Question

Assume that Mr. A and Mr. B have the same utility function for a change, z, in their fortune, given by u(z) = z^1/3. Suppose now that one of the two men receives, as a gift, a lottery ticket which yields either a reward of r dollars (r > 0) or a reward of 0 dollars, with probability 1/2 each. Show that there exists a number b > 0 having the following property: regardless of which man receives the lottery ticket, he can sell it to the other man for b dollars and the sale will be advantageous to both men.

Answer 1

Let's denote the change in fortune for Mr. A and Mr. B after purchasing the lottery ticket as zA and zB, respectively. Initially, both Mr. A and Mr. B have the same fortune, so their initial utilities are equal:

uA = uB = u(0) = 0^1/3 = 0

Now, suppose Mr. A receives the lottery ticket as a gift, and he sells it to Mr. B for b dollars. After the sale, Mr. A's fortune changes by b dollars, while Mr. B's fortune changes by either r - b dollars (if he wins the lottery) or -b dollars (if he loses the lottery), each with a probability of 1/2. The expected utilities for Mr. A and Mr. B after the sale are:

E[uA'] = u(b)
E[uB'] = 1/2 * u(r - b) + 1/2 * u(-b)

For the sale to be advantageous to both men, their expected utilities after the sale should be higher than their initial utilities:

E[uA'] > uA
E[uB'] > uB

Now, let's find a value of b that satisfies both inequalities.

E[uA'] > uA:
u(b) > 0
b^1/3 > 0

Since b > 0, this inequality is always satisfied.

E[uB'] > uB:
1/2 * u(r - b) + 1/2 * u(-b) > 0
1/2 * (r - b)^1/3 + 1/2 * (-b)^1/3 > 0

Now, let's focus on finding a suitable b that satisfies this inequality:

(r - b)^1/3 + (-b)^1/3 > 0
(r - b)^1/3 > b^1/3

Since both sides are positive, we can cube both sides:

r - b > b
r > 2b

Thus, b < r/2.

Therefore, there exists a number b > 0 and b < r/2, such that regardless of which man receives the lottery ticket, he can sell it to the other man for b dollars, and the sale will be advantageous to both men.