Question

A risk-neutral plaintiff in a lawsuit is deciding whether to settle a claim or go to trial. The defendants have offered a $150,000 settlement. If the plaintiff does not settle, she believes that the probability of winning at trial is 75%. If the plaintiff wins, she is awarded X dollars. If she loses her payoff is $0. How large must X be in order for the plaintiff to turn down the settlement offer? Suppose instead that the plaintiff is risk-averse, with a utility function of U(w)=sqrt(w). If the plaintiff does not settle she believes that the probability of winning at trial is 75%. If she wins she is awarded X dollars. If she loses her payoff is $0. How large must X be in order for the plaintiff to turn down the settlement offer? Is this number smaller or larger than that in part a)? Why?

Answer 1

In case Plaintiff goes for trail

Probability of winning=p=0.75

Probability of loosing=1-p=1-0.75=0.25

Payoff if Plaintiff in case of win=X

Payoff if Plaintiff in case of loosing=0

Expected payoff=0.75*X+0.25*0=0.75X

In case plaintiff does not go for trial.

Payoff = 150,000

In order to be indifferent, Expected payoff in case of trial should be equal to payoff in case of settlement.

0.75X=150000

X=150000/0.75=200000

So, Plaintiff would turn down the offer only if X is higher than $200000.

b)

Utility at w=150000 is U(150000)

U(15000)=(150000)^0.5=387.30

Similary U(0)=0

Utility at w=X is X^0.5

Expected utility=0.75*U(X)+0.25*U(0)=0.75*X^0.50+0.25*0=0.75X^0.5

In order to be indifferent Expected utility should be the same in both cases. So,

0.75X^0.5=387.30

X=(387/0.75)²=266669

So, Plaintiff would turn down the offer only if X is higher than $266669.

It is a higher amount than 200000 found earlier.

In this case marginal utility is diminishing. Plaintiff is risk averse. He tries to avoid risk. Higher the amount of w, lower the marginal utility.So, he would settle at sure shot payments early i.e. a risk averse person would enter into gamble if relatively higher payoff is offered.