Demand for
Health Insurance
1
Which Investment will you pick
0.4
$5000
Expected Value
$2600
Choice 1
$1000
0.6
0.6
$5000
$2600
Choice 2
-$1000
0.4
2
Attitude towards risk
• In the absence of any objective criteria,
how an individual or organization deals
with uncertainty depends ultimately on
their attitude towards risk and whether
they are risk averse, risk neutral or a
risk taker.
3
Attitude towards risk
Risk averse • Someone who would prefer, for example,
the certainty of $1,000 rather than a 50%
probability of $3,000.
Risk neutral • Someone who is indifferent, for example,
between the certainty of $1,000 rather
than a 50% probability of $2,000.
Risk taker • Someone who would prefer, for example,
the 50% probability of $5,000 rather than
the certainty of $3,000.
4
Different Approaches to Risk:
•
•
•
•
5
Expected Value
Maximin
Maximax
Hurwicz alpha index rule
Payoff Matrix
2 Choices for investment:
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
6
Expected Value: sum of probabilities  Payoffs
EV1= 0.2 (-1000) + 0.7 (1000) + 0.1 (10,000) = 1500
EV2= 0.1 (0) + 0.6 (1000) + 0.3 (3000) = 1500
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
7
Maximin: Pessimistic/conservative risk attitude
1.
Minimum gain of each choice
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
8
Maximin: Pessimistic/conservative risk attitude
1.
Minimum gain of each choice
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
9
Maximin: Pessimistic/Conservative risk attitude
1.
2.
Minimum gain of each choice
Which is Maximum
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
10
Maximax: Optimistic Criterion
1.
Maximum gain of each choice
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
11
Maximax: Optimistic Criterion
1.
Maximum gain of each choice
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
12
Maximax: Optimistic Criterion
1.
2.
Maximum gain of each choice
Which is Maximum
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
13
Hurwicz alpha index rule:
• The Hurwicz alpha variable is a
measure of attitude to risk. It can range
from  = 1 (optimist) to  = 0
(pessimist). A value of  = 0.5 would
correspond to risk neutrality.
• The Hurwicz criterion = maximum value
x  + minimum value x (1 – )
14
Hurwicz alpha index rule:
1.
Weighted average of min and max for each choice.
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
For  = 0.5 :
The Hurwicz criterion for First Choice: 0.5  (10,000)+ 0.5  (-1000) = 4500
The Hurwicz criterion for Second Choice: 0.5  (3,000)+ 0.5
15  (0) = 1500
Hurwicz alpha index rule:
1.
2.
Weighted average of min and max for each choice.
Select the action with the maximum value
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
For  = 0.5 :
The Hurwicz criterion for First Choice: 0.5  (10,000)+ 0.5  (-1000) = 4500
The Hurwicz criterion for Second Choice: 0.5  (3,000)+ 0.5
16  (0) = 1500
Hurwicz alpha index rule:
1.
2.
Weighted average of min and max for each choice.
Select the action with the maximum value
Probability
Payoff
First Choice
0.2
0.7
0.1
-1000 1000
10,000
Probability
Payoff
Second Choice
0.1
0.6
0.3
0
1000
3,000
For  = 0.1 :
The Hurwicz criterion for First Choice: 0.1  (10,000)+ 0.9  (-1000) = 100
The Hurwicz criterion for Second Choice: 0.1  (3,000)+ 0.9
17 (0) = 300
Hurwicz alpha index rule:
• The maximin strategy equates to the
Hurwicz approach with a value of  = 0.
• The maximax strategy corresponds to 
= 1.
18
Insurance Logic
• The consumer pays insurer a premium
to cover medical expenses in coming
year.
– For any one consumer, the premium will
be higher or lower than medical expenses.
• But the insurer can pool or spread risk
among many insurees.
– The sum of premiums will exceed the sum
of medical expenses.
Characterizing Risk Aversion
• Recall the consumer maximizes utility,
with prices and income given.
– Utility = U (health, other goods)
– health = h (medical care)
• Insurance doesn’t guarantee health, but
provides $ to purchase health care.
• We assumed diminishing marginal
utility of “health” and “other goods.”
Diminishing marginal utility of income
Utility
Income
Utility of Different Income Levels
• Assume that we can assign a numerical
“utility value” to each income level.
• Also, assume that a healthy individual
earns $40,000 per year, but only $20,000
when ill.
Sick
Healthy
Income
Utility
$20,000
70
$40,000
90
Utility of Different Income Levels
Utility when
healthy
Utility
90
A
70
B
Utility when sick
$20,000
$40,000
Income
Probability of Being Healthy or Sick
• Individual doesn’t know whether she will
be sick or healthy.
• But she has a subjective probability of
each event.
– She has an expected value of her utility in the
coming year.
• Define: P0 = prob. of being healthy
P1 = prob. of being sick
P0 + P 1 = 1
Expected Utility as A Function of
Probability
• An individual’s subjective probability of
illness (P1) will depend on her health
stock, age, lifestyle, etc.
• Then without insurance, the individual’s
expected utility for next year is:
• E(U) = P0U($40,000) + P1U($20,000)
= P0•90 + P1•70
Expected Utility & Income As A
Point on AB Line
• For any given values of P0 and P1, E(U) will
be a point on the chord between A and B.
Utility
A
90
70
B
$20,000
$40,000
Income
Expected Utility & Income As A
Point on AB Line
• Assume the consumer sets P1=.20.
• Then if she does not purchase insurance:
E(U) = 0.8 • 90 + 0.2 • 70 = 86
E(Y) = 0.8 • 40,000 + 0.2 • 20,000 =
$36,000
• Without insurance, the consumer has an
expected loss of $4,000.
Expected Utility & Income As Point
C on AB Line
Utility
90
•
86
70
B•
•A
C
$20,000
$40,000
$36,000
Income
Certain Point on Income-Utility
Curve
• The consumer’s expected utility for next
year without insurance = 86 “utils.”
• Suppose that 86 “utils” also represents
utility from a certain income of $35,000.
– Then the consumer could pay an insurer
$5,000 to insure against the probability of
getting sick next year.
– Paying $5,000 to insurer leaves consumer
with 86 utils, which equals E(U) without
insurance.
Certain Point D on Income-Utility
Curve
Utility
90
D
86
70
•
B•
$20,000
$35,000
•
•A
C
$40,000
$36,000
Income
Price of Insurance and Loading Fee
• At most, the consumer is willing to pay
$5,000 in insurance premiums to cover
$4,000 in expected medical benefits.
• $1,000  loading fee  price of insurance
• Covers
– profits
– administrative expenses
– taxes
Thank You !
Any Question ?