Term 2 Lecture 8 Flashcards
(15 cards)
Summary: choice under uncertainty can be modelled similarly in many ways to intertemporal choice, with
Differences in future possible states instead, so individuals are making decisions under uncertainty
- just as people prefer to smooth consumption over time in intertemporal choice, also prefer to reduce uncertainty in consumption across possible future outcomes.
Preferences will depend on
Perceived probabilities of states of the world occurring
1 scenario - facing risk and buying insurance
- individual owns an asset worth A, but faced probability p of losing it
- can purchase insurance coverage K at a cost of yK, K the amount of coverage, y the insurance price per unit of coverage
So the decision is how much K to buy? Why is K>A not allowed?
No loss occurs:
- individual keeps A, but has paid yK for insurance, so c0 = A - yK
Loss occurs:
- individual loses A, but receives a payout K from insurance
- c1 = (1-y)K
Budget constraint:
C0 = A - (y/1-y)c1
- so if person wants higher consumption in the loss state c1, must give up more of their consumption in the no-loss state c0
- the rate at which they trade off consumption between states depends on y, the price of insurance
2nd scenario, betting on an event
- income y, bet an amount B >/ 0
- when does a kink appear
If lose:
- c0 = y - B
- c1 = y + Bz, where z is the odds
Budget constraint:
- c1 = (1+z)y - zc0, where c0 /< c1, the more they bet the lower c0 is, higher z means bigger potential payout, increasing c1
- kink can appear if we are in unfair betting markets, if the tradeoff is worse in one direction, so the budget constraint is not smooth, a kink appears where the odds change
Scenario 3, tax evasion
- income y
- true tax liability is T, but they try evade an amount D by underdeclaring income
- probability p of getting audited, where they then have to pay T + fD to IRS
Not audited, probability 1-p:
- c0 = y - T + D
If audited, probability p:
- c1 = y - T - fD
Budget constraint:
C1 = (1+f)( y - T) - fc0
Why is the budget constraint independent of probability?
- budget constraint only shows possible consumption levels in different states
- trade-off between consumption in different states always remains the same, no matter how often each state occurs
Key Idea: budget constraint shows what’s possible, while probability matters when making choices between risky options
Expected utility theory
U(c0,c1,…,p0,p1,…) = SUM(pi.v(ci))
- pi is the probability of state i occurring
- v(ci) is the Bernoulli utility function
- u(…) is the vNM expected utility function
Sure thing principle ( or the related strong independence axiom)
If two choices have identical outcomes in some states of the world, then the decision should only depend on the states where the outcomes differ.
- imagine you’re picking between two job offers, in both jobs you get 100k when economy is bad, so when you compare the STP says you should only compare when the economy is good
- imagine outcomes within choice 1 and 2 are the same respectively for the 3rd state, you should only make a decision based off the first two states
- this means that if you choose outcome A1, you must also choose A2, as in the second choice, you will disregard the 3rd state and choose from the first two, which are identical to the first two in choice 1
Why is the sure thing principle controversial
IRL, people sometimes violate this principle:
- framing effects - the way options are presented affects decision, Allais paradox
- risk aversion - people might irrationally prefer to avoid uncertainty
- loss aversion - some focus too much on avoiding worst-case scenarios
MRS in the sure thing principle
- STP and EUT assume that the preference trade-off between uncertain outcomes does not depend on unrelated states
Why is v(c) not ordinal, but u(c) is
- ordinal utility means only the ranking of choices matters, not the exact values
- v(c), the within period utility function is not ordinal because if you change how it is measured, like using c or ln(c), it affects choices and risk behaviour
- u(c), expected utility function is ordinal as it just preserves rankings.
Risk aversion formula
- key idea
- formula meaning
- why does this require concavity
Risk-averse people prefer a guaranteed amount of money over a risky gamble with the sample expected value
- v((1-p)c0 + pc1) > (1-p)v(c0) + pv(c1)
- utility of the expected value of the gamble > the expected utility of the gamble itself
- risk averse person doesn’t just look at the expected money amount, they also care about risk
- with concavity, the average of two points on the curve is above the curve itself, so a risk-averse person prefers that, stronger the concavity = stronger the risk aversion
Risk aversion example
- £50 for sure
- Take a gamble where you get £100 or £0 with 50% probability each
- expected value is still £50, so a risk-neutral person would see these as equal
- but a risk-averse person would prefer the guarded £50
Risk aversion under the insurance case:
U(K) = (1-p)v(A-yK) + pv((1-y)K)
U’(K) = (1-p)(-yv’(A-yK)) + p(1-y)v’((1-y)K) = 0
—> (1-p)yv’(A-yK) = p(1-y)v’((1-y)K)
IF INSURANCE IS ACTUARIALLY FAIR, then p = y
—> v’(A-yK) = v’((1-y)K)
- since v’(c) is a decreasing/ concave function, only way for this to hold is if A = K, so consumer is fully insured.
Understanding over insurance and under insurance based on actuarial fairness
If insurance is better than fair, i.e. p>y:
- insurance is too cheap compared to the risk of loss, you would over insure
If insurance is worse than fair, i.e. p<y:
- insurance is overpriced relative to the risk of loss, people underinsure, depending on risk aversion of couse, high risk averse will still buy close to full coverage, less will buy less - depends on concavity.
