Chapter 12 Uncertainty 1

Uncertainty is Pervasive  What is uncertainty in economic systems?  tomorrow’s prices  future wealth  future availability of commodities  present and future actions of other people. 2

Uncertainty is Pervasive  What are rational responses to uncertainty?  A portfolio of contingent consumption goods.  E.g., buying insurance (health, life, auto) 3

12.1 Contingent consumption 或有消費  states of nature: different outcomes of a random event  “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. 4

Contingencies  A state-contingent consumption plan is a specification of what will be consumed in different sates of nature.  E.g. the insurer pays only if there is an accident. 5

12.2 State-Contingent Budget Constraints  Each $1 of accident insurance costs .  Consumer has $m of wealth.  C na is consumption value in the no-accident state.  C a is consumption value in the accident state. 6

State-Contingent Budget Constraints C na CaCa 7

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. 8

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

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

State-Contingent Budget Constraints  Buy $K of accident insurance. C na = m -  K. C a = m - L -  K + K = m - L + (1-  )K.  K = (C a - m + L)/(1-  )  C na = m -  (C a - m + L)/(1-  )  11

State-Contingent Budget Constraints C na CaCa m The endowment bundle. 12

State-Contingent Budget Constraints C na CaCa m The endowment bundle. 13

State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan? 14

12.3 Preferences Under Uncertainty  Think of a lottery.  Win $400 with probability 1/2 and $0 with probability 1/2.  Will you prefer participating the lottery or getting $200 for sure? 15

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 預期效用 16

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 17

Preferences Under Uncertainty  EU = 7 and EM = $45.  U($45) > 7  $45 for sure is preferred to the lottery  risk-aversion. 風險驅避  U($45) < 7  the lottery is preferred to $45 for sure  risk-loving. 風險愛好  U($45) = 7  the lottery is preferred equally to $45 for sure  risk-neutrality. 風險中立 18

Preferences Under Uncertainty Wealth$0$ $45 EU=7 19

Preferences Under Uncertainty Wealth$0$90 12 U($45) U($45) > EU  risk-aversion. 2 EU=7 $45 20

Preferences Under Uncertainty Wealth$0$90 12 U($45) U($45) > EU  risk-aversion. 2 EU=7 $45 MU declines as wealth rises. 21

Preferences Under Uncertainty Wealth$0$ EU=7 $45 22

Preferences Under Uncertainty Wealth$0$90 12 U($45) < EU  risk-loving. 2 EU=7 $45 U($45) 23

Preferences Under Uncertainty Wealth$0$90 12 U($45) < EU  risk-loving. 2 EU=7 $45 MU rises as wealth rises. U($45) 24

Preferences Under Uncertainty Wealth$0$ EU=7 $45 25

Preferences Under Uncertainty Wealth$0$90 12 U($45) = EU  risk-neutrality. 2 U($45)= EU=7 $45 26

Preferences Under Uncertainty Wealth$0$90 12 U($45) = EU  risk-neutrality. 2 $45 MU constant as wealth rises. U($45)= EU=7 27

Preferences Under Uncertainty  State-contingent consumption plans that give equal expected utility are equally preferred. 28

Preferences Under Uncertainty C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3 29

Preferences Under Uncertainty  What is the MRS of an indifference curve?  Get consumption c 1 with prob.  1 and c 2 with prob.  2 (  1 +  2 = 1).  EU =  1 U(c 1 ) +  2 U(c 2 ).  For constant EU, dEU = 0. 30

Preferences Under Uncertainty 31

Preferences Under Uncertainty C na CaCa EU 1 EU 2 EU 3 Indifference curves EU 1 < EU 2 < EU 3 32

Choice Under Uncertainty  Q: How is a rational choice made under uncertainty?  A: Choose the most preferred affordable state-contingent consumption plan. 33

State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan? 34

State-Contingent Budget Constraints C na CaCa m The endowment bundle. Where is the most preferred state-contingent consumption plan? Affordable plans 35

State-Contingent Budget Constraints C na CaCa m Where is the most preferred state-contingent consumption plan? More preferred 36

State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan 37

State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan 38

State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint 39

State-Contingent Budget Constraints C na CaCa m Most preferred affordable plan MRS = slope of budget constraint; i.e. 40

12.4 Insurance  Suppose entry to the insurance industry is free.  Expected economic profit = 0.  I.e.  K -  a K - (1 -  a )0 = (  -  a )K = 0.  I.e. free entry   =  a.  If price of $1 insurance = accident probability, then insurance is fair. 41

Competitive Insurance  When insurance is fair, rational insurance choices satisfy 42

Competitive Insurance  When insurance is fair, rational insurance choices satisfy  I.e. 43

Competitive Insurance  When insurance is fair, rational insurance choices satisfy  I.e.  Marginal utility of income must be the same in both states. 44

Competitive Insurance  How much fair insurance does a risk-averse consumer buy? 45

Competitive Insurance  How much fair insurance does a risk-averse consumer buy?  Risk-aversion  MU(c)  as c .  Hence 46

Competitive Insurance  How much fair insurance does a risk-averse consumer buy?  Risk-aversion  MU(c)  as c .  Hence  full-insurance, K=L 47

“Unfair” Insurance  Suppose insurers make positive expected economic profit.  I.e.  K -  a K - (1 -  a )0 = (  -  a )K > 0. 48

“Unfair” Insurance  Suppose insurers make positive expected economic profit.  I.e.  K -  a K - (1 -  a )0 = (  -  a )K > 0.  Then   >  a  49

“Unfair” Insurance  Rational choice requires 50

“Unfair” Insurance  Rational choice requires  Since  Hence for a risk-averter. 51

“Unfair” Insurance  Rational choice requires  Since  Hence for a risk-averter, K<L.  A risk-averter buys less than full “unfair” insurance. 52

12.5 Diversification  Two firms, A and B. Shares cost $10.  With prob. 1/2 A’s profit is $100 and B’s profit is $20.  With prob. 1/2 A’s profit is $20 and B’s profit is $100.  You have $100 to invest. How? 53

Diversification  Buy 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 54

Diversification  Buy 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 55

Diversification  Buy 5 shares in each firm?  You earn $600 for sure.  Diversification has maintained expected earning and lowered risk. 56

Diversification  Buy 5 shares in each firm?  You earn $600 for sure.  Diversification has maintained expected earning and lowered risk.  Typically, diversification lowers expected earnings in exchange for lowered risk. 57