Hal Varian Intermediate Microeconomics Chapter Twelve

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Hal VarianIntermediate MicroeconomicsChapter
Twelve

• Uncertainty

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Uncertainty is Pervasive

• What is uncertain in economic systems?
• tomorrows prices
• future wealth
• future availability of commodities
• present and future actions of other people.

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Uncertainty is Pervasive

• What are rational responses to uncertainty?
• buying insurance (health, life, auto)
• a portfolio of contingent consumption goods.

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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.

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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.

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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.

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State-Contingent Budget Constraints

• Each 1 of accident insurance costs ?.
• Consumer has m of wealth.
• Cna is consumption value in the no-accident
  state.
• Ca 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
A state-contingent consumptionwith 17
consumption value in the accident state and 20
consumption value in the no-accident state.
20
Ca
17
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State-Contingent Budget Constraints

• Without insurance,
• Ca m - L
• Cna m.

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

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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- ?)

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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- ?)

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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- ?)
• I.e.

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State-Contingent Budget Constraints
Cna
The endowment bundle.
m
Ca
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State-Contingent Budget Constraints
Cna
The endowment bundle.
m
Ca
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State-Contingent Budget Constraints
Cna
The endowment bundle.
m
Where is the most preferredstate-contingentconsu
mption plan?
Ca
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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

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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

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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

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Preferences Under Uncertainty

• EU 7 and EM 45.
• U(45) gt 7 ? 45 for sure is preferred to the
  lottery ? risk-aversion.
• U(45) lt 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.

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Preferences Under Uncertainty
12
EU7
2
Wealth
0
90
45
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Preferences Under Uncertainty
U(45) gt EU ? risk-aversion.
12
U(45)
EU7
2
Wealth
0
90
45
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Preferences Under Uncertainty
U(45) gt EU ? risk-aversion.
12
MU declines as wealth rises.
U(45)
EU7
2
Wealth
0
90
45
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Preferences Under Uncertainty
12
EU7
2
Wealth
0
90
45
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Preferences Under Uncertainty
U(45) lt EU ? risk-loving.
12
EU7
U(45)
2
Wealth
0
90
45
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Preferences Under Uncertainty
U(45) lt EU ? risk-loving.
12
MU rises as wealth rises.
EU7
U(45)
2
Wealth
0
90
45
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Preferences Under Uncertainty
12
EU7
2
Wealth
0
90
45
30
Preferences Under Uncertainty
U(45) EU ? risk-neutrality.
12
U(45)EU7
2
Wealth
0
90
45
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Preferences Under Uncertainty
U(45) EU ? risk-neutrality.
12
MU constant as wealth rises.
U(45)EU7
2
Wealth
0
90
45
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Preferences Under Uncertainty

• State-contingent consumption plans that give
  equal expected utility are equally preferred.

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Preferences Under Uncertainty
Cna
Indifference curvesEU1 lt EU2 lt EU3
EU3
EU2
EU1
Ca
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Preferences Under Uncertainty

• What is the MRS of an indifference curve?
• Get consumption c1 with prob. ?1 andc2 with
  prob. ?2 (?1 ?2 1).
• EU ?1U(c1) ?2U(c2).
• For constant EU, dEU 0.

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Preferences Under Uncertainty
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Preferences Under Uncertainty
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Preferences Under Uncertainty
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Preferences Under Uncertainty
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Preferences Under Uncertainty
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Preferences Under Uncertainty
Cna
Indifference curvesEU1 lt EU2 lt EU3
EU3
EU2
EU1
Ca
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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.

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State-Contingent Budget Constraints
Cna
The endowment bundle.
m
Where is the most preferredstate-contingentconsu
mption plan?
Ca
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State-Contingent Budget Constraints
Cna
The endowment bundle.
m
Where is the most preferredstate-contingentconsu
mption plan?
Affordableplans
Ca
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State-Contingent Budget Constraints
Cna
More preferred
m
Where is the most preferredstate-contingentconsu
mption plan?
Ca
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State-Contingent Budget Constraints
Cna
Most preferred affordable plan
m
Ca
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State-Contingent Budget Constraints
Cna
Most preferred affordable plan
m
Ca
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State-Contingent Budget Constraints
Cna
Most preferred affordable plan
MRS slope of budget constraint
m
Ca
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State-Contingent Budget Constraints
Cna
Most preferred affordable plan
MRS slope of budget constraint
i.e.
m
Ca
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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.

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Competitive Insurance

• When insurance is fair, rational insurance
  choices satisfy

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Competitive Insurance

• When insurance is fair, rational insurance
  choices satisfy
• I.e.

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Competitive Insurance

• When insurance is fair, rational insurance
  choices satisfy
• I.e.
• Marginal utility of income must be the same in
  both states.

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Competitive Insurance

• How much fair insurance does a risk-averse
  consumer buy?

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Competitive Insurance

• How much fair insurance does a risk-averse
  consumer buy?
• Risk-aversion ? MU(c) ? as c ?.

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Competitive Insurance

• How much fair insurance does a risk-averse
  consumer buy?
• Risk-aversion ? MU(c) ? as c ?.
• Hence

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Competitive Insurance

• How much fair insurance does a risk-averse
  consumer buy?
• Risk-aversion ? MU(c) ? as c ?.
• Hence
• I.e. full-insurance.

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Unfair Insurance

• Suppose insurers make positive expected economic
  profit.
• I.e. ?K - ?aK - (1 - ?a)0 (? - ?a)K gt 0.

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Unfair Insurance

• Suppose insurers make positive expected economic
  profit.
• I.e. ?K - ?aK - (1 - ?a)0 (? - ?a)K gt 0.
• Then ? ? gt ?a ?

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Unfair Insurance

• Rational choice requires

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Unfair Insurance

• Rational choice requires
• Since

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Unfair Insurance

• Rational choice requires
• Since
• Hence for a risk-averter.

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Unfair Insurance

• Rational choice requires
• Since
• Hence for a risk-averter.
• I.e. a risk-averter buys less than full unfair
  insurance.

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Uncertainty is Pervasive

• What are rational responses to uncertainty?
• buying insurance (health, life, auto)
• a portfolio of contingent consumption goods.

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Uncertainty is Pervasive

• What are rational responses to uncertainty?
• buying insurance (health, life, auto)
• a portfolio of contingent consumption goods.

?
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Uncertainty is Pervasive

• What are rational responses to uncertainty?
• buying insurance (health, life, auto)
• a portfolio of contingent consumption goods.

?
?
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Diversification

• Two firms, A and B. Shares cost 10.
• With prob. 1/2 As profit is 100 and Bs profit
  is 20.
• With prob. 1/2 As profit is 20 and Bs profit
  is 100.
• You have 100 to invest. How?

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Diversification

• Buy only firm As stock?
• 100/10 10 shares.
• You earn 1000 with prob. 1/2 and 200 with prob.
  1/2.
• Expected earning 500 100 600

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Diversification

• Buy only firm Bs stock?
• 100/10 10 shares.
• You earn 1000 with prob. 1/2 and 200 with prob.
  1/2.
• Expected earning 500 100 600

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Diversification

• Buy 5 shares in each firm?
• You earn 600 for sure.
• Diversification has maintained expected earning
  and lowered risk.

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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.

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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

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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.