Choice Under Risk
Risky Environment: An environment where one of a number of possible outcomes (events) might
occur, and the possibility of an outcome can be described by a probability distribution.
Example: Buy a stock for $100
Outcome A: Stock stays the same
Outcome B: Stock goes up $20
Outcome C: Stock goes down $20
To make choice in such an environment, we first need some tools to describe risk.
Probabilities: P(A)=50%, P(B)=30%, P(C)=20%
Note: a probability is between 0 and 1, and together sum to 1.
Now given outcomes and probabilities, what can you expect to earn if you buy the stock
Expected Value: Measure of average payoff, where weights in average are the probabilities.
EV =
N
!
1
EV = (0.5)(100) + (0.3)(120) + (0.2)(80)
pn xn
Choice Under Risk
Example: Choice between two stocks priced at $100
Stock 1
Stock 2
Outcome
P()
I
P()
I
A
0.5
100
0.2
100
B
0.25
120
0.4
120
C
0.25
80
0.4
80
Variance: Measures “degree of riskiness.” For each outcome, take the square of the value minus
the expected value, multiply by the probability weight, and sum them up.
V ar =
N
!
1
pn (xn − EV )2
Choice Under Risk
While the two previous measures describe the risky environment, are they good description of
how people choose?
St. Petersburg Paradox: I’m going to start flipping a coin, and pay you two dollars a flip until a tails
comes up. How much would you pay to play this gamble?
So we need to include preferences, namely preferences for risk. To keep it simple, suppose people
only care about one good (let’s call it income, I). So u(I) is utility function.
Expected Utility
Similar to expected value, expected utility is “average utility” with weights given by probabilities
N
!
EU =
pn u(xn )
1
Example: Suppose you have $100 and are offered a gamble where you win $50 with p=1/2 or
$100 with p=1/2.
Choice Under Risk
Example: Repeat choice given earlier between two stocks priced at $100
Stock 1
Stock 2
u(x) = 10x
1/2
Outcome
P()
I
P()
I
A
0.5
100
0.2
100
B
0.25
120
0.4
120
C
0.25
80
0.4
80
Risk Aversion, Loving and Neutrality
A person is risk averse (loving, neutral) when the gamble yields less (more, same) utility as the
sure thing with the same EV as the gamble.
We already saw that risk averse behaviour arises when the utility function is concave. Risk loving
behaviour arises when it is convex, and risk neutrality when it is linear.
EV may not equal Sure Thing
A risk averse person will accept a gamble if it is sufficiently in their favour.
Example: Suppose you have $100 and are offered a gamble where you win $50 with p=1/2 or
$200 with p=1/2.
Risk Premium and Certainty Equivalent
Since people are risk averse, they are willing to pay some premium in order to avoid the gamble.
Certainty Equivalent: Max amount you would accept to avoid the gamble at same level of utility.
Risk Premium: Difference between EV and CE
Example: Suppose you have $50 and are facing a gamble where you gain $150 with p=1/2 or $0
with p=1/2.
Risk Pooling
If the probability of outcomes are independent, it is possible for people to pool their risk by relying
on the Law of Large Numbers
Example: Choice between colleges for 1,000,000 people
A: %60 -> 1,000,000 or %40 -> 250,000
B: 690,000
Risk Pooling - General Form
Say there are two farmers who are concerned about their barns burning down. The
probability of one barn burning is p, each farmer has $x, and the cost to repair it is $L
Insurance Markets
You face the following gamble: M = $700, with probability 1/3 you lose $600. An insurance
company offers an insurance contract costing $0.33 per $1 of coverage.
Assumptions:
1) Must be able to fully insure
2) Price of insurance must be actuarially fair (price per $ of coverage = probability)
Economics of Information
Moral Hazard: Since a person is fully insured, they do not find it in their interest to take (costly)
actions to lower their risk. For example, anti-theft devices, or safe driving. To correct for this,
insurance companies require consumers to pay a cost (deductible) if you want to make a claim.
Adverse Selection: Consider a pool of people with differing degrees of risk. The higher the
insurance premium, the less desirable it is for the low-risk people to buy insurance so they may
leave the contract, leaving only the high-risk people. This will end up costing the insurance
company as they will have to pay out larger claims.
Economics of Information
In many economic circumstances, agents must make decision without perfect information
Signalling: Communication that conveys credible information
1) It must be costly-to-fake
2) If one individual reveals information, others must also reveal even if it is less favourable
Consider trying to find a girlfriend/boyfriend...
...or trying to find a job
People often categorize things (trustworthy or untrustworthy etc...). If you do not signal that you
are in one category you are generally assumed to be in the other.
The Market For Lemons
It is possible that the information asymmetries present in a market can lead to market failure.
Why do used cars depreciate in price the minute they’re driven off the lot?