Expected utility

Finance student, Gothenburg university and got recommended to get help here in pluggakuten.

I wonder if someone is able to help me make sure I got the question right on a) and b) and also need help with c) from the beginning.

A consumer’s utility function is u(m) = ln(m). The value of m can be 2 or 16 with
probabilities 0.1 and 0.9, respectively. If she makes no effort to reduce risk, the
probability of m = 2 rises to 0.2. The effort cost is e = 1.

a) Verify that if there is no insurance then the consumer will make the effort to
reduce risk.

My answer: 

E[u(m,e=0) = 0.2 ln(2) + 0.8 ln(16) = 2.36
E[u(m,e=1) = 0.1 ln(2-1) + 0.8 ln(16-1) = 2.44

E[u(m,e=1) > E[u(m,e=0), so the consumer will make efforts.

b) Assume now that an insurance company can observe the effort of the
consumer and charge break-even premium rates accordingly. Find the
consumer’s optimal insurance levels and her expected utilities when she
makes effort and when she does not. Will she make the effort?

 

My answer:

Answer:
E[u(m,e=0) = 0.2 ln(2) + 0.8 ln(16) = 2.36
E[u(m,e=1) = 0.1 ln(2-1) + 0.8 ln(16-1) = 2.44

E[u(m,e=1) > E[u(m,e=0), so the consumer will make efforts.

c) Assume consumers’ efforts are not observable. An insurance company can
charge two different break-even premium rates, but only allow those paying
0.2 rate to fully insure. Find a partial insurance limit F (the larger the better)
for 0.1 rate-payers that induces them to make the effort.

 

I got stuck on this c) question, but not sure if a) and b) is correct