Consider the following gambles, studied by Daniel Kahneman (2012 Nobel laureate) and Amos Tversky in 1977:

Gamble A 50-50 coin flip between

$1,000 and $0.

Gamble B $500 for sure.

Gamble C 50-50 coin flip between $-1,000 (losing $1,000) and $0.

Gamble D $-500 (losing $500) for sure.

a. If you are given $1,000 and then asked to choose between Gamble A and Gamble B, which would you pick if you are a risk-averse expected utility maximizer?

b. If you are given $2,000 and then asked to choose between Gamble C and Gamble D, which would you pick if you are a risk-averse expected utility maximizer?

c. In experiments, Kahneman & Tversky found that most people have B A and C D (with strict preference, not indifference). Prove that this is inconsistent with expected utility theory, regardless of whether the person is risk averse, risk neutral, or risk loving.

Kahneman & Tversky proposed a new theory called Prospect Theory that says the way people evaluate lotteries is that (1) they ignore any initial endowments (like the $1,000 or $2,000 given to them before they make their choice), (2) they are risk averse for gambles with positive payoffs, and (3) they are risk loving for gambles with negative payoffs. Mathematically, they have utility for changes in their current endowment, not their total wealth. And they evaluate gains using a concave function u, and they evaluate losses using a convex function v, where u(0) = v(0) = 0. So, if I give you $10 and then you have a 50-50 coin flip between $1 and $-1, that would be evaluated as 0. 5u(1) + 0.5v(1).

d. Show that BA and CD is consistent with Prospect Theory since u is concave and v is convex.