Lucy and Henry each have $3108. Each knows that with 0.1 probability, they will
ID: 1131743 • Letter: L
Question
Lucy and Henry each have $3108. Each knows that with 0.1 probability, they will lose 85% of their wealth. They both have the option of buying units of insurance, with each unit costing $0.1. Each unit of insurance pays out $1 in the event the loss occurs. The cost of the insurance policy is paid regardless of whether the loss is incurred. Lucy's utility is given by uL(x)=xuL(x)=x, Henry's utility is given by uH(x)=xuH(x)=x.
Answer the following:
Write your answers to 2 decimal places.
a) Without insurance, what is the expected value of the loss?
b) For Henry, facing the "lottery " above without any insurance is as bad as losing how many dollars for sure?
c) Find Lucy's utility maximising choice of . If more than 1 exist, enter the largest .
d) Now suppose insurance costs $0.2. What is Lucy's utility maximising choice of ?
e) What is Henry's utility maximising choice of with the new price?
Lucy and Henry each have $3 108. Each knows hat with O 1 probability they will lose 85% of her eat They both have the option of buyin ts of insurance with each int o n s n a unit of insurance pays out $1 in the event the loss occurs. The cost of the insurance policy is paid regardless of whether the loss is incurred. Lucy's utility is given by u, Henry's utility is given by u(x) Answer the following Write your answers to 2 decimal places. a) Without insurance, what is the expected value of the loss? b) For Henry, facing the "mottery above without any insurance is as bad as losing how many dollars for sure? c) Find Lucy's utility maximising choice of a. If more than 1 exist, enter the largest a d) Now suppose insurance costs $0.2. What is Lucy's utility maximising choice of a? e) What is Henry's utility maxim ising choice of with the new price? CheckExplanation / Answer
a) The value of amount to be lost with a probability of 0.1 = 85% of $3108 = $ 2641.8
Therefore the expected amount of loss = Value of loss * Probability of loss = $2641.8* 0.1 = $ 264.18
b) Here, they want the certain value of loss that is equivalent to the loss of $2641 with 0.1 probability. So, we have to calculate the certainity equivalent of this loss.
The probability is 0.1 that he loses $2641 and 0.9 that he doesn't lose anything. We have to first calculate the expected utility of this loss and then calculate the certain amount that would give him the same utility.
Expected utility of loss = 0.1 * square root of (2641) + 0.9 * square root of zero
= 51.39 *0.1 = 5.139
Since u= square root of the amount . So, the equivalent amount = square of utility.
So, the certain amount of loss = Square (5.139) = $26.4.
3) In order to find the utility maximkizing amount of insurance, we have to equate the expected utility of wealth after purchasing that amount of insurance with the expected utility of wealth without insurance.
The total amount of insurance is given by alpha * 0.1. After purchase of insurance, the remaining wealth is given by 3108- (0.1* alpha).
Utility of Lucy = Amount of wealth (x)
The expected utility of wealth without insurance = 0.9* 3108 + 0.1* (3108- 2641) = 2843.82
Equating these two, we get,
3108 - (0.1* alpha) = 2843.82
Therefore, 0.1 alpha = 264.18 ...(1)
alpha = 2641.8 units
So, the number of units of insurance purchased by Lucy for maximizing her utility is 2641.8.
4) When the cost of insurance increases to 0.2, we simply replace 0.1 by 0.2 in equation (1) of part 3.
We get, 0.2 * alpha = 264.18
alpha = 1320.9 units
So, the amount of insurance purchased reduces to $ 1320.9 when the cost of insurance increases from 0.1 to 0.2.
5) Now, Henry's utility function is given by square root of the amount of wealth (x).
Therefore, his utility after purchase of insurance = square root ( 3108 - 0.2 alpha) and his expected utility without insurance is 0.9 * square root(3108) + 0.1 * square root(466.2)
Equating the two , we get,
square root ( 3108 - 0.2 alpha) = (0.9* 55.75) + ( 0.1* 21.59)
square root ( 3108- 0.2 alpha) = 52.33
Squaring both sides, we get,
3108 - 0.2 alpha = 2738.95
alpha= 1845.23 units
Therefore, henry will purchase $1845.23 units of insurance at the new rate.
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