1、Chapter Twelve,Uncertainty,Main Issue,A state-contingent consumption plan Preferences Under Uncertainty and Expected utility Risk-aversion, risk-loving, and risk-neutralityCompetitive Insurance,Uncertainty is Pervasive,What is uncertain in economic systems? tomorrows prices future wealth future avai
2、lability of commodities present and future actions of other people.,Uncertainty is Pervasive,What are rational responses to uncertainty? buying insurance (health, life, auto) a portfolio of contingent consumption goods.,States of Nature,Possible states of Nature: “car accident” (a) “no car accident”
3、 (na). Accident occurs with probability a, does not with probability na ; a + na = 1. Accident causes a loss of $L.,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.,Contingencies,A state-contin
4、gent consumption plan is implemented only when a particular state of Nature occurs. E.g. take a vacation only if there is no accident.,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 consump
5、tion value in the accident state.,State-Contingent Budget Constraints,Cna,Ca,State-Contingent Budget Constraints,Cna,Ca,20,17,A state-contingent consumption with $17 consumption value in the accident state and $20 consumption value in the no-accident state.,State-Contingent Budget Constraints,Withou
6、t insurance, Ca = m - L Cna = m.,State-Contingent Budget Constraints,Cna,Ca,m,The endowment bundle.,State-Contingent Budget Constraints,Buy $K of accident insurance. Cna = m - K. Ca = m - L - K + K = m - L + (1- )K.,State-Contingent Budget Constraints,Buy $K of accident insurance. Cna = m - K. Ca =
7、m - L - K + K = m - L + (1- )K. So K = (Ca - m + L)/(1- ),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- ),State-Contingent Budget Constraints,Buy $K of accident insurance. C
8、na = 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.,State-Contingent Budget Constraints,Cna,Ca,m,The endowment bundle.,State-Contingent Budget Constraints,Cna,Ca,m,The endowment bundle.,State-Contingent Budget Constraints,Cna,Ca,m,The endo
9、wment bundle.,Where is the most preferred state-contingent consumption plan?,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,Preferences Under Uncertainty,Think of a lottery. Win $90 with prob
10、ability 1/2 and win $0 with probability 1/2. U($90) = 12, U($0) = 2. Expected utility is,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,Preferences Under Uncertainty,EU = 7 and EM = $45. U($45) 7
11、$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.,Preferences Under Uncertainty,Wealth,$0,$90,2,12,$45,EU=7,Preferences Under Uncertainty,Wealth,$0,$90,12
12、,U($45),U($45) EU risk-aversion.,2,EU=7,$45,Preferences Under Uncertainty,Wealth,$0,$90,12,U($45),U($45) EU risk-aversion.,2,EU=7,$45,MU declines as wealth rises.,Preferences Under Uncertainty,Wealth,$0,$90,12,2,EU=7,$45,Preferences Under Uncertainty,Wealth,$0,$90,12,U($45) EU risk-loving.,2,EU=7,$4
13、5,U($45),Preferences Under Uncertainty,Wealth,$0,$90,12,U($45) EU risk-loving.,2,EU=7,$45,MU rises as wealth rises.,U($45),Preferences Under Uncertainty,Wealth,$0,$90,12,2,EU=7,$45,Preferences Under Uncertainty,Wealth,$0,$90,12,U($45) = EU risk-neutrality.,2,U($45)= EU=7,$45,Preferences Under Uncert
14、ainty,Wealth,$0,$90,12,U($45) = EU risk-neutrality.,2,$45,MU constant as wealth rises.,U($45)= EU=7,Preferences Under Uncertainty,State-contingent consumption plans that give equal expected utility are equally preferred.,Preferences Under Uncertainty,Cna,Ca,EU1,EU2,EU3,Indifference curves EU1 EU2 EU
15、3,Preferences Under Uncertainty,What is the MRS of an indifference curve? Get consumption c1 with prob. 1 and c2 with prob. 2 (1 + 2 = 1). EU = 1U(c1) + 2U(c2). For constant EU, dEU = 0.,Preferences Under Uncertainty,Preferences Under Uncertainty,Preferences Under Uncertainty,Preferences Under Uncer
16、tainty,Preferences Under Uncertainty,Preferences Under Uncertainty,Cna,Ca,EU1,EU2,EU3,Indifference curves EU1 EU2 EU3,Choice Under Uncertainty,Q: How is a rational choice made under uncertainty? A: Choose the most preferred affordable state-contingent consumption plan.,State-Contingent Budget Constr
17、aints,Cna,Ca,m,The endowment bundle.,Where is the most preferred state-contingent consumption plan?,State-Contingent Budget Constraints,Cna,Ca,m,The endowment bundle.,Where is the most preferred state-contingent consumption plan?,Affordable plans,State-Contingent Budget Constraints,Cna,Ca,m,Where is
18、 the most preferred state-contingent consumption plan?,More preferred,State-Contingent Budget Constraints,Cna,Ca,m,Most preferred affordable plan,State-Contingent Budget Constraints,Cna,Ca,m,Most preferred affordable plan,State-Contingent Budget Constraints,Cna,Ca,m,Most preferred affordable plan,MR
19、S = slope of budget constraint,State-Contingent Budget Constraints,Cna,Ca,m,Most preferred affordable plan,MRS = slope of budget constraint; i.e.,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
20、 = a. If price of $1 insurance = accident probability, then insurance is fair.,Competitive Insurance,When insurance is fair, rational insurance choices satisfy,Competitive Insurance,When insurance is fair, rational insurance choices satisfyI.e.,Competitive Insurance,When insurance is fair, rational
21、insurance choices satisfyI.e. Marginal utility of income must be the same in both states.,Competitive Insurance,How much fair insurance does a risk-averse consumer buy?,Competitive Insurance,How much fair insurance does a risk-averse consumer buy?Risk-aversion MU(c) as c .,Competitive Insurance,How
22、much fair insurance does a risk-averse consumer buy?Risk-aversion MU(c) as c . Hence,Competitive Insurance,How much fair insurance does a risk-averse consumer buy?Risk-aversion MU(c) as c . Hence I.e. full-insurance.,Competitive Insurance,Cna = m - K. Ca = m - L + (1- )K. Cna = Ca Gives K=L,“Unfair”
23、 Insurance,Suppose insurers make positive expected economic profit. I.e. K - aK - (1 - a)0 = ( - a)K 0.,“Unfair” Insurance,Suppose insurers make positive expected economic profit. I.e. K - aK - (1 - a)0 = ( - a)K 0. Then a ,“Unfair” Insurance,Rational choice requires,“Unfair” Insurance,Rational choi
24、ce requiresSince,“Unfair” Insurance,Rational choice requiresSince Hence for a risk-averter. KL,“Unfair” Insurance,Rational choice requiresSince Hence for a risk-averter. I.e. a risk-averter buys less than full “Unfair” insurance.,Uncertainty is Pervasive,What are rational responses to uncertainty? b
25、uying insurance (health, life, auto) a portfolio of contingent consumption goods.,Uncertainty is Pervasive,What are rational responses to uncertainty? buying insurance (health, life, auto) a portfolio of contingent consumption goods.,Uncertainty is Pervasive,What are rational responses to uncertaint
26、y? buying insurance (health, life, auto) a portfolio of contingent consumption goods.,?,Diversification,Two firms, A and B. Shares cost $10. With prob. 1/2 As profit is $100 and Bs profit is $20. With prob. 1/2 As profit is $20 and Bs profit is $100. You have $100 to invest. How?,Diversification,Buy
27、 only firm As stock? $100/10 = 10 shares. You earn $1000 with prob. 1/2 and $200 with prob. 1/2. Expected earning: $500 + $100 = $600,Diversification,Buy only firm Bs stock? $100/10 = 10 shares. You earn $1000 with prob. 1/2 and $200 with prob. 1/2. Expected earning: $500 + $100 = $600,Diversificati
28、on,Buy 5 shares in each firm? You earn $600 for sure. Diversification has maintained expected earning and lowered risk.,Diversification,Buy 5 shares in each firm? You earn $600 for sure. Diversification has maintained expected earning and lowered risk. Typically, diversification lowers expected earn
29、ings in exchange for lowered risk.,Risk Spreading/Mutual Insurance,100 risk-neutral persons each independently risk a $10,000 loss. Loss probability = 0.01. Initial wealth is $40,000. No insurance: expected wealth is,Risk Spreading/Mutual Insurance,Mutual insurance: Expected loss isEach of the 100 persons pays $1 into a mutual insurance fund. Mutual insurance: expected wealth isRisk-spreading benefits everyone.,
