I get suck on below problem, so I really need heip. I could solve some questions
ID: 1192859 • Letter: I
Question
I get suck on below problem, so I really need heip. I could solve some questions, but I cannot understand other question. Therefore, I want to know how to solve these problem. If it is possible, I also want know these answer.
Suzette works in downtown L.A. as a copy-shop manager, and she earns $100 per day. She drives to work and parking is expensive at the nearby lot: $19 per day. The company that owns the parking lot also owns many others, and it is understaffed. Instead of putting an attendant at each lot it asks lot users to put the money in a special box on the premises along with their car license number. To check on compliance, it sends an employee randomly to its different lots during the week. If a car is found parked in the lot whose owner hasn't paid, a fine of $36 is levied. The chance of being caught if the owner doesn't pay is 20% (i.e., the probability is 0.20). Suzette's net income equals her pay of $100 minus whatever parking costs she incurs (fines or payments). Letting z denote net income per day, Suzette's utility as a function of z is given by z1/2 (i.e., the square root of net income).
1)On a piece of paper, graph Suzette's utility function (plot utility for z = 16,25, 36, 64, 100,121, etc.). Is she risk averse (answer yes or no)?
2)Suzette's utility if she pays the parking fee is equal to (?). Her expected utility if she doesn't pay the fee is equal to (?) [express as a value with one digit following the decimal point]. Remember that in a situation involving risk, expected utility is equal to the probability of one outcome Times its utility plus the probability of the other outcome Times its utility. Should Suzette pay the parking fee (yes or no)?
3)Suppose the fine were $84 instead of $36. Suzette's expected utility if she doesn't pay the fee is now equal to (express as a value with one digit following the decimal point). Should Suzette pay the parking fee (yes or no)?
4)Frank works as a salesman in a store near Suzette's copy shop. He also earns $100 per day, and he parks in Suzette's lot. Frank's utility function, however, is 2z instead of the square root of z (i.e., the utility of a given net income is 2 Times the net income). Graph Frank's utility function. Is he risk averse (yes or no)?
5)When the fine is $36, Frank's utility if he pays the parking fee is equal to . His expected utility if he doesn't pay the fee is equal to (express as a value with one digit following the decimal point). Should Frank pay the parking fee (yes or no)?
6)When the fine is $84. Frank's expected utility if he doesn't pay the fee is equal to (express as a value with one digit following the decimal point). Should Frank pay the parking fee (yes or no)?
Explanation / Answer
1)Suzette's utility function is Z1/2 .Calculating Marginal utility by differentiating this we get 1/2 * z-1/2.For a risk averese person Marginal utility of his wealth decreases as his wealth increases , in other words a risk averse person has a concave utility function when utility is plotted against wealth.Overhere the 2nd derivative of the utility function = -1/4 * z-3/2 is negative , this implies Marginal utility of wealth is falling as wealth increases.
2)Suzette's utility if she pays the fee = (100 - 19)1/2 = 811/2 = 9.
Her expected utility if she doesnt pay the fee is = 0.20 (probability of getting caught) * (100 - 36)1/2 (utility after paying fine) + 0.80 (probability of not getting caught) * (100 - 0)1/2 (utility of not being caught and not paying the fine)
0.20 * (100 - 36)1/2 + 0.80 * (100 - 0)1/2 = 0.20 * 8 + 0.80 * 10
=1.60 + 8 =9.60
As suzette's expected utility from not paying the fee is greater than her utility when she pays the fee so she must not pay the fine.
3)When the fine is = $84 her expected utlity
0.20 * (100 - 84)1/2 + 0.80 * (100 - 0)
= 0.20 * 4 + 8
=8.80.
Now her expected utility fro not paying the fee is less than her utility when she pays the fee as 8.80 < 9.Thus she should pay the fee if fine increases to $84.
4)Frank's utility function is 2z.His Marginal utility of wealth is = 2 which indicates his MU of wealth is constant as wealth increases.Marginal utility of wealth is constant for a risk neutral person.Thus Frank is risk neutral and not risk averse.His utility graph would be a positively sloping straight line through origin with utility and wealth on the y axis and x axis respectively.
5)If the fine is $36 Frank's utility when paying the fee = 2 * (100 - 19) (his utility function = 2z) =162.
His expected utility if he doesnt pay the fees would be = 0.20 * (2 * (100 - 36)) + 0.80 * (2 * (100 - 0))
0.20 * 128 + 0.80 * 200
25.6 + 160
=185.6
Thus Frank's expected utlity from not paying the fee is greater than his utility from paying the fee as 185.6 > 162.Hence he should not pay the fee.
6)When the fine = $84
His expected utility = 0.20 * (2 * (100 - 84) ) + 0.80 * (2 * (100 - 0))
=6.4 + 160
= 166.4.
Again even afer fine is increased to $84 Frank's expected utility from not paying the fee (166.4) > his utility when he pays the fine (162).Thus he should not pay the fee.
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