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?

<p>Preference for certain outcomes even with lower expected value. (B)</p> Signup and view all the answers

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

<p>Always choosing the certain outcome. (B)</p> Signup and view all the answers

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

<p>They switch their preference for option A later in the list. (C)</p> 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?

<p>They display increasing risk aversion. (B)</p> Signup and view all the answers

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

<p>Choices between options remain unchanged regardless of stakes. (A)</p> 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?

<p>Risk preferences remain consistent regardless of stake levels. (D)</p> 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?

<p>$516,000 (B)</p> Signup and view all the answers

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

<p>The utility assigned to wealth x (C)</p> Signup and view all the answers

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

<p>The probability of outcome i (B)</p> Signup and view all the answers

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

<p>EU(A) = Σ pi xi (C)</p> Signup and view all the answers

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

<p>$3.5 (B)</p> 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?

<p>Risk aversion (C)</p> Signup and view all the answers

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

<p>The weighted sum of all possible outcomes based on their probabilities (D)</p> Signup and view all the answers

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

<p>Subjective valuation of risk (D)</p> Signup and view all the answers

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

<p>Walking away (B)</p> 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?

<p>Outcomes are guaranteed to occur in sequence (C)</p> Signup and view all the answers

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

<p>The choice between two gambles remains unchanged if a third gamble is introduced. (A)</p> Signup and view all the answers

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

<p>The expected value can be considered to be infinite. (B)</p> Signup and view all the answers

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

<p>Because individuals experience diminishing returns from additional wealth. (C)</p> 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?

<p>The disparity between average and median suggests risk aversion among the participants. (D)</p> 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?

<p>The decision-maker experiences risk aversion. (B)</p> Signup and view all the answers

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

<p>By multiplying the probabilities with the total possible payoffs. (B)</p> Signup and view all the answers

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

<p>The individual demonstrates risk-seeking behavior. (B)</p> Signup and view all the answers

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

<p>$25 for certain over a 50% chance of $50. (C)</p> Signup and view all the answers

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

<p>The first units of wealth provide greater satisfaction than subsequent units. (D)</p> Signup and view all the answers

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

<p>A is preferred over B, indicating risk aversion. (D)</p> Signup and view all the answers

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

<p>It shows the violation of expected utility theory when preferences do not align. (B)</p> Signup and view all the answers

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

<p>Inconsistency in choices based on the addition of a common consequence. (A)</p> Signup and view all the answers

What is the expected utility of option A as described?

<p>1 * u(100,000,000) (C)</p> Signup and view all the answers

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

<p>D is greater than C, suggesting risk-seeking behavior. (A)</p> Signup and view all the answers

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

<p>The certain outcome of receiving $100 million. (C)</p> Signup and view all the answers

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

<p>Inconsistency in choice patterns due to risk evaluation. (C)</p> Signup and view all the answers

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

<p>45% (D)</p> Signup and view all the answers

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

<p>Certainty leads to lower expected utility. (B)</p> Signup and view all the answers

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

<p>A guaranteed payout of $100 million. (B)</p> Signup and view all the answers

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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.

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