Lecture 4

Environmental Cost - Benefit - Analysis

under risk and uncertainty

The St. Petersburg Paradox Game:

• toss a fair coin

• if head falls up at the first toss, you get 2$, if not the first but at the second toss doubled to 4$, at the third toss doubled again to

•8$, …

How much would you be willing to pay to participate at the game?

Answer: the expected value of the probability weighted outcomes.

The St. Petersburg Paradox

The expected value of the probability weighted outcomes:

w: welfare

p: probability

Would you pay an infinite amount of money to participate in the game?

2 3

1

2 0.5 2 0.5 4 0.5 8 ... 1.0 1.0 1.0...n n

n

E w p

The St. Petersburg Paradox Daniel Bernoulli’s solution involved two ideas that have since revolutionized economics:

(i), that people's utility from wealth, u(w), is not linearly related to wealth (w) but rather increases at a decreasing rate - the idea of diminishing marginal utility, u’(Y) > 0 and u”(Y) < 0;

(ii) that a person's valuation of a risky venture is not the expected return of that venture, but rather the expected utility from that venture. In the St. Petersburg case, the value of the game to an agent (assuming initial wealth is zero) is:

Due to diminishing marginal utility, people would only be willing to pay a finite amount of money to participate in the game.

22

1

2 0.5 2 0.5 2 ...n n

n

E u p u u u

Basic concepts for risk analysis Expected income:

Expected utility of income:

Example:

1 1 2 2E Y p Y p Y

1 2 21E U p U Y p U Y

2 2

1 1 2

1 1 2

1 1 2

, is a fraction of income , U Y 0, U Y 0

expected utility of the gamble: 1

certainty equivalent: 1

in this caser 1 has to be solved for , given ,

a

a a

a a

a a a

U Y a Y

E U p Y p Y

U Y E U p Y p Y

Y p Y p Y Y a

1 2 1 2, , ,p p Y Y

U(Y)

U

U(Y2)

U(E[Y])

E[U]

U(Y1)

Y1 Y* Y** Y2 Y

A

B

C

E

D

Figure 13.1 Risk aversion and the cost of risk bearing (Perman et al.: page 447)

E[Y]=Y**

Y*: certainty equivalent

line AB: convex combinations

p*u(Y1)+(1-p)*u(Y2)

cost of risk bearing (CORB) = Y** -Y*

(also called risk premium)

Y*: certainty equivalent (where utility of a certain payment equals utility of an uncertain payment)

Other rules: maximin rule

C D E

conserve wilderness area 120 50 10

develop a mine 5 30 140

A pay-off matrix

Decision rule: maximize the minimum possible outcome

Other rules: maximax rule

C D E

conserve wilderness area 120 50 10

develop a mine 5 30 140

A pay-off matrix

Decision rule: maximize the maximum possible outcome

Other rules: minimax regret

C D Econserve wilderness area 120 50 10develop a mine 5 30 140

A pay-off matrix

Decision rule: minimize the maximum regret

C D E

conserve wilderness area 0 0 130

develop a mine 115 20 0

The regret matrix

Other rules: assignment of subjective probabilities

Outcomes are weighted by the subjective probabilities of the decision maker.

=> objective probabilities are often not available (never?)

=> subjective probabilities express the value judgement of the decision maker

=> subjective probabilities can be elicited from decision makers (stakeholders)

Other rules: safe minimum standard

F U

conserve wilderness area 130 0

develop a mine 0 z

A regret matrix for the possibility of species extinction

What pay-off should be assigned to having the mine go ahead if state U eventuates?

=> targets need to be set for environmental policy.

Environmental Performance Bond

Technology developers deposit a certain amount of money x that is expected to cover potential environmental damages related to the use of the new technology:

• companies get money back if no harm,

• in the case of damage, damage costs y are deducted.

Decision Analysis with Preferences Unknown

• mean - variance efficiency

• mean - standard deviation

• portfolio analysis

• stochastic efficiency methods

Decision Analysis with Preferences Unknown

• first - degree stochastic dominance (FSD)

• second degree stochastic dominance (SSD)

• third degree stochastic dominance (TSD)

• stochastic dominance with respect to a function (SDRF)

Decision Analysis with Preferences Unknown

First - degree stochastic dominance (FSD) assumptions:

• DM has positive marginal utility

• given two actions A and B, A dominates B in FSD if for the cumulative distribution functions FA(x)FB(x)

Decision Analysis with Preferences Unknown

Second - degree stochastic dominance (SSD)

• DM has decreasing positive marginal utility

• given two actions A and B, A dominates B in SSD if:

* *

,

for all values of *, with at least one strong inequality

x x

A BF x dx F x dx

x

Labor Equipment Yield level

Price

Time Preference

Draws

Model

Calculation ofAnnuityannuity

1000Replications

Cumulative ProbabilityDistribution of Annuity

Possible Uncertain Variables and Distributions

Correlation

Application with

Monte-Carlo

Simulation

Commonly used distributions

a b

f(x)

x

0

1

0

x a

f x a x bb a

X b

0

1

x ax a

F x a x bb a

x b

rectangular distribution

2

, 2 12

b aa bE x V x

Commonly used distributions

a b

f(x)

x

triangular distribution

0

2

2

0

x a

x aa x m

b a m af x

b xm x b

b a b m

x b

2

2

0

1

1

x a

x aa x m

b a m aF x

b xm x b

b a b m

x b

2 2 2

, 3

18

a b mE x

a b m ab am bmV x

m

Commonly used distributions

f(x)

x

normal distribution

2

2

1exp

22

xf x

xz

2, E x V x

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-40000 -30000 -20000 -10000 0 10000 20000 30000

Series1

Series2

Example: Monte - Carlo Simulation

Nr. Zinssätze AKosten-maleAKosten-female Kornpreis Strohpreis NPV Annuität1 0.08 47 24 67 2 71239 177362 0.09 47 25 70 3 76984 198323 0.09 43 25 70 3 81300 211404 0.13 36 27 76 4 99158 283065 0.11 31 26 73 3 95711 260296 0.12 44 26 74 4 90253 250067 0.09 42 25 70 3 80809 208488 0.07 42 24 67 2 73164 180739 0.06 40 22 64 1 66201 1560810 0.05 52 21 63 1 55958 1296711 0.14 54 28 78 5 90626 2651212 0.07 53 23 66 2 64693 1582213 0.07 60 23 67 2 60729 1493714 0.07 44 23 66 2 68902 1669715 0.10 42 25 71 3 85097 2258116 0.11 52 26 73 3 82094 2228217 0.12 50 26 74 4 86048 2388218 0.12 52 26 73 4 83147 2281119 0.14 46 28 77 5 94738 2752020 0.07 46 23 66 2 69438 1701121 0.11 53 26 73 3 81461 2214122 0.11 55 26 73 3 80184 2177823 0.11 48 25 72 3 82577 2214524 0.11 48 26 73 4 85628 2344125 0.09 48 24 69 2 74039 1883126 0.11 48 26 73 3 83926 2274027 0.12 52 26 74 4 83534 2298428 0.12 43 26 75 4 91704 2564729 0.10 41 25 70 3 82911 2163230 0.11 49 26 73 4 84516 2305231 0.10 47 25 71 3 79917 2093032 0.14 41 28 77 5 98007 2850433 0.11 36 26 73 4 93512 2560934 0.07 47 23 66 2 67661 1648335 0.10 46 25 71 3 81608 2152036 0.09 61 24 69 3 66949 1718237 0.13 43 27 76 4 94200 2682738 0.06 41 22 64 1 65921 15541