 A person that chooses the alternative that has the highest Expected Value, when risk is involved, violates the assumptions of expected utility maximisation.
 An expected utility maximizer with u(x) = log(x) rejects all fair gambles (i.e. those with an expected value of zero).
 Prospect Theory can explain the Ellsberg Paradox.
 Sophisticated QuasiHyperbolic Discounters behave time consistently, as they know that they will have a present bias in future decisions and therefore act to eliminate the bias.
 The outcome resulting from a Nash Equilibirum is socially efficient, since in a Nash Equilibrium all players play best responses to each other.
 Rational preferences should only depend on the person’s own outcomes (such as their own consumption, wealth, etc.)
 Jack, who has the utility function u(x, y) = x − γy, where x is his own material payoff, while y is the material payoff of his neighbour, can be described as envious.
 Somebody who plays the lottery cannot be an expected utility maximizer.
 If somebody is taking drugs, then that does not necessarily mean that she/he cannot be an exponential discounter.
 The Allais Paradox demonstrates that many humans evaluate the same lotteries differently if they are combined with the same other lottery.
 Lucy has current wealth of $100, 000. She has just been sent a notice to pay $500 for speeding. Lucy decides not to pay and to go to court. There she will either get off (because of insufficient evidence)and pay nothing or will have to pay the fine plus the court cost which together amount to $1,000. Lucy sees her chances of getting off at 50%.
 Suppose Lucy is an expected utility maximizer. Draw a graph with a utility function that can explain Lucy’s choice. Clearly label the wealth axis. Mark the expected utility of going to court and the utility of paying the fine.
 Can Lucy be an expected utility maximizer if she tells you that she would not only go to court but also reject gambles of winning or loosing $50 with equal probability for all wealth levels.
 Now suppose Lucy is behaving according to prospect theory and sees her current wealth level as her reference point to which she evaluates changes. Draw a graph with a value function that can explain Lucy’s choice to go to court. Clearly label the wealth axis. Mark the expected valuation of going to court and the value of paying the fine (assume that Lucy puts a probability weight of .5 on both winning and losing in court).
 In the same diagram show the expected valuation of the gamble of winning or losing $50 with equal probability. Which feature of Prospect Theory is responsible for Lucy rejecting the gamble?
 Assume a functional form for the value function and show that it can explain Lucy’s choices. (Hint: all typically used functional forms will do.
2.Suppose Tony is deciding on buying an expensive road bike. There are three periods:
Period 0 Tony decides to buy or not to buy the bike. The cost of the bike is equal to P utility units. (u_{0}(buy) = −P )
Period 1 If Tony has bought the bike, then he has to decide if he is going out for a ride. If he goes for a ride, then his effort generates a utility cost of E units of utility. (u_{1}(ride) = −E)
Period 2 If Tony has bought a bike and has ridden on it, then a health benefit equivalent to B utility units accrues. (u_{2}(ride) = B)
 Suppose Tony does not discount future utility. Which choices will he take in periods 0 and 1 if B > E + P ? What will he do if B < E + P ?
 Now suppose Tony is discounting future utility exponentially with a discount factor of δ < 1 per period. Will Tony for any δ buy the bike and then not ride it? Explain why or why not? (You do not have to calculate anything here yet. Use your knowledge about how exponential discounters behave instead.)
 Suppose Tony has a discount factor δ = ^{2}_{3} and P = 2, E = 4 and B = 8. What is he going to do in periods 0 and 1?
 Is it possible that Tony for some values of B, E, P buys the bike and then does not ride it if he is a naive quasihyperbolic discounter. Give an intuition for why or why not instead of calculating.
 What if he is a sophisticated quasihyperbolic discounter? Explain.
 Suppose Tony is naive and absolutely patient in the long term (δ = 1) but has a severe present bias (β = ^{1}_{3} ). Assume again F = 2, E = 4 and B = 8. Is he in period 0 expecting to ride in period 1 if he buys the bike? Is he actually going to ride if he has bought a bike and arrives at period 1? Is he going to buy the bike?
classmate 

you 
work 
shirk 

work 
4, 4 
1, 5 

shirk 
5, 1 
0, 0 
 Find all purestrategy Nash equilibria with purely selfish preferences. [There is also a mixedstrategy NE, which we will pay no attention to here.]
 Drawing on your experience as a student, have you observed outcomes such as those predicted? Have you seen other outcomes? Discuss!
 Suppose, both students like each other and have otherregarding preferences with u(x, y) = x + y/2, where x is the own outcome, while y is the outcome for the other student. What is the Nash prediction under these preferences? Discuss.
 Suppose, both students have competitive preferences and care only about how much better or worse off they are compared to the other student. In other words, u(x, y) = x − y for both students. What is the Nash prediction now! Discuss.
 Suppose both students are inequality averse and lose half a unit of utility for each unit difference in payoffs. In other words u(x, y) = x−(x−y)/2. Find all purestrategy Nash equilibria and discuss.
Tony's Decision Making for Purchasing a Bike and Going for a Ride
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1.
a.Lucy’s current wealth is $ 100 000; the chances of getting off and not paying and the chances of paying the fine plus the court cost are both 50%
Utilities associated with the probabilities are:
The utility of getting off and not paying the fine considering the 0.5 probability is 100.
The utility of paying the fine considering the 0.5 probability is 50.
This contributes to a function 2y + x = 0, the graph drawn represents wealth in the yaxis and fine in the xaxis.
b.Lucy cannot be an expected utility maximize in this case. She is risk averse, she chooses to remain with her wealth rather than gamble and also chooses to pay the fine rather than go to court.
c.By the use of the prospect theory, losses and gains are evaluated considering the both cases; going to court and winning and going to court and losing. The valuation reference point is the current wealth of Lucy which is $100000. It is plotted on the yaxis against the gains and losses in the xaxis.
d.The feature that is responsible for Lucy rejecting the gamble in the prospect theory is the riskaverseness. She chooses to remain with what she has and would rather pay $500 fin than go to court and lose and pay $1000.
e.The functional form for the value function, 2y + x = 0 can be clearly used to explain Lucy’s case. With this functional form, the probabilities for gaining and losing are equal. It is clearly shown that any rise in the probability for winning the case in court increases Lucy’s utility and the more she tends to fulfill her choice of going to court.
2.
Utility measures the consumer preferences over the set of various goods and services. It is a representation of consumer satisfaction obtained from a good or a service. Considering the case of Tony, the utility cost associated with the bike purchase for period 0 is P which is given by u_{0 }(Buy) = P, the utility cost associated with going out for a ride for period 1 is E given by u_{1 }(ride) = E and the utility associated with the health benefit as a result of riding the bike for period 2 is B which is given by u_{2 }(ride) = B.
a.When Tony does not discount future utility, considering the case when B > E+P, the choices that he will take for the periods 0 and 1 are:
 For period 0, Tony will choose to buy the bike at the utility units which equal to p.
 For period 1, Tony will choose to go out for a ride. The two choices are made due to the fact that the utility obtained from ridding the bike which is actually reflected by the health benefit is greater that both utility costs incurred for periods 1 and 2.
Considering the case where B < E + P, the choices that Tony will take for periods 1 and 2 are:
 For period 0, Tony will choose not to buy the bike.
 As a result of choice made in period 0, then it is obvious that Tony will not go for a ride. The choices are made due to the fact that overall utility obtained from the ride for period 2, B, is less that the utilities incurred in periods 0 and 1.
b.Yes, Tony will for any ? be expected to purchase the bike and not ride. This is due to the fact that the expected future utility reflected by the health benefit obtained from riding the bike may be greater in future as compared to the present time.
c.If ? = 2/3, P = 2, E = 4 and B = 8, Tony will not purchase a bike at period 0 and hence he will not go for a ride. This is due to the fact the utility cost incurred in periods 0 and 1 surpass the utility obtained in period 2.
d.Yes, it is possible that for some values of B, E and P, Tony buys the bike and then does not use it if he is a naïve quasihyperbolic discounter. This is due to the fact that the naifs falsely have a belief that today’s preferences will be maximized by the future selves. The naifs are unaware of their selfcontrol problems.
e.Yes, it is possible that for some values of B, E and P, Tony buys the bike and then does not use it if he is a sophisticated quasihyperbolic discounters. Sophisticates have a β < 1. This means that they are timeinconsistent but are aware of the selfcontrol problems and can correctly predict the future unlike the naifs. Therefore based on some values of B, E and P, they can predict the future and if it yields more utility compared to today, they can decide to have the ride in future.
f.The quasihyperbolic function is given by u_{t }= u_{t }+ β(?u_{t+1} + ?^{2} u_{t +2}+ ?^{3} u_{t+3} + ...)? = 1, β = 1/3, P =2, E = 4, B = 8
At period 0, we have 1/3*(82) = 2 > 0, Tony is expecting to ride in period 1 since the utility to be achieved exceeds the utility costs.
At period 1, we have 1/3*(84) = 1.33 > 1, Tony is actually going to ride if he has bought the bike.
Tony is actually going to purchase the bike since the utility achieved (health benefits) from riding the bike exceeds the utility costs incurred in both periods 1 and 2.
3.
 A Nash equilibrium results when the both players chose mutually the best responses. Each player chooses the strategy which maximizes his or her utility in as much as the opponent’s strategies are concerned. The purestrategy Nash equilibria with purely sel?sh preferences are the choice to work both (4, 4) or the choice to shirk both and fail (0, 0).
 Yes, there are other outcomes. The outcomes result to a mixedstrategy Nash equilibria. They are the choice to work and shirk (5, 1) and the choice to shirk and work (1, 5). This is due to the fact that no one is willing to incur the cost and obtain low marks just the same as the one who shirks but does not incur any cost.
 Under this situation, the students are cooperative. They may decide to both work together and obtain a good mark even if they incur costs or decide that one of them should do it and be compensated by the other. The Nash equilibria can either be a pure strategy one at (4, 4) or a mixed one at (1, 5) and (5, 1).
 In this situation, the Nash equilibria is a purestrategy one at (4, 4) where the both students prefer to work hard and get a good mark. The competition effect stirs them up to work for the best as no one wants to be outdone by the other.
 The purestrategy Nash equilibria will be (4, 4) and (0, 0). This is due to the fact that the difference in the each unit of payoffs makes the students to lose half a unit of utility. As each student aims at maximizing utility, no one is willing to lose half unit of utility due to difference in payoffs units and thus adopt similar payoff units.
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