ECON 440: Lecture 5 - Introduction to Risk

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Questions and Answers

What is the expected behavior of a risk-neutral individual concerning the choice between option A and option B as they progress through the price list?

They will prefer option B throughout.

They will switch preference to option A at the bottom.

They will switch preference halfway down after row 4. (correct)

They will have no preference between the two options.

If a participant exhibits increasing relative risk aversion, how does their choice behavior change as stakes increase?

They switch preferences sooner in the list.

They switch preferences later in the list. (correct)

They prefer lower stakes regardless of options.

Their preferences do not change with stakes.

In the Holt and Laury experiment, what was the average switch point for participants regarding their risk preferences?

Row 7

Row 10

Row 3

Row 5 (correct)

What characterizes the Allais Paradox choices when individuals prefer a certain outcome over a gamble with a higher expected value?

Preference for certain outcomes even with lower expected value. Signup and view all the answers

What common choice pattern is observed among participants facing the Allais Paradox?

Always choosing the certain outcome. Signup and view all the answers

How does a risk-averse individual behave on the Holt and Laury price list?

They switch their preference for option A later in the list. Signup and view all the answers

What does it indicate if an individual's preferences switch farther down the list in the Holt and Laury experiment?

They display increasing risk aversion. Signup and view all the answers

What is the implication of constant relative risk aversion in decision-making when stakes increase?

Choices between options remain unchanged regardless of stakes. Signup and view all the answers

Which response accurately describes the relationship between risk and the stake levels according to the findings in the Holt and Laury experiment?

Risk preferences remain consistent regardless of stake levels. Signup and view all the answers

What is the expected value of a gamble that has a 50% chance of winning $32,000 and a 50% chance of winning $1,000,000?

$516,000 Signup and view all the answers

In the context of expected utility theory, what does the variable 'u(x)' represent?

The utility assigned to wealth x Signup and view all the answers

If a gamble is defined as (p1, x1; p2, x2), what does 'pi' represent for each outcome?

The probability of outcome i Signup and view all the answers

How would you summarize the expected utility of a gamble A with multiple outcomes?

EU(A) = Σ pi xi Signup and view all the answers

What is the expected outcome of rolling a fair six-sided die in terms of dollars?

$3.5 Signup and view all the answers

If an individual chooses to walk away from a gamble and secures $500,000, which aspect of decision-making is being demonstrated?

Risk aversion Signup and view all the answers

In expected utility theory, which of the following correctly defines the expected value of a gamble A?

The weighted sum of all possible outcomes based on their probabilities Signup and view all the answers

When assigned a probability of outcome i, what key principle does this reflect in a gamble?

Subjective valuation of risk Signup and view all the answers

In the Millionaire example, which choice represents the highest expected utility if you were to make a decision?

Walking away Signup and view all the answers

Considering a gamble with outcomes indexed by i, which of the following would be incorrect in defining the structure of that gamble?

Outcomes are guaranteed to occur in sequence Signup and view all the answers

What does the independence axiom state in the context of expected utility theory?

The choice between two gambles remains unchanged if a third gamble is introduced. Signup and view all the answers

In the context of the St. Petersburg Paradox, what does the expected value of the game reveal?

The expected value can be considered to be infinite. Signup and view all the answers

Why does utility function u(x) exhibit concavity in the context of risk aversion?

Because individuals experience diminishing returns from additional wealth. Signup and view all the answers

What conclusion can be drawn from the average and median amounts a class is willing to pay to play the St. Petersburg game?

The disparity between average and median suggests risk aversion among the participants. Signup and view all the answers

What does a utility function of the form u(x) = x^α with α < 1 indicate about a decision-maker's preferences?

The decision-maker experiences risk aversion. Signup and view all the answers

How is the expected utility calculated for a gamble involving a coin flip for $25?

By multiplying the probabilities with the total possible payoffs. Signup and view all the answers

What preference does a convex utility function u(x) indicate?

The individual demonstrates risk-seeking behavior. Signup and view all the answers

Which of the following outcomes would likely be preferred by a risk-averse individual?

$25 for certain over a 50% chance of $50. Signup and view all the answers

Which statement aligns with the concept of diminishing marginal utility in economic theory?

The first units of wealth provide greater satisfaction than subsequent units. Signup and view all the answers

What does the typical choice pattern between A and B indicate?

A is preferred over B, indicating risk aversion. Signup and view all the answers

Why is the choice between A' and B' significant in the context of expected utility theory?

It shows the violation of expected utility theory when preferences do not align. Signup and view all the answers

In the context of the Allais Paradox, what does the Common Consequence Problem illustrate?

Inconsistency in choices based on the addition of a common consequence. Signup and view all the answers

What is the expected utility of option A as described?

1 * u(100,000,000) Signup and view all the answers

How does the expected utility of option D compare to C?

D is greater than C, suggesting risk-seeking behavior. Signup and view all the answers

What does the utility function u(100,000,000) represent in this context?

The certain outcome of receiving $100 million. Signup and view all the answers

In the Allais Paradox, what does the option B' demonstrate?

Inconsistency in choice patterns due to risk evaluation. Signup and view all the answers

What percentage of participants chose option A in the Moblab results?

45% Signup and view all the answers

What does the expected utility calculation of EU(A') imply?

Certainty leads to lower expected utility. Signup and view all the answers

In the context of the Allais Paradox, what does option C represent?

A guaranteed payout of $100 million. Signup and view all the answers

Study Notes

Research Group Assignments

Assignments are available on Canvas.
Review guidance for working in research groups located in Module 3.
Group Management Plan due by September 13; one submission required per group.

Expected Utility: The Classic Theory

Expected utility helps make choices between uncertain options or gambles.
Example scenario from "Who Wants to Be a Millionaire":
- Joseph can either guess for a million or walk away with $500,000 guaranteed.

Gambles

A gamble is defined by potential outcomes, indexed by i, with respective probabilities pi and values xi.
Representation of a gamble: (p1, x1; p2, x2; ...; pn, xn).
Examples include outcomes from a die roll or choices in a quiz game.

Expected Value Calculation

Expected value (EV) formula: EV(A) = Σ(pi * xi), where pi is the probability and xi the outcome value.
EV example from Millionaire scenario: $516,000.
For a die roll, EV = 3.5.

Expected Utility Concept

Expected utility is calculated as: EU(A) = Σ(pi * u(xi)).
Consumers choose gambles maximizing their expected utility.

Independence Axiom

A fundamental principle stating preferences between gambles remain consistent when a common gamble is added.
If a preference exists for one option over another, that preference should remain when both are modified by adding the same additional gamble.

Utility Function Characteristics

Utility functions can be concave (risk-averse), convex (risk-seeking), or linear (risk-neutral).
The St. Petersburg Paradox exemplifies diminishing marginal utility; average willingness to pay contrasts with calculated expected value.

Risk Aversion

Utility function: u(x) = x^α; α < 1 indicates risk aversion.
Example calculations show preferences for certain outcomes over risky gambles, reflecting risk-averse behavior.

Experimental Evidence

The Holt and Laury experiment involved 175 students making risky choices.
Results indicate typical risk aversion, showing participants prefer a secure option at lower stakes and switch to riskier choices at higher stakes.

Violations of Expected Utility Theory

Allais Paradox: Demonstrates inconsistencies in choice patterns. For example:
- Choices between certain and probabilistic options show a preference for certainty in A over B, yet preferences flip in A' and B' choices.

Common Consequence Problem

Choices between gambles that add a constant (common consequence) challenge expected utility theory predictions, revealing a common pattern of preference inconsistencies.

Allais Paradox Example 2

Further illustrates preference inconsistencies between two sets of gambles involving certain and variable outcomes, reaffirming the unexpected decision behaviors contradicting traditional utility theory predictions.

MobLab Activities

Highlights experimental tasks evaluating risk preferences among participants, facilitating understanding of real-world choices versus theoretical predictions.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your understanding of expected utility theory and its applications in decision-making under uncertainty. This quiz covers concepts related to gambles, expected value calculations, and practical examples from games and scenarios like 'Who Wants to Be a Millionaire'.