Q2. Asset Economy
Consider the economy that can be in one of the two states of the world tomorrow
and has two financial assets a₁ and a2. Assets' state payoffs are given in the matrix
a1 a2
$1
4
1
$2
3
2
a. Is the above financial market complete? Explain what this means and verify.
Suppose further the prices of the assets (a1, a2) are (q₁ = 5, 92 = 3).
b. What is arbitrage and what are Arrow securities? Explain their role in this
economy. What is the connection between the prices of the Arrow securities and
the existence of arbitrage opportunities?
c. Calculate Arrow security prices in this economy.
d. Suppose the investor's today wealth is wo. Given the Arrow security prices in c.
what levels of wealth (y1, y2) can she transfer into tomorrow? Draw this "possibility
frontier" on the graph with y₁ and y2 on the axis. Mark the assets a₁ and a2 on this
frontier.
e. Suppose the investor is infinitely risk averse and has a utility function U (31, 32) =
min {1, 2}. Determine the investor's optimal portfolio of assets (21, 22). Mark this
portfolio on the graph in d.
f. Suppose the investor is instead risk neutral and has a utility function U (y1, y2) =
T131+2y2, where ; is what she believes is the probability of state i. You may assume
some numerical values for T1 and T2 such that ₁ + T2 = 1. Determine the investor's
optimal portfolio of assets (21, 22). Mark this portfolio on the graph in d.