HW4, Q3 Hello, Need help to solve the attached question. Thank you, HW 4, Questi

Question

HW4, Q3

Hello,

Need help to solve the attached question.

Thank you,

HW 4, Question 3 Suppose your preferences over your wealth Ware described by wx)In(W. You just graduated from college and your initial wealth is Wo $0. You are offered a job and you can choose between a fixed salary or a performance based salary to be earned in t 1. Suppose the performance based salary only depends on tomorrow's (t- 1) market situation, which is either g (good) or b (bad) with equal probability, and the salary is Stla) $150k or S1(b) $50k 1. Would you rather accept the performance based salary or a fixed salary of Fi $100k? hint 1: compare the expected utility from the two salaries hint 1: notice that Fi(G) -F(b) -F 2. What is the lowest fixed salary you are willing to accept over the performance based salary? hint find the certainty equivalent of the random salary S 1

Explanation / Answer

We have been given utlity funtion = ln(W) where W is the wealth being offered. The two scenarios are:

1. Now we have to find out which of these alternatives will be preffered given the utlity function. Therefore we will compare expected utlity in both the scenarios and see which one gives higher expected utlity.

Expected utlity is the sum of all possible utlities multiplied by their respected probabilities. That is, if there are two events x1 and x2 which occur with probability p1 and p2 respectively, then

=> Expected utlity = p1*x1 + p2*x2

Scenario 1: When a fixed salary is offered. When this happens, no matter what the market consition is, the person will receive 100k dollars.

Therefore if F1(g) is the fixed salary in good market situation and F1(b) is the fixed salary in bad market situation, then

F1(g) = F1(b) = $100k and this occurs with probability 1.

Therefore utlity tomorrow = u(100k) = ln(100k)

=> Utlity tomorrow = expected utlity from fixed amount = ln(100k) = 11.51

Scenario 2: When performance based salary is chosen. Let S1(g) = salary when market is good and S1(b) be the salary when market situation is bad. In this case, we will have:

This info is summarised in table:

Therefore expected utlity will be the weighted average of the utlities

=> (1/2 * ln(150000)) + (1/2 * ln(50000)) --------------> (1)

=> (1/2 * 11.92) + (1/2 * 10.82)

=> 11.37

Expected utlity from fixed salary = 11.51 > expected utlity from performance based salary = 11.37

Therefore the person will accept fixed salary. -> answer _____________________________________________________________________________

b) Now we have to find the lowest fixed salary which you are willing to accept over performance based salary. That is, we have to find the lowest fixed salary for which the consumer is indifferent between fixed salry and performance based salary.

This will only happen when the expected utlity from both the salaries become equal.

Now lets assume there is lowest fixed salary for which you are indifferent betwen fixed and performance based salary. Let's call it X dollars.

Then the expected utlity from X amount of fixed salary would be:

EU(F1(g = b)) = U(X) = ln(x)

Also we calculated expected utlity from performance based salary to be 11.37. Or in other words,

Expected utlity from performance based salary =

=> (1/2 * ln(150000)) + (1/2 * ln(50000))

Taking 1/2 common =>

=> 1/2 *(ln(150000)) + ln(50000))

=> 1/2 * ln(150k * 50 k) -----> from property of log as ln(x) + ln(y) = ln(xy)

Therefore eqauting the two expected utlities, we get:

ln(x) = 11.37

Taking exponential on both sides and removing log => (as eln(x) = x)

=> x = e11.37

=> x = $86681.87 approx

Therefore the lowest fixed salary that you are willing to recieve in order to accept it over performance salary is $86681.87 dollars. (Answer)

Market Situation Good = S1(g) Bad = S1(b) Utility 11.92 10.82 Probability 1/2 1/2

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