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.

1. Will she buy any of this security? If yes, how much exactly?
2. If her wealth were 150, would she buy any of this security? If yes, how much?
3. If her wealth were 200, would she buy any of this security? If yes, how much?
4. 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?

Write a few sentences summarizing what you learned from answering the four questions above.

Answer 1

0.2*(100-x+8x)^0l.5+0.8*(100-x)^0l.5=10

0.2*(100+7x)^0l.5+0.8*(100-x)^0.5=10

SOlving using excel we get x <=45.3 units when wealth is 100

Now when wealth is 150

0.2*(150+7x)^0l.5+0.8*(150-x)^0.5>=10

Yes if we are going to buy atmost 132 units of security then we should buy this security

Now when wealth is 200

0.2*(200+7x)^0l.5+0.8*(200-x)^0.5>=10

Yes if we are going to buy atmost 132 units of security then we should buy this security

in this case if we buy atmost 192 units of security then we should buy this security

when 50% tax is applied then we have

When wealth is 100 if we buy x units then we have

0.2*(100-x+8*0.5x)^0.5+0.8(100-0.5x)^0.5>=10 for some x

using excel we get,

x<=67 to n=buy this security

When wealth is 200 if we buy x units then we have

0.2*(200-x+8*0.5x)^0.5+0.8(200-0.5x)^0.5>=10 for some x

then it gives higher utility than 10 for any x ranges from 0 ton 200

therefore she will buy more securities