Customer‘s safety is the future trend of the foodservice and hospitality industry

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Choice under Uncertainty

Part 1. Measuring Uncertainty

Economics 313

Uncertainty

 The General Equilibrium models we have considered follow the simple but strong assumptions familiar from earlier courses, about prices, costs, and preferences.

 In particular, consumers know exactly what buying a specific good entails: there is no uncertainty about what the good is, or when, where or how they will obtain it.

 The GE model can be extended to relax these assumptions, but making goods “state contingent” and having a price for every possible circumstance. An umbrella on a raining day is a different good than an umbrella on a windy, dry day.

Uncertainty

 Later in this lecture, we will explore how to model choice in “state spaces” in a partial equilibrium example of insurance.

 First we ask how people make choices when faced with uncertainty. We introduce the standard Expected Utility framework, and see how it can be used to model attitudes toward risk.

 This model was developed for game theory, and it remains the most basic tool for analysing uncertain choices in strategic settings, as well as insurance markets, investments, contracts and many other situations.

In this lecture…

 Analyzing Decision Making under Uncertainty  Representing Uncertainty by Probability Distributions  Expected Utility Theory  Utility Functions and Attitudes towards Risk  Representing choice under uncertainty in state-contingent space

 Criticisms / paradoxes of Expected Utility Theory

 Application to Insurance

Probability Distributions

 Probability is a deceptively difficult concept. Economists use math to be precise, but what exactly is being modelled is open to a number of interpretations.

 Familiar from your Statistics courses, probability is a number attached to some “state of the world” or “events”. These numbers must follow certain mathematical rules. We’ll review these briefly with a few examples.

 Probabilities are positive; there is no such thing as a negative probability. Over all events, they sum up to one. The probability of no event is zero.

Probability Distributions

 For the example of a coin toss there are two “events”: Head and Tail. Both of these cannot happen: they are “mutually exclusive.”

 If the coin is “fair” these two outcomes are equally likely. So we need two positive numbers that add to one, and that are equal.

0 ≤ Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 = Pr(𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇) and Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 + Pr 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 = 1

Because the coin is fair, we must have Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 = Pr 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 = 1 2

 Can anything else happen? Of course! It could land on edge. But we assume that Pr 𝐿𝐿𝐻𝐻𝐿𝐿𝐻𝐻𝑇𝑇𝐿𝐿𝐿𝐿 𝑜𝑜𝐿𝐿 𝐻𝐻𝐻𝐻𝐿𝐿𝐻𝐻 = 0. Note that “zero probability events” do happen.

Probability Distributions

 A Probability Distribution is a function that tells us how the “probability” of outcomes is “distributed” across the events.

 You could say it shows how likely an event is, but that is just changing words, and doesn’t add much intuition, and anyway is not always true. Thankfully the math is clear! And for this class, so will be the context.

Probability Distributions

 Again consider the toss of a fair coin. The possible outcomes are {Head, Tail, ∅}. Then here is one way to assign probabilities:

Pr ∅ = 0

Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 = 1 2

Pr 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 = 1 2

Because the coin must fall to be either a Head or a Tail, and cannot be both we have

Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑂𝑂𝑂𝑂 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 = 1 Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝐴𝐴𝐴𝐴𝐴𝐴 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 = 0

Probability Distributions

 Generally we will work with “random variables” that map the events into real numbers. This allows us to do calculations.

 We could, for example, define the function 𝑓𝑓(𝑥𝑥) where x is either Head or Tail, by

𝑓𝑓 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 = 1 and 𝑓𝑓 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 = 0.

 Note that random variable are neither random or variable. They are functions that map “events” to real numbers.

 Unlike coin tosses, many interesting economic situations have outcomes that are “naturally” numeric, like quantities and prices. In these cases the random variable is simply the function 𝑓𝑓 𝑥𝑥 = 𝑥𝑥

Distribution of Height in 313

 Take for example “the height of students in this class.” This is “naturally numeric” so we can assume the random variable that maps whatever “random” background factors influence the class membership also determines the height of students in the class, and take those numeric heights as the outcomes of interest.

 If we think of an experiment to choose “randomly” one student from the population of the class, we can think about the “probability” of various heights being chosen. What is a reasonable way to do assign these numbers?

Percentages?

 For example, if 5 people out of 121 are exactly 5.5 feet tall. Then we have 5/121 or 4.13% “chance” of getting a person who is for 5.5 feet tall.

 You could think of repeating drawings a very large number of times, then you would expect to get someone who is 5.5 ft. tall 4.13% of the time.

 This is one way to define probability: long run outcome share – or “limiting frequency” -- in repeated experiments. We can then say “the probability that a randomly picked student in ECON 313 is 5.5 feet is equal to 4.13%”.

 This away to define (or test) a “fair coin”: toss it a large number of times and 50% of results are Heads and 50% Tails.

Percentage or Probability

height

Percentage of people or Probability

5.5 feet

4.13

This graph shows the relative number of students as a function of height.

Statistics: Expected Value, Mean, or Average

 Since we have the data and heights are just numbers we can directly calculate the “average height” of students in the class: add all the heights together, and divide by class size. Let’s make it simple and assume that there are 10 students, 2 are 5 ft, 5 are 5.5 ft and 3 are 6ft.

Then 𝐴𝐴𝐴𝐴𝐻𝐻𝐴𝐴𝐻𝐻𝐿𝐿𝐻𝐻 = 5+5+5.5+5.5+5.5+5.5+5.5+6+6+6 10

= 5.55

 We could use the “population shares instead” 2

10 × 5 +

5 10

× 5.5 + 3

10 × 6 = 5.55

Statistics: Expected Value, Mean, or Average

 If we interpret these population shares as probabilities, this formula is

Pr 5 ∗ 5 + Pr 5.5 ∗ 5.5 + Pr 6 ∗ 6 = 5.5

 We can find out what the average height of a person in this class by multiplying each height by its probability and summing them.

 This kind of Average is called the mean or with probabilities the Expected Value.

More Generally….

 We call π𝑇𝑇 be the probability of event 𝑇𝑇 occurring and 𝐴𝐴𝑇𝑇 the value of event 𝑇𝑇 when it occurs. These events are “mutually exclusive” and “exhaustive”: one and only one of them will occur.

 The two requirements we place on a probability distribution are:

1. 1≥πi ≥0 for all 𝑇𝑇. 2. the sum of all π𝑇𝑇’s must be equal to 1.

Calculating the expected value

 Let πi be the probability that a person is of height vi.

 Suppose further that we find out that there are 101 different heights people have in this class.

 Then the expected value of our random variable, height, is given by

 µ=π1 v1+ π2 v2+π3 v3 +…+π100 v100 +π101 v101.

Some observations:

1. The notation of 𝜋𝜋𝑖𝑖 for probabilities, 𝐴𝐴𝑖𝑖 for outcomes and 𝜇𝜇 for a mean are quite standard.

2. We could work out the Expected Value of a coin toss, given our arbitrary random variable assignment:

Pr 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 ∗ 1 + Pr 𝑇𝑇𝐻𝐻𝑇𝑇𝑇𝑇 ∗ 0 = 1 2

3. It isn’t always clear how to interpret Expected Value… Sometime you might think of it as “a good guess” about the outcome of a random draw, except that no one in the class is 5.55 ft. tall so it’s wrong for sure.

4. What is the probability that as many or more people will visit family for Lunar New Year in 2021 as did in 2018? What is the probability that as many or more people will visit family for Lunar New Year in 2024 as in 2018? What is the limiting frequency in these cases?

Discrete versus Continuous Distributions

Examples

discrete

probability

continuous

Properties of Distributions

 We have talked about the expected value or mean

 Variance is telling you about how spread out values might be

 It is calculated by σ2=Σπi (vi-µ) 2

It is a function of the difference between any given value of the random variable and the mean.

Example: Different variance

Vi (the outcomes of the realizations of the random variable)

probability

Higher variance

Lower variance

Mean and Variance

 If the variance is small, the mean (also called the expected value) is giving you a more accurate prediction what might happen on average from experiments.

 If the variance is large, there is a lot of variability in this distribution. Outcomes are “noisier” and the Expected Value less informative for a single experiment.

 The variance is related to risk, but isn’t exactly the same.

 Both the expected value and the variance play a role in choice

Finding the Mean and Variance

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

Find the expected value of each investment.

Investment A:

.1*10+.3*20+.2*30+.2*40+.2*50=31

Investment B:

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

Find the variance of each investment.

Investment A: .1*(10-31)2+.3*(20-31)2 +.2*(30-31)2 +.2*(40-31)2 +.2*(50-31)2 =169

Investment B: .3*(20-30)2 +.4*(30-30)2 +.3*(40-30)2 =60

Finding the Mean and Variance

-- end of part 1 --

Choice under Uncertainty

Part 2. Choosing by Expected Value

Economics 313

Portfolio Choice

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

If you were faced with choosing one of these investments, what should you do?

This will be a matter of preference, so the answer is, it depends….

How Do People Cope with Uncertainty?

Investment Expected Value (Mean)

Variance (Uncertainty)

A 31 169

B 30 60

 Would you choose investment A or B?  There is a trade off between average payoff and the

uncertainty. A is good “on average” but sometimes it is a lot worse.

The St. Petersburg Paradox

 Consider the following gamble, a “lottery ticket” if you like.

 The Tsar tosses a “fair” coin until the first Head appears. Then the holder of the ticket gets $2n if the head appears on the nth toss.

 So if a head comes up on toss one, you get $2, toss 2, you get $4, etc.

 How much would you pay for the ticket?

How much would you pay?

A: less than $2 B: $2 C: $5 D: $10 E: more than $10

The St. Petersburg Paradox

 Think first about the expected value. After all, it works on average …. The expected value of the lottery ticket is

𝜇𝜇 = 1 2 ∗ 2 +

1 2 ∗ 𝑀𝑀𝑜𝑜𝐴𝐴𝐻𝐻 𝑡𝑡𝑡𝐻𝐻𝐿𝐿 2

𝜇𝜇 = 1 2 ∗ 2 + 1

2 ∗ (1

2 ∗ 4 + 1

2 𝑀𝑀𝑜𝑜𝐴𝐴𝐻𝐻 𝑡𝑡𝑡𝐻𝐻𝐿𝐿 4)

𝜇𝜇 = 1 2 ∗ 2 +

1 2 ∗ (

1 2 ∗ 4 +

1 2 ∗ (

1 2 ∗ 8 +

1 2 𝑀𝑀𝑜𝑜𝐴𝐴𝐻𝐻 𝑡𝑡𝑡𝐻𝐻𝐿𝐿 8)

⋮ 𝜇𝜇 = 1 + 1 + 1 + ⋯

 There is no finite mean. Or it is “infinite”. $2 seems a pretty good bargain….

The Sadistic Philanthropist

 A patient learns he has two more days to live unless he is gets a heart operation

The Sadistic Philanthropist

 He needs $20,000 to get the surgery

 His friends and relatives cannot help him

The Sadistic Philanthropist

 He meets the sadistic philanthropist who makes a proposal

The Proposal

 Gamble A:  Get $10,000 with probability .5  Get $15,000 with probability .5

 Gamble B:  Get $0 with probability .99  Get $20,000 with probability .01

The Patient’s Choice

 Gamble A has a higher expected monetary value than B

 Gamble A: .5*10,000+.5*15,000=12,500

 Gamble B: .99*0+.01*20,000=200.

 But he won’t be able to afford surgery if he chooses gamble A.

-- end of part 2 --

Choice under Uncertainty

Part 3. Expected Utility

Economics 313

Expected Monetary Value vs Expected Utility

 For the St. Petersburg paradox the potential winnings got very large just as the probability got very small. A reasonable response to the gamble was to downplay the large but unlikely outcomes, and focus on the fact that half the time you get $2.

 It’s like the high outcomes are all pretty much the same to a winner. Who cares if you have $16 trillion versus $32 trillion? What does that even mean for your life?

Expected Monetary Value vs Expected Utility

 For the patient Homer, the large payoffs were attractive only if he ignores the fact that he will not be around to enjoy them. Better to stake it all on the small chance that he get’s lucky and can afford the operation. Then money will be worth something.

 It’s like all the low payoffs are pretty much the same (bad). Who cares if you have $15,000 if you are not living. What does it even mean to have money when you are dead?

Expected Monetary Value vs Expected Utility

 In both cases, the “linearity” of Expected Value is making it the wrong tool for the job. With Expected Value every dollar is worth the same thing, one dollar.

 No matter if you already have $100 trillion, or if its one dollar short of a life saving operation. A dollar is a dollar is a dollar.

 What we need is a theory that allows the “utility” of money to vary depending on how much of it you have, or more generally the “state of the world” in which you get it.

Maximizing Expected Utility

 Agents do not simply choose the option that maximizes their expected monetary payoff.

 They evaluate the utility of each payoff or outcome and calculate the average of these.

 The Expected Utility Hypothesis: When choosing under uncertainty, people assess the possible payoffs in terms of utility and then choose the gamble that yields the highest expected utility.

Maximizing Expected Utility

 In the St. Petersburg Paradox, the Expected Value overestimates the Expected Utility because the marginal value of money falls, and the risk of taking another toss isn’t worth the chance you will lose it all.

 In the Sadistic Philanthropist case, the marginal value of money is zero until you have enough for the operation, so the Expected Value underestimates the Expected Utility.

 We seek to generalize this intuition into an operable theory.

Maximizing Expected Utility

 Given a risky situation we find the expected utility of this situation by summing up the utility if event i occurs multiplied with the probability of event i occurring.

( ) ( ) ( )nn n

i ii vUvUvUEU πππ ++== ∑

...11 1

Prob. that event 1 occurs Utility if event 1

occurs

Expected utility

( ) ( ) ( )nn n

i ii vUvUvUEU πππ ++== ∑

...11 1

Maximizing Expected Utility: Cardinal Utility

 Recall that for choice under certainty, we only need ordinal utility: better, worse, faster, slower. The Olympics medals.

 When there is uncertainty in the world, we need a stronger utility concept than ordinal utility called cardinal utility: twice as good, half as good. Length, temperature, speed.

 We will proceed by examples, but you need to know the main implication of cardinality (actually, you already know it…)

Example: Cardinal Utility

 Suppose Edith’s utility over money (denoted by y) is

𝑈𝑈𝐸𝐸 = 𝑦𝑦 1 4. Find the expected utility of investments A

and B and what investment she will choose.

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

Maximizing Expected Utility

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

Find the expected utility of each investment if U=y1/4

Investment A:

.1*(10).25+.3*(20).25+.2*(30).25+.2*(40).25+.2*(50).25=2. 3151

Investment B:

.3*(20).25+.4*(30).25+.3*(40).25=2. 325

maximizes expected utility!

Example: Cardinal Utility

 Suppose Olive’s utility over money (denoted by y) is

𝑈𝑈𝑂𝑂 = 𝑦𝑦 1 2. Find the expected utility of investments A

and B and what investment she will choose.

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

Maximizing Expected Utility

Investment A Plant Wheat in the North

Investment B Plant Wheat in the South

Payoff in $ Probability of Event Payoff Probability of Event 10 .1 10 0 20 .3 20 .3 30 .2 30 .4 40 .2 40 .3 50 .2 50 0

Find the expected utility of each investment if utility over money changes to U=y1/2

Investment A:

.1*(10).5+.3*(20).5+.2*(30).5+.2*(40).5+.2*(50).5=5. 4324

Investment B: .3*(20).5+.4*(30).5+.3*(40).5=5. 4299

maximizes expected utility!

Cardinal Utility

 Notice in this example 𝑈𝑈𝑂𝑂 = 𝑈𝑈𝐸𝐸 2

Both people value prefer more money to less. With ordinal utility that would be everything we need to know.

 With cardinal utility, the shape of the function matters.

-- end of part 3 --

Choice under Uncertainty

Part 4. Preferences and Risk

Economics 313

Marginal Utility of Money

 With the first utility function, Edith prefers investment B to investment A and Olive prefers investment A to B

 Comparing both utility functions over money y, the utility of getting an additional $ is different at any given level of money for the two functions  This difference is not just a matter of scaling

 The shape of the utility function over money tells us about the attitude of risk of an individual

Risk Neutrality

 A person is risk neutral if they will always trade a gamble for a sure payment of the gamble’s Expected Value, and vice versa. For risk neutral people, expected utility is expected value.

 A risk neutral person is willing to take on a “fair” bet – a gamble that costs its Expected Value.

 This implies that a risk neutral person has a constant marginal utility of money.

 A risk neutral person has a linear utility function over money: 𝑈𝑈 = 𝐻𝐻𝑦𝑦 + 𝑏𝑏, where 𝑦𝑦 is money. 𝐻𝐻 > 0 and 𝑏𝑏 are free parameters you can choose for convenience.

Risk Neutrality

 Consider 𝑈𝑈 = 𝑦𝑦 and the gamble: 20 with probability 1/2 and 60 with probability 1/2

 The expected utility is then .5*U(20)+.5*U(60)=.5*20+.5*60=.5(20+60)=40.

 Note that this is the same as U(40) = 40.

 If a person is risk neutral only the expected monetary value matters, but not the risk

 Only for a risk neutral person maximizing expected utility is equivalent to maximizing expected payoff

A Risk Neutral Person in a Graph

money

utility

20 40 .5*20+.5*60

U(20)

U(40) .5*U(20)+.5*U(60)

U(60)

Risk Aversion

 A person who has a decreasing marginal utility of money is risk averse. A risk averse person will always trade a gamble for its expected value.

 Risk aversion means that this person is not willing to take on a fair bet.

 That is, the person needs to be compensated with a higher expected return to take on some risk.

Risk Aversion

 Consider the same bet as before

 A risk averse person is not willing to take on a fair bet: U(40)> .5*U(20)+.5*U(60)

 A risk averse person is indifferent between a lower money value for sure and a gamble with a higher expected return: e.g. U(32.5) = .5*U(20)+.5*U(60).

 32.5 would be called the certainty equivalent of the gamble

Certainty Equivalent Value

 Certainty Equivalent of a gamble: The sum of money for which an individual would be indifferent between receiving that sum and taking the gamble

 The difference between the expected value of a gamble and the certainty equivalent can give us a measure of how risk averse a person is

 The smaller the certainty equivalent is relative to the expected value of a gamble, the more risk averse a person is (the more willing they are to give up wealth in exchange for reduced risk)

A Risk Averse Person in a Graph

money

utility

20 40

𝐸𝐸𝐸𝐸 = .5 ∗ 20 + .5 ∗ 60 = 40

U(20)

𝐸𝐸𝑈𝑈 = .5 ∗ 𝑈𝑈(20) + .5 ∗ 𝑈𝑈(60)

U(60)

U(𝐸𝐸𝐸𝐸)

𝐸𝐸𝑈𝑈

Risk Averse: 𝑈𝑈 𝐸𝐸𝐸𝐸 > 𝐸𝐸𝑈𝑈

A Risk Averse Person in a Graph

money

utility

20 40

𝐸𝐸𝐸𝐸 = .5 ∗ 20 + .5 ∗ 60 = 40

U(20)

𝐸𝐸𝑈𝑈 = .5 ∗ 𝑈𝑈(20) + .5 ∗ 𝑈𝑈(60)

U(60)

U(𝐸𝐸𝐸𝐸)

𝐸𝐸𝑈𝑈 Certainty Equivalent: 𝑈𝑈 𝐶𝐶𝐸𝐸 = 𝐸𝐸𝑈𝑈

Risk Averse: 𝑈𝑈 𝐸𝐸𝐸𝐸 > 𝐸𝐸𝑈𝑈

𝐶𝐶𝐸𝐸

Risk Preference

 People with a risk preference (or risk loving people) have an increasing marginal utility of money

 People with a risk preference are willing to take on an unfair bet

 People with a risk preference are only willing to accept a money value for sure if this value is higher than the expected money value from a gamble

 U(40)< .5*U(20)+.5*U(60)

utility

20 40 .5*20+.5*60

U(20)

.5*U(20)+.5*U(60)

U(60)

U(40)

money

A Risk Loving Person in a Graph

Certainty Equivalent with Risk Loving

 With risk loving individuals, they are willing to give up wealth to take on risk

 In other words, the certainty equivalent of a gamble (how much they would be willing to accept to avoid the risk of a game) is more than the expected value of the gamble

 In other words, they are willing to pay for the risk

 Risk neutral individual’s certainty equivalents are equal to the expected value of a gamble because they are indifference to risk

St. Petersburg Paradox

money

utility A lot more money gives very little utility: risk averse.

The Sadistic Philanthropist

money

utility

20K

Below $20K money is useless: Risk Loving.

Cardinal Utility again

 To preserve the attitudes toward risk represented by a utility function, you can at most multiply it by a positive constant and add a constant:

 If preferences are represented by 𝑈𝑈(𝑌𝑌), then they are represented by 𝐸𝐸 𝑌𝑌 if and only if

𝐸𝐸 𝑌𝑌 = 𝐻𝐻𝑈𝑈 𝑌𝑌 + 𝑏𝑏 for some 𝐻𝐻 > 0 and some 𝑏𝑏

Review Questions

 Which of the following Expected Utility functions displays risk aversion?

A. 𝑈𝑈 𝑀𝑀 = 𝑀𝑀 1 2

B. 𝑈𝑈 𝑀𝑀 = 𝑀𝑀2

C. 𝑈𝑈 𝑀𝑀 = 2𝑀𝑀 D. 𝑈𝑈 𝑀𝑀 = 3

-- end of part 4 --

Choice under Uncertainty

Part 5. Tests and Paradoxes

Economics 313

Question (Tversky and Kahneman 1981)

 There are two treatments for 600 people affected by a deadly disease. Which Treatment would you administer?

Treatment A Treatment B

Saves 200 people A 33% chance of saving all 600 people, 66% possibility of saving no one

Question (Tversky and Kahneman 1981)

 There are two treatments for 600 people affected by a deadly disease. Which Treatment would you administer?

Treatment A Treatment B

400 will die A 33% chance that no people will die, 66% probability that all 600 will die.

Question (Tversky and Kahneman 1981)

 There are two treatments for 600 people affected by a deadly disease. Which Treatment would you administer?

Treatment A Treatment B

Saves 200 people A 33% chance of saving all 600 people, 66% possibility of saving no one

Treatment A Treatment B

400 will die A 33% chance that no people will die, 66% probability that all 600 will die.

Positive Frame: Treatment A chosen by 72% of people

Negative Frame: Treatment A chosen by 22% of people

Criticisms of Expected Utility

 Framing  How you state a problem affects how people choose even if

the underlying problem is the same

 Reference Points  Loss aversion – we weight losses more than gains  The endowment effect – we value an item more if we are

endowed with it  Prospect theory designed to account for this

The Allais Paradox

 Situation 1: A. Get 1 million for sure or B. Get 5 million with probability .10, 1 million with probability

.89, and nothing with probability .01.

 Situation 2: A. Get 1 million with probability .11, nothing with probability

.89. B. Get 5 million with probability .10 and nothing with

probability .90.

The Allais Paradox

 Note that if  1*U(1) > .1*U(5) +.89*U(1) + .01*U(0)

 Subtract .89*U(1) and add .89*U(0) on both sides.  .11*U(1) +.89*U(0) > .1*U(5) + .9*U(0)

 If you accepted A in Situation 1, you should prefer A in the second situation

 This conflicts with expected utility – it implies a fanning out of indifference curves in probability space

The Ellsberg Paradox

 There are two urns each containing a large number of red and black balls.

 The distribution of red and black balls is known for one urn but not for the other.

 In the first urn, half the balls are red and the other half are black.

 People can earn $100 by announcing a color and then drawing a ball with this color from one of the urns.

The Ellsberg Paradox

 Which color would you announce and which urn would you draw the ball from?

The Ellsberg Paradox

 People overwhelmingly choose to draw the ball from the urn with a known set of probabilities, rather than take a chance on the urn with an unknown ratio.

 This is despite the fact that the second urn could have better odds of drawing black marbles, like 99 to 1 or even 100 to no white marbles.

 The fact is, the probability of drawing a black marble from either urn is identical

The Ellsberg Paradox

 To verify this let’s simplify the example: instead of 100 marbles, imagine there are only 2.

 In the known urn, there is 1 black and 1 white

 In the unknown urn it must be that  1/3 of the time there are 2 black marbles,  1/3 of the time there is 2 red  1/3 of the time there is 1 black and 1 red

The Ellsberg Paradox

 The probability of drawing either a black ball from the

urn: 1∗𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

3 + 0∗𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

3 +

1 2 ∗𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

3 = 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

3 + 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

3∗2 =

2∗𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 6

+ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 6

= 3 6 𝑏𝑏𝑇𝑇𝐻𝐻𝑏𝑏𝑏𝑏 = 1

2 𝑏𝑏𝑇𝑇𝐻𝐻𝑏𝑏𝑏𝑏

 The odds of drawing a black ball from either urn is the same thus people behave differently if probabilities are known versus unknown

 The tendency for people to avoid the unknown is known as ambiguity aversion. Note that this is more than just risk aversion

What I Expect You to Know

 How to get the expect value and variance of a bet

 How to calculate expected utility and certainty equivalent

 Understand the difference between expected utility and expected value

 To determine whether an individual is risk neutral, averse or risk loving given their utility functions or information about a bet and its certainty equivalent

 Discuss some of the criticisms of expected utility

-- end of part 5 --

Choice under Uncertainty

Part 6. Risk Pooling

Economics 313

The Desire to Avoid Risk

 Most people get nervous thinking about this dramatic loss

 This is a consequence of risk aversion

 There are several ways to try and protect against this loss

 We will talk about two ways

1. Risk pooling 2. Formal Insurance

1. Risk pooling  Risk pooling is option for risk averse people to reduce risk

often by leveraging the law of large numbers  Individuals aggregating their risks reduces the individual risks

each person faces without changing the expected payoff of a gamble.

2. Formal Insurance  Formal insurance often exists in well established markets with risk  We will discuss it intuitively, formally and with specific examples  But we will start laying down some more theory

Risk Pooling

 Risk pooling an important channel reduce for risk averse people to reduce risk

 One example of risk pooling could be that people facing the same uncertainty share their income equally no matter what happens

 Under certain circumstances, this can reduces the risk without changing the expected payoff of the gamble

Risk Pooling  Risk pooling is extremely valuable for many individuals in

developing countries or circumstances where formal risk reducing markets aren’t available

 But being able to pool risks is fundamental for many economic operations and the smooth functioning of markets

 Risk pooling even helps allow formal insurance markets to exist

Risk Pooling  If a large number of people or firms face risks that are

independent (not correlated with each other) and they pool their incomes and spilt them, they can reduce virtually all their risk

 It works because of the law of large numbers.

 The law of large numbers: if an event happens independently with probability p in each of N instances, the proportion of cases in which the event occurs approaches p as N grows larger

Risk Pooling  But you can reduce risk even by sharing it with a small

number of other people if risks are independent

 Take the following example regarding a simple agricultural risk sharing agreement….

Example

 The Okanagan Valley in British Columbia produces amazing fruit.

 Many farmer sell fruit they pick on the market at a price at $1 a pound for apples and $6 a pound for cherries.

 Because of some nasty little fruit fly, there is a 10% chance that the stock of apples and cherries picked each morning by the farmers will be destroyed.

Concept Check

 If each farmer picks 8 pounds of apples and 2 pounds of cherries a day, we know that their daily income is $20 with probability 0.9 and zero (because of the fruit flies) with probability 0.1

 What is the expected value of a farmer’s income?  0.9*20+0.1*0=18.

 What is the variance?  0.1*(0-18)2+.9(20-18)2=36

Risk Pooling Between Two Farmers

“Let us agree that we will pick our fruits as usual and store them as usual. No matter what happens, we will share the resulting income or loss equally.

Hence, if both our fruit crops destroyed by insects, we will both bear the loss.

If both our fruit crops are undamaged, then, of course, neither of us will suffer a loss.

However, if one of us has bad luck and loses their crop to insects and the other does not, we will share the resulting income equally.”

Probability Tree of This Risk Pooling

Destroyed 0.1

Undamaged 0.9

S 1’

s st

oc k

Probability Tree of This Risk Pooling

Destroyed 0.1

Undamaged 0.9

S 1’

s st

oc k

S 2’

s st

oc k

Destroyed 0.1

Undamaged 0.9

Probability Tree of This Risk Pooling

Destroyed 0.1

Undamaged 0.9

S 1’

s st

oc k

S 2’

s st

oc k

Destroyed 0.1

Undamaged 0.9

0.1*0.1 = 0.01

0.1*0.9 = 0.09

Probability Tree of This Risk Pooling

Destroyed 0.1

Undamaged 0.9

S 1’

s st

oc k

S 2’

s st

oc k

Destroyed 0.1

Undamaged 0.9

Destroyed 0.1

Undamaged 0.9

0.1*0.1 = 0.01

0.1*0.9 = 0.09

Probability Tree of This Risk Pooling

Destroyed 0.1

Undamaged 0.9

S 1’

s st

oc k

S 2’

s st

oc k

Destroyed 0.1

Undamaged 0.9

Destroyed 0.1

Undamaged 0.9

0.1*0.1 = 0.01

0.1*0.9 = 0.09

0.9*0.1 = 0.09

0.9*0.9 = 0.81

Question

 What is the expected monetary value of this deal for each farmer?

. 01 ∗ 0 + .18 ∗ 10 + .81 ∗ 20 = 1.8 + 16.2 = 18

 What is the variance?

. 01 ∗ (0 − 18)2 + .18 ∗ (10 − 18) 2 + .81 ∗ (20 − 18) 2 = 18

Comparison to No Risk Pooling

 Compare the expected monetary payoff of risk pooling with the case were one farmer is by themselves.  Without: .1 ∗ 0 + .9 ∗ 20 = 18  With: 18, hence the same

 Compare the variance of risk pooling with the case were one farmer is by themselves.  Without: .1 ∗ (0 − 18)2 + .9(20 − 18)2 = 36  With: 18, hence variance is lower with risk pooling.

If Each Has: U=y.75

100

 Expected Utility with risk pooling: 0.18 ∗ 10

.75 + 0.81 ∗ 20 .75 = 8. 6727

 Expected Utility without risk pooling: 0.9 ∗ 20

.75 = 8. 5117

 Risk pooling is preferred.

If Each Has: U=y.75

101

 Expected Utility with risk pooling: 0.18 ∗ 10

.75 + 0.81 ∗ 20 .75 = 8. 6727

 Expected Utility without risk pooling: 0.9 ∗ 20

.75 = 8. 5117

 Risk pooling is preferred.

Prob that both are undamaged

Prob that one is damaged and one not

Prob that one is damaged

Risk Pooling Reduces Risk

102

 For any risk averse person, risk pooling is preferred to not risk pooling because the risk is reduced.  Risk pooling decreases the variance in a particular way

that makes a gamble more attractive option for a risk averse person

 You can convince yourselves that with any utility function with decreasing marginal utility of money, risk pooling is preferred. E.g. try U=y.25, U=y.5, U=y.75 etc. (0< exponent <1 works).

The Mean-Preserving Spread Hypothesis

103

If a risk averse agent is faced with two gambles, both of which have the same expected payoff, but different variances and one gamble is a “mean-preserving spread” of the other, the agent will choose the gamble whose variance is lower.

This option more appealing

Which Would You Choose?

104

A B

Lower Variance is Not Always Preferred

105

a) Find the expected payoff and the variance of each gamble.

b) If for a person U = y.25, which gamble would the person choose?

Answer To Part A

106

 Gamble 1:  Mean: .1 ∗ 10 + .3 ∗ 20 + .5 ∗ 30 + .01 ∗ 40 + .07 ∗ 50 + .01 ∗ 60 +

.01 ∗ 70 = 27.2  Variance: .1 ∗ (10 − 27.2)² + .3 ∗ (20 − 27.2)² + .5 ∗ (30 − 27.2)² + .01 ∗

(40 − 27.2)² + .07 ∗ (50 − 27.2)² + .01 ∗ (60 − 27.2)² + .01 ∗ (70 − 27.2)² = 116.16

 Gamble 2:  Mean: .07 ∗ 0 + .2 ∗ 20 + .6 ∗ 30 + .13 ∗ 40 = 27.2  Variance: .07 ∗ (0 − 27.2)² + .2 ∗ (20 − 27.2)² + .6 ∗ (30 −

27.2)² + .13 ∗ (40 − 27.2)² = 88.16

Answer To Part B

107

 Gamble 1: .1 ∗ 10.25 + .3 ∗ 20.25 + .5 ∗ 30.25 + .01 ∗ 40.25 + .07 ∗ 50

.25 + .01 ∗ 60 .25 + .01 ∗ 70

.25 = 2.2505

 Gamble 2: .07 ∗ 0.25 + .2 ∗ 20.25 + .6 ∗ 30.25 + .13 ∗ 40.25 = 2.1541

 The individual will choose gamble 1, although gamble 1 has a higher variance.

What I Expect You to Know  What Risk Pooling Is.

 Why it reduces risk.

 When it is advantageous to pool risk.

 What types of consumers would get an advantage from risk pooling.

-- end of part 6 --

Choice under Uncertainty

Part 7. The Insurance Market

Economics 313

Insurance  In many markets, formal insurance firms develop as a way to take

advantage of the potential to pool individual risks

 You all probability have already encountered different forms of insurance (car, electronics)

 Formal insurance works by making you pay a fee (known as a premium) independent of whether the “bad” state happens

 In return, the insurer promises to compensate you if the bad state does happen

 We will start by setting up more theory in order to discuss formal insurance, then we will discuss it conceptually and do an example

Insurance  The approach we will take is by modelling risk in “state

spaces” where everything of relevance to utility is captured by a “state of the world”.

 For insurance, we think of two states: loss and no loss; or the “good state” and the “bad state”.

 This model is versatile. A similar kind of framework can be applied to investment, or more generally “intertemporal choice” where the states are “today” and “tomorrow” with markets providing a means to move consumption between states.

State-Contingent Wealth Space

113

Wealth in good state (𝑦𝑦1)

Wealth in bad state (𝑦𝑦2)

ICs in State-Contingent Space

114

 To find indifference curves in this space hold expected utility fixed, see what combinations of (𝑦𝑦1, 𝑦𝑦2) give us same expected utility (let 𝜋𝜋1equal the probability of the good state 𝑦𝑦1)

𝐸𝐸𝑈𝑈 𝑦𝑦1, 𝑦𝑦2 = 𝜋𝜋1𝑈𝑈 𝑦𝑦1 + 𝜋𝜋2𝑈𝑈 𝑦𝑦2 = 𝐸𝐸𝑈𝑈0

𝜕𝜕𝑈𝑈(𝑦𝑦1) 𝜕𝜕𝑦𝑦1

𝐻𝐻𝑦𝑦1 + 𝜋𝜋2 𝜕𝜕𝑈𝑈(𝑦𝑦2) 𝜕𝜕𝑦𝑦2

𝐻𝐻𝑦𝑦2 = 0

𝐻𝐻𝑦𝑦2 𝐻𝐻𝑦𝑦1

= − 𝜋𝜋1 𝜕𝜕𝑈𝑈(𝑦𝑦1) 𝜕𝜕𝑦𝑦1

𝜋𝜋2 𝜕𝜕𝑈𝑈(𝑦𝑦2) 𝜕𝜕𝑦𝑦2

Slope of indifference Curves

115

The negative of the ratio of the marginal expected utility in state 1 to the marginal expected utility in state 2.

𝐻𝐻𝑦𝑦2 𝐻𝐻𝑦𝑦1

= − 𝜋𝜋1 𝜕𝜕𝑈𝑈(𝑦𝑦1) 𝜕𝜕𝑦𝑦1

𝜋𝜋2 𝜕𝜕𝑈𝑈(𝑦𝑦2) 𝜕𝜕𝑦𝑦2

Slope of indifference Curves

116

The Marginal utility of your wealth in the good state

𝐻𝐻𝑦𝑦2 𝐻𝐻𝑦𝑦1

= − 𝜋𝜋1 𝜕𝜕𝑈𝑈(𝑦𝑦1) 𝜕𝜕𝑦𝑦1

𝜋𝜋2 𝜕𝜕𝑈𝑈(𝑦𝑦2) 𝜕𝜕𝑦𝑦2

The negative of the ratio of the marginal expected utility in state 1 to the marginal expected utility in state 2.

Slope of indifference Curves

117

𝐻𝐻𝑦𝑦2 𝐻𝐻𝑦𝑦1

= − 𝜋𝜋1 𝜕𝜕𝑈𝑈(𝑦𝑦1) 𝜕𝜕𝑦𝑦1

𝜋𝜋2 𝜕𝜕𝑈𝑈(𝑦𝑦2) 𝜕𝜕𝑦𝑦2

The probability of good state

The Marginal utility of your wealth in the good state

The negative of the ratio of the marginal expected utility in state 1 to the marginal expected utility in state 2.

How to Interpret the MRS in This Context  The Marginal Rate of Substitution in state contingent

wealth space gives us:

The rate at which individuals trade off wealth between the bad state and the good state while remaining indifferent

 Note that the rate individuals are willing to trade off depends on the probability of the states and the marginal utility of wealth in those states

Slope of ICs at 45 Degree Line

119

 If 𝑦𝑦1 = 𝑦𝑦2, then 𝑈𝑈 𝑦𝑦1 = 𝑈𝑈(𝑦𝑦2), which implies that

𝜕𝜕𝑈𝑈(𝑦𝑦1) 𝜕𝜕𝑦𝑦1

= 𝜕𝜕𝑈𝑈(𝑦𝑦2) 𝜕𝜕𝑦𝑦2

This means the slope of the IC at the 45-degree line is equal to the negative of the “risk ratio”

𝐻𝐻𝑦𝑦2 𝐻𝐻𝑦𝑦1

= − 𝜋𝜋1 𝜕𝜕𝑈𝑈 𝑦𝑦1 𝜕𝜕𝑦𝑦1

𝜋𝜋2 𝜕𝜕𝑈𝑈 𝑦𝑦2 𝜕𝜕𝑦𝑦2

= − 𝜋𝜋1 𝜋𝜋2

State-Contingent Wealth Space

120

y1 = y2

wealth in good state (𝑦𝑦1)

wealth in bad state (𝑦𝑦2) 45 degree line is the

certainty line: same wealth in both states of the world

ICs and Attitudes Towards Risk

121

 No matter what the attitude towards risk, the absolute slope of an IC where it intersects with the 45 degree (certainty) line is equal to the risk ratio.

 For a strictly risk averse person, the ICs in state- contingent space have decreasing MRS between states

 For risk neutral person, ICs are straight lines with slope equal to the negative risk ratio

 For a risk loving person, ICs have increasing MRS

Risk Averse Person in State Space

122

y1 = y2

Slope= - π1/ π2

y in good state

y in bad state IC of Risk Neutral Person

Risk Averse Person in State Space

123

y1 = y2

Slope= - π1/ π2

y in good state

y in bad state IC of Risk Neutral Person

To have less wealth in the bad state, a risk averse person needs “extra” compensation in the good state

Risk Averse Person in State Space

124

y1 = y2

Slope= - π1/ π2

y in good state (𝑦𝑦1)

y in bad state (𝑦𝑦2)

IC of Risk Neutral Person

Risk Averse Person in State Space

125

y1 = y2

Slope= - π1/ π2

y in good state

y in bad state IC of Risk Neutral Person

IC of Risk Averse Person

Risk Averse Person in State Space

126

y1 = y2

Slope= - π1/ π2

y in good state

y in bad state

Now That We Have These Tools…. • Now let’s talk about an example of where you may want

insurance

The Desire to Buy Insurance

128

Suppose buy a house and land which has a current value of 100 million.

You think that the house may burn down with a 20% chance in which case you will only have the land which is worth 25 million

80 % Chance 20 % Chance

Insurance

129

 How much would you be willing to pay for insurance?  We can figure this out by calculating your certainty equivalent

 Suppose your utility function over money is given by 𝑈𝑈 = 𝑦𝑦 1 2

 What would be the for-sure amount for which you are indifferent between getting this amount and taking on the gamble? (your certainty equivalent)

 Let’s do this for the gamble of your mansion burning down

Insurance  To find your certainty equivalent set the expected utility of the gamble

equal to your utility of some value (y) with certainty. (y will end up being the certainty equivalent value)

𝑈𝑈 = 𝑦𝑦 1 2 ⇒ 𝐸𝐸𝑈𝑈 = .2 ∗ 25

1 2 + .8 ∗ 100

1 2

𝐸𝐸𝑈𝑈 = .2 ∗ 5 + .8 ∗ 10 = 1 + 8 = 9

𝑦𝑦 = 92 = 81

 This means the amount of 81 million for sure would give you the same utility as the risky gamble where your house burns down with a 20% chance.

 This calculation also tells us about your willingness to insure yourself. If you your payment leaves you better off than your certainty equivalent, you will take the insurance.

<= Invert the utility function

Insurance

131

 Full coverage insurance plans give you the same amount of wealth in both states of the world

 If we can make sure that you end up with 81 million in the case that your house burns down and in the case that your house doesn’t burn down, then you would be as well off as with the gamble

 If you end up with an amount for sure that is higher than 81 million, you will be better off

Insurance

132

 If you had to pay an insurance premium of 19 million in order to get full coverage then you will always be left with a monetary amount of 81 million  In the state of the world your house doesn’t burn down, you

get 100 million minus the premium which is 81 million. If your house burns down and you have 25 million left, you still pay the insurance premium which leaves you with 6 million, but the insurance company would compensate you 75 million so you end up with 81 million again

 If the premium for full coverage would be lower than 19 million you would prefer insurance over no insurance

The Creation of Insurance Markets

133

 Suppose Peter is risk neutral. If Peter would receive 19 million from you and he will have to pay 75 million with a probability of .2, would Peter offer you insurance?

 Peter receives 19 million and does not have to pay anything with probability .8, and he receives 19 million and needs to pay 75 million with probability .2. If he doesn’t insure you, he doesn’t receive anything

 . 8 ∗ 19 + .2 ∗ (19 − 75) = 15.2 − 11.2 = 4 > 0

 Peter would be happy to make this deal. You on the other hand is just indifferent between buying the insurance and not buying it. In general, as long as Peter is less risk averse than you, there will be some gains from trade to be had.

The Fair Premium

134

 Peter made a profit. What it insurance was sold in a perfectly competitive market where risk neutral firms earn zero profit? If the firm charges a premium of 𝑃𝑃, their Expected Profit is given by

𝐸𝐸𝑃𝑃 = 0.8 ∗ 𝑃𝑃 + 0.2 ∗ 𝑃𝑃 − 75

 If competition ensures that 𝐸𝐸𝑃𝑃 = 0, the premium is

𝑃𝑃 = 15

 It is easy to compute that this is the expected loss. A premium equal to the expected loss is called “Actuarially Fair”, or just the fair premium, 𝑃𝑃𝑓𝑓

Insurance: Who Buys or Sells?

135

 Who buys at a fair premium?  A risk averse person will always  A risk neutral person is indifferent between

buying and not buying it  A risk lover would never buy

 Who sells at a fair premium?  A risk averse person will never sell  A risk neutral person is indifferent  A risk lover would always sell

The Demand for Insurance More Generally

136

 Thinking back to our example, let 𝑦𝑦1 represent your wealth in the good state of the world (you have your land and your house)

 Let 𝑦𝑦2 represent your wealth in the bad state of the world (you only have your land)

 Let 𝑦𝑦 be the amount of wealth you have before the uncertainty is realized (in our example this would just be equal to 𝑦𝑦1)

 The bad state occurs with prob. π2 (20%) and the good state with prob. π1. (80%).

The “Budget Constraint”

137

 Budget constraints tell us how we can distribute our purchasing power (wealth or income) across commodities. The price ratio is the rate at which we can shift consumption between goods.

 In the state contingent space we are looking to distribute wealth across states, and we want to know the rate at which we can shift wealth between states through the use of insurance.

 Let $𝑝𝑝 be the unit price of insurance. For $𝑝𝑝 you can buy $1 of “insurance coverage”: a contract that pays $1 if and only if the bad state occurs. Note that you must pay this price in both the bad and good state, but only get the payout if the bad state occurs: i.e. it is a “state-contingent” contract. Assume you can buy any number of these contracts.

The “Budget Constraint”

138

 Assume you buy 𝑞𝑞 units of these contracts. This means you pay $𝑝𝑝𝑞𝑞 in each state, and receive $𝑞𝑞 in the bad state.

 By varying 𝑞𝑞 you can map out the “budget line” that determines the rate you can move wealth between states by these contracts. We know it’s a straight line, because by assumption, the price 𝑝𝑝 is fixed for any number of contracts.

 To see how this “budget constraint” shows up in state space, we can pick two points on it. For example, where 𝑞𝑞 = 0, and 𝑞𝑞 = �𝑞𝑞.

The “Budget Constraint”

139

 If 𝑦𝑦 is the initial wealth and 𝐿𝐿 is the loss in wealth in the bad state, the two points in state space are:

The wealth in each state with 𝑞𝑞 = 0:

𝒚𝒚𝟎𝟎 = 𝑦𝑦1, 𝑦𝑦2 = 𝑦𝑦, 𝑦𝑦 − 𝐿𝐿

The wealth in each state with 𝑞𝑞 = �𝑞𝑞:

�𝒚𝒚 = 𝑦𝑦 − 𝑝𝑝�𝑞𝑞, 𝑦𝑦 − 𝐿𝐿 − 𝑝𝑝�𝑞𝑞 + �𝑞𝑞

 In the first case, you just bear the loss in the bad state. In the second case you pay for the contracts but get �𝑞𝑞 in the bad state.

Good state wealth

Bad state wealth

The “Budget Constraint”

140

 To determine the rate at which these contracts move wealth, just compute the “rise over the run”. Because we have made the vertical axis the bad state, the rise is the difference in wealth in the bad state between contracts and the run is the difference in wealth in the good states:

Slope = rise run

= (𝑦𝑦 − 𝐿𝐿) − (𝑦𝑦 − 𝐿𝐿 − 𝑝𝑝�𝑞𝑞 + �𝑞𝑞)

𝑦𝑦 − (𝑦𝑦 − 𝑝𝑝�𝑞𝑞) = 𝑝𝑝�𝑞𝑞 − �𝑞𝑞 𝑝𝑝�𝑞𝑞

= − 1 − 𝑝𝑝 𝑝𝑝

The Budget Constraint Across States

141

𝑦𝑦 in good state

𝑦𝑦 in bad state

Income bundle w/o insurance

𝑦𝑦 − 𝐿𝐿

𝑦𝑦

The Budget Constraint Across States

142

𝑦𝑦 in good state

𝑦𝑦 in bad state

Income bundle w/o insurance

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

Income bundle with �𝑞𝑞 coverage of insurance at premium 𝑝𝑝

𝑝𝑝�𝑞𝑞

𝑦𝑦

�𝑞𝑞 − 𝑝𝑝�𝑞𝑞

The Budget Constraint Across States

143

𝑦𝑦 in good state

𝑦𝑦 in bad state

Income bundle w/o insurance

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

Income bundle with �𝑞𝑞 coverage of insurance at premium 𝑝𝑝

𝑝𝑝�𝑞𝑞

𝑦𝑦

�𝑞𝑞 − 𝑝𝑝�𝑞𝑞

The Budget Constraint Across States

144

𝑦𝑦 in good state

𝑦𝑦 in bad state

Income bundle w/o insurance

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

Income bundle with �𝑞𝑞 coverage of insurance at premium 𝑝𝑝

𝑝𝑝�𝑞𝑞

𝑦𝑦

�𝑞𝑞 − 𝑝𝑝�𝑞𝑞

Slope of budget line: −(1 − 𝑝𝑝)/𝑝𝑝

The Budget Constraint Across States

145

𝑦𝑦 in good state

𝑦𝑦 in bad state

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

𝑦𝑦

The budget line in state- contingent space identifies all the combinations of money holdings in states 1 and 2, that you can achieve by transferring money from state 1 to state 2 by buying insurance

The Budget Constraint Across States

146

𝑦𝑦 in good state

𝑦𝑦 in bad state

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

𝑦𝑦

Income bundle w/o insurance

Utility without insurance

The Budget Constraint Across States

147

𝑦𝑦 in good state

𝑦𝑦 in bad state

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

𝑦𝑦

Income bundle w/o insurance

Utility without insurance

The Budget Constraint Across States

148

𝑦𝑦 in good state

𝑦𝑦 in bad state

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

𝑦𝑦

Income bundle w/o insurance

Optimal wealth bundle with premium 𝑝𝑝

The Fair Premium Per $ of Coverage

149

 Recall that the actuarially fair premium is equal to expected damages, so the fair contract price per unit is equal to the probability of the bad outcome.

 To see this, the insurance company breaks even if for any amount of coverage 𝑞𝑞 the consumer pays a premium 𝑝𝑝 such that

𝜋𝜋1𝑝𝑝𝑞𝑞 + 𝜋𝜋2 𝑝𝑝𝑞𝑞 − 𝑞𝑞 = 0

𝜋𝜋1𝑝𝑝𝑞𝑞 + (1 − 𝜋𝜋1) 𝑝𝑝𝑞𝑞 − 𝑞𝑞 = 0 because 𝜋𝜋1 = 1 − 𝜋𝜋2

(1 − 𝜋𝜋2)𝑝𝑝 + 𝜋𝜋2 𝑝𝑝 − 1 = 0

𝑝𝑝 = 𝜋𝜋2

Budget Line with Actuarially Fair Premium

150

 The expected payoff for any combination of state 1- wealth and state 2 - wealth on the budget line with actuarially fair premium are always the same.

 Moving wealth across states at this price results in a budget line with the slope

Slope = − 1−𝑝𝑝 𝑝𝑝

= − 1−𝜋𝜋2 𝜋𝜋2

= − 𝜋𝜋1 𝜋𝜋2

Full Insurance at Fair Premium

151

𝑦𝑦 in good state

𝑦𝑦 in bad state

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝜋𝜋2𝐿𝐿

𝑦𝑦

Income bundle w/o insurance

Optimal wealth bundle with premium 𝜋𝜋2:

Full coverage.

Slope −𝜋𝜋1/𝜋𝜋2

Full Insurance at Fair Premium

152

𝑦𝑦 in good state

𝑦𝑦 in bad state

𝑦𝑦 − 𝐿𝐿

𝑦𝑦 − 𝜋𝜋2𝐿𝐿

𝑦𝑦

Income bundle w/o insurance

Slope −𝜋𝜋1/𝜋𝜋2

𝑦𝑦 − 𝑝𝑝�𝑞𝑞

𝑦𝑦 − 𝐿𝐿 + �𝑞𝑞 − 𝑝𝑝�𝑞𝑞

Optimal wealth bundle with not actuarially fair premium 𝑝𝑝 > 𝜋𝜋2: less than full coverage is optimal

Demand for Insurance - Summary

153

 If a person is risk averse, he/she is willing to buy actuarially unfair insurance

 With actuarially fair insurance, it is optimal to be fully covered, 𝑞𝑞 = 𝐿𝐿. So wealth is 𝑦𝑦 − 𝜋𝜋2 in both states.

 With actuarially unfair insurance, it is optimal to be only partially covered, 𝑞𝑞 < 𝐿𝐿.

What I Expect You To Know

154

 Calculate the fair insurance premium and when someone would buy and sell insurance

 Draw diagrams of different risk preferences

 Represent choices and budget constraints in state- contingent space

 Determine whether someone will fully insure or not

-- end of part 7 --