Bob le Flambeur is offered a raffle ticket. With probability p, the raffle ticket wins, and pays 5 dollars. With probability 1-p, the ticket loses and pays nothing. One ticket costs 1 dollar. (a) If Bob has von Neumann-Morgenstern utility u(z) = ³ over change in wealth z, what is the certainty equivalent for Bob of buying one ticket (as a function of p)? What is the risk premium? Solution: If Bob buys a raffle ticket, then with probability p he gains 4 dollars (5 dollars minus the price of the ticket), and with probability 1-p he loses 1 dollar. His expected utility is therefore 64p-(1-p) = 65p-1. The certainty equivalent is therefore CE(p) = 65p-1. The expected value of the lottery is 4p - (1 − p) = 5p – 1, giving a risk premium of R(p) = 5p-1-65p - 1. (b) Now suppose that, instead of the utility function above, Bob is risk-neutral. An un- scrupulous raffle saleswoman makes Bob the following offer. For 50 cents, she will tell him whether the ticket for sale will win or lose before Bob decides whether to buy it. If p = 1, will Bob accept the saleswoman's offer? Assume that there is only one ticket available for him to purchase. Solution: Bob has three options: accept the offer, reject the offer and buy a ticket, or reject the offer and not buy a ticket. Since Bob is risk-neutral, he cares only about maximizing the expected value of the lottery he chooses (i.e. we can suppose that u(x) = x). If he rejects the offer, the expected value associated with buying a ticket is 5-1: 0.250, so he prefers to buy a ticket. If he accepts the offer, then he loses 50 cents for sure, but gains 4 with probability, giving expected value 1-0.5=0.5>0.25. Therefore, Bob will accept the offer.
Bob le Flambeur is offered a raffle ticket. With probability p, the raffle ticket wins, and pays 5 dollars. With probability 1-p, the ticket loses and pays nothing. One ticket costs 1 dollar. (a) If Bob has von Neumann-Morgenstern utility u(z) = ³ over change in wealth z, what is the certainty equivalent for Bob of buying one ticket (as a function of p)? What is the risk premium? Solution: If Bob buys a raffle ticket, then with probability p he gains 4 dollars (5 dollars minus the price of the ticket), and with probability 1-p he loses 1 dollar. His expected utility is therefore 64p-(1-p) = 65p-1. The certainty equivalent is therefore CE(p) = 65p-1. The expected value of the lottery is 4p - (1 − p) = 5p – 1, giving a risk premium of R(p) = 5p-1-65p - 1. (b) Now suppose that, instead of the utility function above, Bob is risk-neutral. An un- scrupulous raffle saleswoman makes Bob the following offer. For 50 cents, she will tell him whether the ticket for sale will win or lose before Bob decides whether to buy it. If p = 1, will Bob accept the saleswoman's offer? Assume that there is only one ticket available for him to purchase. Solution: Bob has three options: accept the offer, reject the offer and buy a ticket, or reject the offer and not buy a ticket. Since Bob is risk-neutral, he cares only about maximizing the expected value of the lottery he chooses (i.e. we can suppose that u(x) = x). If he rejects the offer, the expected value associated with buying a ticket is 5-1: 0.250, so he prefers to buy a ticket. If he accepts the offer, then he loses 50 cents for sure, but gains 4 with probability, giving expected value 1-0.5=0.5>0.25. Therefore, Bob will accept the offer.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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please only do: if you can teach explain steps of how to solve each part
for partb how to solve EV what formula, how to use this formula?
Transcribed Image Text:Bob le Flambeur is offered a raffle ticket. With probability p, the raffle ticket wins, and pays
5 dollars. With probability 1-p, the ticket loses and pays nothing. One ticket costs 1 dollar.
(a) If Bob has von Neumann-Morgenstern utility u(r) = ³ over change in wealth , what
is the certainty equivalent for Bob of buying one ticket (as a function of p)? What is the
risk premium?
Solution: If Bob buys a raffle ticket, then with probability p he gains 4 dollars (5
dollars minus the price of the ticket), and with probability 1-p he loses 1 dollar. His
expected utility is therefore 64p-(1-p) = 65p-1. The certainty equivalent is therefore
CE(p) = 365p 1. The expected value of the lottery is 4p - (1 - p) = 5p - 1, giving a
risk premium of R(p) = 5p-1-65p - 1.
(b) Now suppose that, instead of the utility function above, Bob is risk-neutral. An un-
scrupulous raffle saleswoman makes Bob the following offer. For 50 cents, she will tell
him whether the ticket for sale will win or lose before Bob decides whether to buy it.
If p = 1, will Bob accept the saleswoman's offer? Assume that there is only one ticket
available for him to purchase.
Solution: Bob has three options: accept the offer, reject the offer and buy a ticket,
or reject the offer and not buy a ticket. Since Bob is risk-neutral, he cares only about
maximizing the expected value of the lottery he chooses (i.e. we can suppose that u(x) =
r). If he rejects the offer, the expected value associated with buying a ticket is 5-1=
0.25 > 0, so he prefers to buy a ticket. If he accepts the offer, then he loses 50 cents for
sure, but gains 4 with probability, giving expected value 1-0.5 = 0.5> 0.25. Therefore,
Bob will accept the offer.
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