Estimating Water Demand - Notes

Estimating Water Demand

Learning Outcomes

• Understand factors to consider when estimating residential water demand and water demand as an input in production.

• Model and estimate residential water demand and assess its sensitivity to key factors.

• Model and estimate water demand as an input in production and assess its sensitivity to key drivers.

• Guide policy and water regulation based on the demand modeling and estimation knowledge.

Lecture Outline

• Recap on the concept of demand and its determinants.

• Difference between water demand and water requirement.

• Factors that influence residential water demand.

• Econometric model for estimation.

• Evidence on price sensitivity of residential water demand.

  • Developing countries.

  • Developed countries.

• Demand for water as an input of production (derived demand) - conceptual framework.

• Determinants of water demand as a factor of production.

• Econometric model for estimation.

• Evidence.

• Policy implication of the price elasticity estimates.

Concept of Demand and its Determinants

• Fundamental problem of economics:

  • Resources (including water) are scarce yet our desires / wants are infinite.

  • Society has to allocate these scarce resources among the many competing wants.

• How to allocate / ration these scarce resources:

  • Via a Command System

  • Somebody (somebodies) have to perform the role of allocating scarce resources among competing wants.

  • Via a Price System (sometimes called a Market System)

  • The price system allocates scarce resources among competing wants.

Demand

• If you demand something, then you:

  • Want it.

  • Can afford it.

  • Have made a definite plan to purchase it.

• The quantity demanded of a good or service is the amount that consumers plan or are willing and able to buy during a particular time period, and at a particular price.

Law of Demand

• Ceteris Paribus, the higher the price of a good or service the smaller is the quantity demanded and vice versa.

• Ceteris Paribus: all other things being equal or constant.

Determinants of Demand

• Prices of related good $(P)$

• Expected future prices $(EP)$

• Income $(Y)$

• Expected future income $(EY)$

• Population $(POP)$

• Preferences $(PR)$

• Demand Function: $Q = f(P, EP, Y, EY, POP, PR,)$
  or $Q = a + b<em>1P + b</em>2EP + b<em>3Y + b</em>4EY + b<em>5POP + b</em>6PR + \epsilon$

• Questions:

  • Are there other factors of demand not listed here?

  • Are the above factors also applicable to water demand?

Sensitivity of Demand Determinants

• Sensitivity is measured via elasticities with respect to interest determinant.

  • Own-price sensitivity measure (price elasticity of demand): $E_p = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q}{\Delta P} * \frac{P}{Q}$

  • Income sensitivity measure: $E_y = \frac{\%\Delta Q}{\%\Delta Y} = \frac{\Delta Q}{\Delta Y} * \frac{Y}{Q}$

• If the demand equation is expressed in log-linear form such as $lnQ = a + b<em>1lnP + b</em>2lnEP + b<em>3lnY + b</em>4lnEY + b<em>5lnPOP + b</em>6lnPR + \epsilon$

  • The own-price sensitivity demand is given by the partial derivative of demand with respect to price: $\frac{\partial lnQ}{\partial lnP} = b_1$

  • For income, it is: $\frac{\partial lnQ}{\partial lnY} = b_3$

Water Demand Versus Water Requirement

• Water Requirement:

  • Quantity projected without considering price.

  • Quantity of water necessary to fulfil particular functions or needs under optimal conditions.

  • Generally determined by scientific, technical, or biological standards.

  • Remains independent of economic factors such as price and income.

  • Based on technical or biological needs.

• Water Demand:

  • Quantity of water purchased/used at a given price.

  • Driven by human behaviour, socio-economic factors, and pricing.

• These terms are often used interchangeably, but have distinct meanings in economics.

Demand Curves

• Demand Curve A:

  • Price-sensitive demand.

  • Reflects actual behaviour.

  • Useful for planning and policy making.

• Demand Curve B:

  • Price-insensitive (fixed) "requirement".

  • Assumes no price elasticity.

  • Risky for forecasting and policy making.

• The water requirement approach:

  • Assumes price constancy.

  • Ignores behavioural response to pricing.

  • May lead to overinvestment or underutilization.

  • Often used in engineering-based forecasts.

Estimating Residential Water Demand

• Water is a vital but scarce natural resource.

• Residential water demand = demand from households for water for domestic use.

• Economics helps us understand how and why households use water, and how price and policy influence usage.

Residential Water Use Categories

• Drinking and cooking.

• Bathing and sanitation.

• Laundry and cleaning.

• Gardening and outdoor use.

• These vary with income level, climate, urban vs. rural settings.

Economic Theory of Demand

• Law of demand: as price increases, quantity demanded decreases, ceteris paribus.

• Price elasticity:

  • Necessities - demand inelastic (less sensitive to price changes).

  • Luxury good – demand is elastic (more sensitive to price changes).

  • Water = necessity → demand is inelastic but not perfectly inelastic.

Residential Water Demand Function

• Demand function: $Qd = f(Pw, P_s, Y, Z, X, W)$

  • $Q_d$ = Quantity of water demand by household per year.

  • $P_w$ = Average water price.

  • $P_s$ = Price of a substitute (alternative water source such as a borehole).

  • $Y$ = Household Income.

  • $Z$ = Household demographic factors such as household size, level of education etc.

  • $X$ = households housing characteristics such as number of room, pool, etc.

  • $W$ = climatic factors such as precipitation, temperature.

Econometric Model

• For cross-sectional data (surveys of households for a particular year):

  • $Q<em>{i,t} = \beta</em>0 + \beta<em>1 P</em>{i,t} + \beta<em>2 S</em>{i,t} + \beta<em>3 Y</em>i + \beta<em>4 Z</em>i + \beta<em>5 X</em>i + \epsilon_i$

  • The above model can be estimated using OLS approach

  • But in situation of endogeneity issues, Instrumental Variable approach can be use, including Two-stage least squares (2SLS)

• For panel data (surveys of households for several years):

  • $Q<em>{i,t} = \beta</em>0 + \beta<em>1 P</em>{i,t} + \beta<em>2 S</em>{i,t} + \beta<em>3 Y</em>{i,t} + \beta<em>4 Z</em>{i,t} + \beta<em>5 X</em>{i,t} + \epsilon_{i,t}$

  • The above model can be estimated using panel model approach such as:

  • Pooled OLS

  • Random Effects Model (REM)

  • Fixed Effect model (FEM) and its instrumental variable version

  • Random Parameters /Coefficients Model

Water Pricing Structures

• Uniform Pricing: Same price per unit regardless of quantity used (e.g., $1 /m³ for all consumption).

• Increasing Block Tariff (IBT): Price increases with higher usage to promote conservation (e.g., $1 /m³ (0–10m³), $2 /m³ (10–20m³)).

• Decreasing Block Tariff: Price decreases as usage increases – often used for industries (e.g., $ 2/m³ (0–10m³), $1/m³ (10–20m³)).

• Two-Part Tariff: Fixed charge + variable usage charge (e.g., $10/month + $1/m³).

• Seasonal Pricing: Higher rates during dry or peak seasons (e.g., $ 2/m³ (dry season), $1/m³ (wet season)).

Price Elasticity of Demand

• Price Elasticity $(E_p)$ = percentage change in quantity / percentage change in price.

• Water typically has {E_p} < 1

Price and Quantity Demand Relationship

• Given the demand theory, and water being a normal good, it is expected that price and quantity demand for water to be inversely related.

• The higher the price, the lower the quantity of water demanded by households, ceteris paribus.

• Substitutes price effect is expected to be positive.

Income and Water Demand

• Households' income is expected to be positively related to water demand.

• Since water is a normal good, the higher the income, the higher the demand, all other things held constant.

• Low-income households: Use water mostly for essential needs (drinking, cooking, hygiene).

• High-income households: Higher usage due to discretionary activities (e.g., gardening, pools, laundry).

Case Insight: Cape Town

• A study (Savelli, Elisa, et al., 2023. Urban water crises driven by elites’ unsustainable consumption." Nature Sustainability) showed that wealthier suburbs use significantly more water.

• During drought restrictions, high-income areas reduced usage more sharply — likely due to more discretionary use to cut back on.

• Indigent support policies (free basic water) buffer poor households from income shocks.

Non-Price and Non-Income Determinants

• Education & awareness (Z): influences conservation behaviour and it is expected to have negative relationship.

• Technology (X): efficient toilets, showers reduce use (but there can be a rebound effect that can lead to more use).

• The rebound effect is a phenomenon where improvements in resource efficiency (e.g., energy or water efficiency) lead to increased consumption of that resource, partially or fully offsetting the expected gains.

• Weather & seasonality (W): more use in dry seasons.

• Household size (Z): larger households use more water.

Questions for Discussion

• Why is water demand inelastic? What are the implications for pricing?

• What is a rebound effect when using efficient technologies and its implication on water demand?

• How do income levels influence water use patterns?

• Should water be priced like any other good?

• What policies can promote conservation while ensuring equity?

Estimating Demand for Water as an Input of Production

• Categories of Water Demand

  • Agricultural: Irrigation, livestock

  • Commercial: Offices, retail, hospitality, institutions

  • Industrial: Cooling, manufacturing, processing

  • Recreational & Environmental: Parks, ecosystems

• These uses of water are more a derived demand-thus demand for water to aid the production of another good such as irrigation for agriculture produce or water use in the production of crude oil

Water Use per Sector in South Africa

• Irrigation: 60%

• Municipal/domestic: 30%

• Mining and power generation: 3%

• Commercial forestry plantations: 7%

Production Theory and Water Demand

• Firms use water as part of input bundle: $Y = f(x<em>1, x</em>2,…, x_n)$

• Water inputs include:

  • Intake

  • Recycling

  • Pre-use treatment

  • Discharge

• The input demands can be derived from cost minimization or profit maximization framework

Cost Minimization Framework

• Objective: Minimise cost for given output level $y$

  • subject to $f(x<em>1, x</em>2,…, x_n) \geq y$

  • $min \sum p<em>i x</em>i$

  • $p_i$ = prices of the inputs

• Conditional demand: $x<em>i^* = h(p</em>1, p<em>2,….,p</em>n, y)$

• This suggest that the demand for water or any input is a function of all input prices $(p_i)$ and output $(y)$.

Cost Minimization Framework

• If there are only capital (K), labour (L), Material (M) and water (wd) inputs in the production, the conditional water input demand function $(wd^*)$ can be express as

  • $wd^* = h(p<em>k, p</em>l, p<em>m, p</em>{wd}, y)$

• The econometric model of the above for a single industry or firm for several years can be express as

  • $wd^* = \beta<em>0 + \beta</em>1 p<em>k + \beta</em>2 p<em>l + \beta</em>3 p<em>m + \beta</em>4 p<em>{wd} + \beta</em>5 y + \epsilon$

  • Where $p_i$ note input price

Profit Maximization Framework

• Objective: Maximise profit:

  • $max \Pi = py^* - \sum p<em>i x</em>i$

• Use Hotelling's Lemma for derivation, the optimal inputs functions becomes

  • $x<em>i^* = f(p</em>1, p<em>2,…., p</em>n, p)$

• If the inputs are capital (K), Labour, Material(m) and water (wd), the above can be express as

  • $wd = f(p<em>k, p</em>l, p<em>m, p</em>{wd}, p)$

• The econometric model becomes for a single industry over several years (time series)

  • $wd^* = \beta<em>0 + \beta</em>1 p<em>k + \beta</em>2 p<em>l + \beta</em>3 p<em>m + \beta</em>4 p<em>{wd} + \beta</em>5 p + \epsilon$
    
    $*$ The model can be estimated using OLS or complex models such as ARDL, VECM etc.

Econometric Model

• The econometric model for many industry for one period (e.g., year)- cross-sectional data

  • $wd^* = \beta<em>0 + \beta</em>1 p<em>{k,i} + \beta</em>2 p<em>{l,i} + \beta</em>3 p<em>{m,i} + \beta</em>4 p<em>{wd,i} + \beta</em>5 y<em>i + \epsilon</em>i$

  • Can be estimated using OLS, Instrumental variable (IV) approach

• The econometric model for many industry for many period (e.g., year) - panel data

  • $wd^* = \beta<em>0 + \beta</em>1 p<em>{k,i,t} + \beta</em>2 p<em>{l,i,t} + \beta</em>3 p<em>{m,i,t} + \beta</em>4 p<em>{wd,i,t} + \beta</em>5 y<em>{i,t} + \epsilon</em>{i,t}$

  • Can be estimated using

  • Pooled OLS,

  • Random Effects Model (REM),

  • Fixed Effect model (FEM) and its instrumental variable version

  • Random Parameters /Coefficients Model etc.

Properties of Input Demand

• Dependent on all input and output prices

• Homogeneous of degree zero

• Decreasing in own price

Role of Climate and Institutional Factors

• Either the cost minimization or profit maximization derived water input demand in empirical settings are augmented to capture:

  • climate, and

  • institutional factors

• Temperature and rainfall for the climatic factors

• Regulation for institutional factors

Price and Output Elasticities

• Price elasticity for industrial/commercial users are typically low (–0.1 to –0.5)

  • Due to process rigidity, low cost share

• Output elasticity is often stronger and positive (0.1 to 0.99)

  • Output

Sensitivity of Demand Determinants

• Sensitivity is measured via elasticities with respect to interest determinant.

  • Own-price sensitivity measure (price elasticity of demand): $E_p = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q}{\Delta P} * \frac{P}{Q}$

  • Income sensitivity measure: $E_y = \frac{\%\Delta Q}{\%\Delta Y} = \frac{\Delta Q}{\Delta Y} * \frac{Y}{Q}$

  • If the demand equation is expressed in log-linear form such as $lnQ = a + b<em>1lnP + b</em>2lnEP + b<em>3lnY + b</em>4lnEY + b<em>5lnPOP + b</em>6lnPR + \epsilon$

  • The own-price sensitivity demand is given by the partial derivative of demand with respect to price: $\frac{\partial lnQ}{\partial lnP} = b_1$

  • For income, it is: $\frac{\partial lnQ}{\partial lnY} = b_3$

Residential Water Demand Function

• Demand function: $Qd = f(Pw, P_s, Y, Z, X, W)$

  • $Q_d$ = Quantity of water demand by household per year.

  • $P_w$ = Average water price.

  • $P_s$ = Price of a substitute (alternative water source such as a borehole).

  • $Y$ = Household Income.

  • $Z$ = Household demographic factors such as household size, level of education etc.

  • $X$ = households housing characteristics such as number of room, pool, etc.

  • $W$ = climatic factors such as precipitation, temperature.

Econometric Model

• For cross-sectional data (surveys of households for a particular year):

  • $Q<em>{i,t} = \beta0 + \beta1 P</em>{i,t} + \beta<em>2 S{i,t} + \beta3 Y</em>i + \beta<em>4 Zi + \beta5 X</em>i + \epsilon_i$

  • The above model can be estimated using OLS approach

  • But in situation of endogeneity issues, Instrumental Variable approach can be use, including Two-stage least squares (2SLS)

• For panel data (surveys of households for several years):

  • $Q<em>{i,t} = \beta0 + \beta1 P</em>{i,t} + \beta<em>2 S{i,t} + \beta3 Y</em>{i,t} + \beta<em>4 Z{i,t} + \beta5 X</em>{i,t} + \epsilon_{i,t}$

  • The above model can be estimated using panel model approach such as:

  • Pooled OLS

  • Random Effects Model (REM)

  • Fixed Effect model (FEM) and its instrumental variable version

  • Random Parameters /Coefficients Model

Production Theory and Water Demand

• Firms use water as part of input bundle: $Y = f(x1, x2,…, x_n)$

Cost Minimization Framework

• Objective: Minimise cost for given output level $y$

  • subject to $f(x1, x2,…, x_n) \geq y$

  • $min \sum p<em>i x</em>i$

  • $p_i$ = prices of the inputs

  • Conditional demand: $x_i^* = h(p1, p2,….,pn, y)$

• If there are only capital (K), labour (L), Material (M) and water (wd) inputs in the production, the conditional water input demand function $(wd^*)$

  • $wd^* = h(p<em>k, p</em>l, p<em>m, p</em>{wd}, y)$

  • The econometric model of the above for a single industry or firm for several years can be express as

  • $wd^* = \beta<em>0 + \beta</em>1 p<em>k + \beta2 pl + \beta</em>3 p<em>m + \beta4 p{wd} + \beta</em>5 y + \epsilon$

Profit Maximization Framework

• Objective: Maximise profit:

  • $max \Pi = py^* - \sum p<em>i x</em>i$

  • Use Hotelling's Lemma for derivation, the optimal inputs functions becomes

  • $x_i^* = f(p1, p2,…., pn, p)$

  • If the inputs are capital (K), Labour, Material(m) and water (wd), the above can be express as

  • $wd = f(p<em>k, p</em>l, p<em>m, p</em>{wd}, p)$

  • The econometric model becomes for a single industry over several years (time series)

  • $wd^* = \beta<em>0 + \beta</em>1 p<em>k + \beta2 pl + \beta</em>3 p<em>m + \beta4 p{wd} + \beta</em>5 p + \epsilon$

Econometric Model

• The econometric model for many industry for one period (e.g., year)- cross-sectional data

  • $$wd^* = \beta0 + \beta1 p{k,i} + \beta*2 p{l,