ECO674: Behavioral Economics
Punarjit Roychowdhury
Shiv Nadar University, Delhi NCR
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© Punarjit Roychowdhury
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 Paul Samuelson to colleague: “Would you take a 50-50 bet
where you win $200 if you win but lose $100 if you lose?”
 Colleague: “I won’t bet because I would feel that $100 loss
more than the $200 gain. But I will take you on if you promise to
let me make 100 such bets”
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 Paul Samuelson to colleague: “Would you take a 50-50 bet
where you win $200 if you win but lose $100 if you lose?”
 Colleague: “I won’t bet because I would feel that $100 loss
more than the $200 gain. But I will take you on if you promise to
let me make 100 such bets”
 This is surprising!
 Yes, probability of loss is much smaller with 100 gambles (less than
1%) than with 1 gamble (50%), but the potential magnitude of the
losses is much larger ($10,000 vs $100)
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 In general, this should not have been the case if the person’s
preference followed the traditional expected utility model
 In the expected utility model, preferences exhibit segmentation
independence, i.e., if a person is unwilling (willing) to take one
bet, he should be unwilling (willing) to take two or more
identical bets
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 In general, this should not have been the case if the person’s
preference followed the traditional expected utility model
 In the expected utility model, preferences exhibit segmentation
independence, i.e., if a person is unwilling (willing) to take one
bet, he should be unwilling (willing) to take two or more
identical bets
 Then, how can such behavior be explained?
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 Consider an individual with wealth w trying to maximize his
portfolio
 He/she maximizes the expected utility of investing in the safe
and risky assets will solve
max𝑤𝑥 𝐸 𝑢 𝑤 − 𝑤𝑧 + 𝑤𝑧 𝑧
 where wz is the amount of wealth invested in the risky asset, z is
random variable denoting value of the risky asset at the time of
sale
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 Let 𝜇 > 1 is the expected value of z
 The solution to the problem is
wz
  1


 2 RA
 RA is the coefficient of risk aversion (𝑅𝐴 = −𝑢"(𝑤)/𝑢′(𝑤)) and
the value 𝜎 2 = 𝐸 𝑧 − 𝜇
2
is the variance of z
 Higher RA, lower wz
 Higher 𝜎 2 (riskier asset), lower wz
 Higher 𝜇, higher wz
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 In the US, between 1871 and 1990 stocks (risky) have yielded
6.5% returns, while bonds (safe) have yield only 1%
 Then why did/do people still invest in bonds?
 Commonly explained by the notion that bonds are relatively
less riskier (really? what is if the issuing body goes bankrupt?)
 However if we use the solution to the portfolio problem with
relative risk aversion rate (𝑅𝑅 ) equal to 2 (note, 𝑅𝑅 =𝑤𝑅𝐴 ) and 𝜇
is 7, (recall 6.5% was the average value of stocks), then
wz 
3w
2
 Since the variance of stocks is generally below 3, wz/w > 1
meaning that no wealth should be allocated to bonds
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 In the US, between 1871 and 1990 stocks (risky) have yielded
6.5% returns, while bonds (safe) have yield only 1%
 Then why did/do people still invest in bonds?
 Commonly explained by the notion that bonds are relatively
less riskier (really? what is if the issuing body goes bankrupt?)
 However if we use the solution to the portfolio problem with
relative risk aversion rate (𝑅𝑅 ) equal to 2 (note, 𝑅𝑅 =𝑤𝑅𝐴 ) and 𝜇
is 7, (recall 6.5% was the average value of stocks), then
wz 
3w
2
 Since the variance of stocks is generally below 3, wz/w > 1
meaning that no wealth should be allocated to bonds
 Then why do still people invest in bonds?
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Examples 1 and 2 can be explained by a combination of loss
aversion and decision bracketing
Definition: Bracketing refers to the grouping of decisions
If a set of choices are made so as to take account of the impact of
each individual choice in the set on all other choices, we say they
are bracketed together
Example

If I choose to have a heavy breakfast, I might prefer to have a
lighter lunch

In this case, my decisions are bracketed together since when I
decide to have a heavy breakfast, I anticipate the need for a
lighter lunch  I decide breakfast and lunch as a bundle
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
A decision is narrowly bracketed if it is bracketed together
with only a small set of decisions

A decision is broadly bracketed if it is bracketed together
with a large set of decisions
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 Suppose an individual has the following value function
if
 x
v  x  
2.5 x if
x0
x  0.
 Then expected utility of the gamble is:
𝐸 𝑣 𝑥
= 0.5𝑣 200 + 0.5𝑣 −100 = 100 − 125 = −25
 Thus, the person would turn down the gamble since it would
result in a negative value (potential loss of $100 looms larger
than the potential gain of $200)
 Also, if the person were forced to take one gamble, and then asked
if she would take a second, the calculation would be the same
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 Instead, suppose the person decides to play two of the gambles
jointly, thus leading to a more broadly bracketed choice
 In this case, the expected value is
𝐸 𝑣 2𝑥 = 0.25𝑣 200 + 200 + 0.25𝑣 200 − 100
+0.25𝑣 −100 + 200 + 0.25𝑣 −100 − 100
= 100 + 50 − 125 = 25
 Thus the person would choose to take two gambles jointly,
although she would not choose to take any single gamble
individually
 The same would hold true for 100 of these gambles
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 The equity premium puzzle might also be due to loss aversion
and narrowly bracketed investment decisions
 Length of the planning horizon in investment affects the
variance or the variation in the return of the portfolio
 If the planning horizon is short (e.g., a year), for risky assets,
the variance is much larger relative to the average return
 In fact, over a short horizon, 𝜎 2 could be quite high compared
to 𝜇
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 This could mean that
𝑤𝑧
𝑤
<1
 Thus, if we assume investors bracket their portfolios on an
annual basis (i.e., investors narrowly bracket their portfolios),
then it might be sensible for investors to invest in bonds
 Each year when the investors evaluate the return, the variation and
losses lead them to purchase bonds and ignore their longer
planning horizon
 Instead, if investors don’t evaluate return each years but checks at
the end of every 5 or 10 years (i.e., they broadly bracket their
portfolios and try to adhere to their planning horizon), they
purchase less bonds
 Thaler and others show that the more often we evaluate our
portfolio, more narrowly we bracket, and more we tend to move
more towards safe assets
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Experiment: How does time bracketing affect investment decisions
(Thaler et al., 1997)?
 A group of 80 participants were asked to allocate 100 units of
portfolio between two investments A and B, with B being riskier
and providing much higher mean return than A
 Participants randomly placed into treatments
 Design: Participants make investment choice (invest in A or B) for
200 periods under different conditions:
 Condition 1, participants made investment decision for each period
separately after being able to observe the returns (make decision
200 times)
 Condition 2, participants made decisions for 25 periods at a time
(make decision 8 times)
 In Condition 3 they made them in decisions for 40 periods (make
decision 5 times)
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 All the subjects in all treatments were viewing the same return
history in any decision period
 Result: The fewer the decisions they made, the more likely the
participants were to invest in the risky fund B
 Narrower time bracketing, higher investment in safer asset
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The notion that people would choose long-term investments
(bonds or stocks) based on short term priorities is closely
related to the notion of melioration
Definition: Melioration is a concept from experimental
psychology whereby an individual chooses the option that
yields the highest utility in the current period, ignoring the
impacts of their decision on choices and experiences available
in future periods

Decision bracketing is one explanation of melioration

Investor makes investments that are attractive in the near term,
ignoring the properties of the potential investments in the longer
term
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 So far we have discussed the possibility that bracketing
decisions by time sequence can affect investment choices
 But it is also possible to bracket decisions by individual
investments in the portfolio
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 A broad bracketed investor should maximize her portfolio by
𝑛
max
𝑥1 ,→𝑥𝑛
𝐸 𝐶 + 𝑣 ෍ 𝑥𝑖 𝑧𝑖
𝑖=1
subject to the constraint
n
C  w   xi
i 1
where x represents investment in the asset, z is the random
return, v is the loss averse value function with reference point
equal to amount invested (Σ𝑥𝑖 ), C is the money spent on
consumption, and w is wealth
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 A narrow bracketed investor, on the other hand, would
maximize:
𝑛
max
𝑥1 ,→𝑥𝑛
𝐸 𝐶 + ෍ 𝑣(𝑥𝑖 𝑧𝑖 )
𝑖=1
 Each investment now is evaluated by itself in the value function
with a reference point of 𝑥𝑖
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 If we assume a smooth value function, the solution to the broad
bracketed portfolio choices is:
xi 
i

RA ii

1 
   x j ij 
v ''  j i

 ii
 While the solution to the narrow bracketed portfolio is:
xi 
i
RA ii
1
 
v ''
 
 ii
 Where v” is the slope of the marginal utility function (negative
over gains, positive over losses)
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 These terms differ in that the narrow bracketing ignores the
covariance terms σij (covariance of return from investment i and
j)
 Positive (negative) covariance indicates that 𝑧𝑖 and 𝑧𝑗 are positively
(negatively) correlated
 Implication of this difference?
 When investment choices are narrowly bracketed, the investor will
fail to recognize when the return of investment options are related
 Thus he/she cannot take advantage of diversification (i.e., invest in
different types of asset, some safe and some risky) but rather seeks
that all investments will have minimal risk of return (and possibly
minimal return as well)
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 Observe if covariance is negative (positive), a broad-bracket
investor will invest more (less) in the given asset
 However, since the narrow bracket investor ignores this term,
when covariance is positive, she might end up over-investing in
the assets and thereby end up taking much larger risk without
realizing!
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Experiment: (Read and Lowenstein)
Its Halloween and you were trick-or-treating
You visit three houses that happen to offer exactly the same three
types of candy bars
At the first house, you are allowed to take one candy bar, and you
choose your favorite
At the second house, you are also allowed to choose one, and you
decide to take your favorite
At the third house, you are allowed to choose two, and you decide
to choose one of your favorite and one of the others
What would you do?
Result: Most people pick the same (favorite) candy bar from the
first two houses, but pick two different types at the third
This might be explained by bracketing
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 When people bracket broadly, the person values variety much
more than when bracketing narrowly
 The subjects do not bracket the offered candies in first and
second houses together, i.e., they do not broadly bracket, and
thus they do not value variety
 When the subjects are forced to broadly bracket (in the third
house), they end up choosing variety  diversification bias
 Diversification bias actually can make people worse off
 In this case, diversification bias leads to subjects to not choose both
favorite candy bars; had they chosen both favorite candy bars they
would possibly be better off
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 Evident in many other contexts
 People tend to spread their retirement investments equally
among all possible options in a retirement program, even if
some are relatively unattractive
 In this case, people seek variety when broadly bracketing
choices, even if it might make them worse off when
consumption takes place
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 Cigarettes provide some immediate pleasure or high, making
them highly attractive at any one point in time
 However, smoking may reduce the pleasure from anything that one
has to do for long periods
 Moreover, cigarettes have a diminishing effect on the body as the
smoker builds up a resistance to nicotine implying that someone
who has been smoking for years do not get the same high or
pleasure from nicotine in a single cigarette that a new smoker
receives
 Thus, although there might be some short term ‘benefit’ of
smoking, in long term the loss in utility due to smoking is massive
 Then why do people keep on smoking? What can explain
addiction?
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 Bracketing may play a role
 At any one time, a smoker might consider the high of smoking
(and perhaps other immediate social advantages)
 But narrowly bracketing the choice to the single cigarette
ignore the future impact of the cigarette on the utility of all
future activities
 Thus narrow bracketing could lead to melioration: a focus on
the now at the expense of long term goals
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 Consider a decision problem between choosing to consume
cigarettes x and other consumption y
 A rational consumer would behave as if she has to solve
max u1  x1 , y1   u2  x2 , y2 , x1 
x1 , x2 , y1 y2
 subject to
px  x1  x2   p y  y1  y2   w
 However, narrow bracketed consumer on the other hand would
solve
 max u  x , y    max u  x , y , x 
x1 , y1
1
1
1
x2 , y2
2
2
2
1
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 The narrow bracketed consumer will thus ignore all the future
consequences of her consumption in period 1 (as a smoker
does)
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 The condition for the solution of the narrowly bracketed
individual finds the the point where marginal utility of
consumption divided by the price of consumption is equal for
cigarettes and other consumption in the first period but this
ignores the added cost that cigarettes impose on future
consumption
 If cigarettes reduce the utility of future consumption, then this
cost should be considered
 Thus the proper solution (solution to the first equation) equates
the marginal consumption of cigarettes in period 1 plus the
marginal impact of period 1 cigarette consumption on period 2
utility, all divided by the price of cigarette consumption with
the marginal utility consumption of other items divided by their
price
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 By ignoring the future cost of cigarettes, the smoker consumes
more than optimal number of cigarettes in the first period
 Further, if consumption of cigarettes in the first period makes
consumption of cigarette in the second period more attractive
than consumption of other goods in the second period
(suppose the other goods in sports; cigarette consumption may
make the person extremely unfit to participate in sports in the
second period), then the person also overconsumes cigarettes
in the second period
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 Suppose
𝑢1 𝑥1 , 𝑦1 = 𝑢𝑥1 𝑥1 + 𝑘𝑦1
 If the smoker displays melioration or narrow bracketing,
cigarette consumption in period 1 would be determined by
′
′
𝑢𝑥1
= 𝑢𝑦1
=k
 If that was not the case, period 1 consumption of cigarette
would have been determined by
′
′
𝑢𝑥1
+ 𝑢𝑥2
=k
′
 If 𝑢𝑥2
<0, then the optimal level of cigarette consumption
(without bracketing),𝑥1∗ would be lower than the melioration
optimum (𝑥1𝑀 )
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u1i  
u2 '  x1 < 0
u1 y '  y1 
k
′
𝑢𝑥1
x1 *
+
′
𝑢𝑥2
x1M
u1x '( x1 )
x1
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 People often use narrow bracketing to achieve large goals
 Narrow bracketing could be used as a self control device
 May be easiest to start a difficult task with small steps
 Broad bracketing can make it difficult to verify progress on the
goal
 If one’s goal is to save $2400 annually, it may be easier to overspend
in the interim because “there is so much time left until the goal must
be reached”; on the other hand, if one’s goal is to have $200 per
month, one quickly knows whether one is one track
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 A company has 8 divisions. Each had to choose between a safe
project and a risky project. Each division had a budget of $20
million. Each risky project would gain $20 million with
probability ½ and would lose $10 million with probability ½
 Each division has a budget of $20 million. So each division
manager chooses the safe project instead.
 The president approves of these choices
 The CEO fires the president
 Why? Is the CEO justified?
 Does bracketing play a role in this story?
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