Problem Set #1 Spring 2022
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Problem Set #1
Spring 2022
Reminder: You should feel free to discuss the problems with your fellow students, and in fact are encouraged to do so. However, after discussing the problems, you must write out your answers on your own. Directly copying someone else’s problem set will be considered cheating.
You have plenty of time to complete this problem set, and so I will expect your answers to be clear and concise. It is suggested that you first work out your answers on scratch paper, and then write up (or type up) your answers in a manner that clearly identifies how to solve the problem while not including any extraneous information or discussions.
Question 1:
Suppose that Jennifer evaluates gambles according to prospect theory with a value function that has the three properties suggested by Kahneman & Tversky but with 
(
) = 
. On the other hand, Bruce evaluates gambles according to prospect theory with a value function that has the three properties suggested by Kahneman & Tversky and with a probability weighting function that has the properties of subcertainty, subproportionality, and overweighting of small probabilities. (a) If Jennifer chooses lottery ($2000
1;$900
8;$0
1) over lottery ($900
1), could she also choose lottery ($900
2;$0
8) over lottery ($2000
1;$0
9)? Could Bruce exhibit that pattern of choices? Explain your answers.
(b) If Jennifer chooses lottery ($0
4;$1500
2;$2100
4) over lottery ($1500
2;$1000
8), could she also choose lottery ($0
05;$1000
2;$2000
75) over lottery ($0
15;$2000
75;$2100
1)? Could Bruce exhibit that pattern of choices? Explain your answers.
Question 2
Consider a choice between lottery (
) and lottery (
1), where 
∈ (0
1) and 
= 0. (a) Alice is a risk-averse EU maximizer. How does her choice depend on 
and 
?
(b) Wally evaluates gambles according to prospect theory – and in particular has a value function and a probability-weighting function with the properties suggested by Kahneman & Tversky (1979). How does his choice depend on 
and 
?
Note: You need not provide precise mathematical analyses–rather, it is sufficient to provide an intuitive discussion of Alice’s and Wally’s behavior, and how their behavior depends on 
and 
.
Question 3:
Suppose that people behave according to Koszegi & Rabin’s (2006) model of reference-dependent utility. Preferences are defined over mugs (good 1) and money (good 2), where consumption utility is 
(
) = 
1 + 
2 , and gain-loss utility is
(
) = 
Consider the following modified endowment-effect experiment:
As usual, half the subjects are endowed with mugs (sellers) and half the subjects are shown mugs (choosers). Then, a fair coin is flipped, and if it comes up heads the money option is $4 and if it comes up tails the money option is $6. Finally, each subject decides whether she wants a mug or the money.
(a) First suppose that, when the experiment begins and the mugs are distributed, sellers immedi- ately expect to leave the experiment with a mug and no money, while choosers immediately expect to leave the experiment with no mug and no money – i.e., the reference lotteries are exogenously determined. Solve for the behavior of sellers and choosers as a function of 
, 
, and 
. Who is more likely to choose a mug, sellers or choosers?
(b) Suppose instead that there is enough time between the distribution of mugs and the coin flip that subjects can form expectations over outcomes and adapt to those expectations – that is, they follow a personal equilibrium.
(i) Solve for the set of pure-strategy personal equilibria for sellers and for choosers as a function of 
, 
, and 
.
(ii) Solve for the preferred personal equilibrium for sellers and for choosers as a function of 
, 
, and 
. Who is more likely to choose a mug, sellers or choosers?
Note: For both parts (a) and (b), you should express your answers in terms of a strategy – that is, a plan for what to do if the money option is $4 and what to do if the money option is $6.
Note: For both parts (a) and (b), please express parameter conditions in the form 
≥ __, 
≤ __, or __ ≤ 
≤ __ 
Question 4:
Consider a person with initial wealth 
who evaluates gambles according to the Koszegi-Rabin model with consumption utility 
(
) = 
and gain-loss utility
(
) = ½ 
if 
≥ 0
if 
≤ 0.
Suppose this person is asked between lottery ($
1) versus lottery ($
+ 
;$
1 − 
) for some 
∈ (0
1) and 
0.
(a) Solve for the preferred personal equilibrium (PPE) as a function of 
, and 
. (b) Solve for the choice-acclimating personal equilibrium (CPE) as a function of 
, and 
.
(c) This problem reveals something strange–what is it?
Question 5:
Suppose a person has initial wealth 
but faces an incremental gamble 
≡ (
1
; 
2 
1 − 
) with 
1 
2 ≥ 0 – in other words, 
1 and 
2 are increments to wealth. This person evaluates gambles according to the Koszegi-Rabin (AER 2007) model, with consumption utility 
(
) = 
and gain-loss utility
(
) = ½ 
if 
≥ 0
if 
≤ 0.
For all questions below, let’s define the certainty equivalent 
to be the amount such that, if
the person is offered a binary choice between keeping gamble 
vs. getting 
with certainty, she would be just indifferent.
(a) First, assume that the person’s reference lottery is just 
. Solve for her 
. How does your answer compare to the expected value of 
?
(b) Next, assume that the person’s reference lottery is 
+ 
. Solve for her 
. How does your answer compare to the expected value of 
?
(c) Third, assume that the person follows a choice-acclimating personal equilibrium (CPE). Solve for her 
. How does your answer compare to the expected value of 
? How does your answer compare to your answer in parts (a) and (b)? Discuss.
(Extra Credit) Finally, assume that the person follows a preferred personal equilibrium (PPE). Solve for her 
. How does your answer compare to your answer in parts (a), (b), and (c)?
2022-03-05