PROBLEM (6) A farmer with expected utility preferences with 𝒖(𝒙) = √𝒙 can experience a Bountiful

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PROBLEM (6) A farmer with expected utility preferences
with 𝒖(𝒙) = √𝒙 can experience a Bountiful or
a Dry year with probabilities %80 and %20, and
with $100 and $25 worth of crops,
respectively.
(a) Calculate the expected
value and expected utility of the “lottery” the
farmer is facing. What is the certainty
equivalent and risk premium of this lottery for the
farmer?
(b) The farmer’s risk-neutral friend offers him the
following “insurance” scheme:
“Give me $B if the year is bountiful and I will
compensate you with $D if the year is dry”
What should the numbers B and D be so that
the friend would be willing to offer such a scheme and the farmer
would want to accept it? (Just write down the conditions, that is,
the mathematical inequalities
that B and D should satisfy; don’t try to
solve these equations !)
(c) For which set of (B,D) values: (i)
(19,96) or (ii) (36,56) or (iii)
(19,75) the friend would offer and the farmer would accept the
scheme? (Plug the values into the inequalities you found in (b) to
check)
PLEASE ANSWER ALL THE PARTS!