## Main

### Demographics and ratepayer behaviour

We calculated residential water use in the service area as the sum of household water use across income classes. We estimated household water use separately for 16 income classes using an econometric model that includes price and income elasticity. For a given household class c, we calculated their demand in month t, $${d}_{c}^{t}$$, as:

$${d}_{c}^{t}={\bar{d}}^{t}{m}_{c}{I}_{{\mathrm{P}}}{I}_{{\mathrm{Y}}}(1-{r}^{t})$$
(2)

Each class is an income bin from US$10,000 up to US$250,000+ yr–1, taken from the standard 16-node census income distribution53,54. For each class, we began with a cyclostationary, per-capita water use for the region in month t, $${\bar{d}}{\,}^{t}$$, taken from reported monthly water use in Santa Cruz44. This was multiplied by mc, the average household size for houses of class c, calculated using Integrated Public Use Microdata Series (IPUMS) microdata55. This water use was then adjusted for changes in price by a factor IP using a price elasticity parameter ϵp such that

$${I}_{{\mathrm{P}}}=1+{\epsilon }_{{\mathrm{p}}}\left(\frac{{P}^{t-1}-{P}^{t-2}}{{P}^{t-2}}\right)$$
(3)

where Pt is a household’s water bill in time period t and Pt − 1 and Pt − 2 are the bills for time t − 1 and t − 2, respectively. Similarly, water use was adjusted for changes in income relative to the median by a factor IY such that

$${I}_{{\mathrm{Y}}}=1+{\epsilon }_{{\mathrm{y}}}\left(\frac{{Y}_{c}-{Y}_{{{\mbox{MHI}}}}}{{Y}_{{{\mbox{MHI}}}}}\right)$$
(4)

where Yc is the household income of class c, YMHI is the median income of the population and ϵy is the income elasticity of demand. Curtailment in time period t, rt, is represented as a number from 0 to 1, where rt = 0 indicates no curtailment and rt = 1 indicates 100% curtailment. Curtailment was applied across households uniformly (for example, r does not vary across classes). Finally, the class demand, $$d_c^t$$, was multiplied by the count of households of each income bin in the service region, Hc, which is tabulated in the 2015 American Community Survey56, and summed over all classes to give the total residential utility demand in time t, $${D}_{\,{{\mbox{utility}}}\,}^{t}$$:

$${D}_{\,{{\mbox{utility}}}\,}^{t}=\mathop{\sum}\limits_{c}{d}_{c}^{t}{H}_{c}$$
(5)

Our approach to calculating household water use was based on current literature suggesting that household size is a significant driver of indoor water use and household income a driver of outdoor water use21. We adjusted for household size by calculating the average household size of each income class, mc. Similarly, we adjusted for water use based on income using an income elasticity, ϵy, of 0.15, using a convention for positive income elasticity in which a marginal increase in income leads to a marginal increase in water use. This value was taken from a recent meta-analysis of retail water income elasticity values54 and used for all analyses and figures presented in this paper. For completeness, we also tested two alternative income elasticity values, namely 0 and 0.4, and have included the results in Supplementary Figs. 3 and 4. Except for experiments in which we tested the sensitivity of ϵp, we used a constant value of 0.35 across income classes, taken from a meta-analysis of price elasticities of retail water18. Here, we used a convention in which a positive price elasticity indicates that a marginal increase in price leads to a marginal decrease in water use.

### Sensitivity analyses

We performed a number of additional sensitivity analyses in addition to those presented in the Demand response section. First, we analysed alternative price elasticity estimation approaches. Residential water use differs from many other consumer goods in that water users alter their consumption in response to changes in average rather than marginal prices57,58,59. We included this in our baseline model by calculating price elasticity on the basis of changing average price, which leads to households responding to the addition of flat surcharges. We also modelled an additional scenario in which consumers respond to changes in marginal price, but flat surcharges are applied, altering the average price but eliciting no change in the marginal price of water (Supplementary Fig. 5). In this alternative model, consumers do not respond to price changes. The results show similar dynamics to the price elasticity sensitivity results: a small or non-existent price sensitivity increases bills for all households as they do not lower their water consumption in response to increased rates.

We also tested scenarios in which we varied income elasticity (YED). The results in the main text are reported using an income elasticity value of 0.15, representing a suggested income elasticity value from Havranek et al.54. We also ran sensitivity analyses using income elasticities of 0 and 0.4 (Supplementary Figs. 3 and 4). For a YED value of 0 compared with 0.15, water use is higher for low-income households and lower for high-income households. As Santa Cruz has a higher proportion of high-income households than low-income households, this reduces demand by approximately 10%. When YED was increased to 0.4, water demand was higher by about 20%. At lower YED values, the cost increases due to drought are largely the same for high- and low-income households (Supplementary Fig. 3). This occurs because their baseline water use is only differentiated by differences in household size. More importantly, the lower total residential demand eliminates the need for drought mitigation, which leads to bill increases for low-income households and decreases for high-income households when curtailment is used. At higher elasticity values (Fig. 4), demand is significantly increased, leading to greater curtailment. The disproportionate impact of curtailment on high- and low-income bill increases is exacerbated, and total costs increase for the utility.

Finally, we performed a sensitivity analysis in which we let price elasticity vary with income (Supplementary Figs. 6 and 7). There is a growing body of literature that indicates heterogeneous price responsiveness with respect to income or water use. Some work indicates that low-income customers are more sensitive to price than high-income households59, while other work provides contrary evidence that “price elasticity is largely invariant to household wealth” and high water users are more price responsive60. Given the discrepancy between these findings, we performed two additional analyses in which we let price elasticity vary with income. In the first, we assumed that high-income households have 0 elasticity, and price responsiveness increased linearly as income decreased until the lowest income classes had a PED of 0.35. This demonstrates a finding similar to that of El-Khattabi et al.60 in which low-water-use households respond to price signals and high-water-use households do not. In the second sensitivity analysis, we assumed that low-income households have 0 elasticity, and price responsiveness gradually increased across income classes until the highest income classes had a PED of 0.35. This experiment demonstrates the case in which high-income households respond to price signals but low-income households do not.

### Case study attributes

The aim of this work was to develop process-based insights using realistic and generalizable model assumptions, not to design context-specific solutions for Santa Cruz. While the approach is fully general and can be readily applied to other cities, aspects of the specific case study that we chose limit the generalizability of the results. We describe these attributes here.

The City of Santa Cruz operates as a public utility and is accordingly governed by California Proposition 218 (ref. 50), which strictly governs water rate setting, including drought surcharges. The disproportionate impact of droughts on low-income households will apply when flat drought surcharges, or other regressive surcharge structures, are applied. Flat surcharges comprise the majority of surcharges imposed in California15. As an alternative to public utilities, investor-owned utilities (IOUs) are private utilities and in California are subject to regulation by the California Public Utilities Commission. Rate increases from IOUs must be proposed in advance and justified on the basis of utility expenses, details of any infrastructure improvements and expense projections61. This process happens approximately every 3 years. Given the alternative regulatory structure and utility business model, drought impacts on low-income households in IOU service areas may take the form of gradual rate increases over time rather than short-term impacts.

Current and available water resources also shape the drivers of water affordability. Santa Cruz does not have access to significant groundwater sources (they comprise approximately 5% of available supplies). We hypothesize that utilities with greater access to groundwater resources would be able to use groundwater to mitigate drought impacts by using groundwater when surface water is scarce. Santa Cruz is also a coastal community. Our high-capacity infrastructure option comprises building a costly desalination plant, and while it does not provide the best affordability outcomes in any scenario that we tested, this type of high-capacity, always-available water source may not be available for all utilities. We also assumed that additional supply infrastructure expansion and temporary water sourcing increase the unit cost of supplying water, which is likely the case in water-stressed regions in which existing supplies have been explored, but this may not be the case in areas with significant freshwater supplies.

Finally, many of our results are shaped by the region’s demographics. We used the distribution of household income as inputs to our demand model, and in Santa Cruz there is a high proportion of households in the highest income bracket. We hypothesize that alternative household income distributions would impact affordability as this would change the proportion of utility revenue from high and low water users, thus affecting the revenue losses during curtailment.