Suppose that there are two assets. The first is a riskless asset that pays $1$ dollar. The second pays a and b

with probabilities p and $1-$p$.$ Denote demand for the two assets by $($x$1,$ x$2).$ Suppose that a decision maker$$s

preferences satisfy the axioms of expected utility theory and that he is risk averse. His wealth is $1$ and so

are the prices of the assets, so that his budget constraint is x$1+$ x$2=1.$

$($a$)$ Give a simple necessary condition involving a and b only for the demand for the riskless asset to be

strictly positive.

$($b$)$ Give a simple necessary condition for the demand for the risky asset to be strictly positive.

Now assume the conditions in $($a$)$ and $($b$)$ are satisfied:

$($c$)$ Write down the first order conditions for utility maximization in the asset demand problem.

$($d$)$ Suppose that a $<1.$ Show that $$x$1$

$$a $<0$