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 20 17 A state-contingent consumption with $17 consumption value in the accident state and $20 consumption value in the no-accident state.

9. State-Contingent Budget Constraints u Without insurance, u C a = m - L u C na = m.

10. State-Contingent Budget Constraints C na CaCa m The endowment bundle.

11. 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.

12. State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan?

13. 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

14. 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

15. 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

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

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

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

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

20. Preferences Under Uncertainty C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3

21. 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.

22. Preferences Under Uncertainty

23. C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3

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

25. State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan?

26. State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan

27. State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint; i.e.

28. 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.

29. 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.

30. 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.

31. “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 

32. “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.

33. 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?

34. 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

35. 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

36. 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.