Exam 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:

A group of participants were given the following gambles to choose from: (a) winning $4000 with 80% chance and $0 with 20% chance OR (b) winning $3000 for sure. 80% of respondents chose gamble (b) over gamble (a). The same group of participants were then given a choice between (c) losing $4000 with 80% change and losing $0 with 20% chance OR (d) losing $3000 for sure. Now 92% of the participants choose gamble (c) over (d). (a) Why do these choices represent an anomaly? What kind of preferences over gains and losses can explain this behaviour? (b) What do such preferences imply for the shape of the underlying value (utility) function? Draw a neat diagram to illustrate this value function. A neatly drawn diagram should be sufficient to answer this question. (c) On the diagram you drew for Part (b), show that the increase in value from a gain of a particular size is smaller than the increase in value from avoiding a loss of similar size. Either use coloured pencils or clearly mark (using letters A, B, C etc.) the relevant gains and losses on your diagram. Briefly explain your answer so that it is clear what you are showing on your diagram.

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 choice between this job and another job then he will choose the job with the start-up company as long as the other job pays less than approx. $52,000 per year.

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 expected utility of accepting this gamble is:

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 $4000 with one-half chance but you can lose $4000 with one half-chance. Suppose your utility function is defined as U(W)=2W. To you, the expected utility of this gamble is:

When respondents are given the following two choices A and B, a majority choose B over A. Choice A: $5,000,000 with probability .10, $1,000,000 with probability .89; 0 with probability .01. Choice B: $1,000,000 with probability 1. When respondents are given the following two choices C and D, a majority choose C over D. Choice C: $5,000,000 with probability .10 and 0 with probability .90. Choice D: $1,000,000 with probability .11 and 0 with probability .89. These choices are a violation of the principles of Expected Utility theory, which suggests that if one chooses B over A, then that person should choose D over C. A potential explanation for this pattern of behaviour is the following.

Which of the following is more likely to result in negotiation deadlocks?

Jim Holtzhauer is playing the tables at Vegas. He currently has $4000, which I will denote as 4K in order to keep things simple. He is looking at a bet where with ½ chance he can win 2K 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:

Which of the following is NOT a good description of the phenomenon of Loss Aversion?

It is generally well-known that buying lottery tickets do not make sense since the expected value of the lottery ticket is typically less than the price we pay for the ticket. One way to rationalize the act of buying such tickets is to appeal to:

Suppose you have won $1,000 on a game show. In addition to these winnings, you are now asked to choose between: 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: 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:

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:

Suppose you took a group of people and you gave half of the group a University of Auckland mug. The mugs are valued at $15.99 but the group is not (necessarily) aware of the price. All those who have a mug are called "sellers" while all those without a "mug" are buyers. Now you ask the buyers to name a maximum price at which they are willing to buy the cup. You ask the sellers to name a minimum price at which they are willing to sell a cup. It is highly likely that:

Denote by "W" is the amount of wealth you are holding. Currently you have $5000. Consider a gamble where there is a half-chance of winning $1000 and a half-chance of losing $500. Assuming standard preferences, a risk neutral person will:

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 utility of accepting this gamble is (approx.):

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 Certainty Equivalent (CE) of this gamble is (approx.):

Jim Holtzhauer is playing the tables at Vegas. He currently has $4000, which I will denote as 4K in order to keep things simple. He is looking at a bet where with ½ chance he can win 2K 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:

The rule at Shadows is: If you are drinking, you must be 18. In order to check the validity of this rule, you must check:

Liam currently has $10,000. He is offered a gamble where with one-half chance he can win $2000 but with one-half chance he can lose $2000. (a) Show that Liam is indifferent between accepting this gamble and sticking with this initial endowment of $10,000 if Liam's utility function is given by U(W) = W. (b) However, if Liam's utility function is given by U(W) = Square Root (W), then Liam strictly prefers to stick with his initial endowment of $10,000 rather than accepting this gamble. (Hint: In answering this question, you should denote utilities using 2 decimal points rather than rounding them up to the nearest whole number. Make sure you show your work.) (c) Draw TWO NEAT graphs with wealth (W) on the x-axis and Utility (U) on the y-axis to depict the utility functions in Part (a) and Part (b).

Emilio has just bought a fancy new 65-inch OLED Smart television for $5,000.00. Suppose there is a 10% chance that the television will topple over from his entertainment unit and get smashed to pieces and a 90% chance that nothing will happen. Emilio is risk averse and his utility of wealth is U(W) = W^0.5. What is Emilio's (i) certainty equivalent and (ii) risk premium? How much is the maximum that Emilio should be willing to pay to take out insurance on his TV? How would you answers change if the probability of breakage was 1% rather than 10%?