51-122

Welfare Change and Price Indices

Change in the price of one good

Suppose 1P increases from 0 1

1 1 to P P , other prices remaining constant.

How much compensation is needed to make the consumer as well off as before (same utility)?

compensation needed 1 0 0 01 2 1 2( , , ) ( , , )e P P U e P P U

welfare change (drop in utility):

compensating variation (CV) 1 0 0 0 0 0 1 01 2 1 2 1 2 1 2( , , ) ( , , ) ( , , ) ( , , )e P P U e P P U e P P U e P P U

0 01 1

1 11 1

00 0 1 0 01 2

1 2 1 2 1 1 1 2 1

1

( , , )( , , ) ( , , ) ( , , )

P P

P P

e P P Ue P P U e P P U dP h P P U dP

P

1

P

CV

11P

B

A

01P

01 1 2( , , )h P P U

1x

01x

If the demand is locally linear, then negative

0

1 1 1 1

0 0 11 1 1 1 1 1

1

1CV [ (A B) B] ( )( ) ( )( )

2

1 1 ( )

2 2

U U

U U

U U

area P x P x

xP x x P x P

P

Inverse of slope of compensated demand curve

53

53

Consumer Surplus

Consider an individual with an income of m . Suppose he is only allowed to buy good ( )y OG

initially, at a price of $1 per unit. He will then buy m units of y, which gives him a utility of 0U .

Next, suppose he is allowed to buy x at price$ / unitP , then he is going to buy 1x [the consumption

bundle is 1 1( , )x y ] which gives him a utility of 1U .

Question: What is the maximum amount of money he would have been willing to pay to get 1x ? y

bundle consumed

m

consumer surplus

1y 1U

slope P

2y 0U

1x x

From the above diagram, we can see that the consumer is indifferent between enjoying his initial bundle

(0,m) or enjoying the bundle 1 2( , )x y , hence the maximum amount the consumer is willing to pay for 1x

is 2m y , and he is actually paying 1m y .

Definition: The consumer surplus on a good x

maximum amount a consumer would be willing to pay the amount he actually pays

Consumer surplus 2 1 1 2max $ willing to pay $ actually paid ( ) ( )m y m y y y

Maximum $ willing to pay

1 1

0 0

0 0

( , ) ( ) where ( ) ( , )

x x

yx yxMRS x U dx P x dx P x MRS x U

per y x

( )HD P x

x

The maximum amount a consumer would be willing to

pay is the area under the compensated or Hicksian

demand curve, and not the Marshallian or ordinary

demand curve.

The Marshallian demand curve will be the same as

the Hicksian demand curve if there is no income

effect.

54

54

Remark: Consumer surplus

1

0

0

( , )

x

yxMRS x U P dx (measured in y )

Remark: Consumer surplus cannot be more than his total income.

All-or-nothing demand curve

For 1x , the most the individual would be willing to pay per unit is 2

1

m y

x

, this is the all-or-nothing

demand curve.

Lets take different value of x and compute for each of them the all-or-nothing price.

The all-or-nothing price 0

0

1*( ) ( ) where ( ) ( , )

x

yxP x P q dq P x MRS x Ux

Since ( ) is decreasing in *( ) ( )P x x P x P x .

per y x

1( )P x

( )HD P x

1x x

1 1

1

*( ) ( )

shaded areaP x P x

x

55

55

per y x

all-or-nothing price *( )P x

( )HD P x

x

Note that at any x , *( )P x x area under the demand curve

0

0 0

( ) 1 *( )

*( ) ( ) *( ) ( ) ( )

x

x xd P q dq

d xP xP x P q dq P x x P q dq P x

x dx dx

56

56

IV. Consumer Choice Under Uncertainty

Review of probability theory

Expectation

Suppose that a random variable X has a discrete distribution for which the probability function is f .

Then the expectation is

( ) ( )x

E X xf x

Note: is called the expected value, the average, or the mean.

Example:

X -2 0 1 4

P 0.1 0.4 0.3 0.2

( ) ( 2)(0.1) (0)(0.4) (1)(0.3) (4)(0.2) 0.9E X

Variance (2 ) and Standard Deviation ( )

2 2 2 2

1 -

[( ) ] ( ) ( ) or ( ) ( )n

i i

i

E X X P X X x f x dx

2 2[( ) ]E X

Example:

X 5 7 12

P 1/3 1/3 1/3

2 2 2

5 7 128

3

(5 8) (7 8) (12 8) 9 1 16 26 26var SD 2.94

3 3 3 3

57

57

Covariance

( , ) {[ ( )][ ( )]}i jCov X Y E X E X Y E Y

y y

. . .. .. .. .

. . . . .. .. .

. . . . . .. . . . ..

. .. . . . .. . .. . . . .. . . . . . . . . ..

. . .. .. . . . . . . .. .. . . . . . .. .

. .. . . .. .. . . .. . . .. . . . . .. . . .. . . .. .. . .. .. . .. .

x x y

. .. . . . . . . . . . .

. . . . . . . . . . . . . .. . .. .

. . .. .. ..

.. . . . .. . ..

. . .. .. .. ..

.. .. .. .. .. .. . . . . x

The correlation coefficient ( , )X Y between 2 random variables and i jR R :

( , )( , )

( ) ( )XY

Cov X YX Y

X Y

Properties of Correlation

1. 1 ( , ) 1X Y

2. 1 : perfectly negatively correlated 0: uncorrelated

1 : perfectly positively correlated

0Cov negatively correlated 0Cov positively correlated

0Cov uncorrelated

58

58

Example: 2 random variables with a joint density function

Y 0 1 ( )g Y

X

1 0.24 0.06 0.30 0 0.16 0.14 0.30

1 0.40 0.00 0.40

( )f X 0.80 0.20 1.00

( ) (0.3)( 1) (0.3)(0) (0.4)(1) 0.1

( ) (0.8)(0) (0.2)(1) 0.2

E X

E Y

2 2 2 2

2 2 2

( ) [ ( )] ( 1 0.1) (0.3) (0 0.1) (0.3) (1 0.1) (0.4) 0.69

( ) [ ( )] (0 0.2) (0.8) (1 0.2) (0.2) 0.16

Var X E X E X

Var Y E Y E Y

( , ) {[ ( )][ ( )]} ( 1 0.1)(0 0.2)(0.24) ( 1 0.1)(1 0.2)(0.06)

(0 0.1)(0 0.2)(0.16) (0 0.1)(1 0.2)(0.14) (1 0.1)(0 0.2)(0.40) (1 0.1)(1 0.2)(0) 0.08

Cov X Y E X E X Y E Y

( ) ( , ) 0.69 0.08 Variance-Covariance Matrix

( , ) ( ) 0.08 0.16

Var X Cov X Y

Cov X Y Var Y

59

59

Theorem

1) If Y aX b , then ( ) ( )E Y aE X b

2) 1 1( ... ) ( ) ... ( )n nE X X E X E X

3) If 1 ,..., nX X are n independent random variables, then 1 1( ... ) ( )... ( )n nE X X E X E X .

4) ( ) 0 such that ( ) 1Var X c R P X c

(i.e. we are sure that X must take the value of c )

5) 2( ) ( )Var aX b a Var X

6) 2 2( ) ( ) [ ( )]Var X E X E X

7) If 1 ,..., nX X are n independent random variables, then

i) 1 1( ... ) ( ) ... ( )n nVar X X Var X Var X

ii) 2 2

1 1 1 1( ... ) ( ) ... ( )n n n nVar a X a X a Var X a Var X

Theorem

1. ( , ) ( ) ( ) ( )Cov X Y E XY E X E Y

2. If and X Y are independent random variables, then ( , ) ( , ) 0Cov X Y X Y

[ ( ) ( ) ( )E XY E X E Y if and X Y are independent.]

The converse is not true.

3. Let , 0Y aX b a .

If 0a , then ( , ) 1X Y .

If 0a , then ( , ) 1X Y .

4. ( ) ( ) ( ) 2 ( , )Var X Y Var X Var Y Cov X Y

5. 1 1

( ) ( ) 2 ( , )n n

i i i j

i i i j

Var X Var X Cov X X

60

60

Conditional Probabilities and Statistical Independence

( ) ( )( ) or

( ) ( )

P AB P A BP AB

P B P B

Example:

1(4 and spade) 152(4spade)

13(spade) 1352

1(4 of spade and black) 152(4 of spade black)

26(black) 2652

PP

P

PP

P

Statistical independence and dependence

Independent Events Dependent Events

( | ) ( )P A B P A ( | ) ( )P A B P A

( | ) ( )P B A P B ( | ) ( )P B A P B

( ) ( ) ( )P A B P A P B ( ) ( ) ( )P A B P A P B

Example:

2 cards are selected with replacement, from a standard deck. Find the probability of selecting a king and

a queen.

4( ) ( ) independent

52P QK P Q

4 4 1( ) ( ) ( )

52 52 169P K Q P K P Q

Multiplication Rule

( ) ( ) ( )P A B P AB P B

Example:

Two cards are selected, without replacement, from a standard deck. Find the probability of selecting a

king and then selecting a queen.

Since the first card is not replaced, the events are dependent.

4 4 16( ) ( ) ( ) 0.006

52 51 2652P K Q P Q K P K

61

61

Bayes' Theorem

( ) ( | ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) is the prior probability

( B) is the posterior probability

C C C

P AB P B A P A P B A P AP A B

P B P BA P BA P B A P A P B A P A

P A

P A

or

1 1 2 2 3 3

( | ) ( )( | ) 1, 2, 3

( | ) ( ) ( | ) ( ) ( | ) ( )

i i

i

P B A P AP A B i

P B A P A P B A P A P B A P A

or

1

( | ) ( )( | ) 1, 2, ...,

( | ) ( )

i i

i n

i i

i

P B A P AP A B i n

P B A P A

62

62

Example:

Consider a manufacturing firm that receives shipments of parts from 2 different suppliers.

Let 1A denote the event that a part is from supplier1 and 2A denote the event that a part is from

supplier2. Currently, 65% of the parts purchased by the company are from supplier 1 and the remaining

35% are from supplier 2. The quality of the purchased parts varies with the source of supply. Historical

data suggest that the quality ratings of the 2 supplies are as follow:

1 1

2 2

( | ) 1 ( | ) 0

( | ) 0.9 ( | ) 0.1

P G A P B A

P G A P B A

where

: the event that a part is good

: the event that a part is bad

G

B

Suppose now that the parts from the 2 suppliers are used in the firm's manufacturing process and that a

machine breaks down because it attempts to process a bad part.

Given that the part is bad, what is the probability that it comes from supplier 1? From supplier

2?

1 1

1

1 1 2 2

2 2 2 2

2

2 2 1 1 1 1 2 2

( ) ( ) (0)(0.65)( ) 0

(0)(0.65) (0.1)(0.35)( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

(0.1)(0.35)

(0)(0

P B A P AP A B

P B A P A P B A P A

P B A P A P B A P AP A B

P B A P A P B A P A P B A P A P B A P A

1.65) (0.1)(0.35)

1 1

1

1 1 2 2

2 2

2

1 1 2 2

( ) ( ) (1)(0.65)( G) 0.673575

(1)(0.65) (0.9)(0.35)( ) ( ) ( ) ( )

( ) ( ) (.9)(0.35)( G) 0.326425

(1)(0.65) (0.9)(0.35)( ) ( ) ( ) ( )

P G A P AP A

P G A P A P G A P A

P G A P AP A

P G A P A P G A P A

Suppose the conditional probabilities are as follow:

1 1

2 2

( | ) 0.5 ( | ) 0.5

( | ) 0.9 ( | ) 0.1

P G A P B A

P G A P B A

1 11

1 1 2 2

2 2

2

1 1 2 2

( ) ( ) (0.5)(0.65)( ) 0.902778

(0.5)(0.65) (0.1)(0.35)( ) ( ) ( ) ( )

( ) ( ) (0.1)(0.35)( ) 0.097222

(0.5)(0.65) (0.1)(0.35)( ) ( ) ( ) ( )

P B A P AP A B

P B A P A P B A P A

P B A P AP A B

P B A P A P B A P A

Note that 1 2( | ) ( | ) 1P A B P A B .

63

63

Example: Identifying the Source of a Defective Item

Three different machines M1, M2, and M3 were used for producing a large batch of similar

manufactured items. Suppose that 20% of the items were produced by machine M1, 30% by machine

M2, and 50% by machine M3. Suppose further that 1% of the items produced by machine M1 are

defective, that 2% of the items produced by machine M2 are defective, and that 3% of the items

produced by machine M3 are defective. Finally, suppose that 1 item is selected at random from the

entire batch and it is found to be defective. We shall determine the probability that this item was

produced by the 3 different machines.

Let iM be the event that the selected item was produced by machine iM ( 1,2,3i ), and let D be the

event that the selected item is defective. We must evaluate the conditional probability 1( | )P M D ,

2 3( | ), and ( | )P M D P M D .

The probability ( )iP M that an item selected at random from the entire batch was produced by machine

iM is as follows, for 1,2,3i .

1 2 3( ) 0.2, ( ) 0.3, ( ) 0.5P M P M P M .

Furthermore, the probability ( | )iP D M that an item produced by machine iM will be defective is:

1 2 3( | ) 1%, ( | ) 2%, ( | ) 3%P D M P D M P D M

By Bayes' Theorem, we have

1 1

1

1 1 2 2 3 3

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

(0.2)(0.01) 0.0020.087

(0.2)(0.01) (0.3)(0.02) (0.5)(0.03) 0.023

P M P D MP M D

P M P D M P M P D M P M P D M

2 2

2

1 1 2 2 3 3

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

(0.3)(0.02) 0.0060.261

(0.2)(0.01) (0.3)(0.02) (0.5)(0.03) 0.023

P M P D MP M D


3 3

3

1 1 2 2 3 3

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

(0.5)(0.03) 0.0150.652

(0.2)(0.01) (0.3)(0.02) (0.5)(0.03) 0.023

P M P D MP M D


64

64

Example: Quality Control

Suppose that when a machine is adjusted properly, 50% of the items produced by it are of high

quality and the other 50% are of medium quality. Suppose, however, that the machine is improperly

adjusted during 10% of the time and that, under these conditions, 25% of it are of high quality and 75%

of it are of medium quality.

Suppose that 5 items produced by the machine at a certain time is selected at random and

inspected. If 4 of them are of high quality and 1 item is of medium quality, what is the probability that

the machine was adjusted properly?

( ) 0.9, ( ) 0.1P AP P NP

( | ) 0.5, ( | ) 0.5, ( | ) 0.25, ( | ) 0.75P H AP P M AP P H NP P M NP

1st item (H):

( ) ( ) (0.5)(0.9)( ) 0.947368

(0.5)(0.9) (0.25)(0.1)( ) ( ) ( ) ( )

P H AP P APP AP H

P H AP P AP P H NP P NP

2nd

item (H):

( ) ( ) (0.5)(0.947368)( ) 0.972973

(0.5)(0.947368) (0.25)(0.052632)( ) ( ) ( ) ( )

P H AP P APP AP H


3rd

item (H):

( ) ( ) (0.5)(0.972973)( ) 0.986301

(0.5)(0.972973) (0.25)(0.027027)( ) ( ) ( ) ( )

P H AP P APP AP H


4th

item (H):

( ) ( ) (0.5)(0.986301)( ) 0.993103

(0.5)(0.986301) (0.25)(0.013699)( ) ( ) ( ) ( )

P H AP P APP AP H


5th

item (M):

( ) ( ) (0.5)(0.993103)( ) 0.989691

(0.5)(0.993103) (0.75)(0.006897)( ) ( ) ( ) ( )

P M AP P APP AP M

P M AP P AP P M NP P NP

65

65

Alternatively

1st item (M):

( ) ( ) (0.5)(0.9)( ) 0.857143

(0.5)(0.9) (0.75)(0.1)( ) ( ) ( ) ( )

P M AP P APP AP M

P M AP P AP P M NP P NP

2nd

item (H):

( ) ( ) (0.5)(0.857143)( ) 0.923077

(0.5)(0.857143) (0.25)(0.142857)( ) ( ) ( ) ( )

P H AP P APP AP H


3rd

item (H):

( ) ( ) (0.5)(0.923077)( ) 0.9600

(0.5)(0.923077) (0.25)(0.076923)( ) ( ) ( ) ( )

P H AP P APP AP H


4th

item (H):

( ) ( ) (0.5)(0.96)( ) 0.979592

(0.5)(0.96) (0.25)(0.04)( ) ( ) ( ) ( )

P H AP P APP AP H


5th

item (H):

( ) ( ) (0.5)(0.979592)( ) 0.989691

(0.5)(0.979592) (0.25)(0.020408)( ) ( ) ( ) ( )

P H AP P APP AP H


66

66

Choice under uncertainty

Motivation: Almost every choice involves elements of uncertainty.

Gamble 1: If a coin comes up with a head, you win $100; if with a tail, you lose $1. (0.5) ( $100) (0.5) ( $1)

with probability and

Expected value (0.5)(100) (0.5)( 1) $49.5

Gamble 2: If a coin comes up with a head, you win $1000; if with a tail, you lose $10. (0.5) ( $1000) (0.5) ( $10)

Expected value (0.5)(1000) (0.5)( 10) $495

Gamble 3: If a coin comes up with a head, you win $20,000; if with a tail, you lose $10,000. (0.5) ( $20000) (0.5) ( $10000)

Expected value (0.5)(20000) (0.5)( 10000) $5000

In real world situations, most people will accept Gambles #1 and #2, but not #3. Given that the expected

value of Gamble #3 is larger than #2 and #3, we can conclude that people make their decisions not

according to expected value. An economic theory of choice among uncertain alternatives is established

to explain why. The formal theory was established by John von Neumann and Oskar Morgenstern. Its

central premise is that people choose the alternative that has the highest expected utility. The expected

utility of a gamble is the sum of the expected value of the utilities of each of its possible outcomes.

Example: (A consumer who accepts Gamble #1 and #2, but not #3)

Let 0, 10000U M M

Gamble 1: 0.5 10100 0.5 9999 (0.5)(100.498) (0.5)(99.99) 100.244 100EU

Gamble 2: 0.5 11000 0.5 9990 (0.5)(104.88) (0.5)(99.95) 102.415 100EU

Gamble 3: 0.5 30000 0.5 0 (0.5)(173.205) (0.5)(0) 86.6025 100 (10000)EU U

Definition: A fair gamble is a gamble of which the expected value is equal to 0.

Definition: A favorable gamble is a gamble of which the expected value is larger than 0.

Definition: An unfavorable gamble is a gamble of which the expected value is less than 0.

Definition: A risk-averse individual is an individual which will reject a fair gamble.

Definition: A risk-lover is an individual which will accept a fair gamble.

Definition: A risk-neutral individual is an individual which is indifference between accepting or

rejecting a fair gamble.

[ ( ) (1 ) ( )] ( ) (1 ) ( )A B A BEU p W p W pU W p U W

67

67

Consider a gamble ( ) (1 ) ( ) where EV ( ) (1 ) ( ) 0p G p L p G p L (fair gamble)

i) Consider an individual with a strictly concave utility function and an initial wealth W .

( ) (1 ) ( ) [ ( ) (1 )( )] ( )EU pU W G p U W L U p W G p W L U W

the individual will reject the fair gamblerisk averse individual

W L W W G

ii) Consider an individual with a strictly convex utility function and an initial wealth W .


the individual will accept the fair gamblerisk-seeker or risk-lover

W L W W G

( ) (1 ) ( )EU pU W G p U W L

( )U W

( )U W G

( )U W L

( ) (1 ) ( )EU pU W G p U W L

( )U W

( )U W G ( )U W L

a risk-averse individual is one who has a strictly concave utility function

a risk-lover is one who has a strictly convex utility function

68

68

iii) Consider an individual with a linear utility function and an initial wealth W .


risk-neutral

W L W W G

( ) ( ) (1 ) ( )U W pU W G p U W L

( )U W G ( )U W L

69

69

Numerical example: Risk-averse individual (Decreasing MU)

U(M) U(15000)

U(10000)

U(5000)

M 5000 MCE 10000 15000

Example: 0 10000; 0.5 winning $5000, 1 0.5 losing $5000; ( )M p p U M M

(0.5) 15000 (0.5) 5000 (0.5)(122.47) (0.5)(70.71) 96.59EU

What is the maximum amount of money the individual is willing to pay in order to avoid facing

the gamble?

Let X be the amount.

10000 96.59 10000 9329.63 670.37X X X

10000 670.37 $9329.63CEM

CEM : Certainty Equivalent Income

For a risk-averse individual,

i) 0CEM W for a fair gamble, and

ii) 0( ) ( ) (gamble)CEM E W E W E for other gambles

(0.5) (5000) (0.5) (15000)EU U U

70

70

Numerical example: Risk-seeker/risk-lover (Increasing MU)

U(M)

U(15000)

U(10000)

U(5000)

M 5000 10000 MCE 15000


2 2(0.5)(15000) (0.5)(5000) (0.5)(225,000,000) (0.5)(25,000,000) 125,000,000EU

What is the minimum amount of money the individual is willing to accept in order to give up

facing the gamble?


2(10000 ) 125,000,000 10000 11180.34 1180.34X X X

10000 1180.34 $11180.34CEM

For a risk lover,

i) 0CEM W for a fair gamble.


(0.5) (5000) 0.5 (15000)EU U U

71

71

Numerical example: Risk-neutral (Constant MU)

U(M) U(15000)

U(10000)

U(5000)

M

5000 10000 15000


(0.5)(15000) (0.5)(5000) 10,000 (10000)EU U

For a risk-neutral person,

i) 0CEM W for a fair gamble.


0.5 (5000) 0.5 (15000)EU U U

72

72

Example: Insuring against bad outcomesReservation price of insurance

Suppose a risk-averse individual faces the prospect of a loss. What is the most a consumer would pay

for insurance against the loss?

Let 0 $10000 and ( )W U M M

no accident: loses $0 0.90p

accident: loses $5000 0.10p

(0.9) 10000 0 (0.1) 10000 5000 (0.9)(100) (0.1)(70.711) 97.071EU

Let X be the most a consumer is willing to pay for the insurance against a loss.

Assume full coverage.

no accident: final outcome 10000 0 10000 0.90X X p

accident: final outcome 10000 5000 5000 10000 0.10X X p

10000 97.071 10000 9422.78 $577.22X X X

$10000 $577.22 $9422.78CEM

Note: After an insurance policy is purchased, the outcome will be the same regardless whether there is

an accident or not, i.e. the individual no longer faces uncertainty.

If $577.22I is the actual price of the insurance policy, then the consumer will buy the policy and get a consumer surplus $577.22 I .

Remark: In the above example, we assume that the insurance company provides full coverage to a

risk-averse consumer which is not seen in the real world. There is always coinsurance or

deductible. This is due to the problem of Moral Hazard: the tendency whereby people

spend less effort protecting those goods that are insured against theft or damage. For

example, many people whose cars are insured will not take great care to prevent them

from being damaged or stolen.

Remark: In insurance, there is also the problem of Adverse Selection: it is the process in which

undesirable members of a population of buyers or sellers are more likely to participate in a voluntary exchange. For example, those who know that they are not good drivers

will buy insurance.

Because of this problem, insurance company usually tries to obtain as much information from a

potential policy holder as possible. For example, smokers have to pay higher life insurance

premium and younger drivers have to pay higher auto insurance premium.

73

73

Moral Hazard vs Adverse Selection

1. In a model with adverse selection problem, one player knows some piece of information or

type, but the other player does not. This type is determined by nature, and cannot be affected by

either player. Adverse selection problems involve a hidden type.

2. In a model with moral hazard problem, one player can take an action which is not observed by

the other player. Moral hazard problems involve a hidden action.

Example:

Consider a college hiring a new professor. The professor may spend 20 hours preparing one hour of

lecture or may not prepare at all for the lecture. In this case, the professor takes an action which is

hidden from the university. This is a moral hazard problem.

Example:

Consider the situation between a landlord and a tenant. Before the tenant moves into the apartment, she

knows whether she is a good tenant or a poor tenant. The tenant can take action which determines

whether she is a good tenant or bad tenant, but this action is hidden from the landlord. The

informational asymmetry between the two people involves a hidden action. This is a moral hazard

problem.

There is another moral hazard problem in this relationship. The landlord may be a very good or a very

bad landlord. The landlord has control over whether he is a good landlord or a bad landlord, but before

the tenant moves into the apartment, the tenant does not know whether the landlord is good or bad. In

this case, the hidden action is taken by the landlord.

Example:

Consider the case for a minivan salesman. The salesman knows whether the minivan is a high- or low-

quality vehicle. But whether the minivan is a lemon or not was decided by nature, not by the salesman.

The type of the minivan is known by the salesman but is not known by the consumer. This is an

adverse selection problem, since it involves a hidden type.

74

74

Certainty equivalent adjustment factor ( )


(0.5) 15000 (0.5) 5000 (0.5)(122.47) (0.5)(70.71) 96.59EU


the gamble?


10000 96.59 10000 9329.63 670.37X X X

10000 670.37 $9329.63CEM

9329.630.9329

Expected wealth 10000

CEM

Example: 0 10000; 0.5 winning $5000, 1 0.5 losing $5000; ( ) lnM p p U M M

(0.5) ln(15000) (0.5) ln(5000) (0.5)(9.6158) (0.5)(8.5172) 9.0665EU


the gamble?


9.0665ln(10000 ) 9.0665 10000 8660.26 $1339.74X X e X

10000 1339.74 $8660.26CEM

8660.260.866026

Expected wealth 10000

CEM

Note: Other things being equal, a smaller CEM will lead to a smaller . Hence, more risk-averse

individuals, who have smaller CEM , will have smaller certainty equivalent adjustment factor

.

75

75

Example:

Suppose a risk-averse consumer has an initial wealth of $10,000 (including a car which worth $5000 and

some jewelry which worth $4000). She estimates that her chance of getting into a car accident is 20%.

Assume that a car accident will destroy her car completely. To insure against the potential loss of her

car, she is willing to pay a maximum of $3000 for insurance premium.

Suppose the chance of her jewelry being stolen is 5%. What is the maximum amount of money she is

willing to pay for a theft insurance policy for the jewelry?

Solution:

For the car accident:

with an accident 20% $10000 $5000 $5000W without an accident 80% $10000W Expected wealth (0.2)($5000) (0.8)($10000) $9000

10000 3000 7000 7

Expected wealth 9000 9000 9

CEM

For the loss of jewelry:

stolen 5% $10000 $4000 $6000W not stolen 95% $10000W Expected wealth (0.05)($6000) (0.95)($10000) $9800

7 10000

9 Expected wealth 9800

710000 ( )(9800) $2377.78

9

CEM X

X

76

76

Example:

Mary was suing a fast food restaurant for spilling hot coffee on her. She retained a law firm to file a

lawsuit in state court for $500,000 in damages. Prior to filing suit, the attorney estimated legal, expert

witness, and other litigation costs to be $2,000 for a fully litigated case, for which Mary had a 2%

chance of receiving a favorable judgment. Assume that a favorable judgment will award 100% of the

damage sought, whereas an unfavorable judgment will result in her receiving $0 damages award.

Assume that $5000 is the most Mary would be willing to pay to sue restaurant.

Calculate Marys certainty equivalent adjustment factor ( ) for this investment project. cost of litigation

$5000 $50000.5

Expected wealth ($500,000)(2%) ($0)(98%) $10000

CEM

Now assume that after Mary goes into court, incurring $1000 in litigation costs, a damaging testimony

by an expert witness dramatically changes the outlook of the case in the fast food restaurants favor. Given that Mary now only has a 1% chance of obtaining a favorable judgment of the case, if the fast

food restaurant wants to settlement the case, how much out-of court settlement offer will Mary be

willing to accept?

0.5Expected wealth ($500,000)(1%) ($0)(99%) 5000

(0.5)(5000) $2500

CE CE CE

CE

M M M

M

Since she will save $1000 of litigation cost if she accepts the out-of-court settlement, so long as the fast

food restaurant pays her $1500, she will settle the case.

77

77

Taylor Expansion 2"( *)( *) ( )( *)

( ) ( *) '( *)( *) ... ( , *) 2! !

n nf x x x f a x xf x f x f x x x a x x

n

2"( ) ( )

( ) ( ) '( ) ... ( , ) 2! !

n nf x h f a hf x h f x f x h a x x h

n

Example: Expand ( ) around * 0xf x e x

( ) (0) 1

'( ) '(0) 1

''( ) ''(0) 1

...

x

x

x

f x e f

f x e f

f x e f

2 3 42 3''(0) '''(0)( ) (0) '(0)( 0) ( 0) ( 0) ... 1 ...

2! 3! 2! 3! 4!

f f x x xf x f f x x x x

Example: Expand ( ) ln(1 ) around * 0f x x x .

1

2

( ) ln(1 ) (0) 0

'( ) (1 ) '(0) 1

''( ) 1(1 )

f x x f

f x x f

f x x

3 3

4 4 4

''(0) 1

'''( ) ( 2)(1 ) =(2!)(1 ) '''(0) 2!

( ) 2( 3)(1 ) = (3!)(1 )

f

f x x x f

f x x x

4

5 5 5 5

(0) 3!

( ) (2)(3)( 4)(1 ) (4!)(1 ) (0) 4!

...

f

f x x x f

2 3

2 3 4 5 2 3 4 5 6

''(0) '''(0)( ) (0) '(0)( 0) ( 0) ( 0) ...

2! 3!

( 1) (2!) ( 3!) (4!)( ) 0 ... ...

2! 3! 4! 5! 2 3 4 5 6

f ff x f f x x x

x x x x x x x x xf x x x

78

78

Definition: Cost of risk ( )C expected value of a gambleMCE

0 1

( ) (1 ) ( )EU pU M p U M

M0 MCE EValue M1

cost of risk

1

( ) ( ) 1, 2,..., state of natureN

i i

i

U M C PU M i N

LHS: ( ) ( ) '( )U M C U M U M C

RHS: 2

1 1

1( ) [ ( ) '( )( ) "( )( ) ]

2

N N

i i i i i

i i

PU M P U M U M M M U M M M

2

1 1 1

2

1 1 1

1( ) '( )( ) "( )( ) ]

2

1( ) '( ) ( ) "( ) ( )

2

1( ) '( )(0) "( ) ( )

2

1( ) "( ) ( )

2

N N N

i i i i i

i i i

N N N

i i i i i

i i i

PU M PU M M M PU M M M

U M P U M P M M U M P M M

U M U M U M Var M

U M U M Var M

LHS and RHS 1

( ) '( ) ( ) "( ) ( )2

U M U M C U M U M Var M

1'( ) "( ) ( )

2

( ) "( )[ ]

2 '( )

U M C U M Var M

Var M U MC

U M

The cost of risk is proportional to the variance of M and "( )

'( )

U M

U M .

Note: The formula is not valid for large variances.

79

79

( ) "( )[ ]

2 '( )

Var M U MC

U M

2

( ) "( )[ ]

'( )2

C Var M U M M

M U MM

"( )

'( )

U M

U M : degree of absolute risk aversion

"( )

'( )

U M M

U M : degree of relative risk aversion

Example:

i) 2

1

"( ) 1( ) ln

'( )

U M MU M M

U M MM

ii)

3

21

2

1

2

1

"( ) 14( )'( ) 21

2

MU M

U M MU M M

M

Since 1 1

a person with the utility function ( ) ln2

U M MM M

is more risk averse than a person

with the utility function ( )U M M

80

80

Definition: Cost of risk ( )C expected value of a gambleMCE

( )U M

cost of risk

cost of risk

0 1

( ) (1 ) ( )EU pU M p U M

EU

M0' M0 MCE' MCE EValue M1 M1'

( )U M

cost of risk

cost of risk

0 1

( ) (1 ) ( )EU pU M p U M

M0 MCE' MCE EValue M1

81

81

Risk-pooling and risk-sharing

Example: (risk-pooling)

Suppose there is n individuals, all of whom face the same gamble. Each persons income is a random

variable y with a given distribution, including mean and variance, which is the same for all individuals.

Assume the distribution of each persons income is independent of the distribution of each other persons income.

Suppose these individuals get together and pool their incomes, agreeing that each shall draw the average

income out of the pool.

1

2

... ( ) ( )( ) ( )n

y y y Var y Var yVar nVar n

n n nn

1 ... ( )lim ( ) 0 Cost of risk 0 as nn

y y Var yVar n

n n

Example:

Suppose a student is choosing between 2 colleges.

College A:

1

2

Great job: lifetime income $1,000,000 0.5

Poor job: lifetime income $360,000 0.5

P

P

College B: Adequate job: lifetime income $670,000 1.0P

Let ( )U M M

( ) 0.5 1000000 0.5 360000 800 and ( ) 670000 818.54EU A EU B

Since ( ) ( )EU B EU A , the student will choose college B.

Now suppose 1000 students who are facing this gamble sign a contract agree to attend College A together and share their lifetime income with each other.

According to the Law of Large Number,

(500)(1,000,000) (500)(360,000)lifetime income $680,000

1000

.

In this case, the students will choose College A over College B.

Theorem: The Law of Large Number is a statistical law that says that if an event happens

independently with probability p in each of N instances, the proportion of cases in which the event

occurs approaches p as N .

82

82

Example: Joint ownership of business enterprise

When a new business starts, it may be successful and it may fail.

Let 0 $10000 and ( )W U M M .

Succeed: earns $20000 : 0.5

Fail: loses $10000 : 0.5

succeed

fail

P

P

0

(business) (0.5) 10000 20000 0.5 10000 10000 0.5 30000 0.5 0 86.603

( ) 10000 100

EU

EU W

Because 0(business) ( )EU EU W , therefore the business will not be pursued.

Suppose 100 persons form a joint ownership.

Succeed: earns $200 : 0.5

Fail: loses $100 : 0.5

succeed

fail

P

P

(business) (0.5) 10000 200 0.5 10000 100 0.5 10200 0.5 9900 100.247EU

Because 0(business) ( )EU EU W , therefore the business will be pursued.

83

83

Optimal choice under uncertainty

Example:

Suppose a person has $M of money. If she puts the money in the bank, she can get a return of 10% over a period. If she buys an asset X, she has 50% of chance to get 50% of return and 50% chance to suffer

from a loss of 15%. Determine the portfolio of the consumer if her utility function is lnU W .

Let x be the amount of money put into the risky asset.

max {(0.5) [110%( ) (150%)( )] (0.5) [110%( ) (85%)]}

{(0.5) (1.1 0.4 ) (0.5) (1.1 0.25 )}

(0.5) (1.1 0.4 ) (0.5) (1.1 0.25 )

0.5ln(1.1 0.4

xEU EU M x x M x x

EU M x M x

U M x U M x

M x

) 0.5ln(1.1 0.25 )M x

FOC:

(0.5)( 0.4) (0.5)( 0.25)0

1.1 0.4 1.1 0.25

dEU

dx M x M x

(0.5)( 0.4) (0.5)(0.25)

1.1 0.4 1.1 0.25

( 0.2) (0.125)

1.1 0.4 1.1 0.25

(0.2)(1.1 0.25 ) (0.125)(1.1 0.4 )

0.22 0.05 0.1375 0.05

0.22 0.1375 0.05 0.05

0.0825 0.1

0.08250.825 8

0.1

M x M x

M x M x

M x M x

M x M x

M M x x

M x

x

M

2.5%

84

84

Example:

Suppose a person has $M of money. If she puts the money in the bank, she can get a return of 10% over a period. If she buys an asset X, she has 50% of chance to get 50% of return and 50% chance to suffer

from a loss of 80%. Determine the portfolio of the consumer if her utility function is lnU W .

Let x be the amount of money put into the risky asset.

max {(0.5) [110%( ) (150%)( )] (0.5) [110%( ) (20%)]}

{(0.5) (1.1 0.4 ) (0.5) (1.1 0.9 )}

(0.5) (1.1 0.4 ) (0.5) (1.1 0.9 )

0.5ln(1.1 0.4 ) 0.5

xEU EU M x x M x x

EU M x M x

U M x U M x

M x

ln(1.1 0.9 )M x

FOC: (0.5)( 0.4) (0.5)( 0.9)

01.1 0.4 1.1 0.9

dEU

dx M x M x

(0.5)( 0.4) (0.5)(0.9) ( 0.2) (0.45)

1.1 0.4 1.1 0.9 1.1 0.4 1.1 0.9

(0.2)(1.1 0.9 ) (0.45)(1.1 0.4 )

0.22 0.18 0.495 0.18 0.22 0.495 0.18 0.18

0.2750.275 0.36 0.764

0.36

M x M x M x M x

M x M x

M x M x M M x x

xM x

M

The person is not going to put any money in the risky asset.

Using the Kuhn-Tucker Technique

max 0.5ln(1.1 0.4 ) 0.5ln(1.1 0.9 ) s.t. 0x

EU M x M x x M

L 1 20.5ln(1.1 0.4 ) 0.5ln(1.1 0.9 ) ( )M x M x w x w M x

Kuhn-Tucker conditions:

1 2

(0.5)(0.4) (0.5)( 0.9)0 (1)

1.1 0.4 1.1 0.9xL w w

M x M x

1 1

2 2

0, 0, 0 (2)

1 0, 0, ( ) 0 (3)

x w xw

x w M x w

Case 1: 1 0w

1 2(2) : 0 0 (3) 0w x w

1 1

0.5 0.4 0.5 ( 0.9) 0.25(1) : 0 0 0

1.1 1.1 1.1x w w

M M M

(consistent)

85

85

Case 2: 2 0w

2 1(3) : 0 0 (2) 0w M x x M w

2

2

2

0.5 0.4 0.5 ( 0.9)(1) : 1 0

1.1 0.4 1.1 0.9

0.2 0.45 0.2(0.2 ) 0.45(1.5 ) 0.04 0.675

1.5 0.2 0.3 0.3

0.6350

0.3

x wM M M M

M M M Mw

M M M M

w

(inconsistent)

Hence 0x is the solution.

86

86

Example: Demand for insurance

Suppose a risk-averse consumer initially has monetary value $W . There is a probability p that he will

lose an amount $L . The consumer can however purchase insurance that will compensate him in the

event that he incurs the loss. The premium he has to pay for C of coverage is C . How much coverage will the consumer purchase?

max { [ ] (1 ) [ ]} [ ] (1 ) [ ]C

EU p W L C C p W C pU W L C C p U W C

FOC:

(1 ) '[ ] (1 )( ) '[ ] 0dEU

p U W L C C p U W CdC

'[ ] (1 ) (1)

'[ ] (1 )

U W L C C p

U W C p

If the event occurs, the insurance company receives $( )C C .

If the event does not occur, the insurance company receives $ C .

In a competitive market, the expected profit should be equal to 0 (assuming no administrative cost),

therefore ( ) ( ) (1 ) (1 ) (1 ) 0 (1 ) (1 )E profit p C C p C p C p C p p .

'[ ] (1 )(1) 1

'[ ] (1 )

'[ ] '[ ]

U W L C C p

U W C p

U W L C C U W C

Since " 0

*

U W L C C W C

C L

Note that the optimal amount of compensation is not related to W .

If the expected profit has to be positive (in order to cover administrative cost), then

( ) ( ) (1 ) 0E profit p C C p C

(1 ) (1 ) or (1 ) (1 )p C p C p p

'[ ] (1 )(1) 1

'[ ] (1 )

'[ ] '[ ]

U W L C C p

U W C p

U W L C C U W C

Since " 0 *U W L C C W C C L

87

87

Example: Demand for insurance with Moral Hazard

Now suppose the consumer has some control over the probability of the event in question.

Let X denotes the level of care exercised by the consumer. We assume that the probability that the

accident will occur is a function of the level of care, i.e. ( ) where '( ) 0p p X p X . (i.e. being more

careful will lead to a small probability of having an accident)

However, there is cost involved in being careful. Let us assume that this cost can be represented in

terms of utility so that ( , ) ( ) ( ) where '( ) 0U W X U W H X H X .

,max { ( ) [ ] [1 ( )] [ ]} ( )

( ) [ ] [1 ( )] [ ]} ( )

X CEU p X W L C C p X W C H X

p X U W L C C p X U W C H X

FOC:

( )(1 ) '[ ] [1 ( )] '[ ] 0 (1)

'( ) [ ] '( ) [ ] '( ) 0 (2)

EUp X U W L C C p X U W C

C

EUp X U W L C C p X U W C H X

x

Case 1: *C L (full coverage) (2) '( ) [ ] '( ) [ ] '( ) 0

'( ) [ ] '( ) [ ] '( ) 0

'( ) 0 (contradiction)


p X U W C p X U W C H X

H x

Case 2: *C L (over-insured) (2) '( ) [ ] '( ) [ ] '( ) 0

'( ){ [ ] [ ]} '( )

( ) ( ) ( ) (inconsistent)


p X U W L C C U W C H X

Since both case 1 and case 2 are impossible, we must have *C L (with deductible).

Case 3: *C L (with deductible) * * * *

'[ * *] '[ *]

C L W L C C W C

U W L C C U W C

'[ ] (1 )1

'[ ] (1 )

(1 ) (1 )

(1 ) (1 )

( ) (1 ) (1 ) 0

U W L C C p

U W C p

p p

p C p C

E profit p C p C

88

88

Allocation of wealth to risky assets

Suppose an individual has initial wealth W , which is to be divided between a safe asset whose rate of return is 0 and a risky asset whose rate of return is a random variable R .

If he or she invests $x in the risky asset, final wealth will be ( ) (1 )W x x R W xR .

Problem:

max [ ( )] ( ) ( )x

E U W xR U W xR f R dR

FOC:

( ) ( )[ ( )] ( )

( )

d U W xR f R dRdE U W xR dU W xR

f R dRdx dx dx

'( ) ( ) [ '( ) ] 0 (1)U W xR Rf R dR E U W xR R

SOC:

22

2

2

'( ) ( )[ ( )]

''( ) ( )

[ "( ) ] 0 ( " 0 for a risk-averse individual) (2)

d U W xR Rf R dRdE U W xR

U W xR R f R dRdx dx

E U W xR R U

The FOC defines the amount of investment in the risky asset as a function of initial wealth, *( ).x x W

(1) [ '( ) ] { '[ *( ) ] } '[ *( ) ] ( ) 0 (3)E U W xR R E U W x W R R U W x W R Rf R dR

Differentiating (3) with respect to W , we have

2

'[ *( ) ] ( ){ '[ *( ) ] } '[ *( ) ]

( )

"[ *( ) ][1 *'( ) ] ( ) { "[ *( ) ][1 *'( ) ] }

{ "[ * ] } { "[ * ] *'( )} 0

d U W x W R Rf R dRdE U W x W R R dU W x W R

Rf R dRdW dW dW

U W x W R x W R Rf R dR E U W x W R x W R R

E U W x R R E U W x R R x W

2

[ "( ) ]* '( ) (4)

[ "( ) ]

E U W xR Rx W

E U W xR R

Since the denominator is negative, { *'( )} { [ "( ) ]}sign x W sign E U W xR R

89

89

When absolute risk aversion is decreasing, we have

"( ) "( ) for 0 (5)

'( ) '( )

"( ) "( ) for 0 (6)

'( ) '( )

U W xR U WR

U W xR U W

U W xR U WR

U W xR U W

Multiply (5) by '( ) [ 0 0]

Multiply (6) by '( ) [ 0 0]

"( )"( ) '( )

'( )

U W xR R R

U W xR R R

U WU W xR R U W xR R R

U W

Taking expectation on both sides"( )

{ "( ) } { '( ) } 0 by FOC'( )

U WE U W xR R E U W xR R

U W

2

[ "( ) ](4) * '( ) 0

[ "( ) ]

E U W xR Rx W

E U W xR R

If absolute risk aversion is decreasing in wealth, a rise in wealth will raise the amount of

investment in risky assets.

90

90

Numerical example:

First order condition of the asset allocation problem: [ '( ) ] 0E U W xR R

Let i) 21( ) '( )

2U W W W U W W

ii) 1

( ) for f R a R bb a

(uniform distribution)

[ '( ) ] 0 '( ) ( ) 0

b

a

E U W xR R U W xR Rf R dR

2

2 3

2 3 2 3

2 2 3 3

2 2

23 3

1[ ( )] ( ) 0

1[( ) ] 0

[( ) ] 02 3

[( ) ] [( ) ] 02 3 2 3

( )( ) ( ) 0

2 3

( )( )

3( )( )( )2*2 ( )(

( )3

b

a

b

a

b

a

W xR R dRb a

W R xR dRb a

R RW x

b b a aW x W x

Wb a b a x

Wb a

W b a b ax

b a b ab ab a

2 2 2

3( )( )0 if is not too large

) 2 ( )

W b aW

b ab a

2 2 2 2

* 3 ( ) 3( )0

2 ( ) 2( )

x b a b a

W b ab a b ab a

Higher wealth leads to a smaller x !!

Note that "

'

U

U W W

"as

'

UW

U

absolute risk aversion is increasing in W

91

91

Example: Allocation among different assets

Two assets: 1 2 and e e

1

: inital wealth

: the portion of wealth allocated to

W

x e

1 2max { [ (1 ) ]}x

E U Wxe W x e

FOC:

1 2 1 2 1 2 1 2{ } { '[ (1 ) ]( )} { '[ (1 ) ]( )} 0dEU dU

E E U Wxe W x e We We E U Wxe W x e e edx dx

Let 21( ) [ '( ) ]

2U W aW bW U W a bW quadratic utility function

1 2 1 2

2 2

1 2 1 1 2 1 2 2

2 2

1 2 1 1 2 1 2 2

2

1 2 1 1 2 1 2

{[ [ (1 ) ]( )} 0

{ (1 ) (1 ) } 0

( ) ( ) ( ) ( ) (1 ) ( ) (1 ) ( ) 0

( ) ( ) ( ) ( ) ( ) (

E a b Wxe W x e e e

E ae ae bWxe bWxe e bW x e e bW x e

aE e aE e bWxE e bWxE e e bW x E e e bW x E e

aE e aE e bWxE e bWxE e e bWE e e bWxE

2 2

1 2 2 2

2 2 2

1 2 1 2 2 1 1 2 1 2 2

2

1 2 1 2 2

2 2

1 1 2 2

) ( ) ( ) 0

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )]

( ) ( ) ( ) ( )

( ) 2 ( ) ( )

e e bWE e bWxE e

aE e aE e bWE e e bWE e x bWE e bWE e e bWE e e bWE e

aE e aE e bWE e e bWE ex

bWE e bWE e e bWE e

Note that

1 1

2 2

2 22 2 2

1 1 1 1 1 1 1 1 1 1

2 22 2

1 1 1 1 1

( ) : mean of

( ) : mean of

( ) var( ) [ ( )] var( ) [ ( )] { 2 ( ) [ ( )] }

( ) 2[ ( )] [ ( )] = ( ) [ (

E e e

E e e

E e e E e e E e E e E e e E e E e

E e E e E e E e E e

2

2 2

2 2 2

1 2 1 2 1 2 1 2 1 1 2 2

)]

( ) var( ) [ ( )]

( ) cov( , ) ( ) ( ) cov( , ) [( ( ))( ( ))]

E e e E e

E e e e e E e E e e e E e E e e E e

2

1 2 1 2 2

2 2

1 1 2 2

2

1 2 1 2 1 2 2 2

2 2

1 1 1 2 1 2 2 2

( ) ( ) ( ) ( )

( ) 2 ( ) ( )

( ) ( ) [cov( , ) ( ) ( )] {var( ) [ ( )] }

{var( ) [ ( )] } 2 [cov( , ) ( ) ( )] {var( ) [ ( )] }

aE e aE e bWE e e bWE ex

bWE e bWE e e bWE e

aE e aE e bW e e E e E e bW e E e

bW e E e bW e e E e E e BW e E e

92

92

Mean-Variance Analysis and Portfolio Selection

Traditionally the investors are assumed to care about only the mean and variance of his income.

This is true only under the following situations.

i) The utility function is quadratic

2

2 2

( ) , 0

[ ( )] ( ) ( ) ( ) {[ ( )] var( )}

U M a bM cM b c

E U M a bE M cE M a bE M c E M M

Counter-example:

If 2 3( ) , , 0U M a bM cM dM b c d

then 2 3 2 3[ ( )] ( ) ( ) ( ) ( ) {[ ( )] var( )} ( )E U M a bE M cE M dE M a bE M c E M M dE M

i.e. the investor will care more than the mean and variance of the risky asset

Drawbacks of having a quadratic utility function:

1) When M is very large, marginal utility 0 .

2) If there is only 1 risky asset and one safe one, the investor will hold less of the risky asset when

he becomes richer.

[In the real world, the rich tends to hold riskier and higher yielding portfolios than the poor.]

Proof:

Suppose there are 2 assets each costing $1 per unit and the income yielded by each (per unit) is as

follows ($): Asset A: for sureAM

Asset B: with mean and variance var( )B B BM M M

The individual has wealth $W and spends x on the risky asset and W x on the safe asset.

Individual's problem:

max { [( ) ]}A Bx

E U W x M xM

Assume a quadratic utility function, we have

2 2[ ( )] ( ) ( ) ( ) {[ ( )] var( )}E U M a bE M cE M a bE M c E M M

2Given that ( ) ( ) ( ) and var( ) var( )A B A B BM W x M xM E M W x M xM M x M

93

93

2 2

2 2

max { [( ) ]} [( ) ] [( ) ] var( )

( ) [( ) ] var( )

A B A B A B Bx

A B A B B

E U W x M xM a b W x M xM c W x M xM cx M

a b W x M bxM c W x M xM cx M

FOC:

2 [( ) ]( ) 2 var( ) 0A B A B A B BdEU

bM bM c W x M xM M M cx Mdx

Since each unit of the 2 assets costs $1 each and B is risky, hence 0B AM M .

( ) 2 [( ) ]( ) 2 var( ) 0

( ) 2 ( ) 2 ( ) 2 ( ) 2 var( ) 0

2 ( ) ( )

2 { ( ) ( ) var( )}

2 ( )

2 { ( ) (

B A A B B A B

B A A B A A B A B B A B

A B A B A

A B A B B A B

A B A

A B A B

dEUb M M c W x M xM M M cx M

dx

b M M cWM M M cxM M M xcM M M cx M

cWM M M b M Mx

c M M M M M M M

cM M Mdx

dW c M M M M M

2( )

0) var( )} ( ) var( )

A B A

B A B B A B

M M M

M M M M M

ii) Each security in the portfolio has a normal distribution

Suppose the ith

security 2

( , ) and i iN cov( , )i j ijX X

1 1Let be the number of shares of security , then ( , )

Hence an investor only needs to know the mean and variance of the portfolio!!

n n

i i i i i i j ij

i i i j

a i a X N a a a

94

94

Proposition: Mutual Fund Separation Theorem

If there is a riskless asset and conditions i) and/or ii) are satisfied and all investors have the same

subjective probability distributions, then investors will differ in the amount of wealth they hold in risky

assets, but they will not differ in the fraction of that risky wealth devoted to each particular risky asset.

Proof:

Suppose there are only 2 available securities and x z . Each is perfectly divisible, and 1 unit of each will

cost the investor all his wealth. Assume 2 2( , ) ( , )X X Z ZX N Z N

[We can expect the market to ensure that the riskier security has a higher mean return.]

Now suppose the investor puts half his wealth into each ( and x z ).

1 1

2 2X Z

2 2 2 2 21 1 1 1 ( ) ( ) ( ) ( ) 2 ( , ) ( ) ( ) 2( )( )2 2 2 2 2 2 2 2 2 2 2

X Z XZ

x z x z x z x zVar Var Var Var Cov

Provided and x z are not perfectly and positively correlated, we have 1XZXZX Z

2 2 2 2 2 2 2 2 2 21 1 1 1 1 1 1 1 1 1 ( ) ( ) 2( )( ) ( ) ( ) 2( )( ) ( )2 2 2 2 2 2 2 2 2 2

1 1

2 2

X Z XZ X Z X Z X Z

X Z

z Z

1

( )2

Z X

mean-variance frontier: 2 assets

X X

Z

1 1

2 2X X Z

The above exercise shows that the mean-variance frontier is a concave curve.

95

95

S

P

mean-variance frontier

R

O

Now suppose there is a riskless asset R yielding an income OR for sure if all the investors wealth is invested in that asset.

Now, let the investor combines the riskless asset R with a risky asset. Clearly, any risky asset along the

line RS will be dominated by the line RP . (same , lower ; same , higher )

Together with the ICs, the optimal asset allocation is determined. Notice that no matter where the ICs

are, the point P is in the same position, i.e. the portfolio of risky assets remains the same.

Z P

E

X E

mean-variance frontier

R

O

This person

will put

more money

in the risk-

free asset.

Among the money

these 2 individuals put

in the risky assets, the

ratio of asset X and

asset Z will be the

same.

This person will put less money in

the risk-free asset.

96

96

Increasing Risk (Rothschild and Stiglitz)

Question: What is a random variable Y more variable, riskier, more uncertain than another random variable X ?

4 possible answers:

1. is equal to plus noiseY X

( | ) 0 for all d

Y X Z E Z X X

has the distribution as

Suppose X is a lottery which pays off with probability , 1i i ia p p . Then Y is a lottery ticket which pays off with probability , 1i i ib p p where ib is either a sure payoff of ia or a lottery ticket which has an expected value equals to ia .

Note that and X Y have the same mean.

2. Every risk-averse individual prefers to ( and have the same mean)X Y X Y

( ) ( ) concave EU X EU Y U

i.e. ( ) ( ) ( ) ( )U X f X dX U Y g Y dY

3. has more weight in the tails than Y X

f(x)

s(x)

mean preserving spread (MPS

g(x)=f(x)+s(x)

97

97

4. has a greater variance than Y X

( ) ( ), ( ) ( )Var Y Var X E X E Y

Definition: A partial ordering p on a set is a binary, transitive, reflexive and anti-symmetric relation

if and p pA B B A A B .

Definition of a : is less risky than if d

X Y Y X Z

Definition of I :

1

iff where is MPSN

I i i

i

F G G F S S

.

Definition of U : iff for any bounded concave function, ( ) ( ) ( ) ( )UF G U X dF X U X dG X

Theorem: , , are partial orderingsa I U .

Theorem: I a UF G F G F G

Theorem: is a V complete ordering but it is not equivalent to the other 3.

Definition: A relation P is a binary relation.

Definition: A relation P is transitive if and P P PA B B C A C .

Definition: A relation P is reflective if PA A .

Definition: A relation P is antisymmetric if and P PA B B A A B .

Definition: A partial ordering P on a set is a binary, transitive, reflexive and antisymmetric

relation.

Definition: A complete ordering P on a set is a partial ordering where given any

and , either or P PA B A B B A .

98

98

Effect of increasing risk on the optimal solution

max ( , ) ( , ) ( ) : uncertaintyEU X U X dF X X

FOC:

( , ) ( ) ( , )( ) ( , ) ( ) ( , ) 0

d U X dF XdEU dU XdF X U X dF X EU X

d d d

Let * be the unique soltution to the FOC .

Assume that in the neighborhood of * , is montonic decreasing in U .

Proposition: If ( , ) is a concave function of ,U X X an increase in riskiness will decrease * .

Proof:

a) Risk ( , ) (because is a concave function)EU X U

b) In order to restore the FOC, then has to be lowered as is montonic decreasing in U .

Proposition: If ( , ) is a convex function of ,U X X an increase in riskiness will increase * .

99

99

Example: Savings and uncertainty

Initial wealth: 0W

Each $ saved today yields the random return e .

1 2 0 0max [ ( ) (1 ) ( )] [(1 ) ] (1 ) ( ) : discounts

E U C U C U s W EU sW e rate

FOC:

0 0 0 0'[(1 ) ] (1 ) [ '( ) ]dEU

W U s W E U sW e eWds

0 0 0 0'[(1 ) ] (1 ) [ '( ) ] 0W U s W E U sW e e W

0 0'[(1 ) ] (1 ) [ '( ) ]U s W E U sW e e

Let 2( ) '( )

2

bU C aC C U C a bC

0 0'[(1 ) ] (1 ) [ '( ) ]U s W E U sW e e

0 0[(1 ) ] (1 ) {[ ( )] }a b s W E a b sW e e 2

0 0[(1 ) ] (1 )[ ( ) ( )]a b s W aE e bsW E e 2

0 0 0

2

0 0 0

0

2

0 0

(1 ) ( ) (1 ) ( )

(1 ) ( ) [(1 ) ( ) ]

(1 ) ( )

[(1 ) ( ) ]

a bW bsW aE e bW E e s

a bW aE e bW E e bW s

a bW aE es

bW E e bW

2risk ( )E e s

The above result is not general.

Whether 0risk or depends on whether '( ) is a concave function or s eU sW e a convex function

in e .

Note:

i) 0'( )eU sW e is a concave function in e 2

0 0 0 0

2

2

0 0 0 0 0 0

[ '( )] [ '( ) "( )( )]

"( ) ( ) '''( ) "( )

d eU sW e d U sW e eU sW e sW

de de

U sW e sW e sW U sW e U sW e sW

0 0 0 0

0

[2 "( ) '''( )]

[2 "( ) '''( )] 0 (1)

sW U sW e esW U sW e

sW U C CU C

ii) 0'( )eU sW e is a convex function [2 "( ) '''( )] 0 (2)U C CU C

00 0

( , )( , ) ;

( )'( )

U XU X X e s

U sW eU sW e W e

s

100

100

iii) A non-positive third derivative is sufficient for increasing risk leading to decrease savings.

iv) Arrow-Pratt concept of relative risk aversion: "

'

U CR

U

2

'( " ''') " "'

( ')

U U CU U CUR

U

1 " "' [( " ''') ]

' '

U CUR U CU

U U

1 " 1' [ "(1 ) '''] [ "(1 ) ''']

' ' '

CUR U CU U R CU

U U U

sign( ') sign{ [ "(1 ) ''']}R U R CU

If is non-increasing ( ' 0) and 1 (2) holdsR R R .

If R is non-decreasing ( ' 0R ) and 1R (1) holds .

Example:

1( ) (1 ) ( 0, 1)aU W a W a a

2 2 1'( ) (1 ) "( ) (1 )a aU W a W U W a a W 2 1

2

" [ (1 ) ] constant relative risk aversion

' (1 )

a

a

U C a a W WR a

U a W

If 1 ( 1) risk savings .

If 1 ( 1) risk savings .

a R

a R

' 0 "(1 ) "' 0

" 0 (risk averse), 1 0 "(1 ) "' 2 " "'

R U R CU

Since U R U R CU U CU

' 0 "(1 ) "' 0

" 0 (risk averse), 1 0 "(1 ) "' 2 " "'

R U R CU

Since U R U R CU U CU

101

101

Bayesian Economics

Varian, Hal (1986): Retail Pricing and Clearance Sales, American Economic Review, March, pp.14-32.

One-period model

Assume a risk-neutral firm which will encounter 1 and only 1 buyer whose reservation price is V .

Prior knowledge about V : ( ) density function, ( ) distribution functionf V F V

max [1 ( )] (0) ( )P

P F P F P

( )F V

1

1

( )F P

FOC: [ '( )] [1 ( )] ( ) [1 ( )] 0P F P F P Pf P F P

1 ( )

( )

F PP

f P

0

( )f V

1P

Example:

Let 1 0 1 uniform distribution on [0,1]

( )0 otherwise

Vf V

0 1

( )1 1

V VF V

V

FOC: 1

(1) (1 ) 0 1 2 0 *2

P P P P

Expected profit1 1 1

(1 )2 2 4

Probability that the

reservation price is P Probability that the reservation price is P

The consumer will buy the

good only if P V !!

102

102

Two-period model

Assume if the good is not sold during the period, the seller faces another buyer during the second period

who is identical to the one he met during the first period.

The firm now has 2 chances to sell the good.

Failing to sell the good in period 1 at price 1P provides the seller information about the reservation price

V of the consumer. In this case, this implies that 1V P . It is because if 1V P , the good would have

been sold.

Prior distribution: 1 0 1 uniform distribution on [0,1]

( )0 otherwise

Vf V

Posterior distribution:

(sold| ) ( ) ( | ) ( )( | sold) ( | )

( | ) ( ) ( | ) ( )(sold| ) ( )

(unsold| ) ( )( | unsold)

(unsold| ) ( )

C C

g V f V P B A P Af V P A B

P B A P A P B A P Ag z f z dz

g V f Vf V

g z f z dz

1

1

1 for (sold| )

0 for

P Vg V

P V

1

1

0 for (unsold| )

1 for

P Vg V

P V

1

11

1 1

1

(1) ( ) ( ) 1 for

1 ( ) 1(1) ( )( | sold)

0 for

P

f V f VP V

F P Pf z dzf V

V P

1

1

1

1 10

0 for

(1) ( ) ( ) 1( | unsold) for ( )(1) ( )

P

P V

f V f Vf V V PF P Pf z dz

or

1 1

1

1 1

1

( ) ( ) for

1 ( ) 1( | sold)

0 for

F V F P V PP V

F P PF V

V P

1

1

1 1

1 for

( | unsold) ( )= for

( )

P V

F V F V VV P

F P P

1 :P price charged in period 1

(0,1)V U

103

103

The choice of 1P affects the problem in 2 ways:

1. It affects the probability of a sale in period 1.

2. It determines what the firm can infer from no sale. For example, if 1 1P , then the fact that the

good was not sold in the first period is uninformative, because the firm was certain that 1V at the outset.

Similarly, 1 0P is certain to result in a sale during the first period so that there is no learning

resulted.

1 2

1 1 1 2 2 2,

max [1 ( )] ( )[1 ( )] (1)P P

P F P F P F P P

It is instructive to think of this as a dynamic programming and to consider the firms optimal strategy in

period 2, given that the good is not sold in period 1 at the price 1P .

probability

that the good

is sold in

period 1

probability that

the good is not

sold in period 1

probability that the good is sold

in period 2

104

104

Firms problem in period 2

2

2 2 2max [1 ( )]P

P F P

1 2

2 2 21 2

1

0 if

( ) ( ) if

( )

P P

f P f PP P

F P

1 2

2 2 2

1 2

1

1 if

( ) ( ) if

( )

P P

F P F PP P

F P

FOC:

2 2 2 2 2 2 2 2 2 2[ '( )] [1 ( )] ( ) [1 ( )] 0P F P F P P f P F P

2 2 2 1 1 22 2

1 1 1 2 2

( ) ( ) ( ) ( ) ( ) ( )1 0 [1 ]

( ) ( ) ( ) ( ) ( )

f P F P F P F P F P F PP P

F P F P F P f P f P

Since 2 1 2 1 20 ( ) ( ) 0P F P F P P P a clearance sale is being held

If 1 2 12 2 1 2(0,1) 21 2

P P PV U P P P P

Note:

1. For any given 1P , if the good is not sold during the first period, then the seller can rule out the

probability that 1V P .

2. The distribution that the seller uses in period 2 is 1(0, )U P , so the second periods problem is

equivalent to the one facing a firm with only 1 period to sell and a prior distribution of uniform

distribution. The solution to that problem is to select 122

PP .

Note that the firm will face this

problem only when the good is not

sold in period 1!

105

105

12 (1)

2

PP

1

1 12 2

1 1 1 1

1 1 1 1 1 1 1 1 1

1 1

32 2max [1 ( )] ( ) 1 (1 ) 1 (1 )( ) 2 2 4 4P

P PFP P P P

P F P F P P P P P P PF P P

FOC:

1

1 1 2

1

3 2 11 0

2 3 2 3

PdP P P

dP

2Expected profit

3

Results:

1. Price falls over time.

2. Expected profit is higher.

Deficiencies of the above analysis: 2 important factors are not included.

1. The number of customers who come into the store during the first period.

Intuitively, if only a few customers arrive during the first period, the firm should be less certain

about its influence than if a large number of customers examine the good and reject it at price 1P .

2. Heterogeneity among consumers may be important.

If some consumers are willing to pay V , while others will pay an amount below the firms reservation price, then the problem is more complicated.

The good might not be sold not because the price was too high, but because that periods customers were all of wrong type.

1 2

1 1 1 2 2 2,

max [1 ( )] ( )[1 ( )] (1)P P

P F P F P F P P

106

106

The Risk-Bearing Premium

Suppose there are only 2 states of the world, 1,2s . In the state-claim space, the axes indicate

amounts of the contingent income claims 1 2 and CC .

In a simplified 2-state world,

1 1 2 2 1 2( ) ( ) where 1 EU U q U C q U C q q

is the probability that state happensiq i

For a given level of U , this equation describes an entire set of 1 2 and CC combinations that are equally

preferred, so this is the equation of an indifference curve.

2C certainty

line

1 2

( , )C C

LL line

1 2

( , )C C

45

1C

For any indifference curve, as it crosses the certainty line, it has slope 1

2

q

q as 1 2C C .

A risk-adverse individual will prefer an income with certainty

The dashed line (LL line) through the point 1 2

( , )C C shows all the 1 2( , )C C having the same expected

income as the point 1 2

( , )C C : 1 1 2 2 1 1 2 2

q C q C q C q C c

Along the dashed line, the maximum utility is at the point 1 2

( , )C C as the slope of the dashed line is 1

2

q

q

which is the same as the indifference curve at the point 1 2

( , )C C .

Thus the certainty of having income c is preferred to any other 1 2( , )C C combination whose

mathematical expectation is c .

MRSslope of IC 2 1 1

1 2 2constant

'( )

'( )U

dC q U C

dC q U C

1 1

ln( ) ( )dMRS d MRS

sign signdC dC

1 1 2 2

1 1

1 2 2

1 2 1

1 2 1 1

1 2 2 2

[ln ln '( ) ln ln '( )ln

"( ) "( )

'( ) '( )

"( ) "( ) '( )0 if " 0

'( ) '( ) '( )

d q U C q U Cd MRS

dC dC

U C U C dC

U C U C dC

U C U C q U CU

U C U C q U C

the certainty line connects all the points such that 1 2C C

107

107

Contingent Claim Markets

Suppose an individual is a price-taker in a market where contingent income claims 1 2( , )C C --each of

which offers income if and only if the corresponding state obtains -- can be exchanged in accordance

with the price ratio 1 2P P . The budget line NN goes through the endowment point 1 2( , )C C .

The equation for the budget line NN: 1 1 2 2 1 1 2 2PC P C PC P C

Expected utility is maximized at the point *C . At *C , 1 1 1

2 2 2

'( )

'( )

q U C P

q U C P .

2C certainty line

NN line (slope= 1

2

P

P )

1 2

( , )C C

*C 45

1C

The quantities of state-claims income held are such that

ratio of the probability-weighted marginal utilitiesratio of the state-claim prices

In a statesS situation, we have 1 1 2 2

1 2

'( )'( ) '( )... n n

n

q U Cq U C q U C

P P P

Assuming an interior solution, at the individual's risk-bearing optimum the expected (probability-

weighted) marginal utility per dollar of income will be equal in each and every state.

108

108

Remark:

1. Starting from a certainty position, a risk-averse individual would never accept any gamble at fair

odds.

2C certainty

line

(1 2*, *C C )

LL line

45 1 2

( , )C C

1C

2. If his initial endowment were not a certainty position, when offered the opportunity to transact at

a price ratio corresponding to fair odds he would want to "insure" by moving to a certainty

position-as indicated by the solution C along the fair market line LL.

2C certainty

line

1 2

( , )C C

LL line

1 2

( , )C C

45

1C

Thus an individual with an uncertain endowment might accept a "gamble" in the form of a risky contract

offering contingent income in one state in exchange for income in another. But he would accept only

very particular risky contracts, those that offset the riskiness of his endowed gamble.

109

109

3. If the market price did not represent fair odds, as in the case of market line NN, whether or not

he starts from a certainty endowment the individual would accept some risk; his tangency

optimum would lie off the 45 line at a point like *C in the direction of the favorable odds.

2C certainty line

NN line (slope= 1

2

P

P )

1 2

( , )C C

*C 45

1C

2C certainty

line

NN line

*C

45 1 2

( , )C C

1C

2C certainty

line

*C

NN line

45 1 2

( , )C C

1C

is on the certainty line

NN line is steeper than LL line

is on the certainty line

NN line is flatter than LL line

110

110

Wealth effect

2C D

B

45 certainty line

E1 **C

E2 E3

*C

C 'C

market

lines

1C

In E3, all the points are closer to the

45 line than does *C . The individual reduces his absolute

consumption risk.

In E1 and E2, his "tolerance" for

absolute risk must be increasing with

wealth.

A solution along the dividing line

*C B would represent constant tolerance for absolute risk.

increasing

tolerance for

absolute risk decreasing tolerance

for absolute risk

111

111

Insurance Market

(Rothschild, Michael and Joseph Stiglitz: Equilibrium in Competitive Insurance Markets: An Essay on

the Economics of Imperfect Information)

Consider an individual who will have an income of W if he is lucky enough to avoid accident. In the

event an accident occurs, his income will be only W L . The individual can insure himself against this accident by paying to an insurance company a premium

1 , in return he will be paid 2 if an accident

occurs. Without insurance his income in the 2 states, "accident", "no accident", was ( , )W W L ; with

insurance it is now 1 2( , )W W L , where 2 2 1 .

The vector 1 2( , ) completely describes the insurance contract.

Demand for insurance contracts

1 2 1 2( , , ) (1 ) ( ) ( )V q W W q U W qU W : probability of an accidentq

1 : his income if there is no accidentW

2 : his income if there is accidentW

A contract is worth 1 2

( , ) ( , , )V q V q W W L .

From all the contracts the individual is offered, he chooses the one that maximizes ( , )V q . Since he

always has the option of buying no insurance, an individual will purchase a contract only if ( , ) ( , 0) ( , , )V q V q V q W W L .

We assume that persons are identical in all respects except their probability of having an accident and

that they are risk-averse.

Supply of Insurance Contracts

We assume that companies are risk-neutral, that they are concerned with expected profits, so that

contract when sold to an individual who has a probability of incurring an accident of q , is worth

1 2 1 1 2( , ) (1 ) ( )q q q q

Any contract with non-negative expected profit will be offered.

Definition of Equilibrium

Equilibrium in a competitive insurance market is a set of contracts such that, when customers choose

contracts to maximize expected utility,

(i) no contract in the equilibrium set makes negative expected profits; and

(ii) there is no contract outside the equilibrium set that, if offered, will make a non-negative profit.

112

112

Equilibrium with identical customers

2W

45

*

E

1W

The point 1 2

( , )W W is the typical customers uninsured state. Purchasing the insurance policy

1 2( , ) moves the individual from to the point 1 1 2 2 ( , )W W .

Free entry and perfect competition will ensure that policies bought in competitive equilibrium make zero

expected profits, so that if is purchased, 1 2( , ) (1 ) 0q q q .

The set of all policies that break even is illustrated by the line E in the Figure, which is the fair-odds line. The equilibrium policy * maximizes the individuals expected utility and just breaks even.

* satisfies the 2 conditions of equilibrium: (i) it breaks even;

(ii) selling any contract preferred to it will bring insurance companies expected loss.

Since customers are risk-averse, the point * is located at the

51-122

Documents

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