Economics 1030

Economics 1030
Problem Set 2 Solutions
Question 
Retirement Savings - Decisions and Defaults
(a) Describe the difference between the two treatments in Chart 1. What explanation does
standard economics give for the difference between the two treatments? What explanation
does behavioral economics offer?
The Active Decision group was forced to make an active choice - choose to not participate,
or choose to participate and pick an allocation and savings rate. Doing nothing was not
an option. The Standard Enrollment group could choose to do nothing, resulting in nonparticipation, or it could fill out and submit a form to opt in to the program. Standard
economics says any difference must be due to the costs related to filling out the form, which
intuitively seems too small of a cost to justify such a large difference between the groups.
Behavioral economics explains the difference through default bias - people have a strong
tendency to choose the standard option.
(b) In Chart 2, why do you think the non-automatic enrollment groups show a steady
rise in participation as company tenure increases. Give two reasons why this would be the
case. Why does company tenure (beyond 6 months) have no impact on the participation
rate of the automatic-enrollment group?
The non-automatic enrollment group slowly increases its participation because there are
strong benefits to doing so, such as employer matching of contributions. As employees
get older, and closer to retirement, and as they become more experienced and earn higher
paychecks, these benefits become more valuable. Also, more time with the company increases
the likelihood that they will hear about the benefits of participation from their co-workers.
With the automatic-enrollment group, the 80% level of participation likely reflects the
total number of people for whom it is optimal to participate. The remaining 20% may be
credit-constrained or very impatient, and would prefer not to contribute under the available
policies. (If this is the breakdown of the overall population, we would expect the nonautomatic group’s participation to level off at 80% as tenure becomes large.)
(c) Both the Active-Decision and the Automatic-Enrollment treatments lead to a participation rate near 80%. Does that mean that there is no difference between the two? Suppose
you are advising a firm considering implementing one of these options. List one advantage
and one disadvantage for each approach. (Hint: In the U.S., the average employee changes
jobs approximately once every seven years.)
Active Decision: Employees may value free will and the ability to make their own decision.
However, this program has a lower initial level of enrollment, which makes a big difference
for workers who switch jobs often.
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Automatic Enrollment: Paperwork costs are lower for employees who choose the default,
and participation rates jump to 80% very quickly. However, the default savings rate (2%)
and allocation (money market fund) are unlikely to be optimal for most employees, and with
default bias, many of them are likely to keep this allocation.
Question 
Prospect Theory vs. Expected Utility Theory
Suppose that Peter has prospect-theory utility, while Emily is an expected-utility maximizer. They both have a current wealth of W0√= $1000. Peter’s value function√for gaining or
receiving an amount of dollars X is uP (X) = X if X > 0, and uP (X) = −2 X for X < 0.
Peter’s probability weighting function is π(p) = 21 + 4(p − 12 )3 . (Hint: with this formula,
we know that π(p) + π(1 − p) = 1 for all values of p). Emily’s
utility function is based on
√
her post-gamble wealth W = W0 + X, with uE (W ) = W . She calculates probabilities
accurately.
Consider the following gambles, written in the notation, (x1 , p1 ; x2 , p2 ; ...xn , pn ) where the
gambler wins an amount xi with probability pi .
A = (100, 1)
B = (160, 2/3; 0,1/3)
C = (100, 3/10; 0, 7/10)
D = (160, 2/10; 0, 8/10)
(a) Without doing any math, which gamble do you prefer between A and B? Which do
you prefer between C and D? Explain your reasoning.
Answers will vary, but should be based on expected values and perceived levels of risk.
Note that A and C have lower expected values, but are also less risky than B and D,
respectively.
(b) What is Peter’s preference ordering over these gambles? What is Emily’s? Are they
the same or different?
Peter: π(2/3) = 0.52, π(1/3) = 0.48; π(3/10) = 0.47, π(7/10) = 0.53; π(2/10) = 0.39, π(8/10) =
0.61
uP (0) = 0, uP (100) = 10, uP (160) = 12.65
Therefore: EuP (A) = 10, EuP (B) = 12.65∗0.52 = 6.6; EuP (C) = 0.47∗10 = 4.7, EuP (D) =
0.39 ∗ 12.65 = 4.9
Emily: uE (1000) = 31.62, uE (1100) = 33.17, uE (1160) = 34.06
Therefore: EuE (A) = 33.17, EuE (B) = (2/3) ∗ 34.06 + (1/3) ∗ 31.62 = 33.25
And: EuE (C) = 0.3∗33.17+0.7∗31.62 = 32.09, EuE (D) = 0.2∗34.06+0.8∗31.62 = 32.11
Peter prefers A over B and D over C, while Emily prefers B over A and D over C.
(c) What amount of sure money (gain or loss) would make Peter just as well-off as a
50-50 gamble between a gain of $50 and a loss of $50? What amount would make Emily
indifferent?
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p
√
√ (50) + 0.5 −2 50 = −3.54 = −2 X; X = $3.125
p
√ √
Emily: 0.5
(1050) + 0.5 950 = 31.613 = 1000 − X; X = $0.63
Peter would be indifferent with a loss of $3.125, while Emily would be indifferent with a
loss of $0.63.
Peter: 0.5
Question 
Applications of Prospect Theory
For each of the following anecdotes, briefly explain:
1. Why is the agents’ behavior inconsistent with expected utility theory?
2. Why is the agents’ behavior consistent with prospect theory?
3. Is there any way in which the agents’ behavior could be reconciled with expected utility
theory?
There are many correct answers. The responses below are meant to serve as good examples, but are not comprehensive.
(a) Extended warranties: The price to extend the standard 2-year warranty by an extra
year costs $100 for a laptop worth $1000. The cumulative malfunction rates for laptops by
this manufacturer are 11% through two years, and 18% through three years. Some people
choose to purchase this extended warranty service.
This behavior is not strictly inconsistent with EU theory, but the degree of risk aversion
is quite large: paying $100 now to avoid a loss with an expected value of $70, discounted by
at least two years. If relative risk aversion is extremely high, at a value of 10, and there was
no discounting or inflation, a person with a low wealth of only $25000 would still only pay
$85 for this insurance.
Under Prospect Theory, loss aversion and over-weighting low-probability events combine
to explain this phenomenon.
One possible reconciliation is that some laptop owners use their laptops much more often,
and under much more demanding conditions. In this case, the only people who buy insurance
are the ones with a much higher conditional probability of malfunction, and the price would
be reasonable for them.
(b) Data from a brokerage firm about all the purchases and sales of a large sample of
individual investors shows the “disposition effect”. Investors hold “losing” stocks a median
of 124 days and hold “winning” stocks only 104 days. That is, people hold on to stocks that
have lost value for too long and are too eager to sell stocks that have risen in value.
As long as stock returns are unpredictable an expected utility maximizer should not be
more likely to sell a losing stock than she is to sell a winning one.
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Prospect theory: If the reference point of an investor is the purchase price of a particular
share, once the stock appreciates the investor will evaluate the future returns of the stock
in more risk-averse region of her value function and thus exhibit more risk aversion. This
could push her to decline a gamble she had accepted before. On the other hand, if the stock
price declines, she will evaluate the future payoffs on a more risk-seeking region of her value
function. Then, she will continue to hold the stock.
If investors believed that winning stocks are likely to decline in price and losing stocks
are likely to increase, then their actions would be justified. However, mean reversion in stock
returns occurs on much longer horizons, over 5 to 10 years. Also, they could sell winners to
rebalance their portfolio because the transaction costs of trading at lower prices are relatively
higher. However, Odean (1997) shows that these motives do not make investor behavior
consistent with expected utility. Finally, if investors are trading on private information that
has not yet been revealed to the market, they would want to sell winning stocks that get
close to their private price-target, but hold on to losing stocks that they view as undervalued.
(c) A popular strategy when gambling in a game of roulette is the “Double Down Strategy:” Pick either red or black and bet on whichever color you choose. If you don’t win in
the first game, then double your bet. Repeat this method until your color hits.
The trick is to think of each gamble as a new decision. This strategy asks you to increase
the riskiness of your bet as you lose money, but all standard utility functions assume that
absolute risk aversion is either constant or increasing as income falls. Also, like all casino
games, roulette is a gamble with a negative expected value.
Prospect theory can explain this finding through the fact that the value function is riskseeking in the region of losses - if the gambler views each set of linked gambles as being
linked to the same reference frame, then he will want to take more risk as he loses money.
If a player has access to an infinite line of credit and there are no maximum bets at
the casino, and the game can be played with very low time costs, then this strategy does
generate riskless profits. With sufficiently low risk aversion, it is possible for this approach
to be rational when the assumption of an infinite credit line is relaxed.
(d) Lotteries are widely popular, and per-capita spending on lotteries is higher for lowincome households.
Again, this is a situation which implies that risk aversion is decreasing as income falls,
which is at odds with standard utility functions. Further, this is a case where people take a
gamble with a negative expected return.
Under Prospect Theory, people over-weight low-probability events. Further, if we assume
that low-income households have a reference level above their income, then they are in the
risk-seeking range of the value function.
The question of general popularity can be answered by appealing to a very strong complementarity between high levels of wealth and leisure time, especially in younger people.
Normally, increasing wealth requires one to sacrifice leisure in favor of work, so it is possible
that winning the lottery is much more valuable than the simple monetary award, particularly
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during the early stages of adulthood. One possible explanation for low-income households
is to argue that the distribution of risk aversion is not uniform across all income groups.
It is conceivable that risk-tolerance is correlated with characteristics like impatience, which
lead to lower savings rates and lower levels of upward mobility. Over time, most of the risktolerant and impatient population would naturally end up in the lower part of the income
distribution.
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