Question

Consider a risk averse individual who faces uncertainty with two outcomes: good, bad. The individual has income $360 under good and $90 under bad outcome. The probability of good outcome is 5/9 (so the probability of bad outcome is 1 − 5/9 = 4/9). The individual can buy any non-negative x units of insurance. Every unit of insurance has price $p and it pays $1 in the event of bad outcome.

(a) Suppose the unit price of insurance is p = 1/2. Determine if the insurance market is competitive or not.

(b) Suppose the individual buys x units of insurance. Determine the individual's net income under good income, net income under bad income and the average net income. Draw these three in a diagram as functions of x.

(c) For the individual: (i) compare full insurance with over insurance and (ii) compare full insurance with partial insurance. Then determine best choice of insurance for the individual. (d) Suppose the individual is risk neutral instead of risk averse. Determine best choice of insurance for the individual.

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Answer 1

(A)

An individual is considered to be risk averse when he prefers to have the expected value of the utility from his wealth rather than facing the risk. He always faces a concave utility function.

                  E(u(w)) < u(E(w)) [if consumer is risk averse]

It means that the expected utility of wealth is less than the utility of the expected wealth.Actuarially fair insurance is the one where premium equals the expected claim. An expected claim is the product of the expected probability(p) and the amount(A) that the insurance company pays in bad times, that is, premium = p(A). It is given that probability of bad outcome = 4/9. We also know that the person will claim insurance only when something unfortunate happens to him. Here, the individual is paid $1 by the insurance company. So, actuarially fair insurance is premium = expected claim

                                                   Premium = (4/9)*1

                                                   Premium = (4/9)(1)= $0.44

Also, the unit price of insurance is given to be $p.

(a) P = 1/2, actuarially fair insurance is premium = expected claim  

Premium = (1/2)(1) = $0.5

It means that the insurance is more insurance premium than the desired. The insurance company must charge $0.44 as its premium, whereas it is charging $0.5. Therefore, $0.5 > $0.44.So, the insurance market is not competitive. Insurance companies must charge a premium amount as equal to the expected claim.

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