# 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

\(\sqrt{5}\) 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 \(u(c_s)\)*

Last time: \(c_1\), \(c_2\) represented

consumption in different time periods.

This time: \(c_1\), \(c_2\) 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

# Next Week: Exchange

## Bring two agents, each with an endowment of two goods, into the same model and have them trade.

#### Econ 51 | 04 | Uncertainty and Risk

By Chris Makler

# Econ 51 | 04 | Uncertainty and Risk

Uncertainty and Risk Aversion

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