Problem on consumers marginal utility of income

Question

Consider a consumer with probability p of becoming sick.  Let Is be the consumer’s income if he becomes sick, and let I_ns be his income if he does not become sick, with I_s < I_ns.

Suppose the consumer cares only about his expected utility of income, which is given by:

Expected Utility = p U (I_s) + (1-p) U (I_ns)
 
(1) Suppose that the consumer’s marginal utility of income, ∂U/∂I, increases with I. (That is extra income is valued more when this consumer is richer). What does this say about the consumer’s attitude toward risk?
 
Draw the consumer’s utility curve in a plane with utility on the y-axis and income on the x-axis, showing how utility changes with income.
 
On this same graph, show the consumer’s utility when he is sick and when he is well.
 
Show the consumer’s expected utility when p = 1/2.
 
(2) Suppose now that this consumer is considering the purchase of an insurance plan that will charge α₁ to the consumer when he is healthy, and provide α₂ to the consumer (on net) when he is sick.  Let α = (α₁, α₂) represent this insurance contract.
 
Provide a definition for actuarially fair insurance in this context.  Provide a definition for full insurance in this context. [Use algebra to develop these definitions].
 
What will α1 be (in terms of p, I_s, and I_ns) for an actuarially fair full insurance plan? 
What will α2 be (in terms of p, I_s, and I_ns) for an actuarially fair full insurance plan? 
 
(3) On a fresh copy of your graph from question B(1), show the consumer’s utility level after the purchase of an actuarially fair, full insurance plan.  Also show the consumer’s expected utility level before the purchase of any insurance plan.  How does the purchase of an actuarially fair, full insurance plan affect the consumer’s welfare?
 
What happens to the change in utility from the purchase of actuarially fair, full insurance as the probability of illness approaches zero (the consumer is certainly well)?
 
What happens to the change in utility from the purchase of actuarially fair full insurance as the probability of illness approaches one (the consumer is certainly sick)?