Question

Arnold has $1000 to put toward consumption this month. He believes there is a 40% chance he will fall ill this month, in which case he expects to incur medical costs of $900 (leaving him with $100 to put toward consumption). Arnold's utility over consumption is given by U=log(c)

where c is consumption (in dollars). In the absence of any insurance, the expected value of Arnold's consumption this month is $_____ . In the absence of any insurance, the expected utility Arnold receives from his consumption this month is_____utils (enter only numbers in the blanks, and please round to the second decimal place if necessary).

Consider Arnold from the previous question. What is the actuarially fair premium for $900 worth of insurance coverage to be paid in the event he falls ill this year? (enter only a number in the blank, and please round to the second decimal place if necessary).

Hint: This question is asking for the actuarially fair premium amount in dollars, not the actuarially fair premium share of the insurance coverage (i.e., $bX from class, not b).

Consider Arnold from the previous questions. Suppose that Arnold also has the option of purchasing $450 worth of insurance coverage for falling ill at an actuarially fair price. The expected value of Arnold's consumption with this partial insurance coverage would be $____. Arnold's expected utility with this partial insurance coverage would be___ utils (enter only numbers in the blanks, and please round the the second decimal place).

Answer 1

Consider the given problem here “Arnold” have two possibility that he will be able to consume “$1,000” with probability “0.6” and he will be able to consumer only “$100” with probability “0.4” if he will feel sick.

So, in the absence of the insurance the expected value of “Arnold’s” consumption is “0.6*1,000 + 0.4*100 = 640.

Now, under the 1^st situation he will get utility of “U1 = log(1,000) = 3 and in the 2^nd case the corresponding utility is given by, “U2 = log(100) = 2”. So, the expected utility in the absence of insurance is given below.

=> EU = 3*0.6 + 2*0.4 = 2.6 utils”.

Now a fair premium must be equal to the expected loss, => the fair premium is “0.4*$900 = $360”, for the “$900” of insurance coverage to be paid in the event he falls ill this year.

Now, suppose that “Arnold” also has the option of purchasing “$450” worth of insurance coverage at an actual fair price that at “0.4*900 =$360”.

So, the expected value is given by, “EV = 0.6*(1000 – 360) + 0.4*(1000 – 900 – 360 + 450)

=> 0.6*640 + 0.4*190 = 460.

So, the utility at “1000 – 360 = 640” is “2.81” and from “190” is “2.28”. So, the expected utility is given by, => “0.6*2.81 + 0.4*2.28 = 2.59”.