4 Bob will earn 80000 this year for certain However Dougs in

4. Bob will earn $80,000 this year for certain. However, Doug\'s income depends on whether he gets a big consulting contract - his income is will be $20,000 without the contract but $180,000 if he gets the contract. The probability that he receives the contract is 0.5. The two men are considering an income-sharing agreement. Under this legally binding agreement, the two men will agree to split their combined incomes evenly (a) If each man is risk neutral, will both of them be willing to sign the agreement? Why or why not? (b) Instead, suppose that each man has a utility function over income given by U(M)-M0.5, where M represents income in thousands of dollars. Will both men be willing to sign the agreement? Why or why not? (e) Suppose Bob claims that since his income is risk-free, he deserves a higher percentage of the total income than Doug does. Consider instead a contract that gives x% of the combined income to Doug and the rest to Bob. What is the lowest value of x such that Doug would still accept the contract? That is, for what value of x is Doug indifferent between signing the agreement and not signing the agreement

Solution

Combined income in case of no contract = 80000+20000=100000

Income shared is 50000 each

Combined income in case of contract=80000+180000=260000

Income shared between them is 130000

Expected Combined Income=100000*0.5+260000*0.5=50000+130000=180000

Income shared is 90000 between both.

As Expected income for both is higher than income with certainty hence they both will agree with contract .

Answer for B)

U(M)=M^0.5

Utility from Expected Income =180000^0.5=424.26

E(U)=0.5U(100000)+0.5U(260000)=0.5100000+0.5260000=413.06

Utility from Expected income is higher than Expected Utility and if we plot a graph of sqrt M it shows Diminishing Marginal Utility for every unit increase in Income.This tells us the decreasing utility for every unit increase of income hence they are risk averse and should not sign the agreement.

Answer for C)

X%*0.5((100000)+260000)=180000*X=M

U(M)=M^0.5=(180000X)^0.5>300

180000X>90000

X>0•5 i.e. 50%