In: Economics
Expected
-
utility theory tells us that a risk
-
averse individual would want to be paid
a risk premium to induce
her
to voluntarily accept risk. Or, alternatively, the risk
-
averse individual would have to be offered odds that were in
her
favor in order to
induce h
er
to gamble. Here, “odds” refer to the probability of loss/gain and/or the
magnitude of the loss/gain.
Ms
.
Jane’s
utility function is U(
I
) =
(
10
0
+
I
)
0.
4
5
, where
100 is her initial income
level and
I
is
additional income gained or lost from a gamble. Ms.
Jane
can
choose not to gamble or to gamble.
The gamble involves the roll of a fair dice. If
the number is e
ven Ms.
Jane
wins $100 but if it is odd Ms.
Jane
loses $100.
(i)
Will Ms.
Jane
choose the gamble?
(ii)
In words, briefly describe what would need to change to induce Ms.
Jane
to take the gamble
(
hint
: focus on the characteristics of the gamble not her
preferences).
i)
U(l) = (100 +l)*0.45
Expected Pay off of dice roll = $100*(3/6)-$100*(3/6) = 0
If jane wins: U(100) = (100+100)*0.45 =
For risk averse individual:
U(0) = (100+0)*0.45 = 45
U(-100) = 0
U(100)= (200)*0.45 = 90
Expected Payoff =U(100)*3/6 + U(-100)*3/6 = 90*1/2 + 0 = 45
U(0)= Expected Payoff
Hence the utility derived for jane from not taking the gamble of dice roll and taking the gamble is the same
i.e she will not take the gamble as no incentive is offered .
ii)
In order for Jane to consider the gamble the game has to be modified :
a) Either a more premium should be offered
U(0)<U(a)*3/6 + U(-b)*3/6
45 < (100+a)0.45*1/2 + (100-b)*0.45*1/2
45 < 45/2 + 0.45a/2 + 45/2 - 0.45b/2
0 < 0.45a/2 - 0.45b/2
b<a
For Jane to take part in the dice roll the winning amount a > losing amount b
Example the winning amount is changed to 100 $ (if she gets even on dice) and 50 $ has to be paid if she loses (if she gets odd on dice)
U(0)= 45
Expected Payoff = U(100)*3/6 + U(-50)*3/6 = (100+100)*0.45*0.5 + (100-50)*0.45*0.5 = 45 + 11.25 = 56.25
Therefore U(0) < Expected Payoff