Question

In: Economics

Expected - utility theory tells us that a risk - averse individual would want to be...

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).

Solutions

Expert Solution

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


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