– buying insurance (health, life, auto) – a portfolio of contingent.. consumption goods...[r]
(1)Chapter Twelve
(2)Uncertainty is Pervasive
What is uncertain in economic systems?
– tomorrow’s prices – future wealth
– future availability of commodities – present and future actions of other
(3)Uncertainty is Pervasive
What are rational responses to uncertainty?
– buying insurance (health, life, auto) – a portfolio of contingent
(4)States 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 =
(5)Contingencies
A contract implemented only when a particular state of Nature occurs is
state-contingent.
(6)Contingencies
A state-contingent consumption plan is implemented only when a
particular state of Nature occurs.
(7)State-Contingent Budget Constraints
Each $1 of accident insurance costs
.
Consumer has $m of wealth.
Cna is consumption value in the no-accident state.
(8)State-Contingent Budget Constraints
Cna
(9)State-Contingent Budget Constraints
Cna
C 20
(10)State-Contingent Budget Constraints
Without insurance, Ca = m - L
(11)State-Contingent Budget Constraints
Cna
C m The endowment bundle.
(12)State-Contingent Budget Constraints
Buy $K of accident insurance. Cna = m - K.
(13)State-Contingent Budget Constraints
Buy $K of accident insurance. Cna = m - K.
(14)State-Contingent Budget Constraints
Buy $K of accident insurance. Cna = m - K.
Ca = m - L - K + K = m - L + (1- )K. So K = (Ca - m + L)/(1- )
(15)State-Contingent Budget Constraints
Buy $K of accident insurance. Cna = m - K.
Ca = m - L - K + K = m - L + (1- )K. So K = (Ca - m + L)/(1- )
And Cna = m - (Ca - m + L)/(1- )
I.e. C m L C
(16)State-Contingent Budget Constraints
Cna
Ca m The endowment bundle.
m L
Cna m L Ca
1 1
(17)State-Contingent Budget Constraints
Cna
C m The endowment bundle.
slope
1
Cna m L Ca
1 1
(18)State-Contingent Budget Constraints
Cna
Ca m The endowment bundle.
Where is the most preferred state-contingent
consumption plan?
Cna m L Ca
1 1 slope 1
m L
(19)Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win $0 with probability 1/2
(20)Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win $0 with probability 1/2
U($90) = 12, U($0) = 2. Expected utility is
EU U($90) U($0) 1 2 1 2 1 2 12 1
(21)Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win $0 with probability 1/2
Expected money value of the lottery is
EM 1 $90 $0 2
1
(22)Preferences Under Uncertainty
EU = and EM = $45.
U($45) > $45 for sure is preferred
to the lottery risk-aversion.
U($45) < the lottery is preferred to
$45 for sure risk-loving.
U($45) = the lottery is preferred
(23)Preferences Under Uncertainty
Wealth
$0 $90
2 12
(24)Preferences Under Uncertainty
Wealth
$0 $90
12 U($45)
U($45) > EU risk-aversion.
2 EU=7
(25)Preferences Under Uncertainty
Wealth
$0 $90
12 U($45)
U($45) > EU risk-aversion.
2 EU=7
$45
(26)Preferences Under Uncertainty
Wealth
$0 $90
12
2 EU=7
(27)Preferences Under Uncertainty
Wealth
$0 $90
12
U($45) < EU risk-loving.
2 EU=7
(28)Preferences Under Uncertainty
Wealth
$0 $90
12
U($45) < EU risk-loving.
2 EU=7
$45
MU rises as wealth rises.
(29)Preferences Under Uncertainty
Wealth
$0 $90
12
2 EU=7
(30)Preferences Under Uncertainty
Wealth
$0 $90
12
U($45) = EU risk-neutrality.
2 U($45)= EU=7
(31)Preferences Under Uncertainty
Wealth
$0 $90
12
U($45) = EU risk-neutrality.
2
$45
MU constant as wealth rises.
(32)Preferences Under Uncertainty
(33)Preferences Under Uncertainty
Cna
C
EUEU12 EU3
(34)Preferences Under Uncertainty
What is the MRS of an indifference curve?
Get consumption c1 with prob 1 and c2 with prob 2 (1 + 2 = 1).
EU = 1U(c1) + 2U(c2).
(35)Preferences Under Uncertainty
(36)Preferences Under Uncertainty
EU 1U(c )1 2U(c )2
(37)Preferences Under Uncertainty
EU 1U(c )1 2U(c )2
(38)Preferences Under Uncertainty
EU 1U(c )1 2U(c )2
1MU(c )dc1 1 2MU(c )dc2 2 dEU 1MU(c )dc1 1 2MU(c )dc2 2
(39)Preferences Under Uncertainty
EU 1U(c )1 2U(c )2
dc2 1MU(c )1
dEU 1MU(c )dc1 1 2MU(c )dc2 2
(40)Preferences Under Uncertainty
Cna
Ca
EUEU12 EU3
Indifference curves EU1 < EU2 < EU3
(41)Choice Under Uncertainty
Q: How is a rational choice made under uncertainty?
A: Choose the most preferred affordable state-contingent
(42)State-Contingent Budget Constraints
Cna
Ca m The endowment bundle.
Cna m L Ca
1 1
Where is the most preferred state-contingent consumption plan? slope 1
m L
(43)State-Contingent Budget Constraints
Cna
C m The endowment bundle.
Where is the most preferred state-contingent
consumption plan? Affordable
plans
Cna m L Ca
1 1 slope 1
(44)State-Contingent Budget Constraints
Cna
Ca m
Where is the most preferred state-contingent
consumption plan? More preferred
m L
(45)State-Contingent Budget Constraints
Cna
C m
Most preferred affordable plan
(46)State-Contingent Budget Constraints
Cna
Ca m
Most preferred affordable plan
m L
(47)State-Contingent Budget Constraints
Cna
C m
Most preferred affordable plan MRS = slope of budget constraint
(48)State-Contingent Budget Constraints
Cna
Ca m
Most preferred affordable plan MRS = slope of budget constraint; i.e.
m L
m L
1
(49)Competitive Insurance
Suppose entry to the insurance industry is free.
Expected economic profit = 0.
I.e K - aK - (1 - a)0 = ( - a)K = 0. I.e free entry = a.
(50)Competitive Insurance
When insurance is fair, rational insurance choices satisfy
1 1
(51)Competitive Insurance
When insurance is fair, rational insurance choices satisfy
I.e. MU(c ) MU(ca na ) 1 1
(52)Competitive Insurance
When insurance is fair, rational insurance choices satisfy
I.e.
Marginal utility of income must be the same in both states.
1 1
a a a na MU(c ) MU(c ) a na
(53)Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
(54)Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
(55)Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
Risk-aversion MU(c) as c . Hence
(56)Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
Risk-aversion MU(c) as c . Hence
I.e full-insurance.
(57)“Unfair” Insurance
Suppose insurers make positive expected economic profit.
(58)“Unfair” Insurance
Suppose insurers make positive expected economic profit.
I.e K - aK - (1 - a)0 = ( - a)K > 0. Then > a
1 1
a
(59)“Unfair” Insurance
Rational choice requires
1
a na
MU(c ) MU(c )
(60)“Unfair” Insurance
Rational choice requires
Since
1
a na MU(c ) MU(c ) a na 1 1
a a
(61)“Unfair” Insurance
Rational choice requires
Since
Hence for a risk-averter.
1
a na MU(c ) MU(c ) a na 1 1
a a
(62)“Unfair” Insurance
Rational choice requires
Since
Hence for a risk-averter.
I.e a risk-averter buys less than full
“unfair” insurance.
1
a na MU(c ) MU(c ) a na 1 1
a a
(63)Uncertainty is Pervasive
What are rational responses to uncertainty?
– buying insurance (health, life, auto) – a portfolio of contingent
(64)Uncertainty is Pervasive
What are rational responses to uncertainty?
– buying insurance (health, life, auto) – a portfolio of contingent
consumption goods.
(65)Uncertainty is Pervasive
What are rational responses to uncertainty?
– buying insurance (health, life, auto) – a portfolio of contingent
consumption goods.
(66)Diversification
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.
(67)Diversification
Buy only firm A’s stock? $100/10 = 10 shares.
You earn $1000 with prob 1/2 and $200 with prob 1/2.
(68)Diversification
Buy only firm B’s stock? $100/10 = 10 shares.
You earn $1000 with prob 1/2 and $200 with prob 1/2.
(69)Diversification
Buy shares in each firm? You earn $600 for sure.
Diversification has maintained
(70)Diversification
Buy shares in each firm? You earn $600 for sure.
Diversification has maintained
expected earning and lowered risk. Typically, diversification lowers
(71)Risk 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
(72)Risk 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.
0 01 $ ,10 000 $100