Chapter1 Decision Theory Under Uncertainty

7/29/2019 Chapter1 Decision Theory Under Uncertainty

1/27

Chapter 1. Decision Theory Under

Uncertainty: Applications of the

Expected Utility Theory

Advanced Microeconomics3rd course (2nd semester)Degree in Economics and Finance

Professor: Carmen Arguedas

Departamento de Anlisis Econmico: T Econmica e H Econmica


2/27

CHAPTER 1 OUTLINE

1.1. Fair gambles and the expected utility hypothesis

1.2. The Von-Neumann-Morgenstern Theorem

1.3. Risk aversion

1.4. Applications of the expected utility theory

References: Nicholson (ch.7), Varian (ch.11), Pindyck & Rubinfield (ch.5).


3/27

In this chapter, we consider situations in which individuals (consumers, firms,

the government, ...) make economic decisions underuncertainty. At the timeindividuals make the decision, the do not know the resulting outcome.

Examples:

Gambles, lotteries

Insurance coverage

Making transactions in the stock market

Accepting/ rejecting labor offers

Enrolling on business expansions (on new markets)

1.1. Fair gambles and the expected utility hypothesis


4/27

What are the questions we plan to answer in this chapter?

How do we have to consider uncertainty to make economic decisions (such

as consumption or investment decisions)?

For example, economic agents ask for loans to finance big purchases

(housing, studies, cars, holidays) and plan to reimburse the money with

(uncertain) future rents.

How much risk are individuals willing to assume?

For example, should individuals invest in risky assets or sure bonds?

Should individuals accept a job offer or enrol in a degree or a master program?

Etc.


5/27

What instruments can be used to answer the above questions?

Risk measures (to quantify the risk the of different alternatives)

Preferences for risk

Mechanisms to avoid or reduce risks


6/27

EXAMPLE 1

Suppose your income is 10.000 units and you buy a lottery ticket that offers a

unique prize of 20.000 (out of 100 numbers). The cost of the ticket is 500 units.

Analyze the following questions:

1. How much money you have under the two possible outcomes (winning / losing)?

2. What is the probability (frequency) of each possible outcome?

3. What is your expected money once you have bought the lottery ticket?

4. What is the expected value of the lottery ticket?

5. Do you think the price of the lottery ticket is fair?

6. Would you be willing to pay more or less than 500 to participate in this game?


7/27

EXAMPLE 2

Suppose your income is 100.000 euro, and you own a car valued

20.000 euro. There exists a 1% chance that your car is stolen.

Analyze the following questions:

1. What is the expected value of your total wealth?

2. Supppose that you can buy insurance against robbery and you have two options:

the first reimburses the entire value of the car in case of robbery costs and it costs

300 euro; the second reimburses only half of the value of the car in case of robbery

and it costs 150 euro. Would you purchase any of these options?

3. Propose a fairinsurance. Would you purchase it?


8/27

EXAMPLE 3

St Petersburg paradox (due to Bernouilli)

Suppose you are offered the following game:

You flip a coin as many times are neccesary until it comes up heads. If it

comes up heads in the first try, you receive 2 euro; if it comes up heads in the

second try, you receive 4 euro; if it comes up heads in the third try,

you receive 8 euro; Etc.

How much would you be willing to pay to play this game?

Are individuals always willing to accept fair games?


9/27

EXAMPLE 4

Suppose you receive two labor offers from two different companies.

The first offer is a salary contingent on the companys profits:

a) 2000 euro/ month if profits are large

b) 1000 euro/month if profits are low

The second offer is a fixed salary of 1510 euro/month. In this case, there exists

a 1% chance that the company runs out of business. In this case, you would

receive an unemployment benefit of 510 euro/month.

Analyze the two labor offers in risk terms. Which one would you choose? Why?


10/27

EXAMPLE 5

Assume instead the following two labor offers:

In the first, the salary varies uniformely from 1000 to 2000 (for example,

in increases of 100 euro, with equal probability). In the second, the salary varies

uniformely between 1300 and 1700. Assume also there is no risk of

unemployment in any of the two jobs.

Analyze graphically the two labor offers in risk terms. Which one would you

choose? Why?


11/27

Random variable

Probability density function

Expected value of a random variable

Variance and standard deviation of a random variable

x = {x1, x2, , xn} or x [ x, x }

p = {p1, p2, , pn}, such that pi = 1

or f(x), such thatf(x)dx = 1

E(x) = pixi or E(x) = xf(x)dx

V(x) = pi [xi- E(x)] 2 or E(x) = [x-E(x)] f(x)dx2

MATHEMATICAL CONCEPTS


12/27

In example 5 above, both jobs pay the same salary on average. However, job 1

is riskier than job 2 since the dispersion of salaries (variance) in this case isgreater.


13/27

What is a fairgamble?

A specified set of outcomes and associated probabilities with an expected

value of zero.

Flipping a coin with a friend for a dollar

Gaining 10 euros if a coin comes up heads nd losing 1 euro if it comes up

tails with an entry fee of 4,5 euro.

Examples of fair gambles

Think of examples of unfair gambles

Are individuals willing to take fair gambles? St. Petersburgparadox


14/27

Bernouillis solution to the paradox:

Individuals do not care directly about money (dollar prizes of a gamble) BUT

about the utility these dollars provide.

Assuming decreasing marginal utility, the St Petersburg gamble may converge

to a FINITE expected utility value, even though its expected monetary value is

infinite.

Thus, since the gamble provides a finite expected utility value, individuals would

be willing to pay a FINITE amount of money to play it

EXAMPLE 6

Compute the expeted utility value of the St Petersburg gamble assuming

u(xi) = ln(xi).


15/27

1.2. The Von Neumann-Morgenstern Theorem

Rational individuals would make choices under uncertainty as if they

had a utility function over money U(x) and maximized the expected

value of U(x) (rather than the expected value of the payoff x itself)

Assume: GAMBLE A offers x1 with probability a and x2 with probability 1-a

GAMBLE B offers x3 with probability b and x4 with probability 1-b

The VNM theorem says that a rational individual prefers gamble A to gamble B

If and only if:

a U(x1) + (1-a) U (x2) > b U(x3) + (1-b) U(x4)


16/27

Shortcomings (Varian, pages 192-193)

Allais paradox

Ellsbergs paradox

EXAMPLE 7

EXAMPLE 8


17/27

1.3. Risk aversion

Wo - h Wo + hWo

U(Wo)EU(A)

CEA

Risk

premium

utility

wealth

U(W)

GAMBLE A:gaining or losing h

with equal probability

(fair gamble)


18/27

EXAMPLE 9

Compare gamble A with an alternative gamble B that offers

the possibility of gaining or losing 2h with equal probability

EXAMPLE 10

Comment on example 7.2 from Nicholson, pages 196-197.

(willingness to pay for insurance)


19/27

U U

W W

U(W)

Risk neutrality

For example, U(W) = W

U(W)

Risk loving

For example, U(W) = W, n > 1


20/27

How can we measure risk aversion?

Absolute risk aversion measure: AR (W) = U (W)

U (W)

Relative risk aversion measure: AR (W) = WU (W)

U (W)


21/27

Methods for reducing uncertainty and risk

Insurance

Diversification

Flexibility

Information acquisition

1.4. Applications of the expected utility theory


22/27

Insurance

Consider an individual that has an initial wealth W and a property valued at L

that is subject to a risk fire. In order to protect himself against the risk,

the individual can buy an insurance policy from an insurance company.

The company and the individual have the same prior as to the probability

that the fire will occur (denoted as p). Assume the individual can insure the

entire value of the property. The company demands a price for providing

full coverage should the property burn down. This price is known as the

premium, and we denote it by r.

1) What would be a fairpremium?

2) Would the individual be willing to buy it? Assume the individual is

risk averse and the utility function for wealth is U(W) = ln(W)

3) What is the maximum premium this individual would be willing

to pay for full coverage againt the risk of fire?

EXAMPLE 11


23/27

Problems with insurance (for the insurance company)

Adverse selection: only the worst consumers (those who expect

larger or more likely losses) may end up buying insurance

Moral hazard: having insurance may make consumers less willing

to take steps to avoid losses

MORE ON CHAPTER 3. (ASYMMETRIC INFORMATION MODELS)


24/27

Diversification Dont put all your eggs in one basket

Mainly for investment activities: Diversify your portfolio

EXAMPLE 12

An individual with wealth W can invest in two independent risky assets 1 and 2.

Assume both assets have the same expected payoff and the same variance.

Calculate the risk minimizing portfolio (or combination of assets).


25/27

Flexibility

Applicable to all-or-nothing decisions

The decision maker can obtain some of the benefits of diversification by making

flexible decisions.

State x State x

Payoff Utility

A2

A1

x x

U (A2)

U (A1)


26/27

EXAMPLE 13

Assume there are two alternatives A1 and A2 . The payoffs of these alternatives

depend on the state of the world, which is a uniform variable on [0,1]. Assume

A1 = 1-x and A2 = x. Assume also that the individual is risk neutral.

a) What alternative should the individual choose if he or she did not know the

state of the world at the time the decision has to be made?

b) What is the maximum the individual would pay to have the option to

choose the alternative once the state of the world is known?


27/27

Information acquisition

The value of information can be calculated as the difference between the

utility of the choice made when the information is known and the expected utility

of the choice made when the information is unknown.

It can be modelled similarly to the case of flexibility, studied above

Chapter1 Decision Theory Under Uncertainty

Documents

Transcript of Chapter1 Decision Theory Under Uncertainty