ECON 417: Economics of Uncertainty Contents I Expected Utility Theory 3 1 Lotteries 3 2 St. Petersburg’s Paradox 3 3 von Neumann and Morgenstern Axioms and Expected Utility Form 4 4 Risk Attitudes 5 5 Risk Premium and Certainty Equivalent 6 6 Measures of Risk Aversion 6 II Mean-Variance Optimization 8 7 One Riskfree Asset, One Risky Asset 8 8 Many Risky Assets 9 9 One Riskfree Asset, Many Risky Assets 9 10 Diversification 10 11 Capital Asset Pricing Model 10 III Insurance 11 12 Utility Maximization 11 12.1 Tangency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 12.2 Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 12.3 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 13 State Preference Approach to Insurance 13 14 Overview of Insurance 15 15 Demand for Insurance 15 15.1 Mossin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 15.1.1 Actuarially Fair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 15.1.2 Not Actuarially Fair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 15.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 15.3 Coinsurance and Deductibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 16 Supply of Insurance 18 16.1 Risk pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 16.2 Risk spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 16.3 “Undersupply” of Full Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 17 Asymmetric Information 20 17.1 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 17.1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 17.1.2 Tangency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 17.1.3 Two types of consumers, symmetric information . . . . . . . . . . . . . . . . 22 17.1.4 Two types of consumers, asymmetric information . . . . . . . . . . . . . . . 23 17.1.5 Pooling Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 17.1.6 Separating Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 17.2 Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 IV After the Midterm 27 17.3 Insurance (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 18 The Value of Information 27 19 Options 27 19.1 Financial Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 19.2 Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Part I Ex ...

1. ECON 417: Economics of Uncertainty
Contents
I Expected Utility Theory 3
1 Lotteries 3
2 St. Petersburg’s Paradox 3
3 von Neumann and Morgenstern Axioms and Expected Utility
Form 4
4 Risk Attitudes 5
5 Risk Premium and Certainty Equivalent 6
6 Measures of Risk Aversion 6
II Mean-Variance Optimization 8
7 One Riskfree Asset, One Risky Asset 8
8 Many Risky Assets 9
9 One Riskfree Asset, Many Risky Assets 9
10 Diversification 10
11 Capital Asset Pricing Model 10
III Insurance 11

2. 12 Utility Maximization 11
12.1 Tangency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 11
12.2 Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 12
12.3 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 12
1
13 State Preference Approach to Insurance 13
14 Overview of Insurance 15
15 Demand for Insurance 15
15.1 Mossin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 16
15.1.1 Actuarially Fair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 16
15.1.2 Not Actuarially Fair . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 16
15.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 17
15.3 Coinsurance and Deductibles . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 17

3. 16 Supply of Insurance 18
16.1 Risk pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 18
16.2 Risk spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 19
16.3 “Undersupply” of Full Insurance . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 19
17 Asymmetric Information 20
17.1 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 20
17.1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 20
17.1.2 Tangency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 21
17.1.3 Two types of consumers, symmetric information . . . . . .
. . . . . . . . . . 22
17.1.4 Two types of consumers, asymmetric information . . . . . .
. . . . . . . . . 23
17.1.5 Pooling Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 24
17.1.6 Separating Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 25
17.2 Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. . . . . . . . . 26
IV After the Midterm 27
17.3 Insurance (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 27
18 The Value of Information 27
19 Options 27
19.1 Financial Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 27
19.2 Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 27
2
Part I
Expected Utility Theory
1 Lotteries
For decision making under uncertainty, we consider lotteries.
Lotteries are representations of risky alternatives.
Definition 1 (Simple Lottery). A simple lottery L is a list L =
(p1, ..., p N ) with pi ≥ 0 for all N and∑N
i =1 pi = 1, where pi is the probability of outcome i occurring.
The outcomes that may result

5. are certain: x1, ..., xN .
Definition 2 (Compound Lottery). A compound lottery is a
lottery in which the outcomes are
simple lotteries.
We can find the probability of each outcome (terminal node) in
a compound lottery by multi-
plying the probabilities of the branches leading to the outcome.
Definition 3 (Reduced Lottery). The overall probability measure
R on X is a simple lottery that
we call the reduced lottery of the compound lottery.
R = x1P (x1) + ... + xN P (xN )
The theoretical analysis of expected utility rests on the
consequentialist premise:
We assume that for any risky alternative, only the reduced
lottery over final outcomes is of rel-
evance to the decision maker.
2 St. Petersburg’s Paradox
Suppose someone offers to toss a fair coin repeatedly until it
comes up heads, and
to pay you $1 if this happens on the first toss, $2 if it takes two
tosses to land a head,
$4 if it takes three tosses, $8 if it takes four tosses, etc.
Question: How much is the lottery worth? How much are you
willing to pay to play this lottery?

6. 3
Expected value of the St. Petersburg problem:
E (X ) =
∞∑
i =1
pi xi
= ( 1
2
) · 1 + ( 1
2
)2 · 2 + ( 1
2
)3 · 4 + ...
= ∞
Paradox: If I charged $1 million to play the game, I would
surely have no takers, despite the fact
that $1 million is still considerably less than the expected value
of the game.
Bernoulli:
• argued that individuals do not care directly about the dollar
prizes of a game; rather they

7. care about the utility of prizes
• considered u(x ) = log x , which exhibits diminishing marginal
utility
3 von Neumann and Morgenstern Axioms and Expected Utility
Form
von Neumann and Morgenstern describe necessary and
sufficient conditions for the represen-
tation of a utility function.
In expected utility theory, preference relations, %, are
characterized by 3 axioms:
1. Weak-Order requires that the preference relation be complete
and transitive
• Completeness requires that all elements are comparable.
For L1, L2 ∈ L , the preference relation is complete if either L1
% L2 or L2 % L1
• Transitivity requires that choices be consistent.
For L1, L2, L3 ∈ L , the preference relation is transitive if L1
% L2 and L2 % L3 implies
L1 % L3
2. Continuity means that small changes in probabilities do not
change the nature of the
ordering between two lotteries
If L1, L2 ∈ L are such that L1 % L2, then for all L3 ∈ L ,
there is an α such that 0 < α < 1 and L1 Â (1 −α)L2 +αL3
and there is a β such that 0 < β < 1 and (1 −β)L1 +βL3 Â L2

8. 4
For example, if a “beautiful and uneventful trip by car” is
preferred to “staying home,”
then a mixture of the outcome “beautiful and uneventful trip by
car” with a sufficiently
small but positive probability of “death by car accident” is still
preferred to “staying home.”
3. Independence (of irrelevant alternatives) means that if we
mix each of the two lotteries
with a third one, then the preference ordering of the two
resulting mixtures is indepen-
dent of the particular third lottery used
For all lotteries L1, L2, L3 ∈ L and all α such that 0 < α < 1,
L1 Â L2 ⇐⇒ αL1 + (1 −α)L3 % αL2 + (1 −α)L3
The Allais Paradox is a violation of the Independence Axiom.
Theorem 1 (Expected Utility Theorem). If the decision maker’s
preferences over lotteries satisfy
the weak-order, continuity, and independence axioms, then her
preferences are representable by
a utility function with the expected utility form.
EU (L) =
N∑
i =1

9. pi u(xi ) = p1u(x1) + ... + p N u(xN )
Criteria for Maximization
L1 % L2 ⇐⇒ EU (L1) ≥ EU (L2)
4 Risk Attitudes
Risk aversion captures the idea that individuals dislike risk and
uncertainty.
Definition 4 (Fair bet). A fair bet is a random game with a
specified set of prizes and associated
probabilities that has an expected value of zero.
Definition 5 (Concavity). The function f (x ) is concave if a
straight line joining any two points
on it lies entirely below the function itself. In other words, the
function f (x ) is concave if for
any x1 and x2, and λ : 0 ≤ λ ≤ 1,
f (λx1 + (1 −λ)x2) ≥ λf (x1) + (1 −λ) f (x2)
If f (x ) is a concave (twice differentiable) function, then f ′′(x )
< 0.
A decision maker is risk averse if
5
1. at any level of wealth, he rejects every fair bet
2. he strictly prefers a certainty consequence to any risky

10. prospect whose mathematical ex-
pectation of consequences equals that certainty
3. u(x ) is concave. In other words, for every lottery with
outcomes x1, ..., xN and probabili-
ties p1, ..., p N , respectively
u(
N∑
i =1
pi xi ) ≥
N∑
i =1
pi u(xi )
Concavity of the utility function implies diminishing marginal
utility.
A decision maker is risk loving if
1. at any level of wealth, he accepts every fair bet
2. he strictly prefers the lottery to its mathematical expectation
3. u(x ) is convex. In other words, for every lottery with
outcomes x1, ..., xN and probabilities
p1, ..., p N , respectively
u(
N∑

11. i =1
pi xi ) ≤
N∑
i =1
pi u(xi )
A decision maker is risk neutral if
1. at any level of wealth, he is indifferent to every fair bet
2. he is indifferent between the lottery and its mathematical
expectation
5 Risk Premium and Certainty Equivalent
Definition 6 (Certainty Equivalent). The amount of money, C E
(L), when obtained for certain,
provides the same expected utility as the lottery
Definition 7 (Risk Premium). The maximum amount, π, that an
individual is willing to forego
in order to receive the expected value of the lottery with
certainty
The risk premium is the difference between the expected value
of the lottery and the certainty
equivalent of the lottery.
π = E (L) −C E (L)
6 Measures of Risk Aversion

12. The Arrow-Pratt measures of risk aversion are quantitative
measures of how averse to risk a
person is. It provides a way to measure the degree of concavity
of the utility function (hence,
the strength or intensity of risk aversion).
6
Definition 8 (Absolute Risk Aversion). The absolute risk
aversion measure A(x ) for a utility
function u(x ) is
A(x ) ≡ −u
′′(x )
u(x )
Definition 9 (Relative Risk Aversion). The relative risk
aversion measure R (x ) for a utility func-
tion u(x ) is
R (x ) ≡ −x u
′′(x )
u(x )
= x · A(x )
Definition 10 (DARA, CARA, IARA). The utility function u(·)
has decreasing (constant, increas-
ing) absolute risk aversion if A(x , u) is a decreasing (constant,
increasing) function of x . This

13. depends on the sign of the derivative of A(x ) with respect to x ,
i.e. d A(x )d x .
Empirical evidence supports DARA. The power utility function
exhibits DARA.
Definition 11 (DRRA, CRRA, IRRA). The utility function u(·)
has decreasing (constant, increas-
ing) relative risk aversion if R (x , u) is a decreasing (constant,
increasing) function of x . This
depends on the sign of the derivative of R (x ) with respect to x
, i.e. d R (x )d x .
7
Part II
Mean-Variance Optimization
V (µ,σ)
• µ is the expected return of the asset
• σ is the standard deviation of the asset. Risk is measured by
the standard deviation.
Risk attitudes are determined by the partial derivatives with
respect to risk
• δV
δσ
< 0 risk averse

14. • δV
δσ
> 0 risk loving
• δV
δσ
= 0 risk neutral
Typically, financial economists think of investors as being risk
averse, thus investors trade off
risk and return.
The risk-return tradeoff:
• A risk averse, mean-variance optimizing investor will only
accept a riskier portfolio if the
expected return of that portfolio is appropriately higher
• A risk averse, mean-variance optimizing investor will only
accept a portfolio that has a
lower expected return if the risk of that portfolio is
appropriately lower
Consider the particular functional form for a mean-variance
optimizer:
V (µ,σ) = µ− 1
2
A ·σ2
where µ is expected return, σ2 is the variance, and A is the

15. coefficient of risk aversion ( A > 0).
7 One Riskfree Asset, One Risky Asset
Assume that an investor must decide how to invest all of her
wealth and has only two options:
a riskfree asset, R f and a risky asset. The expected return of the
risky asset is E (Ri ) and its
variance is V ar (Ri ) = σ2i . To determine the optimal fraction
of wealth an investor will allocate
to a risky asset, k∗ , consider the following maximization
problem
max
k
V (µp ,σp ) = µp −
1
2
A ·σ2p
= E (Rp ) −
1
2
A · V ar (Rp )
= R f + k (E (Ri ) − R f ) −
1
2
A · k 2V ar (Ri )

16. 8
Solve the optimization problem. The first-order condition
requires that the derivative, with
respect to k , is equal to zero. We find
k∗ =
(E (Ri ) − R f )
A · V ar (Ri )
= S
A ·σi
Definition 12 (Sharpe Ratio). The Sharpe Ratio, S of the risky
asset is the expected excess re-
turn of the risky asset per unit of its standard deviation. It is the
reward-to-variability ratio of
investing in the risky asset.
S =
E (Ri ) − R f
σi
Definition 13 (Capital Allocation Line). A graph of all possible
expected returns and standard
deviations of a portfolio formed by combining the risky asset
with the riskfree asset.
8 Many Risky Assets
Consider a portfolio of two assets with weights k1 and k2,

17. expected returns E (R1) and E (R2),
and return variances σ21 and σ
2
2.
The portfolio expected return is
E (Rp ) = k1E (R1) + k2E (R2)
The portfolio variance is
V ar (RP ) = k 21σ21 + k 22σ22 + 2 · k1 · k2 ·C ov (R1, R2)
The graph is a hyperbola when volatility is plotted on the x-axis
and expected returns are plotted
on the y-axis.
Definition 14 (Efficient Frontier). A graph of the feasible
investments with the highest expected
returns for all possible portfolio standard deviations. It is the
top part of the graph above the
minimum variance portfolio.
9 One Riskfree Asset, Many Risky Assets
Definition 15 (Capital Market Line). The line from the riskfree
investment through the efficient
portfolio of risky assets when volatility is plotted on the x-axis
and expected returns are plotted
on the y-axis.
9

18. Definition 16 (Tangency Portfolio). The portfolio of risky assets
with the highest Sharpe Ratio.
It is an efficient portfolio and it generates the steepest line
combined with the riskfree asset.
Theorem 2 (Mutual Fund (Separation) Theorem). Investors with
the same beliefs about expected
returns, risks, and correlations all will invest in the portfolio or
“fund” of risky assets that has the
highest Sharpe Ratio, but they will differ in their allocations
between this fund and the riskfree
asset based on their risk tolerance.
10 Diversification
The risk of a stock includes idiosyncratic risk and market risk.
Idiosyncratic risk is also known
as firm-specific, unique, stand-alone, or diversifiable risk.
Market risk is also known as system-
atic or undiversifiable risk.
To limit your exposure to idiosyncratic risk, you can diversify
your portfolio. This means choos-
ing stocks that are imperfectly correlated, i.e. ρ → −1, where ρ
is the correlation coefficient.
The benefit of diversification will increase the further away
from ρ = 1.
Definition 17 (Risk premium). It represents the additional return
that investors expect to earn
to compensate them for a security’s risk. It is the difference

19. between the expected return of the
security minus the riskfree rate of return.
E (Ri ) − R f
11 Capital Asset Pricing Model
Intuition for the Capital Asset Pricing Model (CAPM)
1. Because diversification does not reduce market risk, the risk
premium of a security should
be determined by its market risk.
2. To measure market risk, we need a market portfolio. If all
investors are mean-variance
optimizers, by the Mutual Fund Theorem, they should be
holding the Tangent Portfolio.
Let the Market Portfolio be the Tangent Portfolio.
CAPM relates the security’s risk premium to the market risk
premium.
E (Ri ) − R f = β· (E (Rm ) − R f )
and β, which measures the sensitivity of the security’s return to
the return of the overall market
is
β = C ov (Ri , Rm )
V ar (Rm )

20. 10
Part III
Insurance
12 Utility Maximization
There are three methods you can use to solve the utility
maximization problem:
max
x1 ,x2
u(x1, x2) subject to their budget constraint: I = p1 x1 + p2 x2
12.1 Tangency Condition
The slope of the indifference curve and the slope of the budget
line should be equal at the point
of tangency. It is the point at which the consumer maximizes his
or her utility, given his or her
budget constraint.
slope of indifference curve = slope of budget line
MRSx1 x2 =
p1
p2
MU1
MU2

21. = MRSx1 x2 =
p1
p2
Example 1. Suppose we had the following utility function
max
x1 ,x2
u(x1, x2) = log x1 + log x2 subject to their budget constraint: I
= p1 x1 + p2 x2
slope of indifference curve = slope of budget line
1
x1
1
x2
= p1
p2
⇒ x2 =
p1
p2
x1
Plug into the budget constraint
I = p1 x1 + p2 x2
I = p1 x1 + p2
p1

22. p2
x1
I = 2p1 x1
⇒ x∗ 1 =
I
2p1
and x∗ 2 =
I
2p2
11
12.2 Substitution Method
Example 2. Consider the following constrained problem with
two variables
max
x1 ,x2
log x1 + log x2
s.t
p1 x1 + p2 x2 = I
The idea of the substitution method is to use the constraints to
get rid of some variables. In the

23. example above we can use the constraint to obtain that x2 = I
−p1 x1p2 , and after we plug this into
the objective function we get
ũ(x1) = log x1 + log
I − p1 x1
p2
This becomes an unconstrained maximization problem for a
function of one variable x1. Using
the chain rule we obtain the following first order condition
(FOC)
0 = ũ′(x1) =
1
x1
+ p2
I − p1 x1
(−p1
p2
)
= 1
x1
− p1
I − p1 x1
which yields I − p1 x1 = p1 x1, the solution of this equation is
x∗ 1 = I2p1 .
By the chain rule and the power rule we have

24. ũ′′(c1) = −
1
x 21
− (−1)(−p1)
p1
(I − p1 x1)2
= − 1
x 21
−
p 21
(I − p1 x1)2
Clearly ũ′′(c1) < 0 for any x1 and so the sufficient condition for
a local maximum is satisfied.
Finally, using the constraint p1 x
∗
1 + p2 x∗ 2 = I we get x∗ 2 = I2p2 , so the solution of the
problem is
the consumption bundle (x∗ 1 , x
∗
2 ) = ( I2p1 ,
I
2p2
).
12.3 Lagrange Multipliers

25. Theorem 3. Let f and g be two real-valued continuously
differentiable functions of two vari-
ables. Suppose that (x∗ 1 , x
∗
2 ) is a solution to the following maximization problem
max
x1 ,x2
f (x1, x2)
subject to
g (x1, x2) = 0
and that (x∗ 1 , x
∗
2 ) is not a critical point of g . Then there exists a real number λ
∗ called the lagrange
multiplier, such that (x∗ 1 , x
∗
2 ,λ
∗ ) is a critical point of the following function, called a
Lagrangian
L (x1, x2,λ) = f (x1, x2) +λg (x1, x2)
12
i.e. all three partial derivatives of L are zero

26. ∂L
∂x1
(x∗ 1 , x
∗
2 ,λ
∗ ) = 0
∂L
∂x2
(x∗ 1 , x
∗
2 ,λ
∗ ) = 0
∂L
∂λ
(x∗ 1 , x
∗
2 ,λ
∗ ) = 0
Example 3. Let’s apply the Lagrange Theorem to the
consumer’s problem from previous sec-
tion.
max
x1 ,x2

27. log x1 + log x2
s.t. p1 x1 + p2 x2 = I
The objective function is f (x1, x2) = log x1+log x2, the
constraint function is g (x1, x2) = I −p1 x1−
p2 x2, and the Lagrangian function is
L (x1, x2,λ) = log x1 + log x2 +λ
(
I − p1 x1 − p2 x2
)
From the Lagrange Theorem, the First Order Necessary
Condition is that all partial derivatives
of the Lagrangian are zero, i.e.
∂L
∂x1
(x∗ 1 , x
∗
2 ,λ
∗ ) = 0 ⇒ 1
x∗ 1
−λ∗ p1 = 0
∂L
∂x2
(x∗ 1 , x
∗
2 ,λ

28. ∗ ) = 0 ⇒ 1
x∗ 2
−λ∗ p2 = 0
∂L
∂λ
(x∗ 1 , x
∗
2 ,λ
∗ ) = 0 ⇒ I − p1 x∗ 1 − p2 x∗ 2 = 0
Note that last equation simply says that the constraint in the
maximization problem has to hold.
The above is a system of 3 equations and 3 unknowns (x∗ 1 , x
∗
2 ,λ
∗ ) and is quite easy to solve. We
get:
x∗ 1 =
I
2p1
x∗ 2 =
I
2p2
λ∗ = 2

29. I
13 State Preference Approach to Insurance
Goal: To show that when faced with fair markets in contingent
claims on wealth, a risk averse
person will choose to ensure that he has the same level of
wealth regardless of which state oc-
curs.
Categorize all of the possible things that might happen into a
fixed number of states. We say
that contingent commodities are goods delivered only if a
particular state of the world occurs.
13
Consider the following expected utility model of two contingent
goods: Wg is wealth in good
times and Wb is wealth in bad times.
max EU (Wg , Wb ) = p u(Wb ) + (1 − p )u(Wg )
Initial wealth is W . Assume that this person can purchase a
dollar of wealth in good times for
qg and a dollar of wealth in bad times for qb .
1 The price ratio
qg
qb

30. shows how this person can
trade dollars of wealth in good times for dollars in bad times.
W̃ = qb Wb + qg Wg
We say that prices are actuarially fair if the price ratio reflects
the odds ratio:
qg
qb
= 1 − p
p
Example 4. Consider the following expected utility
maximization problem:
max EU (Wg , Wb ) = p log(Wb ) + (1 − p ) log(Wg ) subject to
W̃ = qb Wb + qg Wg
We can use the tangency condition to solve.
slope of indifference curve = slope of budget line
1−p
Wg
p
Wb
=
qg
qb
1 − p
p

31. Wb
Wg
=
qg
qb
Use the condition that insurance is actuarially fair to simplify,
and we get:
Wg = Wb
The individual is willing to pay an indemnity or cover for
reduced wealth in the “good state" so
that he can have the same level of wealth in the event of a loss.
Wg = Wb
W − qC = W − L − qC +C
where L is the loss, C is the cover, qC is the premium expressed
as the the product of the cover
and a premium rate.
1I use q for prices because I don’t want to confuse it with p for
probability.
14
14 Overview of Insurance
Insurance occurs when one party agrees to pay an indemnity (a

32. promise to pay for the cost of
possible damage, loss, or injury) to another party in case of the
occurrence of a pre-specified
random event generating a loss for the initial risk-bearer.
Definition 18 (Risk transfer). Insurance is the most common
form of risk transfer. The shifting
of risk is of considerable importance for the functioning of our
modern economies.
• Insurance is a particular example of a type of risk-transfer
strategy known as hedging.
Hedging strategies typically involve entering into contracts
whose payoffs are negatively
related to one’s overall wealth or to one component of that
wealth. Thus, for example, if
wealth falls, the value of the contract rises, partially offsetting
the loss in wealth.
The basic characteristics of all insurance contracts are:
• specified loss events
• losses, L
• cover (indemnity), C
• premium, Q . Q = qC is a common, but not universal, way of
expressing the insurance
premium.
15 Demand for Insurance

33. Question: How much insurance would a risk-averse person buy?
What is the demand for cover?
Answer:
max
C
EU = p u(W − L − qC +C ) + (1 − p )u(W − qC )
The first order condition is:
d EU
d C
= p u′(W − L − qC +C )(1 − q ) − (1 − p )u′(W − qC )q = 0
u′(W − L − qC +C )
u′(W − qC ) =
(1 − p )q
p (1 − q ) > 1
15
15.1 Mossin’s Theorem
15.1.1 Actuarially Fair
We say that insurance is actuarially fair if the expected payout
of the insurance company just
equals the cost of the insurance.
expected payout is probability of loss times the cover =

34. expected cost is the insurance premium
pC = qC
p = q
We might expect a competitive insurance market to deliver
actuarially fair insurance. In this
case, the first order condition simplifies to:
u′(W − L − qC +C ) = u′(W − qC )
The consumer should fully insure and set the cover equal to the
loss C ∗ = L (full cover).
Mossin’s Theorem states that a risk averse individual offered
insurance at a fair premium will
always choose full cover.
q = p ⇐⇒ u′(W − qC ∗ ) = u′(W − L − qC ∗ +C ∗ ) ⇐⇒ C ∗ = L
15.1.2 Not Actuarially Fair
Question: What happens if the price of insurance is above the
actuarially fair price, i.e. q > p ?
u′(W − L − qC +C )
u′(W − qC ) =
(1 − p )q
p (1 − q ) > 1
Mossin’s Theorem
With a positive loading, the buyer chooses partial cover;
q > p ⇐⇒ u′(W − qC ∗ ) < u′(W − L − qC ∗ +C ∗ ) ⇐⇒ C ∗ < L

35. With a negative loading the buyer chooses more than full cover;
q < p ⇐⇒ u′(W − qC ∗ ) > u′(W − L − qC ∗ +C ∗ ) ⇐⇒ C ∗ > L
where the last two results follow from the fact that u′(·) is
decreasing in wealth, i.e., from risk
aversion.
16
15.2 Comparative Statics
From the first order condition, we can in principle solve for the
optimal cover as a function of
the exogenous variables of the problem: wealth, the premium
rate, the amount of loss, and the
loss probability. The buyer’s demand for cover function can be
expressed as:
C ∗ = C (L, p, W , Q )
Question: How does the demand for cover change as wealth, the
premium rate (price of insur-
ance), the amount of loss, and the loss probability change?
• Amount of loss, L, i.e. δC
∗
δL . A ceteris paribus increase in L increases the demand for
insur-
ance.

36. • The loss probability, p , i.e. δC
∗
δp . An increase in the risk of loss increases the demand for
cover.
• Wealth, W , i.e. δC
∗
δW
Proposition 1. If p = q , the agent will insure fully at C ∗ = L
for all wealth levels. If p < q ,
the agent’s insurance coverage as a function of wealth, C ∗ (W )
will decrease (increase)
with wealth if the agent has decreasing (increasing) absolute
risk aversion.
• Premium rate, Q , i.e. δC
∗
δQ . The total effect on insurance demand depends on the
relative
magnitudes of the income and substitution effect.
15.3 Coinsurance and Deductibles
Proposition 2. Under a reasonable set of conditions, the optimal
insurance contract always
takes the form of a straight deductible.
Under proportional coinsurance we have cover

37. C = αL, α ∈ [0, 1]
with α = 0 implying no insurance and α = 1 implying full cover.
Under a deductible we have
C =
{
0 for L ≤ D
L − D for L > D
where D denotes the deductible and D = 0 implies full cover.
Given the premium amount Q and wealth W in the absence of
loss, the buyer’s state-contingent
wealth in the case of proportional coinsurance is
Wα = W − L −Q +C = W − (1 −α)L −Q
17
and in the case of a deductible is
WD = W − L −Q +C = W − L −Q + max(0, L − D )
For losses above the deductible, her wealth becomes certain,
and equal to
ŴD = W − L −Q + (L − D ) = W −Q − D
A straight deductible insurance policy efficiently concentrates
the effort of indemnification on
only the largest losses.

38. 16 Supply of Insurance
16.1 Risk pooling
When an insurer enters into insurance contracts with a number
of individuals, or a group of
individuals agrees mutually to provide insurance to each other,
the probability distribution of
the aggregate losses they may suffer differs from the loss
distribution facing any one individual.
Assume
• There are i = 1, 2, ..., N individuals with identically and
independently distributed risks
• C̃i is the loss claim for each individual (the cover paid by the
insurance company in the
event of a loss)
• µ is the expected claims cost (across the population) and σ2 is
the variance of the expected
claims costs
• Each C̃i has the same probability distribution with mean µ and
variance σ
2
Let C
̄ N = 1N
∑N
i =1 Ci be the sample mean or average loss per contract (to the

39. insurance com-
pany).2
Proposition 3 (By the Law of Large Numbers).
lim
N →∞
Pr[|C
̄ N −µ| < ²] = 1
In words, as N becomes increasingly large, the average loss per
contract will be arbitrarily close
to the value µ with probability approaching 1.
2A sample is a (randomly) generated subset of the population
under study. The parameters of the population in-
clude its mean, µ, variance, σ2, and its standard deviation, σ.
The statistics of the sample include the sample mean
(or average), X
̄ = 1N
∑N
i =1 X i , the (unbiased) sample variance is s
2 = 1N −1
∑N
i =1(X i − X
̄ )2, and the sample standard
error is the sample standard deviation divided by the square root
of the sample size, i.e. sp
N
.

40. 18
Stated differently,
for a sufficiently large number of insurance contracts, it is
virtually certain that the loss per
contract is just about equal to the mean of the loss claims
distribution.
Furthermore, the variance of the realized loss per contract about
the mean of loss claims goes
to zero as N becomes increasingly large.
Var(C
̄ N ) = Var(
1
N
N∑
i =1
Ci ) =
1
N 2
· Nσ2 = σ
2
N
16.2 Risk spreading

41. When risks are not covered by insurance companies, the
government can intercede by transfer-
ring money among parties. The government can spread the risk
to increase social welfare.
As a risk is spread over an increasing number of individuals, the
total cost of the risk tends to
zero and the price individuals are willing to pay for the risky
prospect tends to the expected
value of the project.
Theorem 4 (Arrow-Lind). Under certain assumptions, the social
cost of risk moves to zero as
the population tends to infinity, so that projects can be
evaluated on the basis of expected net
benefit alone. A necessary condition for the results is that the
covariance between the individual’s
wealth from the insurance business and his marginal utility of
wealth, if he does not share in this
business, must be zero.
The Arrow-Lind Theorem provides a basis for the assumption
that the insurer is risk neutral.
16.3 “Undersupply” of Full Insurance
1. Transactions (or insurance) costs include: drawing up and
selling new insurance con-

42. tracts, administering the stock of existing contracts, processing
claims, estimating loss
probabilities, calculating premiums, and administering the
overall business. The Raviv
model shows how the existence of deductibles and coinsurance
in the (equilibrium) in-
surance contract is related to the nature of insurance costs.
2. Nondiversifiable risks cannot be insured.
3. Adverse selection: individuals know their risk better than the
insurance company
4. Moral hazard: individuals can take certain actions to reduce
the probability of loss
19
17 Asymmetric Information
Markets may not be fully efficient when one side has
information that the other side does not
(asymmetric information). Carefully designed contracts may
reduce such problems by provid-
ing incentives to reveal one’s information and take appropriate
actions.
Principal-Agent Model
There are two economic agents in this model: the informed

43. party and the uninformed party.
One party will propose a “take it or leave it” (TIOLI) contract
and therefore request a “yes or no”
answer; the other party is not free to propose another contract.
The principal is the one who
proposes the contract and the agent is the party who just has to
accept or reject the contract.
Hidden Type
The uninformed party is imperfectly informed of the
characteristics of the informed party; the
uninformed party moves first. The agent has private information
about the state of the world
before signing the contract with the principal. The agent’s
private information is called his type.
For historical reasons stemming from its application in the
insurance context, the hidden-type
model is also called an adverse selection model.
Hidden Action
The uninformed party moves first and is imperfectly informed
of the actions of the informed
party. The agent’s actions taken during the term of the contract
affect the principal, but the
principal does not observe these actions directly. The principal
may observe outcomes that

44. are correlated with the agent’s actions but not the actions
themselves. For historical reasons
stemming from the insurance context, the hidden-action model
is called a moral hazard model.
17.1 Adverse Selection
Adverse selection is defined as the situation where the insured
has better information about
her risk type than the insurer. We then say that the individual
risk is her private information.
We will consider the Rothschild and Stiglitz (1976) model of
adverse selection in competitive
insurance markets.
17.1.1 Basic model
Basic Model
• The individual is risk averse
• Individual is endowed with wealth, W
• In the event of a loss, the individual will have W − L
• p is the probability of the loss
20
• He can insure himself by paying a premium Q = qC in return
for a cover C , if a loss occurs

45. • The pair (Q , C ) completely describes the insurance contract
• Insurance contracts are exclusive: each individual can take on
only a single insurance
contract
Demand for Insurance
EU = p u(Wb ) + (1 − p )u(Wg )
where u(x ) is the utility of money income; u(x) is an increasing
concave function. An individual
chooses the insurance contract that maximizes his expected
utility.
Supply of Insurance
• Companies are risk neutral and are concerned only about
expected profits:
π = Q − pC
• A perfectly competitive market ⇒ zero economic profits
• Zero administrative costs
• Free entry
• Each firm can offer only one contract
Equilibrium in a competitive insurance market is a set of
contracts such that when individuals
choose contracts to maximize expected utility
1. No contract in the equilibrium set makes negative expected
profits

46. 2. No contract outside the equilibrium set that, if offered, will
make a nonnegative profit
Every firm makes zero profits and no firm (existing or new) can
make positive profits by offering
a new contract.
17.1.2 Tangency Condition
Budget Line
Final wealth in the two states of the world are
W̃ =
{
Wg = W − qC in “good” state
Wb = W − L + (1 − q )C in “bad” state
To find the budget line, multiply Wg by (1 − q ) and Wb by (q ).
Then add the two equations.
(1 − q )Wg + qWb = W − qC − qW + q 2C + qW − q L + qC − q
2C
= (1 − q )W + q (W − L)
= W − q L
21
Solve for Wb and you’ll get
Wb =
W − q L

47. q
− 1 − q
q
Wg
This is a straight line passing through the point (W , WL ), i.e.
the no insurance point, and having
a negative slope equal to
1−q
q in absolute value.
Marginal Rate of Substitution (MRS)
Recall from microeconomics that the marginal rate of
substitution is the slope of the indiffer-
ence curve, i.e. M R S = − x1x2 . It describes how much x2 a
person is willing to give up in order
to get more x1 and remain indifferent between the two
consumption bundles. For example, if
M R S = 5 then the consumer is willing to give up 5 units of x2
to get one unit of x1. The M R S is
also equal to the ratio of the marginal utilities.
From the expected utility maximization function, we find that
M R S =
MUW g
MUW b
= (1 − p )
p

48. u′(Wg )
u′(Wb )
17.1.3 Two types of consumers, symmetric information
Suppose that the market consists of two kinds of customers:
• low risk individuals with loss probability, pL
• high risk individuals with loss probability, p H
• Note that 1 > p H > pL > 0
22
The MRS for each type is
M R SL =
(1 − pL )
pL
u′(Wg )
u′(Wb )
andM R S H =
(1 − p H )
p H
u′(Wg )
u′(Wb )
The slope of the indifference curve of low risks is steeper than
that of high risks.

49. In the first-best, symmetric information case, the insurance
company can observe the individ-
ual’s risk type and offer a different policy to each. In the
competitive market, each type can get
a separate contract with an actuarially fair premium and chooses
full coverage.
17.1.4 Two types of consumers, asymmetric information
Question: What happens when the individual has private (not
observable or verifiable) infor-
mation about his type?
Intuition: If the same full insurance contracts for each group
were offered, but types are not
observable, then all individuals would choose the low type
insurance contract. This could lead
to negative profits for the firm. Why? Insurers break even
serving only the low-risk types, so
adding individuals with a higher probability of loss would push
the company below the break-
even point. Therefore, we cannot offer full insurance to both
types.
23
There are two types of equilibria to consider: pooling and
separating.

50. Definition 19 (Pooling equilibrium). Pooling equilibrium in a
competitive screening model is
an equilibrium where each type of agent chooses the same
contract.
Definition 20 (Separating equilibrium). A separating
equilibrium is a competitive screening
model is where different types purchase different contracts.
17.1.5 Pooling Equilibrium
Proposition 4. There cannot be a pooling equilibrium.
Intuition: The pooling equilibrium cannot be a final equilibrium
because there exist insurance
policies that are unattractive to high-risk types, attractive to
low-risk types, and profitable to in-
surers. These policies will involve “cream-skimming” behavior:
the policies will attract low-risk
types away from the pooling contract. The insurers that
continue to offer the pooling contract
are left with individuals whose risk is so high that insurers
cannot expect to earn any profit by
serving them.
24
17.1.6 Separating Equilibrium

51. Proposition 5. The separating equilibrium will involve
actuarially fair full insurance for the
high risk types and low risk individuals will be offered the best
possible partial insurance con-
tract at a fair price, conditional on that contract being
unattractive to high-risk individuals.
Definition 21 (Incentive compatibility constraints). The
incentive compatibility (IC) constraints
state that each consumer prefers the contract that was designed
for him.
Intuition: We need to consider incentive compatibility
constraints. There is no reason to distort
the choice of insurance for the high-risk types, because low risk
individuals do not have any
incentive to “pretend” to be high risk. But we need to make sure
the high risk types don’t pretend
to be low risk types. The incentive compatibility constraint for
the high type requires that the
contract designed for the low risk type be below or on the
indifference curve of the high risk
type that goes through the full insurance contract.
25
Existence of a separating equilibrium:
“An equilibrium will not exist if the costs to the low-risk
individual of pooling are low (because

52. there are relatively few of the high-risk individuals who have to
be subsidized, or because the
subsidy per individual is low, i.e. when the probabilities of the
two groups are not too different),
or if their costs of separating are high” (Rothschild and Stiglitz,
1976).
17.2 Moral Hazard
In the moral hazard model of insurance, the probability of the
loss state may depend on the
behavior of the insured individual. This creates an incentive
problem that leads to less than full
insurance, so that the insured retains some incentive to behave
differently.
Suppose
• a risk-averse individual faces the possibility of incurring a
loss, L, that will reduce his
wealth, W
• the probability of loss is p and is a decreasing convex function
of effort, e (or level of care)
• exerting effort is costly, i.e. c (e ) in an increasing function in
effort; let c (e ) = e (The insur-
ance company cannot monitor the individual’s level of effort, e
).

53. • u(x ) is the individual’s utility given wealth
The individual’s expected utility as a function of the effort or
level of care chosen is
EU = p (e )u(Wb ) + (1 − p (e ))u(Wg )
= p (e )u(W − e −Q − L +C ) + (1 − p (e ))u(W − e −Q )
26
The expected profit of the (risk-neutral) insurance company is
π = Q − p (e )C
An actuarially fair insurance contract would set a premium
equal to the expected coverage, i.e.
Q = p (e )C .
The timing is as follows:
• The principal offers an insurance contract (Q , C )
• The individual decides to accept or reject the contract
• The individual chooses an effort level, e
Definition 22 (Participation Constraint). The participation, or
individually rational (IR), con-
straint guarantees that the consumer will accept the contract.
The individual must be at least
as well off as he would be if he accepted the next best
alternative. (No insurance may be the

54. next best alternative).
In our setting, the optimal contract must
• satisfy the zero-profit constraint (the IR constraint for the
firm)
• satisfy the IR or participation constraint for the individual
• ensure that the effort level in the contract is credible in the
sense that it will be chosen by
the agent under the incentives provided by the rest of the
contract.
Part IV
After the Midterm
17.3 Insurance (cont.)
18 The Value of Information
19 Options
19.1 Financial Options
19.2 Real Options
27
Journal of Economic Perspectives—Volume 25, Number 1—
Winter 2011—Pages 115–138

55. FF rom the large-scale social insurance programs of Social
Security and Medi-rom the large-scale social insurance
programs of Social Security and Medi-care to the heavily
regulated private markets for property and casualty care to the
heavily regulated private markets for property and casualty
insurance, government intervention in insurance markets is
ubiquitous. The insurance, government intervention in insurance
markets is ubiquitous. The
fundamental theoretical reason for such intervention, based on
classic work from fundamental theoretical reason for such
intervention, based on classic work from
the 1970s, is the problem of adverse selection. But despite the
age and infl uence the 1970s, is the problem of adverse
selection. But despite the age and infl uence
of the theory, systematic empirical examination of selection in
actual insurance of the theory, systematic empirical examination
of selection in actual insurance
markets is a relatively recent development. Indeed, in awarding
the 2001 Nobel markets is a relatively recent development.
Indeed, in awarding the 2001 Nobel
Prize for the pioneering theoretical work on asymmetric
information to George Prize for the pioneering theoretical work
on asymmetric information to George
Akerlof, Michael Spence, and Joseph Stiglitz, the Nobel
committee noted this Akerlof, Michael Spence, and Joseph
Stiglitz, the Nobel committee noted this
paucity of empirical work (Nobelprize.org, 2001).paucity of
empirical work (Nobelprize.org, 2001).
Over the last decade, however, empirical work on selection in
insurance markets Over the last decade, however, empirical
work on selection in insurance markets
has gained considerable momentum, and a fairly extensive (and
still growing) has gained considerable momentum, and a fairly
extensive (and still growing)

56. empirical literature on the topic has emerged. This research has
found that adverse empirical literature on the topic has
emerged. This research has found that adverse
selection exists in some insurance markets but not in others. It
has also uncovered selection exists in some insurance markets
but not in others. It has also uncovered
examples of markets that exhibit “advantageous selection”—a
phenomenon not examples of markets that exhibit
“advantageous selection”—a phenomenon not
considered by the original theory, and one that has different
consequences for considered by the original theory, and one that
has different consequences for
equilibrium insurance allocation and optimal public policy than
the classical case equilibrium insurance allocation and optimal
public policy than the classical case
of adverse selection. Researchers have also taken steps toward
estimating the welfare of adverse selection. Researchers have
also taken steps toward estimating the welfare
consequences of detected selection and of potential public
policy interventions.consequences of detected selection and of
potential public policy interventions.
Selection in Insurance Markets: Theory
and Empirics in Pictures
■ ■ Liran Einav is Associate Professor of Economics, Stanford
University, Stanford, California. Liran Einav is Associate
Professor of Economics, Stanford University, Stanford,
California.
Amy Finkelstein is Professor of Economics, Massachusetts
Institute of Technology, Cambridge, Amy Finkelstein is
Professor of Economics, Massachusetts Institute of Technology,
Cambridge,
Massachusetts. Both authors are also Research Associates,
National Bureau of Economic Massachusetts. Both authors are
also Research Associates, National Bureau of Economic

57. Research, Cambridge, Massachusetts. Their e-mail addresses are
Research, Cambridge, Massachusetts. Their e-mail addresses are
⟨ ⟨ [email protected]@stanford.edu⟩ ⟩ and and
⟨ ⟨ afi [email protected][email protected]⟩ ⟩ ..
doi=10.1257/jep.25.1.115
Liran Einav and Amy Finkelstein
116 Journal of Economic Perspectives
In this essay, we present a graphical framework for analyzing
both theoretical In this essay, we present a graphical framework
for analyzing both theoretical
and empirical work on selection in insurance markets. This
graphical approach, and empirical work on selection in
insurance markets. This graphical approach,
which draws heavily on a paper we wrote with Mark Cullen
(Einav, Finkelstein, and which draws heavily on a paper we
wrote with Mark Cullen (Einav, Finkelstein, and
Cullen, 2010), provides both a useful and intuitive depiction of
the basic theory of Cullen, 2010), provides both a useful and
intuitive depiction of the basic theory of
selection and its implications for welfare and public policy, as
well as a lens through selection and its implications for welfare
and public policy, as well as a lens through
which one can understand the ideas and limitations of existing
empirical work on which one can understand the ideas and
limitations of existing empirical work on
this topic.this topic.
We begin by using this framework to review the “textbook”
adverse selection We begin by using this framework to review
the “textbook” adverse selection

58. environment and its implications for insurance allocation, social
welfare, and public environment and its implications for
insurance allocation, social welfare, and public
policy. We then discuss several important extensions to this
classic treatment that are policy. We then discuss several
important extensions to this classic treatment that are
necessitated by important real-world features of insurance
markets and which can necessitated by important real-world
features of insurance markets and which can
be easily incorporated in the basic framework. Finally, we use
the same graphical be easily incorporated in the basic
framework. Finally, we use the same graphical
approach to discuss the intuition behind recently developed
empirical methods approach to discuss the intuition behind
recently developed empirical methods
for testing for the existence of selection and examining its
welfare consequences. for testing for the existence of selection
and examining its welfare consequences.
We conclude by discussing some important issues that are not
well-handled by this We conclude by discussing some important
issues that are not well-handled by this
framework and which, perhaps relatedly, have been little
addressed by the existing framework and which, perhaps
relatedly, have been little addressed by the existing
empirical work; we consider these fruitful areas for additional
research. Our essay empirical work; we consider these fruitful
areas for additional research. Our essay
does not aim at reviewing the burgeoning empirical literature on
selection in insur-does not aim at reviewing the burgeoning
empirical literature on selection in insur-
ance markets. However, at relevant points in our discussion we
point the interested ance markets. However, at relevant points in
our discussion we point the interested
reader to recent papers that review or summarize recent fi
ndings.reader to recent papers that review or summarize recent
fi ndings.

59. Adverse and Advantageous Selection: A Graphical
FrameworkAdverse and Advantageous Selection: A Graphical
Framework
The Textbook Environment for Insurance MarketsThe Textbook
Environment for Insurance Markets
We start by considering the textbook case of insurance demand
and cost, in We start by considering the textbook case of
insurance demand and cost, in
which perfectly competitive, risk-neutral fi rms offer a single
insurance contract which perfectly competitive, risk-neutral fi
rms offer a single insurance contract
that covers some probabilistic loss; risk-averse individuals
differ only in their that covers some probabilistic loss; risk-
averse individuals differ only in their
(privately-known) probability of incurring that loss; and there
are no other fric-(privately-known) probability of incurring that
loss; and there are no other fric-
tions in providing insurance, such as administrative or claim-
processing costs. tions in providing insurance, such as
administrative or claim-processing costs.
Thus, more in the spirit of Akerlof (1970) and unlike the well-
known environment Thus, more in the spirit of Akerlof (1970)
and unlike the well-known environment
of Rothschild and Stiglitz (1976), fi rms compete in prices but
do not compete of Rothschild and Stiglitz (1976), fi rms
compete in prices but do not compete
on the coverage features of the insurance contract. We return to
this important on the coverage features of the insurance
contract. We return to this important
simplifying assumption later in this essay.simplifying
assumption later in this essay.
Figure 1 provides a graphical representation of this case and

60. illustrates the Figure 1 provides a graphical representation of
this case and illustrates the
resulting adverse selection as well as its consequences for
insurance coverage and resulting adverse selection as well as its
consequences for insurance coverage and
welfare. The fi gure considers the market for a specifi c
insurance contract. Consumers welfare. The fi gure considers
the market for a specifi c insurance contract. Consumers
in this market make a binary choice of whether or not to
purchase this contract, and in this market make a binary choice
of whether or not to purchase this contract, and
fi rms in this market compete only over what price to charge for
the contract.fi rms in this market compete only over what price
to charge for the contract.
The vertical axis indicates the price (and expected cost) of that
contract, and The vertical axis indicates the price (and expected
cost) of that contract, and
the horizontal axis indicates the quantity of insurance demand.
Since individuals the horizontal axis indicates the quantity of
insurance demand. Since individuals
face a binary choice of whether or not to purchase the contract,
the “quantity” face a binary choice of whether or not to
purchase the contract, the “quantity”
of insurance is the fraction of insured individuals. With risk-
neutral insurance of insurance is the fraction of insured
individuals. With risk-neutral insurance
providers and no additional frictions, the social (and fi rms’)
costs associated with providers and no additional frictions, the
social (and fi rms’) costs associated with
Liran Einav and Amy Finkelstein 117
providing insurance are the expected insurance claims—that is,

61. the expected providing insurance are the expected insurance
claims—that is, the expected
payouts on policies.payouts on policies.
Figure 1 shows the market demand curve for the insurance
contract. Because Figure 1 shows the market demand curve for
the insurance contract. Because
individuals in this setting can only choose the contract or not,
the market demand individuals in this setting can only choose
the contract or not, the market demand
curve simply refl ects the cumulative distribution of
individuals’ willingness to pay curve simply refl ects the
cumulative distribution of individuals’ willingness to pay
for the contract. While this is a standard unit demand model that
could apply to for the contract. While this is a standard unit
demand model that could apply to
many traditional product markets, the textbook insurance
context allows us to link many traditional product markets, the
textbook insurance context allows us to link
willingness to pay to cost. In particular, a risk-averse
individual’s willingness to pay willingness to pay to cost. In
particular, a risk-averse individual’s willingness to pay
for insurance is the sum of the expected cost and risk premium
for that individual.for insurance is the sum of the expected cost
and risk premium for that individual.
In the textbook environment, individuals are homogeneous in
their risk aver-In the textbook environment, individuals are
homogeneous in their risk aver-
sion (and all other features of their utility function). Therefore,
their willingness to sion (and all other features of their utility
function). Therefore, their willingness to
pay for insurance is increasing in their risk type—that is, their
probability of loss, or pay for insurance is increasing in their
risk type—that is, their probability of loss, or
expected cost—which is privately known. This is illustrated in

62. Figure 1 by plotting expected cost—which is privately known.
This is illustrated in Figure 1 by plotting
the marginal cost (MC) curve as downward sloping: those
individuals who are willing the marginal cost (MC) curve as
downward sloping: those individuals who are willing
to pay the most for coverage are those that have the highest
expected cost. This to pay the most for coverage are those that
have the highest expected cost. This
downward-sloping MC curve represents the well-known adverse
selection property of downward-sloping MC curve represents
the well-known adverse selection property of
insurance markets: the individuals who have the highest
willingness to pay for insur-insurance markets: the individuals
who have the highest willingness to pay for insur-
ance are those who are expected to be the most costly for the fi
rm to cover.ance are those who are expected to be the most
costly for the fi rm to cover.
The link between the demand and cost curve is arguably the
most important The link between the demand and cost curve is
arguably the most important
distinction of insurance markets (or selection markets more
generally) from traditional distinction of insurance markets (or
selection markets more generally) from traditional
Figure 1
Adverse Selection in the Textbook Setting
Quantity
P
ri
ce
Demand curve

63. MC curve
A
B
C
D
E
F
G
JPeqm
AC curve
Q eqm Q max
118 Journal of Economic Perspectives
product markets. The shape of the cost curve is driven by the
demand-side customer product markets. The shape of the cost
curve is driven by the demand-side customer
selection. In most other contexts, the demand curve and cost
curve are independent selection. In most other contexts, the
demand curve and cost curve are independent
objects; demand is determined by preferences and costs by the
production technology. objects; demand is determined by
preferences and costs by the production technology.
The distinguishing feature of selection markets is that the
demand and cost curves The distinguishing feature of selection

64. markets is that the demand and cost curves
are tightly linked, because the individual’s risk type not only
affects demand but also are tightly linked, because the
individual’s risk type not only affects demand but also
directly determines cost.directly determines cost.
The risk premium is shown graphically in the fi gure as the
vertical distance The risk premium is shown graphically in the
fi gure as the vertical distance
between expected cost (the MC curve) and the willingness to
pay for insurance between expected cost (the MC curve) and the
willingness to pay for insurance
(the demand curve). In the textbook case, the risk premium is
always positive, since (the demand curve). In the textbook case,
the risk premium is always positive, since
all individuals are risk averse and there are no other market
frictions. As a result, all individuals are risk averse and there
are no other market frictions. As a result,
the demand curve is always above the MC curve, and it is
therefore effi cient for all the demand curve is always above the
MC curve, and it is therefore effi cient for all
individuals to be insured (individuals to be insured (Q effeff
== Q maxmax). Absent income effects, the welfare loss from ).
Absent income effects, the welfare loss from
not insuring a given individual is the risk premium of that
individual, or the vertical not insuring a given individual is the
risk premium of that individual, or the vertical
difference between the demand and MC curves.difference
between the demand and MC curves.
When the individual-specifi c loss probability (or expected cost)
is private infor-When the individual-specifi c loss probability
(or expected cost) is private infor-
mation to the individual, fi rms must offer a single price for
pools of observationally mation to the individual, fi rms must
offer a single price for pools of observationally

65. identical but in fact heterogeneous individuals. Of course, in
practice fi rms may identical but in fact heterogeneous
individuals. Of course, in practice fi rms may
vary the price based on some observable individual
characteristics (such as age or vary the price based on some
observable individual characteristics (such as age or
zip code). Thus, Figure 1 can be thought of as depicting the
market for coverage zip code). Thus, Figure 1 can be thought of
as depicting the market for coverage
among individuals who are treated identically by the fi
rm.among individuals who are treated identically by the fi rm.
The competitive equilibrium price will be equal to the fi rms’
average cost at The competitive equilibrium price will be equal
to the fi rms’ average cost at
that price. This is a zero-profi t condition; offering a lower
price will result in nega-that price. This is a zero-profi t
condition; offering a lower price will result in nega-
tive profi ts, and offering higher prices than competitors will
not attract any buyers. tive profi ts, and offering higher prices
than competitors will not attract any buyers.
The relevant cost curve the fi rm faces is therefore the average
cost (AC) curve, The relevant cost curve the fi rm faces is
therefore the average cost (AC) curve,
which is also shown in Figure 1. The (competitive) equilibrium
price and quantity is which is also shown in Figure 1. The
(competitive) equilibrium price and quantity is
given by the intersection of the demand curve and the AC curve
(point given by the intersection of the demand curve and the AC
curve (point C ).).
The fundamental ineffi ciency created by adverse selection
arises because The fundamental ineffi ciency created by adverse
selection arises because
the effi cient allocation is determined by the relationship
between the effi cient allocation is determined by the

66. relationship between marginal cost cost
and demand, but the equilibrium allocation is determined by the
relationship and demand, but the equilibrium allocation is
determined by the relationship
between between average cost and demand. Because of adverse
selection (downward sloping cost and demand. Because of
adverse selection (downward sloping
MC curve), the marginal buyer is always associated with a
lower expected cost than MC curve), the marginal buyer is
always associated with a lower expected cost than
that of infra-marginal buyers. Therefore, as drawn in Figure 1,
the AC curve always that of infra-marginal buyers. Therefore,
as drawn in Figure 1, the AC curve always
lies above the MC curve and intersects the demand curve at a
quantity lower than lies above the MC curve and intersects the
demand curve at a quantity lower than
Q maxmax. As a result, the equilibrium quantity of insurance
will be less than the effi cient . As a result, the equilibrium
quantity of insurance will be less than the effi cient
quantity (quantity (Q maxmax) and the equilibrium price () and
the equilibrium price (Peqmeqm) will be above the effi cient
price, ) will be above the effi cient price,
illustrating the classical result of under-insurance in the
presence of adverse selec-illustrating the classical result of
under-insurance in the presence of adverse selec-
tion (Akerlof, 1970; Rothschild and Stiglitz, 1976). That is, it is
effi cient to insure tion (Akerlof, 1970; Rothschild and Stiglitz,
1976). That is, it is effi cient to insure
every individual (MC is always below demand) but in
equilibrium the every individual (MC is always below demand)
but in equilibrium the Q maxmax – – Q eqmeqm
individuals who have the lowest expected costs remain
uninsured because the individuals who have the lowest expected
costs remain uninsured because the
AC curve is not always below the demand curve. These
individuals value the insur-AC curve is not always below the

67. demand curve. These individuals value the insur-
ance at more than their expected costs, but fi rms cannot insure
these individuals ance at more than their expected costs, but fi
rms cannot insure these individuals
and still break even.and still break even.
The welfare cost of this under-insurance depends on the lost
surplus (the The welfare cost of this under-insurance depends
on the lost surplus (the
risk premium) of those individuals who remain ineffi ciently
uninsured in the risk premium) of those individuals who remain
ineffi ciently uninsured in the
Selection in Insurance Markets: Theory and Empirics in Pictures
119
competitive equilibrium. In Figure 1, these are the individuals
whose willingness to competitive equilibrium. In Figure 1, these
are the individuals whose willingness to
pay is less than the equilibrium price, pay is less than the
equilibrium price, Peqmeqm. Integrating over all these
individuals’ . Integrating over all these individuals’
risk premia, the welfare loss from adverse selection in this
simple framework is given risk premia, the welfare loss from
adverse selection in this simple framework is given
by the area of the deadweight loss trapezoid by the area of the
deadweight loss trapezoid DCEF ..
Even in the textbook environment, the amount of under-
insurance generated Even in the textbook environment, the
amount of under-insurance generated
by adverse selection, and its associated welfare loss, can vary
greatly. Figure 2 illus-by adverse selection, and its associated
welfare loss, can vary greatly. Figure 2 illus-

68. trates this point by depicting two specifi c examples of the
textbook adverse selection trates this point by depicting two
specifi c examples of the textbook adverse selection
environment, one that produces the effi cient insurance
allocation and one that environment, one that produces the effi
cient insurance allocation and one that
produces complete unraveling of insurance coverage. The effi
cient outcome is produces complete unraveling of insurance
coverage. The effi cient outcome is
depicted in panel A. While the market is adversely selected
(that is, the MC curve depicted in panel A. While the market is
adversely selected (that is, the MC curve
is downward sloping), the AC curve always lies below the
demand curve. This leads is downward sloping), the AC curve
always lies below the demand curve. This leads
to an equilibrium price to an equilibrium price Peqmeqm , that,
although it is higher than marginal cost, still , that, although it
is higher than marginal cost, still
produces the effi cient allocation (produces the effi cient
allocation (Q eqmeqm == Q effeff == Q maxmax). This
situation can arise, for ). This situation can arise, for
example, when individuals do not vary too much in their
unobserved risk (that is, example, when individuals do not vary
too much in their unobserved risk (that is,
the MC and consequently AC curve is relatively fl at) and/or
individuals’ risk aver-the MC and consequently AC curve is
relatively fl at) and/or individuals’ risk aver-
sion is high (that is, the demand curve lies well above the MC
curve).sion is high (that is, the demand curve lies well above
the MC curve).
Figure 2
Specifi c Examples of Extreme Cases
A: Adverse Selection with No Efficiency Cost

69. Quantity
P
ri
ce
Demand curve
MC curve
AC curve
Peqm
Q max
C
(continued on next page)
120 Journal of Economic Perspectives
The case of complete unraveling is illustrated in panel B of
Figure 2. Here, the The case of complete unraveling is
illustrated in panel B of Figure 2. Here, the
AC curve always lies above the demand curve even though the
MC curve is always AC curve always lies above the demand
curve even though the MC curve is always
below it.below it.11 As a result, the competitive equilibrium is
that no individual in the market As a result, the competitive
equilibrium is that no individual in the market
is insured, while the effi cient outcome is for everyone to have
insurance. One could is insured, while the effi cient outcome is

70. for everyone to have insurance. One could
also use panel B to illustrate the potential death spiral dynamics
that may lead to also use panel B to illustrate the potential death
spiral dynamics that may lead to
such unraveling. For example, if insurance pricing is naively set
but dynamically such unraveling. For example, if insurance
pricing is naively set but dynamically
adjusted to refl ect the average cost from the previous period
(which is, in fact, a adjusted to refl ect the average cost from
the previous period (which is, in fact, a
fairly common practice in many health insurance settings), the
market will gradu-fairly common practice in many health
insurance settings), the market will gradu-
ally shrink until it completely disappears. This convergent
adjustment process is ally shrink until it completely disappears.
This convergent adjustment process is
illustrated by the arrows in panel B. Cutler and Reber (1998)
provide an empirical illustrated by the arrows in panel B. Cutler
and Reber (1998) provide an empirical
case study of a death spiral of this nature in the context of a
health insurance plan case study of a death spiral of this nature
in the context of a health insurance plan
offered to Harvard University employees.offered to Harvard
University employees.
Public Policy in the Textbook CasePublic Policy in the
Textbook Case
Our graphical framework can also be used to illustrate the
consequences of Our graphical framework can also be used to
illustrate the consequences of
common public policy interventions in insurance markets. The
canonical solution common public policy interventions in
insurance markets. The canonical solution
to the ineffi ciency created by adverse selection is to mandate
that everyone purchase to the ineffi ciency created by adverse

71. selection is to mandate that everyone purchase
insurance. In the textbook setting, this produces the effi cient
outcome in which insurance. In the textbook setting, this
produces the effi cient outcome in which
everyone has insurance. However, the magnitude of the welfare
benefi t produced everyone has insurance. However, the
magnitude of the welfare benefi t produced
1 This can happen even within the textbook example if the
individuals with the greatest risk are certain to
incur a loss, so their risk premium is zero and their willingness
to pay is the same as their expected costs.
Figure 2 (continued)
B: Adverse Selection with Complete Unraveling
P
ri
ce
Quantity
Q max
Demand curve
MC curve
AC curve
Liran Einav and Amy Finkelstein 121

72. by an insurance purchase requirement can vary dramatically
depending on the by an insurance purchase requirement can
vary dramatically depending on the
specifi cs of the market. The two extreme examples presented in
Figure 2 illustrate specifi cs of the market. The two extreme
examples presented in Figure 2 illustrate
this point, but even in intermediate cases captured by Figure 1,
the magnitude of this point, but even in intermediate cases
captured by Figure 1, the magnitude of
the welfare loss (area the welfare loss (area CDEF ) is highly
sensitive to the shape and location of the cost ) is highly
sensitive to the shape and location of the cost
and demand curves and is therefore ultimately an empirical
question.and demand curves and is therefore ultimately an
empirical question.22
Another commonly discussed policy remedy for adverse
selection is to subsi-Another commonly discussed policy
remedy for adverse selection is to subsi-
dize insurance coverage. We can use Figure 1 to illustrate.
Consider, for example, dize insurance coverage. We can use
Figure 1 to illustrate. Consider, for example,
a lump sum subsidy toward the price of coverage. This would
shift demand out, a lump sum subsidy toward the price of
coverage. This would shift demand out,
leading to a higher equilibrium quantity and less under-
insurance. The welfare loss leading to a higher equilibrium
quantity and less under-insurance. The welfare loss
would still be associated with the area between the original
(pre-subsidy) demand would still be associated with the area
between the original (pre-subsidy) demand
curve and the MC curve, and would therefore unambiguously
decline with any posi-curve and the MC curve, and would
therefore unambiguously decline with any posi-
tive subsidy. A large enough subsidy (greater than the line
segment tive subsidy. A large enough subsidy (greater than the

73. line segment GE in Figure 1) in Figure 1)
would lead to the effi cient outcome, with everybody
insured.would lead to the effi cient outcome, with everybody
insured.
A fi nal common form of public policy intervention is
regulation that imposes A fi nal common form of public policy
intervention is regulation that imposes
restrictions on the characteristics of consumers over which fi
rms can price discrimi-restrictions on the characteristics of
consumers over which fi rms can price discrimi-
nate. Some regulations require “community rates” that are
uniform across all nate. Some regulations require “community
rates” that are uniform across all
individuals, while others prohibit insurance companies from
making prices contin-individuals, while others prohibit
insurance companies from making prices contin-
gent on certain observable risk factors, such as race or gender.
For concreteness, gent on certain observable risk factors, such
as race or gender. For concreteness,
consider the case of a regulation that prohibits pricing on the
basis of gender. Recall consider the case of a regulation that
prohibits pricing on the basis of gender. Recall
that Figure 1 can be interpreted as applying to a group of
individuals who must that Figure 1 can be interpreted as
applying to a group of individuals who must
be treated the same by the insurance company. When pricing
based on gender is be treated the same by the insurance
company. When pricing based on gender is
prohibited, males and females are pooled into the same market,
with a variant of prohibited, males and females are pooled into
the same market, with a variant of
Figure 1 describing that market. When pricing on gender is
allowed, there are now Figure 1 describing that market. When
pricing on gender is allowed, there are now
two distinct insurance markets—described by two distinct

74. variants of Figure 1—one two distinct insurance markets—
described by two distinct variants of Figure 1—one
for women and one for men, each of which can be analyzed
separately. A central for women and one for men, each of which
can be analyzed separately. A central
issue for welfare analysis is whether, when insurance companies
are allowed to price issue for welfare analysis is whether, when
insurance companies are allowed to price
on gender, consumers still have residual private information
about their expected on gender, consumers still have residual
private information about their expected
costs. If they do not, then the insurance market within each
gender-specifi c segment costs. If they do not, then the
insurance market within each gender-specifi c segment
of the market will exhibit a constant (fl at) MC curve and the
equilibrium in each of the market will exhibit a constant (fl at)
MC curve and the equilibrium in each
market will be effi cient. In this case, policies that restrict
pricing on gender are market will be effi cient. In this case,
policies that restrict pricing on gender are
unambiguously welfare decreasing since they create adverse
selection where unambiguously welfare decreasing since they
create adverse selection where
none existed before. However, in the more likely case that
individuals have some none existed before. However, in the
more likely case that individuals have some
residual private information about their risk that is not captured
by their gender, residual private information about their risk
that is not captured by their gender,
each gender-specifi c market segment would look qualitatively
the same as Figure 1 each gender-specifi c market segment
would look qualitatively the same as Figure 1
(with downward sloping MC and AC curves). In such cases, the
welfare implica-(with downward sloping MC and AC curves). In
such cases, the welfare implica-
tions of restricting pricing on gender could go in either

75. direction; depending on tions of restricting pricing on gender
could go in either direction; depending on
the shape and position of the gender-specifi c demand and cost
curves relative to the shape and position of the gender-specifi c
demand and cost curves relative to
the gender-pooled ones, the sum of the areas of the deadweight
loss trapezoids in the gender-pooled ones, the sum of the areas
of the deadweight loss trapezoids in
2 Although in the specifi c examples in Figure 2, the welfare
cost of adverse selection is increasing with the
amount of under-insurance it creates, this does not have to be
the case in general.
122 Journal of Economic Perspectives
the gender-specifi c markets could be larger or smaller than the
area of the single the gender-specifi c markets could be larger
or smaller than the area of the single
deadweight loss trapezoid in the gender-pooled
market.deadweight loss trapezoid in the gender-pooled
market.33
Departures from the Textbook EnvironmentDepartures from the
Textbook Environment
Although the textbook treatment of insurance markets may give
rise to dramat-Although the textbook treatment of insurance
markets may give rise to dramat-
ically different magnitudes of the welfare costs arising from
adverse selection, the ically different magnitudes of the welfare
costs arising from adverse selection, the
qualitative fi ndings are robust. Under the textbook
assumptions, private informa-qualitative fi ndings are robust.

76. Under the textbook assumptions, private informa-
tion about risk never produces over-insurance relative to the effi
cient outcome, tion about risk never produces over-insurance
relative to the effi cient outcome,
and mandatory insurance coverage is always a (weakly) welfare-
improving policy and mandatory insurance coverage is always a
(weakly) welfare-improving policy
intervention. However, these robust qualitative results only hold
in this textbook intervention. However, these robust qualitative
results only hold in this textbook
case. They may be reversed with the introduction of two
important features of actual case. They may be reversed with
the introduction of two important features of actual
insurance markets: 1) insurance “loads” or administrative costs
of providing insur-insurance markets: 1) insurance “loads” or
administrative costs of providing insur-
ance, and 2) preference heterogeneity.ance, and 2) preference
heterogeneity.
Consider fi rst a loading factor on insurance, for example in the
form of addi-Consider fi rst a loading factor on insurance, for
example in the form of addi-
tional administrative cost associated with selling and servicing
insurance, perhaps tional administrative cost associated with
selling and servicing insurance, perhaps
due to costs associated with advertising and marketing, or with
verifying and due to costs associated with advertising and
marketing, or with verifying and
processing claims. Many insurance markets display evidence of
nontrivial loading processing claims. Many insurance markets
display evidence of nontrivial loading
factors, including markets for long-term care insurance (Brown
and Finkelstein, factors, including markets for long-term care
insurance (Brown and Finkelstein,
2007), annuities (Friedman and Warshawsky, 1990; Mitchell,
Poterba, Warshawsky, 2007), annuities (Friedman and

77. Warshawsky, 1990; Mitchell, Poterba, Warshawsky,
and Brown, 1999; Finkelstein and Poterba, 2002), health
insurance (Newhouse, and Brown, 1999; Finkelstein and
Poterba, 2002), health insurance (Newhouse,
2002), and automobile insurance (Chiappori, Jullien, Salanié,
and Salanié, 2006).2002), and automobile insurance (Chiappori,
Jullien, Salanié, and Salanié, 2006).44
The key implication of such loads is that it is now not
necessarily effi cient to The key implication of such loads is
that it is now not necessarily effi cient to
allocate insurance coverage to all individuals. Even if all
individuals are risk averse, allocate insurance coverage to all
individuals. Even if all individuals are risk averse,
the additional cost of providing an individual with insurance
may be greater than the additional cost of providing an
individual with insurance may be greater than
the risk premium for certain individuals, making it socially effi
cient to leave such the risk premium for certain individuals,
making it socially effi cient to leave such
individuals uninsured. This case is illustrated in Figure 3, which
is similar to Figure 1, individuals uninsured. This case is
illustrated in Figure 3, which is similar to Figure 1,
except that the cost curves are shifted upward refl ecting the
additional cost of insur-except that the cost curves are shifted
upward refl ecting the additional cost of insur-
ance provision.ance provision.55
Figure 3 is drawn in a way that the MC curve crosses the
demand curve “inter-Figure 3 is drawn in a way that the MC
curve crosses the demand curve “inter-
nally” (that is, at a quantity lower than nally” (that is, at a
quantity lower than Qmaxmax), at point ), at point E , which
depicts the socially , which depicts the socially
effi cient insurance allocation. It is effi cient to insure everyone
to the left of point effi cient insurance allocation. It is effi cient

78. to insure everyone to the left of point E
(because their willingness to pay for insurance exceeds their
expected cost), but (because their willingness to pay for
insurance exceeds their expected cost), but
3 An example illustrates how pricing on gender can increase
deadweight loss. Consider three types of
individuals. Type 1 individuals (representing 10 percent of the
population) have expected cost of 20
and willingness to pay for insurance of 30. Type 2 individuals
(60 percent) have expected cost of 5 and
willingness to pay of 20, and type 3 (30 percent) have expected
cost of 4 and willingness to pay of 7.5. The
competitive (zero-profi t) price in this market is 6.2, leading to
an effi cient allocation in which everyone
is insured (this case is similar to that of panel A in Figure 2).
Suppose now that type 2 individuals are all
females and type 1 and 3 individuals are all males, and gender
can be priced. In this case, the competitive
price for women is 5 and they are all insured. However, the
competitive price for men is 8, leaving all
type 3 individuals ineffi ciently uninsured.
4 Admittedly, most of these papers lack the data to distinguish
between loading factors arising from
administrative costs to the insurance company and those arising
from market power (insurance company
profi ts). Still, it seems a reasonable assumption that it is not
costless to run an insurance company.
5 We note that Figure 3 could also describe a market with no
frictions, but in which a fraction of the
individuals are risk loving.
Selection in Insurance Markets: Theory and Empirics in Pictures
123

79. socially ineffi cient to insure anyone to the right of point
socially ineffi cient to insure anyone to the right of point E
(because their willing- (because their willing-
ness to pay is less than their expected cost). In this situation, it
is effi cient to keep ness to pay is less than their expected cost).
In this situation, it is effi cient to keep
Q maxmax – – Q effeff individuals uninsured. individuals
uninsured.
The introduction of loads does not affect the basic analysis of
adverse selection, The introduction of loads does not affect the
basic analysis of adverse selection,
but it does have important implications for its standard public
policy remedies. but it does have important implications for its
standard public policy remedies.
The competitive equilibrium is still determined by the zero profi
t condition, or the The competitive equilibrium is still
determined by the zero profi t condition, or the
intersection of the demand curve and the AC curve (point
intersection of the demand curve and the AC curve (point C in
Figure 3), and in in Figure 3), and in
the presence of adverse selection (and thus a downward sloping
MC curve), this the presence of adverse selection (and thus a
downward sloping MC curve), this
leads to under-insurance relative to the social optimum (leads to
under-insurance relative to the social optimum (Q eqmeqm <<
Q effeff), and to a ), and to a
familiar deadweight loss triangle familiar deadweight loss
triangle CDE ..
However, with insurance loads, the textbook result of an
unambiguous welfare However, with insurance loads, the
textbook result of an unambiguous welfare
gain from mandatory coverage no longer obtains. As Figure 3
shows, while a mandate gain from mandatory coverage no

80. longer obtains. As Figure 3 shows, while a mandate
that everyone be insured “regains” the welfare loss associated
with under-insurance that everyone be insured “regains” the
welfare loss associated with under-insurance
(triangle (triangle CDE ), it also leads to over-insurance by
covering individuals whom it is ), it also leads to over-
insurance by covering individuals whom it is
socially ineffi cient to insure (that is, whose expected costs are
above their willingness socially ineffi cient to insure (that is,
whose expected costs are above their willingness
to pay). This latter effect leads to a welfare loss given by the
area to pay). This latter effect leads to a welfare loss given by
the area EGH in Figure 3. in Figure 3.
Therefore whether a mandate improves welfare over the
competitive allocation Therefore whether a mandate improves
welfare over the competitive allocation
depends on the relative sizes of triangles depends on the
relative sizes of triangles CDE and and EGH ; this in turn
depends on the ; this in turn depends on the
specifi c market’s demand and cost curves and is therefore an
empirical question.specifi c market’s demand and cost curves
and is therefore an empirical question.
Figure 3
Adverse Selection with Additional Cost of Providing Insurance
Source: Einav, Finkelstein, and Cullen (2010), fi gure 1.

81. P
ri
ce
Demand curve
MC curve
A
B
C
D E
F
G
Peqm
Q eqm Q max
AC curve
H
Q eff
Peff
Quantity

82. 124 Journal of Economic Perspectives
A second important feature of real-world insurance markets not
captured by A second important feature of real-world insurance
markets not captured by
the textbook treatment is preference heterogeneity: that is, the
possibility that the textbook treatment is preference
heterogeneity: that is, the possibility that
individuals may differ not only in their risk but also in their
preferences, such as individuals may differ not only in their risk
but also in their preferences, such as
their willingness to bear risk (risk aversion). The classical
models (like Rothschild their willingness to bear risk (risk
aversion). The classical models (like Rothschild
and Stiglitz, 1976) make the simplifying and theoretically
attractive assumption that and Stiglitz, 1976) make the
simplifying and theoretically attractive assumption that
individuals have the same preferences and may vary only in
their (privately known) individuals have the same preferences
and may vary only in their (privately known)
expected costs. As a result, willingness to pay for insurance is
an increasing function expected costs. As a result, willingness
to pay for insurance is an increasing function
of expected costs.of expected costs.
In practice, of course, individuals may differ not only in their
expected cost but In practice, of course, individuals may differ
not only in their expected cost but
also in their preferences. Indeed, recent empirical work has
documented substan-also in their preferences. Indeed, recent
empirical work has documented substan-
tial preference heterogeneity in different insurance markets,
including automobile tial preference heterogeneity in different
insurance markets, including automobile

83. insurance (Cohen and Einav, 2007), reverse mortgages
(Davidoff and Welke, 2007), insurance (Cohen and Einav,
2007), reverse mortgages (Davidoff and Welke, 2007),
health insurance (Fang, Keane, and Silverman, 2008), and long-
term care insur-health insurance (Fang, Keane, and Silverman,
2008), and long-term care insur-
ance (Finkelstein and McGarry, 2006). The existence of
unobserved preference ance (Finkelstein and McGarry, 2006).
The existence of unobserved preference
heterogeneity opens up the possibility of heterogeneity opens up
the possibility of advantageous selection, which produces
selection, which produces
opposite results to the opposite results to the adverse selection
results just discussed. selection results just discussed.66
Consider for example heterogeneity in risk aversion in addition
to the original Consider for example heterogeneity in risk
aversion in addition to the original
heterogeneity in risk (expected cost). All else equal, willingness
to pay for insurance heterogeneity in risk (expected cost). All
else equal, willingness to pay for insurance
is increasing in risk aversion and in risk. If heterogeneity in risk
aversion is small, is increasing in risk aversion and in risk. If
heterogeneity in risk aversion is small,
or if those individuals who are high risk are also more risk
averse, the main insights or if those individuals who are high
risk are also more risk averse, the main insights
from the textbook analysis remain. But if high-risk individuals
are less risk averse from the textbook analysis remain. But if
high-risk individuals are less risk averse
and the heterogeneity in risk aversion is suffi ciently large,
advantageous selection and the heterogeneity in risk aversion is
suffi ciently large, advantageous selection
may emerge. Namely, the individuals who are willing to pay the
most for insurance may emerge. Namely, the individuals who
are willing to pay the most for insurance

84. are those who are the most risk averse, and in the case
described, these are also are those who are the most risk averse,
and in the case described, these are also
those individuals associated with the lowest (rather than the
highest) expected cost. those individuals associated with the
lowest (rather than the highest) expected cost.
Indeed, it is natural to think that in many instances individuals
who value insurance Indeed, it is natural to think that in many
instances individuals who value insurance
more may also take action to lower their expected costs: drive
more carefully, invest more may also take action to lower their
expected costs: drive more carefully, invest
in preventive health care, and so on.in preventive health care,
and so on.
Figure 4 provides our graphical illustration of such
advantageous selection and Figure 4 provides our graphical
illustration of such advantageous selection and
its consequences for insurance coverage and welfare. In contrast
to adverse selection, its consequences for insurance coverage
and welfare. In contrast to adverse selection,
advantageous selection is defi ned by an advantageous selection
is defi ned by an upward sloping MC (and AC) curve. sloping
MC (and AC) curve.77 As price As price
is lowered and more individuals opt into the market, the
marginal individual opting is lowered and more individuals opt
into the market, the marginal individual opting
in has higher expected cost than infra-marginal individuals.
Since the MC curve is in has higher expected cost than infra-
marginal individuals. Since the MC curve is
6 Another important (and more nuanced) aspect of preference
heterogeneity is that it complicates the
notion of effi ciency. With preference heterogeneity, the
mapping from expected cost to willingness to
pay need no longer be unique. That is, two individuals with the

85. same expected cost may have different
valuations for the same coverage, or two individual with the
same willingness to pay for the coverage
may have different underlying expected costs. This possibility
does not affect our earlier and subsequent
analysis, except that one needs to recognize that it requires a
weaker sense of effi ciency. Specifi cally, it
requires us to think of a constrained effi cient allocation that
maximizes welfare subject to a uniform
price. In such cases, the (constrained) effi cient allocation need
not coincide with the fi rst-best allocation.
Bundorf, Levin, and Mahoney (2010) discuss and empirically
analyze this issue in more detail.
7 More generally, once we allow for preference heterogeneity,
the marginal cost curve needs not be
monotone. However, for simplicity and clarity we focus our
discussion on the polar cases of monotone
cost curves.
Liran Einav and Amy Finkelstein 125
upward sloping, the AC curve will lie everywhere below it. If
there were no insurance upward sloping, the AC curve will lie
everywhere below it. If there were no insurance
loads (as in the textbook situation), advantageous selection
would not lead to any loads (as in the textbook situation),
advantageous selection would not lead to any
ineffi ciency; the MC and AC curves would always lie below
the demand curve, and in ineffi ciency; the MC and AC curves
would always lie below the demand curve, and in
equilibrium all individuals in the market would be covered,
which would be effi cient.equilibrium all individuals in the
market would be covered, which would be effi cient.

86. With insurance loads, however, advantageous selection
generates the mirror With insurance loads, however,
advantageous selection generates the mirror
image of the adverse selection case, also leading to ineffi
ciency, but this time due to image of the adverse selection case,
also leading to ineffi ciency, but this time due to
over-insurance rather than under-insurance. Figure 4 depicts
this case. The effi cient over-insurance rather than under-
insurance. Figure 4 depicts this case. The effi cient
allocation calls for providing insurance to all individuals whose
expected cost is allocation calls for providing insurance to all
individuals whose expected cost is
lower than their willingness to pay—that is, all those who are to
the left of point lower than their willingness to pay—that is, all
those who are to the left of point E
(where the MC curve intersects the demand curve) in Figure 4.
Competitive equilib-(where the MC curve intersects the demand
curve) in Figure 4. Competitive equilib-
rium, as before, is determined by the intersection of the AC
curve and the demand rium, as before, is determined by the
intersection of the AC curve and the demand
curve (point curve (point C in Figure 4). But since the AC curve
now lies below the MC curve, in Figure 4). But since the AC
curve now lies below the MC curve,
equilibrium implies that too many individuals are provided
insurance, leading to equilibrium implies that too many
individuals are provided insurance, leading to
over-insurance: there are over-insurance: there are Q eqmeqm –
– Q effeff individuals who are ineffi ciently provided
individuals who are ineffi ciently provided
insurance in equilibrium. These individuals value the insurance
at less than their insurance in equilibrium. These individuals
value the insurance at less than their
expected costs, but competitive forces make fi rms reduce the
price, thus attracting expected costs, but competitive forces
make fi rms reduce the price, thus attracting

87. these individuals together with more profi table infra-marginal
individuals. Again, these individuals together with more profi
table infra-marginal individuals. Again,
the area of the deadweight loss triangle the area of the
deadweight loss triangle EDC quantifi es the extent of the
welfare loss quantifi es the extent of the welfare loss
from this over-insurance.from this over-insurance.
Figure 4
Advantageous Selection
Source: Einav, Finkelstein, and Cullen (2010), fi gure 2.
Quantity
P
ri
ce
Demand curve
MC curve
A
B
C
D
E
F

88. G
Peqm
AC curve
Q eqm Q max
Peff
H
Q eff
126 Journal of Economic Perspectives
From a public policy perspective, advantageous selection calls
for the opposite From a public policy perspective, advantageous
selection calls for the opposite
solutions relative to the tools used to combat adverse selection.
For example, given solutions relative to the tools used to
combat adverse selection. For example, given
that advantageous selection produces “too much” insurance
relative to the effi cient that advantageous selection produces
“too much” insurance relative to the effi cient
outcome, public policies that tax existing insurance policies
(and therefore raise outcome, public policies that tax existing
insurance policies (and therefore raise
Peqmeqm toward toward Peffeff) or outlaw insurance coverage
(mandate no coverage) could be ) or outlaw insurance coverage
(mandate no coverage) could be
welfare-improving. Although there are certainly taxes levied on
insurance policies, welfare-improving. Although there are
certainly taxes levied on insurance policies,

89. to our knowledge advantageous selection has not yet been
invoked as a rationale to our knowledge advantageous selection
has not yet been invoked as a rationale
in public policy discourse, perhaps refl ecting the relative
newness of both the theo-in public policy discourse, perhaps
refl ecting the relative newness of both the theo-
retical work and empirical evidence. To our knowledge,
advantageous selection was retical work and empirical evidence.
To our knowledge, advantageous selection was
fi rst discussed by Hemenway (1990), who termed it
“propitious” selection. De Meza fi rst discussed by Hemenway
(1990), who termed it “propitious” selection. De Meza
and Webb (2001) provide a theoretical treatment of
advantageous selection and its and Webb (2001) provide a
theoretical treatment of advantageous selection and its
implications for insurance coverage and public
policy.implications for insurance coverage and public policy.
Advantageous selection is not merely a theoretical possibility.
It has recently Advantageous selection is not merely a
theoretical possibility. It has recently
been documented in several insurance markets, with different
sources of been documented in several insurance markets, with
different sources of
individual heterogeneity that give rise to it. Finkelstein and
McGarry (2006) individual heterogeneity that give rise to it.
Finkelstein and McGarry (2006)
document advantageous selection in the market for long-term
care insurance and document advantageous selection in the
market for long-term care insurance and
provide evidence that more cautious individuals invest more in
precautionary provide evidence that more cautious individuals
invest more in precautionary
behavior and are less likely to use a nursing home but at the
same time are more behavior and are less likely to use a nursing
home but at the same time are more

90. likely to purchase long-term care insurance. Fang, Keane, and
Silverman (2008) likely to purchase long-term care insurance.
Fang, Keane, and Silverman (2008)
document advantageous selection in the market for Medigap
coverage, which document advantageous selection in the market
for Medigap coverage, which
provides private health insurance that supplements Medicare for
the elderly, but provides private health insurance that
supplements Medicare for the elderly, but
show that in the case of Medigap, cognition may be the driving
force: individuals show that in the case of Medigap, cognition
may be the driving force: individuals
with higher cognitive ability are often able to make better
decisions, which can with higher cognitive ability are often able
to make better decisions, which can
translate into both greater coverage and at the same time lower
healthcare translate into both greater coverage and at the same
time lower healthcare
expenditures.expenditures.
Advantageous selection provides a nice example of the interplay
in the selec-Advantageous selection provides a nice example of
the interplay in the selec-
tion literature between theory and empirical work. The original
adverse selection tion literature between theory and empirical
work. The original adverse selection
theory motivated empirical work testing for the existence of
adverse selection. This theory motivated empirical work testing
for the existence of adverse selection. This
empirical work in turn provided examples of advantageous
selection (which the empirical work in turn provided examples
of advantageous selection (which the
original theory had precluded), suggesting the need for
important extensions to original theory had precluded),
suggesting the need for important extensions to
the theory. We now turn to a more detailed discussion of how

91. the existing empirical the theory. We now turn to a more
detailed discussion of how the existing empirical
work can be viewed through the graphical framework we have
developed.work can be viewed through the graphical framework
we have developed.
Empirical Work on SelectionEmpirical Work on Selection
Empirical research on selection in insurance markets has fl
ourished over the Empirical research on selection in insurance
markets has fl ourished over the
last decade. This empirical literature began, quite naturally, by
asking how we can last decade. This empirical literature began,
quite naturally, by asking how we can
test for whether the classic adverse selection models apply in
real-world insurance test for whether the classic adverse
selection models apply in real-world insurance
markets. In other words, what would selection look like in the
data, when or if it markets. In other words, what would
selection look like in the data, when or if it
exists? Empirical research has now progressed from trying to
detect the existence exists? Empirical research has now
progressed from trying to detect the existence
(and nature) of selection toward attempts to quantify its welfare
consequences and (and nature) of selection toward attempts to
quantify its welfare consequences and
those of potential public policy interventions. We can use our
graphical framework those of potential public policy
interventions. We can use our graphical framework
to understand the intuition and limitations of this research
program.to understand the intuition and limitations of this
research program.
Selection in Insurance Markets: Theory and Empirics in Pictures