity of water demand are population, per capita water use, median household income, and average size of the household. In addition, precipitation data and water pricing structures are also needed. The population, median household income, and housing units data are obtained from the United States Census Bureau²⁴ and municipal sources, with a 1% annual population growth rate. Monthly total precipitation data was taken between 2000 and 2006 from Pullman 2 NW, WA (Coop ID: 456789, 46.75 N 117.18 W, elevation 2545 ft.)²⁵ (see Appendix B)”. In summer, more water is needed for irrigation to maintain vegetation if there is inadequate precipitation. A study conducted by Linaweaver et al²⁶ used evapotranspiration in place of precipitation. Available moisture, or moisture defined by the difference between precipitation and evapotranspiration, can be used as an alternative variable for precipitation. The monthly water price and water extraction data were acquired from the City of Pullman for the years 2000 to 2006 (Appendix B). Table 1 shows the sample data for calculating the price elasticity of water demand of a single-family households. (Appendix B presents the comprehensive data for the City of Pullman's economic analysis). Equation (5) shows the water use per household per 100 cubic feet of single-family households:

Table 1.

Sample data for economic analysis of single-family households, Pullman, Washington.

Scenarios for the price elasticity of water demand.

While calculating the price elasticity of water demand, a regression analysis is carried out based on different population dynamics. Single-family households, total residential households, and total population water use are the three different economic parameterizations developed for calculating the price elasticity of water demand. Total residential households include single, duplex, multiple, group, and mobile homes. These water consumption estimates do not include industrial sites, commercial building, schools, or offices in the area because of lack of data. Based on the three economic scenarios, five different cases are developed by adjusting the variables input into the regression analysis. These scenario adjustments are essential to understanding the influence of individual variables on the regression equation in estimating the final price elasticity of water demand.

In economic scenario 1, the regression analysis is carried out for monthly water use of a single-family household per 100 cubic feet, as a dependent variable. In case 1 of economic scenario 1, all five independent variables are used, whereas in case 2, the regression is carried out without household size. In economic scenario 2, regression analysis is carried out for monthly water use per household per 100 cubic feet with the total residential sector replacing single-family household. In case 3, all the independent variables in the regression equations are used as in case 1 in scenario 1. In case 4, regression analysis is conducted without household size and in case 5 without a fixed price. Finally, in economic scenario 3, the dependent variable is the mean monthly household water use of the total population of the City of Pullman. In this scenario, total water consumption is divided by service area population to compute the proxy of mean household water consumption. The economic scenarios 1, 2, and 3 discussed above are shown in Table 2.

Table 2.

Economic scenarios for calculating price elasticity of water demand, Pullman, Washington.

System dynamics model.

Computation of the price elasticity of water demand can be limited by different determinants and assumptions (Michelsen et al,²⁷ Schleich and Hillenbrand,²⁸ Klaiber et al²⁹ etc.). Validation of the elasticity values computed from the regression equation is important to compare an accurate interpretation of the relationship among the variables of water demand. Previous studies of the price elasticity of water demand estimate the interrelation between the variables of regression equation, but usually not applied these results in the modeling purpose. The system dynamics approach, utilized in this study, has facilitated the validation of the elasticity values by comparing the estimated water demand from the economic model to the present water extraction trend. The economic model, based on the system dynamics approach, is further applied for conducting a sensitivity analysis to forecast future water demand. Systems thinking for education and Research (STELLA 9³⁰) modeling and simulation software is used to develop a system dynamics model to study the sustainability of the Palouse Basin. As mentioned earlier, use of system dynamics approach can be a pertinent approach to utilize the developed regression equation. Appendix A shows a demand model, a simple exponential population forecast model, and an economic model developed using the system dynamics approach. Water demand is forecasted based on the demand model and economic model using a system dynamics approach. The first water demand forecast (demand model, Fig. 2A) is based on population, growth rate, and water use per capita per day (∽160 gallons) calculated in billions of gallons (Appendix A—Section 1).

Figure 2.

(A) System dynamics model for water demand forecast using the demand model, where Total Pullman Demand is in billions of gallons, and Per Capita Water Use Pullman is in gallons per day (∽160 gallons). (B) System dynamics model for water demand forecast using the economic model, where Pullman Demand by Economic Model (Q_Pullman) is the total water demand per annum using regression equation (in billions of gallons), Total Pullman Population is the estimated total population of the City of Pullman, Per Capita Per Day Water Use Regression (Q_c-d) is in gallons. Housing Units is the total number of households in Pullman (single, duplex, multi, group, and mobile homes).

The second is a simple economic model developed from the regression equation (Eqn. (8), Fig. 2B). The multiple regression equation developed from the price elasticity of water demand is used in the economic model, which assumed that the independent variables of the regression equation behave linearly for the projected time period, i.e. until 2025. A trend line is developed from the study period (2000 to 2005) and linearly extrapolated. A shorter time period (2025) is chosen to simulate the water demand forecast based on the economic model because the linear extrapolation of the variables might not be accurate for extended periods of time. The number of households (H) is also linearly extrapolated using a similar approach. In the economic model, population is indirectly calculated inside the model based on the number of households and the average number of people per household using linearly extrapolated results. The indirect calculation of population (P₁) in the economic model should closely match the population from the population (P₀) forecast model (Appendix A—Section 2). Any discrepancy in the population forecast between these two approaches can create differences in the water demand forecast.

In the developed economic model, total calculated water demand is converted into per capita daily water use (Q_c-d) in the designated years by dividing the population (Eqn. (6)) (Appendix A, Section 3). Q_c-d is further compared to the actual per capita daily water use value (∽160 gallons). To reduce bias while calculating Q_c-d, population computed in the economic model (P₁) needs to be used. In the simulation process, both P₀ and P₁ are used to understand the variations in Q_c-d.

Limitations.

There are some specific assumptions in this study. Because of a lack of exact service connections, the single-family household data are adopted from the published literature. The household level survey data are precise and effective when calculating price elasticity of water demand. The population, household size, and median household income are generally calculated annually, and some are calculated on a decade basis. There is difficulty in collecting these data on a monthly basis or for smaller time periods, so the data used in these analyses are linearly interpolated to get monthly figures. Because of these difficulties, the monthly time series data are used to calculate the price elasticity of water demand. These types of aggregated time series data have lots of complications, as it is difficult to understand the behavior of individual households. The linearity in the variables in the economic model while forecasting water demand is another key assumption.

Price elasticity of water demand.

The price elasticity of water demand is calculated based on detailed water-pricing structure and water extraction data from City of Pullman groundwater wells, between 2000 to 2006 (Appendix B). Figure 3 shows the annual water consumption for the residential sector of City of Pullman (single, duplex, multi, group, and mobile homes) and marginal price for the period 2000 to 2006. Figure 3 also shows a constant increase in water consumption, with a slight decrease in 2003. Water consumption in the year 2000 is about 715 million gallons, and reached about 755 million gallons in 2006 with increasing demand. The City of Pullman has both marginal and fixed price water costs. Up to 500 ft³ for any kind of user class, no marginal price is paid, but certain fixed price is paid whether water is used or not. The City of Pullman has an increasing block rate of marginal water price, varying in the peak (summer) and off-peak (winter) months, and it also differs according to user class. The marginal price increased from $0.32/100 ft³ to $1.19/100 ft³ between 1971 and 2006 (Table 3). Also, there is more than a 20 percent increase in marginal price between 2000 and 2006 in a single household family size. The fixed price charge also varies according to the user class and size of the water meter (a larger meter relates to a larger service line) (Lamar T and Weppner S³¹). Between 1971 and 2006, the ready-to-serve charge (fixed price) of a one-inch water meter size increased from $2.75 to $26.93. The fixed price increased by approximately 22 percent between 2000 and 2006. A one-inch water meter is taken as the representative in the calculation, assuming that the majority of the single-family households uses this water meter size (Table 3). There is about 6 percent increase in water consumption between 2000 and 2006 in the residential sector of the City of Pullman, Washington. Table 3 shows the detailed water pricing structures of the City of Pullman for the years 1971-2006.

Table 3.

Marginal and fixed price rate structures of the City of Pullman, Washington.

Figure 3.

Annual water consumption for the residential sector and marginal price of the City of Pullman, Washington, between 2000 and 2006.

Even if both the marginal and fixed prices are consistently increasing, water consumption of the City of Pullman is also increasing with some seasonal variation within the year. The trends presented in Figure 3 indicate that water consumption and water pricing are hardly related. It is expected that once the water consumption trend falls as a result of marginal price increase, the trend will never rise again, unless there are other factors influencing the relation. These facts indicate that because of the constant demand for water and the limited resources, the current price structures of water are not directly influencing water consumption. As the demand for water will grow as the population and industry increase, alternative sources of water will need to be obtained (Dhungel).¹⁰

Household size exhibits a slightly decreasing trend from 2.24 to 2.21 between 2000 and 2006, while annual median household income increased from $21,600 to $24,300. The means of P, H, I, FP, MP, and Q are 1.53 in, 2.23, $22,993, $24.18, $1.17, and 10.04/100 ft³, respectively. Similarly, the standard deviations of P, H, I, FP, MP, and Q are 1.16 in, 0.01, $792.6, $1.51, $0.17, and 5.41/100 ft³ (single-family household), respectively (Appendix B). Figure 4 shows the monthly water consumption trend of the residential sector and precipitation of Pullman during the study period. In general, summer water consumption is relatively higher than in winter. The maximum residential water consumption is about 110 million gallons in summer and 40 million gallons in winter.

Figure 4.

Monthly water consumption of the residential sector and precipitation of the City of Pullman, Washington, between 2000 and 2007.

In the following section, the results of the ordinary least squares (OLS) of the log linear regression (Eqn. (8)) of the economic scenarios are presented. The expected signs of the independent variables are: household size positive, marginal price negative, fixed price negative, median household income positive, and precipitation negative.

Economic scenario 1.

In case 1 of scenario 1, the results show high elasticity values for household size and median household income (Table 4). In this case, the household size had an elasticity of about 355, while that of the median household income is about 41. The elasticity of marginal price and fixed price is about +2.98 and -6.94, respectively, while precipitation is -0.09. In case 2, price elasticity of marginal price still has high elasticity, ie +2.96. In case 2, the elasticity of precipitation, household income, and fixed price is -0.09, 5.83, and -7.18, respectively. Most of the attained elasticity values of independent variables are larger than those published in the literature. The coefficient of determination (R²) in both cases is 0.77. The F statistics are about 50.6 and 63.5 for case 1 and 2, respectively (Table 4). The constant term of the regression equation in case 1 and 2 becomes negative with larger numbers. The t stat and probability value (P-value) are also shown in Table 4. The negative and positive values in Tables 4 and 5 are the expected signs of the variables in the regression equations. It is difficult to explain the higher elasticity values for these independent variables, which is possibly because of the weak relationship among the variables.

Table 4.

Regression coefficients for price elasticity curve for single-family and residential households.

Economic scenarios 2 and 3.

In economic scenario 2, the results show that there is a slight decrease in the price elasticity of marginal price in all cases compared to economic scenario 1, but it still attains a positive sign. The elasticity of the marginal price of case 3, 4, and 5 is +1.6, +1.58, and +1.62, respectively (Table 5). The fixed price elasticity is about -5.07 and -5.21 for case 3 and 4, respectively. The R² is 0.68 in both case 3 and 4, and 0.62 in case 5 (Table 5). As in economic scenario 1, the constant term of the regression equation has larger negative values in these economic scenarios. The rest of the results of the regression analysis are presented in Table 5. The trend of the elasticity of median household income is similar in both scenarios 1 and 2. In general, these two scenarios show similar elasticity trends among all applied independent variables. The results of economic scenario 3 are not statistically different from scenarios 1 and 2, so the results are not presented or discussed.

Table 5.

Regression coefficients for price elasticity curve for residential households.

The results of all the above scenarios show a positive sign in the marginal price (case 1 to 5). The fixed price, median household income, and precipitation signs obtained the expected signs in all cases. In all scenarios, the elasticity of household size is high. Equation (7) shows the results of the multiple regression equations in logarithmic form for case 3 of economic scenario 2, and Eqn. (8) in exponential form.

The +1.6 marginal price elasticity of demand means that a 1% increase in marginal price will increase water use by 1.6%, while -5.07 fixed price elasticity means a 1% increase in fixed price will decrease water use by 5.07%. The results show that the marginal price does not directly influence water demand, while fixed price has a large impact in all cases. The result of this study contradicts the results of Lyman,⁶ where marginal price shows elasticity to water demand in the City of Moscow, ID. The studies by Rode⁷ and Peterson⁸ were inconclusive because of the insignificant marginal variable and other independent variables in the city nearby the City of Pullman, WA. The regression equations (7) and (8) are chosen in the economic model to forecast water demand, as this scenario represents the total residential household of the City of Pullman and elasticity of marginal and the fixed price attained smaller values compared to the other cases.

In the next section, the results of the developed economic model using a regression equation are discussed. This study incorporates a system dynamics approach in price elasticity of water demand in order to forecast demand.

System dynamics model.

The first part of this section shows the results of the demand model (Fig. 2A). Using a growth rate of about 1% per year, the total forecasted population of Pullman will be about 31,000 in 2025, and about 65,000 in 2100 (can vary with students enrollment). Cheng Q and Chang N³² synthesized the various approaches to forecast short-and long-term municipal water demands since 1960s. They characterized these approaches as the regression analysis, the time series analysis, the computational intelligence approach, the hybrid approach, and the Monte Carlo simulation approach. Figure 5 shows a water demand projection for the Palouse Basin cities, using a demand model based on a system dynamics approach (Appendix A—Section 1). In the year 2000, the PBAC estimated that approximately 160 gallons of water per person per day are used in the Palouse Basin, encompassing all the cities. The estimated demand in the initial year (2005) was 0.16 billion gallons for Colfax, 1.27 billion gallons for Moscow, 0.04 billion gallons for Potlatch, 0.02 billion gallons for Viola, and 1.47 billion gallons for Pullman. The exact water extraction in 2005 from major cities was 1.05, 1.38, and 0.266 billion gallons for Moscow, Pullman, and Colfax, respectively.

Figure 5.

Water demand forecast by the demand model for the major cities of the Palouse Basin region using system dynamics approach between 2005 and 2100.

The second part of this section discusses the results of the economic model. Figure 6 shows the linear extrapolation of the variables up to the year 2025 using system dynamics approach (except precipitation, which uses a constant mean areal precipitation of entire Palouse Basin). Areal mean precipitation was computed between 1971 and 2000 (consistent with widely available climate normals). The Parameter-elevation Regressions on Independent Slopes Model (PRISM)³³ precipitation maps (developed at Oregon State University) were utilized in this study. Linear extrapolation of the marginal price, fixed price, median household, and household numbers shows an increasing trend, while the household size shows a decreasing trend (Fig. 6). The constant term of the regression equation was also kept constant while forecasting water demand from the economic model (Appendix A-Section 3). This might create bias in the simulation, which probably needs to be adjusted after analyzing the final water demand forecast. Appendix A in section 3 shows the equations of the trend line used in the economic model.

Figure 6.

Linear extrapolation of independent variables for regression equation using system dynamics approach between 2005 and 2025.

The forecasted population in the economic model (P₁) is about 23,000, which is 7,000 less than the population forecast model (P₀) at the end of the simulation period (2025) based on the total residential households. This indicates that these variables of the regression equation may not necessarily behave linearly. The economic model projects about 0.75 billion gallons of water consumption for the City of Pullman in 2005 based on the total residential households (Eqn. (8)), about half of that in the demand model. Because of the lack of the exact number of service connections in different user classes and variability in the coefficients of regression equation, the water demand obtained from the economic model may have differed from that of the demand model in 2005. The water demand projected from the demand model, as well as the actual water extraction, shows a constant increase in water demand for the City of Pullman (Fig. 5). Figure 7 shows the annual water demand forecast using an economic model using Eqn. (8) (in billions of gallons). The results of the economic model project a decreasing water demand in the coming 20 years. At the end of the simulation period, the water demand unrealistically declined to 0.3 billion gallons. Per capita per day water use (Q_c-d) decreased from 81 gallons to 26 gallons when P₀ is used, and to 32 gallons when P₁ was used at the end of the simulation period. This projection indicates that the applied regression equation is unable to simulate realistic water demand forecast. As discussed earlier, there can be various uncertainties regarding linear extrapolation of the regression equation variables and the computed elasticity of the variables. The linear extrapolation of fixed price reached $41 (Fig. 6) at the end of the simulation period, with a high elasticity value. This high elasticity can be one of the reasons for the rapid declining trend in water demand. As discussed above, the population of the economic model projected a smaller number than the population forecast model.

Figure 7.

Water demand projection by the economic model for the City of Pullman using system dynamics approach between 2005 and 2025 (Scenario 2- Case 3).

Figure 8 shows the results of a scenario where all the variables of the regression equation 8, as in the previous scenario, kept the fixed price of water constant at $26.93. This sensitivity analysis is important in understanding the role of the high elasticity value of fixed price in the economic model. With this change, the economic model predicted an increasing trend in water demand (Fig. 8) similar to the demand model (Fig. 5). The water demand increased up to 2.45 billion gallons in 2025 with an over-prediction of water demand while compared to the demand model. Finally, Figure 9 shows the simulation results of scenario 2- case 4 where the household size is excluded from the regression equation in the economic model (Eqn. 9). In case 4, the income elasticity and constant term in the regression equation is significantly smaller than scenario 2- case 3 (Eqn. 8).

Figure 8.

Water demand projection by the economic and demand model for the city of Pullman using system dynamics approach with constant fixed price in linear regression equation between 2005 and 2025 (Scenario 2- Case 3).

Figure 9.

Water demand projection by the economic and demand model for the City of Pullman using system dynamics approach between 2005 and 2100 (Scenario 2- Case 4).

Both the economic and demand models projected similar water demand trend during the extended simulation period, ie 2100 years. This result confirmed that the developed economic model and the approach utilized in this study can play a vital role in understanding and validating the regression equation of the price elasticity of water demand. It should be noted that the developed economic model fundamentally differs from the demand model. The high elasticity value of the household size in the regression equation (Eqn. 8) is one of the reasons why the economic model has difficulty to forecasting reasonable water demand. The water demand from the economic (Eqn. 9) and demand model are about 2.6 and 3.8 billion gallons, respectively at the end of the simulation period.

Based on the limited simulation analysis, it is challenging to generate a realistic water demand forecast for all the scenarios because of the possible weak relationship among the variables of the regression equation. To understand the future implication of water demand and pricing of this region, a set of sensitivity analysis can be done by using economic scenarios (case 1 to 5 and possibly others) and adjusting the elasticity values in the regression equation. The main objective here is to demonstrate a system dynamics approach as an appropriate tool to develop an economic model which need to be further explored in the areas where water pricing and demand has a strong relationship.