Download presentation
Presentation is loading. Please wait.
1
Chapter Twelve Uncertainty
2
Uncertainty is Pervasive u What is uncertain in economic systems? –tomorrow’s prices –future wealth –future availability of commodities –present and future actions of other people.
3
Uncertainty is Pervasive u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods.
4
States of Nature u Possible states of Nature: –“car accident” (a) –“no car accident” (na). u Accident occurs with probability a, does not with probability na ; a + na = 1. u Accident causes a loss of $L.
5
Contingencies u A contract implemented only when a particular state of Nature occurs is state-contingent. u E.g. the insurer pays only if there is an accident.
6
Contingencies u A state-contingent consumption plan is implemented only when a particular state of Nature occurs. u E.g. take a vacation only if there is no accident.
7
State-Contingent Budget Constraints u Each $1 of accident insurance costs . u Consumer has $m of wealth. u C na is consumption value in the no- accident state. u C a is consumption value in the accident state.
8
State-Contingent Budget Constraints C na CaCa
9
State-Contingent Budget Constraints C na CaCa 20 17 A state-contingent consumption with $17 consumption value in the accident state and $20 consumption value in the no-accident state.
10
State-Contingent Budget Constraints u Without insurance, u C a = m - L u C na = m.
11
State-Contingent Budget Constraints C na CaCa m The endowment bundle.
12
State-Contingent Budget Constraints u Buy $K of accident insurance. u C na = m - K. u C a = m - L - K + K = m - L + (1- )K.
13
State-Contingent Budget Constraints u Buy $K of accident insurance. u C na = m - K. u C a = m - L - K + K = m - L + (1- )K. u So K = (C a - m + L)/(1- )
14
State-Contingent Budget Constraints u Buy $K of accident insurance. u C na = m - K. u C a = m - L - K + K = m - L + (1- )K. u So K = (C a - m + L)/(1- ) u And C na = m - (C a - m + L)/(1- )
15
State-Contingent Budget Constraints u Buy $K of accident insurance. u C na = m - K. u C a = m - L - K + K = m - L + (1- )K. u So K = (C a - m + L)/(1- ) u And C na = m - (C a - m + L)/(1- ) u I.e.
16
State-Contingent Budget Constraints C na CaCa m The endowment bundle.
17
State-Contingent Budget Constraints C na CaCa m The endowment bundle.
18
State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan?
19
Preferences Under Uncertainty u Think of a lottery. u Win $90 with probability 1/2 and win $0 with probability 1/2. u U($90) = 12, U($0) = 2. u Expected utility is
20
Preferences Under Uncertainty u Think of a lottery. u Win $90 with probability 1/2 and win $0 with probability 1/2. u U($90) = 12, U($0) = 2. u Expected utility is
21
Preferences Under Uncertainty u Think of a lottery. u Win $90 with probability 1/2 and win $0 with probability 1/2. u Expected money value of the lottery is
22
Preferences Under Uncertainty u EU = 7 and EM = $45. u U($45) > 7 $45 for sure is preferred to the lottery risk-aversion. u U($45) < 7 the lottery is preferred to $45 for sure risk-loving. u U($45) = 7 the lottery is preferred equally to $45 for sure risk- neutrality.
23
Preferences Under Uncertainty Wealth$0$90 2 12 $45 EU=7
24
Preferences Under Uncertainty Wealth$0$90 12 U($45) U($45) > EU risk-aversion. 2 EU=7 $45
25
Preferences Under Uncertainty Wealth$0$90 12 U($45) U($45) > EU risk-aversion. 2 EU=7 $45 MU declines as wealth rises.
26
Preferences Under Uncertainty Wealth$0$90 12 2 EU=7 $45
27
Preferences Under Uncertainty Wealth$0$90 12 U($45) < EU risk-loving. 2 EU=7 $45 U($45)
28
Preferences Under Uncertainty Wealth$0$90 12 U($45) < EU risk-loving. 2 EU=7 $45 MU rises as wealth rises. U($45)
29
Preferences Under Uncertainty Wealth$0$90 12 2 EU=7 $45
30
Preferences Under Uncertainty Wealth$0$90 12 U($45) = EU risk-neutrality. 2 U($45)= EU=7 $45
31
Preferences Under Uncertainty Wealth$0$90 12 U($45) = EU risk-neutrality. 2 $45 MU constant as wealth rises. U($45)= EU=7
32
Preferences Under Uncertainty u State-contingent consumption plans that give equal expected utility are equally preferred.
33
Preferences Under Uncertainty C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3
34
Preferences Under Uncertainty u What is the MRS of an indifference curve? u Get consumption c 1 with prob. 1 and c 2 with prob. 2 ( 1 + 2 = 1). u EU = 1 U(c 1 ) + 2 U(c 2 ). u For constant EU, dEU = 0.
35
Preferences Under Uncertainty
40
C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3
41
Choice Under Uncertainty u Q: How is a rational choice made under uncertainty? u A: Choose the most preferred affordable state-contingent consumption plan.
42
State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan?
43
State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan? Affordable plans
44
State-Contingent Budget Constraints C na CaCa m Where is the most preferred state-contingent consumption plan? More preferred
45
State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan
46
State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan
47
State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint
48
State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint; i.e.
49
Competitive Insurance u Suppose entry to the insurance industry is free. u Expected economic profit = 0. u I.e. K - a K - (1 - a )0 = ( - a )K = 0. u I.e. free entry = a. u If price of $1 insurance = accident probability, then insurance is fair.
50
Competitive Insurance u When insurance is fair, rational insurance choices satisfy
51
Competitive Insurance u When insurance is fair, rational insurance choices satisfy u I.e.
52
Competitive Insurance u When insurance is fair, rational insurance choices satisfy u I.e. u Marginal utility of income must be the same in both states.
53
Competitive Insurance u How much fair insurance does a risk- averse consumer buy?
54
Competitive Insurance u How much fair insurance does a risk- averse consumer buy? u Risk-aversion MU(c) as c .
55
Competitive Insurance u How much fair insurance does a risk- averse consumer buy? u Risk-aversion MU(c) as c . u Hence
56
Competitive Insurance u How much fair insurance does a risk- averse consumer buy? u Risk-aversion MU(c) as c . u Hence u I.e. full-insurance.
57
“Unfair” Insurance u Suppose insurers make positive expected economic profit. u I.e. K - a K - (1 - a )0 = ( - a )K > 0.
58
“Unfair” Insurance u Suppose insurers make positive expected economic profit. u I.e. K - a K - (1 - a )0 = ( - a )K > 0. u Then > a
59
“Unfair” Insurance u Rational choice requires
60
“Unfair” Insurance u Rational choice requires u Since
61
“Unfair” Insurance u Rational choice requires u Since u Hence for a risk-averter.
62
“Unfair” Insurance u Rational choice requires u Since u Hence for a risk-averter. u I.e. a risk-averter buys less than full “unfair” insurance.
63
Uncertainty is Pervasive u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods.
64
Uncertainty is Pervasive u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods.
65
Uncertainty is Pervasive u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods. ?
66
Diversification u Two firms, A and B. Shares cost $10. u With prob. 1/2 A’s profit is $100 and B’s profit is $20. u With prob. 1/2 A’s profit is $20 and B’s profit is $100. u You have $100 to invest. How?
67
Diversification u Buy only firm A’s stock? u $100/10 = 10 shares. u You earn $1000 with prob. 1/2 and $200 with prob. 1/2. u Expected earning: $500 + $100 = $600
68
Diversification u Buy only firm B’s stock? u $100/10 = 10 shares. u You earn $1000 with prob. 1/2 and $200 with prob. 1/2. u Expected earning: $500 + $100 = $600
69
Diversification u Buy 5 shares in each firm? u You earn $600 for sure. u Diversification has maintained expected earning and lowered risk.
70
Diversification u Buy 5 shares in each firm? u You earn $600 for sure. u Diversification has maintained expected earning and lowered risk. u Typically, diversification lowers expected earnings in exchange for lowered risk.
71
Risk Spreading/Mutual Insurance u 100 risk-neutral persons each independently risk a $10,000 loss. u Loss probability = 0.01. u Initial wealth is $40,000. u No insurance: expected wealth is
72
Risk Spreading/Mutual Insurance u Mutual insurance: Expected loss is u Each of the 100 persons pays $1 into a mutual insurance fund. u Mutual insurance: expected wealth is u Risk-spreading benefits everyone.
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.