Chat with AIExam 4: Expected Utility Theory and Prospect Theory
Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Based on this information, you would conclude that:
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Suppose you have won $1,000 on a game show. In addition to these winnings, you are now asked to choose between:
Alternative 1. Gamble A: Win $1000 with 0.50 probability or Gamble B: Win $500 for sure.
On the other hand, suppose you have won $2,000 on a game show and are then asked to choose between:
Alternative 2. Gamble C: Lose $1,000 with 0.50 probability or Gamble D: Lose $500 for sure.
It is frequently observed that a majority choose Gamble B for Choice 1 while they choose Gamble C in Alternative 2.
Show that the final wealth levels are not different for the two alternatives. If the final wealth levels are not different then why do people choose Gamble B over Gamble A in Alternative 1, while they choose Gamble C over D for Alternative 2?
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Answered by Examlex AI CopilotThe final wealth levels for the two alternatives are not different. In both cases, the expected value of the gambles is the same.
For Alternative 1:
- Gamble A: Win $1000 with 0.50 probability
- Gamble B: Win $500 for sure
The expected value of Gamble A is 0.50 * $1000 = $500. So, on average, you would win $500 with Gamble A.
The expected value of Gamble B is simply $500. So, on average, you would win $500 with Gamble B.
For Alternative 2:
- Gamble C: Lose $1000 with 0.50 probability
- Gamble D: Lose $500 for sure
The expected value of Gamble C is 0.50 * (-$1000) = -$500. So, on average, you would lose $500 with Gamble C.
The expected value of Gamble D is simply (-$500). So, on average, you would lose $500 with Gamble D.
Since the expected values are the same for both alternatives, the final wealth levels are not different.
However, people may choose Gamble B over Gamble A in Alternative 1 because they are risk-averse. Even though the expected value is the same, the uncertainty of winning $1000 with 0.50 probability may be less appealing than the certainty of winning $500.
Similarly, people may choose Gamble C over Gamble D in Alternative 2 because they are risk-seeking. Even though the expected value is the same, the uncertainty of losing $1000 with 0.50 probability may be more appealing than the certainty of losing $500.
In both cases, people's risk preferences drive their choices, even though the final wealth levels are not different.
Suppose Ana's current wealth (W) is $8000 and Ana obeys the principles of expected utility theory. Suppose she is offered a gamble where she can win $5000 with one-half chance but she can lose $5000 with one half-chance. Suppose her utility function is defined as U(W) = W^0.5. (The square root of her wealth). What is the expected monetary payoff of this gamble? What is the expected utility of the gamble? What is Ana's certainty equivalent? What is her risk premium? If Ana were offered this gamble, then on the basis of the utility function defined above, would she accept the gamble or would she prefer on to hang on to her $8000 endowment?
(36) Answered by Examlex AI CopilotTo find the expected monetary payoff of the gamble, we calculate the weighted average of the possible outcomes. The expected monetary payoff (EM) is given by:
EM = (0.5 * $5000) + (0.5 * (-$5000)) = $2500 - $2500 = $0
The expected utility of the gamble is found by calculating the utility of each possible outcome and taking the weighted average. The expected utility (EU) is given by:
EU = (0.5 * (8000 + 5000)^0.5) + (0.5 * (8000 - 5000)^0.5)
= (0.5 * 13000^0.5) + (0.5 * 3000^0.5)
= (0.5 * 113.86) + (0.5 * 54.77)
= 57.43 + 27.39
= 84.82
To find Ana's certainty equivalent, we solve for the value of the certain amount of wealth that would give her the same level of utility as the gamble. Let CE be the certainty equivalent. We solve for CE in the equation:
U(CE) = EU
CE^0.5 = 84.82
CE = 84.82^2
CE = 7189.47
The risk premium is the difference between the expected value of the gamble and the certainty equivalent. The risk premium (RP) is given by:
RP = EU - W
RP = 84.82 - 8000
RP = -7915.18
Since the risk premium is negative, Ana would prefer to hang on to her $8000 endowment rather than accept the gamble. This is because the expected utility of the gamble is lower than the utility of her current wealth, indicating that she would be better off sticking with her current wealth rather than taking the gamble.
Suppose your current wealth (W) is $10,000. Suppose you are offered a gamble where you can win $4000 with one-half chance but you can lose $4000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the Certainty Equivalent (CE) of this gamble is approximately:
(30) Consider two sides negotiating over how to split a surplus. It is more likely that they will arrive at a mutually agreeable outcome if the two sides:
(44) Ken Jennings has just been offered a job with a start-up company. The job pays $30,000 guaranteed per year. On top of that that there is a ½ probability that he can get a $50000 performance bonus making a total of $80,000. Ken operates under the assumptions of expected utility theory and in general, has a strong preference for a job that pays a fixed amount per year. Ken's utility of wealth function is given by U(W) = W^0.5. (U(W) is equal to the Square Root of W). Show that if Ken has a choice between this job and another job that pays $60,000 per year then he will choose the other job over the job with the start-up company.
(37) Assume, Elizabeth's utility function is: U(W) = W^0.5 and she operates under the tenets of expected utility theory. She is considering two job proposals:. Alternative 1: take a job at a bank with a certain salary of $54,000 per annum. Alternative 2: take a job with a start-up company, get a base salary of $4,000 per annum a plus a bonus of $100,000 per annum a with probability 0.5 (otherwise bonus = $0). Show that Elizabeth would prefer Alternative 1 over Alternative 2 based on expected utility calculations.
(40) Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $5000 with one-half chance but you can lose $5000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). For you the risk premium of this is (approx.):
(42) Todd has $1000. He is given a choice of flipping a coin, heads wins $1000 while tails loses $500. Todd refuses to accept the gamble. A possible explanation is that:
(38) Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Should Jim accept this gamble? If yes, why? If not, then why not? Does Jim have a certainty equivalent in this case? If yes, then what is this amount? Briefly explain whether this will imply Jim actually giving up money or Jim having to be given extra money in order to forego the gamble.
(34) For the Value Function in Prospect Theory, the magnitude of value increase from a gain of a particular size is smaller than the magnitude of the value decrease from an equivalent loss. Draw a neat diagram with the Value Function and explain what this means.
(35) Respondents are given the following choices:
1) Choose between Gamble A: Win $4000 with probability 0.8 or Gamble B: Win $3000 for sure.
2) Choose between Gamble C: Lose $4000 with probability 0.8 or Gamble D: Lose $3000 for sure.
80% of respondents choose Gamble B over Gamble A; i.e. they prefer to win $3000 for sure over $3200 in expectation. But 92% of those same respondents chose Gamble C over Gamble D; i.e., they prefer to lose $3200 in expectation than lose $3000 for sure.
A potential explanation of this pattern of choices is that:
(31) Assume, Elizabeth's utility function is: U(W) = W^0.5 and she operates under the tenets of expected utility theory. She is considering two job proposals:. Alternative 1: take a job at a bank with a certain salary of $54,000 per annum. Alternative 2: take a job with a start-up company, get a base salary of $4,000 per annum a plus a bonus of $100,000 per annum a with probability 0.5 (otherwise bonus = $0).
(37) Ken Jennings has just been offered a job with a start-up company. The job pays $30,000 guaranteed per year. On top of that that there is a ½ probability that he can get a $50000 performance bonus making a total of $80,000. Ken, operates under the assumptions of expected utility theory and in general, has a strong preference for a job that pays a fixed amount per year. Ken's utility of wealth function is given by U(W) = W. (U(W) is equal to the Square Root of W). Based on this information, you would conclude that:
(31) Suppose your current wealth (W) is $8000 and you obey the principles of expected utility theory. Suppose you are offered a gamble where you can win $5000 with one-half chance but you can lose $5000 with one half-chance. Suppose your utility function is defined as U(W) = W^0.5. (The square root of your wealth). To you, the expected payoff from this gamble is:
(39) The fact that people often ask for a much higher price for a good they possess than they are willing to pay to buy the same good is an example of:
(40) Suppose I bought some Harvard mugs valued at $10.98. I gave half of the class a mug. These are the sellers. The other half do not have any mugs. They are the buyers. Neither side knows that true value of the mugs. Now supposed I asked the sellers to name a price at which they are willing to sell the mugs. At the same I asked the buyers to name a price at which they are willing to buy the mugs. It is likely that on average:
(39) Gamble A: Win $1000 with 0.50 probability or Gamble B: Win $500 for sure.
On the other hand, suppose you have won $2,000 on a game show and are then asked to choose between:
Gamble C: Lose $1,000 with 0.50 probability or Gamble D: Lose $500 for sure.
These two gambles are obviously identical in terms of final wealth states and probabilities. However, subjects are much more likely to choose the risk averse B and the risk seeking C. This suggests that participants:
(36) Jim Holtzhauer is playing the tables at Vegas. He currently has $10,000, which I will denote as 10K in order to keep things simple. He is looking at a bet where with ½ chance he can win 6K while with ½ chance he will lose 2K. Jim's utility of wealth function is given by U(W) = (W)^2. (U(W) is equal to the W Squared). Based on this information, you would conclude that:
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