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程序代写案例-ECON3124

时间：2021-04-11

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

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.

学霸联盟

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

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.

学霸联盟