Suppose Joe has preferences over income given by U(I) = ln(I).

Furthermore, suppose there are two states of the world and that Joe believes the state 1 will occur with probability equal to 0.25 and state 2 will occur with probability 0.75.

1. If Joe's income endowment in the two states is given by (I₁, I₂) = (80, 110) and he is able buy z₁= 40 extra income in state 1 by selling z₂= -20 income in state 2, what is his expected utility?

2. What is the |slope| of Joe's indifference curve at the income allocation from problem #1?

3. If Joe decided to buy and sell A-D securities so that he faced no income risk at all in the future, what would be the |slope| of his indifference curve at that point?