ECUE 3: Economics under Uncertainty

• ECUE 3 examines economics under uncertainty.
• The course outline includes uncertainty and risk, attitudes, utility models, insurance demands, and portfolio choices.

Introduction

• Uncertainty is inseparable from economic life.
• Early economists favored an expansive concept of uncertainty.
• The universe can be represented by finite possible situations with limited knowledge.
• A core question is how agents behave in uncertain situations.
• Uncertainty features in both private and public decision-making.
• Analyzing such decisions formalizes the consideration of random consequences.
• Quantifying and comparing these random factors is essential.
• The course focuses on providing concepts and tools for decision-making under uncertainty, especially probabilized uncertainty.

Chapter 1: Uncertainty and Risk: Utility Expectation

• Theorists emphasize that economics shouldn't be seen through a mechanical lens.
• Incorporating randomness requires new definitions for uncertainty and risk.

Distinction Between Risk and Uncertainty

• Knight (1921) distinguished between risk and uncertainty.
• The key difference is that risk involves a known distribution of outcomes, either calculated or statistically observed.
• Uncertainty lacks this known distribution due to the impossibility of grouping cases with high singularity.

Risky vs. Uncertain Situations

• A situation is risky when predictions can be made using mathematical or frequentist probabilities.
• Mathematical probabilities are calculated a priori.
• Chance games where outcomes are equally likely
• Frequentist probabilities are derived from numerous observations of a repeating event.
• Observation example: average rainy days per year
• Both types are considered objective probabilities.
• An uncertain situation is unique and can't be reduced to similar cases, making it non-probabilizable.
• Predictions rely on separate judgment exercises:
• First, forming a personal estimation (experience, intuition).
• Second, assessing the quality of that judgment, depending on confidence.
• This confidence is epistemic probability, from the Greek "episteme" (knowledge).
• Probabilities allows for calculation and optimization in uncertain environments.
• Singular situations limit the use of probabilities, complicating behavior modeling.
• The text focuses on choice in risky environments.

Representing the Uncertain Future

• Traditional utility is revisited.
• Under certainty, choices maximize satisfaction or utility functions.
• Under uncertainty, expected utility theory is used.
• Choices are based on lotteries instead of goods, using expected utility (EU).
• Expected utility addresses the St. Petersburg paradox.

Expected Utility and the St. Petersburg Paradox

• Daniel Bernoulli (1700-1782) proposed calculating the expectation of utility of gains.

• Instead of the expectation of gains

• The St. Petersburg Paradox involves a game where a player flips a coin until "heads" appears, stopping when "tails" appears.

• Payouts increase exponentially (2, 2², 2ⁿ ducats).

• A rational player determines the maximum entry fee they'd pay.

• Mathematicians propose using the expectation of gains, which leads to an infinite expected value if the amount of coups is not limited.

• Bernoulli noted that most players would only pay a small amount to play.

• To resolve it, Bernoulli suggested considering the utility associated with different gains rather than their mathematical expectation.

• Players consider the cost of playing against the game's potential reward.

• Marginal Utility declines where the utility of a hypothetical gain of 2ⁿ relative to 2ⁿ⁺¹ decreases.

• This is due to aversion of risk, but especially from risk itself when n is high

• Considering the satisfaction provided by a gain is a foundation of the expected utility model.

• Bernoulli transforms the game by using expected utility rather than monetary value.

• To express diminishing marginal utility, Bernoulli used the Log function to represent utility.

• This results in a bounded sum as a utility measure.

• The paradox disappears if players evaluate results non-linearly, allowing notion of utility.

The Notion of Lottery

• The simplest lottery in the EU model involves two outcomes (gain or nothing) with associated probabilities summing to 1.
• Without this form, the lottery concept is broader than consumer theory's goods bundles.
• Lottery refers to possible consequences of a decision (contingent gain).
• Parachute or umbrellas sales depending on the weather
• Uncertainty in sales is expressed as a lottery.
• Example: L[(prob, C₁); (prob, C2)]
• C₁ represents consequences of the lottery.
• Consider L₁ representing umbrella sales.
• L₁[(prob1, C11); (1 – prob1, C12)], where prob1 is possibility of good weather.
• C11 is gain if the weather is good
• C12 is gain if the weather is plesent.
• Sun umbrella lottery is L2

The Rule of Decision

• L₁[(prob1, C11); (1 – prob1, C12)]'s utility relies on all consequences.

• Each lottery's utility is the mathematical expectation of weighted consequence utilities.

• The expected utility of a lottery Lj (EU(Lj)) is calculated by the shown formula.

  • The shown formula is where probi is the associated probability to i with U(Cji) utility awarded to the lottery, if i is realized

• The expected utility function or VNM function (Von Neumann-Morgenstern) is the simplest form.

• To extend the discussion to the collection of lotteries if an agent are met with a choice.

• The greatest expected utility is selected.

• This reveals the maximization principle.

Examples

• Example one with a motorist choosing a bridge to cross , FHB or HKB, which each involves 3 states of traffic; liquid, dense and congested.
• Table data and expected utility formulas can be used.
• A subjective probability also analyzes copies in imperfect information (Harshgyi and Harsanyi).

Schematics

• A Rational decision-maker needs to pick among multiple alternatives.
• Every element of the natural condition is considered along with the associated subjective probabilities.
• The probability causes an emotion, or an intuitive decision or belief.
• Rate utility based on their orders of perference
• The sum of the conditional beliefs depends of the probability of the natural states.
• Follow the actor with them maximum predicted utility.
• A workers selects among 2 jobs with U(x)=x¹/².