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# Incertidumbre

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### Incertidumbre

1. 1. Chapter TwelveUncertainty
2. 2. Uncertainty is PervasiveWhat is uncertain in economicsystems?–tomorrow’s prices–future wealth–future availability of commodities–present and future actions of otherpeople.
3. 3. Uncertainty is PervasiveWhat are rational responses touncertainty?–buying insurance (health, life,auto)–a portfolio of contingentconsumption goods.
4. 4. States of NaturePossible states of Nature:–“car accident” (a)–“no car accident” (na).Accident occurs with probability πa,does not with probability πna ;πa +πna = 1.Accident causes a loss of \$L.
5. 5. ContingenciesA contract implemented only when aparticular state of Nature occurs isstate-contingent.E.g. the insurer pays only if there isan accident.
6. 6. ContingenciesA state-contingent consumption planis implemented only when aparticular state of Nature occurs.E.g. take a vacation only if there isno accident.
7. 7. State-Contingent BudgetConstraintsEach \$1 of accident insurance costsγ.Consumer has \$m of wealth.Cna is consumption value in the no-accident state.Ca is consumption value in theaccident state.
8. 8. State-Contingent BudgetConstraintsCnaCa
9. 9. State-Contingent BudgetConstraintsCnaCa2017A state-contingent consumptionwith \$17 consumption value in theaccident state and \$20 consumptionvalue in the no-accident state.
10. 10. State-Contingent BudgetConstraintsWithout insurance,Ca = m - LCna = m.
11. 11. State-Contingent BudgetConstraintsCnaCamThe endowment bundle.m L−
12. 12. State-Contingent BudgetConstraintsBuy \$K of accident insurance.Cna = m - γK.Ca = m - L - γK + K = m - L + (1- γ)K.
13. 13. State-Contingent BudgetConstraintsBuy \$K of accident insurance.Cna = m - γK.Ca = m - L - γK + K = m - L + (1- γ)K.So K = (Ca - m + L)/(1- γ)
14. 14. State-Contingent BudgetConstraintsBuy \$K of accident insurance.Cna = m - γK.Ca = m - L - γK + K = m - L + (1- γ)K.So K = (Ca - m + L)/(1- γ)And Cna = m - γ (Ca - m + L)/(1- γ)
15. 15. State-Contingent BudgetConstraintsBuy \$K of accident insurance.Cna = m - γK.Ca = m - L - γK + K = m - L + (1- γ)K.So K = (Ca - m + L)/(1- γ)And Cna = m - γ (Ca - m + L)/(1- γ)I.e. Cm LCna a=−−−−γγγγ1 1
16. 16. State-Contingent BudgetConstraintsCnaCamThe endowment bundle.m L−γγCm LCna a=−−−−γγγγ1 1m L−
17. 17. State-Contingent BudgetConstraintsCnaCamThe endowment bundle.slope = −−γγ1Cm LCna a=−−−−γγγγ1 1m L−γγm L−
18. 18. State-Contingent BudgetConstraintsCnaCamThe endowment bundle.Where is themost preferredstate-contingentconsumption plan?Cm LCna a=−−−−γγγγ1 1slope = −−γγ1m L−γγm L−
19. 19. Preferences Under UncertaintyThink 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. 20. Preferences Under UncertaintyThink of a lottery.Win \$90 with probability 1/2 and win\$0 with probability 1/2.U(\$90) = 12, U(\$0) = 2.Expected utility isEU U(\$90) U(\$0)= × + ×= × + × =12121212122 7.
21. 21. Preferences Under UncertaintyThink of a lottery.Win \$90 with probability 1/2 and win\$0 with probability 1/2.Expected money value of the lotteryisEM \$90 \$0= × + × =121245\$ .
22. 22. Preferences Under UncertaintyEU = 7 and EM = \$45.U(\$45) > 7 ⇒ \$45 for sure ispreferred to the lottery ⇒ risk-aversion.U(\$45) < 7 ⇒ the lottery is preferredto \$45 for sure ⇒ risk-loving.U(\$45) = 7 ⇒ the lottery is preferredequally to \$45 for sure ⇒ risk-neutrality.
23. 23. Preferences Under UncertaintyWealth\$0 \$90212\$45EU=7
24. 24. Preferences Under UncertaintyWealth\$0 \$9012U(\$45)U(\$45) > EU ⇒ risk-aversion.2EU=7\$45
25. 25. Preferences Under UncertaintyWealth\$0 \$9012U(\$45)U(\$45) > EU ⇒ risk-aversion.2EU=7\$45MU declines as wealthrises.
26. 26. Preferences Under UncertaintyWealth\$0 \$90122EU=7\$45
27. 27. Preferences Under UncertaintyWealth\$0 \$9012U(\$45) < EU ⇒ risk-loving.2EU=7\$45U(\$45)
28. 28. Preferences Under UncertaintyWealth\$0 \$9012U(\$45) < EU ⇒ risk-loving.2EU=7\$45MU rises as wealthrises.U(\$45)
29. 29. Preferences Under UncertaintyWealth\$0 \$90122EU=7\$45
30. 30. Preferences Under UncertaintyWealth\$0 \$9012U(\$45) = EU ⇒ risk-neutrality.2U(\$45)=EU=7\$45
31. 31. Preferences Under UncertaintyWealth\$0 \$9012U(\$45) = EU ⇒ risk-neutrality.2\$45MU constant as wealthrises.U(\$45)=EU=7
32. 32. Preferences Under UncertaintyState-contingent consumption plansthat give equal expected utility areequally preferred.
33. 33. Preferences Under UncertaintyCnaCaEU1EU2EU3Indifference curvesEU1 < EU2 < EU3
34. 34. Preferences Under UncertaintyWhat is the MRS of an indifferencecurve?Get consumption c1 with prob. π1 andc2 with prob. π2 (π1 + π2 = 1).EU = π1U(c1) + π2U(c2).For constant EU, dEU = 0.
35. 35. Preferences Under UncertaintyEU U(c ) U(c )1 2= +π π1 2
36. 36. Preferences Under UncertaintyEU U(c ) U(c )1 2= +π π1 2dEU MU(c )dc MU(c )dc1 1 2 2= +π π1 2
37. 37. Preferences Under UncertaintyEU U(c ) U(c )1 2= +π π1 2dEU MU(c )dc MU(c )dc1 1 2 2= ⇒ + =0 01 2π πdEU MU(c )dc MU(c )dc1 1 2 2= +π π1 2
38. 38. Preferences Under UncertaintyEU U(c ) U(c )1 2= +π π1 2⇒ = −π π1 2MU(c )dc MU(c )dc1 1 2 2dEU MU(c )dc MU(c )dc1 1 2 2= +π π1 2dEU MU(c )dc MU(c )dc1 1 2 2= ⇒ + =0 01 2π π
39. 39. Preferences Under UncertaintyEU U(c ) U(c )1 2= +π π1 2⇒ = −dcdcMU(c )MU(c )2112ππ12.dEU MU(c )dc MU(c )dc1 1 2 2= +π π1 2dEU MU(c )dc MU(c )dc1 1 2 2= ⇒ + =0 01 2π π⇒ = −π π1 2MU(c )dc MU(c )dc1 1 2 2
40. 40. Preferences Under UncertaintyCnaCaEU1EU2EU3Indifference curvesEU1 < EU2 < EU3dcdcMU(c )MU(c )naaana= −ππana
41. 41. Choice Under UncertaintyQ: How is a rational choice madeunder uncertainty?A: Choose the most preferredaffordable state-contingentconsumption plan.
42. 42. State-Contingent BudgetConstraintsCnaCamThe endowment bundle.Cm LCna a=−−−−γγγγ1 1Where is themost preferredstate-contingentconsumption plan?slope = −−γγ1m L−γγm L−
43. 43. State-Contingent BudgetConstraintsCnaCamThe endowment bundle.Where is themost preferredstate-contingentconsumption plan?AffordableplansCm LCna a=−−−−γγγγ1 1slope = −−γγ1m L−γγm L−
44. 44. State-Contingent BudgetConstraintsCnaCamWhere is themost preferredstate-contingentconsumption plan?More preferredm L−γγm L−
45. 45. State-Contingent BudgetConstraintsCnaCamMost preferred affordable planm L−γγm L−
46. 46. State-Contingent BudgetConstraintsCnaCamMost preferred affordable planm L−γγm L−
47. 47. State-Contingent BudgetConstraintsCnaCamMost preferred affordable planMRS = slope of budgetconstraintm L−γγm L−
48. 48. State-Contingent BudgetConstraintsCnaCamMost preferred affordable planMRS = slope of budgetconstraint; i.e.m L−γγm L−γγππ1−= anaMU(c )MU(c )ana
49. 49. Competitive InsuranceSuppose entry to the insuranceindustry is free.Expected economic profit = 0.I.e. γK - πaK - (1 - πa)0 = (γ - πa)K = 0.I.e. free entry ⇒ γ = πa.If price of \$1 insurance = accidentprobability, then insurance is fair.
50. 50. Competitive InsuranceWhen insurance is fair, rationalinsurance choices satisfyγγππππ1 1−=−=aaanaMU(c )MU(c )ana
51. 51. Competitive InsuranceWhen insurance is fair, rationalinsurance choices satisfyI.e. MU(c ) MU(c )a na=γγππππ1 1−=−=aaanaMU(c )MU(c )ana
52. 52. Competitive InsuranceWhen insurance is fair, rationalinsurance choices satisfyI.e.Marginal utility of income must bethe same in both states.γγππππ1 1−=−=aaanaMU(c )MU(c )anaMU(c ) MU(c )a na=
53. 53. Competitive InsuranceHow much fair insurance does a risk-averse consumer buy?MU(c ) MU(c )a na=
54. 54. Competitive InsuranceHow much fair insurance does a risk-averse consumer buy?Risk-aversion ⇒ MU(c) ↓ as c ↑.MU(c ) MU(c )a na=
55. 55. Competitive InsuranceHow much fair insurance does a risk-averse consumer buy?Risk-aversion ⇒ MU(c) ↓ as c ↑.HenceMU(c ) MU(c )a na=c c .a na=
56. 56. Competitive InsuranceHow much fair insurance does a risk-averse consumer buy?Risk-aversion ⇒ MU(c) ↓ as c ↑.HenceI.e. full-insurance.MU(c ) MU(c )a na=c c .a na=
57. 57. “Unfair” InsuranceSuppose insurers make positiveexpected economic profit.I.e. γK - πaK - (1 - πa)0 = (γ - πa)K > 0.
58. 58. “Unfair” InsuranceSuppose insurers make positiveexpected economic profit.I.e. γK - πaK - (1 - πa)0 = (γ - πa)K > 0.Then ⇒ γ > πa ⇒ γγππ1 1−>−aa.
59. 59. “Unfair” InsuranceRational choice requiresγγππ1−= anaMU(c )MU(c )ana
60. 60. “Unfair” InsuranceRational choice requiresSinceγγππ1−= anaMU(c )MU(c )anaγγππ1 1−>−aa, MU(c ) > MU(c )a na
61. 61. “Unfair” InsuranceRational choice requiresSinceHence for a risk-averter.γγππ1−= anaMU(c )MU(c )anaγγππ1 1−>−aa, MU(c ) > MU(c )a nac < ca na
62. 62. “Unfair” InsuranceRational choice requiresSinceHence for a risk-averter.I.e. a risk-averter buys less than full“unfair” insurance.γγππ1−= anaMU(c )MU(c )anaγγππ1 1−>−aa, MU(c ) > MU(c )a nac < ca na
63. 63. Uncertainty is PervasiveWhat are rational responses touncertainty?–buying insurance (health, life,auto)–a portfolio of contingentconsumption goods.
64. 64. Uncertainty is PervasiveWhat are rational responses touncertainty?–buying insurance (health, life,auto)–a portfolio of contingentconsumption goods.
65. 65. Uncertainty is PervasiveWhat are rational responses touncertainty?–buying insurance (health, life,auto)–a portfolio of contingentconsumption goods.?
66. 66. DiversificationTwo firms, A and B. Shares cost \$10.With prob. 1/2 A’s profit is \$100 andB’s profit is \$20.With prob. 1/2 A’s profit is \$20 andB’s profit is \$100.You have \$100 to invest. How?
67. 67. DiversificationBuy only firm A’s stock?\$100/10 = 10 shares.You earn \$1000 with prob. 1/2 and\$200 with prob. 1/2.Expected earning: \$500 + \$100 = \$600
68. 68. DiversificationBuy only firm B’s stock?\$100/10 = 10 shares.You earn \$1000 with prob. 1/2 and\$200 with prob. 1/2.Expected earning: \$500 + \$100 = \$600
69. 69. DiversificationBuy 5 shares in each firm?You earn \$600 for sure.Diversification has maintainedexpected earning and lowered risk.
70. 70. DiversificationBuy 5 shares in each firm?You earn \$600 for sure.Diversification has maintainedexpected earning and lowered risk.Typically, diversification lowersexpected earnings in exchange forlowered risk.
71. 71. Risk Spreading/Mutual Insurance100 risk-neutral persons eachindependently risk a \$10,000 loss.Loss probability = 0.01.Initial wealth is \$40,000.No insurance: expected wealth is0 99 40 000 0 01 40 000 10 00039 900⋅ × + ⋅ −=\$ , (\$ , \$ , )\$ , .
72. 72. Risk Spreading/Mutual InsuranceMutual insurance: Expected loss isEach of the 100 persons pays \$1 intoa mutual insurance fund.Mutual insurance: expected wealth isRisk-spreading benefits everyone.0 01 10 000 100⋅ × =\$ , \$ .\$ , \$ \$ , \$ , .40 000 1 39 999 39 900− = >