Econ 440 Lecture 5 PDF

Econ 440: Lecture 5 Prof. Ragan Petrie Texas A&M University Sept 3, 2024 Research groups ▶ Research group assignments are posted on Canvas ▶ Read Guidance for working in research groups (posted in Module 3) ▶ Submit Group Management Plan by Sept 13 (GMP form to complete posted...

Econ 440: Lecture 5 Prof. Ragan Petrie Texas A&M University Sept 3, 2024 Research groups ▶ Research group assignments are posted on Canvas ▶ Read Guidance for working in research groups (posted in Module 3) ▶ Submit Group Management Plan by Sept 13 (GMP form to complete posted in Module 1, one submitted per group) Expected Utility: The Classic Theory Expected Utility: The Classic Theory  Expected Utility: The Classic Theory Motivating Example ▶ Suppose you are on the last round of the TV show Who Wants to be a Millionaire? ▶ You have narrowed down to two possible answers ▶ Guess wrong: go home with $32,000 ▶ Guess right: go home with $1,000,000 ▶ Walk away: go home with $500,000 for certain ▶ What do you do? Go to slido.com, #ec440  Expected Utility: The Classic Theory Gambles ▶ We need a way to make choices between uncertain options, e.g. gambles ▶ Consider a gamble called A, for example ▶ Possible outcomes are indexed by i = 1, 2, 3,... , n ▶ Probability of outcome i: pi ▶ Value of outcome i: xi ▶ Gamble is then summarized by (p1 , x1 ; p2 , x2 ;... ; pn , xn ) ▶ Examples: ▶ Guess from Millionaire example: 21 , $32, 000; 12 , $1, 000, 000
▶ Walk away: (1, $500, 000) ▶ Roll die, get paid the amount of the roll in dollars: 1 1 1 1 1 1
6 , $1; 6 , $2; 6 , $3; 6 , $4; 6 , $5; 6 , $6  Expected Utility: The Classic Theory Expected Value ▶ Expected value of gamble A: n X EV (A) = pi xi = p1 x1 + p2 x2 +... + pn xn i ▶ Examples: ▶ Guess from Millionaire:.5 * 32,000 +.5 * 1,000,000 = 516,000 ▶ Die roll: 1/6(1 + 2 + 3 + 4 + 5 + 6) = 3.5  Expected Utility: The Classic Theory Expected Utility ▶ Expected utility ▶ Consumer assigns utility u(x) to wealth x ▶ Expect utility theory says that n X EU(A) = pi u(xi ) = p1 u(x1 ) + p2 u(x2 ) +... + pn u(xn ) i ▶ Consumers will choose the gamble that maximizes expected utility  Expected Utility: The Classic Theory The Importance of Independence ▶ Expected utility theory satisfies the independence axiom ▶ Formally, the axiom states that for any gambles A, B, C such that A ⪰ B and any p ∈ (0, 1], we must have pA + (1 − p)C ⪰ pB + (1 − p)C ▶ Informally, this means that if you like apples more than bananas, you like a lottery where you win an apple with 90% probability and $1 million with 10% probability more than you like a gamble where you win a banana with 90% probability and $1 million with 10% probability  Expected Utility: The Classic Theory What Shape Should u(x) Have? ▶ Consider the following game: I will flip a coin until the first heads comes up. If the first heads is on flip number n, then I’ll pay you $2n. How much would you pay to play this game? ▶ Originally proposed by Bernoulli (1738) ▶ Known as the St. Petersburg Paradox ▶ What is class willing to pay? $3,261 (avg), $2 (median) ▶ What is the expected value of this game? EV = ½(2) + ¼(4) + ⅛(8) + 1/16(16) + … = 1 + 1 + 1 +... = infinity ▶ It is clear that there is a diminishing marginal utility of money ▶ Intuition: an extra $1000 is massive windfall for a very poor person but not even noticeable for very rich person ▶ Means that u(x) is concave, which represents risk-averse preferences ▶ Can also have risk-seeking preferences (convex u(x)) or risk-neutral preference (linear u(x))  Expected Utility: The Classic Theory Risk Aversion ▶ One possible family of functions: u(x) = x α √ ▶ Example: u(x) = x, ie α = 12 ▶ Expected utility of $9 for certain? EU(1,$9) = u(9) = sqrt(9) = 3 ▶ Expected utility of a fair coin flip for $25? EU(.5,$25;.5,$0) =.5*sqrt(25) +.5*sqrt(0) =.5 * 5 = 2.5 ▶ Would decision-maker prefer $9 for certain or a coin flip for $25? $9 for certain ▶ Note: EU of St Petersburg Paradox gamble using this utility function  Expected Utility: The Classic Theory Lab Evidence ▶ Subjects: 175 university students ▶ Choose either option A or B in each row: ▶ Repeated for 20x, 50x, 90x payoffs Source: Holt and Laury (2002)  Expected Utility: The Classic Theory Expected Results ▶ How should responses change as subject progresses through price list from top to bottom? ▶ Note that option B is always riskier than option A ▶ Should prefer option A at top of price list ▶ By bottom row, should switch to preferring option B ▶ Where do you switch if risk-neutral? halfway down after row 4 ▶ What if risk-averse? switch farther down the list ▶ What if risk-seeking? switch farther up the list ▶ How should responses change with stakes? Three possibilities: 1. Constant relative risk aversion: choices between options A and B should not depend on stakes 2. Increasing relative risk aversion: choices are more risk averse as stakes go up (i.e. switch later) 3. Decreasing relative risk aversion: choices are less risk averse as stakes go up (i.e. switch earlier)  Expected Utility: The Classic Theory Results: Holt and Laury ▶ Is the average participant risk averse, risk neutral, or risk loving? risk averse: note average switch point is well past row 5 ▶ What is type of relative risk aversion? increasing risk aversion: note lines move out as stakes increase Source: Holt and Laury (2002)  Expected Utility: The Classic Theory MobLab Activity - Risk Preferences (HL task)  Expected Utility: The Classic Theory MobLab Activity - Risk Preferences (HL task) Violations of Expected Utility Theory Violations of Expected Utility Theory  Violations of Expected Utility Theory The Allais Paradox: Version 1 1. Choose your preferred option: A: Receive $100 million for certain B: 10% chance of $500 million, 89% chance of $100 million, 1% chance of no money 2. Choose your preferred option: A′ : 11% chance of $100 million, 89% chance of no money B ′ : 10% chance of $500 million, 90% chance of no money ▶ Typical choice pattern? A ⪰ B; B ′ ⪰ A′ ▶ EU(A) = 1 * u(100,000,000) ▶ EU(B) =.1*u(500,000,000) + 0.89*u(100,000,000) +.01*u(0) ▶ EU(A′ ) =.11*u(100,000,000) +.89*u(0) ▶ EU(B ′ ) =.1*u(500,000,000) +.9*u(0)  Violations of Expected Utility Theory Moblab Results - Allais Paradox 1 ▶ Choose your preferred option: A: Receive $100 million for certain B: 10% chance of $500 million, 89% chance of $100 million, 1% chance of no money ▶ Choose your preferred option: A′ : 11% chance of $100 million, 89% chance of no money B ′ : 10% chance of $500 million, 90% chance of no money ▶ Typical choice pattern? A ⪰ B; B ′ ⪰ A′ ▶ Moblab results: 45% chose A; 22% chose A′  Violations of Expected Utility Theory Common Consequence Problem ▶ Suppose you choose A ⪰ B ▶ Then expected utility theory says you must choose A′ ⪰ B ′ EU(A′ ) ≥ EU(B ′ ) ⇐⇒.11u(100) +.89u(0) ≥.1u(500) +.9u(0) ⇐⇒.11u(100) +.89u(0) ≥.1u(500) +.89u(0) +.01u(0) ⇐⇒.11u(100) +.89u(100) ≥.1u(500) +.89u(100) +.01u(0) ⇐⇒ u(100) ≥.1u(500) +.89u(100) +.01u(0) ⇐⇒ EU(A) ≥ EU(B) ▶ Typical choice pattern is incompatible with expected utility theory ▶ Called common consequence version of the Allais Paradox, because I added the.89 chance of $100 million to both sides  Violations of Expected Utility Theory The Allais Paradox: Version 2 1. Choose your preferred option: C : Receive $100 million for certain D: 98% chance of $500 million, 2% chance of no money 2. Choose your preferred option: C ′ : 1% chance of $100 million, 99% chance of no money D ′ : 0.98% chance of $500 million, 99.02% chance of no money ▶ Typical choice pattern? C ⪰ D; D ′ ⪰ C ′ ▶ EU(C ) = u(100,000,000) ▶ EU(D) =.98*u(500,000,000) +.02*u(0) ▶ EU(C ′ ) =.01*u(100,000,000) +.99*u(0) ▶ EU(D ′ ) =.0098*u(500,000,000) +.9902*u(0)  Violations of Expected Utility Theory Moblab Results - Allais Paradox 2 ▶ Choose your preferred option: C : Receive $100 million for certain D: 98% chance of $500 million, 2% chance of no money ▶ Choose your preferred option: C ′ : 1% chance of $100 million, 99% chance of no money D ′ : 0.98% chance of $500 million, 99.02% chance of no money ▶ Typical choice pattern? C ⪰ D; D ′ ⪰ C ′ ▶ Moblab results: 30% chose C ; 22% chose C ′  Violations of Expected Utility Theory Common Ratio Problem ▶ Suppose we observe C ⪰ D ▶ Then expected utility theory says we must have C ′ ⪰ D ′ EU(C ) ≥ EU(D) ⇐⇒ u(100) ≥.98u(500) +.02u(0) ⇐⇒ 0.01u(100) ≥.0098u(500) +.0002u(0) ⇐⇒ 0.01u(100) + 0.99u(0) ≥.0098u(500) +.0002u(0) + 0.99u(0) ⇐⇒ 0.01u(100) + 0.99u(0) ≥.0098u(500) +.9902u(0) ⇐⇒ EU(C ′ ) > EU(D ′ ) ▶ Called common ratio version of the Allais Paradox, because I multipled both sides of the equation by 0.01  Violations of Expected Utility Theory What Is Going On? ▶ Expected utility theory says we should have A ⪰ B ⇐⇒ A′ ⪰ B ′ and C ⪰ D ⇐⇒ C ′ ⪰ D ′ ▶ So if actual behavior doesn’t follow these results, expected utility theory must not represent people’s true preferences? ▶ Next time we will see a theory that does explain these choice patterns better