Question

Your current wealth is M=$100 and your utility function is U=M². You have received a ticket that allows you to participate in the following fair gamble: in a coin flip, you will receive $40 if the coin lands on heads and lose $40 if it lands on tails.

A. What is the expected value of this gamble? What is the expected utility of this gamble?

B. What is the lowest price for which you would sell your ticket?

Answer 1

Initial Wealth (M) = $100

Utility (U) = M^{​​​​​​2}

So there will be two cases with equal probability as it's a coin toss :

• If heads profit of $40. So, New Wealth = $140 with probability = 1/2 (0.5)
• If tails loss of $40. So, New Wealth = $60 with probability = 1/2 (0.5)

A. We know Expected Value of gamble

$Expected \; Value = \sum _{i} w_{i}P(w_{i}})$

So, Expected Value = (0.5 * $140) + (0.5 * $60)

= $70 + $30

= $100

So, Expected Value of gamble is $100

Now, Expected Utility is given by :

$Expected \; Utility = \sum _{i} U(w_{i})P(w_{i}})$

So, Expected Utility = (0.5 * U($140)) + (0.5 * U($60))

= (0.5 * 140² ) + (0.5 * 60²)

= (0.5 * 19600) + (0.5 * 3600)

= 9800 + 1800

= 11600

So, Expected Utility of gamble is 11600 utils

B. Lowest Price at which to sell the ticket = Income at Expected utility level - Initial Income

As U = M^{​​​​​​2}

^{​​​​​​}So, M = √U

So, Income at Expected Utility level = √Expected Utility = √11600 = $107.7032

So, Lowest Price at which to sell ticket = $107.7032 - $100 = $7.7032 ~ $7.70

Hence, Lowest Price for which I will sell the ticket is $7.70