Risky business
ECO61 Microeconomic Analysis
Udayan Roy
Fall 2008
Choices can have unpredictable results
• The possible outcomes of a risky decision are
called:
– States of nature
Probability
• Each state of nature has a probability
– Probability is a number between 0 and 1, a fraction
– The sum of the probabilities of all states of nature is
one (1)
– Probability of a state of nature may be its relative
frequency if and when the risky decision is repeated a
large number of times (objective probability)
– Or it might simply be the gut feeling of the person
making the risky decision (subjective probability)
Payoffs
• The financial reward in each state of nature
(to the person who takes a risky decision) is
called its payoff
• The possible outcomes of a risky decision
could also be thought of as the payoffs from
the decision (instead of the states of nature)
• The full list of all possible payoffs and the
probability of each payoff is called the
probability distribution of the risky decision
Expected payoff
• Let P1, P2, … , PN be the N possible payoffs of a
risky decision, and
• Let π1, π2, … , πN be the probabilities of those
payoffs
• Then the expected payoff of the risky decision
is
EP = π1  P1 + π2  P2 + … + πN  PN
• This is what the average payoff would be if the
risky decision is, hypothetically, repeated
numerous times
Variance: a measure of riskiness
• The variance of a risky decision is defined as
Variance = π1  (P1 – EP)2 + π2  (P2 – EP)2 + … + πN 
(PN – EP)2
• Note that if the payoffs of a risky decisions, P1, P2, … ,
PN, are all close to the expected payoff, EP, then the
variance is small.
• And this is precisely a case in which the risk is small,
because the payoffs, though unpredictable, are all
quite similar
• Therefore, the variance is an excellent measure of risk
• The square root of the variance is called the standard
deviation
– It is also a popular measure of the riskiness of a risky
decision
Example
• New Stuff, Inc. is deciding whether to build a
large factory or a small factory for its new
product
• The risk is two-fold:
– A competitor has filed for a patent on a similar
product. Royalties must be paid if patent is upheld
– The demand for the product may be low or high
• Here are the payoffs for New Stuff, Inc.
Example: probability distribution
• New Stuff, Inc.’s payoffs and probability distribution
• The expected payoff is what would be the average
payoff if, hypothetically, New Stuff, Inc.’s risky decision
is repeated numerous times.
Example: variance and std. dev.
Risk Preferences
• One can think of a consumption bundle as a list of the
quantities of each good consumed in each possible state of
nature
– That is, each consumption bundle can be thought of as a lottery ticket
• The guaranteed consumption line shows the consumption
bundles for which the level of consumption does not depend
on the state
• For bundles that do not lie on this line, the consumer’s payoff
is uncertain
– One can compute expected consumption for any particular bundle
• A constant expected consumption line shows all risky
consumption bundles with the same level of expected
consumption
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Figure 11.3: Consumption Bundles Example
Π, probability of
sunny weather,
is 2/3.
Therefore, 1 – Π
is the probability
of hurricane. It
is 1/3.
Exercise:
Calculate the
expectation and
variance of
bundles A, B,
and C.
Could you have
calculated Π from the
diagram alone?
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Preferences and
Indifference Curves
• If one bundle guarantees more of every good than
a second bundle, a consumer should prefer the first
– Reflects the More-Is-Better Principle
– Does not have to guarantee a particular level of
consumption
• Slope of an indifference curve indicates willingness
to shift consumption from one state of nature to
another
– Depends on the probabilities of the states
– Change in probabilities changes slopes of indifference
curves
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Figure 11.4: Preferences for Risky
Consumption Bundles
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Risk Aversion
• A person is risk averse if, in comparing a riskless
bundle to a risky bundle with the same level of
expected consumption, he prefers the riskless bundle
– Risk averse individuals do not avoid risk at all costs
– Usually willing to accept some risk provided they receive
adequate compensation in the form of higher expected
consumption
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Risk Premium
• The certainty equivalent of a risky bundle is the
amount of consumption which, if provided with
certainty, would make the consumer equally well off
– This is the consumer’s willingness to pay for the risky bundle
• For a risk-averse person, the certainty equivalent of a
risky bundle is always less than expected consumption
– Providing the same expected consumption with no risk would
make the individual better off
• The risk premium of a risky bundle is the difference
between its expected consumption and the consumer’s
certainty equivalent
– Certainty equivalent = expected payoff + risk premium
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Figure 11.6: Risk Aversion
The lottery B is riskier
than the lottery A, and
has the same expected
consumption.
Therefore, B is on a lower
indifference curve than A
is.
This reasoning also shows
why any indifference
curve must be tangent to
the constant expected
consumption line at
precisely the guaranteed
consumption line.
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Expected Utility Functions
• Recall from our discussion of consumer
behavior without uncertainty, that a
consumer’s preferences can be represented by
– A map of indifference curves, or
– A utility function
• For consumer choice under uncertainty,
preferences can be represented by
– A map of indifference curves, or
– An expected utility function
Expected Utility Functions
• An expected utility function:
– Assigns a benefit level to each possible state of nature
based only on what is consumed
– Then takes the expected value of those benefits
– It is a weighted average of all possible benefit levels
using the probability of each level as its weight
• Sample expected utility function:
U Fs , FH    W FS   1   W FH 
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Expected Utility and Risk Aversion
• Can determine the consumer’s attitude toward risk
from the shaper of her benefit function, W(F):
– If W(F) is concave (flattens as F increases), she’s risk averse
• Example: W(F) = F0.5.
– If it’s convex (gets steeper as F increases), she’s risk loving
• Example: W(F) = F5.
– If it’s linear, she’s risk neutral
• Example: W(F) = 5F.
• The greater the concavity of the benefit function, the
greater the risk aversion
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Figure 11.10: Expected Utility for a RiskAverse Consumer
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The Nature of Insurance
• People address a wide range of risks by purchasing insurance
policies
• An insurance policy is a contract that reduces the financial
loss associated with some risky event, such as burglary
• The purchaser of an insurance policy is essentially placing a
bet
• Having paid M, the premium, the policy holder receives B, the
benefit, if a loss occurs, for a net gain of B – M
• If a loss doesn’t occur the consumer loses the premium M
– The probability of this (favorable, to the insurance buyer) outcome is Π
– The probability of the unfavorable outcome is 1 – Π
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The Nature of Insurance
• Suppose you will get bundle/lottery A if you have no
insurance.
• This lottery gives
– FS units of food with probability Π, and
– FH units of food with probability 1 – Π
• So, without insurance, expected consumption is ECA = Π 
FS + (1 – Π)  FH .
• After buying insurance, you will get the bundle C, which
gives
– FS – M units of food with probability Π, and
– FH + B – M units of food with probability 1 – Π
• So, with insurance, expected consumption is ECC = Π  (FS –
M) + (1 – Π)  (FH + B – M) .
The Nature of Insurance
•
hurricane
Guaranteed
Consumption Line
•
•
Insurance budget line
FH + B – M
C
FH
•
A
FS – M
FS
sunny
When the insurance
contract is actuarially
fair, both A and C have
the same expected
consumption,
And, therefore, the
insurance budget line
lies on the constant
expected consumption
line through A
When the insurance
contract is actuarially
unfair, the expected
consumption of C is less
than the expected
consumption of A,
And, therefore, the
insurance budget line
lies below the constant
expected consumption
line through A
Actuarial Fairness
• An insurance policy is actuarially fair if its expected net
payoff is zero
• Actuarial fairness requires:
   M   1    B  M   0
1    B    M  1    M  M
  0  1    B  M
• So an actuarially fair insurance premium equals the
promised benefit times the probability of a loss
• Insurance policies are usually less than actuarially fair
because insurance companies must cover their costs of
operation
• On average purchasing such a policy reduces the
purchaser’s expected consumption
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Actuarial Fairness
• Recall that, without insurance, expected
consumption is ECA = Π  FS + (1 – Π)  FH .
• And, with insurance, expected consumption is ECC
= Π  (FS – M) + (1 – Π)  (FH + B – M)
= Π  FS + Π  (– M) + (1 – Π)  FH + (1 – Π)
 (B – M)
= ECA + Π  (– M) + (1 – Π)  (B – M).
• Therefore, for actuarially fair insurance, ECC = ECA.
• Therefore, A and C must lie on the same constant
expected consumption line
Demand for Insurance
• Risk-averse consumers are willing to purchase
insurance because it cancels out other risks
• If the insurance is actuarially fair, a risk averse
consumer will purchase full insurance
– With full insurance, the promised benefit equals the
potential loss
– This does not depend on degree of risk aversion
• If the insurance is less-than-actuarially fair, the
amount of insurance purchased depends on degree
of risk aversion
– Risk neutral consumer will purchase none
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Figure 11.12: Demand for Insurance
The probability of sunny weather is Π = 2/3.
Without insurance, the outcome is A
The insurer promises to pay 3 kg of food in
case of a hurricane for every 1 kg of food
paid as premium
This is actuarially fair insurance , because
the insurance company’s expected payoff is
zero
The consumer’s best choice is B. That is,
when actuarially fair insurance is available,
the consumer will buy full insurance (and,
therefore, end up on the guaranteed
consumption line)
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Figure 11.13: Demand for
Unfair Insurance
Now the insurer offers to pay 2 kg
of food in case of a hurricane for
every 1 kg of food paid as
premium
Now the insurer’s expected
payoff is positive. That is, the
insurance contract is actuarially
unfair
The consumer chooses E, which is
not on the guaranteed
consumption line. That is, the
consumer’s uncertainty will be
only partially insured
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Other Methods of Managing Risk
• Four other strategies for managing risk
• Object of risk management is to make risky activities more
attractive by reducing the potential losses while preserving
much of the potential gains
• Risk sharing involves dividing a risky prospect among
several people
• Hedging is the practice of taking on two risky activities with
negatively correlated financial payoffs
• Diversification is the practice of undertaking many risky
activities each on a small scale
• People also often try to reduce risk through information
acquisition
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Insurance: actuarially fair and unfair
Note that the value of actuarially fair
insurance = risk premium.
Figure 11.15: Risk Sharing
• Bundle A is initial lottery, and is
riskless
• By investing, consumer can
move to B with higher expected
consumption
• Consumer prefers to avoid risk
associated with B
• With partners to split
investment and profits, can
reach points on the green line
• D is most preferred bundle with
risk sharing
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Hedging
• Hedging is the practice of taking on two risky
activities with negatively correlated financial payoffs
– Two variables are negatively correlated if they tend to
move in the opposite direction
• Bad news on one investment tends to be offset by
good news on the other
• Insurance is a form of hedging
– Benefit paid by a flood insurance policy is perfectly
negatively correlated with a loss from flooding
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Figure 11.16: Hedging a
Risky Venture
Maria earns $600 regardless of the weather.
She has the opportunity to buy a sunscreen
concession at the local beach for $300.
If the weather turns out to be sunny, she will
earn $600 profit—revenues of $900 minus
cost of $300. Her consumption will,
therefore, be $1200.
If there’s a hurricane, she will have no sales
and she will lose her $300 investment. Her
consumption will, therefore, be $300.
This is bundle A.
The probability of sunny weather is 2/3.
Maria gets F if she buys both investments. This shows
the benefits of hedging. Also, Maria gets G if she has
accurate weather info. The value of such info is $300.
Maria also has the opportunity to invest $300
in a portable generator distributorship.
For this business, the payoffs are exactly the
opposite of the sunscreen concession.
This is bundle B.
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Diversification
• Diversification is the practice of undertaking many
risky activities, each on a small scale, rather than a
few risky activities on a large scale
– “Don’t put all your eggs in one basket”
– Dividing investments among many activities reduces risk
• As correlation between the payoffs on the
investments increases, the risk-reducing effect of
diversification decreases
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