4 Multiple Assets
Suppose an agent is trying to maximize utility
$U = \log c_1 + \beta(\pi \log c_b + (1 - \pi) \log c_r)$
log is the natural log. Exogenous output is given by:
$y_1 = 2000$, $y_b = 2240$, $y_r = 1400$,
$\pi = .8$
The other parameters are given by: $\beta = .8$ and $1 + r = 1.25$. There are two assets. One is a
riskless bond, and one is a stock that pays more in the boom.
If you buy one unit of the riskless bond, you get $1 + r$ in the second period no matter
what.
If you buy one unit of the stock, you get $p_b$ in the boom, or $p_r$ in the recession.
The budget constraints are that:
$y_b = c_b + x_b$,
2
$y_r = c_r + x_r$,
$y_1 = c_1 + x_1$
But now, you can choose to allocate your capital account between stocks and bonds:
$x_1 = b + s$
where $b$ is your bonds and $s$ is your stocks. In period 2, the captial account/current account
is given by:
$-x_b = b(1 + r) + s p_b$, $-x_r = b(1 + r) + s p_r$
Assume that the stock pays 1.5 in the boom and 0.25 in the recession, so
$p_b = 1.5$, $p_r = 0.25$
1. Rewrite the problem as an unconstrained maximization problem in terms of $b$ and $s$.
2. What is the first-order condition with respect to $b$? What about with respect to $s$?
3. Solve for $b$ and $s$. What sign does $s$ have? Can you give an economic intuition as to
why it has that sign?
4. What is $x_1$? How does it compare to your answers to the last two questions?
