Question
In: Economics
(15) There are two groups of individuals, each with a utility function given by U(M)=√M, where...
(15) There are two groups of individuals, each with a utility
function given by U(M)=√M, where M=8,100 is the initial wealth
level for every individual. The fraction of group 1 is 3/4 . Each
member of group faces a loss of 7,200 with probability 1/2. Each
member of group faces a loss of 3,200 with probability 1/2. Assume
that the insurance market is perfectively competitive. And the
insurance company does not know who belongs to which group.
(5) What is the most a member of each group would be willing
to pay to insure against this loss?
(5) Suppose that the insurance market is perfectively
competitive. And the insurance company does not know who belongs to
which group. What will be the equilibrium level of insurance
premium when both groups are expected to purchase the insurance? At
this premium, will both groups indeed purchase the insurance?
Explain.
(5) Now suppose that the fraction of group 1 is α instead of
3/4. Then find the range of α for which the adverse selection
problem does not occur.
Solutions
Expert Solution
5)
In case of group 1,
Expected utility=1/2*(8100)^0.5+1/2*(8100-7200)^0.5=60 utils
WTP for insurance, X, can be determined as under
U(8100-X)=60
(8100-X)^0.5=60
8100-X=3600
X=8100-3600=$4500
In case of group 2,
Expected utility=1/2*(8100)^0.5+1/2*(8100-3200)^0.5=80 utils
WTP for insurance, Y, can be determined as under
U(8100-Y)=80
(8100-Y)^0.5=80
8100-Y=6400
Y=8100-6400=$1700
6)
Equilibrium insurance price=expected WTP=(3/4)*4500+(1/4)*1700=$3800
Since equilibrium insurance price is higher than the WTP of group 2. Group 2 will not buy the insurance.
Only group 1 will buy the insurance.
7)
Equilibrium insurance price is given as
Equilibrium insurance price=expected WTP=
*4500+(1-)*1700=4500+1700-1700
Equilibrium insurance price=2800+1700
Equilibrium insurance cost for company for group 1=expected payout=(1/2)*7200=$3600
Equilibrium insurance cost for company for group 2=expected payout=(1/2)*3200=$1600
Problem of adverse selection will not be there if
Equilibrium price
3600
2800+17003600
28001900
0.6786
Range of
=[0.6786]
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