Economic Analysis of Alternative Water Plans:

Water Resource EconomicsWater Resources Planning and

ManagementDaene C. McKinney

River Basin Planning

• Municipal Water Supply– Benefits: Try to meet targets

• Irrigation Water Supply– Benefits: Try to meet targets

• Hydropower– Benefits: Try to meet targets

minimum

penalty for missing target in month t

target releaseXI,tTI,t

ZI

wM weight for Municipal demandwI weight for Irrigation demandwP weight for Recreation TM,t monthly target for municipal demandTI,t monthly target for irrigation demandTP,t monthly target for recreation

𝑴𝒊𝒏𝒊𝒎𝒊𝒛𝒆 𝒁=∑𝒕=𝟏

𝑻 [𝑾𝑴(𝑻𝑴 , 𝒕 −𝑿𝑴 ,𝒕𝑻 𝑴 , 𝒕 )

𝟐

+𝑾 𝑰 (𝑻 𝑰 ,𝒕 −𝑿 𝑰 ,𝒕𝑻 𝑰 , 𝒕 )

𝟐

+𝑾 𝑷 (𝑻 𝑷 , 𝒕−𝑿 𝑷 ,𝒕𝑻 𝑷 ,𝒕 )

𝟐 ]

Municipal

GainsLosses

EnvironmentalFlow

Reservoir

Irrigation

Inflow

Rt

Qt

St K1

S.T.

Decision Making• Developing and managing water resources systems involves making

decisions.• Modeling and data management tools can contribute to the information

needed to make informed decisions.• Decisions in water resources management inevitably involve making

tradeoffs – compromising – among competing opportunities, goals or objectives.

• One of the tasks of water resources system planners or managers involved in evaluating alternative designs and management plans or policies is to identify the tradeoffs, if any, among competing opportunities, goals or objectives.

• It is then up to a largely political process involving all interested stakeholders to find the best compromise decision.

System Performance Criteria• Measures indicating just how well different management plans and

policies serve the interests of all stakeholders are typically called system performance criteria

• B/C framework – Convert impacts into a single monetary metric– Find the plan that maximizes benefits vs costs. – Does not address distributional issues of who benefits and who pays, and by how much.

• Water resources planning and management takes place in a multi-criteria environment

• Stakeholders – individuals or interest groups who have an interest in the outcome of any plan

• Quantification of an objective is the adoption of some quantitative scale that provides an indicator for how well the objective would be achieved

Economic Criteria• Water resources system development and management is often motivated

by economic criteria.• Two economic concepts: scarcity and substitution.

– Scarcity - supplies of natural, synthetic and human resources are limited. Hence people are willing to pay for them. They should therefore be used in a way that generates the greatest return, i.e. they should be used efficiently.

– Substitution - individuals, social groups and institutions are generally willing to trade a certain amount of one objective value for more of another

• Maximize economic profit from water supply for irrigation, M&I water use, and hydroelectric power generation, subject to institutional, physical, and other constraints

Objective for Ag, Muni and Hydropower Water Use

• Profit from agricultural demand sites = equal to crop revenue minus fixed crop cost, irrigation technology improvement cost, and water supply cost

Objective for Agricultural Water Use

A harvested area (ha)p crop price (US$/mt)FC fixed crop cost (US$/ha)TC technology cost (US$/ha)Cw water price (US$/m3)wag water delivered to demand sites in growing season (m3)

Objective for Municipal Water Use

• Benefit from industrial and municipal demand sites is calculated as water use benefit minus water supply cost

Muni(w) benefit from M&I water use (US$),wmuni,t municipal water withdrawal in period t(m3)w0 maximum water withdrawal (m3)p0 willingness to pay for additional water at full use (US$)e price elasticity of demand (estimated as -0.45) 1/e

Objective Function for Hydropower Water Use

• The profit from power generation

Pt Power production for each period (KWh)wturbine,t Water passing turbines for each period (m3)Ppower Price of paid for power (US$/KWh)Cp Cost of producing power (US$/KWh)

Consumers

• Purchase “goods” and “services”• Have “preferences” expressed by “utility” function

),...,,( 21 nxxxx

),...,,()( 21 nxxxuu x x 2

x 1

Indifference curve ),( 21 xxu

Increasing utility

u

Better Bundles

Worse Bundles

Good 2

Good 1

Consumer’s Budget

• Consumers have a “budget”, expressed by a budget constraint

m/p2

m/p1

x1

x2

Unaffordable bundles

Affordable bundles

Budget line p1x1+p2x2=m

Slope = -p1/p2

mxpxp 2211

Good 2

Good 1

Consumer’s Problem

0

tosubject)(Maximize

xxp

x

m

u

K

kkk xpmuL

1)(),( xx

0

,...,1,0

1

K

kkk

kkk

xpmL

Kkpxu

xL

Kkp

xu

k

k ,...,1

m

u

Purchase so that the ratio of marginal benefit (marginal utility) to marginal cost (price) is equal among all purchases

The ratio (in dimensions of $/unit or shadow price) is the Lagrange multiplier, the change in utility for a change in consumer income

Consumer’s Problem (2 goods)

x 2

x 1 x 1 *

x 2 * Budget line slope = -p1/p2

Optimal choice MRS12 = -p1/p2

Indifference curve slope = MRS12

Increasing utility 0)(

0

0

2211

22

11

xpxpm

pxu

pxu

mxpxp

xxu

1111

21tosubject

),(Maximize

Solution: slope of budget line equals slope of indifference curve

2

2

1

1p

xu

px

u

Good 2

Good 1

Demand

• Solution to Consumer’s Problem gives puschase amounts which aggregate to demand

m,* * pxx

Price, p

Quantity, x

Demand curve x(p,m)

Willingness-to-Pay• Value - What is someone willing to pay?• Suppose consumer is willing to pay:

– $38 for 1st unit of water– $26 for 2nd unit of water– $17 for 3rd unit of water– And so on

• If we charge p* = $10– 4 units will be purchased for $40– But consumer is willing to pay $93 – Consumer’s surplus is $53

Price, p

Quantity, x 1 2 3 4

p*=10

20

30

40 38

26

17

12

5

WTP = Gross Benefit = 93

CS = Net Benefit = 53

Total cost = 40

Price, p

Quantity, q

Willingness-to-Pay

Market Prices – Revealed WTP • Some goods or services are traded in markets

– Value can estimated from consumer surplus (e.g., fish, wood)

• Ecosystem services used as inputs in production (e.g., clean water)– Value can be estimated from contribution to profits made from the

final good

• Some services may not be directly traded in markets– But related goods that can be used to estimate their values are trade in

markets • Homes with oceanviews have higher price• People will take time to travel to recreational places• Expenditures can be used as a lower bound on the value of the view or the

recreational experience

Firms

• Firms produce outputs from inputs (like water) • Firm objective: maximize profit

x

y

Production function y = f(x)

Slope = df/dx

Input, x1

Input, x2

Isoquant

Increasing output

y0

y1

y2

0)( yf x

input

output

input 1

input 2

Production Function

Ymax= maximum yield (mt/ha)b0 … b8 = coefficients,x = irrigation water applied

(mm)Emax = Max ET (mm)s = irrigation water salinity

(dS/m)u = irrigation uniformity

)]/ln()/([ max2max10max ExaExaaYY

sbubba 2100

sbubba 5431 sbubba 8762

0.00

2.00

4.00

6.00

8.00

0 5,000 10,000 15,000 20,000Input, x (m3/ha)

Out

put,

y (to

n/ha

)

I II III

Profit

• Output• Input• Revenue• Cost

• Profit

N

nnnxwC

1

pyR

)(xfy

CR

x

N

nnnxwpf

1)(x

The Firm’s Problem

Nnwxfp

x nnn

,...,1,0

Nnp

wxf n

n,...,1

x

y

Prod. Fcn. y = f(x)

slope = df/fx

Isoprofit line = py – wx slope = w/p

x*

y*

/w

df/dx= w/p

Revenue (1) Price-setting Firm

Revenue

• Marginal Revenue

pyR

dydpyp

dydp

pR

yR

dydR

• Increase in output (dy) has two effects 1. (1) Adds revenue from sale of more units, and2. (2) Causes value of each unit to decrease

(1) (2)

Revenue (2) Price-taking Firm

Revenue

• Marginal Revenue– derivative WRT y

pyR

pdypyd

dydR )(

• Competitive firm: p is constant

ExampleLinear demand function

byayp )(

• Marginal revenue– slope is twice that of demand

byadydR 2

y

Demand function p = a - by

b

Revenue py = ay – by2

y*

2 b

a

a/2b

p Marginal Revenue

= a – 2by

a/b y y

a

• Revenue2byaypyR

y

Demand function p = a - by

b

Revenue py = ay – by2

y*

2 b

a

a/2b

p Marginal Revenue = a – 2by

a/b y y

a

y

Demand function p = a - by

b

Revenue py = ay – by2

y*

2 b

a

a/2b

p Marginal Revenue = a – 2by

a/b y y

a

Cost Functions

01

1

),...,(

tosubject

Minimize

yxxf

xw

N

N

nnn

0

,...,10

0

yfL

Nnxfw

xL

nn

n

n

m

n

m

xf

xf

ww

Cost FunctionsTotal Cost (fixed and variable costs)

Average cost (cost per unit to produce y units)

)(:min)( xxw fyyTC

)()( yVCFCyTC

yyTCAC )(

Marginal cost (cost to produce additional unit)

dydVC

dydTCMC

Example (1) Price-taking Firm• How much water should a water company produce

)()(Maximize yTCpyy

dydTCpy

dydp

dyd 0

MCp

)()( yMCpyMR

y* Product

Price &

p = MC

Example (2) Price-setting Firm• Firm influences market price • Choose production level and price to maximize profit

)()(Maximize yTCpyy

dydTCpy

dydp

dyd 0

y

Price & Cost

p = MC

MC

AC

Demand

p*

y*

MR

MR = MC

pm

ym

)()( yMCpydydpyMR

)()( yMCyMR

• Profit from agricultural demand sites = equal to crop revenue minus fixed crop cost, irrigation technology improvement cost, and water supply cost

Objective Function for Agricultural Water Use

A harvested area (ha)p crop price (US$/mt)FC fixed crop cost (US$/ha)TC technology cost (US$/ha)Cw water price (US$/m3)wag water delivered to demand sites in growing season (m3)

Objective Function for Municipal and Industrial Water Use

• Benefit from industrial and municipal demand sites is calculated as water use benefit minus water supply cost

Muni(w) benefit from M&I water use (US$),wmuni,t municipal water withdrawal in period t(m3)w0 maximum water withdrawal (m3)p0 willingness to pay for additional water at full use (US$)e price elasticity of demand (estimated as -0.45) 1/e

Objective Function for Hydropower Water Use

• The profit from power generation

Pt Power production for each period (KWh)wturbine,t Water passing turbines for each period (m3)Ppower Price of paid for power (US$/KWh)Cp Cost of producing power (US$/KWh)

• Maximize economic profit from water supply for irrigation, M&I water use, and hydroelectric power generation, subject to institutional, physical, and other constraints

Combined Objective Function for Ag, M&I and Hydropower Water Use