Problem set II (Reference Dependence)
Instructions:
i) When in doubt, round numeric responses to two decimal places
ii) It is okay to discuss the course material with your peers, but please solve
all the questions independently and write all your responses in your own words.
On the coversheet, please write and sign the following honor pledge: “I pledge
on my honor that I have not given or received any unauthorized assistance on
this problem set.”
iii) Please submit your assignment by the deadline of March 20, 11:59
PM. Late assignments would not be accepted.
Problems:
For problems (1)-(2)-(3), assume that the utility of consuming c1 mugs and c2
money is
m((c1, c2)) = c1 + c2
1) Show that if the consumer expects to buy at price p with probability
q ∈ [0, 1],
i) the utility from buying is 1 − p + (1 − q)η(1 − λp), and her utility if she does
not buy is qη(p − λ).
ii) She is indifferent between buying and not buying if
q =
(1 + ηλ)p − (1 + η)
η(λ − 1)(p + 1)
1
2) Under λ = 2, η = 1, in the notes, we have derived pmax =
1+ηλ
1+η =
3
2
,
pmin =
1+η
1+ηλ =
2
3
. Take any 2
3 < p < 3
2
. Show that at such a price, there are
two Personal Equilibria.
Equilibrium 1: If you start by expecting to buy at price p, then it is your best
response to indeed buy the mugs.
Equilibrium 2: If you start by not expecting to buy at price p, then it is your
best response to not buy the mugs.
What is the utility at each equilibrium outcome? Which outcome has higher
utility? (your answer should depend on whether p ≥ 1 or p ≤ 1)
3) Suppose in an experiment with 2 subjects, one is randomly assigned a mug
(Seller), and the other is not (Buyer). Buyer and seller are told that with
probability, p, exchange will be forced at price x, and with probability 1 − p
they have the autonomy to decide..
If Seller doesn’t sell when under autonomy, they face the stochastic reference
point p
|{z}
forced
(0, x)
| {z }
no mug,$x
+ (1 − p)
| {z }
autonomy
(1, 0)
| {z }
mug,no money
.
Show that the reference-dependent utility of not selling when under autonomy,
with this stochastic reference point is
px + (1=p)
| {z }
utility from forced exchange
+p(1=p)η(1=λ)(1 + x)
If Seller instead does sell when under autonomy, they get the certain outcome
(0, x)
| {z }
no mug,$x
.
Show that the reference-dependent utility of selling given the stochastic
reference point, is x + (1=p)η(x=λ).
Take η = 1, λ = 2 to simplify. Write down the inequality condition for not
selling under autonomy to be a personal equilibrium. Solve for x to show that
the maximum price S¯
p at which the seller doesn’t want to sell the mug in
equilibrium is 3−p
2+p
.
2