Optimal Reliability for a Water Authority

Richard T. Carson,� Je¤rey K. O�Haray and Garey Rameyz

October 2012

Preliminary Version

Abstract

Water planners determine system reliability through supply side investment in water

infrastructure, and demand side management via rate schedules and rationing rules.

Previous literature on peak-load pricing is inapplicable to the case of water because it

does not capture the joint e¤ects of rainfall on water supply and demand. This paper

develops a model of water provision in which supply and demand variations are driven by

random rainfall �uctuations. We provide a complete characterization of socially optimal

price, system capacity and reliability levels, and we show that capacity provides social

bene�ts only for a middle range of rainfall realizations. We also consider the e¤ects of

policy constraints, including restrictions on pricing and rationing by end use, on the

optimal level of reliability.

�University of California San Diego. E-mail: rcarson@ucsd.edu.

yUnion of Concerned Scientists. E-mail: je¤reyohara@gmail.com.

zUniversity of California San Diego. E-mail: gramey@ucsd.edu.

1

1 Introduction

Municipal water planners design water systems to achieve a speci�ed level of reliability, which

is the probability that supply is su¢ cient to serve demand fully. Like managers in other

public utility sectors, water managers can in�uence reliability through both the supply and

demand sides. The contribution of this paper is to extend the theoretical peak-load pricing

literature by deriving analytically the optimal levels of price, capacity and reliability for a

municipal water system.

It is estimated that the cost of maintaining water system reliability in the U.S. in the

coming decades will be upwards of $1 trillion (AWWA (2012)). Current and future strains on

water systems include climate change, population growth, economic development, depleted

groundwater aquifers, environmental considerations, and aging municipal water infrastruc-

ture. These remarks highlight the importance of incorporating economic criteria into the

management of water systems, as reliability targets are typically determined on an ad hoc

basis. Moreover, municipal water agencies often set rates equal to the expected amortized

cost of distribution and infrastructure due to the di¢ culties associated with calculating mar-

ginal cost, possible revenue constraints, and political considerations. Investment in water

supply infrastructure is infrequent as it entails signi�cant scale economies. This means water

will be signi�cantly underpriced relative to marginal cost, in that the historical average cost

of capacity can be far less than the future replacement cost of capacity (Hanemann (2006)).

The standard peak-load pricing model was developed to address these kinds of issues in

the context of electric power systems, but the model is inappropriate for municipal water

applications since it does not capture the manner in which rainfall a¤ects water supply and

demand. Levels of rainfall form a constraint on available water supplies, irrespective of the

size of the system�s delivery capacity. At the same time, outdoor precipitation serves as a

substitute for delivered water, so that water demand is more easily satis�ed when rainfall

levels are high. In other words, water supply and demand respond jointly and in opposite

directions to rainfall.1

1Although we develop our analysis in the context of municipal water supply, the results also apply to other

2

We develop a theoretical model of water system reliability that captures these key e¤ects

of rainfall. Our model classi�es water consumers according to whether they are indoor or

outdoor users. Indoor users rely exclusively on supplies delivered by the water authority, but

outdoor users can augment these supplies with point-of-use collection. Rainfall is determined

randomly. Realized rainfall constrains water supply, so that supply is increasing in the

random rainfall variable up to the point at which rainfall exceeds system delivery capacity.

Moreover, for any given water price, water demand is a decreasing function of the rainfall

variable, since higher rainfall increases point-of-use collection and thus decreases outdoor

users�demand.

The water authority chooses price and capacity to maximize expected social welfare.

Our analysis demonstrates that capacity contributes to welfare only when realized rainfall is

su¢ ciently high to make full use of available capacity, and also su¢ ciently low to generate

adequate demand. For very low rainfall realizations, a water shortage ensues, but there is

also excess capacity; increments to capacity would not alleviate the shortage, so they provide

zero social bene�t. For very high rainfall realizations, excess capacity arises due to reduced

demand among outdoor users, and again increments to capacity would provide zero bene�t.

Thus, capacity contributes to social welfare only for a middle range of rainfall realizations.

The water authority must take account of both the low- and high-rainfall margins in order

to obtain a correct assessment of welfare.

By taking careful account of the structure of the social welfare function, we provide a

complete characterization of solutions to the water authority�s social welfare maximization

problem. Necessary conditions for optimal price and capacity follow along the lines of

previous peak-load pricing results. We extend the literature by providing general su¢ cient

conditions for optimality. Moreover, we demonstrate that the optimal capacity is unique,

public utility sectors that face similar joint supply-demand responses. For example, under climate change

scenarios electricity systems may exhibit this characteristic, as higher temperatures increase the demand for

electricity, while simultaneously constraining supply (Van Vliet et al. (2012)) . Supply responses are driven

by reduced water availability for hydropower production and cooling water intake for coal-�red and nuclear

plants, among other factors.

3

and the optimal price is unique whenever optimal reliability is strictly positive.

In practical applications, key policy variables may be restricted by long-term contracts

or political considerations, and water planners must treat such restrictions as additional

constraints on welfare maximization. Our model can be applied to evaluate how water sys-

tems should best be modi�ed in the presence of such policy constraints. First, we show

that optimal capacity is reduced whenever the authority is forced to set the water price

suboptimally. This implies that e¢ cient pricing provides the greatest social returns to ca-

pacity. When price is too high, increments to capacity become less bene�cial since demand

is reduced by the high price, leading to excess capacity for a greater range of rainfall realiza-

tions. When price is too low, demand becomes excessive for a range of rainfall realizations,

and reducing capacity provides a way to restrict ine¢ cient overconsumption. The impact of

higher prices on reliability is ambiguous a priori, since higher prices directly reduce demand

and indirectly reduce optimal capacity levels; we demonstrate, however, that the former

e¤ect dominates, so that reliability is strictly increasing in price, irrespective of whether the

price is above or below its optimum.

A critical issue in the peak-load pricing literature is that the optimal levels of price and

capacity must be jointly determined with the rationing scheme employed by the utility in

the event of a supply shortage. Spot market pricing, which achieves e¢ cient rationing, is

seldom implemented to curtail shortages in practice. Like other public utilities, municipal

water authorities rely on quantity-based rationing schemes, but rationing is typically tied

to end-use restrictions. This di¤ers from the willingness-to-pay, proportional rationing and

random rationing schemes considered in the peak-load pricing literature.

We derive new results that are relevant to the case of water by considering a stylized

suboptimal rationing rule based on end-use restrictions. In particular, we consider a prefer-

ential rationing rule wherein indoor users receive priority in the event of shortage. We show

that, in response to this rule, the water authority optimally raises both price and system ca-

pacity. Higher price serves to restrict indoor users�demand disproportionately, while higher

capacity makes greater supplies available. Both of these e¤ects serve to direct supplies to

4

outdoor users in the event of a shortage. The reliability of the water system is increased as

a consequence.

The paper proceeds as follows. The related literature is reviewed in Section 2, the model

is presented in Section 3, and optimal reliability is characterized in Section 4. Section 5

considers policy constraints, Section 6 analyzes a numerical example and Section 7 concludes.

2 Literature review

The theoretical peak-load pricing literature, surveyed in Crew, Fernando, and Kleindorfer

(1995), focuses on how to establish price, capacity and reliability optimally for a non-storable

commodity with �uctuating demand. More recent contributions have examined the impli-

cations of consumers� expectations on system reliability (Schroyen and Oyenuga (2011)).

Building on previous e¤orts, Chao (1983) and Kleindorfer and Fernando (1993) develop

general theoretical peak-load price and reliability results for stochastic supply and demand

functions. They assume that the stochastic components of supply and demand are indepen-

dent, which may be reasonable for some public utility sectors. However, this assumption

is inappropriate for municipal water systems since supply and demand are jointly a¤ected

by random rainfall.2 In contrast to previous �ndings, we show that an increase in capacity

does not provide bene�ts when demand is at its highest, since this arises in situations where

available rainfall is low.

Our analysis establishes conditions for existence and uniqueness of jointly optimal price

and capacity, and of optimal capacity for a given price. This is, to our knowledge, the �rst

general characterization of solutions in the theoretical peak-load pricing literature. Some

papers in the literature establish second-order conditions for special cases (e.g., Brown and

Johnson (1969), Chao (1983)), but the general model developed by Kleindorfer and Fernando

(1993) and Crew, Fernando, and Kleindorfer (1995) does not establish su¢ cient conditions.

We adopt a static framework, in line with the bulk of previous research on peak-load

2As mentioned in footnote 1, stochastically independent supply and demand may also be an inappropriate

assumption for electricity systems under climate change scenarios.

5

pricing. Although e¤ectively managing stored supplies is a central aspect of maintaining

water system reliability, treating in�ow as a stochastic variable in a dynamic context is a

challenging undertaking, as it also requires accounting for optimal reservoir management.

Theoretical peak-load pricing models incorporating storage have relied on deterministic spec-

i�cations (e.g., Nguyen (1976), Gravelle (1976)). With respect to water systems, Riley and

Scherer (1979) and Manning and Gallagher (1982) use a dynamic model to derive opti-

mal price and capacity when in�ows are deterministic, which can be useful for examining

the implications of temporal variations in supply and demand on optimal storage capacity.

While our treatment of supply and demand o¤ers greater insight into how uncertainty im-

pacts price and capacity, our results are less informative with respect to how changes in the

amplitude of supply and demand impact storage capacity.

Another strand of theoretical research has focused on calculating optimal prices, short-

run marginal costs, or social welfare from various pricing regimes of a municipal water

system (Fakhraei, Narayanan, and Hughes (1984), Swallow and Marin (1988), Zarnikau

(1994), Elnaboulsi (2001), Gri¢ n (2001), Elnaboulsi (2009), Reynaud (2010)). This research

has considered a greater range of institutional details of water distribution systems, but it

has not attempted to solve simultaneously for both price and capacity, and as consequence

optimal reliability levels are not derived.

There has been a great deal of empirical work by economists on municipal water issues,

including numerous articles on water price elasticity (for meta-analyses of the water demand

literature, see Espey, Espey, and Shaw (1997) and Dalhuisen et al. (2003)). Factors that

in�uence in�uence water demand include both the magnitude and frequency of precipitation,

as well as season (e.g., Klaiber et al. (2012)). In this literature, demand is often assumed to

be lognormally distributed because water demand can have signi�cant right-skewness even

when the independent variables have no skewness (Hewitt and Hanemann (1995)); log-log

speci�cations are commonly adopted (Olmstead, Hanemann, and Stavins (2007), Olmstead

(2009) and Mansur and Olmstead 2011). Importantly, precipitation has been found to

have no in�uence on indoor demand, while outdoor demand increases signi�cantly in drier

6

conditions (Mansur and Olmstead (2011)).3

3 Model

Although water managers make multiple types of investments in developing water supply

systems, in this paper we focus on overall system delivery capacity. Speci�c supply com-

ponents may also operate to increase system reliability; therefore, much of the analysis will

be relevant to these components also. We also consider the case in which the rate schedule

charges a constant price per unit of water purchased.4

3.1 Structure

Suppose there are two end-uses for water, indoor and outdoor. Let SI and SO be the quan-

tities of water supplied for indoor and outdoor use, respectively, and let S = SI +SO denote

total water supply. Supply is random, in the sense that rainfall must be collected and stored

before it is supplied. Storable precipitation is given by R, which is a nonnegative random

variable having distribution function F (R) and continuous density f(R), with f(R) > 0 for

all R. System capacity is denoted by K, so that S is bounded above by minfR;Kg. Note

that the capacity of the system is determined prior to precipitation occurring and cannot be

adjusted ex post, which is consistent with previous approaches in the peak-load pricing liter-

3Empirical research has also measured the willingness-to-pay by consumers to avoid water shortages

(Carson (1991), Barakat & Chamberlain (1994), Howe, et al. (1994) and Gri¢ n and Mjelde (2000)) and the

welfare implications of alternate rationing schemes (Woo (1994) and Mansur and Olmstead (2011)). Olmstead

and Stavins (2009) summarize the considerations involved with selecting the appropriate rationing method

for urban water supply. In addition, there is an engineering literature that has concentrated on supply-side

issues such as capacity expansion and reservoir operations (e.g., Loucks, et al. (1981) and Oliveira and

Loucks (1997)) as a means of maintaining reliability, where the reliability targets used for evaluation are

chosen arbitrarily and demand is �xed.

4 In practice, some water rates are set in increasing or decreasing block schedules, meaning that the per-

unit rate for water paid by consumers is a function of the quantity of water consumed. We focus on the case

of constant per-unit rates for simplicity. Moreover, we assume that all users face the same rate schedule.

7

ature. The cost of capacity is � per unit. In addition, the water authority incurs treatment

and transmission costs of b per unit of water supplied. Assume that no �ow constraints exist

on system capacity.

Let UI(SI) and UO(SO; R) indicate the utility functions of indoor and outdoor water

users, respectively, when SI and SO are allocated e¢ ciently within each end-use. Note that

storable rainfall a¤ects the utility of outdoor water consumption, since outdoor consumers

also obtain water from point-of-use collection. Let derivatives be indicated by subscripts,

i.e., UIS(SI) denotes the derivative of UI(SI) with respect to SI . We impose the following

assumptions on the utility functions:

Assumption 1. UIS(SI) > 0 > UISS(SI), limSI!1 UIS(SI) = 0 and limSI!0 UIS(SI) =1.

Assumption 2. UOS(SO; R) > 0 > UOSS(SO; R), UOR(SO; R) > 0 > UOSR(SO; R),

limSO!1 UOS(SO; R) = 0 and limSO!0 UOS(SO; R) =1.

Assumption 3. There exist constants M and m, with M > m > 0, such that �M <

UOij(SO; R) < �m for all SO; R and all i; j = S;R.

Assumptions 1 and 2 impose standard monotonicity, curvature and boundary conditions.

Assumption 2, also states that an increase in rainfall reduces the marginal utility of water

supplied to outdoor users.5 Assumption 3 is a technical condition that rules out irregular

behavior as R grows without bound.6

Water for either end-use is supplied at a constant price of P per unit. As with system ca-

pacity, the price is determined prior to the realization of rainfall. In the absence of rationing,

indoor and outdoor consumers demand DI(P ) and DO(P;R) units of water, respectively,

from the water authority. Under the above assumptions, these demand functions are always

5For simplicity, we assume rationing and supply disruptions do not impose additional costs. Moreover,

there are no spillage costs from reservoir overtopping in this static framework.

6As an alternative to Assumption 3, it could be assumed that F (R) has bounded support. The latter

assumption makes the analysis somewhat less elegant, however.

8

strictly positive, strictly decreasing in all arguments, and are uniquely determined by

UIS(DI(P )) = P; (1)

UOS(DO(P;R); R) = P: (2)

Total demand is given by

D(P;R) = DI(P ) +DO(P;R): (3)

Note that D(P;R) is strictly decreasing in both of its arguments. Let D(P ) =

limR!1D(P;R) indicate the limiting level of total demand as R increases without bound.

3.2 Precipitation thresholds and reliability

Realized water supply can be linked to threshold levels of precipitation. First, de�ne Ra(P )

by

D(P;Ra(P )) = Ra(P ): (4)

As long as R � Ra(P ), D(P;R) � R will hold, meaning that rainfall is su¢ cient to meet

demand given that adequate system capacity is available. If R < Ra(P ), however, then

D(P;R) > R, so that water will be in short supply no matter how great is system capacity.

Thus, Ra(P ) represents a rainfall shortage threshold. Note that Ra(P ) is strictly positive

and strictly decreasing in P .

Even when rainfall is adequate, system capacity may be insu¢ cient to meet demand.

For K 2 (D(P ); D(P; 0)), de�ne Rb(P;K) by

D(P;Rb(P;K)) = K: (5)

In this instance, R � Rb(P;K) assures that D(P;R) � K, so capacity is su¢ cient to meet

demand, whereas R < Rb(P;K) implies D(P;R) > K, meaning capacity is insu¢ cient.

It follows that Rb(P;K) constitutes a capacity shortage threshold. It can be veri�ed that

Rb(P;K) is strictly positive and strictly decreasing in P and K for K 2 (D(P ); D(P; 0)).

Moreover, the capacity shortage threshold may be extended continuously to the full range

of K by setting Rb(P;K) = 0 for K � D(P; 0) and Rb(P;K) =1 for K � D(P ).

9

The total amount of water supplied as a function of price, precipitation and capacity

is indicated by S(P;R;K). This quantity is determined as follows. As long as minfR;Kg

exceeds D(P;R), demand can be fully met, so that S(P;R;K) = D(P;R). If minfR;Kg is

strictly less than D(P;R), then there is a shortage, and only S(P;R;K) = minfR;Kg will

be supplied. Thus:

S(P;R;K) = minfR;K;D(P;R)g: (6)

The following lemma links the form of S(P;R;K) to the rainfall and capacity shortage

thresholds. (The proofs of all lemmas and propositions are given in the Appendix.)

Lemma 1. a. If K � D(P;K), then K � Ra(P ) � Rb(P;K) and

S(P;R;K) =

8<: R; R � Ra(P );

D(P;R); R > Ra(P ):(7)

b. If K < D(P;K), then Rb(P;K) > Ra(P ) > K and

S(P;R;K) =

8>>><>>>:R; R � K;

K; K < R � Rb(P;K);

D(P;R); R > Rb(P;K):

(8)

The lemma is illustrated in Figures 1 and 2, where the bold-faced lines depict realized

supply S(P;R;K) as a function of R. The rainfall shortage threshold Ra(P ) is determined

by the intersection of D(P;R) with the 45� line. In Figure 1, it may be observed that

K > D(P;K) assures K > Ra(P ). This means capacity never constrains supply: when

R < Ra(P ) there is a shortage due to insu¢ cient rainfall, whereas R � Ra(P ) means that

rainfall is adequate to meet demand. In addition, it may be observed that K > Ra(P )

implies Ra(P ) > Rb(P;K).

In the K < D(P;K) case, shown in Figure 2, Ra(P ) > K must hold. Moreover,

Rb(P;K) > Ra(P ) follows from Ra(P ) > K. Thus, capacity constrains supply for an

intermediate range of rainfall, where R falls between K and Rb(P;K). For R < K, a

shortage occurs due to insu¢ cient rainfall. For R � Rb(P;K), both rainfall and capacity

becomes adequate since high levels of rainfall reduce outdoor demand for water.

10

The reliability of the water system is de�ned as the probability that shortage does not

occur. Let reliability be denoted by �(P;K), so that

�(P;K) = ProbfS(P;R;K) = D(P;R)g:

Based on the previous discussion, it follows that reliability equals the probability that re-

alized rainfall exceeds the relevant shortage threshold, i.e., �(P;K) = 1 � F (Ra(P )) when

K � D(P;K), and �(P;K) = 1 � F (Rb(P;K)) when K < D(P;K). In the latter case,

reliability will be strictly positive when K > D(P ), since Rb(P;K) <1, but reliability will

be zero when K � D(P ), since Rb(P;K) =1.

3.3 Social welfare under e¢ cient rationing

As our benchmark case, we make the assumption that total water supplies are allocated

e¢ ciently across end-uses in the event of a shortage.7 In particular, let U(S;R) be given by

U(S;R) = maxSI ;SO

fUI(SI) + UO(SO; R)g; (9)

subject to SI ; SO � 0; SI + SO � S:

Solutions to this problem are denoted by (SEI ; SEO ). The following lemma derives properties

of solutions.

Lemma 2. Problem (9) has a unique solution. For S > 0, the solution satis�es SEI ; SEO > 0

and

US(S;R) = UIS(SEI ) = UOS(S

EO ; R): (10)

Moreover, USS(S;R) < 0, and S = D(P;R) implies SEI = DI(P ) and SEO = DO(P;R).

Observe in equation (10) that e¢ cient rationing equates marginal utilities across end-

uses.

7The case of ine¢ cient rationing across end-uses will be considered in Section 5.2 below.

11

Expected social welfare under e¢ cient rationing is given by

W (P;K) =

Z 1

0[U(S(P;R;K); R)� bS(P;R;K)] f(R)dR� �K: (11)

The water authority determines optimal price and capacity by solving the following problem:

maxP;K

W (P;K) subject to P;K � 0: (12)

4 Optimal reliability

4.1 Optimal capacity subproblem

The solution to problem (12) can be conveniently analyzed by �rst considering the following

capacity choice subproblem, in which price is treated as parameter:

maxKW (P;K) subject to K � 0: (13)

Let K�(P ) be a solution to problem (13).

The following proposition provides necessary conditions for optimal system capacity.

Proposition 1. a. There exists a solution K�(P ) for each P .

b. Any solution satis�es K�(P ) 2 (0; D(P;K�(P ))) andZ Rb(P;K�(P ))

K�(P )[US(K

�(P ); R)� b] f(R)dR = �: (14)

Observe that optimal capacity must satisfyK�(P ) < D(P;K�(P )), which is the situation

depicted in Figure 2. This is because increments of capacity in excess of D(P;K�(P )) have

zero social value, as they never a¤ect realized supply. Moreover, Rb(P;K�(P )) > Ra(P )

must hold, which implies �(P;K�(P )) < 1� F (Ra(P )); optimal reliability must be strictly

less than the level that would obtain in the absence of a capacity constraint.

Equation (14) gives the �rst-order condition for problem (13). The left-hand side of

(14) represents the expected marginal utility of capacity net of treatment and transmission

costs. Capacity provides zero net marginal bene�t when R lies below K�(P ), since rainfall

12

constrains supply at a level below capacity. Net marginal bene�t is also zero when R exceeds

Rb(P;K�(P )), since high rainfall reduces demand below capacity.

These results are analogous to �ndings in the peak load pricing literature, with one key

di¤erence: In the earlier literature, capacity provides marginal bene�ts only when realized

demand is su¢ ciently high, re�ected here by a low realization of R. In the case of water,

however, low values of R are also associated with low supply, and capacity is not needed

if supply is su¢ ciently low. Thus, in contrast to previous �ndings, the marginal bene�t of

system capacity derives from a middle range of outcomes, where both available supply and

demand are large relative to capacity. Overlooking either the low-supply or low-demand

margins will lead the bene�ts of system capacity to be overstated.

Uniqueness of solutions is investigated in the next proposition.

Proposition 2. There exists a price P o < b such that (14) determines a unique solution for

each P > P o.

Uniqueness can be assured for a region of prices that includes b in its interior. In this

region, W (P;K) is strictly concave in K at any point that satis�es the �rst order condition,

so that (14) determines a unique maximum. For prices P � P o, on the other hand, (14)

may determine a local minimum, and multiple solutions may exist. These possibilities arise

because adjustments in the capacity shortage threshold Rb(P;K) can introduce convexity

into the social welfare function when the price lies below P o.

To understand the latter point, suppose the �rst order condition for problem (13) is

satis�ed at (P;K) with P � P o. The derivative of the �rst order condition with respect to

K includes the term hUS(K;R

b)� bif(Rb)RbK = [P � b] f(Rb)RbK ;

where the equality follows from (5), Lemma 2 and (1). This term is strictly positive, since

P � P o < b and RbK < 0. Intuitively, the capacity shortage threshold Rb falls as K rises,

so that smaller quantities of water will be supplied at a price that lies below treatment and

transmission costs. This means an increase inK may have a positive e¤ect on marginal social

13

welfare. Due to this e¤ect, social welfare may become strictly convex in K, particularly if

Rb is very large as a consequence of K approaching D(P ). Assumption 3 serves to bound

the latter e¤ect and to ensure that social welfare is strictly concave whenever P does not lie

too far below b.

4.2 Optimal price and capacity

Now consider the complete problem (12), and let solutions be denoted by (P �;K�). The set

of solutions is characterized in the following proposition.

Proposition 3. a. (P �;K�) = (b;K�(b)) is a solution.

b. If K�(b) > D(b), then the solution is unique.

c. If K�(b) � D(b), then the set of solutions is given by K� = K�(b) and P � 2 [0; P �],

where P � = maxfP : K�(b) � D(P )g.

As in the peak-load pricing literature, the optimal price is necessarily set equal to mar-

ginal treatment and transmission costs, and optimal capacity equates marginal social bene�t

at this price to marginal capacity cost. Proposition 3 extends the literature by establishing

that the necessary conditions are su¢ cient for optimality, and also by providing a complete

characterization of solutions. In dealing with the possible convexity of the social welfare

function, the proof of Proposition 3 utilizes a two-step procedure, whereby P � = b is �rst

established as an optimal price choice, and then the argument of Proposition 2 is invoked

to show that K� = K�(b) is the optimal capacity choice given this price.

Observe that the solution is unique for K�(b) > D(b), in which case Rb(b;K�(b)) < 1.

For K�(b) � D(b), the solution is supported by a range of prices that satisfy D(P ) � K�(b).

Whether K�(b) lies above or below D(b) depends on the relationship between marginal

utility and marginal costs. The following corollary links the two possibilities to the capacity

cost parameter �.

14

Corollary 1. K�(b) > D(b) holds if and only ifZ 1

D(b)[US(D(b); R)� b]f(R)dR > �: (15)

According to (15), the high capacity outcome is associated with a range of low marginal

capacity costs.

4.3 Implications for reliability and water shortages

We now interpret the solution in terms of reliability. Any solution to problem (12) must sat-

isfyRb(b;K�(b)) > 0, in view of Proposition 1; this means �(b;K�(b)) = 1�F (Rb(b;K�(b))) <

1. Moreover, K�(b) > D(b) implies Rb(b;K�(b)) < 1, so that �(b;K�(b)) > 0, whereas

K�(b) � D(b) implies Rb(b;K�(b)) = 1 and �(b;K�(b)) = 0. These observations are sum-

marized in the following corollary.

Corollary 2. a. If K�(b) > D(b), then optimal reliability satis�es �(b;K�(b)) 2 (0; 1).

b. If K�(b) � D(b), then optimal reliability satis�es �(b;K�(b)) = 0.

It follows that water shortages occur with positive probability under the optimal policy.

Moreover, in view of Proposition 3 it follows that the optimal price is uniquely determined

if reliability is strictly positive. Finally, the two corollaries show that optimal reliability is

strictly positive if and only if marginal capacity costs are su¢ ciently small. In other words,

for very large capacity costs, water shortages will be optimal for all levels of rainfall.

These results shed light on the interaction between realized rainfall, system capacity and

the nature of water shortages. The solution (b;K�(b)) partitions the rainfall realizations

according to the ranking 0 < K�(b) < Ra(b) < Rb(b;K�(b)). When R < Ra(b), a water

shortage necessarily occurs due to the inadequacy of rainfall. For R � K�(b) capacity plays

no role in the shortage, whereas for R 2 (K�(b); Ra(b)) the shortage is exacerbated by

insu¢ cient capacity. When R 2 [Ra(b); Rb(b;K�(b))), realized rainfall is adequate to meet

demand, but system capacity is insu¢ cient. For these realizations, the water shortage is

driven entirely by limited capacity.

15

5 Policy constraints

In practical contexts, the water authority may have limited scope for determining water

rates and the rationing rule. For example, long-term contracts or political constraints may

impose ex ante restrictions on price, or convey privileged access to certain classes of users.

This section considers how the optimal policy is altered when constraints of this sort are

present. For brevity, we restrict attention to situations in which optimal reliability is strictly

positive, i.e., K�(b) > D(b).

5.1 Suboptimal pricing

First consider the possibility that the water authority is constrained to set the price at a

suboptimal level, meaning that P 6= b must be chosen. The next proposition shows that

this restriction will reduce the optimal level of capacity.

Proposition 4. If K�(b) > D(b), then K�(P ) < K�(b) for all P > P o satisfying P 6= b.

The proof demonstrates that the marginal social bene�t of capacity declines whenever

price departs from its optimal level. If P is raised above b, holding capacity constant at

K�(b), then Rb(b;K�(b)) falls to Rb(P;K�(b)), and the capacity constraint ceases to bind

for realizations of R that lie in the interval (Rb(P;K�(b)); Rb(b;K�(b))). On this range of

R, capacity ceases to a¤ect social welfare, and thus expected social welfare increases when

K is reduced. If P is lowered below b, holding capacity constant, then the marginal utility

of water consumption lies below marginal treatment and transmission costs for values of R

lying close to Rb(P;K�(b)). In this case, reducing K serves to raise expected social welfare

by restricting supply for rainfall realizations such that demand is excessively high. In this

way, e¢ cient pricing provides the greatest social returns to capacity.

Next we consider the e¤ects of suboptimal pricing on reliability.

Proposition 5. If K�(P ) > D(P ) and P > P o, then �(P;K�(P )) is strictly increasing in

P .

16

The proposition shows that higher prices are associated with greater reliability, irrespec-

tive of whether the price lies above or below b. In particular, an increase in price exerts

a positive direct e¤ect on reliability, by reducing demand for water; this is re�ected in the

fact that Rb(P;K) is strictly decreasing in P . The capacity choice also in�uences reliability,

however, and this e¤ect depends on where P lies in relation to b. As the proof shows, K�(P )

is strictly increasing in P when P < b, and strictly decreasing when P > b. Thus, starting

at P < b, a small price increase raises K�(P ), and when combined with the reduction in

demand this yields an unambiguous increase in reliability. Starting at P � b, on the other

hand, a price increase serves to reduce K�(P ), and the demand and capacity e¤ects work in

opposite directions. The proof shows that the demand e¤ect dominates, so that any increase

in P will lead to greater reliability.

Proposition 5 establishes the existence of a price-reliability tradeo¤ in optimal water

policymaking. If water consumers are charged a lower price, they must also accept lower

reliability. In the case of a price reduction below below treatment and transmission costs,

the optimal capacity is reduced along with the price, which works to steepen the tradeo¤.

5.2 Preferential access for indoor users

Consider the situation in which indoor users obtain preferential access in the event of a water

shortage. In particular, given the realized value of R, demand for indoor use is fully served

at price P before demand for outdoor use is served. Under this preferential rationing rule,

realized quantities supplied for indoor and outdoor use, denoted by SNI and SNO , respectlvely,

are given by

SNI = minfR;K;DI(P )g; (16)

SNO = S(P;R;K)� SNI : (17)

Note that the preferential rationing rule assigns zero supply to outdoor users for a nontrivial

range of R. Thus, we make a slight modi�cation in the assumptions and specify the utility of

17

outdoor users as UO(SNO +�;R), where � is an arbitrarily small positive constant.8 Expected

social welfare under the preferential rationing rule is

WN (P;K) =

Z 1

0[UI(S

NI ) + UO(S

NO + �;R)� bS(P;R;K)]f(R)dR� �K; (18)

When P is �xed exogenously, the water authority solves

maxKEWN (P;K) subject to K � 0: (19)

Let KN�(P ) be a solution. The next proposition compares the optimal capacity choice in

this case to optimal capacity under e¢ cient rationing, derived in Proposition 1.

Proposition 6. If KN�(P ) > D(P ), then KN�(P ) > K�(P ).

Observe that the water authority responds to the rationing ine¢ ciency by choosing

strictly larger capacity at the given price. When shortages occur, the preferential rule

channels available supplies to indoor users, who end up having relatively low marginal utility

of consumption, as compared to e¢ cient rationing which equates marginal utilities across

all users. Thus, increases in capacity provide greater marginal social bene�t under the

preferential rationing rule, since they work to direct supply to the relatively high-utility

outdoor users.

Now consider the choice of price. The water authority solves:

maxP;K

EWN (P;K) subject to P;K � 0: (20)

Let (PN�;KN�) be a solution. The next proposition considers optimal price and reliability.

Proposition 7. If KN� > D(PN�), then PN� > b and �(PN�;KN�) > �(b;K�(b)).

In conjunction with expansion of capacity, the water authority responds to the prefer-

ential rationing rule by raising the optimal price. This serves to reduce the consumption

8 Inclusion of the constant e¤ectively introduces a lower bound to UOS(SO; R) and ensures that the social

welfare function is well de�ned.

18

of low-valuation indoor users when shortages occur, leaving more water available for high-

valuation outdoor users. The optimal price balances this bene�cial e¤ect of price increases

against the losses that occur when demand is fully served, but marginal utility exceeds

marginal treatment and transmission costs.

The combination of higher price and greater capacity serves to raise optimal reliability,

both by reducing demand and by increasing available supply. In e¤ect, the water author-

ity assists the outdoor users, who bear the costs of the misallocation, by raising system

reliability. Reliability for indoor use is also increased as a byproduct.

6 Numerical example

Quantitative dimensions of the theoretical results may be illustrated by means of a numerical

example. Assume that rainfall is determined by a generalized gamma distriution, which has

the density function

f(R) =pRd�1e�(R=a)

p

ad�(d=p):

Utility functions are speci�ed as follows:

UI(SI) = ��I

1� �IS�(1��I)=�II ;

UO(SO + �;R) = ��O

1� �O�(SO + �)

�(1��O)=�Oe�!R=�O ;

where �I ; �O 2 (0; 1) and �; !; � > 0. The parameters �I and �O indicate the price elastici-

ties of demand for indoor and outdoor users, respectively; !R gives the rainfall elasticity of

outdoor demand; and � serves to scale the outdoor users in the social welfare function, as a

re�ection of the relative sizes of the two consumer populations.9

The parameter values used in the example are listed in Table 1. The implied rainfall

density, depicted in Figure 3, implies a mean rainfall level of 6:093. Given that b = 1, the

9 It is straightforward to verify that these functions satisfy Assumptions 1 and 2. Although Assumption

3 is violated, we have established numerically that the latter assumption is not needed in this instance, as

(14) possesses a unique solution for all P under the parameters considered.

19

unique optimal system capacity is K� = K�(1) = 1:808, in accordance with Proposition 3.

Moreover, since the example satis�es D(1) = 1:00 < K�(1), it follows that P � = 1 is the

unique optimal price. Optimal reliability is equal to �(1;K�) = 0:628.

Figure 4 plots realized water supply as a function of R under the optimal policy (P;K) =

(1;K�). A water shortage occurs whenever R < Rb(1;K�). With probability 0:291, the out-

come R 2 (K;Rb(1;K�)) occurs, and there is a shortage with a binding capacity constraint.

When R � K, there is a shortage in which low rainfall is the binding constraint; such an out-

come occurs with probability 0:081. Additional capacity would be of no help in alleviating

this type of shortage.

The latter range of outcomes plays an important quantitative role in determining the

optimal capacity. Suppose that the water authority ignores the possibility of a low-rainfall

shortage, i.e., the authority incorrectly views the supply function as S0(P;R;K) =

minfK;D(P;R)g, as would be appropriate in a standard peak-load pricing formulation.

The expected social welfare function would be reformulated as

W 0(P;K) =

Z 1

0

�U(S0(P;R;K); R)� bS0(P;R;K)

�f(R)dR� �K:

In the numerical example, this welfare function is maximized by P 0� = 1 and K0� = 2:466.

The resulting supply function S(1; R;K0�), depicted as a dashed line in Figure 4, yields

reliability of �(1;K0�) = 0:774. Thus, omission of the low-rainfall margin causes capacity

to be 36 percent too high, and reliability to be 23 percent too high.

Figure 5 graphs K�(P ) and �(P;K�(P )) as functions of P .10 The condition K�(P ) >

D(P ) holds as long as P > 0:38. Panel A of the �gure shows that K�(P ) < K�(1) for all

P 6= 1, and panel B shows that reliability strictly increases in P when K�(P ) > D(P ), in

line with Propositions 4 and 5. At a suboptimally low price of P = 0:67, optimal capacity

and reliability fall to 1:786 and 0:498, respectively. Note that lowering P to 0:67 while

holding capacity constant at K�(1) would give �(0:67;K�(1)) = 0:506, i.e., almost all of the

reduction in reliability is accounted for by the increase in water demand caused by the lower

10 It was established numerically that P o = 0 in this example under the parameters considered.

20

price. It is important to note, however, that the water authority does not optimally respond

by accomodating the higher demand with greater system capacity. In fact, it is best for the

authority to reduce capacity slightly, which leads to a further reduction in reliability.

The preferential rationing case is illustrated in Figure 6. The solutions KN�(P ) and

K�(P ) are plotted together in panel A, and the corresponding reliability levels �(P;KN�(P ))

and �(P;K�(P )) are plotted in panel B. In response to indoor users�preferential access, the

water authority chooses a higher capacity for each given P , and reliability is greater as a

result. When P = 1, capacity is increased by about 14 percent, to 2:056, and reliability by

about 10 percent, to 0:693.11

7 Conclusion

This paper has extended the peak-load pricing literature by developing a model suitable for

municipal water systems. The key modi�cation is that random rainfall �uctuations a¤ect

both the supply and demand components of the planner�s problem. We provide a complete

characterization of the optimal price, system capacity and reliability. Capacity provides

social bene�ts only for a middle range of rainfall realizations, where rainfall is su¢ ciently

great to utilize capacity fully, but not so great that demand falls short of capacity. Both

the low- and high-rainfall margins are important for determining optimal reliability. Using

a numerical example, we show that a naive application of the standard peak-load pricing

formulation, which ignores the low-rainfall margin, would lead to a quantitatively signi�cant

overinvestment in capacity, and an excessively high level of reliability.

We also consider the implications of policy constraints on the water planner�s decisions.

Optimal capacity is necessarily reduced when the price is set either above or below its optimal

level, but optimal reliability is strictly increasing in price. This points to the existence of

11 In this example, the authority would set an extremely high optimal price of PN� = 426:29 in order

to reduce greatly the demand of indoor users, and thereby to limit the large losses of social welfare when

outdoor users receive zero suppply. The implied optimal capacity and reliability levels are KN� = 0:329 and

�(PN�;KN�) = 0:997, respectlvely.

21

a price-reliability tradeo¤, wherein a lower price is accompanied by lower reliability. A

preferential rationing rule that favors indoor users leads to an increase in both the optimal

price and capacity, which serves to channel additional supplies to underserved outdoor users

in the event of a shortage. Overall system reliability is increased as a consequence.

In practice, municipal water planners typically rely on ad hoc reliability targets in making

their decisions. This paper is intended to supplant such practices by providing a �rm

theoretical foundation for the optimal determination of reliability. In future work we mean

to provide a more complete empirical assessment of the framework, which will be of relevance

to its application. In addition, we plan to develop a dynamic version of the model, which

will allow us to derive optimal release schedules and other decision rules that are needed for

water system operation.

22

Appendix

Proof of Lemma 1. K � D(P; 0) implies Rb(P;K) = 0 < Ra(P ), and also Ra(P ) =

D(P;Ra(P )) < D(P; 0) � K. Next, K 2 [D(P;K); D(P; 0)) implies D(P;Rb(P;K)) =

K � D(P;K), and thus Rb(P;K) � K. From this we have, for all R < Rb(P;K),

D(P;R) > D(P;Rb(P;K)) = K � Rb(P;K) > R, so that Ra(P ) � Rb(P;K) must hold.

Moreover, Ra(P ) > K would imply Ra(P ) = D(P;Ra(P )) < D(P;K) � K, which is a con-

tradiction; thus Ra(P ) � K. It follows that K � D(P;K) implies K � Ra(P ) � Rb(P;K).

Now consider: (i) R � Ra(P ) implies R � Ra(P ) = D(P;Ra(P )) � D(P;R) and R � K;

thus S(P;R;K) = R. (ii) R > Ra(P ) implies R > Ra(P ) = D(P;Ra(P )) > D(P;R) and

D(P;R) < D(P;Rb(P;K)) = K; thus S(P;R;K) = D(P;R). This establishes (7).

As for part b, K < D(P;K) implies K < D(P;R) for all R � K, so that K < Ra(P ).

Moreover, R � Ra(P ) implies D(P;R) � D(P;Ra(P )) = Ra(P ) > K, so that Rb(P;K) >

Ra(P ).

Now supposeD(P ) < K < D(P;K). (i)R � Rb(P;K) impliesD(P;R) � D(P;Rb(P;K))

= K; thus S(P;R;K) = minfR;Kg. (ii)R > Rb(P;K) impliesD(P;R) < D(P;Rb(P;K)) =

K < R; thus S(P;R;K) = D(P;R). For K � D(P ), we have D(P;R) > K for all R, and

thus S(P;R;K) = minfR;Kg. This establishes (8).

�

Proof of Lemma 2. Since the objective function is strictly concave in SI ; SO, and the con-

straint set is convex in SI ; SO, (9) has a unique solution. Moreover, any solution satis�es

SEO = S � SEI . Kuhn-Tucker conditions are given by

UIS(SEI )� UOS(S � SEI ; R)� � � 0; SEI � 0;

[UIS(SEI )� UOS(S � SEI ; R)� �] � SEI = 0;

S � SEI � 0; � � 0; [S � SEI ] � � = 0;

where � is the Lagrange multiplier for the constraint S � SEI � 0. Assuming that S > 0,

SEI = 0 implies S�SEI > 0, so that � = 0. This in turn implies UIS(SEI )�UOS(S�SEI ; R) =

23

UIS(0) � UOS(S;R) = 1, which contradicts the �rst condition. Thus, the solution must

satisfy SEI > 0 and UIS(SEI )�UOS(S�SEI ; R) = � � 0. On the other hand, SEI = S implies

UIS(SEI )�UOS(S�SEI ; R) = UIS(S)�UOS(0; R) = �1 < 0, which contradicts the preceding

inequality. Thus, the solution must satisfy SEI < S, which yields SEO > 0. It follows that

any solution satis�es SEI ; SEO > 0. Moreover, S

EI < S implies � = 0, and thus

UIS(SEI )� UOS(S � SEI ; R) = 0: (21)

Total di¤erentiation of (21) and the maximized objective function yields:

SEIS =UOSS(S � SEI ; R)

UISS(SEI ) + UOSS(S � SEI ; R);

US(S;R) = UIS(SEI )S

EIS + UOS(S � SE1 ; R)

�1� SEIS

�:

Combine these equations and use (21) to obtain (10). Moreover, totally di¤erentiate

US(S;R) = UIS(SEI ) to obtain

USS(S;R) = UISS(SEI )S

EIS :

SEIS > 0 then assures USS(S;R) < 0.

Finally, suppose S = D(P;R). From (1) and (2) we have UIS(DI(P )) =

UOS(DO(P;R); R) = P . Thus, SEI > DI(P ) implies SEO < D0(P;R) and UIS(SEI ) <

UIS(DI(P )) = UOS(DO(P;R); R) < UOS(S � SEI ; R), contradicting (10). Similarly, SEI <

DI(P ) implies UIS(SEI ) > UIS(DI(P )) =

UOS(DO(P;R); R) > UOS(S � SEI ; R), contradicting (10). Conclude that SEI = DI(P ),

and thus SEO = DO(P;R), if S = D(P;R).

�

Proof of Proposition 1. Using Lemma 1, the derivative of W (P;K) with respect to K may

be expressed as

WK(P;K) =

Z maxfK;Rb(P;K)g

K[US(K;R)� b] f(R)dR� �: (22)

24

Continuity of Rb(P;K) assures that WK(P;K) is a continuous function for all P and K.

Kuhn-Tucker necessary conditions for problem (13) are

WK(P;K�(P )) � 0; K�(P ) � 0; WK(P;K

�(P )) �K�(P ) = 0: (23)

Moreover, (22) indicates thatWK(P;K) = �� < 0 for allK � D(P;K), sinceK � Rb(P;K)

in this case. Since D(P;K) > 0, it follows that K � D(P;K) implies WK(P;K) < 0 and

K > 0, in violation of (23). Conclude that K�(P ) < D(P;K�(P )) must hold. This in turn

implies that solutions must lie within the compact interval [0; Ra(P )], since K < D(P;K)

implies K < Ra(P ). This establishes part a.

Now �x R1; R2 with 0 < R1 < R2, and choose " 2 (0; R1) so that K < " implies

Rb(P;K) > R2 and UIS(K;R) > (b+�)=(F (R2)�F (R1)) for all R 2 [R1; R2]. Thus, K < "

implies, using Lemma 2:

WK(P;K) �Z Rb(P;K)

KUS(K;R)f(R)dR� b� �

=

Z Rb(P;K)

KUIS(S

EI )f(R)dR� b� � >

Z R2

R1

UIS(K)f(R)dR� b� � > 0:

It follows that limK!0WK(P;K) = WK(P; 0) > 0, so that K�(P ) = 0 violates (23). Con-

clude that K�(P ) > 0 must hold, and thus (23) implies WK(P;K�(P )) = 0. Moreover,

since K�(P ) < D(P;K�(P )), we know that Rb(P;K�(P )) > K�(P ) must hold; thus (22)

evaluated at K = K�(P ) yields (14). This establishes part b.

�

The proofs of Propositions 2 through 5 make use of the following lemma.

Lemma 3. There exists a price P o < b such that P > P o and WK(P;K) = 0 imply

WKK(P;K) < 0 if K 6= D(K), and the right and left derivatives of WK(P;K) with respect

to K exist and are strictly negative if K = D(K).

Proof. For K 2 (D(P ); D(P;K)), the derivative of WK(P;K) with respect to K may be

expressed as

WKK(P;K) =

Z Rb(P;K)

KUSS(K;R)f(R)dR� [US(K;K)� b] f(K) (24)

25

+hUS(K;R

b(P;K))� bif(Rb(P;K))RbK(P;K):

Using (5), Lemma 2 and (1):

US(K;Rb(P;K)) = US(D(P;R

b(P;K)); Rb(P;K))

= UIS(DI(P )) = P:

Furthermore, (5), (3) and (2) may be used to obtain

RbK(P;K) = �UOSS(DO(P;R

b(P;K)); Rb(P;K))

UOSR(DO(P;Rb(P;K)); Rb(P;K)): (25)

It follows that (24) may be expressed as

WKK(P;K) =

Z Rb(P;K)

KUSS(K;R)f(R)dR� [US(K;K)� b] f(K) (26)

� [P � b] f(Rb(P;K))UOSS(DO(P;Rb(P;K)); Rb(P;K))

UOSR(DO(P;Rb(P;K)); Rb(P;K)):

Since USS < 0, the �rst term on the right-hand side of (26) is strictly negative. Moreover,

WK(P;K) = 0 implies

0 =

Z Rb(P;K)

K[US(K;R)� b] f(R)dR� �

< [US(K;K)� b] (F (Rb(P;K))� F (K))� �;

so that [US(K;K)� b] > 0. Since f(K) > 0, it follows that the second term is strictly

negative whenever WK(P;K) = 0.

As for the third term, de�ne the set P by

P = fP :WK(P;K) = 0 �WKK(P;K) for some K 2 [D(P ); D(P;K))g :

That is, P gives the set of prices such that WKK(P;K) < 0 fails to hold at some K 2

[D(P ); D(P;K)) such thatWK(P;K) = 0. Suppose P is nonempty. From (26) we have that

P � b implies WKK(P;K) < 0 for all K 2 [D(P ); D(P;K)) such that WK(P;K) = 0; thus

P is bounded above by b, and we can de�ne P o = supP � b.

26

Now suppose P o = b. In this case, there exist sequences fPng1n=1 � P and fKng1n=1,

with Pn ! b, such that for all n, Kn 2 [D(Pn); D(Pn;Kn)) and WK(Pn;Kn) = 0 �

WKK(Pn;Kn); in addition, let Rbn = Rb(Pn;Kn). In view of (22) and Kn < Rbn,

WK(Pn;Kn) = 0 impliesZ Rbn

Kn

[US(Kn; R)� b] f(R)dR = � > 0 for all n: (27)

We know that, for any given " 2 (0; b), there exists a positive integer N such that Pn >

b�" > 0 for all n > N , while Pn � b for all n. Since Kn < D(Pn;Kn) implies Kn < Ra(Pn),

we have Kn < Ra(Pn) < Ra(b � ") < 1 for all n > N , and also Kn � D(Pn) � D(b).

Thus, fKng1n=1 � [D(b); Ra(b�")]. Passing to a subsequence if necessary, there exist values

K̂ 2 [D(b); Ra(b� ")] and R̂b � 1, with K̂ � R̂b, such that (Kn; Rbn)! (K̂; R̂b). (27) then

implies Z R̂b

K̂

hUS(K̂;R)� b

if(R)dR > 0;

so that K̂ < R̂b must hold. Moreover, using Assumption 3:����[Pn � b] f(Rbn)UOSS(DO(Pn; Rbn); Rbn)UOSR(DO(Pn; Rbn); Rbn)

���� < jPn � bj f(Rbn)Mm :Thus:

limn!1

[Pn � b] f(Rbn)UOSS(DO(P;R

bn); R

bn)

UOSR(DO(P;Rbn); Rbn)= 0;

so that

limn!1

WKK(Pn;Kn) =

Z R̂b

K̂USS(K̂;R)f(R)dR�

hUS(K̂; K̂)� b

if(K̂) < 0;

which contradicts the fact that WKK(Pn;Kn) � 0 for all n. Conclude that P o < b. If P is

empty, then set P o = 0.

For K < D(P ), WKK(P;K) may be expressed as

WKK(P;K) =

Z 1

KUSS(K;R)f(R)dR� [US(K;K)� b] f(K): (28)

Since the �rst term on the right-hand side is strictly negative, while WK(P;K) = 0 im-

plies that the second term is strictly negative, it follows that WKK(P;K) < 0 whenever

27

WK(P;K) = 0. Finally, for K = D(P ) we have, in view of (26):

1

"(WK(P;D(P ) + ")�WK(P;D(P ))) =WKK(P;K");

for small " > 0 and K" 2 (D(P ); D(P ) + "). Assumption 3 assures that WKK(P;K")

converges to a strictly negative limit as "! 0. A similar result holds for " < 0, using (28).

�

Proof of Proposition 2. According to Proposition 1, any solution to (13) satis�es

WK(P;K�(P )) = 0. Thus, as long as P > P o, Lemma 3 ensures that the following in-

equalities hold for all su¢ ciently small " > 0:

WK(P;K�(P )� ") > 0 > WK(P;K

�(P ) + "): (29)

Since WK(P;K) is continuous, it is not possible for (29) to hold at more than one value of

K�(P ); thus, WK(P;K) > 0 for all K < K�(P ) and WK(P;K) < 0 for all K > K�(P ).

�

Proof of Proposition 3. Making use of Lemmas 1 and 2 and equation (1), the derivative of

W (P;K) with respect to P may be expressed as

WP (P;K) =

Z 1

maxfRa(P );Rb(P;K)g[P � b]DP (P;R)f(R)dR: (30)

Choose any P 6= b, and let p be a price between P and b. (30) and DP (P;R) < 0 imply

WP (p;K) � (p� b) =Z 1

maxfRa(p);Rb(p;K)g[p� b]2DP (p;R)f(R)dR � 0:

Moreover, the inequality is strict for K > D(p), since maxfRa(p); Rb(p;K)g < 1 in this

case. Thus:

W (P;K)�W (b;K) =Z P

bWP (p;K)dp � 0; (31)

with strict inequality for K > D(P ). It follows that W (P;K) �W (b;K) �W (b;K�(b)), so

that (b;K�(b)) is a solution.

Now suppose K�(b) > D(b). According to Lemma 3, WKK(b;K�(b)) < 0 must hold,

meaning that K�(P ) is di¤erentiable, and thus continuous, at P = b. Moreover, it is easy

28

to verify that D(P ) is a continuous function, as a consequence of Assumption 3. Since both

K�(P ) and D(P ) are continuous at P = b, we have K�(P̂ ) > D(P̂ ) for all P̂ su¢ ciently

close to b, so that (31) implies W (P̂ ;K�(P̂ )) < W (b;K�(P̂ )). Moreover, arguing as above,

it can be shown that P > P̂ > b or P < P̂ < b imply

W (P;K�(P ))�W (P̂ ;K�(P )) =

Z P

P̂WP (p;K

�(P ))dP � 0:

It follows that W (P;K�(P )) � W (P̂ ;K�(P )) � W (P̂ ;K�(P̂ )) < W (b;K�(P̂ )) �

W (b;K�(b)). This argument is valid for any P 6= b, so that (b;K�(b)) gives the unique

solution.

Next, suppose K�(b) � D(b), and consider any P that satis�es K�(P ) � D(P ). Accord-

ing to (14), K�(P ) must satisfyZ 1

K�(P )[US(K

�(P ); R)� b] f(R)dR = �:

This equation determines a unique value of K�(P ) which does not depend on P . Thus,

K�(P ) = K�(b) whenever K�(P ) � D(P ). Moreover, (30) indicates that WP (P;K�(b)) = 0

for any P such thatK�(b) � D(P ), since Rb(P;K�(b)) =1 in this case; thusW (P;K�(b)) =

W (b;K�(b)) on this range of P . It follows that (P �;K�(b)) gives a solution for any P � that

satis�es K�(b) � D(P �).

Finally, consider P such that K�(P ) > D(P ). In this case, (31) implies

W (P;K�(P ))�W (b;K�(P )) =

Z P

bWP (p;K

�(P ))dp < 0:

It follows that W (P;K�(P )) < W (b;K�(P )) � W (b;K�(b)), so that (P;K�(P )) cannot be

a solution.

�

Proof of Corollary 1. Given b > P o and WK(b;K�(b)) = 0, the proof of Lemma 3 es-

tablishes that WK(b;K) > 0 for K < K�(b) and WK(b;K) < 0 for K > K�(b). Thus

K = D(b) < K�(b) implies WK(b;D(b)) > 0, which is (15), whereas K = D(b) � K�(b)

implies WK(b;D(b)) � 0, which is the negation of (15).

29

�

Proof of Proposition 4. For K 2 (D(P ); D(P;K)), making use of (30), Lemma 1 and (25),

the derivative of WP (p;K) with respect to K may be expressed as

WPK(P;K) = � [P � b] f(Rb(P;K))DP (P;R

b(P;K))

DR(P;Rb(P;K)): (32)

In this instance, for any P 6= b, and any price p between P and b:

WPK(p;K) � (p� b) = � [p� b]2 f(Rb(P;K))DP (P;R

b(P;K))

DR(P;Rb(P;K))< 0:

For K =2 (D(p); D(p;K)), (22), (30) and Lemma 1 yield

WPK(p;K) � (p� b) = 0:

Thus:

WK(P;K)�WK(b;K) =

Z P

bWPK(p;K)dp � 0; (33)

with strict inequality for K 2 (D(P ); D(P;K)).

Proposition 2 shows that (14) de�nes K�(b) uniquely, while the proof establishes that

WK(b;K) < 0 for allK > K�(b). Thus, (33) impliesWK(P;K) �WK(b;K) < WK(b;K�(b))

= 0 for all K > K�(b), which implies K�(P ) � K�(b). Moreover, since K�(b) < D(b;K�(b))

must hold, (33) and K�(b) > D(b) imply WK(P;K�(b)) < WK(b;K

�(b)) = 0, so that

K�(P ) < K�(b).

�

Proof of Proposition 5. According to Lemma 3, K�(P ) > D(P ) and P > P o imply

WKK(P;K�(P )) < 0, so that K�(P ) is di¤erentiable. Implicitly di¤erentiate the equation

WK(P;K�(P )) = 0 and use (32) to obtain

WPK(P;K�(P )) +WKK(P;K

�(P ))K�P (P ) = 0

K�P (P ) = �

WPK(P;K�(P ))

WKK(P;K�(P ))

= [P � b] f(Rb(P;K)) DP (P;Rb(P;K))

DR(P;Rb(P;K))WKK(P;K�(P ))

30

K�P (P ) = [P � b] f(Rb(P;K))

DP (P;Rb(P;K))

DR(P;Rb(P;K))WKK(P;K�(P )): (34)

It follows that K�P (P ) > 0 for P 2 (P o; b) and K�

P (P ) < 0 for P > b, when K�(P ) 2

(D(P ); D(P;K�(P ))). Furthermore, for K 2 (D(P ); D(P;K)), (5) may be used to obtain

RbP (P;K) = �DP (P;R

b(P;K))

DR(P;Rb(P;K));

and

RbK(P;K) =1

DR(P;Rb(P;K)): (35)

Using the latter equations and (34) yields

d

dPRb(P;K�(P )) = RbP (P;K

�(P )) +RbK(P;K�(P ))K�

P (P ) (36)

= �DP (P;Rb(P;K))

DR(P;Rb(P;K))

�1� [P � b] f(Rb(P;K))

DR(P;Rb(P;K))WKK(P;K�(P ))

�:

The term in braces is strictly positive if

WKK(P;K�(P )) <

[P � b] f(Rb(P;K))DR(P;Rb(P;K))

;

which is implied by (24) and (35). Thus (36) yields dRb(P;K�(P ))=dP < 0, which in turn

impliesd

dP�(P;K�(P ))) = �f(Rb(P;K�(P ))

d

dPRb(P;K�(P )) > 0: (37)

�

The proofs of Propositions 6 and 7 make use of the following two lemmas.

Lemma 4. Suppose quantities supplied are determined by (16) and (17).

a. If K > DI(P ), then SNO = S(P;R;K) � DI(P ) for R > DI(P ), and SNO = 0 for

R � DI(P ).

b. If K � DI(P ), then SNO = 0 for all R.

Proof. For K > DI(P ), (16) becomes SNI = minfR;DI(P )g. R � DI(P ) implies SNI =

R = minfR;K;D(P;R)g = S(P;R;K); thus SNO = 0 follows from (17). R > DI(P ) implies

31

SNI = DI(P ) < S(P;R;K), and (17) gives SNO = S(P;R;K)�DI(P ). This establishes part

a.

For K � DI(P ), (16) becomes SNI = minfR;Kg; moreover, S(P;R;K) = minfR;Kg

must also hold. Thus S(P;R;K)� SNI = 0 for all R. This establishes part b.

�

Lemma 5. Suppose K 2 (DI(P ); D(P;K)). If R 2 (DI(P ); Rb(P;K)), then SEI < DI(P )

and SEO > minfR;Kg �DI(P ) > 0.

Proof. IfK 2 (D(P ); D(P;K)), thenR < Rb(P;K) impliesD(P;R) > D(P;Rb(P;K)) = K,

whereas K � D(P ) implies D(P;R) > K for all R; thus R < Rb(P;K) implies S(P;R;K) =

minfR;Kg < D(P;K). Suppose SEI � DI(P ) for some R 2 (DI(P ); Rb(P;K)). Then SEO =

minfR;Kg � SEI < D(P;K) �DI(P ) = DO(P;K), which implies UIS(SEI ) � UIS(DI(P ))

= P = UOS(DO(P;R) + �;R) < UOS(SEO + �;R). Since this contradicts (10), conclude that

SEI < DI(P ), and thus, for all R 2 (DI(P ); Rb(P;K)), SEO > minfR;Kg �DI(P ) > 0.

�

Proof of Proposition 6. For K > DI(P ), the derivative of WN (P;K) with respect to K

may be expressed as

WNK (P;K) =

Z maxfK;Rb(P;K)g

K[UOS(K �DI(P ) + �;R)� b] f(R)dR� �: (38)

For K 2 (DI(P ); D(P;K)), (38) and (22) may be combined to obtain

WNK (P;K)�WK(P;K) (39)

=

Z Rb(P;K)

K[UOS(K �DI(P ) + �;R)� US(K;R)] f(R)dR:

In view of Lemma 5, R 2 (K;Rb(P;K)) implies SEO > K �DI(P ) > 0, and thus UOS(SEO +

�;R) < UOS(K�DI(P )+�;R); (10) in turn implies UOS(SEO +�;R) = US(K;R). It follows

from (39) that WNK (P;K) > WK(P;K).

Now consider solutions to Problem (19). As in the e¢ cient rationing case, a solution

exists for all P . Moreover, since D(P ) > DI(P ), the hypothesis KN�(P ) > D(P ) assures

32

that (38) de�nes WNK (P;K) at K = KN�(P ), and WN

K (P;KN�(P )) = 0 holds as well.

Now, K � K�(P ) implies D(P;K) � D(P;K�(P )), so that K � K�(P ) < D(P;K�(P )) �

D(P;K). Thus K�(P ) > DI(P ) and K 2 (DI(P );K�(P )] imply WNK (P;K) > WK(P;K) �

0, which in turn implies KN�(P ) > K�(P ). If K�(P ) � DI(P ), then KN�(P ) > K�(P )

follows directly from the hypothesis.

�

Proof of Proposition 7. For K > DI(P ), making use of Lemmas 1 and 2 and equation (1),

the derivative of WN (P;K) with respect to P may be expressed as

WNP (P;K) =

Z maxfRa(P );Rb(P;K)g

DI(P )[P � UOS(minfR;Kg �DI(P ) + �;R)]DIP (P )f(R)dR

(40)

+

Z 1

maxfRa(P );Rb(P;K)g[P � b]DP (P;R)f(R)dR:

Now, KN� < D(PN�;KN�) must hold in any solution, while the hypothesis KN� > D(PN�)

assures that (40) de�nes WNP (P;K) at (P;K) = (PN�;KN�). Thus, necessary conditions

for a solution include

WNP (P

N�;KN�) =

Z Rb(PN�;KN�)

DI(PN�)[PN� � UOS(minfR;KN�g �DI(PN�) + �;R)] (41)

� DIP (PN�)f(R)dR+Z 1

Rb(PN�;KN�)[PN� � b]DP (PN�; R)f(R)dR = 0.

Moreover, KN� 2 (D(PN�); D(PN�;KN�)) impliesDI(PN�) < KN� < Rb(PN�;KN�) <1:

Using Lemma 5, R 2 (DI(PN�); Rb(PN�;KP�)) implies SEI < DI(P

N�) and SEO >

minfR;KN�g � DI(PN�) > 0, and thus PN� = UIS(DI(PN�)) < UIS(S

EI ) = UOS(S

EO +

�;R) < UOS(minfR;KN�g�DI(PN�) + �;R). This means the �rst term on the right-hand

side of (41) is strictly positive, so the second term must be strictly negative; that is, PN� > b

must hold.

Since PN� > b, (37) implies �(PN�;K�(PN�)) > �(b;K�(b)). Moreover:

d�(P;K)

dK= �f(Rb(P;K))RbK(P;K) > 0:

33

Proposition 6 implies KN� = KN�(PN�) > K�(PN�), so that �(PN�;KN�) >

�(PN�;K�(PN�)) must hold.

�

34

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37

Table 1

Parameter Values for Numerical Example

Parameter Description Value

a Water density parameter 2.5

d Water density parameter 2.5

p Water density parameter 1.0

��������� Price elasticity of indoor demand 0.5

�� Price elasticity of outdoor demand 0.5

� Rainfall sensitivity of outdoor demand 0.5

� Scale parameter for outdoor demand 50

b Unit treatment and transmission costs 1.0

� Unit storage capacity cost .33

� Minimum quantity parameter ��