$4.$ 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$1<1$ 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 pz$1+(1$ 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 KuhnTucker 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 pz$1+(1$ p$)$z$2>1).$ This exercise o$$ers an explanation of why risk averse consumers are empirically observed to invest in the stock mar $($b$)$ Comparative Statics Across Consumers: Now suppose there is another consumer with Bernoulli utility function v$($w$)$ who is more risk averse than the consumer with utility u$($w$)$ for each w$,$ that is v$($w$)=($u$($w$))$ where is a strictly increasing and strictly concave function. Show that the consumer who is more risk averse invests less in the risky asset. $($c$)$ Comparative Statics Across Wealth: Consider a consumer with the Bernoulli utility function over wealth given by u$($w$)=$ log w$.$ Drawing upon your computation from Part a$,$ calculate $.$ Show that is increasing with wealth, i$.$e$.$ the risky asset is a normal good.n