Uncertainty
Varian: Intermediate Microeconomics, 8e, chapter 12
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In this lecture you will learn
• how standard tools of consumer choice can be used for analysing
decisions under risk and what is special about these decisions
• how to model different attitudes towards risk
• whether it is better to bet on favourites or outsiders
• what makes health insurance valuable
• what we are willing to pay in order to prevent catastrophes
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What do we choose?
Probability distributions with different consumptions (= lotteries).
Probability distribution (lottery) = list of possible consumption bundles
with probabilities that I get them
L = {π1, π2, ..., πn},
where πn ≥ 0 is the probability to get bundle n where n πn = 1.
Example:
I bet my last 100 CZK on a toss of a coin.
If I win, I have 200 CZK. If I lose, I have 0 CZK.
What does this lottery look like?
The lottery in this case is L = {π1, π2} = {1/2, 1/2},
where result 1 is 200 CZK and result 2 is 0 CZK.
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Contingent consumption
States of nature are different outcomes of some random event.
Examples:
• a bet (toss of a coin) – 2 states of nature: head, tail
• car insurance – 2 states of nature: car stolen, car not stolen
For simplicity we will study the decisions of consumers
• facing one random event with a few states of nature,
• with consumption measured in monetary units.
Contingent consumption plan is a specification of what will be
consumed in each different state of nature—each different outcome
of one random process.
Difference to lottery: Contingent consumption plan shows only
consumption and not probabilities.
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Example – insurance
A consumer plans to spend 35 000. Her car will be destroyed
in an accident with probability 1% – the damage of 10 000.
Her contingent consumption plan is (cb, cg ) = (25 000, 35 000):
• a bad state of nature b occurs with probability 1%
• a good state of nature g occurs with probability 99%
Insurance offers a way to change this probability distribution.
If she pays a premium of γK she gets an insurance payment of K.
The consumer chooses between the following consumption plans:
(cb, cg ) = (25 000 + K − γK, 35 000 − γK) (1)
By eliminating K from (1) we get the budget line (BL):
cg = 35 000 +
γ
(1 − γ)
25 000 −
γ
(1 − γ)
cb
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Example – insurance (cont’d)
Budget line (BL): cg = 35 000 +
γ
(1 − γ)
25 000
vertical intercept
−
γ
(1 − γ)
slope of BL
cb.
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Example – insurance (cont’d)
For the choice of the contingent consumption plan we can use the
consumer theory we have developed in previous lectures:
• budget constraint is given (e.g. by insurance choice)
• preferences defined over different consumption plans
The consumer chooses the best consumption plan she can afford.
What is the optimal premium K? It depends on preferences.
E.g. on the consumer’s attitudes towards risk:
• If she is conservative, she chooses a high K.
• If she likes risk, she might not buy any insurance.
Before we continue our insurance example, we explain
1 how preferences under risk are represented by utility functions,
2 what the properties of these functions are,
3 how to use utility functions to represent attitudes to risk.
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Representing preferences using utility functions
Choice under uncertainty does add a special structure to the problem. How
a person values consumption in different states will depend on the
probabilities that the states occur.
E.g. the probability of an accident π influences my marginal rate of
substitution. The higher the π, the more of cg I am willing to sacrifice
for an additional unit of cb (a more expensive insurance).
The utility function for consumption in states 1 and 2 is given by
u(c1, c2, π1, π2)
where
• c1 and c2 is consumption in states 1 and 2,
• π1 and π2 are probabilities of states 1 and 2.
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Examples of utility functions
• Perfect substitutes:
u(c1, c2, π1, π2) = π1c1 + π2c2
π1c1 + π2c2 is the expected value of a given event.
• Cobb-Douglas utility function:
u(c1, c2, π1, π2) = cπ1
1 cπ2
2
Or sometimes a more convenient monotonic transformation:
ln u(c1, c2, π1, π2) = π1 ln c1 + π2 ln c2
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Von Neumann-Morgenstern utility function
Von Neumann-Morgenstern utility function is given by
u(c1, c2, π1, π2) = π1v(c1) + π2v(c2),
where v(c1) and v(c2) are utilities in individual states of nature.
In examples on the previous slide:
• perfect substitutes: v(c) = c
• Cobb-Douglas utility function: v(c) = ln c
This function is also called the expected utility function –
u(c1, c2, π1, π2) equals to the expected utility of consumption
in individual states of nature π1v(c1) + π2v(c2).
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Positive affine transformation
Consumer preferences represented by the expected utility function,
which has the additive form described above.
Any monotonic transformation describes the same preferences,
but the additive form representation is especially convenient.
E.g. the functions π1 ln c1 + π2 ln c2 and cπ1
1 cπ2
2 describe the same
Cobb-Douglas preferences but cπ1
1 cπ2
2 does not have the additive form.
Positive affine transformation t(u) – a type of monotonic
transformation that preserves the expected utility property:
t(u) = au + b where a > 0.
A positive affine transformation simply means multiplying by a positive
number a and adding a constant b.
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Why is expected utility reasonable?
Let us have a random event with 3 states of nature:
• my house burns down with probability πf – consumption cf
• my house does not burns down with probability πn – consumption cn
• I sell the house this year with probability πs – consumption cs
Under uncertainty only one state of nature is actually going to occur.
=⇒ There is a natural independence among different states.
The independence is well represented by the additive utility function:
u(cf , cn, cs, πf , πn, πs) = πf v(cf ) + πnv(cn) + πsv(cs)
MRS between cf and cn is independent from cs:
MRSfn = −
πf
∂v(cf )
∂cf
πn
∂v(cn)
∂cn
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A comparison to the decision-making under certainty
My preferences for 3 goods (tee, coffee, milk) = (t, c, m) can be
represented by a utility function
u(t, c, m) = 2t + cm.
The marginal rate of substitution between t and c is
MRStc = −
2
m
.
The MRS between tee and coffee depends on the quantity of milk m.
Decision-making under certainty: one can consume combinations
of goods at the same time. =⇒ We cannot a priori exclude any
functional forms of the utility function.
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SUPPLEMENT: Independence assumption
Preferences can be represented by the expected utility function only
if the independence assumption holds.
Preference relation satisfies the independence assumption if for all
triples of lotteries L, L a L and for the parameter α ∈ (0, 1) it holds that
L L
if and only if
αL + (1 − α)L αL + (1 − α)L .
In other words:
If we mix any two lotteries with a third one, preferences between the two
lotteries will stay the same (are not influenced by the third one).
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SUPPLEMENT: Example of a choice under risk
Peter can go for a dinner to a restaurant A, or to a restaurant B.
There are three different food qualities: good G, average A, or bad B.
Restaurant A has a good cook who often has a bad day:
With probability 50% Peter gets G and with probability 50% B.
The cook in restaurant B is more average:
With probability 90% Peter gets A and with probability 10% B.
What restaurant does Peter choose? It depends on preferences.
Let us assume that Peter’s preferences are represented by such an
expected utility function that he chooses restaurant A, i.e.:
0.5v(G) + 0.5v(B) > 0.9v(A) + 0.1v(B) (2)
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SUPPLEMENT: Example of a choice under risk (cont’d)
Does Peter choose differently if he finds out that in both restaurants
Jamie Oliver prepares a perfect food P for him with probability 50%?
No. This information increases the expected utility from restaurants A and
B by the same amount. If (2) holds, then it must also hold that:
1
2
v(P) +
1
2
0.5v(G) + 0.5v(B) >
1
2
v(P) +
1
2
0.9v(A) + 0.1v(B)
The independence assumption says that if both restaurants offer P with
the same probability, Peter’s choice does not change.
The independence assumption sounds reasonable. =⇒ It seams
reasonable to represent preferences under uncertainty using the expected
utility function.
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SUPPLEMENT: TV competition
Suppose that you have won two weekly editions of a TV competition and
each time you were given a choice between two lotteries:
1st week:
1 100% – 500 000 CZK
2 1% – 0 CZK
10% – 1 000 000 CZK
89% – 500 000 CZK
2nd week:
1 11% – 500 000 CZK
89% – 0 CZK
2 10% – 1 000 000 CZK
90% – 0 CZK
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SUPPLEMENT: TV competition – Allais paradox
If the independence assumption holds, the consumer should have chosen
the same option in both weeks, either 1 or 2. Why?
The choices 1 and 2 in both weeks are the same:
Probabilities
1/100 10/100 89/100
1st week
1 500 000 500 000 500 000
2 0 1 000 000 500 000
2nd week
1 500 000 500 000 0
2 0 1 000 000 0
By mixing 0.89×500 000 (1st week) and 0.89×0 (2nd week) to both
lotteries 1 and 2, we get the choices from the previous slide.
In this situation people usually violate the independence assumption
(Allais paradox).
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SUPPLEMENT: Prospect theory
Prospect theory (Kahneman and Tversky, Econometrica, 1979) = the
most cited behavioral alternative to the expected utility theory
that is able to explain i.a. the Allais paradox.
Utility from a lottery (x1, x2, π1, π2):
V (x1, x2, p1, p2) = w(π1)v(x1) + w(π2)v(x2)
• value function v(·) – S-shaped around reference + loss aversion
• weighting function w(·) – Inverted S = people underweight high and
overweight high probabilities; 0% and 100% are perceived correctly
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SUPPLEMENT: Explaining Allais paradox
Choices leading to Allais paradox:
1st week:
1 100% – 500 000 CZK
2 1% – 0 CZK
10% – 1 000 000 CZK
89% – 500 000 CZK
2nd week:
1 11% – 500 000 CZK
89% – 0 CZK
2 10% – 1 000 000 CZK
90% – 0 CZK
Choices in the 1st week 1 and 2nd week 2 (violation of the independence
assumption) are in line with prospect theory if
w(0.1)
1 − w(0.89)
<
v(500 000)
v(1 000 000)
<
w(0.1)
w(0.11)
.
An intuitive explanation:
• 1st week: 1 – 89% underweighted vs. 100% perceived correctly
• 2nd week: 2 – 10 and 11% overweighted to a similar extent
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Attitudes toward risk
The consumer has a wealth of $10,
• with probability 50 % she wins $5,
• with probability 50 % she loses $5.
Her wealth has
• the expected value EV = 0.5 × 5 + 0.5 × 15 = 10,
• the expected utility EU = 0.5 × u(5) + 0.5 × u(15).
The consumer
• is risk averse if u(EV ) > EU – a concave u(c),
• is risk seeking if u(EV ) < EU – a convex u(c),
• is risk neutral if u(EV ) = EU – a linear u(c).
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Risk aversion
A concave utility function =⇒ u(EV ) > EU
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Risk seeking
A convex utility function =⇒ u(EV ) < EU
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Attitudes toward risk – specific utility functions
The consumer with the initial wealth of $10
• wins $5 with probability 50%,
• looses $5 with probability 50%.
What is her attitude toward risk for the following utility functions?
• Utility function u(c) =
√
c:
u(EV ) =
√
EV =
√
10 = 3.16
EU = 0.5 ×
√
5 + 0.5 ×
√
15 = 3.05
u(EV ) > EU =⇒ The consumer is risk-averse.
• Utility function u(c) = c2:
u(EV ) = EV 2 = 102 = 100
EU = 0.5 × 52 + 0.5 × 152 = 125
u(EV ) < EU =⇒ The consumer is risk-seeking.
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Certainty equivalent
Certainty equivalent (CE) = what is the (minimal) certain amount of
money I am willing to exchange for a given lottery.
Example:
Consumer’s wealth is 10.
A lottery: L = {π1, π2} = {0.5, 0.5}, where c1 = 15 a c2 = 5
Expected utility function: u = π1
√
c1 + π2
√
c2
What is the certainty equivalent and the expected value?
Utility of the lottery: uL = EU = 0.5 ×
√
15 + 0.5 ×
√
5 = 3.05
Utility from CE equals to uL: uL =
√
CE ⇐⇒ CE = u2
L = 9.33
The expected value: EV = 0.5 × 15 + 0.5 × 5 = 10
If:
• CE < EV , the consumer is risk-averse.
• CE > EV , the consumer is risk-seeking.
• CE = EV , the consumer is risk-neutral.
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Example – choice of the optimal insurace (graph)
Linie rozpoˇctu (BL): cg = 35 000 + γ
(1−γ) 25 000 −
γ
(1 − γ)
sklon BL
cb
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Example – fair insurance
Consumption in a bad state: cb = 25 000 + K − γK
Consumption in a good state: cg = 35 000 − γK
Probability of the bad state (accident) is π.
We assume that the insurer offer the fair insurance.
What is the optimal insurance premium for a risk-averse consumer?
Fair insurance – the insurer chooses such a premium ratio γ so that
its profit is zero: γK − πK = 0 ⇐⇒ γ = π.
By substituting γ = π into the equation
MRS = −
π∆u(cb)
∆cb
(1 − π)
∆u(cg )
∆cg
= −
γ
1 − γ
we get
∆u(cb)
∆cb
=
∆u(cg )
∆cg
.
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Example – fair insurance (cont’d)
Marginal utility of consumption has to be the same in both states.
A risk-averse consumer has a diminishing MU of consumption.
If e.g. cb < cg , than it would have to hold: ∆u(cb)
∆cb
>
∆u(cg )
∆cg
If we want to get ∆u(cb)
∆cb
=
∆u(cg )
∆cg
, then it must hold:
cb = cg
25 000 + K − γK = 35 000 − γK
K = 10 000
Conclusion:
If a risk-averse consumer faces a fair insurance, she fully insures.
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Numerical example – insurance
Consumption in a bad state: cb = 25 000 + K − γK = 25 000 + 0.9K
Consumption in a good state: cg = 35 000 − γK = 35 000 − 0.1K
Utility function: u(cb, cg , πb, πg ) = 0.1 ln cb + 0.9 ln cg
What is the optimal insurance payment K?
We solve the equation cg for K:
K = (35 000 − cg )/0.1 = 350 000 − cg /0.1
We substitute into cb:
cb = 25 000 + 0.9(350 000 − cg /0.1)
The budget line:
cb + 9cg = 340 000
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Numerical example – insurance (cont’d)
We look for the bundle, at which MRS = the slope of BL:
MRS = −
γ
1 − γ
or MRS = −
pb
pg
−
0.1cg
0.9cb
= −
0.1
0.9
cb = cg
By substituting into the budget line we get:
cg + 9cg = 340 000
cg = 34 000
The optimal insurance payment:
K = (35 000 − cg )/0.1 = 10 000 $
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APPLICATION: Diversification
The investor has $100 which she can invest in
• firm S (sun glasses) – price of 1 share pS = $10
• firm U (umbrellas) – price of 1 share pU = $10
Summer will be rainy with 50% and sunny with 50% probability:
• rainy: pU = $20 and pS = $5
• sunny: pU = $5 a pS = $20
What should a risk-averse investor do?
Two options:
• $100 in firm U, contingent consumption plan (cU, cS ) = (200, 50),
EV = 0.5 × 200 + 0.5 × 50 = 125 $,
• 50 $ in each firm, contingent consumption plan (cU, cS ) = (125, 125),
EV = 0.5 × 125 + 0.5 × 125 = 125 $.
Diversification reduces the risk, but the EV remains the same.
As long as asset price movements are not perfectly positively correlated,
there will be some gains from diversification.
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APPLICATION: Risk spreading
A village: 1 000 risk-averse farmers with a wealth 3 500 000 CZK
The risk of fire: 1 % (houses far apart = the risk is independent)
The cost of fire: 1 000 000 CZK
How do farmer insure if a standard insurance is not available?
Risk spreading: Fire victims get 1 000 CZK from all farmers.
On average 10 farmers are affected – farmer’s expected wealth
• if she is not affected: 3 500 000 − 10 × 1 000 = 3 490 000 CZK
• if she is affected: 2 500 000 + 990 × 1 000 = 3 490 000 CZK
The expected wealth of the farmer without insurance:
0.99 × 3 500 000 + 0.01 × 2 500 000 = 3 490 000 CZK
The expected wealth is the same, but the risk is lower.
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APPLICATION: Favourite-longshot bias (FLB)
Is it more profitable to bet on favourites or outsiders? On favourites.
Source: Snowberg a Wolfers, JPE, 2010, Fig. 1
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APPLICATION: Favourite-longshot bias (FLB) (cont’d)
Snowberg and Wolfers (JPE, 2010) study FLB in the US racetracks.
Two explanations why people prefer betting on outsiders:
1 neoclassical approach – risk-seeking:
higher odd for the outsider =⇒ higher risk (riskier is better)
2 behavioral approach – people misperceive probabilities:
overweighting low probabilities of outsiders and a possible
effect of underweighting of high probabilities of favourites
Racetrack odds in the US correspond more to the behavioral explanation.
Still an open question.
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CASE: The insurance value of Medicare
Medicare (health insurance to people older than 65) was introduced in
1965 – the largest expansion of health insurance in the 20th century.
Finkelstein and McKnight (J Publ. Econ., 2008) study the effect of the
expansion: Medicare has almost no effect on mortality.
But Medicare reduces cash expenditures – for the quartile of people with
the highest health costs reduced cash expenditures by 40%.
For risk-averse patients it is valuable. According to Finkelstein and
McKnight the insurance value equals to 2/5 of the cost of Medicare.
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APPLICATION: Insurance against global warming
What percentage of your yearly income is equivalent to eliminating
the risk of a catastrophe that happens once every 1,000 years at random,
and when it occurs, there is a 1% chance you will be killed?
Becker, Murphy and Topel (BEJEAP, 2010) estimate that we should be
willing to pay 4.5% of the income. If the event happened once in every
100 years, then 36% of the income.
Why that much?
1 Death is permenent and bad for utility.
2 People are risk-averse.
3 We expect the growth in income in future, so are willing to pay a big
share of today’s income to secure our growing future incomes.
Weitzman (RES, 2009) estimates 1% probability that global warming
raises the temperature by more than 20◦ Celsius = a huge problem.
Others view it more optimistically.
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What should you know?
• We can use the tools of consumer choice also for studying
the decision-making under uncertainty.
• The difference to decision-making under certainty (utility function
can have any form) is that due to the independence of the states
of nature we can use a utility function with an additive form =
• = Expected utility function: EU = π1v(c1) + π2v(c2)
• Expected value: EV = π1c1 + π2c2
• Certainty equivalent CE = willingness to pay for a lottery
• The consumer is
• risk averse: concave v(c), u(EV ) > EU, EV > CE
• risk seeking: convex v(c), u(EV ) < EU, EV < CE
• risk neutral: linear v(c), u(EV ) = EU, EV = CE
• If a risk-averse consumer faces fair insurance, she fully insures.
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