Consider the following portfolio selection problem: a consumer has an

initial wealth of $w$ which has to be allocated between a risky and a riskless

asset. For each dollar invested in the risky asset, the consumer gets a

return $z_1<mn\; id="EditorElement\_2092682">1</mn>$ with probability $p$ and $z_2>1$ with probability $1-p.$ It

is assumed that the asset has an expected return greater than $1,$ that is

$p{z}_1+(1-p)z_2>1.$ The riskless asset yields a dollar for each dollar

invested. Let $$ and $$ be the part of wealth allocated to the risky and

riskless asset respectively. We refer to $(,)$ as the consumer's portfolio.

The consumer has a Bernoulli utility function over wealth given by $u(w)$

which is strictly increasing and strictly concave; further, the consumer is

assumed to be an expected utility maximizer.

$($a$)$ Set up the consumer's expected utility maximization problem as an

unconstrained problem with choice variable $$ and write the Kuhn$-$

Tucker condtions. Show that ${}^{**},$ the optimal holdings of the risky

asset, is strictly positive. That is$,$ even though the consumer is risk

averse, the consumer would still invest some amount in the risky asset

if the risk is actuarially fair $($in the sense that $p{z}_1+(1-p)z_2>1).$

This exercise offers an explanation of why risk averse consumers are

empirically observed to invest in the stock market.