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.