Uncertainty and Risk - DARP
Uncertainty and Risk - DARP Uncertainty and Risk - DARP
Microeconomics CHAPTER 8. UNCERTAINTY AND RISK Exercise 8.4 Suppose you are asked to choose between two lotteries. In one case the choice is between P 1 and P 2 ;and in the other case the choice o¤ered is between P 3 and P 4 , as speci…ed below: P 1 : $1; 000; 000 with probability 1 8 < P 2 : P 3 : P 4 : $5; 000; 000 $1; 000; 000 : $0 $5; 000; 000 $0 $1; 000; 000 $0 with probability 0.1 with probability 0.89 with probability 0.01 with probability 0.1 with probability 0.9 with probability 0.11 with probability 0.89 It is often the case that people prefer P 1 to P 2 and then also prefer P 3 to P 4 . Show that these preferences violate the independence axiom. Outline Answer: Let there be only three possible states of the world: red, blue and green, with probabilities 0.01, 0.10, 0.89 respectively. Then the payo¤s in the four prospects can be written red blue green P 1 1 1 1 P 2 0 5 1 P 3 0 5 0 P 4 1 1 0 where all the entries in the table are in millions of dollars. Note that P 1 and P 2 have the same payo¤ in the green state; P 3 and P 4 form a similar pair, except that the payo¤ in the green state is 0. Axiom 8.2 states that if P 1 is preferred to P 2 than any other similar pair of prospects (1; 1; z) and (0; 5; z) ought also to be ranked in the same order, for arbitrary z: but this would imply that P 4 is preferred to P 3 , the opposite of the preferences as stated. Note also that if the preferences had been such that P 4 was preferred to P 3 then the independence axiom would imply that P 1 was preferred to P 2 . cFrank Cowell 2006 118
Microeconomics Exercise 8.5 This is an example to illustrate disappointment. Suppose the payo¤s are as follows x 00 weekend for two in your favourite holiday location x 0 book of photographs of the same location x …sh-and-chip supper Your preferences under certainty are x 00 x 0 x . Now consider the following two prospects 8 < x 00 P 1 : : x 0 with probability 0 P 2 : 8 < with probability 0:99 x with probability 0:01 x 00 with probability 0:99 x 0 with probability 0:01 : x with probability 0 Suppose a person expresses a preference for P 1 over P 2 . Brie‡y explain why this might be the case in practice. Which of the three axioms State Irrelevance, Independence, Revealed Likelihood, is violated by such preferences? Outline Answer: It is possible that, given the information that the …rst event (with payo¤ x 00 ) has not happened you would then prefer x to x 0 : photographs of your favourite holiday spot may be too painful once you know that the holiday is not going to happen. So you may prefer P 1 over P 2 . These preferences violate the independence axiom. To see this, note that, by the revealed likelihood axiom, since x 0 is strictly preferred to x , it must be the case that P2 0 is strictly preferred to P1, 0 where P 0 1 : P 0 2 : x 0 with probability 0 x with probability 1 x 0 with probability 0:01 x with probability 0:99 But P1 0 and P2 0 can be written equivalently as 8 < x P1 0 : x 0 with probability 0 : P 0 2 : 8 < with probability 0:99 x with probability 0:01 x with probability 0:99 x 0 with probability 0:01 : x with probability 0 By the independence axiom if P 0 2 is strictly preferred to P 0 1, then P 2 must be strictly preferred to P 1 . cFrank Cowell 2006 119
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Microeconomics CHAPTER 8. UNCERTAINTY AND RISK<br />
Exercise 8.4 Suppose you are asked to choose between two lotteries. In one<br />
case the choice is between P 1 <strong>and</strong> P 2 ;<strong>and</strong> in the other case the choice o¤ered is<br />
between P 3 <strong>and</strong> P 4 , as speci…ed below:<br />
P 1 : $1; 000; 000 with probability 1<br />
8<br />
<<br />
P 2 :<br />
P 3 :<br />
P 4 :<br />
$5; 000; 000<br />
$1; 000; 000<br />
:<br />
$0<br />
$5; 000; 000<br />
$0<br />
$1; 000; 000<br />
$0<br />
with probability 0.1<br />
with probability 0.89<br />
with probability 0.01<br />
with probability 0.1<br />
with probability 0.9<br />
with probability 0.11<br />
with probability 0.89<br />
It is often the case that people prefer P 1 to P 2 <strong>and</strong> then also prefer P 3 to P 4 .<br />
Show that these preferences violate the independence axiom.<br />
Outline Answer:<br />
Let there be only three possible states of the world: red, blue <strong>and</strong> green,<br />
with probabilities 0.01, 0.10, 0.89 respectively. Then the payo¤s in the four<br />
prospects can be written<br />
red blue green<br />
P 1 1 1 1<br />
P 2 0 5 1<br />
P 3 0 5 0<br />
P 4 1 1 0<br />
where all the entries in the table are in millions of dollars. Note that P 1 <strong>and</strong> P 2<br />
have the same payo¤ in the green state; P 3 <strong>and</strong> P 4 form a similar pair, except<br />
that the payo¤ in the green state is 0. Axiom 8.2 states that if P 1 is preferred<br />
to P 2 than any other similar pair of prospects (1; 1; z) <strong>and</strong> (0; 5; z) ought also<br />
to be ranked in the same order, for arbitrary z: but this would imply that P 4<br />
is preferred to P 3 , the opposite of the preferences as stated.<br />
Note also that if the preferences had been such that P 4 was preferred to P 3<br />
then the independence axiom would imply that P 1 was preferred to P 2 .<br />
cFrank Cowell 2006 118
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