ch07 [Sola lettura] [modalità compatibilità] · 2 Mathematical Statistics • The expected value of a random variable is the outcome that will occur “on average” i n i i EX

1

Chapter 7

Uncertainty andUncertainty andInformation

Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved.

Mathematical Statistics• A random variable is a variable that

records, in numerical form, the possible outcomes from some random event

• The probability density function (PDF)shows the probabilities associated with pthe possible outcomes from a random variable

2

Mathematical Statistics• The expected value of a random variable

is the outcome that will occur “on average”

in

ii xfxXE

1

︶︵

dxxfxXE

︶︵

Mathematical Statistics• The variance and standard deviation

measure the dispersion of a random variable about its expected value

in

iix xfxEx

2

1

2

dxxfxExx

22

2xx

3

Expected Value

• Games which have an expected value ofGames which have an expected value of zero (or cost their expected values) are called fair games– a common observation is that people would

prefer not to play fair games

St. Petersburg Paradox• A coin is flipped until a head appearspp pp

• If a head appears on the nth flip, the player is paid $2n

x1 = $2, x2 = $4, x3 = $8,…,xn = $2n

• The probability of getting of getting aThe probability of getting of getting a head on the ith trial is (½)i

1=½, 2= ¼,…, n= 1/2n

4

St. Petersburg Paradox• The expected value of the St. Petersburg p g

paradox game is infinitei

i i

iii xXE

1 1 2

12)(

1111 ...)(XE )(

• Because no player would pay a lot to play this game, it is not worth its infinite expected value

Expected Utility• Individuals do not care directly about the y

dollar values of the prizes– they care about the utility that the dollars

provide

• If we assume diminishing marginal utility of wealth the St Petersburg game maywealth, the St. Petersburg game may converge to a finite expected utility value– this would measure how much the game is

worth to the individual

5

Expected Utility

• Expected utility can be calculated in theExpected utility can be calculated in the same manner as expected value

)()(1

n

iii xUXE

• Because utility may rise less rapidly than y y p ythe dollar value of the prizes, it is possible that expected utility will be less than the monetary expected value

The von Neumann-Morgenstern Theorem

• Suppose that there are n possible• Suppose that there are n possible prizes that an individual might win (x1,…xn) arranged in ascending order of desirability– x1 = least preferred prize U(x1) = 0

– xn = most preferred prize U(xn) = 1

6


• The point of the von Neumann• The point of the von Neumann-Morgenstern theorem is to show that there is a reasonable way to assign specific utility numbers to the other prizes available


• The von Neumann-Morgenstern methodThe von Neumann Morgenstern method is to define the utility of xi as the expected utility of the gamble that the individual considers equally desirable to xi

U( ) U( ) (1 ) U( )U(xi) = i · U(xn) + (1 - i) · U(x1)

7


• Since U(x ) = 1 and U(x ) = 0• Since U(xn) = 1 and U(x1) = 0

U(xi) = i · 1 + (1 - i) · 0 = i

• The utility number attached to any other prize is simply the probability of winning it

• Note that this choice of utility numbers is• Note that this choice of utility numbers is arbitrary

Expected Utility Maximization

• A rational individual will choose among• A rational individual will choose among gambles based on their expected utilities (the expected values of the von Neumann-Morgenstern utility index)

8


• Consider two gambles:g– first gamble offers x2 with probability q and

x3 with probability (1-q)

expected utility (1) = q · U(x2) + (1-q) · U(x3)

– second gamble offers x5 with probability tand x6 with probability (1-t)

expected utility (2) = t · U(x5) + (1-t) · U(x6)


• Substituting the utility index numbers g ygives

expected utility (1) = q · 2 + (1-q) · 3

expected utility (2) = t · 5 + (1-t) · 6

• The individual will prefer gamble 1 to p ggamble 2 if and only if

q · 2 + (1-q) · 3 > t · 5 + (1-t) · 6

9


• If individuals obey the von Neumann• If individuals obey the von Neumann-Morgenstern axioms of behavior in uncertain situations, they will act as if they choose the option that maximizes the expected value of their von N M ili i dNeumann-Morgenstern utility index

Risk Aversion• Two lotteries may have the same

expected value but differ in their riskiness– flip a coin for $1 versus $1,000

• Risk refers to the variability of the outcomes of some uncertain activity

• When faced with two gambles with the same expected value, individuals will usually choose the one with lower risk

10

Risk Aversion• In general, we assume that the marginal g , g

utility of wealth falls as wealth gets larger– a flip of a coin for $1,000 promises a small

gain in utility if you win, but a large loss in utility if you lose

– a flip of a coin for $1 is inconsequential asa flip of a coin for $1 is inconsequential as the gain in utility from a win is not much different as the drop in utility from a loss

Risk AversionUtility (U)

U(W) is a von Neumann-Morgensternutility index that reflects how the individualy ( )

U(W)

utility index that reflects how the individualfeels about each value of wealth

The curve is concave to reflect theti th t i l tilit

Wealth (W)

assumption that marginal utilitydiminishes as wealth increases

11


Suppose that W* is the individual’s currentl l f i

y ( )

U(W)

level of income

U(W*) is the individual’scurrent level of utility

U(W*)

Wealth (W)W*

y

Risk Aversion

• Suppose that the person is offered two• Suppose that the person is offered two fair gambles:

– a 50-50 chance of winning or losing $h

Uh(W*) = ½ U(W* + h) + ½ U(W* - h)

50 50 h f i i l i $2h– a 50-50 chance of winning or losing $2h

U2h(W*) = ½ U(W* + 2h) + ½ U(W* - 2h)

12


The expected value of gamble 1 is Uh(W*)y ( )

U(W)U(W*)Uh(W*)

Wealth (W)W* W* + hW* - h


The expected value of gamble 2 is U2h(W*)y ( )

U(W)U(W*)

U2h(W*)

Wealth (W)W* W* + 2hW* - 2h

13

Risk AversionUtility (U) U(W*) > Uh(W*) > U2h(W*)y ( )

U(W)U(W*)

U(W ) U (W ) U (W )

U2h(W*)

Uh(W*)

Wealth (W)W* W* + 2hW* - 2h W* - h W* + h

Risk Aversion

• The person will prefer current wealth toThe person will prefer current wealth to that wealth combined with a fair gamble

• The person will also prefer a small gamble over a large one

14

Risk Aversion and Insurance

• The person might be willing to payThe person might be willing to pay some amount to avoid participating in a gamble

• This helps to explain why some individuals purchase insurance

Risk Aversion and insurance

Utility (U)W ” provides the same utility asparticipating in gamble 1y ( )

U(W)U(W*)Uh(W*) The individual will be

willing to pay up toW* - W ” to avoid

participating in gamble 1

Wealth (W)W*W* - h W* + h

participating in thegamble

W ”

15

Risk Aversion and Insurance

• An individual who always refuses fair• An individual who always refuses fair bets is said to be risk averse– will exhibit diminishing marginal utility of

income

– will be willing to pay to avoid taking fair g p y gbets

Willingness to Pay for Insurance

• Consider a person with a current wealthConsider a person with a current wealth of $100,000 who faces a 25% chance of losing his automobile worth $20,000

• Suppose also that the person’s von Neumann-Morgenstern utility index isg y

U(W) = ln (W)

16


• The person’s expected utility will beThe person s expected utility will beE(U) = 0.75U(100,000) + 0.25U(80,000)

E(U) = 0.75 ln(100,000) + 0.25 ln(80,000)

E(U) = 11.45714

• In this situation, a fair insurance premium would be $5,000 (25% of $20,000)


• The individual will likely be willing to pay• The individual will likely be willing to pay more than $5,000 to avoid the gamble. How much will he pay?

E(U) = U(100,000 - x) = ln(100,000 - x) = 11.45714

100,000 - x = e11.45714

x = 5,426

• The maximum premium is $5,426

17

Measuring Risk Aversion• The most commonly used risk aversion

measure was developed by Pratt

)('

)(" )(

WU

WUWr

• For risk averse individuals, U”(W) < 0– r(W) will be positive for risk averse

individuals

– r(W) is not affected by which von Neumann-Morganstern ordering is used

Measuring Risk Aversion

• The Pratt measure of risk aversion is• The Pratt measure of risk aversion is proportional to the amount an individual will pay to avoid a fair gamble

18

Measuring Risk Aversion• Let h be the winnings from a fair betLet h be the winnings from a fair bet

E(h) = 0

• Let p be the size of the insurance premium that would make the individual exactly indifferent between taking the fair bet h and paying p with certainty to avoid the gamble

E[U(W + h)] = U(W - p)

Measuring Risk Aversion• We now need to expand both sides ofWe now need to expand both sides of

the equation using Taylor’s series

• Because p is a fixed amount, we can use a simple linear approximation to the right-hand side

U(W - p) = U(W) - pU’(W) + higher order terms

19

Measuring Risk Aversion• For the left-hand side, we need to use a ,

quadratic approximation to allow for the variability of the gamble (h)

E[U(W + h)] = E[U(W) - hU’(W) + h2/2 U” (W)

+ higher order terms+ higher order terms

E[U(W + h)] = U(W) - E(h)U’(W) + E(h2)/2 U” (W)

+ higher order terms

Measuring Risk Aversion

• Remembering that E(h)=0 dropping the

)(")()(')( WkUWUWpUWU

• Remembering that E(h)=0, dropping the higher order terms, and substituting kfor E(h2)/2, we get

)()('

)(" Wkr

WU

WkUp

20

Risk Aversion and Wealth• It is not necessarily true that risk aversion

declines as wealth increases– diminishing marginal utility would make

potential losses less serious for high-wealth individuals

– however, diminishing marginal utility also o e e , d s g a g a ut ty a somakes the gains from winning gambles less attractive

• the net result depends on the shape of the utility function

Risk Aversion and Wealth• If utility is quadratic in wealth

U(W) = a + bW + cW 2

where b > 0 and c < 0

• Pratt’s risk aversion measure is

cWU 2)("

cWb

c

WU

WUWr

2

2

)(

)( )(

• Risk aversion increases as wealth increases

21

Risk Aversion and Wealth• If utility is logarithmic in wealth

U(W) = ln (W )

where W > 0


WU 1)("

WWU

WUWr

1

)(

)(" )(

• Risk aversion decreases as wealth increases

Risk Aversion and Wealth• If utility is exponential

U(W) = -e-AW = -exp (-AW)

where A is a positive constant


eAWU AW2)("A

Ae

eA

WU

WUWr

AW

AW

2

)(

)(" )(

• Risk aversion is constant as wealth increases

22

Relative Risk Aversion

• It seems unlikely that the willingness to• It seems unlikely that the willingness to pay to avoid a gamble is independent of wealth

• A more appealing assumption may be that the willingness to pay is inversely proportional to wealth

Relative Risk Aversion

• This relative risk aversion formula is

)('

)(" )()(

WU

WUWWWrWrr

23

Relative Risk Aversion• The power utility function

U(W) = WR/R for R < 1, 0

exhibits diminishing absolute relative risk aversion

W

R

W

WR

WU

WUWr

R

R )1()1(

)('

)(")(

1

2

WWWU )(

but constant relative risk aversion

RRWWrWrr 1)1()()(

The Portfolio Problem

• How much wealth should a risk averse• How much wealth should a risk-averse person invest in a risky asset?– the fraction invested in risky assets should

be smaller for more risk-averse investors

24


• Assume an individual has wealth (W )• Assume an individual has wealth (W0) to invest in one of two assets– one asset yields a certain return of rf

– one asset’s return is a random variable, rr


• If k is the amount invested in the risky• If k is the amount invested in the risky asset, the person’s wealth at the end of one period will be

W = (W0 – k)(1 + rf) + k(1 + rr)

W = W (1 + r ) + k(r r )W = W0(1 + rf) + k(rr – rf)

25


• W is now a random variable• W is now a random variable– it depends on rr

• k can be positive or negative– can buy or sell short

• k can be greater than W0k can be greater than W0

– the investor could borrow at the risk-free rate

The Portfolio Problem• If we let U(W) represent this investor’sIf we let U(W) represent this investor s

utility function, the von Neumann-Morgenstern theorem states that he will choose k to maximize E[U(W)]

• The FOC is

0'10

frfrf rrUE

krrkrWUE

kWUE

26

The Portfolio Problem• As long as E(r – rf) > 0 an investorAs long as E(rr rf) > 0, an investor

will choose positive amounts of the risky asset

• As risk aversion increases, the amount of the risky asset held will fall

– the shape of the U’ function will change

The State-Preference Approach

• The approach taken in this chapter hasThe approach taken in this chapter has not used the basic model of utility-maximization subject to a budget constraint

• There is a need to develop new techniques to incorporate the standard choice-theoretic framework

27

States of the World• Outcomes of any random event can be y

categorized into a number of states of the world– “good times” or “bad times”

• Contingent commodities are goods d li d l if ti l t t f thdelivered only if a particular state of the world occurs– “$1 in good times” or “$1 in bad times”

States of the World• It is conceivable that an individual could

purchase a contingent commodity– buy a promise that someone will pay you

$1 if tomorrow turns out to be good times

– this good will probably sell for less than $1

28

Utility Analysis

• Assume that there are two contingentAssume that there are two contingent goods– wealth in good times (Wg) and wealth in bad

times (Wb)

– individual believes the probability that good times will occur is times will occur is

Utility Analysis

• The expected utility associated with theseThe expected utility associated with these two contingent goods is

V(Wg,Wb) = U(Wg) + (1 - )U(Wb)

• This is the value that the individual wants to maximize given his initial wealth (W)to maximize given his initial wealth (W)

29

Prices of Contingent Commodities

• Assume that the person can buy $1 of p y $wealth in good times for pg and $1 of wealth in bad times for pb

• His budget constraint isW = pgWg + pbWb

• The price ratio pg /pb shows how this person can trade dollars of wealth in good times for dollars in bad times

Fair Markets for Contingent Goods

• If markets for contingent wealth claims are well-developed and there is general agreement about , prices for these goods will be actuarially fair

pg = and pb = (1- )

• The price ratio will reflect the odds in favor• The price ratio will reflect the odds in favor of good times

1b

g

p

p

30

Risk Aversion• If contingent claims markets are fair, a g ,

utility-maximizing individual will opt for a situation in which Wg = Wb

– he will arrange matters so that the wealth obtained is the same no matter what state occursoccurs

Risk Aversion• Maximization of utility subject to a budget

constraint requires that

b

g

b

g

b

g

p

p

WU

WU

WV

WVMRS

)(')1(

)('

/

/

• If markets for contingent claims are fair

1)('

)('

b

g

WU

WU

bg WW

31

Risk Aversion

Wb

The individual maximizes utility on thet i t li h W W

Since the market for contingentclaims is actuarially fair, thel f th b d t t i t 1

certainty line

Wb certainty line where Wg = Wb

slope of the budget constraint = -1

WgWg*

Wb*

U1

Risk Aversion

Wb

In this case, utility maximization may not occur on the certainty line

If the market for contingent

certainty line

Wb occur on the certainty line

gclaims is not fair, the slope ofthe budget line -1

Wg

U1

32

Insurance in the State-Preference Model

• Again consider a person with wealth ofAgain, consider a person with wealth of $100,000 who faces a 25% chance of losing his automobile worth $20,000– wealth with no theft (Wg) = $100,000 and

probability of no theft = 0.75

lth ith th ft (W ) $80 000 d– wealth with a theft (Wb) = $80,000 and probability of a theft = 0.25


• If we assume logarithmic utility then• If we assume logarithmic utility, then

E(U) = 0.75U(Wg) + 0.25U(Wb)

E(U) = 0.75 ln Wg + 0.25 ln Wb

E(U) = 0.75 ln (100,000) + 0.25 ln (80,000)

E(U) = 11.45714

33


• The budget constraint is written in terms ofThe budget constraint is written in terms of the prices of the contingent commodities

pgWg* + pbWb* = pgWg + pbWb

• Assuming that these prices equal the probabilities of these two states

0.75(100,000) + 0.25(80,000) = 95,000

• The expected value of wealth = $95,000


• The individual will move to the certainty lineThe individual will move to the certainty line and receive an expected utility of

E(U) = ln 95,000 = 11.46163

– to be able to do so, the individual must be able to transfer $5,000 in extra wealth in good times into $15 000 of extra wealth in bad timesinto $15,000 of extra wealth in bad times

• a fair insurance contract will allow this

• the wealth changes promised by insurance (dWb/dWg) = 15,000/-5,000 = -3

34

A Policy with a Deductible• Suppose that the insurance policy costs

$4,900, but requires the person to incur the first $1,000 of the loss

Wg = 100,000 - 4,900 = 95,100

Wb = 80,000 - 4,900 + 19,000 = 94,100

E(U) = 0 75 ln 95 100 + 0 25 ln 94 100E(U) 0.75 ln 95,100 + 0.25 ln 94,100

E(U) = 11.46004

• The policy still provides higher utility than doing nothing

Risk Aversion and Risk Premiums

• Consider two people, each of whom p p ,starts with an initial wealth of W*

• Each seeks to maximize an expected utility function of the form

WWWWV

Rb

Rg )()( 1

RRWWV bg

bg )(),( 1

• This utility function exhibits constant relative risk aversion

35


WWWWV

Rb

Rg )()( 1

RRWWV bg

bg )(),( 1

• The parameter R determines both the degree of risk aversion and the degree of curvature of indifference curves implied bycurvature of indifference curves implied by the function– a very risk averse individual will have a large

negative value for R

A very risk averse person will have sharply curved


Wb

A person with more tolerance for risk will have flatter indifference curves such as U2

A very risk averse person will have sharply curvedindifference curves such as U1

certainty line

Wb

W*

U2

U1

WgW*

W

36

Suppose that individuals are faced with losing h


Wbpp g

dollars in bad times

certainty line

Wb

W*

The difference between W1

and W2 shows the effect of risk aversion on thewillingness to accept risk

U2

U1

WgW*

W willingness to accept riskW* - h

W2 W1

Properties of Information• Information is not easy to definey

– difficult to measure the quantity of information obtainable from different actions

– too many forms of useful information to permit the standard price-quantitypermit the standard price quantity characterization used in supply and demand analysis

37

Properties of Information

• Studying information also becomesStudying information also becomes difficult due to some technical properties of information– it is durable and retains value after its use

– it can be nonrival and nonexclusive• in this manner it can be considered a public

good

The Value of Information• Lack of information represents a problem p p

involving uncertainty for a decision maker– the individual may not know exactly what the

consequences of a particular action will be

• Better information can reduce uncertainty and lead to better decisions and higherand lead to better decisions and higher utility

38

The Value of Information• Assume an individual forms subjective j

opinions about the probabilities of two states of the world– “good times” (probability = g) and “bad

times” (probability = b)

I f ti i l bl b it h l• Information is valuable because it helps the individual revise his estimates of these probabilities

The Value of Information

• Assume that information can be• Assume that information can be measured by the number of “messages” (m) purchased– these messages can take on a value of 1

(with a probability of p) or 2 (with a b bilit f (1 )probability of (1 – p)

39


• If the message takes the value of 1 the• If the message takes the value of 1, the person believes the probability of good times is 1

g so 1b = 1 - 1

g

• If the message takes the value of 2, the person believes the probability of goodperson believes the probability of good times is 2

g so 2b = 1 - 2

g


• Let V = indirect maximal utility when the• Let V1 = indirect maximal utility when the message =1

• Let V2 = indirect maximal utility when the message =2

• The expected utility is thene e pec ed u y s eEwith m = pV1 + (1 – p)V2

40


• What if the person does not purchase the• What if the person does not purchase the message?

• Let V0 = indirect maximal utility without the message

• Now, expected utility will beo , e pec ed u y beEwithout m = V0 = pV0 + (1 – p)V0


• Therefore information has value as long• Therefore, information has value as long as

Ewith m > Ewithout m

41

The Value of Information on Prices

• In chapter 4 we learned that is an• In chapter 4, we learned that is an individual has a utility function of U(x,y) = x0.5y0.5, the indirect utility function is

5.05.02︶,,︵ yx pp

ppV II

2 yx pp


• If p = 1 p = 4 and I = 8 then V = 2• If px = 1, py = 4, and I = 8, then V = 2

• Suppose that y is a can of brand-name tennis balls– can be found at a price of $3 or $5 fromcan be found at a price of $3 or $5 from

two stores• the consumer does not know which store

charges which price

42


• Before shopping expected utility is• Before shopping, expected utility is

8,5,15.08,3,15.0︶,,︵ VVppVE yx I

049.2894.0155.1︶,,︵ Iyx ppVE


• To calculate the value of information• To calculate the value of information, we solve the equation

049.23122︶

,,︵ 5.05.05.0

*

pp*ppV

yxyx

III

• Solving this yields I* = 7.098– the information is worth 8 – 7.098 = 0.902

43

Flexibility and Option Value

• It may be beneficial to postpone• It may be beneficial to postpone decisions until information is available– but this type of flexibility may involve costs

of its own• may have to give up interest on investments if

the investor waits to make his portfolio decisionthe investor waits to make his portfolio decision

• financial options add flexibility to this type of decision making

Asymmetry of Information

• The level of information that a person buysThe level of information that a person buys will depend on the price per unit

• Information costs may differ significantly across individuals– some may possess specific skills for acquiring

information

– some may have experience that is relevant

– some may have made different former investments in information services

44

Important Points to Note:• The most common way to model

behavior under uncertainty is to assume that individuals seek to maximize the expected utility of their options

Important Points to Note:

• Individuals who exhibit diminishingIndividuals who exhibit diminishing marginal utility of wealth are risk averse– they generally refuse fair bets

45

Important Points to Note:• Risk averse individuals will wish to

insure themselves completely against uncertain events if insurance premiums are actuarially fair– they may be willing to pay more than

actuarially fair premiums to avoid takingactuarially fair premiums to avoid taking risks

Important Points to Note:• Two utility functions have been y

extensively used in the study of behavior under uncertainty– the constant absolute risk aversion

(CARA) function

the constant relative risk aversion– the constant relative risk aversion (CRRA) function

46

Important Points to Note:• One of the most extensively studied y

issues in the economics of uncertainty is the “portfolio problem”– asks how an investor will split his wealth

between risky and risk-free assets

in some cases it is possible to obtain– in some cases, it is possible to obtain precise solutions to this problem

• depends on the nature of the risky assets that are available

Important Points to Note:• The state-preference approach allows p pp

decision making under uncertainty to be approached in a familiar choice-theoretic framework– especially useful for looking at issues that

arise in the economics of informationarise in the economics of information

47

Important Points to Note:• Information is valuable because it

permits individuals to make better decisions in uncertain situations– information can be most valuable when

individuals have some flexibility in their decision makingdecision making

Documents

ch07 [Sola lettura] [modalità compatibilità] · 2 Mathematical Statistics • The expected value of a random variable is the outcome that will occur “on average” i n i i EX