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Chapter 5
Uncertainty and Consumer Behavior
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©2005 Pearson Education, Inc. Chapter 5 2
Q: Value of Stock
Investment in offshore drilling exploration:
Two outcomes are possibleSuccess – the stock price increases from
$30 to $40/shareFailure – the stock price falls from $30 to
$20/share
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©2005 Pearson Education, Inc. Chapter 5 3
Expected Value
failure) of )(valuePr(failure
success) of )(valuePr(success EV
)($20/share43)($40/share41 EV
$25/share EV
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©2005 Pearson Education, Inc. Chapter 5 4
Expected Value
In general, for n possible outcomes:Possible outcomes having payoffs X1, X2, …
Xn
Probabilities of each outcome is given by Pr1, Pr2, … Prn
nn2211 XPr...XPrXPr E(X)
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©2005 Pearson Education, Inc. Chapter 5 5
Describing Risk
Variability
The extent to which possible outcomes of an uncertain event may differ
How much variation exists in the possible choice
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©2005 Pearson Education, Inc. Chapter 5 6
Q: Which Job?
Suppose you are choosing between two part-time sales jobs that have the same expected income ($1,500)
The first job is based entirely on commission.The second is a (almost) salaried position.The third is a salaried position.
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©2005 Pearson Education, Inc. Chapter 5 7
There are two equally likely outcomes in the first job--$2,000 for a good sales job and $1,000 for a modestly successful one.
The second pays $1,510 most of the time (.99 probability), but you will earn $510 if the company goes out of business (.01 probability).
The third pays $1,500.
Variability
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©2005 Pearson Education, Inc. Chapter 5 8
Variability
Outcome 1 Outcome 2
Prob. Income Prob. Income
Job 1: Commission .5 2000 .5 1000
Job 2:
Fixed Salary.99 1510 .01 510
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©2005 Pearson Education, Inc. Chapter 5 9
Variability
Expected Income
EV1 = ½ $2,000 + ½ $1,000 = $1,500
EV2 = (0.99)$1,510+(0.01)$510 = $1,500
EV3 = $1,500
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©2005 Pearson Education, Inc. Chapter 5 10
Variability
Greater variability from expected values signals greater risk.
Variability comes from deviations in payoffsDifference between expected payoff and
actual payoff
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©2005 Pearson Education, Inc. Chapter 5 11
Variability – An Example
Deviations from Expected Income ($)
Outcome 1
Deviation Outcome 2
Deviation
Job 1 $2000 $500 $1000 -$500
Job 2 1510 10 510 -990
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©2005 Pearson Education, Inc. Chapter 5 12
Variability
Average deviations are always zero.
We must adjust for negative numbersWe can measure variability with standard
deviation
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©2005 Pearson Education, Inc. Chapter 5 13
Variability
The standard deviation is written:
2222
11 )(Pr)(Pr XEXXEX
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©2005 Pearson Education, Inc. Chapter 5 14
Standard Deviation – Example 1
Deviations from Expected Income ($)
Outcome 1
Deviation Outcome 2
Deviation
Job 1 $2000 $500 $1000 -$500
Job 2 1510 10 510 -990
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©2005 Pearson Education, Inc. Chapter 5 15
Standard Deviation – Example 1
Standard deviations of the two jobs are:
500000,250
)000,250($5.0)000,250($5.0
1
1
50.99900,9
)100,980($01.0)100($99.0
2
2
2222
11 )(Pr)(Pr XEXXEX
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©2005 Pearson Education, Inc. Chapter 5 16
Q: Revised
What if the outcome probabilities of two jobs have unequal probability of outcomes Job 1: greater spread & standard deviationYou will choose job 2 again
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©2005 Pearson Education, Inc. Chapter 5 17
Unequal Probability Outcomes
Job 1
Job 2
The distribution of payoffsassociated with Job 1 has a greater spread and standard
deviation than those with Job 2.
Income
0.1
$1000 $1500 $2000
0.2
Probability
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©2005 Pearson Education, Inc. Chapter 5 18
Q: Re-Revised
Suppose we add $200 to each payoff in Job 1 which makes the expected payoff = $1700.Job 1: expected income of $1,700 and a
standard deviation of $500.Job 2: expected income of $1,500 and a
standard deviation of $99.50
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©2005 Pearson Education, Inc. Chapter 5 19
Unequal Probability Outcomes
Job 1
Job 2
Income
0.1
$1000 $1500 $2000
0.2
Probability
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©2005 Pearson Education, Inc. Chapter 5 20
St. Petersburg Paradox
Game: Toss a coinPayoff:
If H at the 1st toss: 21 = 2If H at the 2nd toss: 22 = 4…If H at the nth toss: 2n
The fee for the game: 15What is the EV of the game?
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©2005 Pearson Education, Inc. Chapter 5 21
Preferences Toward Risk
Can expand evaluation of risky alternative by considering utility that is obtained by riskA consumer gets utility from incomePayoff measured in terms of utility
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©2005 Pearson Education, Inc. Chapter 5 22
Example
A person is earning $15,000 and receiving 13.5 units of utility from the job.
She is considering a new, but risky job.0.50 chance of $30,0000.50 chance of $10,000
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©2005 Pearson Education, Inc. Chapter 5 23
Example
Utility at $30,000 is 18Utility at $10,000 is 10Must compare utility from the risky job
with current utility of 13.5To evaluate the new job, we must
calculate the expected utility of the risky job
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©2005 Pearson Education, Inc. Chapter 5 24
Preferences Toward Risk
The expected utility of the risky option is the sum of the utilities associated with all her possible incomes weighted by the probability that each income will occur.
E(u) = (Prob. of Utility 1) *(Utility 1)
+ (Prob. of Utility 2)*(Utility 2)
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©2005 Pearson Education, Inc. Chapter 5 25
Example
The Expected Utilility is:E(u) = (1/2)u($10,000) + (1/2)u($30,000)
= 0.5(10) + 0.5(18)
= 14E(u) of new job is 14 which is greater than
the current utility of 13.5 and therefore preferred.
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©2005 Pearson Education, Inc.
Example
Chapter 5 26Chapter 5 26
Income ($1,000)
UtilityE
10
10
14
18
0 30
A
15 20
13.5
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©2005 Pearson Education, Inc.
Example 2 Game: Toss a fair coin
Game 1H: +$100 T: -$0.5
Game 2H: +$200 T: -$100
Game 3H: +$20,000 T: -$10,000
Chapter 5 27
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©2005 Pearson Education, Inc.
Chapter 5 28
Expected Values
EV1 = (1/2)$100 + (1/2)(-$0.5) = $49.75
EV2 = (1/2)$200 + (1/2)(-$100) = $50
EV3 = (1/2)$20,000 + (1/2)(-$10,000)
= $5,000
Chapter 5 28
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©2005 Pearson Education, Inc.
Chapter 5 29
Expected UtilitySuppose U(M) = M1/2 , M = $10,000
U(M) = 10,0001/2 = 100EU1 = (1/2) 10,1001/2 + (1/2) 9,999.51/2 = 100.248EU2 = (1/2) 10,2001/2 + (1/2) 9,9001/2
= 100.247
EU3 = (1/2) 30,0001/2 + (1/2) 01/2
= 86.603Chapter 5 29
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©2005 Pearson Education, Inc.
Chapter 5 30
Example 3
Q: Your utility function is U(M) = M1/2 and your initial wealth is 36. Will you play a gamble in which you win 13 with probability of ½ and lose 11 with probability of ½ ?U(M) = 360.5 = 6EV = ½ (36+13) + ½ (36-11) = 37 EU = ½ (36+13)0.5 + ½ (36–11)0.5
= ½ 7 + ½ 5 = 6
Chapter 5 30
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©2005 Pearson Education, Inc. Chapter 5 31
Preferences Toward Risk
Risk AverseA person who prefers a certain given income
to a risky income with the same expected value.
The person has a diminishing marginal utility of income
Most common attitude towards risk
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©2005 Pearson Education, Inc.
Chapter 5 32
Risk Averse
Chapter 5 32Chapter 5 32
Income ($1,000)
Utility
The consumer is risk averse because she would prefer a certain income of
$20,000 to an uncertain expected income =
$20,000
E
10
10
14
18
0 30
A
20
16
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©2005 Pearson Education, Inc. Chapter 5 33
Preferences Toward Risk
A person is said to be risk neutral if they show no preference between a certain income, and an uncertain income with the same expected value.
Constant marginal utility of income
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©2005 Pearson Education, Inc. Chapter 5 34
Income ($1,000)10 20
Utility
0 30
6A
E
C
12
18
The consumer is riskneutral and is indifferentbetween certain eventsand uncertain events
with the same expected income.
Risk Neutral
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©2005 Pearson Education, Inc. Chapter 5 35
Income ($1,000)
Utility
0 10 20 30
The consumer is riskloving because she
would prefer the gamble to a certain income.
Risk Loving
3A
E
C8
18
F10.5
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©2005 Pearson Education, Inc. Chapter 5 36
Preferences Toward Risk
The risk premium is the maximum amount of money that a risk-averse person would pay to avoid taking a risk.
The risk premium depends on the risky alternatives the person faces.
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©2005 Pearson Education, Inc. Chapter 5 37
Risk Premium – Example
From the previous exampleA person has a .5 probability of earning
$30,000 and a .5 probability of earning $10,000
The expected income is $20,000 with expected utility of 14.
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©2005 Pearson Education, Inc. Chapter 5 38
Risk Premium – Example
Point F shows the risky scenario – the utility of 14 can also be obtained with certain income of $16,000
This person would be willing to pay up to $4000 (20 – 16) to avoid the risk of uncertain income.
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©2005 Pearson Education, Inc. Chapter 5 39
Income ($1,000)
Utility
0 10 16
Here, the risk premium is $4,000 because a certain income of $16,000 gives the person
the same expected utility as the
uncertain income with expected value
of $20,000.
10
18
30 40
14
A
CE
20
Risk Premium
F
Risk Premium – Example
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©2005 Pearson Education, Inc. Chapter 5 40
Reducing Risk
Consumers are generally risk averse and therefore want to reduce risk
Three ways consumers attempt to reduce risk are:
1. Diversification
2. Insurance
3. Obtaining more information