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Chapter Twelve

Chapter Twelve. Uncertainty. Uncertainty is Pervasive. What is uncertain in economic systems? tomorrow’s prices future wealth future availability of commodities present and future actions of other people. Uncertainty is Pervasive. What are rational responses to uncertainty?

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Chapter Twelve

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  1. Chapter Twelve Uncertainty

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

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. State-Contingent Budget Constraints Cna Ca

  9. 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

  10. State-Contingent Budget Constraints • Without insurance, • Ca = m - L • Cna = m.

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

  12. State-Contingent Budget Constraints • Buy $K of accident insurance. • Cna = m - K. • Ca = m - L - K + K = m - L + (1- )K.

  13. 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- )

  14. 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- )

  15. 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.

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

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

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

  19. 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

  20. 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

  21. 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

  22. Preferences Under Uncertainty • EU = 7 and EM = $45. • U($45) > 7  $45 for sure is preferred to the lottery risk-aversion. • U($45) < 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.

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

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

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

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

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

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

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

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

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

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

  33. Preferences Under Uncertainty Cna Indifference curvesEU1 < EU2 < EU3 EU3 EU2 EU1 Ca

  34. 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.

  35. Preferences Under Uncertainty

  36. Preferences Under Uncertainty

  37. Preferences Under Uncertainty

  38. Preferences Under Uncertainty

  39. Preferences Under Uncertainty

  40. Preferences Under Uncertainty Cna Indifference curvesEU1 < EU2 < EU3 EU3 EU2 EU1 Ca

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

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

  43. State-Contingent Budget Constraints Cna The endowment bundle. m Where is the most preferredstate-contingentconsumption plan? Affordableplans Ca

  44. State-Contingent Budget Constraints Cna More preferred m Where is the most preferredstate-contingentconsumption plan? Ca

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

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

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

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

  49. 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.

  50. Competitive Insurance • When insurance is fair, rational insurance choices satisfy

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