Lucy and Henry each have $2159. Each knows that with 0.1 probability, they will
ID: 1132506 • Letter: L
Question
Lucy and Henry each have $2159. 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 S0.1. 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 u (, Henry's utility is given by u(x) VT 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 S0.2. What is Lucy's utility maximising choice of a? e) What is Henry's utility maximising choice of a with the new price?Explanation / Answer
a) The value of amount to be lost with a probability of 0.1 = 85% of $2159 = 1835.15.
Therefore, the expected amount of loss = Value of loss * Probability of loss = $1835.15*0.1 = $183.51
b) Here, they want the certain value of the loss that is equivalent to the loss of $1836.15 with 0.1 probability. So, we have to calculate the certainty equivalent of this loss.
The probability is 0.1 that he loses $1836 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.
Expeced utility of loss = 0.1* squarerootof(1836) + 0.9* squarerootof(zero)
= 0.1* 42.84 = 4.28
Since u= square root of the amount .So the equivalent amount = square of utility.
So, the certain amount of loss = square(4.28) = 18.31
c) In order to find the utility maximizing 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 2159 - (0.1*alpha)
Utility of Lucy = Amount of wealth(x)
The expected utility of wealth without insurance = 0.9*2159 + 0.1* (2159- 1836) = 1943.1 +32.3 = 1975.4.
Equating these two, we get,
2159 -(0.1*alpha) = 1975.4
Therefore 0.1 alpha= 183.6 (1)
alpha = 1836 units
so, the number of units of insurance purchased by Lucy for maximizing her utility is 1836.
d) 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 = 183.6
alpha = 918 unitsSo, te amount of insurance purchased reduces from 1836 to 918 when the cost of insurance increases from 0.1 to 0.2.
e) Now Henry's utility function is given by square root of amount of wealth x.
Therefore, his utility after purchase of insurance = squareroot(2159 - 0.2alpha) and his expected utility without insurance is 0.9* squareroot(2159) + 0.1* squareroot(323)
Equating the two, we get,
squareroot(2159 - 0.2alpha) = (0.9* 46.46) + 0.1 *17.97
squareroot(2159- 0.2alpha) = 43.604
Squaring both sides, we get,
2159 - 0.2 alpha = 1901.3
alpha = 1288.5 units
Therefore, Henry will purchase 1288.5 units of insurance at the new rate.
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