Question

An investor's utility function for money (Bernoulli utility function) is the square root of money: u(x)=√x. Her decision making can be modeled by assuming that she maximizes her expected utility. Her current wealth is 100. (All quantities are in hundreds of dollars.)

She has the opportunity to buy a security that either pays 8 (the "good outcome") or loses 1 (the "bad outcome"). She can buy as many units as she wishes. For example, if she buys 5 units, she gets 40 in the good outcomes, but loses 5 in the bad outcome. The probability of the good outcome is 0.2, and the probability of the bad outcome is 0.8.

In answering the questions below, you may use Excel to find your answers, if you wish.

Will she buy any of this security? If yes, how much exactly?

If her wealth were 150, would she buy any of this security? If yes, how much?

If her wealth were 200, would she buy any of this security? If yes, how much?

Suppose that a tax of 50% is imposed on this security. This means that whenever she gains 8 from the security, she gets to keep only 4. However, whenever she loses 1, she actually gets back 0.5, i.e. she only loses 0.5 (because her capital loss is tax deductible). If her initial wealth is 200, will she buy more or less of this security than in question 3?

Answer 1

U(X)=sqrt(X)

W=100

Expected Utility=0.2*sqrt(90+7X)+0.8*sqrt(100-X)

Exepcted Utility from Wealth =sqrt(100)=10

therefore if EU from wealth< Expected utillity fromm security then he would buy this security

0.2sqrt(90+7X)+0.8sqrt(100-X)=10

Using EXCECL solver we get

X>42.6 hundreds of dollars to purchase security he would purchase security when heh buys mre than 42.6 units of security

If wealth is 150

sqrt(150)<0.2sqrt(150+7X)+0.8sqrt(150-X)
when Wealth is 150 then for X=30 we will have maximum utility

Hence he should purchase 30 units of security

If wealth is 200

then as per the same logiv he needs to buy 42 units of security

If taxes of 50% is considered then

E(U)<0.5(E(U_s)

E(U)=sqrt(200)=14.142

14.142<0.5*(0.2sqrt(200+7X)+0.8sqrt200-X)

14.142<0.1*sqrt(200+7X)+0.4*sqrt(200-X)

At any level of X units of security he can maximisehis utility therefore he wont buy any units of security when W=200