The probability of getting sick is 0.05-0.03x and the expenditure is 200 if being sick. Moreover, health condition x depends on effort. The relation between effort and x is x = x0 exp(e-0.5). XO is a random variable that follows uniform distribution U[0,1] and e is the effort level that can take three values: 0, 0.3 and 0.5. The private cost of effort is c(e) = 10e^2. 1. Please provide the range of XO that will optimally pay effort 0, 0.3, and 0.5 respectively when they correctly calculate their probability of being sick. 2. If we consider an insurance policy that covers the loss in full, and the premium is 6.5. Assume x0 = 0.8. (a) If buyers are risk neutral, will they purchase the policy? Given the optimal decision of whether or not to purchase the policy, what is the resulting probability of getting sick, and what is the expected medical expenditure (before insurer reimburses, if any)? (b) Is the insurer's profit positive, negative, or zero? Does there exist a premium level that satisfies the following two conditions: (1) the buyers are willing to purchase the insurance; (i) the insurer earns a nonnegative profit? Explain. (c) What contractual changes will you make to let a market exist such that the two conditions hold?
The probability of getting sick is 0.05-0.03x and the expenditure is 200 if being sick. Moreover, health condition x depends on effort. The relation between effort and x is x = x0 exp(e-0.5). XO is a random variable that follows uniform distribution U[0,1] and e is the effort level that can take three values: 0, 0.3 and 0.5. The private cost of effort is c(e) = 10e^2. 1. Please provide the range of XO that will optimally pay effort 0, 0.3, and 0.5 respectively when they correctly calculate their probability of being sick. 2. If we consider an insurance policy that covers the loss in full, and the premium is 6.5. Assume x0 = 0.8. (a) If buyers are risk neutral, will they purchase the policy? Given the optimal decision of whether or not to purchase the policy, what is the resulting probability of getting sick, and what is the expected medical expenditure (before insurer reimburses, if any)? (b) Is the insurer's profit positive, negative, or zero? Does there exist a premium level that satisfies the following two conditions: (1) the buyers are willing to purchase the insurance; (i) the insurer earns a nonnegative profit? Explain. (c) What contractual changes will you make to let a market exist such that the two conditions hold?
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
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![The probability of getting sick is 0.05-0.03x and the expenditure is 200 if being sick. Moreover,
health condition x depends on effort. The relation between effort and x is x = x0 exp(e-0.5). XO is a
random variable that follows uniform distribution U[0,1] and e is the effort level that can take three
values: 0, 0.3 and 0.5. The private cost of effort is c(e) = 10e^2.
1. Please provide the range of XO that will optimally pay effort 0, 0.3, and 0.5 respectively when they
correctly calculate their probability of being sick.
2. If we consider an insurance policy that covers the loss in full, and the premium is 6.5. Assume x0 =
0.8.
(a) If buyers are risk neutral, will they purchase the policy? Given the optimal decision of whether or
not to purchase the policy, what is the resulting probability of getting sick, and what is the expected
medical expenditure (before insurer reimburses, if any)?
(b) Is the insurer's profit positive, negative, or zero? Does there exist a premium level that satisfies
the following two conditions: (i) the buyers are willing to purchase the insurance; (ii) the insurer
earns a nonnegative profit? Explain.
(c) What contractual changes will you make to let a market exist such that the two conditions hold?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1710f1d-64f0-431d-a33d-5715f3afd81b%2F733a28be-6f57-457f-9554-806d49151c87%2Ftwm2wi5_processed.png&w=3840&q=75)
Transcribed Image Text:The probability of getting sick is 0.05-0.03x and the expenditure is 200 if being sick. Moreover,
health condition x depends on effort. The relation between effort and x is x = x0 exp(e-0.5). XO is a
random variable that follows uniform distribution U[0,1] and e is the effort level that can take three
values: 0, 0.3 and 0.5. The private cost of effort is c(e) = 10e^2.
1. Please provide the range of XO that will optimally pay effort 0, 0.3, and 0.5 respectively when they
correctly calculate their probability of being sick.
2. If we consider an insurance policy that covers the loss in full, and the premium is 6.5. Assume x0 =
0.8.
(a) If buyers are risk neutral, will they purchase the policy? Given the optimal decision of whether or
not to purchase the policy, what is the resulting probability of getting sick, and what is the expected
medical expenditure (before insurer reimburses, if any)?
(b) Is the insurer's profit positive, negative, or zero? Does there exist a premium level that satisfies
the following two conditions: (i) the buyers are willing to purchase the insurance; (ii) the insurer
earns a nonnegative profit? Explain.
(c) What contractual changes will you make to let a market exist such that the two conditions hold?
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