Uncertainty and Risk
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 4
Uncertainty and Risk
Lecture 4
Up to now: no uncertainty about what is going to happen in the world.
In the real world: lots of uncertainty!
We'll model this by thinking about preferences over consumption lotteries in which you consume different amounts in different states of the world.
We're just going to think about preferences, not budget constraints.
Today's Agenda
Part 1: Preferences over Risk
Part 2: Market Implications
[worked example]
Lotteries
Expected Utility
Certainty Equivalence and Risk Premium
Insurance
Risky Assets
Option A
Option B
I flip a coin.
Heads, you get 1 extra homework point.
Tails, you get 9 extra homework points.
I give you 5 extra homework points.
pollev.com/chrismakler
Heads, you get 5 extra homework points.
Tails, you get 5 extra homework points.
Option A
Option B
I flip a coin.
Heads, you get 1 extra homework point.
Tails, you get 9 extra homework points.
I give you 5 extra homework points.
Heads, you get 5 extra homework points.
Tails, you get 5 extra homework points.
Expected Utility
Suppose your utility from points is given by
Expected value of the homework points (c) is:
Expected value of your utility (u) is:
Risk Aversion
Expected value of the homework points (c) is:
Utility from expected points is:
Expected value of your utility (u) is:
Because the utility from having 5 points for sure is higher than the expected utility of the lottery, this person would be risk averse.
Utility from expected points:
Expected utility (u):
Indifference curve for 2 utils
Indifference curve for
5 utils
Option A
Option B
I flip a coin.
Heads, you get 1 extra homework point.
Tails, you get 9 extra homework points.
I give you 4 extra homework points.
pollev.com/chrismakler
Heads, you get 4 extra homework points.
Tails, you get 4 extra homework points.
Certainty Equivalent
Suppose your utility from points is given by
Expected value of your utility (u) is:
What amount CE, if you had it for sure, would give you the same utility?
Indifference curve for 2 utils
Certainty Equivalent
Certainty Equivalent = 4
Expected Points = 5
Risk Premium = 1
Risk Premium
How much would you be willing to pay to avoid risk?
What is Sixt offering me here?
Example 1: Betting on a Coin Toss
Example 2: Deal or No Deal
Start with $250 for sure
Would you do it?
If you bet $150 on a coin toss,
you would face the lottery:
50% chance of 100, 50% chance of 400
Two briefcases left: $200K and $1 million
Would you accept that offer? What's the highest offer you would accept?
The "banker" offers you $561,000 to walk away; or you could pick one of the cases.
A lottery is a set of outcomes,
each of which occurs with a known probability.
Lotteries
Example 1: Betting on a Coin Toss
Start with $250 for sure
If you bet $150 on a coin toss,
you would face the lottery:
50% chance of 100,
50% chance of 400
We can represent a lottery as a "bundle" in "state 1 - state 2 space"
Suppose the way you feel about money doesn't depend on the state of the world.
Independence Assumption
Payoff if don't take the bet: u(250)
Payoff if win the bet: u(400)
Payoff if lose the bet: u(100)
Expected Utility
"Von Neumann-Morgenstern Utility Function"
Probability-weighted average of a consistent within-state utility function u(cs)
Last time: c1, c2 represented
consumption in different time periods.
This time: c1, c2 represent
consumption in different states of the world.
Comparison to intertemporal consumption
Marginal Rate of Substitution
You prefer having E[c] for sure to taking the gamble
You're indifferent between the two
You prefer taking the gamble to having E[c] for sure
Risk Aversion
You prefer having E[c] for sure to taking the gamble
You're indifferent between the two
You prefer taking the gamble to having E[c] for sure
Certainty Equivalence
"How much money would you need to have for sure
to be just as well off as you are with your current gamble?"
Risk Premium
"How much would you be willing to pay to avoid a fair bet?"
Market Implications
How do markets allow us to shift our consumption across states of the world?
Case 1: Insurance
Case 2: Risky Assets
Money in good state
Money in bad state
35,000
25,000
Suppose you have $35,000. Life's good.
If you get into a car accident, you'd lose $10,000, leaving you with $25,000.
You might want to insure against this loss by buying a contingent contract that pays you $K in the case of a car accident.
Money in good state
Money in bad state
35,000
25,000
You want to insure against this loss by buying a contingent contract that pays you $K in the case of a car accident. Suppose this costs $P.
Now in the good state, you have $35,000 - P.
In the bad state, you have $25,000 - P + K.
35,000 - P
25,000 + K - P
Budget line