Behavioural Economics ECON5324
Problem Set 3

Due 18:00 (Sydney time), 8 April 2022




1. [40pts] This question asks you to think about narrow bracketing and risky choice using
actual reference-dependent utility functions from earlier in the class. Suppose an investor has
a utility function given by ( − ), where is the money she has at the end of the day and
is her reference point in money. Her reference point is her wealth at the start of the day,
normalized to zero. The function v satisfies () = for > 0 and () = 2 for ≤ 0.

(a) Show that the investor would reject a 40% gain $250 or 60% lose $100 gamble, but would
take two independent plays of the same gamble. Explain the intuition behind the result.

(b) Now suppose that the investor is already facing some contemporaneous risk; for example,
she might own stocks that could go up or down today (and she cannot sell these stocks
today). Specifically, she has an equal probability of losing $100 or gaining $100 in her
financial investments today. Show that in this case, she would accept a single, independent
40% gain $250 or 60% lose $100 gamble.

(c) What is the minimum gain $x such that the above investor with stocks accepts a 40-60
gain $x or lose $100 gamble?

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


3. [60pts] Consider the “Freddy” model of the representativeness heuristic from
class, and let N = 8. Suppose Freddy observes quarters of performance by mutual-fund
manager Helga. Helga may be skilled, mediocre, or unskilled. A skilled mutual-fund manager
has a 3/4 chance of beating the market each quarter, a mediocre manager has a 1/2 chance of
beating the market each quarter, and an unskilled manager has a 1/4 chance of beating the
market each quarter (and Freddy knows all this). In reality, the performance of a manager is
independent from quarter to quarter.

(a) Suppose first that Freddy thinks Helga is mediocre. What does Freddy think is the
probability that Helga beats the market in the first quarter? Suppose that she does actually
beat the market in the first quarter. What does Freddy think is the probability she does it
again? Suppose that she beats the market again. What does Freddy think is the probability
that she will do so a third time?

(b) How do the three probabilities in part (a) compare to their true probabilities? What
phenomenon does this reflect?

(c) Now suppose that Freddy does not know whether Helga is skilled, mediocre, or unskilled.
He has just observed three consecutive quarters of below-market performance by Helga. Can
he conclude which type of manager Helga is? Can he rule out any type? Explain
the intuition.

(d) How many more quarters of below-market performance does Freddy
need to observe to be sure of Helga’s type?

(e) This part asks you to derive what Freddy concludes about the proportion of skilled,
mediocre, and unskilled managers in the population when he observes the performance of a
large sample of mutual-fund managers over two quarters. Suppose that in reality,
all managers are mediocre.

i. What proportion of managers will have two above-market performances? Two
below-market performances? Mixed performances? This is what Freddy observes.

ii. Suppose Freddy thought that the proportion of skilled, mediocre, and unskilled
managers in the population was ̃, 1−2̃, and ̃, respectively. What does Freddy
expect should be the proportion of managers who show two above-market
performances in a row?

iii. Given your answers to the previous two parts, what does Freddy deduce is the
proportion ̃ of skilled managers in the population? What does someone who does
not believe in the Law of Small Numbers deduce is the proportion of skilled managers
in the population? Give an intuition for your answer.

(f) Explain intuitively how part (e) relates to the difficulty of explaining to a basketball fan
that there is no such thing as a hot hand.