Choice Under Uncertainty IIContingent Consumption

Preferences Under Uncertainty Expected utility using states of nature outside individuals control (been using) Actions can be taken to alter the consumption bundles(/wealth) available in different states of nature

o Ex . if we back up computer files to external discs, we reduce our loss in the event of a hard disc crash, but at a cost (in time, effort, and the external medium itself)

o Ex . fire an theft insurance has a cost (premium), but changes the “consumption bundle” available in the event of burglary or a fire

Consumer can choose substitute consumption in one state of nature (no fire) for consumption in another (house burns down), to increase expected utility

State-contingent consumption plans that give = expected utility are equally preferred

State-Continent Budget ConstraintsEx. accident insurance with variable face value (insurance buyer can choose coverage amount)

o Each $1 of accident insurance costs ‘ ’γo Consumer has $m of wealtho Cna = consumption value in no-accident stateo Ca = consumption value in accident state

o Without insurance Ca = m – L Cna = m where L = $amount of loss

o If L is larger relative to “m” consumption may be very different in the 2 states

Buy $K of accident insurance at a premium rate of per dollar of face value (coverage)γo Cna = m – Kγo Ca = m – L – K + K = m – L + (1 – )Kγ γ

Consumption in event of insured loss is:o Base income ‘m’ lesso Loss pluso The net proceeds of insurance (claim less premium)

Consumption is there is no loss is:o ‘m’ less insurance premium paid

Buy $K of accident insurance at rate γo Cna = m – [ins. Prem.] = m - Kγo Ca = m – L – K + K = m – L + (1 – )Kγ γo Solving for K …

K = (Ca – m + L) /(1 – )γo Substituting

Ca = m – (Cγ a – m + L) /(1 – )γ

Numerical Example Initial wealth = $50,000 Amount of loss = $40,000 Premium = 10% of face Amounts of consumption in accident and no-accident states when different face

amounts of insurance are bought are:o CNA = 50,000 – (0.1)(Face)o CA = 50,000 – 40,000 + (0.9)(Face)

Preferences Under Uncertainty What is the MRS of an indifference curve? Get consumption C1 w/ prob π1 and C2 with prob π2

(π1 + π2) = 1EU = π1U(c1) + π2U(c2)

For constant EU, dEU = 0

Choice Under Uncertainty How is rational choices made under uncertainty?

o Choose the most preferred affordable state-contingent consumption plan

State-Contingent Budget Constraints

Competitive Insurance Suppose entry to insurance industry is free Expected economic profit = 0

K - γ πaK – (1 – πa)0 = ( - γ πa)K = 0 free entry = γ πa

if price f $1 insurance = accident probability then insurance = fair(insurer’s premium revenue just =s expected claims payments)

when insurance is fair (premium per $ = probability of claim), rational insurance choices satisfy

Marginal utility of income must be the same in both states How much fair insurance does a risk-averse consumer buy? MU(ca) = MU(cna)

o Risk-aversion MU(c) decrease and c increaseso Risk averse consumer buys full insurance

“Unfair” Insurance Suppose insurers make +ve expected economic profit

K - γ πaK – (1 – πa)0 = ( - γ πa)K > 0 Then > γ πa

rational choice requires:

o since

for risk-avertero a risk-averter buys less than full “unfair” insurance

Example Jimmy owns racing car enters in competitions Utility of wealth function U(W) = (W)0.5

If car crashes = destroyed 0 wealth Not in a crash 252,000 wealth (value of car); he owns no other wealth Due to nature of race-car driving and the skill of he drivers he hires prob of his car being destroyed in a year = 1/6 Can buy $K of insurance on his car ad rate of ($2/7) for every dollars worth of insurance he buys How much insurance will Jimmy buy, if he maximizes expected utility over the states (crash,no crash)?

Budget Constraint NO insurance

o Wealth in state N (no crash) = 252,000o Wealth in state C (crash) = 0

FULL insurance boughto Pays premium (2/7)(252,00) = $72000o Wealth at state N 252,000 – 72000 = $180,000o Wealth in state C 252,000 (ins. Claim) – 72,000 (prem) = $180,000

Form:PNACNA + PACA < B

Can determine B when we choose PNA

When CNA = 252,000 CA = 0 (no insurance When CNA = 180,000 CA = $180,000 (full insurance) PA = 2/7 (ins prem rate per $1 face)

o Choose PNA = (1 _ PA) = 5/7o Then B 180,000 = (5/7)252,000 + (2/7)0o And B 180,000 = (5/7)180,000 = (2/7)180,000

“Budget Constraint”o (5/7)CNA + (2/7)CA < 180,000

each $1 unit of CN costs $(5/7) [pay $5, get $5 + $2 save premium] each $1 unit of CC costs $(2/7) [pay $2 premium, get $7 of claim] “budget”constraint” across 2 states:

o (5/7)CN + (2/7)CC = 180,000 for Cc < 180,000o he cant buy more than 100% insurance and will buy less than 100% since prem is unfair

prob of loss = 1/6 (16.67%, prem = 2/7 = 28% tangency condition:

Tangency combined with Budget Constraint sub optimal amount fo state C consumption into budget constraint

(5/7)CN + (2/7)CC = 180,000 (5/7)[4 CC] + (2/7)CC = 180,000 (20/7)CC + (2/7)CC = (22/7) CC = 180,000

CC = [180,000](7/22) = $57,272 CN = $229,091 (4CC)

o 252,000 – 229,092 = $22,909 is spent on prem, to buy (7/22)22,909 = $80,181 of face value in state C, Cc = $80,181 (claim) - $22,909 (premium) = $57,272

Uncertainty is pervasive what are rational responses to uncertainty?

o A portfolio of contingent consumption goods

Diversification Two firms A and B Shares cost $10 Prob ½ A’s profit = 100 B’s profit = $20 Prob ½ A’s profit = 20 B’s profit = 100 You have 100 to invest. How?

o Buy only firm A’s stock? $100/10 = 10 shares you earn $1000 with prob ½ and $200 with prob ½ expected earning: $500 + $100 = $600

o buy only B’s stock? $100/10 = 10 shares you earn $1000 with prob ½ and $200 with prob ½ expected earning: $500 + $100 = $600

o buy 5 shares each firm earn $600 for sure

diversification has maintained expected earnings and lowered risk typically, diversification lowers expected earnings in exchange for lowered risk

Example buy shares in: Shell Oil, Day & Ross Trucking state 1: Low Oil price

o trucking profits high but oil profits low state 2: High Oil Price

o trucking profits low (cost of fuel high), but oil profits high

Risk Eliminates (Generally Reduced) Through Risk Pooling principle is that if we have enough investment, the returns from which are not perfectly positively correlated …

o not all of them lose at the same time when some have bad days, others have good/less bad the “pool” of risks has a smaller risk (technically measured by variance) than does any single asset (share)

Risk Spreading/ Mutual Insurance 100 risk-averse persons each independently risk a $10,000 loss loss prob = 0.01 (1%) initial wealth is $40,000 no insurance: expected wealth

o 0.99 x $40,000 + 0.01(40,000 – 10,000) = $39,900o expected utility < utility $39,900 with certainty (by risk aversion)

Mutual Insurance each of the 100 persons pays $100 into a mutual insurance fund total premium = $100*$100 = $10,000 premium collections = to loss to one (unlucky) contributor on average 1/100 experiences a $10,000 loss each period – is compensated fully funds paid in premium by all (including

loser) certain wealth for each of 99 w/o loss = $39,900 loss of $10,000 fully compensated, loser has wealth $39,900 - $10,000 = $39,900 iff each consumer = risk-averse, utility of $39,900 with certainty > expected utility when expected wealth = $39,900 risk-spreading implies every contributor has higher utility than with no mutual insurance

Risk Pooling many independent gambles, some win, some lose all gains and losses are shared if gambles perfectly –vely correlated, risk eliminated (there is no loss to anyone)

o imperfectly correlated risk reduced

Risk Spreading each member of mutual insurance scheme made better off because small certain payment (loss) implies higher VNM

utility than risky prospect with = Expected wealth there WILL be a loss all member in group share it