What value of a would imply that the offered policy was actuarially fair?

Continuous demand for insuarance: What fraction of a person’s potential losses
will they choose to insure if they are free to choose any level of insurance?
Consider the following model. Sam has an income of W , and with probability p
experiences a loss of L ≤ W . An insurance company offers a range of insurance

policies. A policy that pays Sam I in the event of a loss can be purchased for a
premium of a × I . Sam must choose an insurance level I . After Sam makes his

choice, the loss is realized or not, and Sam consumes his available resources.
Sam’s utility from consumption C is ln(C ).
(a) Leaving I undetermined for now (i.e. just as a variable I , and not an
optimal choice), write down expressions for Sam’s consumption if the loss
occurs and if the loss does not occur.
(b) Using these expressions, write down Sam’s expected utility.
(c) Find the insurance level I that maximizes Sam’s expected utility.
(d) What value of a would imply that the offered policy was actuarially fair?
(e) If offered insurance at this actuarially fair price, what insurance level I
would Sam choose?
(f ) If a is higher than the actuarially fair level, will Sam choose full insurance,
partial insurance, or no insurance?