Ec 1030: Problem Set 7 Suggested Solutions
Spring 2017
Problem 1
Tomasz and David both have the following expected utility preferences over wealth:
EU (P ) =
n
X
pi v(wi − w∗ )
i=1
v(x) =


x
if x ≥ 0

4x
if x < 0
(a) Let’s consider Tomasz’s utility if he buys the stock. He pays a price p no matter what,
and then either receives $3300 or loses $300. We can write his expected utility as:
2
1
t
= v(3300 − p) + v(−300 − p)
EUbuy
3
3
Note that −300 − p < 0 for all p ≥ 0, so Tomasz will be in the loss domain of the
value function when the stock returns a loss. When the stock returns a gain though,
3300 − p can be positive or negative depending on p. We can quickly dismiss the case
where p > 3300 because Tomasz would never pay that much for this lottery. He would
be ensuring that he loses money regardless of the outcome! So we proceed assuming
p ≤ 3300.
1
2
t
EUbuy
= (3300 − p) + · 4 · (−300 − p)
3
3
8p
p
= 1100 − − 800 −
3
3
= 300 − 3p
1
If he does not buy the stock, there will be no change in his wealth, so his utility will
be 0. Tomasz will be willing to buy if:
t
t
EUbuy
> EUdon
0t
300 − 3p > 0
p < 100
For any price lower than $100, he will choose to buy the stock. Note: we have confirmed
our earlier assumption that p ≤ 3300.
(b) David already owns the stock, so if he decides to sell, he gets a price of p with certainty.
d
= v(p) = p
EUsell
If he decides to keep the stock, he gets the following expected utility:
1
2
d
EUkeep
= v(3300) + v(−300)
3
3
1
2
= (3300) + (−1200)
3
3
= 1100 − 800
= 300
So he will sell if p > 300.
(c) David’s selling price for the stock is much higher than the price Tomasz is willing to
pay to buy it. You can think of this as a type of endowment effect, but notice that it
is subtly yet importantly different than the endowment effect we saw with mugs (and
money). David and Tomasz have the same reference point here. Neither is endowed
with any more money than the other (and there are no mugs!). David is endowed with
2
stock, but stock does not appear in his utility function so he can’t experience a loss
when parting with it.
The key is that p enters as a gain with certainty for David, but as a loss for Tomasz
in the case where the stock pays out -$300. This loss then gets counted more heavily
because of loss aversion, making Tomasz less willing to buy the stock.
3