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8 Cards in this Set
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1. You have $64 and the utility function U = x^(1/2) where x is money. Jimbo, a well known bully, comes up and says "Guess which hand I am holding my knife in. If you guess right you can keep all your money. If you guess wrong, I will take $60. Or, if you give me enough money right now, I'll just walk away and let you keep the rest."
a) Using a diagram and certainty equivalents, show and calculate how much money you are willing to give Jimbo to get him to leave you alone. |
See picture
EUg=EUc
(1/2)*(sqrt(4))+(1/2)*(sqrt(64))= 1*sqrt(64-x)
(1+4)^2= 64-x x=39 |
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Use diminishing marginal utility to explain in words why this is more than $30, the EV of the loss. |
DMU means that the utility costs of losing your last bits of money will be very high. On the other hand, it is not such a loss to give up money when you have a lot. So, people will give up some money, for certainty, to guard the chance of losing almost all of it |
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In reality, people are often risk loving over losses, not risk averse. How would this change the amount of money you'd offer Jimbo, and why? |
A risk lover over losses would have increasing, rather than decreasing MU. Paying the insurance would be costly to them in utility terms, while losing all of their money would not cost much extra utility. |
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Consider a person with $100 to start and a utility function given by u(x)=x½, where x is their consumption in dollars. They are given a chance to take a gamble with a 50% chance of gaining $100, and a 50% chance of gaining $0.
a. According to expected utility maximization, what amount of money, for certain, would be equivalent to being allowed to play this gamble? Sketch the appropriate figure, and show your math. (Hint: Set expected utility from the gamble equal to the utility of some unknown x, and then solve for x.) |
See picture.
EUg=EUc
(1/2)*(sqrt(100))+(1/2)*(sqrt(200))= 1*sqrt(x)
x=145.71=CE |
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Consider a person with $100 to start and a utility function given by u(x)=x½, where x is their consumption in dollars. They are given a chance to take a gamble with a 50% chance of gaining $100, and a 50% chance of gaining $0.
b. Now suppose that their utility function is given by u(x)=x^2. Recalculate the certainty equivalent. |
EUg=EUc
(1/2)*((100)^2)+(1/2)*((200)^2)= 1*(x)^2
x=158.11=CE |
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UO President Coltrane has appointed you head of UO parking enforcement. Your mission is to eliminate illegal parking. Car drivers are completely rational expected utility maximizers. They have utility functions given by u=x½ where x is $ of consumption on things other than parking. But of course, everyone must park, whether legally or illegally. People start with $144. A legal parking permit costs $63, and if you catch someone parking illegally you are allowed to fine them $80.
a. Catching illegal parkers is expensive. Exactly how often do you have to catch illegal parkers, before they will buy the $63 parking permit instead? (Hint: Set EU legal = EU illegal and solve for p, the probability). |
EUI= EUL p*√(144-80)+(1-p)*√144=√(144-63) p*√64+(1-p)*√144=√81 p*8+(1-p)*12=9 p*8+12-12*p=9 -4*p=-3 p=3/4 |
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UO President Coltrane has appointed you head of UO parking enforcement. Your mission is to eliminate illegal parking. Car drivers are completely rational expected utility maximizers. They have utility functions given by u=x½ where x is $ of consumption on things other than parking. But of course, everyone must park, whether legally or illegally. People start with $144. A legal parking permit costs $63, and if you catch someone parking illegally you are allowed to fine them $80.
b. Now go back to catching illegal parkers ½ the time. Assume you are also in charge of selling legal parking permits – so the price is no longer fixed at $63. If you can raise the price of a ticket for illegal parking from $80 to $108, how high can you set the price of a parking permit? Do the math and explain. |
EUI= EUL
(1/2)*(sqrt(144-108))+(1/2)*(sqrt(144))= 1*sqrt(144-x)
(3+6)^2= 144-x 81=144-x x=63 |
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The De Martino paper argued that many people are actually risk-loving over losses – they prefer the risk of a big loss to the certain expected value of the loss. If this is true, what effect will it have on your explanation in a) about stopping illegal parkers using fines? Explain. |
You will have to increase the probability of catching illegal parkers, because risk lovers will not be willing to pay much to avoid the risk of a loss. |
