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Lecture4-RiskandUncertainty.pdf

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Price Theory

Lecture 4: Risk and Uncertainty

Topics for today’s lecture . . .

1. Risky alternatives

2. Expected utility

3. Risk preferences

4. Insuring against risk

Risky alternatives

Discussion: Which job would you prefer?

Imagine that you have just graduated from your degree. You have two job offers to choose

from:

• The first job with ‘Alpha Corp.’ has a salary of $50,000.

• The second job with ‘Beta Inc.’ has a salary of $30,000, and pays you a bonus of $40,000 if you achieve your performance targets. (You think that you have a 50%

probability of reaching these targets.)

Which job would you choose?

What factors would influence your decision?

Definition: Lottery

Any choice (alternative) with uncertain consequences.

In microeconomics we use lotteries to model uncertain prospects such as investment returns.

Describing a lottery

The first step in describing a lottery is to list all of the possible outcomes.

• Outcomes must be described such that they are mutually exclusive.

The second step is to determine the probability of each outcome occurring.

• A probability is a number that lies between zero and one.

• The sum of the probabilities of all outcomes must be equal to one.

The third step is to determine the consequences of each outcome for the decision-maker.

• We will typically consider situations in which the consequences are monetary payoffs.

Describing Alpha Corp.’s job offer as a lottery

0.0

50k

0 I

Outcome 1: You are employed.

• Probability pA = 1.

• You receive the salary IA = $50,000.

This is called a sure-thing payoff because

it occurs with certainty.

Describing Beta Inc.’s job offer as a lottery

0.0

0.5

30k

IB1

70k

IB2

0 I

Outcome 1: You miss your targets.

• Probability pB1 = 0.5.

• You receive the salary IB1 = $30,000.

Outcome 2: You achieve your targets.

• Probability pB2 = 0.5.

• You receive the salary $30,000 and a bonus $40,000 (total IB2 = $70,000).

Comparing lotteries: Expected value

An expected value can be calculated for any lottery that gives rise to monetary payoffs.

To calculate the expected value, multiply each payoff by the probability it will occur, and add

them together.

• The expected value can be interpreted as the average (mean) payoff the lottery would generate if repeated many times.

The expected value of the two job offers are:

• Alpha Corp.: EV A = pA × IA = 1 × $50,000 = $50,000.

• Beta Inc.: EV B = pB1 × IB1 + pB2 × IB2 = 0.5 × $30,000 + 0.5 × $70,000 = $50,000.

Comparing lotteries: Risk

0.0

50k

IA0.5

30k

IB1

70k

IB2

0 I

Risk is a measure of the degree to which

a lottery’s payoffs are ‘spread out’.

• Both jobs have the same expected value.

• Beta Inc.’s job offer is more risky that Alpha Corp.’s job offer.

It is common to measure a lottery’s risk

using the variance of its payoffs.

Subjective probabilities

In reality, the probabilities of events are rarely objectively well defined.

• On the one hand, the probability of flipping ‘heads’ on a fair coin is (close to) 0.5.

• On the other hand, it is hard to construct an objective measure of the probability that there will be a recession next year.

Subjective probabilities are a decision-maker’s assessment of the risks she/he faces.

• Different decision-makers may hold different beliefs concerning the probabilities of the outcomes of a risky event.

• So long as a decision-maker has (or behaves as though she/he has) subjective probabilities, the results of today’s lessons apply.

Exercise: Describing an investment as a lottery

Suppose that you purchase $10,000 of shares in agricultural company that farms corn.

• There is a 50% probability of a good harvest, which will increase the value of your shares by $4000.

• There is a 20% chance of a recession that will halve the value of your shares.

You should assume that the probabilities of the two events are independent, and that (where

applicable) the effects of a recession are applied after the effects of a good harvest.

1. Describe this investment as a lottery, and illustrate the lottery on a graph.

2. What is the expected value of this lottery?

Exercise solutions

1. This lottery has four possible outcomes:

• Outcome 1: Normal harvest and no recession (NN ).

• Outcome 2: Normal harvest with a recession (NR ).

• Outcome 3: Good harvest and no recession (GN ).

• Outcome 4: Good harvest with a recession (GR ).

Given that the probability of the two events are independent, we can find the probability

of each outcome by multiplying the probabilities of the constituent events.

Exercise solutions

0.0

0.4

10k

INN

0.1

INR

0 I

1. (continued)

Outcome 1: NN .

• Probability pNN = 0.5 × 0.8 = 0.4.

• Your shares are worth INN = $10,000.

Outcome 2: NR .

• Probability pNR = 0.5 × 0.2 = 0.1.

• Your shares are worth INR = 0.5 × $10,000 = $5000.

Exercise solutions

0.0

0.4

10k

INN

0.1

INR

14k

IGN

IGR

0 I

1. (continued)

Outcome 3: GN .

• Probability pGN = 0.5 × 0.8 = 0.4.

• Your shares are worth IGN = $10,000+$4000 = $14,000.

Outcome 4: GR .

• Probability pGR = 0.5 × 0.2 = 0.1.

• Your shares are worth IGR = 0.5 × $14,000 = $7000.

Exercise solutions

2. The expected value of the lottery is the sum of the possible payoffs, weighted by the

probability that they occur:

EV = pNN × INN + pNR × INR + pGN × IGN + pGR × IGR = 0.4 × $10,000 + 0.1 × $5000 + 0.4 × $14,000 + 0.1 × $7000

= $4000 + $500 + $5600 + $700

= $10,800.

Expected utility

Preferences over lottery outcomes

A decision-maker’s preferences over outcomes will not generally be sufficient to determine

her/his preferences over lotteries.

• The job offer from Alpha Corp. will allow you to purchase 50,000 units of the composite good.

• The job offer from Beta Inc. will allow you to either purchase 30,000 or 70,000 units of the composite good.

If more-is-better with respect to the composite good, then Beta Inc.’s job offer will produce

either the most-preferred, or least preferred, outcome.

We need to add more structure to a decision-maker’s preferences over lotteries if we are to

understand decision-making under uncertainty.

Definition: Independence axiom

The axiom that states: A decision-maker’s preferences over lotteries should depend only on

the way in which two lotteries differ.

The independence axiom is a requirement of rationality in the presence of uncertainty. The

axiom is positive in the sense that it describes an aspect of rationality; and normative in the

sense that your preferences must satisfy the independence axiom in order to be rational.

Discussion: Job offers with the threat of recession

Consider the following variation on the job offers by Alpha Corp. and Beta Inc.:

Suppose that their is a 10% chance of a recession, in the event of which you will lose your job

and receive an income of $0.

• Alpha Corp.’s job offer now gives you IA1 = $50,000 with probability pA1 = 0.9, and IA2 = $0 with probability pA2 = 0.1.

• Beta Inc.’s job offer now gives you IB1 = $30,000 with probability pB1 = 0.45, IB2 = $70,000 with probability pB2 = 0.45, and IB3 = $0 with probability pB3 = 0.1.

Which job offer would you choose?

An example of the independence axiom at work

The independence axiom states that your choice of job should be independent of whether or

not there is a risk of recession.

Both lotteries deliver you a payoff IA2 = IB3 = $0 with probability pA2 = pB3 = 0.1. Because

this feature is present in both lotteries, it should not affect your choice.

Conditional on no recession:

• The Alpha Corp. job delivers you $50,000 with certainty.

• The Beta Inc. job gives you an equal chance of $30,000 and $70,000.

This is exactly the lottery you face when there is no risk of a recession at all.

Definition: Expected utility

The sum of the utilities that a decision-maker would derive from all possible outcomes of a

lottery, weighted by the probability that each outcome occurs.

If a decision-maker’s preferences over lotteries satisfy the independence axiom, then her/his

preferences can be represented by expected utility.

The expected utilities of the job offers

Suppose that Harry’s preferences over income levels are represented by the utility function

U(I ) = √

I .

Harry’s expected utility from Alpha Corp.’s job offer, which delivers IA = $50,000 with

probability pA = 1, is,

EUA = pA × √

IA = 1 × √

50,000 ≈ 224.

Harry’s expected utility from Beta Inc.’s job offer, which delivers IB1 = $30,000 with

probability pB1 = 0.5, and IB2 = $70,000 with probability pB2 = 0.5, is,

EUB = pB1 × √

IB1 + pB2 × √

IB2 = 0.5 × √

30,000 + 0.5 × √

70,000 ≈ 219.

Harry prefers the job with Alpha Corp.

Exercise: Expected utility

Suppose that Harry’s preferences over income levels are represented by the utility function

U(I ) = √

I .

1. Calculate the expected utility of Alpha Corp.’s job offer: IA1 = $50,000 with pA1 = 0.9,

and IA2 = $0 with pA2 = 0.1.

2. Calculate the expected utility of Beta Inc.’s job offer: IB1 = $30,000 with pB1 = 0.45,

IB2 = $70,000 with pB2 = 0.45, and IB3 = $0 with pB3 = 0.1.

3. Which job will Harry choose?

Exercise solutions

1. The expected utility of Alpha Corp.’s job offer is,

EUA = pA1 × √

IA1 + pA2 × √

IA2

= 0.9 × √

50,000 + 0.1 × √

≈ 0.9 × 224 + 0.1 × 0 = 201.6.

2. The expected utility of Beta Inc.’s job offer is,

EUB = pB1 × √

IB1 + pB2 × √

IB2 + pB3 × √

IB3

= 0.45 × √

30,000 + 0.45 × √

70,000 + 0.1 × √

≈ 0.45 × 173 + 0.45 × 265 + 0.1 × 0 = 197.1.

3. The job with Alpha Corp. has a higher expected utility (201.6) than the job with Beta

Inc. (197.1). Therefore, Harry will choose the Alpha Corp. job.

Risk preferences

Stock options or a bonus check

0.0

0.6

0.4

16k

0 I

Sally must choose between the following

two bonus schemes:

• Stock options that will be worth $1000 with probability p1 = 0.6, and

$16,000 with probability p2 = 0.4.

• A (sure-thing) cash bonus of $7000.

Note: Both alternatives have the same

expected value.

The utilities of the possible outcomes

0.0

16k

2.6

U(I ) = √

I/1000

0 I

Sally’s preferences are represented by the

utility function U(I ) = √

I/1000.

The utilities associated with the possible

outcomes are:

• U(1000) = 1.

• U(16,000) = 4.

• U(7000) ≈ 2.646.

The expected utilities of the two alternatives

0.0

16k

2.6

2.2

U(I ) = √

I/1000

0 I

The expected utility of the stock options

is,

EU = 0.6 × 1 + 0.4 × 4 = 2.2.

Note: For a two-outcome lottery, the

expected utility is the height of the line

connecting the utilities of the two

outcomes, at the expected value.

Sally is risk-averse because she prefers a

sure-thing to a lottery with the same

expected value.

Risk aversion and diminishing marginal utility

0.0

16k

2.6

2.2

U(I ) = √

I/1000

0 I

downside upside

Sally’s utility function displays a

diminishing marginal utility; the function

becomes flatter as income increases.

• Sally places increased weight on the downside, where the marginal utility

of income is high.

• Sally places reduced weight on the upside, where the marginal utility of

income is low.

Risk loving

0.0

16k

1.6

0.8

U(I ) = (I/8000)2

0 I1k

A decision-maker is risk-loving if she/he

prefers a lottery to a sure-thing with the

same expected value.

Risk-loving behaviour results from

increasing marginal utility.

• The weight on the downside is reduced, as the marginal utility of

income is low.

• The weight on the upside is increased, as the marginal utility of income is

high.

Quiz 1

Charlie’s utility function is U(I ) = I/2000. He has to choose between:

• A lottery paying $1000 with probability 0.6, and $16,000 with probability 0.4.

• A sure-thing payoff of $7000.

From this information we can conclude that,

(a) Charlie prefers the lottery over the sure-thing.

(b) Charlie prefers the sure-thing over the lottery.

(d) Charlie’s preferences cannot be deduced without a graph.

Risk neutrality

0.0

16k

3.5

0.5

U(I ) = I/2000

0 I

A decision-maker is risk-neutral if she/he

is indifferent between a lottery and a

sure-thing with the same expected value.

Risk neutral decision-makers have linear

utility functions,

U(I ) = α + βI,

where α is a constant, and β is a positive

constant.

When will a risk-averse decision-maker take a risk?

0.0

16k

2.2

7k4k

U(I ) = √

I/1000

0 I

Despite being risk-averse, Sally is willing

to take a risk if the potential reward is

great enough.

Suppose that instead of $7000, the cash

bonus is $4000.

Sally’s utility from the sure-thing payoff is

now U(4000) = 2; less than the expected

utility of the stock options.

Risk premium

0.0

Risk

premium

16k

2.2

7k4.8k

U(I ) = √

I/1000

0 I

A risk premium is the difference between

the expected value of a lottery, and the

sure-thing payoff with the same expected

utility.

For this lottery, Sally’s risk-premium is

(approximately) $2200.

To find the risk premium of a lottery,

solve the equation EU = U(EV − RP) to find RP .

Exercise: Risk premium

Lola’s preferences over income levels are described by the utility function U(I ) = 2 √

I .

Consider the lottery that delivers Lola I1 = $2500 with probability p1 = 0.5, I2 = $1600 with

probability p2 = 0.4, and I3 = $400 with probability p3 = 0.1.

1. Find the expected value of this lottery.

2. Find Lola’s expected utility from this lottery.

3. Find Lola’s risk premium.

Exercise solutions

1. The expected value of the lottery is,

EV = p1 × I1 + p2 × I2 + p3 × I3 = 0.5 × 2500 + 0.4 × 1600 + 0.1 × 400 = $1930.

2. Lola’s utility function is U(I ) = 2 √

I thus her expected utility is,

EU = p1 × 2 √

I1 + p2 × 2 √

I2 + p3 × 2 √

= 0.5 × 2 √

2500 + 0.4 × 2 √

1600 + 0.1 × 2 √

400

= 0.5 × 2 × 50 + 0.4 × 2 × 40 + 0.1 × 2 × 20 = 86.

Exercise solutions

3. We need to solve the equation EU = U(EV − RP). Using Lola’s utility function,

EU = 2 √

EV − RP.

Substituting for EV and EU from questions 1 and 2,

86 = 2 √

1930 − RP, and dividing both sides by 2, 43 = √

1930 − RP.

Squaring both sides of the equation,

1849 = 1930 − RP or RP = $81.

Insuring against risk

The risk of an accident

Sena drives to work every day. Each year there is a 24% probability that Sena will have a car

accident, causing $40,000 worth of damages.

• Sena’s income is $62,500.

• Sena’s utility function is U(I ) = √

I .

Sena’s situation can be described as a lottery:

• I1 = $62,500 with probability p1 = 0.76.

• I2 = $62,500 − $40,000 = $22,500 with probability p2 = 0.24.

• The expected value is EV = 0.76 × 62,500 + 0.24 × 22,500 = $52,900.

Insurance

Sena has the option of purchasing an insurance policy that will compensate him in the event

of an accident.

• The insurance premium is a fee F , which Sena must pay regardless of whether or not he has an accident.

• The policy pays Sena an amount $40,000 if he has an accident.

The insurance policy provides Sena with a sure-thing (risk-free) payoff:

• If Sena does not have an accident then his income will be I = $62,500 − F .

• If Sena has an accident then I = $62,500 − $40,000 + $40,000 − F = $62,500 − F .

Definition: Fairly priced insurance policy

An insurance policy in which the insurance premium is equal to the expected value of the

promised insurance payment.

A fairly priced insurance policy, which completely compensates a decision-maker for losses

they incur, provides a sure-thing payoff equal to the expected value of the lottery the

decision-maker would face without insurance.

Quiz 2

A decision-maker would be willing to purchase a fairly priced insurance policy, that completely

compensates her/him in the event that her/his house is damaged by a flood, if the

decision-maker is,

(a) risk-averse (but not risk-neutral or risk-loving).

(b) risk-neutral (but not risk-averse or risk-loving).

(d) either risk-averse or risk-neutral (but not risk-loving).

The sure-thing payoff from fairly priced insurance

The fair price for the insurance policy is the sum of any payments the insurer may be required

to make, weighted by the probability that the payment will be required.

• For Sena’s insurance policy the fair price is 0.24 × $40,000 = $9600.

The fair priced insurance policy provides Sena with the sure-thing payoff

I = 62,500 − 9600 = $52,900.

This is exactly the expected value of the lottery Sena was facing without insurance.

Sena’s preference for insurance

Without insurance, Sena’s expected utility is,

EU = p1 × √

I1 + p2 × √

= 0.76 × √

62,500 + 0.24 × √

22,500

= 0.76 × 250 + 0.24 × 150 = 226.

With a fairly priced insurance policy, Sena’s utility is,

U(52,900) = √

52,900 = 230.

Therefore, Sena prefers to purchase insurance.

The problems with fairly priced insurance policies

The fair price of an insurance policy can be interpreted as the average payment made by an

insurance company, if it insures many decision-makers with independently distributed risks.

• The ‘fair price’ does not allow for an insurance company’s cost of doing business.

• The ‘fair price’ does not allow the insurance company to build the capital reserves necessary to fund payments in the event of a correlated risk.

For these reasons (and others) insurance premiums will typically exceed the fair price.

Discussion: Moral hazard

Suppose that you purchased a fairly priced insurance policy for your car, which fully

reimburses you for damage suffered in an accident.

• How would the insurance policy affect the way you drive, and the way you maintain your car?

• What are the consequences of this change in behaviour for the fair price of the policy?

• Would the problem remain if your insurance policy included a deductible, requiring you to pay for part of the damage in the event of an accident?

Sena’s willingness-to-pay for insurance

The maximum price Sena would be willing to pay for the insurance policy is the fair price

($9600) plus his risk premium.

To find Sena’s risk premium we need to solve the equation EU = U(EV − RP).

Substituting for Sena’s utility function, EV and EU , we get,

226 = √

52,900 − RP.

Squaring both sides of the equation,

51,076 = 52,900 − RP or RP = $1824.

Thus the maximum price Sena is willing to pay is 9600 + 1824 = $11,424.

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Lottery

• Risk

• Expected value

• Independence axiom

• Expected utility

• Sure thing payoff

• Risk premium

• Insurance

• Fairly price insurance policy

• Deductible