EC2101 Week5

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WEEK 5
CONSUMER SURPLUS
RISK
April 4, 2012
Consumer Surplus
2

Valuation
 The
maximum amount the consumer is willing to pay
for a product


Consumer surplus (CS) for an individual consumer
is the difference between the consumer’s valuation
for a product and the cost of purchasing the
product
Aggregate consumer surplus is the sum of all
individual consumer surplus
EC2101 Semester 1 AY 2012/2013
WEEK 5
Example: Individual Demand for Houses
3
Prefers to buy
0

Prefers not to buy
V
P
Consumer’s demand for the product is
ìï
1
Q=í
ïî 0
if
if
P£V
P>V
EC2101 Semester 1 AY 2012/2013
WEEK 5
Example: Market Demand for Houses
4

Suppose there are 4 consumers with different
valuations for houses
 Consumer
1 values the house at 0.9 million
 Consumer 2 values the house at 0.7 million
 Consumer 3 values the house at 0.5 million
 Consumer 4 values the house at 0.1 million

Suppose each consumer buys at most 1 house
EC2101 Semester 1 AY 2012/2013
WEEK 5
5
Example: Market Demand for Houses in
Graph
P
0.9
0.7
0.5
0.1
0
1
2
3
EC2101 Semester 1 AY 2012/2013
4
WEEK 5
Q
Example: Calculating Consumer Surplus
6
P
Suppose market price is 0.3 million
CS= 0.9 - 0.3+ 0.7- 0.3+ 0.5- 0.3 =1.2
0.9
0.7
0.5
P = 0.3
0.1
0
1
2
3
EC2101 Semester 1 AY 2012/2013
4
WEEK 5
Q
Notes on Market Demand for Houses
7



Represents consumers’ valuation for the product in
decreasing order
As the number of consumers with different
valuations increase, market demand curve will
become “smoother”
With a very large number of consumers, market
demand can be modeled as a smooth curve
EC2101 Semester 1 AY 2012/2013
WEEK 5
8
Consumer Surplus with Smooth Demand
Curve
P
Suppose market demand is Q=100-P
100
The current market price is 20
CS=0.5*80*(100-20)=3,200
20
0
A
Total expenditure
Q
80
EC2101 Semester 1 AY 2012/2013
WEEK 5
Notes on Consumer Surplus
9

CS measures how much economic value is
captured by consumers through their market
transactions

CS is the area below the demand curve and above
the price
EC2101 Semester 1 AY 2012/2013
WEEK 5
How does CS change with price?
10
P
100
30
B
A
20
0
70
Q
80
EC2101 Semester 1 AY 2012/2013
WEEK 5
Network Externality
11


We have network externality if the amount of good
demanded by one consumer depends on other
consumers’ demand
Positive network externality
 Quantity
demanded by a typical consumer increases
as demand from other people grows

Negative network externality

Quantity demanded by a typical consumer decreases
as demand from other people grows
EC2101 Semester 1 AY 2012/2013
WEEK 5
Positive Network Externality
12
P
10
8
D200
A
B
C
D
0
200250
Q
400
EC2101 Semester 1 AY 2012/2013
WEEK 5
Negative Network Externality
13
P
1500
1000
A
C
B
D200
0
D
200 260
Q
400
EC2101 Semester 1 AY 2012/2013
WEEK 5
14
Part 1
Uncertainty and Risk
EC2101 Semester 1 AY 2012/2013
WEEK 5
What is risk?
15

If consumers do no know the consequences of a
decision for sure, we have uncertainty
 E.g.


stock price could go up or down
If consequences of a decision are uncertain, we
face risk
Our goal
 How
to describe/model risk?
 How to manage risk?
EC2101 Semester 1 AY 2012/2013
WEEK 5
How to describe risk?
16

Suppose you have $1000
 Option
 No
1: put it in your savings account
risk
 Option
2: buy a lottery
 Involves risk

All possible outcomes for option 2
 Outcome
1: win
 Outcome 2: lose
EC2101 Semester 1 AY 2012/2013
WEEK 5
Probability
17

How likely is each outcome?
 Outcome
1: 10%
 Outcome 2: 90%


Probability measures the likelihood that each
outcome occurs
Where does probability come from?
 Objective:
past experience
 Subjective: personal judgment
EC2101 Semester 1 AY 2012/2013
WEEK 5
Expected Value
18

Payoff for each outcome of option 2
 Outcome
1 (win): 10000
 Outcome 2 (lose): 0


Expected value is the probability-weighted average
of the payoffs associated with all possible
outcomes
Expected value of buying a lottery is
0.9*0+0.1*10000=1000
EC2101 Semester 1 AY 2012/2013
WEEK 5
Calculating Expected Value
19




Suppose there are n possible outcomes
Outcome i’s payoff is Xi
Outcome i’s probability is Pri
Expected value is
n
E(X) = å Pri Xi = Pr1 X1 + Pr2 X2 +
where
+ Prn Xn
i=1
n
å Pr = Pr + Pr +
i
1
2
+ Prn =1
i=1
EC2101 Semester 1 AY 2012/2013
WEEK 5
Degree of Risk
20


Suppose you have option 3: buy a stock
Outcomes, probabilities, and payoffs
 Price
increases, probability 50%, payoff $1500
 Price decreases, probability 50%, payoff $500


Expected value
0.5*1500+0.5*500=1000
Are option 2 and option 3 equally risky?
EC2101 Semester 1 AY 2012/2013
WEEK 5
Deviation
21


Deviation is the difference between actual payoff
and expected value
Option 2 (lottery)
 If
win, deviation=10000-1000=9000
 If lose, deviation=0-1000=-1000

Option 3 (stock)
 If
price increases, deviation=1500-1000=500
 If price decreases, deviation=500-1000=-500
EC2101 Semester 1 AY 2012/2013
WEEK 5
How to measure degree of risk?
22


We use standard deviation to measure the degree
of risk
Option 2
0.1´ (9000)2 + 0.9 ´ (-1000)2 = 3000

Option 3
0.5´ (500)2 + 0.5´ (-500)2 = 500

Option 2 is more risky
 Same
expected value
 Higher standard deviation
EC2101 Semester 1 AY 2012/2013
WEEK 5
Alternative Way: Variance
23



We can also use variance to measure the degree of
risk
Variance = standard deviation squared
Option 2
0.1´(9000)2 + 0.9 ´(-1000)2 = 9000000

Option 3
0.5´(500)2 + 0.5´(-500)2 = 250000
EC2101 Semester 1 AY 2012/2013
WEEK 5
24
Part 2
Preferences towards Risk
EC2101 Semester 1 AY 2012/2013
WEEK 5
Which one will you choose?
25

Option 1 (bank)
 Expected
value: 1000
 Standard deviation: 0

Option 2 (lottery)
 Expected
value: 1000
 Standard deviation: 3000
EC2101 Semester 1 AY 2012/2013
WEEK 5
Evaluating Risky Outcomes Using Utility
Function
26


Consider a utility function of income U(I)
Consumer’s utility from option 1 is
 U(1000)

Consumer’s utility from option 2 is
 U(0)
if consumer loses
 U(10000) if consumer wins

Consumer’s expected utility from option 2 is
 0.9U(0)+0.1U(10000)
EC2101 Semester 1 AY 2012/2013
WEEK 5
Expected Utility
27






Expected utility is the probability-weighted sum of
all utilities associated with all possible outcomes
Suppose there are n possible outcomes
Consumer’s utility function is U(X)
Outcome i’s payoff is Xi
Outcome i’s probability is Pri
Expected utility is
n
E(U) = å Pri U(Xi ) = Pr1 U(X1 ) + Pr2 U(X2 ) +
i=1
EC2101 Semester 1 AY 2012/2013
WEEK 5
+ Prn U(Xn )
Expected Utility vs. Expected Value
28


Suppose U(I)=I2
Consider a risky income: 50% chance 5, 50%
chance 15
 Expected
value=0.5*5+0.5*15=10
 Expected utility
0.5U(5)+0.5U(15)=0.5*25+0.5*225=125
EC2101 Semester 1 AY 2012/2013
WEEK 5
Which one gives you the highest EU?
29

Option 1 (bank)
 Expected

Option 2 (lottery)
 Expected

utility: U(1000)
utility: 0.9U(0)+0.1U(10000)
It depends on your preference (utility function)
EC2101 Semester 1 AY 2012/2013
WEEK 5
Risk Averse
30

A consumer is risk averse if the consumer prefers a
certain income to a risky income with the same
expected value
 The
consumer’s utility of a certain income is higher
than the expected utility of a risky income with the
same expected value

Risk averse consumer will choose option 1
 Both
options have the same expected value: 1000
 Option 1 has no risk
EC2101 Semester 1 AY 2012/2013
WEEK 5
Utility Function of Risk Averse Consumer
31
U(I )
U(1000) > 0.9U(0)+ 0.1U(10000)
B
U(10000)
U(I )
Expected utility
A
U(1000)
0.9U(0)+ 0.1U(10000)
C
U(0)
1000
Expected value
10000
EC2101 Semester 1 AY 2012/2013
WEEK 5
I
Expected Utility in Graph
32
U(I )
aU(0) + (1- a)U(10000), 0 £ a £1
a= 0
B
U(10000)
D
A
0.5U(0) + 0.5U(10000)
a = 0.5
U(1000)
0.9U(0)+ 0.1U(10000)
C
U(0)
1000
a =1
U(I )
5000
10000
EC2101 Semester 1 AY 2012/2013
WEEK 5
I
What about two options with different
expected value?
33
U(I )
A risk averse consumer may or may not prefer a
certain income to a risky income with higher
expected value
B
U(10000)
E
A
U(1000)
U(I )
0.5U(0) + 0.5U(10000)
D
0.84U(0)+ 0.16U(10000)
0.9U(0)+ 0.1U(10000)
C
U(0)
1000 1600
5000
10000
EC2101 Semester 1 AY 2012/2013
WEEK 5
I
Risk Neutral
34

A consumer is risk neutral if the consumer is
indifferent between a certain income and a risky
income with the same expected value
 The
consumer’s utility of a certain income is the same
as the expected utility of a risky income with the same
expected value

Risk neutral consumer think both options are the
same
EC2101 Semester 1 AY 2012/2013
WEEK 5
Utility Function of Risk Neutral Consumer
35
U(I )
U(1000) = 0.9U(0) + 0.1U(10000)
U(I )
U(10000)
U(1000)
U(0)
B
A
1000
0.9U(0)+ 0.1U(10000)
10000
EC2101 Semester 1 AY 2012/2013
WEEK 5
I
Risk Loving
36

A consumer is risk loving if the consumer prefers a
risky income to a certain income with the same
expected value
 The
consumer’s utility of a certain income is lower
than the expected utility of a risky income with the
same expected value

Risk neutral consumer will choose option 2
EC2101 Semester 1 AY 2012/2013
WEEK 5
Utility Function of Risk Loving Consumer
37
U(I )
U(1000) < 0.9U(0)+ 0.1U(10000)
B
U(10000)
U(I )
C
U(1000)
U(0)
1000
0.9U(0)+ 0.1U(10000)
A
I
10000
EC2101 Semester 1 AY 2012/2013
WEEK 5
38
Part 3
How to Manage Risk?
EC2101 Semester 1 AY 2012/2013
WEEK 5
What do risk averse consumers do with
risk?
39


Most consumers are risk averse
Risk averse consumer may still prefer a risky option
 Consumer
will bear risk if there is enough reward to
compensate for the risk


When to bear risk and when to eliminate risk?
How to eliminate risk?
EC2101 Semester 1 AY 2012/2013
WEEK 5
Risk Premium
40

Risk premium is the maximum amount of money a
risk averse consumer is willing to pay to avoid
taking a risk
 Difference between
the expected value of a risky
option and a certain income to make the consumer
indifferent between the two
EC2101 Semester 1 AY 2012/2013
WEEK 5
Risk Premium in Graph
41
U(I )
Risk premium=60-45=15
U(I )
U(80)
U(45)
U(10)
B
D
C
2
5
U(10) + U(80)
7
7
A
10
45
60
I
80
EC2101 Semester 1 AY 2012/2013
WEEK 5
Calculating Risk Premium
42

Suppose the utility function of a consumer is
U(I ) = I

Consumer can buy a risky asset
 Payoff=900
with probability 60%
 Payoff=400 with probability 40%

What is the risk premium associated with this
asset?
EC2101 Semester 1 AY 2012/2013
WEEK 5
Calculating Risk Premium Cont’
43

Expected utility of the asset
0.6 900 + 0.4 400 = 26

To get the same utility, consumer needs a certain
income of
I = 26 2 = 676

Expected value of the asset
0.6 ´ 900 + 0.4´ 400 = 700

Risk premium is 700-676=24
EC2101 Semester 1 AY 2012/2013
WEEK 5
Understanding Risk Premium
44


The consumer is willing to pay 24 to avoid risk
Compared to a certain income of 676
 Consumer
will choose a risky asset only when the
expected value of the risky asset is at least 24 higher
than the certain income
EC2101 Semester 1 AY 2012/2013
WEEK 5
Managing Risk: Insurance
45



Suppose you own a car
Your annual income is $50000
You face risk
 With
probability 95% nothing happens to your car,
your income is $50000
 With probability 5% you have an accident and it will
cost you $10000 (your income is $40000)

Expected value
0.95*50000+0.05*40000=$49500
EC2101 Semester 1 AY 2012/2013
WEEK 5
Actuarially Fair Insurance
46

Consider the following insurance policy
 Insurance premium
 Coverage


$500
$10000
Expected payout
0.95*0+0.05*10000=$500
This insurance is actuarially fair because insurance
premium is equal to the expected payout
EC2101 Semester 1 AY 2012/2013
WEEK 5
Full Insurance
47

If you buy the insurance
 No
accident, income=50000-500=$49500
 Accident, income=50000-500-10000+10000=$49500

The insurance provides full coverage
 You
are fully insured
 Insurance eliminates all risk

Any risk averse consumer should buy an actuarially
fair insurance that provides full coverage
EC2101 Semester 1 AY 2012/2013
WEEK 5
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