Your utility function is U = square root of C, where C is the amount of consumption that you have in any given period. Y

[answerhappygod]

Your utility function is U = square root
of C, where C is the amount of consumption that you have in
any given period. Your income is $90,000 per year, and there is a
2% chance that you will be involved in a catastrophic accident that
will cost you $80,000 next year. Note that you spend all of your
income in any state of the world so that income and consumption are
the same in any given state of the world (i.e., accident/ no
accident).
a) What is your Utility if there is no
accident?
b) What is your (statistically) Expected
Utility?
c) What is your statistically expected or average income
over different states of the world?
d) Calculate an actuarially fair insurance premium. This
is the amount you would pay every year to cover the average cost of
the accident based on its probability over the years; it is the
difference between the no-accident income and your actual expected
or average income from part c). This is the premium that insurance
companies would collect and pay out entirely to those unfortunate
enough to have an accident, with no profit or other funds left to
the insurance company. In other words it is just a redistribution
from the fortunate to the unfortunate.
e) What is the most that you would be willing to pay for
insurance, given your utility function? Do this in two steps: i)
calculate the income (or consumption) received with certainty which
has the same utility as you would have in a world in which you are
uninsured and face the possibility of an accident -- that is, your
expected utility from part b). ii) Calculate the difference between
this and your no-accident income this is the most you would be
willing to pay to remove all risk because you would at least keep
the same level of utility or satisfaction after paying it as you
have in the actual risk-filled world.
The answer to e) is more than that in d)
which shows insurance companies can obtain premiums in excess of
what they pay out to the insured because of the risk-aversion of
the insured individuals.