The document discusses several concepts related to sunk costs and rational decision making. It provides examples of how sunk costs can lead to irrational decisions through the sunk cost fallacy. Specifically, it discusses two hypothetical examples given by Richard Thaler where people decide to continue with plans or purchases even when it is no longer rational due to having already incurred some initial cost. The document also proposes models for incorporating sunk costs and transaction utility into rational decision making frameworks to better explain when the sunk cost fallacy occurs.
2. Axiom of Substitution
“If a person has two options, x and y, and s/he is indifferent
between x and y, then s/he should be indifferent between two
lotteries that have x and y as their price”.
One corollary of this particular axiom is something called the
cancellation principle.
If one has got two options, and s/he remove something identical
from those two options, s/he should still now have the same
preference between those two options as s/he had before.
3. • The independence over lotteries axiom.
• Let p be a probability, and X, Y, and Z be outcomes or lotteries over
outcomes.
• Independence says that if an individual prefers X to Y, s/he must also
prefer the lottery of X with probability p and Z with probability 1 – p
to the lottery of Y with probability p and Z with probability 1 – p.
4. Allis Paradox
• See the survey results first
1
A: Rs 1 million
B: 10% - win Rs. 2.5million
89% - win Rs 1 million
1% - win 0
2
A: 11% - win Rs 1 million
89% - win nothing
B: 10% - win Rs. 2.5million
90% - win 0
5. Transaction Utility and Behavioral Anomalies
• Transaction utility: the utility one receives for feeling one has
received greater value in a transaction than one has given away
in paying for the good. (Acquisition Utility: Utility from
consumption)
• This leads to three prominent anomalies: the sunk cost fallacy,
flat-rate bias, and reference-dependent preferences.
1. Sunk Cost Fallacy
• It occurs when one tries to recover sunk costs by continuing an
activity for which there is a negative return.
6. Typical Firm Problem
• Firms maximize profits given by:
• The firm chooses input quantity x* by taking the derivative to find the
solution condition where price times the slope of the production
function is equal to the cost of inputs:
max
x
pf x rx C
p f x x r
6
7. BUT… Fixed Costs may matters
• Consider the case of a consumer facing a two part tariff:
• Part 1: A fixed cost to access the good
• Part 2: Some amount for each piece
• Special Cases:
• Linear pricing (or charging a fixed amount for each unit) can be thought of as
a two part tariff with the fixed part equal to zero
• Flat Rate Pricing (consumers can consume as much as they like for a fixed fee)
is a two part tariff where the piece rate is zero
7
8. Two Part Tariff Problem for Rational Consumers
1 2
1 2
,
max ,
x x
U x x
0 1 1 1 2 2
ˆ
p x p x p x y
8
9. Purchasing Both Goods
• Shown by dashed BL and IC. Consumer’s utility is 𝑢 𝑥1
∗
, 𝑥2
∗
Not Purchasing the Two Part Tariff Good
• In this case, BL will shift up and consumer will purchase only x2. Consumer’s
utility is 𝑢 𝑥1
∗
, 𝑥2
∗
.
X1
X2
𝑥2
𝑢 0, 𝑥2
𝑢 𝑥1
∗
, 𝑥2
∗
𝑥1
∗
𝑥2
∗
X1
X2
𝑥2
𝑢 0, 𝑥2
𝑢 𝑥1
∗
, 𝑥2
∗
𝑥1
∗
𝑥2
∗
Consumer 1
Consumer 2
o If both the flat fee and the linear
price are positive, consumers
will only purchase two part tariff
good if doing so allows them to
obtain a higher level of utility
(Cons 2).
o For this, changing the fee can
alter the consumption of both
goods.
10. Figure 2.3 Utility Maximization with Flat Rate
Pricing
x1
x2
2 2
x y p
1 2
*, *
U U x x
2 0 2
x y p p
1 *
x
2 *
x
1 2
*, *
U U x x
1 2
*, *
U U x x
10
x1
x2
2 2
x y p
1 2
*, *
U U x x
2
0,
U U x
2 0 2
x y p p
2
x
11. • Suppose you plan to go out for a concert /
movie show / a match (think about
whatever you like the most). To avoid any
hassle, you purchased the ticks worth Rs.
1500 beforehand . On the day, just when
you were ready to leave for the event,
heavy rainfall started all of the sudden. It
looks like it is not going to stop sooner so
reaching the venue is quite tough, not
impossible though. What will you do?
• The bar of our interest, which represents sunk
cost is “Have Paid”
Class Response
12. Examples of Sunk Cost Fallacies
Richard Thaler, gave the following two hypothetical examples of the
sunk cost fallacy:
•“A family pays $40 for tickets to a basketball game to be played 60
miles from their home. On the day of the game there is a snowstorm.
They decide to go anyway, but note in passing that had the tickets
been given to them, they would have stayed home.”
12
13. Example 1: Theater Tickets and Pricing
Programs
Experiment (Arkes and Blumer):
•Design: Randomly offered 60 people different prices for theater
tickets. Some received a $2 discount. Others received a $7 discount.
•Results: Those who paid full price attended more plays.
Questions:
•What might explain their results?
•Why is this an example of sunk cost fallacy?
•What else could explain their results?
13
14. Example 2: The Concorde Paradox and Public
Policy
• Governments spent billions of dollars beginning in the 1960’s on the
supersonic Concorde. Only 14 were ever used.
• John McCain on why we should explore Mars…
“There’s too much invested there. There’s infrastructure that’s very
expensive and very extensive.”
Question:
• How is government decision making different from individual decision
making?
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15. Modeling the Sunk Cost fallacy
• To test for rationality, individuals choice of x1* should depend only on
their remaining budget y, or if the effect of the budget is small
relative to total consumption, then x1* should be independent of
wealth
• A behavioral model could insert the level of costs p0 as a parameter of
the optimal x1* such that
𝑑𝑥1
𝑑𝑝0
> 0
15
16. Thaler’s model of transaction utility
• Consumer Problem:
• Subject to:
• Where z represents the perception of a good deal
• Then consumers’ bliss point for x1 depends on the solution condition:
1 2 1 0 1 2 1 0
, , , , ,
U x x z x p u x x z x p
0 1 2 2
ˆ
p x p x y
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17. Various Explanations of Sunk Cost Fallacy
• Transition v/s Acquisition Utility
• Loss Aversion
• Mental Accounting
• Framing
