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12
Uncertainty

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Uncertainty is Pervasive
uWhat is uncertain in economic systems?
–tomorrow’s prices
–future wealth
–future availability of commodities
–present and future actions of other people.

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Uncertainty is Pervasive
uWhat are rational responses to uncertainty?
–buying insurance (health, life, auto)
–a portfolio of contingent consumption goods.

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States of Nature
uPossible states of Nature:
–“car accident” (a)
–“no car accident” (na).
uAccident occurs with probability pa,  does not with probability pna ;
               pa + pna = 1.
uAccident causes a loss of $L.
u

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Contingencies
uA contract implemented only when a particular state of Nature occurs is state-contingent.
uE.g. the insurer pays only if there is an accident.

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Contingencies
uA state-contingent consumption plan is implemented only when a particular state of Nature occurs.
uE.g. take a vacation only if there is no accident.

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State-Contingent Budget Constraints
uEach $1 of accident insurance costs g.
uConsumer has $m of wealth.
uCna is consumption value in the no-accident state.
uCa is consumption value in the accident state.

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State-Contingent Budget Constraints
Cna
Ca

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State-Contingent Budget Constraints
Cna
Ca
20
17
A state-contingent consumption
with $17 consumption value in the accident state and $20 consumption value in the no-accident
state.

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State-Contingent Budget Constraints
uWithout insurance,
uCa = m - L
uCna = m.

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State-Contingent Budget Constraints
Cna
Ca
m
The endowment bundle.

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State-Contingent Budget Constraints
uBuy $K of accident insurance.
uCna = m - gK.
uCa = m - L - gK + K = m - L + (1- g)K.

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State-Contingent Budget Constraints
uBuy $K of accident insurance.
uCna = m - gK.
uCa = m - L - gK + K = m - L + (1- g)K.
uSo K = (Ca - m + L)/(1- g)

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State-Contingent Budget Constraints
uBuy $K of accident insurance.
uCna = m - gK.
uCa = m - L - gK + K = m - L + (1- g)K.
uSo K = (Ca - m + L)/(1- g)
uAnd Cna = m - g (Ca - m + L)/(1- g)

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State-Contingent Budget Constraints
uBuy $K of accident insurance.
uCna = m - gK.
uCa = m - L - gK + K = m - L + (1- g)K.
uSo K = (Ca - m + L)/(1- g)
uAnd Cna = m - g (Ca - m + L)/(1- g)
uI.e.

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State-Contingent Budget Constraints
Cna
Ca
m
The endowment bundle.

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State-Contingent Budget Constraints
Cna
Ca
m
The endowment bundle.

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State-Contingent Budget Constraints
Cna
Ca
m
The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?

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Preferences Under Uncertainty
uThink of a lottery.
uWin $90 with probability 1/2 and win  $0 with probability 1/2.
uU($90) = 12,   U($0) = 2.
uExpected utility is

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Preferences Under Uncertainty
uThink of a lottery.
uWin $90 with probability 1/2 and win  $0 with probability 1/2.
uU($90) = 12,   U($0) = 2.
uExpected utility is

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Preferences Under Uncertainty
uThink of a lottery.
uWin $90 with probability 1/2 and win  $0 with probability 1/2.
uExpected money value of the lottery is

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Preferences Under Uncertainty
uEU = 7 and EM = $45.
uU($45) > 7 Þ $45 for sure is preferred to the lottery Þ risk-aversion.
uU($45) < 7 Þ the lottery is preferred to $45 for sure Þ risk-loving.
uU($45) = 7 Þ the lottery is preferred equally to $45 for sure Þ risk-neutrality.

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Preferences Under Uncertainty
Wealth
$0
$90
2
12
$45
EU=7

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Preferences Under Uncertainty
Wealth
$0
$90
12
U($45)
U($45) > EU Þ risk-aversion.
2
EU=7
$45

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Preferences Under Uncertainty
Wealth
$0
$90
12
U($45)
U($45) > EU Þ risk-aversion.
2
EU=7
$45
MU declines as wealth
rises.

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Preferences Under Uncertainty
Wealth
$0
$90
12
2
EU=7
$45

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Preferences Under Uncertainty
Wealth
$0
$90
12
U($45) < EU Þ risk-loving.
2
EU=7
$45
U($45)

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Preferences Under Uncertainty
Wealth
$0
$90
12
U($45) < EU Þ risk-loving.
2
EU=7
$45
MU rises as wealth
rises.
U($45)

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Preferences Under Uncertainty
Wealth
$0
$90
12
2
EU=7
$45

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Preferences Under Uncertainty
Wealth
$0
$90
12
U($45) = EU Þ risk-neutrality.
2
U($45)=
EU=7
$45

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Preferences Under Uncertainty
Wealth
$0
$90
12
U($45) = EU Þ risk-neutrality.
2
$45
MU constant as wealth
rises.
U($45)=
EU=7

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Preferences Under Uncertainty
uState-contingent consumption plans that give equal expected utility are equally preferred.

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Preferences Under Uncertainty
Cna
Ca
EU1
EU2
EU3
Indifference curves
EU1 < EU2 < EU3

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Preferences Under Uncertainty
uWhat is the MRS of an indifference curve?
uGet consumption c1 with prob. p1 and
c2 with prob. p2   (p1 + p2 = 1).
uEU = p1U(c1) + p2U(c2).
uFor constant EU, dEU = 0.

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Preferences Under Uncertainty

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Preferences Under Uncertainty

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© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty

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© 2010 W. W. Norton & Company, Inc.
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Preferences Under Uncertainty

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Preferences Under Uncertainty

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Preferences Under Uncertainty
Cna
Ca
EU1
EU2
EU3
Indifference curves
EU1 < EU2 < EU3

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Choice Under Uncertainty
uQ: How is a rational choice made under uncertainty?
uA: Choose the most preferred affordable state-contingent consumption plan.

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State-Contingent Budget Constraints
Cna
Ca
m
The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?

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State-Contingent Budget Constraints
Cna
Ca
m
The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?
Affordable
plans

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State-Contingent Budget Constraints
Cna
Ca
m
Where is the
most preferred
state-contingent
consumption plan?
More preferred

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State-Contingent Budget Constraints
Cna
Ca
m
Most preferred affordable plan

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State-Contingent Budget Constraints
Cna
Ca
m
Most preferred affordable plan

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State-Contingent Budget Constraints
Cna
Ca
m
Most preferred affordable plan
MRS = slope of budget
            constraint

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State-Contingent Budget Constraints
Cna
Ca
m
Most preferred affordable plan
MRS = slope of budget
            constraint; i.e.

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Competitive Insurance
uSuppose entry to the insurance industry is free.
uExpected economic profit = 0.
uI.e.  gK - paK - (1 - pa)0 = (g - pa)K = 0.
uI.e. free entry Þ g = pa.
uIf price of $1 insurance = accident probability, then insurance is fair.

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Competitive Insurance
uWhen insurance is fair, rational insurance choices satisfy
u
u

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Competitive Insurance
uWhen insurance is fair, rational insurance choices satisfy
u
u
uI.e.
u
u

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Competitive Insurance
uWhen insurance is fair, rational insurance choices satisfy
u
u
uI.e.
uMarginal utility of income must be the same in both states.
u
u

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Competitive Insurance
uHow much fair insurance does a risk-averse consumer buy?

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Competitive Insurance
uHow much fair insurance does a risk-averse consumer buy?
u
uRisk-aversion Þ MU(c) ¯ as c .

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Competitive Insurance
uHow much fair insurance does a risk-averse consumer buy?
u
uRisk-aversion Þ MU(c) ¯ as c .
uHence

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Competitive Insurance
uHow much fair insurance does a risk-averse consumer buy?
u
uRisk-aversion Þ MU(c) ¯ as c .
uHence
uI.e. full-insurance.

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“Unfair” Insurance
uSuppose insurers make positive expected economic profit.
uI.e.  gK - paK - (1 - pa)0 = (g - pa)K > 0.

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“Unfair” Insurance
uSuppose insurers make positive expected economic profit.
uI.e.  gK - paK - (1 - pa)0 = (g - pa)K > 0.
uThen Þ g > pa Þ

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“Unfair” Insurance
uRational choice requires
u
u
u

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“Unfair” Insurance
uRational choice requires
u
u
uSince
u

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“Unfair” Insurance
uRational choice requires
u
u
uSince
uHence                 for a risk-averter.
u
u
u

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“Unfair” Insurance
uRational choice requires
u
u
uSince
uHence                 for a risk-averter.
uI.e. a risk-averter buys less than full “unfair” insurance.
u

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Uncertainty is Pervasive
uWhat are rational responses to uncertainty?
–buying insurance (health, life, auto)
–a portfolio of contingent consumption goods.

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© 2010 W. W. Norton & Company, Inc.
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Uncertainty is Pervasive
uWhat are rational responses to uncertainty?
–buying insurance (health, life, auto)
–a portfolio of contingent consumption goods.
ü

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© 2010 W. W. Norton & Company, Inc.
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Uncertainty is Pervasive
uWhat are rational responses to uncertainty?
–buying insurance (health, life, auto)
–a portfolio of contingent consumption goods.
ü
?

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Diversification
uTwo firms, A and B.  Shares cost $10.
uWith prob. 1/2 A’s profit is $100 and B’s profit is $20.
uWith prob. 1/2 A’s profit is $20 and B’s profit is $100.
uYou have $100 to invest.  How?
u

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Diversification
uBuy only firm A’s stock?
u$100/10 = 10 shares.
uYou earn $1000 with prob. 1/2 and $200 with prob. 1/2.
uExpected earning: $500 + $100 = $600

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Diversification
uBuy only firm B’s stock?
u$100/10 = 10 shares.
uYou earn $1000 with prob. 1/2 and $200 with prob. 1/2.
uExpected earning: $500 + $100 = $600

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Diversification
uBuy 5 shares in each firm?
uYou earn $600 for sure.
uDiversification has maintained expected earning and lowered risk.

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Diversification
uBuy 5 shares in each firm?
uYou earn $600 for sure.
uDiversification has maintained expected earning and lowered risk.
uTypically, diversification lowers expected earnings in exchange for lowered risk.

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Risk Spreading/Mutual Insurance
u100 risk-neutral persons each independently risk a $10,000 loss.
uLoss probability = 0.01.
uInitial wealth is $40,000.
uNo insurance: expected wealth is

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Risk Spreading/Mutual Insurance
uMutual insurance: Expected loss is
u
uEach of the 100 persons pays $1 into a mutual insurance fund.
uMutual insurance: expected wealth is
u
uRisk-spreading benefits everyone.