1.There are 2 securities and 3 states. Security payos are x₁ = (1, 1, 1) and x₂ = (3, 1, 5). Prices are p₁ = 0.9 and p₂ = 3.5.

(i)Find all strictly positive state prices and all risk-neutral probabilities. Are security prices arbitrage free?

(ii)Consider a contingent claim z = (2, 2, 0). Find the upper and the lower bounds on the value of contingent claim z using state prices from part (i). Suppose that a security with payo z is added to the market at price p_Z = 1.2. Explain why there is arbitrage opportunity in the market with securities 1, 2, and z. You are not asked to find an arbitrage portfolio, but you may if you need to.

2.There are three states, s = 1, 2, 3, and three agents. Agents have expected utility functions with the same von Neumann-Morgenstern utility v(y) = ln(y), and probabilities ¹₃ for each state. There is no consumption at date 0. Date-1 endowments are as follows: agent 1 gets L units of the good in state 1 and zero in states 2 and 3, agent 2 gets M units in state 2 and zero in states 1 and 3, and agent 3 gets H units in state 3 and zero in states 1 and 2. Assume that H > M > L > 0.

(i)What can you say about Pareto optimal consumption allocations in this en-vironment, in particular, about the comparison of consumption across states? If you use a general result about Pareto optimal allocations, state that result clearly and make sure that it does apply to this environment.

(ii)Consider complete security markets. In an equilibrium in security markets, does agent 1 consume more in state 1 (where he gets the larger endowment) than in states 2 and 3? Does agent 2 consume more in state 2 than in states 1 and 3? Justify your answer. .

(iii)Let the market portfolio of securities be such that its payo equals the aggregate endowment. Show that the expected return on the market portfolio in an

equilibrium in complete markets is strictly greater than the risk-free return.

(iv)Would your answers to (i), (ii), and (iii) change if probabilities of states were

¹₂for state 1 and ¹₄for each states 2 and 3, the same for all agents?

3.Consider Example 30.6.2 (page 50 of Class Notes) of equilibrium price bubbles on a security with zero dividends. Take equilibrium prices p_T and allocation (c^I_T, h^I_T) as provided in text - you don't need to verify that they are an equilibrium.

Show that event prices (for a single event at every date) are a constant sequence equal to 1. Next, show that the present value of the aggregate endowment is infinite. Lastly, show that price bubbles are strictly positive at every date, and satisfy the "roll-over" property (30.8) on page 49.