Derive the equilibrium price ratio of the contingent claims

Question

Question 1

Suppose we have an economy with 1 comodity, two possible states, and two consumers, and no trades at date 0. Agent 1 is risk-netral (i.e., Agent 1's utility function satisfies: u" (x) = 0) while Agent 2 is risk-averse(u''(x) < 0).

Assume:

β¹ = { 6 w.p. Π

      { 3 w.p. (1- Π)

β² = { 4 w.p. Π

       { 2 w.p. (1- Π)

1 a) Derive the equilibrium price ratio of the contingent claims contract for this economy.

1 b) Is there a value for the probability it, such that can Agent two achieve complete insurance in equilibrium?

If so, describe how it will be achieved. What is the value of Π, in this instance?

1c) What assumption did we make in this problem that was key to the result you obtained in 1a) and 1 b) Explain.