Estimating the Supply Function of a Water Utility

Learning Objectives

• Understand what determines the supply of water by a utility, the institutional landscape, and pricing.

• Formulate a theoretical supply function.

• Identify variables affecting water supply.

• Estimate a supply function using econometric methods.

• Provide empirical example(s).

Supply

• If a firm supplies a good or service, then the firm:

  • Has the resources/factors of production and technology to supply it.

  • Can profit from producing the good or service.

  • Has made a definite plan to produce and sell the good.

• The quantity supplied of a good or a service is the amount that producers plan to sell during a given time period at a particular price.

• Law of supply: Ceteris Paribus, the higher the price of a good, the greater is the quantity supplied.

What is Water Supply?

• Quantity of water a public utility or a private company is willing and able to produce and deliver to users.

• It is a process of sourcing, treating, storing, and distributing water to users in a reliable and safe manner.

• Water supply depends on price, costs of inputs, weather, infrastructure, regulation/policy, and technology.

Components of Water Supply System

• Source:

  • Surface water: Rivers, lakes, reservoirs

  • Groundwater: Wells, aquifers

  • Desalination: Seawater

  • Rainwater harvesting

• Treatment:

  • Ensures the removal of contaminants and pathogens for safe drinking

  • Includes filtration, chlorination, and sometimes advanced purification

• Storage:

  • The use of Tanks, reservoirs, or elevated towers in storing the treated water to meet demand

• Distribution:

  • A network of pipes, pumps, and valves to deliver water to homes, businesses, and farms

Water Supply vs Water Demand

Cost Structure of a Water Utility

• Fixed Costs (FC): Infrastructure, capital cost or repayment

• Variable Costs (VC): Energy, chemicals, labor

• Total Cost (TC): TC = FC + VC(Q)

• FC is fixed cost – costs that do not vary per unit of output produced.

• VC(Q) is variable cost – cost that changes with a change in output.

Marginal Cost and Supply

• Marginal Cost (MC): Additional cost to produce one more unit

• MC(Q) = \frac{\Delta TC}{\Delta Q}

• Under perfect competitive case, output is produced when P = MC(Q)

  • P is price

  • MC(Q) is marginal cost

• Under Monopoly, output is produced when MR = MC(Q)

  • MR is marginal revenue

Output Condition Under Perfect Competition

• In a perfectly competitive market, the firm maximizes profit where: MR = MC(Q), but since P = MR under perfect competition, the condition becomes: P = MC(Q).

• Meaning: Supply is the quantity at which marginal cost equals market price.

Output Condition Under Monopoly

• A monopoly produces output at where marginal revenue equals marginal cost: MR(Q) = MC(Q)

• Since a monopoly faces a downward-sloping demand curve and has to reduce price to sell more, P(Q) > MR(Q)

• First-order condition for monopoly profit maximization [P(Q)* - MC(Q)] = 0

• MR(Q) = MC(Q)

• Supply curve for the monopoly: Quantity supplied depends on the demand curve and cost structure.

Monopoly: MR and MC Curves

• MR is the marginal revenue curve

• MC is the marginal cost curve

• If you impose the monopoly demand curve, you get the amount that will be supplied at a corresponding price, determined by the demand curve.

Institutional Landscape of Water Utilities

• In developing countries, water utilities are mostly:

  • Public, municipal, or regional government utilities

• Few countries globally have some private ownership such as:

  • USA: ~25% investor-owned utilities

  • The reason is due to the importance and the huge infrastructure cost for water supply systems

• In South Africa, it is characterized by a mix of national, regional, and local (municipal) entities.

Pricing Structures by Consumer Type

Why Current Pricing is Inefficient

• Excludes Full Economic Cost:

  • Water is undervalued.

  • Capital costs measured via debt service, not user cost of capital.

• Ignores Resource Scarcity:

  • Prices do not signal scarcity.

  • Demand treated as exogenous "requirement."

• Relies on Average Cost:

  • Prices based on arbitrary allocations of joint capital costs.

Utilities Pricing for Water

• Theory suggests that for efficient water supply, set P = MC(Q)

• Issue: May lead to a financial deficit.

• In practice, utilities use average cost pricing, thus P = TC/Q

• The reason being that MC pricing will not generate enough revenue to cover the total cost of water supply due to the high fixed cost in the industry.

• Average-cost pricing ignores marginal cost (MC) and elasticity.

• The problem of Average – Cost pricing is that it results in excess demand during peak periods and poor investment signals.

Efficient Pricing Options for Utilities

Two key options:

• Ramsey Pricing

• Seasonal pricing

• Two-Part Tariffs

• Ramsey Pricing: Set prices inversely proportional to price elasticities to minimize welfare loss (CS + PS).

  • How should a utility set prices when marginal cost pricing does not cover total costs (e.g., due to fixed costs)?

  • Ramsey pricing provides an answer to the above question

  • It maximizes social welfare subject to a revenue sufficiency constraint for a public utility.

  • The key idea of Ramsey Pricing is that prices should deviate from marginal cost in inverse proportion to the price elasticity of demand for each consumer group or time period.

Ramsey Pricing Formula

\frac{Pi - MCi}{Pi} = \frac{\lambda}{\epsiloni}
Where:

• P_i = is the price for group i

• MC_i = is the marginal cost for supply group i

• \lambda = is the Ramsey multiplier that ensures breakeven revenue for the utility company

• \epsilon_i = is demand elasticity for group i.

• The Ramsey pricing scheme implies that price markups should be greater for consumer groups with less elastic demand (price-insensitive groups).

Seasonal Pricing

• Seasonal Pricing: Adjust prices according to marginal cost variations over time (higher in summer, lower in winter).

• Water demand and supply often fluctuate throughout the year due to climatic and usage patterns.

• Seasonal pricing aims to reflect the marginal cost of water provision during peak and off-peak periods.

  • Summer (Peak Season): Higher demand due to irrigation, outdoor use, and higher temperatures.

  • Winter (Off-Peak Season): Lower demand as most outdoor use ceases.

Two-Part Tariff Pricing

• Efficient recovery of fixed costs: A = (C - R) / N, P=MC

  • Where:

    • A: Annual fixed charge per user

    • C: Total costs

    • R: Revenues from marginal cost pricing

    • N: Number of consumers

• Condition: Annual fee must not exceed consumer surplus (why?).

Policy Simulations On Pricing Options

• Base Case: Average-cost pricing (uniform rate)

• Scenario 1: Ramsey Pricing

  • Higher price for inelastic users (lower welfare loss)

  • Cross-subsidize elastic demand groups (e.g., low-income households)

• Scenario 2: Seasonal Pricing

  • Price reflects higher summer marginal costs

  • Reduces overuse in peak season, aligns with infrastructure stress

• Scenario 3: Two-part tariff

  • Effective and efficient recovery of fixed cost

Key Takeaways for Water Tariff Policy

• Marginal-Cost Pricing Is Ideal but politically difficult

• Ramsey, Seasonal Pricing, and Two-part Tariffs are practical second-best options

• Efficient pricing aligns consumption with infrastructure and environmental limits

• Accurate demand and cost estimation are essential for reform

• Reforms should be complemented by communication strategies to address equity concerns

Theoretical Water Supply Function

• General functional form: Q = f(P, C, CV)

  • Where:

    • Q: Quantity supplied

    • P: Price or tariff per m^3

    • C: Input costs (electricity, chemicals, labor)

    • CV: Control variables (rainfall, capacity, policy changes)

Econometric Specification (Linear Form)

ln Q{it} = \beta0 + \beta1 P{it} + \beta2 C{it} + \beta3 ln Z{it} + \epsilon_{it}

• Where:

  • Q_{it}: Water supplied by utility i at time t

  • P_{it}: Price or average tariff

  • C_{it}: Cost variables (e.g., electricity cost per m^3)

  • ln Z_{it}: Other exogenous factors, control variables

  • \epsilon_{it}: Error term

Sensitivity Analysis

• The sensitivity of any of the determinants of water supply can be assessed via supply elasticity

• For instance, from the water supply econometric model ln Q{it} = \beta0 + \beta1 P{it} + \beta2 C{it} + \beta3 ln Z{it} + \epsilon_{it}

  • The sensitivity of water supply to price is given by \epsilonP = \beta1 where \epsilonP is the percentage change in water supply due to a percentage change in price. \beta1

  • The sensitivity of water supply to input cost (C{it}) is given by \epsilonC = \beta2 where \epsilonC is the percentage change in water supply due to a percentage change in input cost \beta_2 - thus input cost elasticity.

Variable Expectations

Example Data for Water Supply Estimation

Estimation and Interpretation

• Given the general functional form: Q = f(P, C, CV)

  • Where

    • Q = Output(m^3)

    • P = Price or tariff ($/m^3)

    • C = Electricity cost

    • CV = Rainfall

• Output = \beta0 + \beta1Price + \beta2Electricity cost + \beta3Rainfall + \epsilon

Estimated Coefficients

• Output = -162243 + 557073 Price −102096 Electricity cost −250.0 Rainfall

  • Can you obtain the price elasticity of water supply evaluated at April price and output?

  • What is the interpretation for Price coefficient (\beta_1)? 557073

  • What is the interpretation for Electricity cost coefficient (\beta_2)? −102096

  • Could we have estimated the supply function in a way that we obtain the elasticities directly? How should this be done?

Common Challenges in Supply Estimation

• Endogeneity: Price may be influenced by output

• Multicollinearity: High correlation among cost variables

• Omitted Variable Bias: Missing infrastructure or regulation variables

Remedies and Extensions

• Use Instrumental Variables (IV) for price

• Include Fixed Effects for utilities in panel data

• Use Log-Log Specification

Policy Relevance

• Helps utilities forecast supply response to tariff changes

• Supports investment planning and cost recovery

• Enables better drought and climate adaptation strategies

Summary

• Water utility supply depends on price, cost, and constraints

• Econometric models can estimate supply response

• Reliable data and correct specification are key

• Current water pricing often inefficient: ignores scarcity and full cost.

• Efficient pricing requires:

  • Marginal cost-based pricing

  • Ramsey pricing if deficits arise

  • Two-part tariffs for equity and efficiency

  • Seasonal/peak pricing under capacity constraints

Discussion Questions

• Why is marginal cost pricing considered efficient?

• What challenges do utilities face in implementing Ramsey pricing?

• How could seasonal pricing improve resource management?

• Can smart metering and digital billing enable more dynamic pricing structures?

• What are the key determinants of water supply?

• Why is price elasticity positive?