2. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk
◮ Many economic choice is risky in the sense that the
outcome is not certain.
◮ Project choice
◮ Investment decision
◮ Occupational choice
◮ Marriage decision
3. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risky choice as lottery
◮ A decision problem under risk can be interpreted as a
choice of “lotteries”.
◮ Project choice
◮ If the future economic condition is good, your project
yields $1000
◮ If it is bad, your project yields $0.
◮ If the probability of having a good economy is 20%, it is
a lottery such that $1000 with 20% and $0 with 80%
4. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Notation
◮ If the probability of having a good economy is 20%, it is
a lottery such that $1000 with 20% and $0 with 80%
◮ Formally, the above lottery can be written as
L = (x1,x2;p1,p2) = (1000,0;0.2,0.8)
5. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Example
◮ Which is your favorite lottery?
◮ L1 = (10,10;0.5,0.5)
◮ L2 = (0,20;0.5,0.5)
◮ L3 = (5,10;0.2,08)
6. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Preference over lotteries
◮ As Lecture 1, define a preference over lotteries, e.g.,
L1 ≻ L2 ≻ L3.
◮ If your taste is transitive and complete, it is rational
preference.
◮ Then, you have a utility function U(L) that represents
your taste for lotteries.
◮ However, you might want more intuitive and useful
representation.
7. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Expected value
◮ If it is a lottery, how about the average or expected
value?
◮ That is,
U(L) = p(x1)x1 +p(x2)x2
◮ More generally, if there are N possible outcomes,
U(L) = ∑
n
p(xn)xn
8. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Problem
◮ Note that if L1 = (10,10;0.5,0.5) and
L2 = (0,20;0.5,0.5), then U(L1) = U(L2) if U is the
expected value.
◮ Thus, unless you find L1 and L2 indifferent, this is not
right way to represent your preference.
◮ In other words, the expected value cannot reflect your
attitude toward risk.
9. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Expected utility
◮ There is an alternative way.
◮ Expected utility theory represents your preference with
the following form
U(L) = p(x1)u(x1)+p(x2)u(x2)
◮ More generally, if there are many possible outcomes
U(L) = ∑
n
p(xn)u(xn)
10. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Example
◮ Suppose you cast a dice. If your get x ∈ {1,2,..,6}, you
receives $x.
◮ The expected utility is
EU(x) =
1
6
[u(1)+u(2)+u(3)+u(4)+u(5)+u(6)]
◮ For example, if u(x) =
√
x, then EU(x) = 1.805303682
11. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
When does EU exist?
◮ We learn that a preference can be represented by a
utility if and only if the preference is rational.
◮ When can a preference over lotteries be represented by
an expected utility?
◮ In other words, what is the condition that guarantees
existence of u(x) such that
L′
L′′
if and only if
∑
n
pL′ (xn)u(xn) ≥ ∑
n
pL′′ (xn)u(xn)
12. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Continuity
Definition
A preference over lotteries satisfies continuity assumption if,
for any lotteries L′,L′′,L′′′ such that L′ L′′ L′′′, there
exists α ∈ (0,1) such that αL′ +(1−α)L′′′ ∼ L′′
◮ Roughly put, a small change of a probability can make a
small change in your preference.
13. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Independence
Definition
A preference over lotteries satisfies independence
assumption if, for any lotteries L′,L′′,L′′′ such that L′ L′′,
αL′ +(1−α)L′′′ αL′′ +(1−α)L′′′ for any α ∈ (0,1].
◮ Roughly put, if you expand two lotteries with a common
lottery, it should not affect preference over the two
lotteries.
14. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
The condition
◮ The following result is known.
Proposition
There exists an expected utility that represents a preference
over lotteries if and only if the preference is rational and
satisfies the continuity and independence assumptions.
15. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk attitude
◮ The expected value cannot capture a risk attitude.
◮ If L1 = (10,10;0.5,0.5) and L2 = (0,20;0.5,0.5), then
what can we say about U(L1) vs. U(L2) if U is the
expected utility?
◮ Can the expected utility capture a risk attitude?
16. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk aversion
◮ Suppose u(x) = xα (other functional form can be also
possible)
◮ Suppose you prefer L1 to L2, that is, you are risk
averse.
◮ Then choose α ∈ (0,1), say 1/2
◮ Then,
1
2101/2 + 1
2101/2 = 101/2 = 3.162 > 1
2201/2 = 2.236.
◮ So α < 1 represents that you are risk averse.
17. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk aversion and concavity
◮ If u(x) is strictly concave, then the agent is always risk
averse.
◮ u(x) = ln(x)
◮ u(x) =
√
x
18. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk loving
◮ Suppose you prefer L2 to L1, that is, you are a risk
lover.
◮ Then choose α > 1 in your xα.
◮ Then, 1
2102 + 1
2102 = 102 = 100 < 1
2202 = 200.
◮ So α > 1 represents that you are a risk lover.
19. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk loving and convexity
◮ If u(x) is strictly convex, then the agent is always risk
averse.
◮ u(x) = x3
◮ u(x) = ex
20. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk neutrality
◮ If α = 1, then the expected utility becomes the
expected value.
◮ 1
210+ 1
210 = 10 = 1
220.
◮ That is, L1 and L2 are indifferent.
◮ So α = 1 means that you are risk neutral.
21. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk neutrality and linearity
◮ if u(x) is linear, then the agent is always risk averse.
◮ u(x) = 4x
◮ u(x) = 0.3x
22. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Certainty equivalent
◮ L = (x,y;p,1−p)
◮ Consider z such that u(z) = pu(x)+(1−p)u(y)
◮ That is, you are indifferent between certain z dollar and
lottery L.
◮ Then, z is called certainty equivalent of L
◮ If you are risk averse, z < px +(1−p)y.
◮ If you are risk neutral, z = px +(1−p)y.
23. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk premium
◮ Risk premium of L is an extra money that can be paid
to compensate the risk of L.
◮ Formally, we can define risk premium as
px +(1−p)y −z.
◮ Note that if the agent is risk neutral z = px +(1−p)y
and thus, the risk premium is always 0.
24. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Example
◮ L = (10,60;0.5,0.5) and u(x) = 100x − x2
2
◮ pu(x)+(1−p)u(y) = 1
2(1000−50)+ 1
2 (6000−1800) =
2625
25. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Example
◮ Certainty equivalence z solves u(z) = 2625 or
0 = z2
−200z +5250
◮ The certainty equivalent is
z =
200−
√
40000−21000
2
≈ 31
◮ Risk premium is
1
2
10+
1
2
60−31 = 4
26. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Insurance
◮ Consider the following simple insurance market.
◮ Suppose that the probability of having an accident is
known as p.
◮ The accident could make your income from 100 to 0.
◮ Insurance guarantees you to have 100 all the time as
long as you pay $q.
27. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Expected utility
◮ So with the insurance, your utility is
U(100−q)
.
◮ On the other hand, without the insurance, your
expected utility is
(1−p)U(100)+pU(0)
.
28. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Buyer’s problem
◮ You buy the insurance if your utility from buying is
higher than your expected utility from facing
uncertainty.
◮ That is,
(1−p)U(100)+pU(0) ≤ U(100−q)
29. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Seller’s problem
◮ The insurance company has to make sure that
◮ she can sell the product (Sale’s condition)
◮ the profit is not negative (Profitability condition)
30. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Sale’s condition
◮ Recall the certainty equivalent of L = (0,100;p,1−p) is
z such that
U(z) = (1−p)U(100)+pU(0)
◮ Then, the consumer prefers to buy the insurance as long
as 100−q ≥ z or
q ≤ 100−z
◮ Thus, the insurance company has to choose q so that it
satisfies the above condition to sell the insurance.
31. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Profitability condition
◮ The insurance company gets q but they have to pay
100 with probability p. Thus, their expected profit
given q is q −100p.
◮ Thus, q has to be q ≥ 100p to keep her profit
non-negative.
32. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Trading condition
◮ The trade occurs if and only if the sale’s condition and
the profitability condition are both satisfied.
◮ In other words, q has to be
100−z ≥ q ≥ 100p
33. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk neutral buyer
◮ If you are risk neutral, z = (1−p)100. Then,
100−z = 100p
◮ Thus, q = 100p is the only value that satisfies the
trading condition
100−z ≥ q ≥ 100p
◮ Then, the expected profit is always 0.
34. Intermediate
microeconomics:
Lecture 2
Risk and Expected
utility
Risk attitude
Application to
insurance market
Risk aversion
◮ Recall that if you are risk averse, z < (1−p)100. Then,
100−z > 100p
◮ Thus, by choosing q such that
100−z > q > 100p,
the company earns a positive profit and the consumer
buys the insurance.
