$4$ Multiple Assets

Suppose an agent is trying to maximize utility

$U=log{c}_1+(log{c}_b+(1-)log{c}_r)$

$log$ is the natural log$.$ Exogenous output is given by:

$y_1=2000,y_b=2240,y_r=1400,=\mathrm{.}8$

The other parameters are given by: $=\mathrm{.}8$ and $1+r=1\mathrm{.}25.$ 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:

$y_b=c_b+x_b,y_r=c_r+x_r,y_1=c_1+x_1$

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 $1\mathrm{.}5$ in the boom and $0\mathrm{.}25$ in the recession, so

$p_b=1\mathrm{.}5,p_r=0\mathrm{.}25$

Rewrite the problem as an unconstrained maximization problem in terms of $b$ and $s.$

What is the first$-$order condition with respect to $b?$ What about with respect to $s?$

Solve for $b$ and $s.$ What sign does $s$ have? Can you give an economic intuition as to

why it has that sign?

What is $x_1?$ How does it compare to your answers to the last two questions?