Water Resources Planning and Management

Daene C. McKinney

Water Resource Economics

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

x

xp

x

m

u

K

kkk xpmuL

1)(),( xx

0

,...,1,0

1

K

kkk

kkk

xpmL

Kkpx

u

x

L

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

px

u

px

u

mxpxp

xxu

1111

21

tosubject

),(Maximize

Solution: slope of budget line equals slope of indifference curve

2

2

1

1

px

u

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)

Ou

tpu

t, y

(to

n/h

a)

I II III

Profit

• Output• Input• Revenue• Cost

• Profit

N

nnnxwC

1

pyR

)(xfy

CR

x

N

nnnxwpf

1)(x

The Firm’s Problem

Nnwxf

px n

nn,...,1,0

Nnp

w

x

f 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 – Monopolistic Firm

Revenue

• Marginal Revenue

pyR

dy

dpyp

dy

dp

p

R

y

R

dy

dR

• Increase in output (dy) has two effects 1. (1) Adds revenue from sale of more units, and

2. (2) Causes value of each unit to decrease

(1) (2)

Revenue – Competitive Firm

Revenue

• Marginal Revenue– derivative WRT y

pyR

pdy

pyd

dy

dR )(

• Competitive firm: p is constant

ExampleLinear demand function

byayp )(

• Marginal revenue– slope is twice that of demand

byady

dR2

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

Firm’s Problem – 2nd Way

01

1

),...,(

tosubject

Minimize

yxxf

xw

N

N

nnn

0

,...,10

0

yfL

Nnx

fw

x

L

nn

n

n

m

n

m

xf

xf

w

w

Cost FunctionsTotal Cost (fixed and variable costs)

Average cost (cost per unit to produce y units)

)(:min)( xxw fyyTC

)()( yVCFCyTC

y

yTCAC

)(

Marginal cost (cost to produce additional unit)

dy

dVC

dy

dTCMC

Example – Competitive Firm• How much water should a water company produce

y

Price & Cost

p = MC

MC

AC

Demand

p*

y*

)()(Maximize yTCpyy

dy

dTCpy

dy

dp

dy

d 0

MCp

)()( yMCpyMR

Example – Monopolistic Firm• Firm influences market price • Choose production level and price to maximize profit

)()(Maximize yTCpyy

dy

dTCpy

dy

dp

dy

d 0

y

Price & Cost

p = MC

MC

AC

Demand

p*

y*

MR

MR = MC

pm

ym

)()( yMCpydy

dpyMR

)()( yMCyMR

Consumers' and Producers’ Surpluses• Consumers' Surplus

– amount consumer willing to pay minus amount actually paid, but didn’t have to– Reflects benefit to consumer

• Producers' Surplus– amount producer would have been willing to accept (cost of production) plus

additional amount from consumer– Reflects benefit to producer

Price, p

Output, y

Demand

p1

x0

A

0

Supply

p2

p0

Consumer Surplus

Producer Surplus

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

Benefit Function for Ag 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)

Benefit Function for M&I 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

Benefit Function for Hydropower

• 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

Objective Function

Pricing– Demand affected by price of water

• price elasticity of demand: – %change in demand for %change in price

– Conservation• Non-Price methods (education, etc.)• Price methods

– “declining block rates” - the more water used, the lower the price for the last units of use (discourages conservation)

– “alternative rate structures” - encourage users to reduce their consumption

» Increasing (or inverted) block: Rates increase at set usage level intervals • Seasonal block: Two different rate structures are set (one in the summer and one in the winter) • Baseline block: A baseline usage water usage amount is set based on a customer's winter use and a surcharge is then imposed for any use over the baseline during the summertime

Price$3/1,000 gallons

= 174 gal/day/person

= 870 million gal/day

Dallas-Fort Worth area (5 million)

ElasticityIn Texas elasticity = -0.32Consumption will decline 3.2% for every 10% rise in price

Prices for WaterLocation Date US$/m3 US$/1000 gal

Cario 1993 $0.04 $0.15Syria NA $0.07 $0.26Egypt (Beheira) 1993 $0.07 $0.26Gaza City 1998 $0.12 $0.45Morocco (Meknes) 2000 $0.20 $0.76Algiers 1998 $0.21 $0.79Amman 1999 $0.23 $0.87Sana'a 1999 $0.25 $0.95Gaza (Khan Younis) 1998 $0.29 $1.10Tunisia 1999 $0.32 $1.21Lebanon 1998 $0.32 $1.21Morocco) Marrakech 2000 $0.39 $1.48Canada 1999 $0.41 $1.55US 1999 $0.50 $1.89Morocco (Safi) 2000 $0.53 $2.01Spain 1999 $0.54 $2.04Ramallah 1994 $0.91 $3.44UK 1999 $1.15 $4.35France 1999 $1.17 $4.43Netherlands 1999 $1.19 $4.50Denmark 1999 $1.61 $6.09Germany 1999 $1.81 $6.85

Source: World’s Water 2002-2003, Table 17.