# Anton and Klara each have the reference-dependent preferences for

money outcomes c that we introduced in class. In particular, both have the following utility c +v(c − r) where v(x) = x for x ≥ 0, and v(x) = λx for x< 0. λ> 1 is a preference parameter, possibly different for the two, and r is the reference level which we assume is the status quo.

(a) (8 points) One day at lunch Ralf asks them each if they would take a single 50/50 lose $100 / gain $200 gamble. They both say no. What is the range of values of λ consistent with this answer?

(b) (13 points) Ralf offers them each n = 2 of these gambles played independently, and paid out together and immediately after lunch. Both Anton and Klara turn this offer down as well. Before doing so, Anton says "Give me a moment while I calculate what the implied aggregate bet is", and concludes silently that the newly offered bet is a ¼ chance of winning $400, ½ of winning $100, and ¼ of losing $200. But Klara declares "I am not going to figure that out! I am only turning this compound bet down because I would turn down each component bet individually. Why should I feel any different about the bets just because you are repeating them?" (Anton is broadly bracketing, Klara is narrow bracketing.) Now what is the range of values of λ consistent with each of their answers? (It is different for the two of them.)

(c) (14 points) Suppose that the next day, after they have talked about narrow bracketing in their P&E class, the three of them were eating lunch again. Klara suddenly blurts out "My investments pay me out either positive or negative $150! Moreover, the chance my investments will lose me $150 today is ½ and the chance that they'll gain me $150 is ½." Ralf then again offers Klara a single 50/50 gain $200-lose $100 bet. Klara now responds: "Yesterday I would have answered that question based on the outcome alone. But today I'm going to apply my preferences to the outcome from your proposed gamble integrated with my other changes in wealth today." Klara proceeds to calculate how she feels about the bet. Now what is the range of values of λ for which Klara would turn down Ralf's bet? Give a brief intuition for this result.

(d) (6 points) Carefully explain how this question is related to the discussion of narrow bracketing in class.