Uncertainty and Risk - DARP
Uncertainty and Risk - DARP
Uncertainty and Risk - DARP
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Microeconomics<br />
Exercise 8.5 This is an example to illustrate disappointment. Suppose the<br />
payo¤s are as follows<br />
x 00 weekend for two in your favourite holiday location<br />
x 0 book of photographs of the same location<br />
x …sh-<strong>and</strong>-chip supper<br />
Your preferences under certainty are x 00 x 0 x . Now consider the following<br />
two prospects<br />
8<br />
< x 00<br />
P 1 :<br />
:<br />
x 0 with probability 0<br />
P 2 :<br />
8<br />
<<br />
with probability 0:99<br />
x with probability 0:01<br />
x 00 with probability 0:99<br />
x 0 with probability 0:01<br />
:<br />
x with probability 0<br />
Suppose a person expresses a preference for P 1 over P 2 . Brie‡y explain why<br />
this might be the case in practice. Which of the three axioms State Irrelevance,<br />
Independence, Revealed Likelihood, is violated by such preferences?<br />
Outline Answer:<br />
It is possible that, given the information that the …rst event (with payo¤ x 00 )<br />
has not happened you would then prefer x to x 0 : photographs of your favourite<br />
holiday spot may be too painful once you know that the holiday is not going to<br />
happen. So you may prefer P 1 over P 2 .<br />
These preferences violate the independence axiom. To see this, note that,<br />
by the revealed likelihood axiom, since x 0 is strictly preferred to x , it must be<br />
the case that P2 0 is strictly preferred to P1, 0 where<br />
P 0 1 :<br />
P 0 2 :<br />
x<br />
0<br />
with probability 0<br />
x with probability 1<br />
x<br />
0<br />
with probability 0:01<br />
x with probability 0:99<br />
But P1 0 <strong>and</strong> P2 0 can be written equivalently as<br />
8<br />
< x <br />
P1 0 : x 0 with probability 0<br />
:<br />
P 0 2 :<br />
8<br />
<<br />
with probability 0:99<br />
x with probability 0:01<br />
x with probability 0:99<br />
x 0 with probability 0:01<br />
:<br />
x with probability 0<br />
By the independence axiom if P 0 2 is strictly preferred to P 0 1, then P 2 must be<br />
strictly preferred to P 1 .<br />
cFrank Cowell 2006 119