Question

Consider an agent who has an original income of 100. The gent now faces an uncertain situation with two states, each state occurring with probability in state 1, the weat faces a safe situation and incurs no cost to his original income. In state 2, the agent suffers an accident that imposes a loss of L=75. Hence in state 1, the agent's income is 100 while in state 2. his income is 100 - 75 = 25. The agent's utility for money (Bernoulli utility function) is given by (w) - (a) Find the expected utility, under the umption that the cent cannot be any insurance inst the risk of accident recall that the two states are ally likely one that now at a cost of 16, the great could reduce the probability of a lomie, reduce the ability of state 2 tot llence, the agents et income we be 8 in state I (ie, 100- 16 = 84). state 2 (ie, 100 - 15 - 16 -9). For example, then may install smoke alarm that make catastrophic fire le my s tem in her home stop smoking or skydiving, and so a the got pay the cost to reduce the risk of A wer this question by calculating and comparing the relevant expected utilities terwying the 16 to reduce the rela t ie

thank you

Answer 1

a)

In case of no insurance,

Probability of accident=p=1/2=0.50

Wealth in case of accident=100-75=$25

Utility in case of accident=U(25)=25^1/2=5 utils

Probability of no accident=1-p=1-1/2=0.50

Wealth in case of no accident=$100

Utility in case of no accident=U(100)=100^1/2=10 utils

Expected Utility=p*U(25)+(1-p)*U(100)=0.50*5+0.50*10=7.50 utils

b)

Probability of accident=p=1/16=0.0625

Wealth in case of accident=100-75-16=$9

Utility in case of accident=U(25)=9^1/2=3 utils

Probability of no accident=1-p=1-0.0625=0.9375

Wealth in case of no accident=100-16=$84

Utility in case of no accident=U(84)=84^1/2=9.1652 utils

Expected Utility=p*U(9)+(1-p)*U(84)=0.0625*3+0.9375*9.1652=8.7799 utils

We can see that

Expected Utility (in case of incurring a cost of $16)>Expected utility (no insurance)

So, agent would incur a cost of $16 to reduce the probability of accident.