(Use a spreadsheet or similar computational tool.) Consider a one-period economy with 5 possible states and 5 assets traded. The state-contingent dividends and prices of the assets and the state probabilities are as follows:

State 1 State 2 State 3 State 4 State 5 Price asset 1 1 1 1 1 1 0.9 asset 2 0 2 4 6 8 1.7 asset 3 4 0 2 4 2 2.3 asset 4 10 0 0 2 2 4.3 asset 5 4 4 0 4 4 2.8 Probability 0.25 0.25 0.2 0.2 0.1

(a) Verify that the market is complete and find the unique state-price deflator.
Consider an individual investor, Alex, with access to the above financial market with the given prices. Suppose Alex has time-additive expected utility with a time preference rate of δ = 0.03 and a constant relative risk aversion of γ = 2. Suppose his optimal consumption at time 0 is 5 and that he will receive an income of 5 at time 1 no matter which state is realized.

(b) What is Alex’s optimal time 1 consumption? What does it cost him to finance that consumption? What is the optimal portfolio for Alex?
Suppose now that there is only one other individual, Bob, in the economy. Bob also has time-additive expected utility with a time preference rate of 0.03 but a relative risk aversion of 5. Bob’s optimal time 0 consumption is also 5.

(c) What is Bob’s optimal time 1 consumption?

(d) What is the aggregate time 0 consumption and the state-dependent aggregate time 1 consumption?

(e) What is Bob’s time 0 endowment and state-dependent time 1 endowment? What is Bob’s optimal portfolio?

(f) Verify that the markets clear.