© 2010 W. W. Norton & Company, Inc. 12 Uncertainty.

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Presentation transcript:

© 2010 W. W. Norton & Company, Inc. 12 Uncertainty

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 3 Uncertainty is Pervasive u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 8 State-Contingent Budget Constraints C na CaCa

© 2010 W. W. Norton & Company, Inc. 9 State-Contingent Budget Constraints C na CaCa A state-contingent consumption with $17 consumption value in the accident state and $20 consumption value in the no-accident state.

© 2010 W. W. Norton & Company, Inc. 10 State-Contingent Budget Constraints u Without insurance, u C a = m - L u C na = m.

© 2010 W. W. Norton & Company, Inc. 11 State-Contingent Budget Constraints C na CaCa m The endowment bundle.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 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-  )

© 2010 W. W. Norton & Company, Inc. 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-  )

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 16 State-Contingent Budget Constraints C na CaCa m The endowment bundle.

© 2010 W. W. Norton & Company, Inc. 17 State-Contingent Budget Constraints C na CaCa m The endowment bundle.

© 2010 W. W. Norton & Company, Inc. 18 State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan?

© 2010 W. W. Norton & Company, Inc. 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

© 2010 W. W. Norton & Company, Inc. 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

© 2010 W. W. Norton & Company, Inc. 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

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 23 Preferences Under Uncertainty Wealth$0$ $45 EU=7

© 2010 W. W. Norton & Company, Inc. 24 Preferences Under Uncertainty Wealth$0$90 12 U($45) U($45) > EU  risk-aversion. 2 EU=7 $45

© 2010 W. W. Norton & Company, Inc. 25 Preferences Under Uncertainty Wealth$0$90 12 U($45) U($45) > EU  risk-aversion. 2 EU=7 $45 MU declines as wealth rises.

© 2010 W. W. Norton & Company, Inc. 26 Preferences Under Uncertainty Wealth$0$ EU=7 $45

© 2010 W. W. Norton & Company, Inc. 27 Preferences Under Uncertainty Wealth$0$90 12 U($45) < EU  risk-loving. 2 EU=7 $45 U($45)

© 2010 W. W. Norton & Company, Inc. 28 Preferences Under Uncertainty Wealth$0$90 12 U($45) < EU  risk-loving. 2 EU=7 $45 MU rises as wealth rises. U($45)

© 2010 W. W. Norton & Company, Inc. 29 Preferences Under Uncertainty Wealth$0$ EU=7 $45

© 2010 W. W. Norton & Company, Inc. 30 Preferences Under Uncertainty Wealth$0$90 12 U($45) = EU  risk-neutrality. 2 U($45)= EU=7 $45

© 2010 W. W. Norton & Company, Inc. 31 Preferences Under Uncertainty Wealth$0$90 12 U($45) = EU  risk-neutrality. 2 $45 MU constant as wealth rises. U($45)= EU=7

© 2010 W. W. Norton & Company, Inc. 32 Preferences Under Uncertainty u State-contingent consumption plans that give equal expected utility are equally preferred.

© 2010 W. W. Norton & Company, Inc. 33 Preferences Under Uncertainty C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 35 Preferences Under Uncertainty

© 2010 W. W. Norton & Company, Inc. 36 Preferences Under Uncertainty

© 2010 W. W. Norton & Company, Inc. 37 Preferences Under Uncertainty

© 2010 W. W. Norton & Company, Inc. 38 Preferences Under Uncertainty

© 2010 W. W. Norton & Company, Inc. 39 Preferences Under Uncertainty

© 2010 W. W. Norton & Company, Inc. 40 Preferences Under Uncertainty C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 42 State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan?

© 2010 W. W. Norton & Company, Inc. 43 State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan? Affordable plans

© 2010 W. W. Norton & Company, Inc. 44 State-Contingent Budget Constraints C na CaCa m Where is the most preferred state-contingent consumption plan? More preferred

© 2010 W. W. Norton & Company, Inc. 45 State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan

© 2010 W. W. Norton & Company, Inc. 46 State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan

© 2010 W. W. Norton & Company, Inc. 47 State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint

© 2010 W. W. Norton & Company, Inc. 48 State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint; i.e.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 50 Competitive Insurance u When insurance is fair, rational insurance choices satisfy

© 2010 W. W. Norton & Company, Inc. 51 Competitive Insurance u When insurance is fair, rational insurance choices satisfy u I.e.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 53 Competitive Insurance u How much fair insurance does a risk- averse consumer buy?

© 2010 W. W. Norton & Company, Inc. 54 Competitive Insurance u How much fair insurance does a risk- averse consumer buy? u Risk-aversion  MU(c)  as c .

© 2010 W. W. Norton & Company, Inc. 55 Competitive Insurance u How much fair insurance does a risk- averse consumer buy? u Risk-aversion  MU(c)  as c . u Hence

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 57 “Unfair” Insurance u Suppose insurers make positive expected economic profit. u I.e.  K -  a K - (1 -  a )0 = (  -  a )K > 0.

© 2010 W. W. Norton & Company, Inc. 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 

© 2010 W. W. Norton & Company, Inc. 59 “Unfair” Insurance u Rational choice requires

© 2010 W. W. Norton & Company, Inc. 60 “Unfair” Insurance u Rational choice requires u Since

© 2010 W. W. Norton & Company, Inc. 61 “Unfair” Insurance u Rational choice requires u Since u Hence for a risk-averter.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 63 Uncertainty is Pervasive u 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 u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods.

© 2010 W. W. Norton & Company, Inc. 65 Uncertainty is Pervasive u What are rational responses to uncertainty? –buying insurance (health, life, auto) –a portfolio of contingent consumption goods. ?

© 2010 W. W. Norton & Company, Inc. 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?

© 2010 W. W. Norton & Company, Inc. 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

© 2010 W. W. Norton & Company, Inc. 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

© 2010 W. W. Norton & Company, Inc. 69 Diversification u Buy 5 shares in each firm? u You earn $600 for sure. u Diversification has maintained expected earning and lowered risk.

© 2010 W. W. Norton & Company, Inc. 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.

© 2010 W. W. Norton & Company, Inc. 71 Risk Spreading/Mutual Insurance u 100 risk-neutral persons each independently risk a $10,000 loss. u Loss probability = u Initial wealth is $40,000. u No insurance: expected wealth is

© 2010 W. W. Norton & Company, Inc. 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.