Question

Suppose there is a 10% chance that a risk-averse individual with a current wealth of $20,000 will contract a debilitating disease and suffer a loss of $10,000.

A. Calculate the premium (P), for actuarially fair insurance in this situation and use a utility of wealth graph to show that the individual will prefer fair insurance against this loss to accepting the risk uninsured.

B. Show on your graph how much the consumer would be willing to pay and still buy the insurance , Label it P' .

Answer 1

Let's assume that utility function of a risk-averse individual with a current wealth of $20,000 is W^1/2, where W stands for wealth. This is actually a typical utility function for risk-averse individuals. Now, because of the uncertainty regarding the health of consumer we find expected utility of the individual from the given wealth.

Expected Wealth = probability of getting disease* ( remaining wealth )+probability of not getting the disease*( the total wealth)

= 0.1(10000)+0.9(20000) = 1000+18000=19000

Expected Utility = probability of getting disease* (Utility from remaining wealth )+probability of not getting the disease*(utility from the total wealth)

= .1(10000^1/2)+.9(20000^1/2) = .1(100)+0.9(141.4) = 136.26 call it equation 1

(a) Actuarially fair insurance premium is calculated with the following formula: (Probability of loss)*(size of loss)

= 0.1*10000=1000

utility of consumer after paying for insurance premium = probability of getting disease* (Utility from remaining wealth )+probability of not getting the disease*(utility from the remaning wealth)

Wealth remaining in case the individual does not get the disease = 20000-1000=19000

Wealth remaining in case the individual does get the disease and recover full money back from insurance= 20000-10000-1000+20000=29000

utility of consumer after paying for insurance premium=0.1(29000^1/2)+0.9(19000^1/2)

= 0.1*170.2+0.9*137.8= 141.02 call it equation 2

So, we see that utility of individual is higher in case of insurance rather than not having insurance as value in equation 2 is greater than value in equation 1.

B. Maximum willingness to pay is maximum the individual pays to avaoid uncertainty. In the diagram it is represented by the point E*.

E* is found by plugging the value 136.2 in utility function U= W^1/2

  136.2=W^1/2

W = 18496

So. the maximum willingness to pay will be 20000-18496 = 1504