4. AUTO INSURANCE. Consider a standard auto insurance problem in which we assume wealth levels XGOOD=100 and x BAD=0, probabilities Pr[xGOOD]=Pr[]=0.7 and Pr[xBAD]=Pr[]=0.3, administrative costs of 9, the utility function U[x] = √x, and an insurance policy price set exactly halfway between the Insurer's willingness to accept and the Driver's willingness to pay. TIP: Sketch the diagram and follow the lecture slides. a) Find the expected utility if the Driver is uninsured. Remember that EU is a weighted average of the two utility levels, U[x GOOD] and U[xBAD], based on their probabilities. b) Find the certainty equivalent, which is the xCE such that U[xCE] = sqrt[xCE] = EU. c) Find the expected claim, which is how much an insurer expects (on average) to pay to fix or replace the car. EClaim = Pr[](xGOOD – XGOOD) + Pr[](xGOOD – xBAD)
4. AUTO INSURANCE. Consider a standard auto insurance problem in which we assume wealth levels XGOOD=100 and x BAD=0, probabilities Pr[xGOOD]=Pr[]=0.7 and Pr[xBAD]=Pr[]=0.3, administrative costs of 9, the utility function U[x] = √x, and an insurance policy price set exactly halfway between the Insurer's willingness to accept and the Driver's willingness to pay. TIP: Sketch the diagram and follow the lecture slides. a) Find the expected utility if the Driver is uninsured. Remember that EU is a weighted average of the two utility levels, U[x GOOD] and U[xBAD], based on their probabilities. b) Find the certainty equivalent, which is the xCE such that U[xCE] = sqrt[xCE] = EU. c) Find the expected claim, which is how much an insurer expects (on average) to pay to fix or replace the car. EClaim = Pr[](xGOOD – XGOOD) + Pr[](xGOOD – xBAD)
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![4. AUTO INSURANCE. Consider a standard auto insurance problem in which we assume
wealth levels XG0OD =100 and XBAD=0, probabilities Pr[x600D]=Pr[O]=0.7 and
Pr[XBAD]=Pr[6]=0.3, administrative costs of 9, the utility function U[x]= Vx , and an insurance
policy price set exactly halfway between the Insurer's willingness to accept and the Driver's
willingness to pay. TIP: Sketch the diagram and follow the lecture slides.
a) Find the expected utility if the Driver is uninsured. Remember that EU is a weighted average
of the two utility levels, U[xc00D] and U[X®AD], based on their probabilities.
b) Find the certainty equivalent, which is thexCE such that U[xCE] = sqrt[xCE] = EU.
c) Find the expected claim, which is how much an insurer expects (on average) to pay to fix or
replace the car. EClaim = Pr[ OJ(xc00D – x600D) + Pr[®](x©00D – XBAD)
d) What is the Driver's willingness to pay for an insurance policy? WTP = XG0OD – xCE
e) How much of that WTP is a risk premium (the extra amount the Driver will pay “because he
hates risk")? WTP = EClaim + Risk Premium.
f) What is the Insurer's willingness to accept? This is the lowest amount the Insurer must
charge in order to break even, on average. Remember that there are two expenses the Insurer
must cover: WTA = E[Claim] + Admin
g) Find the insurance policy price, assumed to split the economic pie equally between the
Driver and the Insurer (i.e., a price halfway between WTP and WTA sets consumer surplus
equal to producer surplus).
h) Find the insurer's expected profit (or PS) and its actual profit if there is an accident.
i) MORAL HAZARD REDO! Show the effect on the Insurer's profit if, after the insurance
policy price has been set and an insurance policy purchased, the fully insured Driver behaves
recklessly, resulting in Pr[XG00D] = ½. Ie., the Insurer thinks an accident occurs 30% of the
time, but it actually occurs 50% of the time. TIP: You essentially have to redo parts (a)-(h)
with the new probabilities, but then assume the uninformed Insurer still sets the price found
in part (g).
= 1/
GOOD
„GOOD](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6871a655-6df9-491d-9e6a-d916a0d3d66e%2Fb9d2d96d-5b85-466e-8abc-f7e3c403e655%2Fsx764z_processed.png&w=3840&q=75)
Transcribed Image Text:4. AUTO INSURANCE. Consider a standard auto insurance problem in which we assume
wealth levels XG0OD =100 and XBAD=0, probabilities Pr[x600D]=Pr[O]=0.7 and
Pr[XBAD]=Pr[6]=0.3, administrative costs of 9, the utility function U[x]= Vx , and an insurance
policy price set exactly halfway between the Insurer's willingness to accept and the Driver's
willingness to pay. TIP: Sketch the diagram and follow the lecture slides.
a) Find the expected utility if the Driver is uninsured. Remember that EU is a weighted average
of the two utility levels, U[xc00D] and U[X®AD], based on their probabilities.
b) Find the certainty equivalent, which is thexCE such that U[xCE] = sqrt[xCE] = EU.
c) Find the expected claim, which is how much an insurer expects (on average) to pay to fix or
replace the car. EClaim = Pr[ OJ(xc00D – x600D) + Pr[®](x©00D – XBAD)
d) What is the Driver's willingness to pay for an insurance policy? WTP = XG0OD – xCE
e) How much of that WTP is a risk premium (the extra amount the Driver will pay “because he
hates risk")? WTP = EClaim + Risk Premium.
f) What is the Insurer's willingness to accept? This is the lowest amount the Insurer must
charge in order to break even, on average. Remember that there are two expenses the Insurer
must cover: WTA = E[Claim] + Admin
g) Find the insurance policy price, assumed to split the economic pie equally between the
Driver and the Insurer (i.e., a price halfway between WTP and WTA sets consumer surplus
equal to producer surplus).
h) Find the insurer's expected profit (or PS) and its actual profit if there is an accident.
i) MORAL HAZARD REDO! Show the effect on the Insurer's profit if, after the insurance
policy price has been set and an insurance policy purchased, the fully insured Driver behaves
recklessly, resulting in Pr[XG00D] = ½. Ie., the Insurer thinks an accident occurs 30% of the
time, but it actually occurs 50% of the time. TIP: You essentially have to redo parts (a)-(h)
with the new probabilities, but then assume the uninformed Insurer still sets the price found
in part (g).
= 1/
GOOD
„GOOD
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