Risk and Uncertainty.

## Topics in Risk and Uncertainty

Econ 395

Problem One

The President of the United States is interviewing candidates for the position of Surgeon

General. To determine their aptitude for the job, the President asks each candidate the

following two questions.

Question One

A flu epidemic will ravage the country within the next twelve months. Under

current arrangements, it is expected that the flu will kill 600 people. All others

affected by the flu will recover completely. There are two vaccination programs

available that may offer some relief:

(i) This program will save 400 people.

(ii) This program will save 0 people with 1/3 probability and save 600 people

with 2/3 probability.

Which vaccination program would you choose?

Question Two

A flu epidemic will ravage the country within the next twelve months. Two

vaccination programs are available to combat the epidemic.

(i) Under this program, 200 people are certain to die from the flu.

(ii) Under this program, there is a 2/3 chance that nobody will die, but a 1/3

chance that 600 people will die.

Which vaccination program would you choose?

The President tells you that his favored candidate strictly preferred program 1 in

Question One and program 2 in Question Two. Do you think that this candidate’s

preferences satisfy all the typical axioms of decision making (i.e. completeness,

reflexivity and transitivity)?

Econ 395

Fall 2013

Problem Two

Consider a set of monetary prizes, V = {$0, $600, $1000} and the following four lotteries

over those prizes:

α1 = (0.1, 0, 0.9)

α2 = (0, 1, 0)

α3 = (0.55, 0, 0.45)

α4 = (0.5, 0.5, 0)

where the lottery αi

= (αi

1, αi

2, αi

3) means that the lottery delivers $0 with probability αi

1,

$600 with probability αi

2 and $1000 with probability αi

3.

(a) Given a choice between lotteries α1 and α2

, which would you choose? Given a

choice between lotteries α3 and α4

, which would you choose?

(b) When presented with the choice between α1 and α2

, Ilene quickly and

emphatically chose lottery α2

. So, for Ilene, α2 ≥ α1

. If Ilene’s preferences satisfy

the independence axiom, how will she choose between a pair of compound

lotteries constructed in the following manner:

�(�) = � �) + 1 − � �.

�(�) = � �0 + 1 − � �.

for some � ∈ 0,1 . Recall, �. is a compound lottery that delivers the worst

prize (prize 1, $0) with certainty. Therefore, � and � are compound lotteries that

both deliver �. with the same probability. But with probability �, � delivers the

lottery �) and � delivers the lottery �0.

(c) Can you find a value for � that makes the compound lottery �(�) equivalent to

to lottery �5?

(d) Using the value of � you derived in part (c), determine the probabilities with

which compound lottery �(�) delivers prizes of $0, $600 and $1000.

(e) Do your answers to the last three questions offer any insight into how Ilene

might choose between lotteries �5 and �6?

(f) If Ilene announces that she prefers �5 to �6, do you think her preferences satisfy

the Independence Axiom? Do your preferences satisfy the Independence

Axiom?

(g) Draw a diagram with α1 (i.e. the probability of receiving $0) on the horizontal

axis and α2 (i.e. the probability of receiving $600) on the vertical axis. Draw the

straight line

Econ 395

Fall 2013

α1 + α2 = 1.

Shade the area that represents the set of all possible lotteries over the three

prizes, V = {$0, $600, $1000}. Explain how a lottery can be associated with a

point in this diagram, when each lottery is defined by three probabilities, but

only two probabilities can be seen in the diagram. Which point represents the

most desirable lottery possible? Which point represents the least desirable

lottery possible? Show on your diagram the direction of increasing preference.

(h) An agent’s indifference curve in this diagram will be a collection of all the

lotteries that deliver the same level of satisfaction for that agent. In other

words, if lotteries � and η lie on the same indifference curve, then that agent is

indifferent between � and η (i.e. � ~i η). In which direction would you anticipate

an indifference curve to slope?

(i) Mark the points on the diagram associated with the lotteries, α1

, α2

, α3 and α4

.

(j) Draw some indifference curves that are consistent with the choices you made in

part (a). Draw another diagram that is consistent with the preferences of Ilene.

Do you think that either you or Ilene might be an expected utility maximizer?

Problem Three

Every time Alvin walks into his local grocery store, he is presented with the opportunity

to buy a $10 scratch-and-win lottery ticket. He never does so. Clearly, the certainty of

neither winning nor losing is preferred to the uncertain outcome associated with the

lottery.

One day, Alvin and some friends attend a charity event, and they are presented with the

opportunity to put their tickets into the drawing for one of two door prizes. One prize is

$10. The other prize is a $10 scratch-and-win lottery ticket. Alvin decides to put his

ticket in the drawing for the $10 scratch-and-win lottery ticket.

When asked why he didn’t enter the drawing for $10, Alvin explained, “I’ll only win a

door prize if today is my lucky day. And if today is my lucky day, then I may as well play

the scratch-and-win lottery.”

(a) Do you think Alvin is rational in making this decision?

Econ 395

Fall 2013

(b) What if Alvin had said, “I usually don’t like gambling because I don’t like to risk

losing money. But in this case, I am gambling with someone else’s money. There

is no way I can lose.”

Do you think Alvin would seem more rational in this case, or less? Do you think

it is possible to produce a rational explanation for Alvin’s choice?

(c) Do Alvin’s choices satisfy the independence axiom? Do you think that the

independence axiom is necessarily satisfied by real, rational decision makers?

Put another way: if you discovered that your own choices failed to satisfy the

independence axiom, would you want to revise your decisions?

Problem Four

Jack manufactures snowboards and Jill manufactures skateboards. During the

winter both Jack and Jill face uncertainty over income. If the snowfalls are

significant then there is high demand for Jack’s products but low demand for Jill’s.

On the other hand, if the snows are light then demand for Jill’s products remain

relatively high and demand for Jack’s product is low.

Jack and Jill are both risk-averse expected utility maximizers.

Suppose the chance of light snowfall is 1/3, and the chance for heavy snowfall is 2/3.

In the event of heavy snowfall, Jack’s income is $1100/week while Jill’s income is

$500/week. In the event of light snowfall, Jack’s income is $200/week and Jill’s

income is $1100/week.

(a) If Jack can insure against poor snowfalls, what would be the “fair” price of

$1/week in coverage?

(b) If Jack can buy insurance at the fair price, how much coverage will he buy? What

income levels will he attain in each state of the world under this contract?

(c) If Jill can insure against heavy snowfalls, what would be the “fair” price of

$1/week in coverage? Is coverage for Jill more or less expensive than coverage

for Jack? Why?

(d) If Jill can buy insurance against heavy snowfalls at the fair price, how much

coverage will she buy? What income levels will she attain in each state of the

world if she buys that contract?

Econ 395

Fall 2013

(e) Explain why there is no difference between buying fairly priced insurance against

heavy snowfalls and selling fairly priced insurance against light snowfalls.

(f) If Jill sells insurance to Jack, how much coverage does Jack demand at the fair

price? How much coverage does Jill wish to supply at the fair price? Is the fair

price an equilibrium price in the competitive market? If not, do you expect that

Jack would have to pay more or less than the fair price if the market price were

determined competitively?

(g) Now suppose that Jill is risk-neutral rather than risk averse. Consider her supply

of, or demand for, insurance at various prices. What price should prevail in

competitive equilibrium (i.e. when Jack and Jill are both price-takers, and supply

and demand are equated)?

Problem Five

Amy wishes to hire a painter to paint her house. She knows that some painters are

better than others. But she cannot tell how good a painter is without hiring her.

Amy knows that there are three types of painter available in the market.

1. Excellent painters. An excellent painter will accept work only if paid at least

$50/hour. Amy would be prepared to pay an excellent painter $55/hour.

These painters make up 20% of all painters in the market.

2. Good painters. A good painter will accept work only if paid at least $30/hour.

Amy would be prepared to pay a good painter as much as $40/hour. These

painters make up 40% of the market.

3. Mediocre painters. A mediocre painter will not work for less than $15/hour.

Amy would pay a mediocre painter as much as $18/hour. These painters

make up 40% of the market.

(a) If the market for painters were to allocate resources efficiently, then Amy would

hire a painter whose services maximize the amount of surplus that can be shared

between the two contracting parties. Which type of painter provides the

greatest surplus? In the socially optimal world, which painter would Amy hire?

(b) What is the most Amy would pay a painter picked at random from the

population? (Assume Amy is risk-neutral. Assume also that she simply chooses

a name at random from the list available on Angie’s List, or in the Yellow Pages,

or in some equivalent list).

Econ 395

Fall 2013

(c) If Amy offered the price you calculated in part (b), which types of painter would

accept the job? Explain why Amy would never offer such a price.

(d) If Amy were to select a painter randomly from the population of “good” and

“mediocre” painters, what price would she be prepared to pay that painter?

Which painters would accept the job at this price? (Again, assume that Amy is

risk-neutral).

(e) If Amy is risk-neutral, what price do you think she will offer a painter? Which

type or types of painter will accept such an offer? Is this outcome efficient?

(f) If Amy is risk-averse, would your earlier answers change at all? If your answers

change, would you expect the outcome to be more or less efficient than the

outcome when Amy is risk-neutral?

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