– buying insurance (health, life, auto) – a portfolio of contingent.. consumption goods...[r]
Trang 1Chapter Twelve
Uncertainty
Trang 4States of Nature
Possible states of Nature:
– “car accident” (a)
– “no car accident” (na).
Accident occurs with probability a , does not with probability na ;
a + na = 1
Accident causes a loss of $L.
Trang 6 A state-contingent consumption plan
is implemented only when a
particular state of Nature occurs.
E.g take a vacation only if there is no accident.
Trang 7State-Contingent Budget
Constraints
Each $1 of accident insurance costs
.
Consumer has $m of wealth.
C na is consumption value in the accident state.
no- C a is consumption value in the
accident state.
Trang 8State-Contingent Budget
Constraints
Trang 19Preferences Under Uncertainty
Trang 20Preferences Under Uncertainty
1 2 1
12 1 2 7.
Trang 21Preferences Under Uncertainty
Trang 22Preferences Under Uncertainty
U($45) > 7 $45 for sure is preferred
to the lottery risk-aversion
U($45) < 7 the lottery is preferred to
$45 for sure risk-loving
U($45) = 7 the lottery is preferred
equally to $45 for sure
risk-neutrality
Trang 23Preferences Under Uncertainty
Trang 24Preferences Under Uncertainty
Trang 25Preferences Under Uncertainty
Trang 26Preferences Under Uncertainty
Trang 27Preferences Under Uncertainty
Trang 28Preferences Under Uncertainty
U($45)
Trang 29Preferences Under Uncertainty
Trang 30Preferences Under Uncertainty
Trang 31Preferences Under Uncertainty
U($45)=
EU=7
Trang 32Preferences Under Uncertainty
State-contingent consumption plans that give equal expected utility are equally preferred.
Trang 33Preferences Under Uncertainty
Trang 34Preferences Under Uncertainty
What is the MRS of an indifference curve?
Get consumption c 1 with prob 1 and
c 2 with prob 2 ( 1 + 2 = 1).
EU = 1 U(c 1 ) + 2 U(c 2 ).
For constant EU, dEU = 0.
Trang 35Preferences Under Uncertainty
EU 1 U(c ) 1 2 U(c ) 2
Trang 36Preferences Under Uncertainty
EU 1 U(c ) 1 2 U(c ) 2
dEU 1 MU(c )dc 1 1 2 MU(c )dc 2 2
Trang 37Preferences Under Uncertainty
EU 1 U(c ) 1 2 U(c ) 2
dEU 0 1 MU(c )dc 1 1 2 MU(c )dc 2 2 0 dEU 1 MU(c )dc 1 1 2 MU(c )dc 2 2
Trang 38Preferences Under Uncertainty
EU 1 U(c ) 1 2 U(c ) 2
1 MU(c )dc 1 1 2 MU(c )dc 2 2
dEU 1 MU(c )dc 1 1 2 MU(c )dc 2 2
dEU 0 1 MU(c )dc 1 1 2 MU(c )dc 2 2 0
Trang 39Preferences Under Uncertainty
EU 1 U(c ) 1 2 U(c ) 2
dc 2 1 MU(c ) 1
dEU 1 MU(c )dc 1 1 2 MU(c )dc 2 2
dEU 0 1 MU(c )dc 1 1 2 MU(c )dc 2 2 0
1 MU(c )dc 1 1 2 MU(c )dc 2 2
Trang 40Preferences Under Uncertainty
MU(c ) MU(c )
na a
a na
a na
Trang 41Choice Under Uncertainty
Q: How is a rational choice made under uncertainty?
A: Choose the most preferred
affordable state-contingent
consumption plan.
Trang 43Affordable plans
Trang 44More preferred
m L
m L
Trang 47m L
Trang 48MU(c ) MU(c )
a na
Trang 49 I.e free entry = a
If price of $1 insurance = accident
Trang 50a na
MU(c ) MU(c )
a na
Trang 51a na
MU(c ) MU(c )
a na
Trang 52Competitive Insurance
When insurance is fair, rational
insurance choices satisfy
a na
MU(c ) MU(c )
a na
MU(c ) MU(c a na )
Trang 53Competitive Insurance
How much fair insurance does a averse consumer buy?
risk-MU(c ) risk-MU(c a na )
Trang 57“Unfair” Insurance
Suppose insurers make positive
expected economic profit.
I.e K - a K - (1 - a )0 = ( - a )K > 0.
Trang 58“Unfair” Insurance
Suppose insurers make positive
expected economic profit.
Trang 59MU(c ) MU(c )
a na
Trang 60MU(c ) MU(c )
a na
Trang 61MU(c ) MU(c )
a na
Trang 62“Unfair” Insurance
Hence for a risk-averter.
I.e a risk-averter buys less than full
MU(c ) MU(c )
a na
Trang 66 Two firms, A and B Shares cost $10.
With prob 1/2 A’s profit is $100 and B’s profit is $20.
With prob 1/2 A’s profit is $20 and
B’s profit is $100.
You have $100 to invest How?
Trang 69 Buy 5 shares in each firm?
You earn $600 for sure
Diversification has maintained
expected earning and lowered risk.
Trang 70 Buy 5 shares in each firm?
You earn $600 for sure
Diversification has maintained
expected earning and lowered risk.
Typically, diversification lowers
expected earnings in exchange for lowered risk.
Trang 71Risk Spreading/Mutual Insurance
100 risk-neutral persons each
independently risk a $10,000 loss.
Loss probability = 0.01.
Initial wealth is $40,000.
No insurance: expected wealth is
0 99 $ ,40 000 0 01 40 000 ($ , $ ,10 000)
Trang 72Risk Spreading/Mutual Insurance
Mutual insurance: Expected loss is
Each of the 100 persons pays $1 into
a mutual insurance fund.
Mutual insurance: expected wealth is
Risk-spreading benefits everyone.