Chapter 7: Choice under Uncertainty and the Economics of Information

What is choice under uncertainty?

Making decisions without knowing which outcome will occur

• Bc most economic decisons made w/o complete info

• Cant know all possible outcomes when making a decision

What are economic decisions under uncertainty +why?

Gambles

• Bc outcomes are uncertain

Proability

Measures the likelihood of an outcome occurring.

What is the probability of heads on a fair coin?

50%

• Ie over many tosses, each outcome should occur approximately 50% of the time.

Important property of a gamble

• Its expected value

Expected value

The sum of all possible outcomes, weighted by its respective probability of occurrence

• ie weighted average of all possible outcomes (weights = respective probabilities)

What does expected value measure?

Average payoff expected over many repetitions of the gamble

Expected value formula

EV = Σ (probability x outcome)

Does a higher EV mean a gamble is more attractive?

NOT necessarily

• People also consider how desirable or unpleasant the outcomes are

Positive vs negative expected value

• Positive EV = expected net gain.

• Negative EV = expected net loss.

What does a gamble’s attractiveness depend on?

1. Its expected value

2. AND the desirability of its possible outcomes

Fair coin toss:

1. Which gamble has the highest EV?

2. Which gamble is most likely to be accepted?

3. Why do most people reject Gamble 3?

Which gamble has the highest EV?	Gamble 3 (€5,000).
Which gamble is most likely to be accepted?	Gamble 1, because its potential loss is very small.
Why do most people reject Gamble 3?	The potential loss of €10,000 is extremely unpleasant.

The von Neumann-Morgenstern Expected Utility Model

People choose the alternative that has the highest expected utility

• EU maximisation

Expected utility

EU of a gamble = expected value of utility over all possible outcomes

• The weighted average of the utility of all possible outcomes

What does expected utility maximisation assume?

• A utility function U that assigns a numerical measure to the satisfaction associated with different outcomes.

According to the Expected Utility Model, what option do people choose?

Option w the highest expected utility (EU)

• Not necessarily the highest EV

What does utility measure?

The satisfaction associated with an outcome.

Expected utility formula

EV = Σ (probability x utility)

When should you accept vs reject a gamble?

• Accept: EU _gamble > U(M₀)

• Reject: EU _gamble < U(M₀)

• Indifferent: EU gamble = U(M₀)

M₀ = initial wealth

If you refuse a gamble what is your utility

Same utility as initial wealth

• U(M₀)

Example for a fair coin toss

Gamble	Heads	Tails	Expected value
1	+€100	-€0.50	€49.75
2	+€200	-€100	€50
3	+€20,000	-€10,000	€5,000

Do expected values have the same rankings as expected utilities for a set of alternatives?

Not neccessarily

• Bc utility is usually a non linear function of wealth

Expected utility vs expected value

• EV calculates the average monetary outcome.

• EU calculates the average utility of the outcomes.

Is utility function usually concave or convex + explain?

Concave

• Concave UF = curves downwards

• As wealth increases, slope decreases

• Diminishing marginal utility

• ie curve lies above the chord connecting any two points on it.

Feature of marginal utility for a concave utility function

Diminishing marginal utility

• As wealth increases, MU decreases

Diminishing marginal utility of wealth

• The more wealth a consumer has, the additional utility from 1 extra unit of wealth decreases

• Each additional unit of wealth provides less additional utility

What is marginal utility?

Slope of the utility function

What type of person is someone with a concave utility function (of wealth)?

Risk averse

Risk averse

Preferences described by a utility function with diminishing marginal utility of wealth (concave UF)

• Prefer certainty over a gamble with the same expected value

• Always refuse a fair gamble (EV = 0)

What type of utility function represents risk aversion?

Concave utility function

Fair gamble

Gamble with an expected value of 0

• EV of your wealth if you accept the gamble = certain value of your wealth if you refuse it

• ie Expected wealth from gambling = wealth from refusin

Why do risk-averse people reject fair gambles?

Bc utility lost from losing exceeds the utility gained from winning

• Due to diminishing marginal utility

• So EU gambling < U (initial wealth)

Geometric interpretation of the expected utility of a gamble

1. 2 points in the utility function: winning and losing

2. Join point with straight line (chord)

3. EU lies on this chord

4. Concave UF: curve is above the chord

5. So EU gamble < U(EV wealth)

6. For a fair gamble EU gamble <U(M0)

If the probability of winning is half, where does EU lie?

Halfway between the chord joining the winning and losing gamble

Utility function for risk averse person

Where does expected utility lie geometrically?

On the chord joining the utilities of the possible outcomes.

Where does EU lie if probabilities are unequal?

At a weighted position along the chord.

Will risk-averse people accept every positive-EV gamble?

No.

• If the potential loss makes the EU too low → reject

Example- unequal probabilities

Remember: Probabilities do not have to be 50/50. Always multiply each utility by its respective probability.

What do you need to do to initial wealth BEFORE applying the utility function to calculate EU?

Add/subtract the gamble's winnings/losses from initial wealth

If probability of winning is p, what is the probability of losing?

1-p

Expected utility with unequal probabilities

What determines the position of EU on the chord?

Depends on probabilities of winning or losing

Where will EU lie if there is a higher probability of winning?

Closer to the winning endpoint

Are most people risk averse or risk seeking?

Averse

• Increase wealth → DMU (concave)

Risk seeking

Someone whose preferences are described by a utility function with increasing marginal utility of wealth.

• Convex UF

• Slope increases as wealth increases.

• Prefer taking a fair gamble over receiving the certain outcome.

• EU fair gamble > U(M0)

• EU of accepting fair gamble > U of refusing it

Utility function for risk averse people

Convex

• Chord lies above curve

Risk neutral

Preferences described by a utility function with constant marginal utility of wealth

• Linear UF

• Indifferent betw accepting + refusing a fair gamble

• EU fair gamble = U(M0) (aka U of refusing it)

Utility function for a risk neutral person

How many possible outcomes can a gamble have?

Many

What must all probabilities in a gamble add up to?

1 (100%)

Risk averse vs risk seeking vs risk neutral

Why do concave utility curves make risk feel costly?

Bc diminishing marginal utility

Information that reduced uncertainty has?

Economic value

Why does info that reduces uncertainty have economic value?

Helps ppl make better decisons

• By revealing / predicting possible outcomes

• Ppl make better decisions based on the outcomes they learn about

• Eg career guidance

Does only perfect info have economic value?

No. Even imperfect

• Anything that reduces uncertainty (even a little) has economic value

Why is information valuable?

It allows people to choose more effectively between alternatives.