1. In class, we studied the famous St. Petersberg Game. In that game, Daniel Bernoulli proposed the utility-of-wealth function uo(w) = vw. However, what if people had a different utility function? (a) (5 points) Suppose if instead of using the function in class, we used the function U1(w) = log(w). What would a person with this utility function be willing to pay to play the St. Petersberg Game? Show your work. (Hint: You can either figure this out using some calculus, or you can calculate it term-by-term using a calculator or a program like Excel.) (b) (5 points) Now, suppose another person had the utility function uz(w) = e2.log(w). Repeat (a), under this assumption. (c) (3 points) How do the results for uo, ul, and up compare to one another? What does this say about the preferences that create these three utility functions. (d) (2 points) An important property of utility functions is that they are equivalent under positive affine transformations (that is, being transformed by equations of the type y = mx +b where m > 0). What is the relationship between ui and uz? How does it related to your answers above? (e) (5 points) Is the utility function representation of an individual's preferences unique? Explain why or why not, with reference to the above questions. Total for Question 1: 20 2. (10 points) Consider the following statement: We have a large number of consumers (labelled i = 1, 2, ...), with utility functions ui(2) where x is an allocation of goods in the economy. To maximize social welfare, we find the maximum of U where U = L; Ui(x) over the possible values of x. Using your knowledge of utility and choices, what are the issues with this kind of "social welfare" function (often called utilitarian). Discuss, being specific and relating to the concepts developed in this course. 3. Consider the following description of a game: An auctioneer offers an object up for a sale: specifically, a $100 dollar bill. There are two player, who make decisions in alternative turns. Each turn, a player may either place bid on object, or drop out. A bid must be in made in increments of a penny ($0.01), and the lowest possible bid is a penny. A new bid must also be higher than any past bids, if any exist. The game ends with either (a) a player drops out, or (b) a player bids an amount equal to or greater than some predetermined "maximum" number (say, $1 million dollars). When the game ends, the person with the highest bid gets the object, and pays their bid. However, the other player (the loser) also must pay their (the loser's) most recent bid, if any. If the game ends with no active bids, no one gets the object. (a) (10 points) Model this strategic situation as a strategic game, explicitly describing the four parts of a game. Define any of your notation, if you use it, and be clear. Explain your choices. (b) (5 points) Suppose you are in a situation where you have placed a bid 2 > 0. If your opponent places a bid y = x + 0.01 > X, should you raise your bid to beat them? Explain, with reference to your model. Total for Question 3: 15 4. Suppose Homer is willing to pay $500 for a sofa, and Marge is willing to pay $300 for the same sofa. Both of them value not having a sofa at $0, and they can't share (Homer is large). (a) (5 points) Is it fair to say Homer values the sofa more than Marge? Why or why not? Explain. (b) (5 points) Is it efficient for Homer to get the sofa, and Marge not? What about the opposite arrangement? Explain. (c) (5 points) Now, suppose Homer and Marge both could exchange money (which they value), along with the sofa. Repeat (b) under this assumption. (Hint: What are the outcomes in this situation?) (d) (5 points) Can you see why, in a standard supply and demand model, we often just consider the willingness pay? How does it relate to your answer to (c), above? Total for Question 4: 20 5. Consider the example of a Prisoners' Dilemma, as described in Figure 1 Often, in real- world explanations the outcomes are not expressed in utility. Suppose these represented changes in wealth (e.g. so -0.50 means "lose 50 cents”). Player 2 X R X -0.50, -0.50 Player 1 +1.50, -1 R -1, +1.50 +1, +1 Figure 1: A Prisoner's Dilemma (a) (5 points) Suppose that both players value money. What would the game matrix look like, if they had an initial wealth of $100 and displayed diminishing marginal utility of wealth. You can pick an example, or argue in general. (b) (5 points) What if they were billionaires, with the same utility-of-wealth function as in (a). What is the difference? (C) (5 points) What could their payoffs look like if they were loss averse; i.e. they dislike losing money much more than gaining money. (d) (5 points) What could their payoffs look like if they were other regarding, that is they valued the outcomes of their "opponent” as themselves? (e) (5 points) What do these examples tell us about interpreting strategic situations in the real world? Total for Question 5: 25 6. (15 points) Thinking ahead to your group project, take a real-world situation that is your own, describe it in clear terms, then model it as a strategic game. 7. (5 points (bonus)) (*) In class, a typical student valued the St Petersberg game at about $2. Find a utility function that represents these preferences, and show how it does.