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?

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.

Basic concepts for risk analysis Expected income: Expected utility of income: Example:

U(Y) U U(Y 2 ) U(E[Y]) E[U] U(Y 1 ) Y1Y1 Y*Y* Y ** Y2Y2 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(Y 1 )+(1-p)*u(Y 2 ) 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 A pay-off matrix Decision rule: maximize the minimum possible outcome

Other rules: maximax rule A pay-off matrix Decision rule: maximize the maximum possible outcome

Other rules: minimax regret A pay-off matrix Decision rule: minimize the maximum regret 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 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 F A (x)  F B (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:

Application with Monte-Carlo Simulation

Commonly used distributions ab f(x) x rectangular distribution

Commonly used distributions ab f(x) x triangular distribution m

Commonly used distributions f(x) x normal distribution 

Example: Monte - Carlo Simulation