12
© 2010 W. W. Norton & Company, Inc.
Uncertainty
Uncertainty is Pervasive
 What
is uncertain in economic
systems?
– tomorrow’s prices
– future wealth
– future availability of commodities
– present and future actions of other
people.
© 2010 W. W. Norton & Company, Inc.
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Uncertainty is Pervasive
 What
are rational responses to
uncertainty?
– buying insurance (health, life, auto)
– a portfolio of contingent
consumption goods.
© 2010 W. W. Norton & Company, Inc.
3
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 = 1.
 Accident causes a loss of $L.
© 2010 W. W. Norton & Company, Inc.
4
Contingencies
A
contract implemented only when a
particular state of Nature occurs is
state-contingent.
 E.g. the insurer pays only if there is
an accident.
© 2010 W. W. Norton & Company, Inc.
5
Contingencies
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.
© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
$1 of accident insurance costs .
 Consumer has $m of wealth.
 Cna is consumption value in the noaccident state.
 Ca is consumption value in the
accident state.
 Each
© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
Ca
© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
A state-contingent consumption
with $17 consumption value in the
accident state and $20 consumption
value in the no-accident state.
20
17
© 2010 W. W. Norton & Company, Inc.
Ca
9
State-Contingent Budget
Constraints
 Without
insurance,
 Ca = m - L
 Cna = m.
© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
m
The endowment bundle.
mL
© 2010 W. W. Norton & Company, Inc.
Ca
11
State-Contingent Budget
Constraints
 Buy
$K of accident insurance.
 Cna = m - K.
 Ca = m - L - K + K = m - L + (1- )K.
© 2010 W. W. Norton & Company, Inc.
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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- )
© 2010 W. W. Norton & Company, Inc.
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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- )
© 2010 W. W. Norton & Company, Inc.
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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- )
m  L

 I.e.
Cna 

Ca
1
1
© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
m  L

Cna 

Ca
1
1
Cna
m
The endowment bundle.
mL
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
m  L

Cna 

Ca
1
1
Cna
m
The endowment bundle.
slope  
mL

1
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
m  L

Cna 

Ca
1
1
Cna
m
The endowment bundle.
slope  
mL

1
Where is the
most preferred
state-contingent
consumption plan?
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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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
© 2010 W. W. Norton & Company, Inc.
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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
1
1
EU   U($90)   U($0)
2
2
1
1
  12   2  7.
2
2
© 2010 W. W. Norton & Company, Inc.
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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
1
1
EM   $90   $0  $45.
2
2
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
 EU
= 7 and EM = $45.
 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  riskneutrality.
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
12
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
23
Preferences Under Uncertainty
U($45) > EU  risk-aversion.
12
U($45)
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
24
Preferences Under Uncertainty
U($45) > EU  risk-aversion.
12
U($45)
MU declines as wealth
rises.
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
25
Preferences Under Uncertainty
12
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
26
Preferences Under Uncertainty
U($45) < EU  risk-loving.
12
EU=7
U($45)
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
27
Preferences Under Uncertainty
U($45) < EU  risk-loving.
12
MU rises as wealth
rises.
EU=7
U($45)
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
28
Preferences Under Uncertainty
12
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
29
Preferences Under Uncertainty
U($45) = EU  risk-neutrality.
12
U($45)=
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
30
Preferences Under Uncertainty
U($45) = EU  risk-neutrality.
12
MU constant as wealth
rises.
U($45)=
EU=7
2
$0
$45
© 2010 W. W. Norton & Company, Inc.
$90
Wealth
31
Preferences Under Uncertainty
 State-contingent
consumption plans
that give equal expected utility are
equally preferred.
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
Cna
Indifference curves
EU1 < EU2 < EU3
EU3
EU2
EU1
Ca
© 2010 W. W. Norton & Company, Inc.
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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).
 For constant EU, dEU = 0.
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
EU   1U(c1 )   2U(c 2 )
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
EU   1U(c1 )   2U(c 2 )
dEU   1MU(c1 )dc1   2MU(c 2 )dc 2
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
EU   1U(c1 )   2U(c 2 )
dEU   1MU(c1 )dc1   2MU(c 2 )dc 2
dEU  0   1MU(c1 )dc1   2MU(c 2 )dc 2  0
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
EU   1U(c1 )   2U(c 2 )
dEU   1MU(c1 )dc1   2MU(c 2 )dc 2
dEU  0   1MU(c1 )dc1   2MU(c 2 )dc 2  0
  1MU(c1 )dc1   2MU(c 2 )dc 2
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
EU   1U(c1 )   2U(c 2 )
dEU   1MU(c1 )dc1   2MU(c 2 )dc 2
dEU  0   1MU(c1 )dc1   2MU(c 2 )dc 2  0
  1MU(c1 )dc1   2MU(c 2 )dc 2
dc 2
 1MU(c1 )


.
dc1
 2MU(c 2 )
© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty
Cna
Indifference curves
EU1 < EU2 < EU3
dcna
 a MU(ca )

dca
 na MU(cna )
EU3
EU2
EU1
Ca
© 2010 W. W. Norton & Company, Inc.
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Choice Under Uncertainty
 Q:
How is a rational choice made
under uncertainty?
 A: Choose the most preferred
affordable state-contingent
consumption plan.
© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
m  L

Cna 

Ca
1
1
Cna
m
The endowment bundle.
slope  
mL

1
Where is the
most preferred
state-contingent
consumption plan?
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
m  L

Cna 

Ca
1
1
Cna
m
The endowment bundle.
slope  
Affordable
plans
mL

1
Where is the
most preferred
state-contingent
consumption plan?
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
More preferred
m
Where is the
most preferred
state-contingent
consumption plan?
mL
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
Most preferred affordable plan
m
mL
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
Most preferred affordable plan
m
mL
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
Most preferred affordable plan
MRS = slope of budget
constraint
m
mL
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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State-Contingent Budget
Constraints
Cna
Most preferred affordable plan
MRS = slope of budget
constraint; i.e.
m

 a MU(ca )

1    na MU(cna )
mL
m  L Ca

© 2010 W. W. Norton & Company, Inc.
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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.
 If price of $1 insurance = accident
probability, then insurance is fair.
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 When
insurance is fair, rational
insurance choices satisfy

a
 a MU(ca )


1   1   a  na MU(cna )
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 When
insurance is fair, rational
insurance choices satisfy

a
 a MU(ca )


1   1   a  na MU(cna )
 I.e.
MU(ca )  MU(cna )
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 When
insurance is fair, rational
insurance choices satisfy

a
 a MU(ca )


1   1   a  na MU(cna )
MU(ca )  MU(cna )
 Marginal utility of income must be
the same in both states.
 I.e.
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 How
much fair insurance does a riskaverse consumer buy?
MU(ca )  MU(cna )
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 How
much fair insurance does a riskaverse consumer buy?
MU(ca )  MU(cna )
 Risk-aversion  MU(c)  as c .
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 How
much fair insurance does a riskaverse consumer buy?
MU(ca )  MU(cna )
 Risk-aversion  MU(c)  as c .
ca  cna .
 Hence
© 2010 W. W. Norton & Company, Inc.
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Competitive Insurance
 How
much fair insurance does a riskaverse consumer buy?
MU(ca )  MU(cna )
 Risk-aversion  MU(c)  as c .
ca  cna .
 Hence
 I.e. full-insurance.
© 2010 W. W. Norton & Company, Inc.
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“Unfair” Insurance
 Suppose
insurers make positive
expected economic profit.
 I.e. K - aK - (1 - a)0 = ( - a)K > 0.
© 2010 W. W. Norton & Company, Inc.
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“Unfair” Insurance
 Suppose
insurers make positive
expected economic profit.
 I.e. K - aK - (1 - a)0 = ( - a)K > 0.

a
 Then   > a 

.
1 1a
© 2010 W. W. Norton & Company, Inc.
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“Unfair” Insurance
 Rational
choice requires

 a MU(ca )

1    na MU(cna )
© 2010 W. W. Norton & Company, Inc.
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“Unfair” Insurance
 Rational
choice requires

 a MU(ca )

1    na MU(cna )
 Since

a

, MU(ca ) > MU(cna )
1 1a
© 2010 W. W. Norton & Company, Inc.
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“Unfair” Insurance
 Rational
choice requires

 a MU(ca )

1    na MU(cna )
a

, MU(ca ) > MU(cna )
1 1a
 Hence ca < cna for a risk-averter.
 Since

© 2010 W. W. Norton & Company, Inc.
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“Unfair” Insurance
 Rational
choice requires

 a MU(ca )

1    na MU(cna )
a

, MU(ca ) > MU(cna )
1 1a
 Hence ca < cna for a risk-averter.
 Since

 I.e.
a risk-averter buys less than full
“unfair” insurance.
© 2010 W. W. Norton & Company, Inc.
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Uncertainty is Pervasive
 What
are rational responses to
uncertainty?
– buying insurance (health, life, auto)
– a portfolio of contingent
consumption goods.
© 2010 W. W. Norton & Company, Inc.
63
Uncertainty is Pervasive
 What
are rational responses to
uncertainty?
 – buying insurance (health, life, auto)
– a portfolio of contingent
consumption goods.
© 2010 W. W. Norton & Company, Inc.
64
Uncertainty is Pervasive
 What
are rational responses to
uncertainty?
 – buying insurance (health, life, auto)
? – a portfolio of contingent
consumption goods.
© 2010 W. W. Norton & Company, Inc.
65
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.
 You have $100 to invest. How?
© 2010 W. W. Norton & Company, Inc.
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Diversification
 Buy
only firm A’s stock?
 $100/10 = 10 shares.
 You earn $1000 with prob. 1/2 and
$200 with prob. 1/2.
 Expected earning: $500 + $100 = $600
© 2010 W. W. Norton & Company, Inc.
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Diversification
 Buy
only firm B’s stock?
 $100/10 = 10 shares.
 You earn $1000 with prob. 1/2 and
$200 with prob. 1/2.
 Expected earning: $500 + $100 = $600
© 2010 W. W. Norton & Company, Inc.
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Diversification
 Buy
5 shares in each firm?
 You earn $600 for sure.
 Diversification has maintained
expected earning and lowered risk.
© 2010 W. W. Norton & Company, Inc.
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Diversification
 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.
© 2010 W. W. Norton & Company, Inc.
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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
0  99  $40,000  0  01($40,000  $10,000)
 $39,900.
© 2010 W. W. Norton & Company, Inc.
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Risk Spreading/Mutual Insurance
 Mutual
insurance: Expected loss is
0  01  $10,000  $100.
 Each of the 100 persons pays $1 into
a mutual insurance fund.
 Mutual insurance: expected wealth is
$40,000  $1  $39,999  $39,900.
 Risk-spreading benefits everyone.
© 2010 W. W. Norton & Company, Inc.
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