Behavioural Economics ECON3124
Problem Set 3

Due 6pm (AEST), 16 April 2021

1. [30pts] Explain each of the following findings/curiosities briefly using concepts from the
Probability Judgments and Making It Up parts of the course.

(a) Lottery bettors are less likely to choose numbers that have won in recent weeks than
numbers that have not won in recent weeks.

(b) Residents of New Mexico tend to overestimate the number of deaths in the US per year
caused by dehydration, and underestimate the number caused by hypothermia. Residents of
Minnesota make the opposite mistake.

(c) As a result of being on the dollar menu, the price of 2 apple pies at McDonald’s is \$1.00.
At some McDonald’s restaurants, the option to buy one apple pie for \$.95 is also available.
Nobody buys just one apple pie at these restaurants. Explain why the restaurants might still
offer this option.

(d) Williams-Sonoma is a mail-order business. It used to sell two kinds of breadmakers,
lower-end (\$200) and higher-end (\$275). They then decided to add an even higher-end
breadmaker (\$429) to their product line, hoping to sell more units. As it turned out,
almost nobody purchased this new breadmaker. Explain why Williams-Sonoma decided to
keep carrying the \$429 breadmaker at a loss.

(e) When a mammogram returns a positive result (that is, suggests breast cancer), the patient
undergoes a more thorough test for breast cancer. These second tests often reveal that the
mammogram was a false positive, and cancer is actually not present. The patient is then told
that the mammogram was essentially a fluke, and that she is completely fine. But even after
being told that they are fine, women who have had false positive mammograms worry more
about breast cancer, and believe they are more likely to get breast cancer, than women whose
mammogram was normal.

(f) In a study, students agreed to fill out a short questionnaire for \$1.50. At the end of the
survey, respondents were offered a chance to exchange their \$1.50 for a gift. Some students
could exchange the money for a metal zebra pen, and some students could exchange
the money for their choice of a metal zebra pen or two plastic pilot pens. Among the students
who had two options for exchange, 53 percent chose to keep their money. Of the students
who had only one option, 25 percent chose to keep their money.

2. [30pts] 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 money and is her reference point in
money. Her reference point is her current wealth, normalized to zero. The function v satisfies
() = for > 0 and () = 2 for ≤ 0.
(a) Show that the investor would reject a fifty-fifty gain \$160 or 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. 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 an additional independent fifty-fifty gain \$160 or lose \$100
gamble.
[Note: In fact, with a mere \$10,000 invested in the stock market, there is a greater than 85%
chance that one's wealth swings by more than \$100 in a single day. So many people face far
more financial risk than the above.]

(c) What is the minimum gain \$x such that the above investor with stocks accepts a 50-50
gain \$x or lose \$100 gamble?

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

3. [40pts] 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

(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.