UncertainityWalters & Layard CH !31 INDIVIDUAL BEHAVIOR TOWARDS RISK A- structure of the uncertainty...

uncertainity Walters & Layard CH !3 1

INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK

A- structure of the uncertainty ;Suppose that a farmer is faced with a fertilizer project . STATE OF WORLD 1 2 PLAN RAIN NO RAIN

Project A Fertilizer YA1= 50 YA2=10

Project B No Fertilizer YB1= 30 YB2 = 30

probability (0.5) Π1 (0.5) Π2 Prospect A = (50 , 10 , 0.5 , 0.5 )Prospect B = (30 , 30 , 0.5 , 0.5 ) the two prospects is shown in the figure ;


U(YU(Y1 1 , Y , Y2 2 , ∏ , ∏11 , ∏ , ∏2 2 ) )

(Y with ∏(Y with ∏11 )=Y )=Y11

(Y with ∏(Y with ∏2 2 )= Y )= Y22



Farmer choice depends on his preferences . He is able to order all feasible prospects . We could represent his ordering by an ex-ante utility index → V = V ( Y1 , Y2 ; Π1 , Π2 )

Certainty lineCertainty line

450

Project A

C

50

10

10

50

Y1

Y2 What will be the farmer’s decision

We could calculate the expected value of each alternative .

E(Y|A) = E(Y|A) = ΠΠ1 1 YYA1A1 + + ΠΠ22 Y YA2A2 = (0.5)(50)+(0.5)(10)= 30 = (0.5)(50)+(0.5)(10)= 30

E(Y|B) = E(Y|B) = ΠΠ1 Y1 YB1B1 + + ΠΠ22 Y YB2B2 =(0.5)(30)+(0.5)(30)= 30 =(0.5)(30)+(0.5)(30)= 30

=certain prospect =certain prospect

If he decides based on expected If he decides based on expected value , he will be indifferent value , he will be indifferent between choosing A or B . Just like between choosing A or B . Just like taking a fair gamble .taking a fair gamble .

Indifference curve

30

30

Proj

ect B



however most people are influenced by the spread of possible outcome as well as by the average outcome to be expected . In other words they care about risk .Risk averse person prefer certain outcome to a fair gamble Risk averse person prefer certain outcome to a fair gamble with equal expected value with certain outcome . with equal expected value with certain outcome . Risk lover person prefers a gamble to certain outcome with Risk lover person prefers a gamble to certain outcome with equal expected value .equal expected value .Risk neutral person is indifferent between a gamble and a Risk neutral person is indifferent between a gamble and a certain outcome with equal expected value certain outcome with equal expected value . If needs are the same (taste is the same in rainy and no rain state ) and each state is equally likely , the indifference curve must be symmetric around the 45 degree line . symmetric around the 45 degree line .



If person is risk averse his indifference curve must be convex to the origin , because only this will ensure that he will not take a fair gamble (point A or C ). In other words this will guarantee that point B (certain outcome) is preferred to points A or C . THE EXPECTED UTILITY APPROACHAlternative prospects are ranked according to the expected utility they provide . Our previous general function is now assumed to be in the form of expected utility form ;V(y1 , y2 ; Π1 , Π2) = Π1 U(y1) + Π2U(y2)E(U|A) = Π1 U(yA1) + Π2 U(yA2)= (0.5)U(50) + (0.5)U(10)E(U|B) = (0.5) U(30) + (0.5) U(30) = U(30)He selects point B if E(U|B) > E(U|A) which depends on the shape of the utility function .



if individual is risk averse . Point B will be preferred.

Y10 30 50

U(Y)

U(10)

U(30)

U(50)

(0.5)U(50) + (0.5)U(10)

Point BPoint A or C

Y1 Y2

Risk loverR

S

T

Risk neutral



INSURANCE AND GAMBLING

Risk averse person will be willing to pay to avoid risk.

Suppose that the current income is 50 . The person will loose 40 if his house burns down . There is a chance of 50-50 of fire.

If the person do not insure the house expected utility is measured by the height of point R . The guaranteed income of (30 – TR) will provide him with the equal happiness . A

The maximum that he is willing to pay to have a guaranteed income of (30-TR) is (20+TR)=50-(30-TR) when accident does not happen. When accident happens he will receive an insurance payment equal to (20 - TR). In other words with his initial income of 10 he will have (30 – TR) again .

So, by paying a premium equal to (20+TR) he will transfer the uncertain prospect A : ( 50 , 10 ; 0.5 ,0.5 ) to a certain prospect of ( 30 – TR , 30 – TR ; 0.5 , 0.5 ) .



a risk lover person is willing to take fair gamble in contrast to certain outcome . However he will also willing to take unfair gamble provided it is not very unfair.

Suppose that the person’s income is 30 and he is offered a bet of 20 units . He will be willing to take it according to the following figure provided that he believes that the probability of wining exceed or equal to (PT/PQ).

Suppose that the probability of wining is (PT/PQ) , by taking the bet he will transform the certain prospect of ( 30 , 30 ; 0.5 , 0.5 ) to a uncertain prospect ( 50 , 10 ; PT/PQ , TQ/PQ )

So if probability of winning is between (PT/PQ) and (PN/PQ) the game is unfair (PN/PQ <1/2) but still he accepts the game and prefers it to having of 30 income for certain.(Since he obtains more utility when he accepts uncertain prospect) .



Y

U(YA)

P

Q

10 5030

U(10)

U(50)

U(30)

(0.5)U(10) + (0.5)U(50) N

ST

The risk lover person will pay an amount equal to ST to enter the game .

Probability of wining should be greater than (PT/PQ) and smaller than (PN/PQ) for him to accept the game .


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK Why people both insure and gamble

Fridman and Savage postulated a utility function that was concave at low incomes (risk averse at the beginning) and convex at middle income levels (risk lover at middle ages for earning income) and again concave at high income levels risk averse at the final years of life . This kind of utility function could show the following behavior

Y

U

Y0 = current income

(Y0 – 0.3L)= Y0 - SY0 - L Y0 + 3S

U(Y0 + 3S)

U(Y

0 )U

(Y0–L

)

0.75U(Y0- S) + 0.25U(Y0+3S)

0.3(Y0 – L ) + 0.7Y0

U(Y0 – S)

With a small risk (Π=0.3) of a large loss ( house burning ) he will accept a fair insurance. Since

U(Y0 – 0.3L ) >[ 0.7u(Y0)+0.3U(Y0 – L)]

This does not mean that he will not also take a fair gamble if probability of gain ( ab /ac)=0.25( ab /ac)=0.25 is less than half and bet is fair {E(x)=3S(0.25) - S(0.75)=0} and the winning (+3s) if successful is larger than the loss ( - s ) if unsuccessful,( stock market )

a

b

c


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK does this really explain the behavior in question ?

If we assume that all individuals have the same utility function it implies that gambling will be concentrated in the middle income groups.

One alternative approach is to suppose that individuals differ in their utility functions, but each individual has a utility function that is first concave , then convex and after that concave , with convex section beginning at about the level of his current income (Y0). What happens if the convex section extended for any distance .

How else we could explain the gambling ? There are two obvious possibilities .

First people may overestimate the probability of success.

Second , they may enjoy the sensation of gambling for its own sake , as well as the outcomes to which it give rise . Other explanation mat also possible.



COST OF RISK

What is the cost of risk ? How much of his expected income a person faced with a risky prospect would be willing to sacrifice to an insurance company in order to achieve certainty ? We have to find the certainty-equivalent income that gives the same utility as expected utility of uncertain prospect .

The cost of risk is the difference between the expected value of a risky prospect and its certainly-equivalent income.

The cost of risk will be equal to the distance equal to TR in slide no 5 . The more utility curve is concave (the higher is the degree of risk aversion) , the greater is the cost of risk .

It is useful to have a appropriate measure for the cost of risk .


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK Cost of risk could be defined as following ;

U(YE – C ) = ∑i=1N Πi U (Yi) N = possible state of nature

∑i=1N Πi U (Yi) (BR in slide 6 )

C= (TR in slide 6 )

YE = expected income (point B slide 6)

(YE – C ) = the certain income giving utility equal to the expected utility of the risky prospect .If C is reasonably small , and using the Taylor series ;

U(YE – C ) ≈ U(YE) – U’(YE) C + U”(YE)(C2/2)

C = ( YE – Y ) , Y =(YE – C )= certain income

U( YE – C ) = U(Y)

U(YE – C )=U(Y)≈U(YE)-U’(YE) (YE-Y)+(1/2)U”(YE)(YE-Y)2

E{U(Yi)}=∑i=1N Πi U (Yi)=∑i=1

N Πi U(YE)+U’(YE){∑i=1N Πi (Yi-YE)} +

(1/2)U”(YE) ∑i=1N Πi (Y-YE)2

∑i=1N Πi U (Yi) = U(YE)+U’(YE) (0) + (1/2) U”(YE) VAR (Y)

Very small



∑i=1N Πi U (Yi) = U( YE – C) = U(YE) – U’(YE) C + U”(YE)(C2/2)

If C is very small then C2 ≈ 0 , so ;

∑i=1N Πi U (Yi) = U(YE) – U’(YE) C , then ;

U(YE) – U’(YE) C = U(YE) + (1/2) U”(YE) VAR (Y)

C ≈ - (1/2) U”(YE) VAR (Y) / U’(YE) Risk premium = (C/YE) = fraction of expected income that a person would be wiling to sacrifice for the sake of certainty .

(C/YE ) ≈ { - U”(YE )YE / 2U’ (YE ) } { VAR (Y) / YE2 }

C ≈ { - U”(YE ) / 2U’ (YE ) } { VAR (Y) }

For small VAR ( where YE is small enough) , the cost of risk ( C ) is proportional to the variance of income and risk premium ( C/YE ) is proportional to the coefficient of variation squared .



-(U”/U’) = degree of absolute risk = Pratt measure. -(U”/U’)YE =degree of relative risk aversion=(dU’/dYE )(YE/U’)Which is equal to the elasticity of the marginal utility of income . The faster the marginal utility of income falls , the more risk averse is the person , so the greater is the cost of risk .RISK POOLING AND RISK SPREADINGRISK POOLING AND RISK SPREADING There are two major mechanism by which the cost of risk could reduce . Risk spreading and risk pooling .1-RISK POOLING1-RISK POOLING there is a large number of individuals (n) all whom faces the same risky prospect . Each person income is a random variable with a given distribution and the distribution is the same for all individuals. the distribution of each person’s income is independent of the distribution of each other person’s income.( every one face's the same probability of his house being set into fire , but the accident of one’s house being set into fire is independent of the other one.)


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK Suppose that n individual get together and pool their income, agreeing that each shall draw the average income. ΣYi /n = (Y1 + Y2 + ….+ Yn )/n → Yi = ΣYi /n , Yi = the ith individual income distribution i=1,2,3….n The variance in their total income is the same whether the income are pooled or not . Since all income distributions are independent from each other and have the same variance.Pooled ; Var(Y1 +.. Yn )=Var (Σ Yi /n +….. Σ Yi /n ) Var ( n Σ Yi /n ) = Var (Σ Yi) = n Var ( Yi ) Unpooled Var (Y1 + Y2 + Y3 ) = Var (Σ Yi ) = n Var ( Yi ), But the variation in individual income is reduced . Originally individual i receives Yi and his variance in income is Var (Yi) After pooling the income, each receive ΣYi /n and the variance of each individual income is Var (Σ Yi /n) = 1/n2 Var Σ Yi = (1/n2 ) n Var (Yi ) =n/n2 Var(Yi )= 1/n Var(Yi ) When n → ∞ , then Var (Σ Yi /n) =0



If n identically distributed and independent income distribution are pooled , the variance of average income tends to converge to zero as n tends to infinity in other words ;

C = cost of risk ≈ {- U”(YE )/ 2U’(YE )} VAR (Y) = 0

example of risk pooling are friendly societies and business merges .

2 - RISK SPREADING2 - RISK SPREADING

This is spreading one given income distribution over more than one person . If a risky project is undertaken by one person , the cost of risk is ; C = (-U”/2U’)Var(Y) .

But if the same risky project is undertaken jointly by a group of n person who agrees to divide the proceeds equally, then,

Y/n = income of each one who participate in the project .



Ci = cost of risk for each individual= (-U”/2U’)Var(Y/n) =(-U”/2U’n2) Var(Y)

Total cost of risk= C = n Ci = (-U”/2U’ n){Var(Y)}

If n → ∞ , then C = 0 .

This is the base of joint stock company . The net benefit of most public sector projects can be spread over a large enough number of people , for the cost of risk is negligible .

3-THE COST OF COVARIANC

So far only one source of uncertainty is postulated . More realistic picture of the problem is as following , in which there are more than one source of uncertainty .


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK STATE 1 STATE 2 PLAN RAIN NO RAIN A ; FERTILIZER 50 1050 10B ; NO FERTILIZER 30 2030 20How does this affect the cost of risk .Farmer’s income (Y) = income without fertilizer (X) + return to fertilizer project ( Z)Project cost of risk = what amount of certain return to project ( ZE – C ) would give the same expected utility as the actual project . Σi Πi U(Xi + ZE – C) = Σi Πi U(Xi + Zi ) X1 = 30 , X2=20 , Z1 =20 , Z2 = -10 , ZE = ½(20) + ½(-10)= 5

Σi Πi U( Xi + Zi ) = expected utility of the total income of actual project (like BR in slide 6 )


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK Σi Πi U(Xi + ZE – C) = expected utility of certain value of total equivalent income (income without fertilizer in each state plus expected return to the project minus cost of risk . Like YE – C in slide 6 ). As it is noted the procedure is the same as before (slide 6).The only difference is that we have two states of the world ( x=1 , x=2 ) and the same analysis should be repeated in each state noting that in each state (which is one source of uncertainty ) income without fertilizer is given and return to fertilizer is another source of uncertainty.Using the Taylor series as before it could be shown that we will get ;C ≈ ( -U”/2U’) [ VAR(Z) + 2 COV(X , Z) ]If COV(X , Z) =0 , C ≈ ( -U”/2U’) [ VAR(Z)] as beforeIf COV(X , Z) >0 ,|C| is higher than before .Cost of risk is greater if COV(X , Z) <0 , ,|C| is lower than before . Cost of risk is lower



Consider the following example ;

RAIN NO RAIN

PLAN

A; FERTILIZER 50 1050 10

B; TUBE WELL 20 4020 40

C; NUTRAL 30 2030 20

PROBABILITY 0.5 0.50.5 0.5

Return to project A 20 -10

Return to project B -10 20

Neutral (neither) 30 20


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK as it is seen the project B will be chosen since COV(X,Z) <0 . When certain income is high , return is low and vice versa .The cost of risk of a project depends on its contribution to the variance of total income , and hence not only on its own variance but on its covariance with other elements of income .MEAN VARIANCE ANALYSIS AND PORTFOLIO SELECTIONThis section concerns with single investor choice of optimum portfolio . We assume that the investor only cares about the mean and variance of his income . V = V ( YE , var (y) ) , where YE is expected income or mean income .Using Taylor expansion ;∑i=1

N Πi U (Yi) ≈ U(YE) + (1/2) U”(YE) VAR (Y), (slide no. 13)Utility function V could be written in an expected form ;V=f{EU(Y)}= f{ ∑i=1

N Πi U (Yi) }=f{ U(YE)+(1/2) U”(YE)VAR (Y) ,}



For exact reconciliation of V function with E(U) , one or both of the following conditions must hold ;1 – utility function should be quadratic ,2 – each security in the portfolio has a normal distribution.1- quadratic utility function U(Y) = a + bY – cY2

E[U(Y)]=a + b E(Y) – cE(Y2) , var (y) = E(y – yE)2 =E(y2) – yE2

E[U(Y)] = a + b YE – c [ yE 2 + var (y) ]

This implies that for sufficiently high incomes utility falls as income rise (cY2 > a+bY ). If Y is high → (cY2 > a+bY) → U(Y) ↓ This means that U’ (MUy = b – 2cy) should be decreasing or U” should be negative . This means that if there is one risky asset and a safe one, the investor will hold less of the risky asset as he gets richer. If Y is high → ( cE(Y2) > bE(Y) ) → E[U(Y)] ↓


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK Common observation does not suggest that risk is inferior in the sense that the rich people tends to hold less of the risky asset ( higher yield portfolio ) than poor. So the quadratic function may not be a good example .2- normal distribution for each security The ith security with normal distribution ;µi = mean of the ith security , σi

2 = variance of ith security ,σij = the covariance with the jth security .ai = number of the security of type i µ = ∑i aiµi = mean total income the investor expect .σ2 = ∑i ∑j aiajσij = variance of total income .Distribution of total income is normal with µ and σ2 as mean and variance. There will be different ai with different portfolio , but all will have normal distribution . So all possible distribution will have the same shape except for the mean and variance .so, The investor only concerned with mean and variance of the portfolio . From the two portfolio with the same variance ,σ2, the one with higher mean (µ) will be chosen by a risk averse person . To define the opportunity set the investor needs to know for each To define the opportunity set the investor needs to know for each σσ2 2 , the maximum µ available . , the maximum µ available .



suppose that there are two divisible securities ,X,Z. one unit of each will cost the investor his whole wealth. We will expect that the market to insure that the more risky security has a higher mean return . . Suppose that the investor puts halfhalf of his wealth on on XX and the other half onhalf on ZZ security . In this way ,the mean and variance of the portfolio is as following; µ = (µx + µZ)/2 ,σxz

2 = (1/2)2 σx2 + (1/2)2 σZ

2 + 2(1/2)(1/2)σxz

Rxz = (σxz) / (σx σz) < 1 2(1/2)(1/2) (σxz) < 2(1/2)(1/2) (σx σz) (1/2)2 σx

2+(1/2)2 σZ2+2(1/2)(1/2)σxz<(1/2)2 σx

2+(1/2)2 σZ2+2(1/2)(1/2) (σx σz)

σxz2 < (1/2 σx

+ 1/2 σZ)2 → σσxzxz < (1/2 < (1/2 σσxx + 1/2 + 1/2 σσZZ) →) → looking

at the figure we will see that the frontier of the opportunity frontier of the opportunity set is concave . set is concave .



X

Z

T

σxzσx σz

(½)(σx+σz)

μx

μz

μ

σxz < (1/2 σx + 1/2 σz )

Suppose that there is one risk less asset R , yielding OR for sure if all investor’s wealth is placed in that asset . Suppose that he puts half of his wealth on risk less asset , R(μ,0) , and half of his wealth on risky one , P(μp , σp) , then point Q (μq , σq ) will be resulted which is the utility maximization point .

σp

μp

Q

σQ = (½)σp + 0

R

P

σR=0

μR=

Set of risky portfolioSet of risky portfolio

Frontier of the opportunity setFrontier of the opportunity set

MM



He will never mix point R with any other point (like M) inside the set of risky portfolio except point P (point of tangency). Because mixing point R with point P will expand the opportunity set by the whole dashed area which is larger than any other point .A point on the straight line through point R and P is A point on the straight line through point R and P is preferred to any other point below it because it lies on a preferred to any other point below it because it lies on a higer opportunity set.higer opportunity set. It follows that the investor will always choose a point on line PR . In other words he will put a fraction of his wealth into the risk less asset and a fraction into the risky one (like a1 and a2). This policy will yield him more This policy will yield him more satisfaction (utility) compared to when he puts all of his satisfaction (utility) compared to when he puts all of his income to risky asset ( point P )income to risky asset ( point P ) . So whatever his utility function is, he will always mix his risky assets in the unique ratio ( a1 , a2 , corresponding to points P , and R ) with risk less one .

INDIVIDUAL BEHAVIOR TOWARDS RISK Separation theorem ;• If there is a riskless asset and conditions 1 and/or 2 is

satisfied (1 – utility function should be quadratic , 2 – each security in the portfolio has a normal distribution) and all investors have the same subjective probability distributions, then investors will differ in the amount of wealth they hold in the risky assets, but they will not differ in the fraction of that “risky” wealth devoted to each particular risky asset ( a1 , a2 , corresponding to points P , and R ).

• Experiences do not support this ,because ;• A- people may guess differently about mean and variance.• B- they may start from different historically portfolios and

locked in by transaction costs or tax problems.• C- different asset might offer different tax advantages.• D- mean-variance approach might have deficiencies.



INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK THE EXPECTED UTILITY THEOREM .So far we have assumed that people maximize expected utility and have treated this as a hypothesis to be confirmed or disapproved on the base of evidence . Prospect A ; ( 50 , 10 , 0.5 , 0.5 )Prospect B ; ( 30 , 30 , 0.5 , 0.5 ) We could write the above prospects in the following way ; Prospect A ; ( 50 , 30 , 10 ; 0.5 , 0 , 0.5 )Prospect B ; ( 50 , 30 , 10 ; 0 , 1 , 0 ) more generally ; prospect A = ( yprospect A = ( y11 , y , y22 , y , y33 ; ; ΠΠ11

AA , , ΠΠ22AA , , ΠΠ33

A A ) ) prospect B = ( yprospect B = ( y11 , y , y22 , y , y33 ; ; ΠΠ11

BB , , ΠΠ22BB , , ΠΠ33

B B ) ) 1-First assumption ; there is a preference ordering among all outcomes ( yi) which is complete and transitive .2-Second ; there is also a preference ordering among all prospects which is complete and transitive .Since the prospects differ only in their probabilities , we can write the utility function as follows , V = V (V = V (ΠΠ11 , , ΠΠ22 , , ΠΠ33

).).We will show that it is possible to write the above utility function as follows


INDIVIDUAL BEHAVIOR TOWARDS RISK INDIVIDUAL BEHAVIOR TOWARDS RISK V=V (V=V (ΠΠ11UU11 + + ΠΠ22UU22 + + ΠΠ33UU33 ) ) Where Ui can be defined as utility of outcome yi and also the probability as shown in the following ;If (y ; y1…..yn ) yn is best and y1 is worst ,then by Von-Nweman & Morgeneshtern theorem ;For each yi , there exist probability Πi in such a way that ;

U(yi) = Πi u(yn ) + (1- Πi ) u(Y1 )Yi = certain income ,{ Πi u(y1) + (1- Πi ) u(yn)} = the utility of the lottery winning y1 with probability Πi and yn with probability (1 – Πi ) .

If u(y1) = 0,u(yn) = 1 , then U(yi)= Πi , { { UUii = = ΠΠi i , utility index, utility index }}Lets take the best and worst possible outcome (y1 , y3 ) and ask for each outcome (y1 , y2 , y3 ), the following question ;

At what probability (Ui = Πi )would he indifferent between Yi ( i=1,2,3 ) for certain and a lottery offering y1 with probability Ui= Πi and offering y3 with a probability (1-Ui)

U(yi) = { (Ui) U( y1 ) + (1-Ui) U( y3 ) } , { Ui = Πi } i = 1,2,3 Insisted of having yi ( i= 1,2,3) for certain we could have a lottery yielding y1 with probability Ui ( i= 1,2,3 ) and y3 with probability (1-Ui) , (i=1,2,3) . As we have mentioned Ui is the probability ( ∏ j) which makes the both sides of the above equation equall.



utility

Income YY3 Y1

{{ U Uii yy11 + + (1-U(1-Uii ) ) yy33 } , { } , { UUii = = ΠΠii } }

{Πi u(y3) + (1- Πi ) u(y1 )}

Yi

U(Y1)

U(Y3)V = V (Π1 , Π2 , Π3 ).

Instead of having yInstead of having y22 with probability with probability ΠΠ2 , 2 , we we

could replace Ycould replace Y1 1 with probability with probability ΠΠ22UU22 and and

YY33 with probability with probability ΠΠ22(1-u(1-u22), so), so

ΠΠ22 U(yU(y2 2 ) = ) = ΠΠ22 { (U { (U 22)) U( yU( y11 ) + (1-U) + (1-U 2 2) U( y) U( y3 3 ) } ) }

V (V (ΠΠ11,,ΠΠ22,,ΠΠ33 )=V( )=V(ΠΠ11++ΠΠ22UU2 2 , , ΠΠ22(1-U(1-U22)+ )+ ΠΠ33 ) )

Instead of having yInstead of having y11 with probability with probability ΠΠ11

we could replace Ywe could replace Y11 with probability with probability ΠΠ11UU11

and Yand Y33 with probability with probability ΠΠ11(1-U(1-U11) , so) , so

ΠΠ11U(yU(y11) = ) = ΠΠ11{ (U{ (U11)) U( yU( y11 ) + (1-U) + (1-U1 1 ) U( y) U( y3 3 ) } ) }

V (V (ΠΠ11,,ΠΠ22,,ΠΠ33 )=V( )=V(ΠΠ11UU11++ΠΠ22UU22 ,0, ,0, ΠΠ11(1-U(1-U11) + ) + ΠΠ2(1-U2)+ 2(1-U2)+ ΠΠ3).3).Instead of having yInstead of having y33 with probability with probability ΠΠ33 , we could replace Y , we could replace Y11 with probability with probability

ΠΠ33UU33 and Y and Y33 with Probability with Probability ΠΠ33(1-U(1-U33), so), so

ΠΠ 3 3 U(yU(y33 ) = ) = ΠΠ3 3 { (U{ (U3 3 )) U( yU( y11 ) + (1-U) + (1-U3 3 ) U( y) U( y3 3 ) } ) }

V (Π1,Π2,Π3 )=V(Π1 U1 + Π2 U2 + Π3 U3 , Π1 (1-U1 ) + Π2 (1-U2 ) + Π3 (1-U3 ) V (Π1,Π2,Π3 )=V(Π1 U1 + Π2 U2 + Π3 U3 , 1 – Π1U1 - Π2U2 – Π3U3 )


INDIVIDUAL BEHAVIOR TOWARDS RISKINDIVIDUAL BEHAVIOR TOWARDS RISKSo prospects are ranked according to their probability of happening times their values summed over all outcomes . In other words according to their expected value .We have to make three more assumptions as follows ;3- if Yi is preferred to Yj a prospect offering a higher probability of Yi and a lower one for Yj , other things equal , will be preferred . if U(Yi)>U(Yj) and Πi>Πj and E[U(Y)] = Πi U(Yi) + Πj U(Yj) → will be higher 4- Assumption of continuity .for any particular Y1 preferred to Y2 , preferred to Y3 , there is one unique probability (U2) , at which the sure outcome Y2 is indifferent to to the lottery with probability U2 of Y1 and (1-U2) of y3 . What this really means is that For every certain prospect (for example , Y2) , there exist an equivalent probability distribution involving Y1 and Y3


INDIVIDUAL BEHAVIOR TOWARDS RISKINDIVIDUAL BEHAVIOR TOWARDS RISK

5- Assumption of independence . The evaluation of prospect is not affected if some element in it is replaced by another element which is indifferent to it . In particular , any certain element can be replaced by an uncertain prospect of equal value .

This assumption has been criticized . For example , it is said , it may require a high probability (U2) to induce a person to sacrifice Y2 , if Y2 is really certain . But if Y2 is not certain anyway , a smaller probability U2 might be sufficient to compensate him . This argument seems to imply that people are averse to risk not because of their concern about the outcomes of various alternatives, but because because they dislike the condition of uncertaintythey dislike the condition of uncertainty . .


INDIVIDUAL BEHAVIOR TOWARDS RISKINDIVIDUAL BEHAVIOR TOWARDS RISK

Clearly it can be observed that the ranking of the projects according to Σi Πi (a+bUi) will be the same as Σi Πi Ui . Equally the utility indicator can not be subjected to any can not be subjected to any nonlinearnonlinear increasing transformation without running the risk of obtaining different ranking .

EXPECTED UTILITY THEOREM

Given assumption 1-5 , an individual chooses among risky prospects according to ΣiΠiUi where Πi is the probability of outcome i and Ui ( a utility index for outcome) is invariant between prospects and is unique up to a linear transformation.


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMIC


MARKET EQILIBRIUM AND WELFARE ECONOMICSMARKET EQILIBRIUM AND WELFARE ECONOMICSMarketMarket even under uncertainty could produce efficient outcomeeven under uncertainty could produce efficient outcome . Assumptions are as follows ; the output (Y) is given in each state of the world , Y1 = 100 , when whether is rainy Y2 = 50 , when whether is not rainy Two individuals with utility function Vi = Vi ( Y1

i , Y2i ), i = A, B

The problem is to find the efficient consumption bundleThat is ; Max VMax VA A = V= VAA ( Y ( Y11

AA , Y , Y22AA ), ),

ST VST VB0 B0 = V= VBB ( Y ( Y11BB , Y , Y22

BB ), ), YY22 = Y = Y22

AA + Y + Y22BB

YY11 = Y = Y11AA + Y + Y11

BB

If we could imagine a market for output in each state , then by the knowledge of chapter one , the Pareto efficiency can be resulted . In this way of analysis social welfare can be judged entirely in terms of individual preferences as they are known before the state of nature is known . So we assume that most people put importance on ex-ante utility as well as ex-post utility .




So in this chapter we have used the notion of ex-ante utility function with two components ; first ;Y1 as income in the bad year (when there is no rain with probability of happening equal to 1/2 ) , second ; Y2 as income in the good year (when there is rain , with probability of happening equal to 1/2) . In this way we can define an indifference curve for individuals A and B . Both Both individuals are risk-averse and possess convex individuals are risk-averse and possess convex indifference curve (slide no. 2 ) and individual A who is indifference curve (slide no. 2 ) and individual A who is more risk-averse having the more convex one . more risk-averse having the more convex one .

In order to find out whether the theory of welfare works in the framework of uncertainty we will examine this in the next slide in the context of Edgeworth-Box diagramm.


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICSUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICS

OA

OBCertainty line

Y2=

50

Y1=100

450

450

β

αα

ICA

ICB

a

b

c

e

f

g

αα > > ββ → →

(Y(Y22/Y/Y11))AA > (Y > (Y22 /Y /Y11))BB

A who is more risk-A who is more risk-averse than B will averse than B will receives more in receives more in the bad year (no the bad year (no rain , Yrain , Y22 ) than B . ) than B .

This happens This happens because ICbecause ICA A is is

more convex and more convex and closer to certainty closer to certainty line and point line and point EE lies above the lies above the certainty line. certainty line.

So theory works.So theory works.

αα > > ββ → →

(Y(Y22/Y/Y11))AA > (Y > (Y22 /Y /Y11))BB

A who is more risk-A who is more risk-averse than B will averse than B will receives more in receives more in the bad year (no the bad year (no rain , Yrain , Y22 ) than B . ) than B .

This happens This happens because ICbecause ICA A is is

more convex and more convex and closer to certainty closer to certainty line and point line and point EE lies above the lies above the certainty line. certainty line.

So theory works.So theory works.

Contract curve = efficiency locus

RTSY1 Y2

EE



MARKET FOR CONTINGENT COMMODITIES

Let us suppose a market for each of these contingent commodities(Y1,Y2) Let us also suppose that each of the individuals A , and B , have an equal equal initial endowmentinitial endowment of money(mA , mB ). A is more risk-averse than B , so efficiency locus , and efficiency equilibrium point ( E) lies above the diagonal . Now we should look for a set of prices which could equalize demand and supply in each state under prefect competition.

OA

OB

E

Y2 =

50

Y1 =100

P

Common budget lineCommon budget linemA = mBmA = mB

PY1/PY2 = RTSY1Y2<1

As it is clear from the As it is clear from the figure , price in good figure , price in good year relative to bad year year relative to bad year (P(PY1Y1/P/PY2 Y2 ) is less than one ) is less than one

which means that output which means that output is cheaper in good year is cheaper in good year (Y(Y11) when there is not ) when there is not

shortage of Y .shortage of Y .

Certainty linesCertainty lines

454500

454500



Do these state contingent commodities exist ? Do these state contingent commodities exist ? Insurance and Stock markets are two good examples .

To begin with take the following example with two commodities instead of one commodity .

STATE 1 STATE 2

COMMODITY

X X1 X2

Y Y1 Y2

In order to find the efficient allocation of X and Y between the two individuals A and B , we could solve the welfare maximization optimization as indicated by the following equations ;


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICMax VA = Π1 U1

A (x1 A,y1

A) + Π2 U2 A (x2

A,y2 A)

S.T. VB0 = Π1 U1B (x1

B,y1B) + Π2 U2

B (x2 B,y2

B )

Px1 x1A + Py1 y1

A = m1 A

Px2 x2 A + Py2 y2

A = m2 A

Px1 x1B + Py1 y1

B = m1B

Px2 x2B + Py2 y2

B = m2B

this could be clearly be done through first ; contingent commodity market and second ; equally in theory the efficient allocation could be found using security market only . Under first schemefirst scheme the individuals are faced with four prices for the individuals are faced with four prices for contingent commoditiescontingent commodities (Px1 , Py1 , Px2 , Py2 ) and model could be optimized as mentioned above, and under second second oneone the individuals are faced with ;


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMIC

1- a price today for a dollar to be delivered in state 1 , 1- a price today for a dollar to be delivered in state 1 , namely Pnamely P11 , and another price for a dollar to be delivered , and another price for a dollar to be delivered in state 2 , namely Pin state 2 , namely P22 . .2 – correct forecast of four future spot prices ; 2 – correct forecast of four future spot prices ; (P(P**

x1 x1 , P, P**y1y1 , P , P**

x2x2 , P , P**y2y2)) .

Let us consider the second scheme ;second scheme ;The individual first of all allocates his initial wealth between the two securities ; buying q1 securities of the first kind to be spent in the first period , and q2 securities of the second kind to be spent in the second period.. So ; pp11qq1 1

AA + p + p22qq22AA = m = mAA

pp11qq1 1 BB + p + p22qq22

BB = m = mBB

in each state the individuals are trying to maximize their utility subject to the budget constraint ;


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMIC for individual A ; max U1

A ( x1 A , y1

A ) S.T. q1

A = Px1* X1

A + Py1* y1

A max U2

A ( x2 A , y2

A ) S.T. q2

A = Px2 * X2

A + Py2* y2

A For individual B ;max U1

B ( x1 B , y1

B ) S.T. q1

B = Px1* X1

B + Py1* y1

B max U2

B ( x2 B , y2

B ) S.T. q2

B = Px2 * X2 B + Py2

* y2 B

under what circumstances will the actual bundle consumed actual bundle consumed be the same under both schemesbe the same under both schemes. We need to have the following equalities ;


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMIC(Px1

*) P1 = Px1 , Px1 = actual price of x in state 1 (Px2

*) P2 = Px2 Px2 = actual price of x in state 2 (Py1

*) P1 = Py1 (Py2

*) P2 = Py2 These equalities are required to fulfill the budget constraint ; p1 q1

A = p1 Px1* X1

A + p1 Py1* y1

A = Px1 x1A + Py1 y1

A

p2 q2 A = p2 Px2

* X2 A + p2 Py2

* y2 A = Px2 x2

A + Py2 y2 A

p1 q1 B = p1 Px1

* X1 B + p1 Py1

* y1 B = Px1 x1

B + Py1 y1B

p2 q2 B = p2 Px2

* X2 B +p2 Py2

* y2 B = Px2 x2

B + Py2 y2 B

mA = p1 q1 A + p2 q2

A mB = p1 q1

B + p2 q2 B



There is no obvious reason why individuals could forecast the There is no obvious reason why individuals could forecast the future prices correctly . future prices correctly . In other words it can be said that the In other words it can be said that the .

Pareto-Optimality will be achieved if there is Pareto-Optimality will be achieved if there is perfect market perfect market for state contingent for state contingent money payments provided that individuals money payments provided that individuals correctly forecast the structure of product pricescorrectly forecast the structure of product prices..

If we forget the diversity of commodities and assume that utility depends on income in each state of nature , clearly the security market can do the same job as contingent commodities market . But we need security markets that can provide each individual with whatever income he chooses in each possible state.

Does stock market or security market can do the job ? In order to find out the answer we have to work out the stock market function from the view point of stock holders who are controlling the production decision of the firm. .



PRODUCTION DECISION AND STOCK MARKET.

What will be the production criteria under uncertainty. The question can be reduced to the one that will ask how will a how will a productive enterprise (firm) in fact behave so as to maximize productive enterprise (firm) in fact behave so as to maximize the welfare of its owners (share holders or stock holders)the welfare of its owners (share holders or stock holders). Suppose that for a firm there exist a number of possible projects and each combination yields the following net income for firm .

NET RETURN

STATE 1 STATE 2

PROSPECTS

A 50 10

B 60 12

….. …. ….


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICThe firm’s decision is to maximize the owner’s utility . Let us see what the owner’s utility depends upon.The ith individual is maximizing his expected utility ; MAX : EU(Y) = Σk ΠikUi(yik) , yik = Σj Sij Rjk for each K.Subject to his budget constraint which is equal to S.T. : Σj Sij Mj = Σj Sij

0 Mj ;

k ; index for the state of the worldyik = income of individual i in the state of K . Sij = individual i th share from j th firm’s income. Rjk = income of j th company in the kth state of the world. Sij

0 = the initial share of j th firm belongs to individual i . Mj = the market value of all shares of firm j company .


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE CONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE CONOMICIt is clear that the shareholders of a firm will gain in welfare from an increase in the market value Mj of the firm . So how can a firm decide whether a given investment will raise its market value . The answer is clear taking into account the following example ; NET RETURN STATE 1 STATE 2 PROSPECTS A 50 10 B 60 12 …….. …… ……..

It is noted from the example that whatever the project does , its income in the two states of the world are in the ratio of 5:1.There will be a determinate market price for an income determinate market price for an income prospect ( 5 ; 1 )prospect ( 5 ; 1 ) which will not change if our firm invest in project A or B ( or may be in any other project ) . Thus the firm can use this price as a basis for its evaluation of the project .


UNCERTAINITY,MARKET EQILIBRIUM,WELFARE ECONOMICUNCERTAINITY,MARKET EQILIBRIUM,WELFARE ECONOMIC

In cases where a pattern of returns is common enough in the economy to have a well defined price , independent of the firm’s decision , we say that the firm belongs to a risk class . In this case, it maximizes the total value of the firm’s business by evaluating prospects at the price relevant to their risk-class.

When one considers the number of possible states of nature When one considers the number of possible states of nature and different possible patterns of returns,and different possible patterns of returns, it seems unlikely it seems unlikely that the condition would be generally satisfied .that the condition would be generally satisfied .

For example if my house burns down , there might be no if my house burns down , there might be no productive enterprise that experiences an increase in productive enterprise that experiences an increase in profit to compensate me at the time my house burns profit to compensate me at the time my house burns down . down . So I can not insure myself against fire by any So I can not insure myself against fire by any pattern of share . pattern of share .



For the share market to be a fully efficient instrument for the allocation of risk , it must be possible to assemble a prospect of whatever proportions is optimal , given the initial wealth . This requires that , if there are n states if there are n states of nature , there are at least n firms whose incomes are linearly of nature , there are at least n firms whose incomes are linearly independent independent

To solve this assume that I wish to buy the following particular vector of incomes in each state of nature : (Y1

0 , Y20 , Y3

0,…Yn0) . Rjk is the net

income of j th firm in the k th state of nature . If there are n firms , and there are n states of nature , this gives us an n by n matrix whose element Rjk show the income of each firm in each state of nature . If this matrix is nonsingular , there must exist a vector of share holdings such as ( S1 , S2 , …. Sn ) where Sj is the fraction of j th firm owned by me , such that I guarantee myself the specified income in each state of nature .this pattern of share holdings is got by solving for (S1 , S2 ,,,, Sn) in the following ;


UNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMICUNCERTAINITY ,MARKET EQILIBRIUM , WELFARE ECONOMIC (YY11

00 , Y , Y2200 , …Y , …Ynn

00) =(S) =(S11 , S , S22 ,,,, S ,,,, Snn) R) R11 11 ,,,,,,,,, R,,,,,,,,, R1n1n

RR21 21 ,,,,,,,,, R,,,,,,,,, R2n2n

……..................................

RRn1n1 ,,,,,,,,, R ,,,,,,,,, Rnnnn It is essential that at least n firms have linearly independent It is essential that at least n firms have linearly independent income vectorsincome vectors . If all productive enterprises have the same income in any two of the relevant states of the world , (whether or not my house burns down ) , then this condition is not satisfied . It seems most unlikely that it would be satisfied . For this reason insurance against personal For this reason insurance against personal misadventure cannot be achieved fully through the security misadventure cannot be achieved fully through the security marketmarket , because the stock market is limited by the requirement that each individual gets the same fraction of a given firm’s income in every state of nature .The “perfect insurance market”is free of this constraintThe “perfect insurance market”is free of this constraint will do will do the trick : they will allow us to ensure for ourselves any the trick : they will allow us to ensure for ourselves any income in any state of nature .income in any state of nature .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION

THE INSURANCE MARKET AND MORAL HAZARDTHE INSURANCE MARKET AND MORAL HAZARD

Perfect market insurance Perfect market insurance is an ideal market for perfectly insuring the risk which never realized. In a perfect insurance market an individual would be able to bet an unlimited extent at fixed odds on any particular state of nature coming to pass. But we can not find an insurance company to act like this . Most insurance contracts take the following form ;Most insurance contracts take the following form ;

You would be told that you could insure only for the cost of medical bills , and there would be usually a maximum cover.maximum cover.

Why there does not exist a perfect insurance market ?Why there does not exist a perfect insurance market ?

Usually there are two reasons ;

1- Moral Hazard , 2 – Adverse Selection1- Moral Hazard , 2 – Adverse Selection


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION 1- Moral Hazard ;

moral hazard arises when the customers of an customers of an insurance company can affect the liabilities of the company insurance company can affect the liabilities of the company without the company knowing it .without the company knowing it . Thus whether you are illill or have a fire fire depends in part on your own actionsyour own actions and not only on the state of naturestate of nature. There are two main cases ;

First ; the event is exogenous (being ill ) but the exact extent First ; the event is exogenous (being ill ) but the exact extent and nature of illness is not observable by the insurance and nature of illness is not observable by the insurance companycompany . . If the company has undertaken to cover all If the company has undertaken to cover all medical costs , the effective price of the medical care to the medical costs , the effective price of the medical care to the individual will be zeroindividual will be zero . He will demand too much from medical care . Suppose that demand for medical care is shown by the following figure ;



SDemand for medical care Demand for medical care conditional on having conditional on having particular illnessparticular illness

P

Qx0 x1

Marginal cost of providingMarginal cost of providing

the medical service the medical service

D

D

1

If there is no insurance , the If there is no insurance , the medical care bought is Xmedical care bought is X00. if . if there is unlimited insurance there is unlimited insurance medical care bought will be medical care bought will be equal to Xequal to X11 . . The ideal The ideal arrangement is to insure the arrangement is to insure the person for Xperson for X00 to be paid to him to be paid to him conditionally on having this conditionally on having this particular diseaseparticular disease. Because at . Because at XX00 we have efficiency condition , we have efficiency condition , MRT=MC=P=MRS. MRT=MC=P=MRS.

But in practice the insurance But in practice the insurance company pays any medical bills company pays any medical bills arising from a wide rage of arising from a wide rage of illness , though payment may be illness , though payment may be subject to a maximum payment . subject to a maximum payment .

Q=medical care

a

b c


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION To see this assume that every insurance policy provide unlimited cover . So the medical care (insurance) bought will So the medical care (insurance) bought will be xbe x11 . Probability of being seek is . Probability of being seek is ΠΠ . Premiums(P) are paid . Premiums(P) are paid only when person is wellonly when person is well . And it is equal to ; P = [ XP = [ X11ΠΠ /(1- /(1- ΠΠ) ]) ] → X1Π = P(1- Π)→ expected value = 0 (fair)expected value = 0 (fair) P =the amount that the insurance company receive when person is not ill.( insurance premium )

X1 = the amount that insurance company pay when he is ill .

utility of being ill having insurance = Us =Us( money income (m) + value of medical care ( unlimited medical care = a+b+c ))utility of not being ill having insurance=UN = UN(m – premium=x1(Π/(1- Π))We should calculate the expected value of having insurance and not having it as follows


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION Having insurance ;

E (Ui)= ΠUs( m+a+b+c ) + (1- Π) UN (m-x1(Π/(1- Π))

No insurance ;

E (Uni) = ΠUs( m + a ) + (1- Π) UN (m)

If E (Ui ) > E (Uni ) → he insure himself

If E (Ui ) < E (Uni ) → he does not insure himself

There is no reason why E (Ui ) be always greater than E (Uni). This is why the insurance markets are not complete .

To cover the moral hazard the insurance companies practice coinsurance in which the insured has to pay a fraction of any loss . Also common are deductions in which the insurance pay only the excess of loss over some fixed amount .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION Some times individuals are demanding more than the efficient amount of the medical care ,individuals will force the insurance company to raise the premium for all the customers . This is the externalities which will be produced by the behavior of individuals .

It should be kept in mind that when we conclude that free markets are subject to market failure , when compared to idealized free market , does not mean that the state intervention will do a better job than actual free market .

SecondSecond ; ; when the insurance company face a well defined when the insurance company face a well defined cost if the event occurs, but the individual customer can affect cost if the event occurs, but the individual customer can affect the probability of event occurring and the company can not the probability of event occurring and the company can not monitor itmonitor it . . For example the car accident insurance could affect how hard I try to avoid minor damages to the car . Externalities just like the

previous example could increase the insurance premium.



Suppose the individuals get together and set up a cooperative cooperative insurance schemeinsurance scheme . For the insurance scheme to be balanced we should have P = P = ΠΠY/ ( 1- Y/ ( 1- ΠΠ ) ) , in which ;

PP = the premium which the individuals should pay when there is no disaster.

YY = the amount which will be paid by insurance company to insured in disaster year .

The individuals then agree collectively on the amount of agree collectively on the amount of preventive effort “ e “preventive effort “ e “ which somehow could be measured by individuals.

U1 = utility in disaster year .

U2 = utility in the year when there is no disasters .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION d= the amount of loss when there is a disaster .Π = probability of happening a disaster .The individual is trying to maximize the E(U) as follows ; E(U) = E(U) = ΠΠ U U11(m – d + y) + (1-(m – d + y) + (1-ΠΠ) U) U2 2 (m – (m – ΠΠY/(1- Y/(1- ΠΠ) ) -e) ) -e What should be the amount of Y for maximizing the E(U) .∂ ∂ E(U)/∂ Y = E(U)/∂ Y = ΠΠ U U11’’ - (1 – - (1 – ΠΠ )U )U22’ (’ (ΠΠ/(1- /(1- ΠΠ)=0 → )=0 → ΠΠ U U11’’ = = ΠΠ U U22’’ Ui’ = marginal utility of income in state i . If utility functions in each state is independent from each other , if ΠΠ U U11’’ = = ΠΠ U U22’ , income in each state should be equal’ , income in each state should be equal .But the cooperative insurance company often breaks down , since the private insurance may feel unable to prescribe the private insurance may feel unable to prescribe the level of preventive effortlevel of preventive effort .Allowing for the fact that people with higher cover have higher probabilities of disaster, insurance company may suggest a schedule where premium per dollar is a rising function of Y .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION P = Y f (Y) . f ’ > 0

Max E(U) =Π U1(m–d+y)+(1–Π)U2[m–y f( Y) ]

∂ E(U)/∂ Y = Π U1’ +(1 – Π )U2’ (- f(Y) – y f ’ ) = 0

For the insurance company to balance, (fair game)

P = y f (y) = ΠY/ (1- Π)

Substituting this in ∂ E(U)/∂ Y → we will get ;

ΠU1’ = ΠU2’(1+ Yf’/f ) > ΠU2’

consequently people do not achieve certainty that consequently people do not achieve certainty that they would choose if they could agree on a they would choose if they could agree on a common plan .common plan .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION ADVERSE SELECTIONADVERSE SELECTIONSuppose that there are two groups of people ;1- high risk people with H number of them. 2- low risk people with L number of them .Both groups risk a disaster costing d but the probability is ΠH for high risk people and ΠL (< ΠH ) for low risk people .If both groups participate in insurance policy and the company company can not distinguish between themcan not distinguish between them , the premium will be calculated as follows ;Expected premium for full coverage = P P =d { ΠH [ H/(H+L) ] + ΠL [ L/(H+L) ] }Expected loss of low risk people = d ΠL

d d ΠΠLL < d { ΠH [ H/(H+L) ] + ΠL [ L/(H+L) ] } since ΠH > ΠL Expected loss of high risk people = d ΠH

d d ΠΠHH > d { ΠH [ H/(H+L) ] + ΠL [ L/(H+L) ] } since ΠH > ΠL


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION If individuals know their own risk , those with low risk may not If individuals know their own risk , those with low risk may not be willing to take out the policybe willing to take out the policy . If there were many classes of risk, one can imagine situations in which the only surviving market would attract the very worst class of risk. In some cases no market at all may exist , as for mental health insurance or chronic disease.

in other words the problem of adverse selection arises if in other words the problem of adverse selection arises if individuals know their own risk but the insurance company do individuals know their own risk but the insurance company do not .not . In these cases the policies that are offered are then less In these cases the policies that are offered are then less efficient than would exist if efficient than would exist if individually tailored policiesindividually tailored policies could could be offeredbe offered .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION UNPRODUCTIVE SCREENINGUNPRODUCTIVE SCREENING suppose there are two kinds of people ;1- geniuses , with GG number of them 2- idiot , with I I number of them .

PG = productivity of geniuses people.

PI = productivity of idiots people. PPII < P < PGG The employers do not know who is genius and who is idiot beforehand . the only way employers can discriminate between geniuses and idiots is on the base of external characteristics visible before the contract of employment is made . If employers can see no differentiating characteristics If employers can see no differentiating characteristics they will pay each group the same real wage W ;they will pay each group the same real wage W ;

W = PW = PGG [ G/(G+I) ] + P [ G/(G+I) ] + PI I [ I/(G+I) ] = P[ I/(G+I) ] = P

Since PSince PGG > P > PII , we will have W> P , we will have W> PII and W< P and W< PG G

One way to distinguish between them is acquiring a signal . One way to distinguish between them is acquiring a signal .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION Suppose that in a competitive equilibrium competitive equilibrium ( equal ( equal opportunities ) , all G people are educated and no idiots opportunities ) , all G people are educated and no idiots werewere . If PG is the wage paid to the educated ( G ) people and PI is the wage paid to the non educated ( I ) people , under what condition would such an equilibrium exist in which there such an equilibrium exist in which there would not be the possibility of any idiots be educatedwould not be the possibility of any idiots be educated . For this to happen we should have ;

cost of education to I people = CCI I > P> PGG - P - PII . It must not privately worthwhile for an idiot to become educated and worthwhile for an genius to become educated.

cost of education to G people = CCGG < P < PG G – P– PII .

Though the signal is privately worthwhile for the genius , Though the signal is privately worthwhile for the genius , is it is it socially worthwhile? Certainly notsocially worthwhile? Certainly not


ECONOMICS . OF INFORMATION ECONOMICS . OF INFORMATION All that education does is to raise the wage of genius by separating them from the mass of people who is receiving weighted average wage of P. At the same time it lowers the wage of those ( idiot) with whom his productivity was previously averaged . And it lowers their wage (wage of idiot people) by more than it raises his (idiot) own wage net of education cost . This might be so since the cost of education is now being born by somebody.

We have described the separating equilibrium . But there might be another one . Suppose that we start with no one being educated . It is quite possible that a small group of genius separated themselves and got educated. They would get paid PG and otherwise they would only get P . But nothing so far implies that this is impossible PPGG – P < C – P < CGG . In this case , if no genius were getting educated , it would not be worth any small group of them attempting to do so. So there exist a pooling equilibrium in which no one get educated pooling equilibrium in which no one get educated .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION Productive screening So far , information had no social value So far , information had no social value . Individuals could use resources to transmit information about themselves that is privately valuable to them but did not increase the social product . This was

because it did not matter who did which job. But in reality some But in reality some people have comparative advantage in some jobspeople have comparative advantage in some jobs . Some people would enjoy some jobs more than the others . This may benefit the whole society in form of inventing something or making the innovations by those geniuses who have found their interest .

So up to a point the resources ought to be devoted to the So up to a point the resources ought to be devoted to the process by which employers learn about the workers and process by which employers learn about the workers and workers learn about the jobs workers learn about the jobs . It is this productive role of information that has been stressed particularly by writers of Chicago school like Stigler and by many of those who believe that the natural rate of unemployment is also optimal , since it leads to just the right amount of labor-market search .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION Information about the natural world

The information which generates knowledge about the natural world is also valuable . For example consider the return to weather forecasting . Farmers may be informed of deluge and harvest their crops few days earlier . There may be cases that knowing the weather makes no difference to the social product . For example, an individual by using the resources may learn that a hurricane may devastate his apple crop and that nothing can be done about it . The information is socially valueless. But the individual can make money from it by selling his crop forward to someone who knows nothing about the hurricane . Thus again it is possible that too many resources will be devoted to the production of information .


ECONOMICS OF INFORMATION ECONOMICS OF INFORMATION When one turns to basic research , matters are even more complicated . It is not clear whether free enterprise would lead to too much money or too little being spent on research .

There are two problems here ;

information about the natural world is a public good . Once it has been discovered it can be transmitted relatively cheap . However if it is sold at marginal cost of distributing it those who produce the research gets no return H. So patent laws exist to ensure that there are incentives to generate knowledge .Since patents have limited life span and anti-trust laws limit the profit to which they give rise , there may be under investment in research . If so this could be in principle remedied by public subsidy , but it is not all easy to know on what principles the subsidy should be allocated .

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UncertainityWalters & Layard CH !31 INDIVIDUAL BEHAVIOR TOWARDS RISK A- structure of the uncertainty...

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