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Chapter 12
Uncertainty

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.

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

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

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.

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.

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.

State-Contingent Budget Constraints
Cna
Ca

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.

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

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

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

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)

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)

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.

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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


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

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.

Preferences Under Uncertainty



Preferences Under Uncertainty



Preferences Under Uncertainty



Preferences Under Uncertainty



Preferences Under Uncertainty



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

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

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

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

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

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

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

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

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

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.

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

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

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

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


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

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

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.

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

“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

“Unfair” Insurance
uRational choice requires
u
u
u

“Unfair” Insurance
uRational choice requires
u
u
uSince
u

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

“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

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

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

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

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

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

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

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

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.

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

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.