12 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 In this lecture we introduce Different attitudes towards risk Indifference curves for a risk-averse consumer State-contingent budget constraint Optimal choice of insurance

Preferences Under Uncertainty Different attitudes towards risk Risk-aversion Risk-neutrality Risk-loving

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 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 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  $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 12 EU=7 2 $0 $45 $90 Wealth

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

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

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

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

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

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

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

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

Preferences Under Uncertainty State-contingent consumption plans that give equal expected utility are equally preferred. Indifference curve for contingent consumption Now, we have two possible states: “accident” and “no accident” Contingent consumptions: & Ca Cna

Preferences Under Uncertainty Cna Indifference curves: EU1 < EU2 < EU3 EU3 EU2 EU1 Ca

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 Uncertainty

Preferences Under Uncertainty

Preferences Under Uncertainty Cna Indifference curves EU1 < EU2 < EU3 EU3 EU2 EU1 Ca Slope:

Uncertainty QUESTION 1 Convex preferences shown above apply for risk-averse consumer. Can you show why?

Uncertainty Preferences are convex when for any X, Y and Z where X Z Y Z And for every 𝜽 ϵ[𝟎,𝟏]: 𝜽𝑿+(𝟏−𝜽)Y≽𝒁 ~ f ~ f

Uncertainty For risk aversion person utility is concave. Utility is concave when: 𝐔(𝜽𝑿+(𝟏−𝜽)Y)≥𝜽𝑼 𝑿 +(𝟏−𝜽)U(Y) For all X and Y and for every 𝜽 ϵ[𝟎,𝟏]:

Uncertainty f f ~ ~ Now we have X,Y,Z X Z and Y Z And concave utility U() that represents preferences: 𝐔(𝜽𝑿+(𝟏−𝜽)Y)≥𝜽𝑼 𝑿 +(𝟏−𝜽)U(Y) ≥𝜽𝑼 𝒁 +(𝟏−𝜽)U(Z) ≥𝑼 𝒁 For all X and Y and for every 𝜽 ϵ[𝟎,𝟏]: ~ f ~ f

Uncertainty 𝐔(𝜽𝑿+(𝟏−𝜽)Y)≥𝜽𝑼 𝑿 +(𝟏−𝜽)U(Y) ≥𝜽𝑼 𝒁 +(𝟏−𝜽)U(Z) ≥𝑼 𝒁 Means that: 𝑼(𝜽𝑿+(𝟏−𝜽)Y)≥𝑼 𝒁 Where follows: 𝜽𝑿+(𝟏−𝜽)Y≽𝒁  That means preferences are convex!

Insurance State contingent budget constraint

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.

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.

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.

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

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

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 = 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. 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.

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

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

State-Contingent Budget Constraints Cna The endowment bundle. m Where is the most preferred state-contingent consumption plan? Ca

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 Constraints Cna The endowment bundle. m Where is the most preferred state-contingent consumption plan? Affordable plans Ca

State-Contingent Budget Constraints Cna More preferred m Where is the most preferred state-contingent consumption plan? Ca

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

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

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

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

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.

Competitive Insurance When insurance is fair, rational insurance choices satisfy

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

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

“Unfair” 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 choice requires Since

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

“Unfair” Insurance Rational choice requires Since Hence for a risk-averter. I.e. a risk-averter buys less than full “unfair” insurance.