Insurance Market and Adverse Selection
Sam Hwang April 12, 2016 1 / 191 Why Study Market for Insurance? Spending on insurance accounts for no small part of
household’s expenditures
As of 2014, 10.7% of the average household expenditure in
the U.S. is spent on personal insurance payments and pension
contribution 2 / 191 Why Study Market for Insurance? As of 2014, 5.2% of the average household expenditure is
spent on personal insurance payments and pension
contribution
In some countries like Canada, most people do not even have
options to opt out of insurance
Why???? And more...
3 / 191 Review of Consumer’s Problem
For reasons that will become clear soon, let’s review basics of
consumer’s problem
Suppose that there are n consumption goods x1 , x2 , . . . , xn
pk : market price of good xk
Your income is $I
Your budget set is
n pk xk ≤ I
k=1 Consumer’s problem: how much of each consumption good to
consume? 4 / 191 Review of Consumer’s Problem
Your utility function is u(x1 , x2 , . . . , xn )
Often we assume that u(x1 , x2 , . . . , xn ) is strictly quasiconcave
This means that given two bundles of goods a = (a1 , . . . , an )
and b = (b1 , . . . , bn ), if
u(a) ≥ u(b)
then you prefer a ”MIX” c of the bundles a and b to b
A ”mix” of two bundles of goods a and b, or a ”convex
combination” of two bundles a and b is
αa + (1 − α)b
=(αa1 + (1 − α)b1 , . . . , αan + (1 − α)bn )
where α is some number between 0 and 1
Then strict quasiconcavity implies
u(αa1 + (1 − α)b1 , . . . , αan + (1 − α)bn ) > u(b1 , . . . , bn )
5 / 191 Two-Good Case
There are two goods
A strictly quasiconcave utilty function u(x, y ) has indifference
curves that look like 6 / 191 Two-Good Case
With strict quasiconcavity, given two bundles a = (a1 , a2 ) and
b = (b1 , b2 ) where u(a) ≥ u(b), you always prefer a mix of A
and B
u(αa1 + (1 − α)b1 , αa2 + (1 − α)b2 ) > u(b1 , b2 ) 7 / 191 Two-Good Case
If we assume that the utility function u(x, y ) are differentiable
(i.e., we can take first derivative of it), then strict
quasiconcavity implies that the marginal rate of substitution
between X and Y is decreasing
MRS at some particular consumption bundle (x0 , y0 ) is
MRS(x0 , y0 ) = ∂u(x0 ,y0 )
∂X
∂u(x0 ,y0 )
∂Y = Marginal utility of X at (x0 , y0 )
Marginal utility of Y at (x0 , y0 ) The interpretation of MRS(x0 , y0 ) is the value of good X TO
YOU in terms of good Y when you consume the bundle
(x0 , y0 )
For example, if MRS(x0 , y0 ) = 5, then when you consume the
bundle (x0 , y0 ), the value of good X to you is five times that
of good Y to you
Along any indifference curve of a strictly quasiconcave utility
function, MRS decreases as X increases, meaning that as you
consume more and more of X , it is worth less and less to you
in terms of good Y 8 / 191 Budget Set in Two Good Cases
The budget set in the two-good case is
{(x, y ) : px x + py y ≤ I } What is the slope of the budget line?
− px
Market price of good X
=−
= −Market value of good X in ter
py
Market price of good Y For example, if px = $5 and py = $1, then the value of good
X in the market is five units of good Y
9 / 191 Optimal Consumption Bundle At optimal bundle, the marginal rate of substitution =
Price of good X
Price of good Y 10 / 191 Optimal Consumption Bundle Why are A not optimal?
At A, marginal rate of substitution is larger than the price
ratio, or the value of X in terms of good Y to you is larger
than the market value of good X in terms of good Y
11 / 191 Optimal Consumption Bundle
This means that if you sell some of your good Y to buy good
X in the market, your utility would rise
px
For example, if at A, MRS=5 and py = 4, then you can sell 1
unit of Y to get 1 units of X in the market
4
However, because your value of X = 5 × your value of Y at
A, you gain more utility
1
-value of a unit of Y + value of a unit of X
4
5
= − value of a unit of Y + value of unit of Y
4
1
= value of a unit of Y
4
P is optimal because your value of X is the same as the
market value of X; so there is no trade you can make with the
market to increase your utility
12 / 191 Demand for Insurance
Let’s consider a typical situation that faces the consumer of
insurance
Suppose your initial wealth is w
In the next time period, you know that
Some bad events occur and you will lose $L with probability p
Nothing happens and no change in your wealth with
probability 1 − p v (x) : your Bernoulli utility function
Then your expected utility is...
pv (w − L) + (1 − p)v (w )
That is, this is your expected from the lottery
(w − L, p; w , (1 − p)) 13 / 191 Demand for Insurance Then Sam comes along and makes an offer to you
”If you pay me q dollars now, I will pay you a dollar in the next
period only if your wealth decreases”
”But if nothing happens, you do not get anything from me” In effect, Sam is offering insurance to you
Then how will you determine your optimal consumption of
insurance? 14 / 191 Demand for Insurance
We can formulate the problem of choosing an optimal
consumption of insurance as the problem of choosing an
optimal consumption bundle of two goods
What are the two goods?
1. Wealth when something bad happens X
2. Wealth when nothing bad happens Y And the utility function u(X , Y ) is equal to the expected
utility for the lottery (X , p; Y , (1 − p)) given the same
Bernoulli utility function v (·)
u(X , Y ) = pv (X ) + (1 − p)v (Y )
Then what about the budget set?
If you do not buy any insurance from Sam, what consumption
bundle will you end up with?
(w − L, w )
15 / 191 Demand for Insurance
First, without Sam and his offer of insurance, what is your
budget set?
Without the insurance, there is no way you can consume more
of Y, wealth when bad things happen
Therefore, without insurance, your budget set is a point
(w − L, w ) 16 / 191 Demand for Insurance
How does Sam’s insurance change the budget set?
If you were to increase one unit of consumption of X (wealth
in the bad state), how does your Y (wealth in the good state)
change?
To increase a unit of X, what do you need to do?
How much insurance do you need to buy?
If you buy z units of insurance of Sam, then
Your wealth in bad state: w − L − qz + z
Your wealth in good state: w − qz To increase one unit of X , we want z-qz=1
z= 1
1−q 17 / 191 Demand for Insurance 1
So you need to buy 1−q units of insurance from Sam to
increase one unit of X How much did your wealth in good state decrease because
1
you bought 1−q units of insurance?
1
So as a result of buying 1−q units of insurance, your
consumption bundle changes as follows: Your wealth in bad state: w − L + 1
q
Your wealth in good state: w − 1−q 18 / 191 Drawing the Budget Set We have just found another point on the budget line
We can connect these two dots to get the budget line 19 / 191 Drawing the Budget Set Why is there no line left of (w-L,w)?
Because Sam is not offering an insurance as follows:
”If you pay me r dollars now, I will pay you a dollar in the
next period only if nothing bad happens” There is no way for you to increase Y unless such insurance is
offered
20 / 191 Drawing the Budget Set 21 / 191 Drawing the Budget Set What is the slope of the budget line?
∆Y
q
=−
∆X
1−q
Remember the slope of the budget line in the consumer’s
pX
problem: − pY
So, in the insurance problem, the relative price between two
goods is:
q
Price of wealth in bad state
=
Price of wealth in good state
1−q 22 / 191 Utility Function Now that we have the budget line, we only need to draw our
indifference curve to find the optimal consumption bundle
Our utility function u(X , Y ) is just the expected utility from
the lottery (X , p; Y , (1 − p)) given the Bernoulli utility
function v (·)
u(X , Y ) = pv (X ) + (1 − p)v (Y ) 23 / 191 Utility Function
Suppose you are risk neutral. Then give me an example of
v (·) that represents your preference
v (x) = x
With this Bernoulli utility function, your utility function over
the consumption bundle (X , Y ) is
u(X , Y ) = pX + (1 − p)Y
With this utility function, how does your indifference curve
look like?
The indifference curve on which you get a utility level of u1 is
pX + (1 − p)Y = u1 ⇒ Y = − u1
p
X+
1−p
1−p So the slope of the indifference curve is p
1−p
24 / 191 Indifference Curve for Risk Neutral DM
Indifference curve for u(X , Y ) = pX + (1 − p)Y Which indifference curve represents highest utility level?
What is the marginal rate of substitution? 25 / 191 Optimal Consumption for Risk Neutral DM
Now that we have
Budget line
Indifference curves We can find the optimal consumption bundle for risk neutral
DM
The optimal consumption bundle will depend on the slopes of
the budget line and indifference curves
q
The slope of the budget line − 1−q depends on q (decreases in
q)
p
The slope of the indifference curve − 1−p depends on p
(decreases in p) 26 / 191 Optimal Consumption When q > p In this case, you do not buy any insurance
In this case, a dollar in the bad state costs
good state q
1−q dollars in the p
However, since MRS is 1−p , the value of a dollar in the bad
p
state is worth 1−p times the one in the good state
q
p
Since 1−q > 1−p , a dollar in the bad state is not worth is
relative market price 27 / 191 Optimal Consumption When q < p In this case, you spend all your money on the insurance
In this case, a dollar in the bad state costs
good state q
1−q dollars in the p
Since MRS is 1−p , the value of a dollar in the bad state is
p
worth 1−p times the one in the good state
q
p
Since 1−q < 1−p , a dollar in the bad state is worth more than
its relative market price 28 / 191 Optimal Consumption When q = p In this case, you are indifference between buying any amount
of insurance or not buying at all
When q = p, then we say the price of insurance q is
actuarially fair 29 / 191 Recap of the Lecture on March 22nd Typical situation where you might want insurance
Bernoulli utility function: v (x)
Initial wealth: $w
With prob. p, suffers loss of $L
With prob. 1 − p, nothing happens Insurance offered
Pay $q now, then you will get a dollar only when you suffer the
loss Question: how much insurance will you want to purchase? 30 / 191 Recap of the Lecture on March 22nd We could have solved it algebraically
Expected utility from buying z units of insurance is
pv (w − L − qz + z) + (1 − p)v (w − qz)
Take the first order condition with respect to z
Solve for z
But instead we solved this problem by reformulating it with
familiar graphical tools of economics: budget line and
indifference curves 31 / 191 Recap of the Lecture on March 22nd What are the two ”goods” to be consumed?
X: wealth in bad state
Y: wealth in good state What is your utility function over the two goods, u(X , Y )?
u(X , Y ) = pv (X ) + (1 − p)v (Y )
How to draw budget line?
Calculate how much Y you should give up to get a unit (or one
dollar) of X 32 / 191 Recap of the Lecture on March 22nd (X0 , Y0 ): your current consumption of the two goods
If you want to increase your consumption of X from X0 to
X0 + 1, what should you do?
You buy Buying
Y by... 1
1−q q
1−q 1
1−q units of insurance units of insurance decreases your consumption of units (or dollars) So from (X0 , Y0 ) to (X0 + 1, Y0 − q
1−q ) 33 / 191 Recap of the Lecture on March 22nd 34 / 191 Recap of the Lecture on March 22nd Risk neutral decision maker whose v (x) = x
Utility function u(X , Y ) = pX + (1 − p)Y
Is his/her indifference curve
1. Convex to the origin?
2. Concave to the origin?
3. Linear? The slope of each indifference curve is?
Question: what does his/her optimal consumption bundle
depend on? 35 / 191 Recap of the Lecture on March 22nd
When q > p
When q < p
When q = p
When q = p, we say that the price of the insurance is...
actuarially fair What is special about actuarially fair price? When q = p, what
are insurer’s expected profits from a unit of the insurance?
With probability p, the insurer has to pay out a dollar, so he
suffers loss of 1-p dollars
With probability (1-p), the insurer makes the profit of p dollars
So his expected profit is -p(1-p)+(1-p)p=0 So when the price of insurance is actuarially fair, the insurer
makes expected profit of zero 36 / 191 Risk Neutral Averse Maker
Now suppose you are a risk averse decision maker
Most individuals should be risk averse
Then what is the sign of the second derivative of your
Bernoulli utility function?
Your utility function over the consumption bundle (X,Y) is
u(X , Y ) = pv (X ) + (1 − p)v (Y )
MRS is equal to...
∂u
∂X
∂u
∂Y = pv (X )
(1 − p)v (Y ) you can check for yourself that MRS decreases in X 37 / 191 Shape of the Indifference Curve for Risk Averse DM What is the optimal consumption bundle when the price of
insurance is actuarially fair?
That is, q = p At the optimum, the MRS = the slope of the budget line
pv (X )
q
p
=
=
(1 − p)v (Y )
1−q
1−p
where the second equality comes from the price being
actuarially fair
38 / 191 Shape of the Indifference Curve for Risk Averse DM
p
pv (X )
=
(1 − p)v (Y )
1−p
⇒v (X ) = v (Y ) ⇒ X = Y
This means that if the price of insurance is actuarially fair,
risk averse DM optimally choose equalize X with Y
How much insurance do you buy in the optimum? Suppose
you buy z units and let’s solve for z
Then your wealth in the good state is w − pz
Your wealth in the bad state is w − pz − L + z
Then since in the optimum your wealth level across states is
equalized,
w − pz = w − pz − L + z ⇒ z = L
So you buy L units of insurance
39 / 191 Change in Expected Utility for Risk Averse DM
We say that the risk averse DM is fully insured if X=Y
Then is the risk averse DM better off by purchasing the
insurance? Note that the expected wealth level decreases relative to
status quo
Expected wealth without insurance: w-pL
Expected wealth with optimal level of insurance: w-L So by getting fully insured and removing risk (which was
costly), risk averse DMs are better off
40 / 191 Optimal Consumption When q > p
Again, at the optimum the MRS = the slope of the budget
line
q
pv (X )
=
(1 − p)v (Y )
1−q In this case, you do not get full insurance (i.e., X=Y) because
if X = Y , this condition is not met
When X = Y , the LHS (MRS) is smaller than RHS (the slope
of the budget line)
You can show graphically that at the optimum, X < Y
This means that when q > p, risk averse DM is less than fully
insured
Intuition: bad event is not very likely, so the dollar in the bad
state is not worth as much as when q = p, so you get less
than full insurance
41 / 191 Market for Insurance So risk averse decision maker will demand for insurance
because they get higher utility than by ”removing” the risk
with insurance
Now let’s look at the supply side of the insurance
There can be two cases depending on bad events are
independent across people or not
1. Large number of insurees; the event of each suffering loss is
independent
2. Small number of insurees; the event of each suffering loss is
NOT independent; large number of people immune to loss 42 / 191 Case 1: Large number of insurees; Independent Loss
Let’s recall what ”independence of two events” mean
Experiment: a procedure can be repeated infinitely, have a
fixed set of outcomes
1. Sam lives in his house
2. Anujit lives in his house Sample space: the set of outcomes of an experiment
1. {((Sam’s house burns down), (Sam’s house is intact)}
2. {((Anujit’s house burns down), (Anujit’s house is intact)} An event: a subset of the sample space
Two events A and B are independent if
Pr (A ∩ B) = Pr (A)Pr (B) 43 / 191 Case 1: Large number of insurees; Independent Loss It is likely that the events (Sam’s house burns down) and
(Anujit’s house burns down) are independent
Unless Sam and Anujit live next door or live together
We will talk about non-independent case Similarly, the event that an individual’s house burns down is
likely to be independent from the event that another
individual’s house burns down
Suppose that the probability of each individual’s house
burning down is 0.005
Each individual’s house is worth $1 mil.
In case of fire, the value of house is zero 44 / 191 Case 1: Large number of insurees; Independent Loss
Suppose that everyone’s Bernoulli utility function is
√
v (x) = x
Then, without insurance, the expected utility is
√
0.995 × 1, 000, 000 + 0.005 × 0 = 995
Now suppose that ICBC offers a full insurance at the price of
$5,500
What would be actuarially fair price?
So ICBC is not offering an actuarially fair price
As we have seen already, risk averse decision maker will get
less than full insurance if the price is not actuarially fair
However, for simplicity, let’s assume that everyone purchases
full insurance
45 / 191 Case 1: Large number of insurees; Independent Loss
Even without actuarially fair price, the insurees are better off
with this full insurance because
√
1000000 − 5500 = 997.246
Everyone’s happy; but will ICBC make nonnegative profit so
that it is willing to offer this insurance?
Suppose there are 500 insurees. Then ICBC collects
$5, 500 × 500 = $2, 750, 000
If 3 or more house burns down (or the fraction of houses burnt
down exceeds 0.55%), ICBC will make negative profits
We can simulate the probability that at least three house
burns down 46 / 191 Case 1: Large number of insurees; Independent Loss
With 500 insurees, the distribution of the fraction of
population whose house burns down is Pr(negative profit) ≈ 0.4540
What happens if we have more insurees?
47 / 191 Case 1: Large number of insurees; Independent Loss
With 50000 insurees, the distribution of the fraction of
population whose house burns down is Pr(negative profit) ≈ 0.051
What if more insurees?
48 / 191 Case 1: Large number of insurees; Independent Loss
With 5000000 insurees (roughly the population of BC), the
distribution of the fraction of population whose house burns
down is Pr(negative profit) ≈ 0
49 / 191 Case 1: Large number of insurees; Independent Loss As we can see, as the number of insurees increases, the
probability of negative profit gets closer to zero
So as the pool of insurees becomes large, the insurer can be
guaranteed to make positive profit
Everyone who is insured is better off; the insurer makes
positive profit
Introducing insurance leads to Pareto improvement
Pareto improvement occurs when no one is worse off and
someone is better off 50 / 191 Case 2: Small number of insurees; Non Independent Loss
Some bad events hit a ”region” or ”neighborhood” in a larger
society
Earthquake
Flood
Epidemic Suppose Sam and Anujit now lives in a city, say LA, where the
probability of earthquake is 0.005
The two events
Sam’s house breaks down due to earthquake
Anujit’s house breaks down due to earthquake is not independent because, most likely
Pr (Sam & Anujit’s houses break down due to earthquake) = 0.005
rather than 0.0052
51 / 191 Case 2: Small number of insurees; Non Independent Loss
Suppose the value of all houses in LA is $1, 000, 000
Without insurance, the expected utility is 995 utils
Now an insurer comes along and say,
”Pay x dollars now; If earthquake, only then I will give you x
dollars; otherwise I will keep the x dollars”
Will anyone buy this insurance?
The expected utility from this insurance is
√
√
1000000 − x × 0.995 + 1000000 − 1000000 − x + x × 0.005
√
=0.995 1000000 − x
which is less than 995 utils; so no one buys this insurance
The reason this insurer cannot offer more than x dollars in
case of Earthquake is because earthquake affects everyone in
LA
No demand for this kind of insurance, so no suppliers
52 / 191 Case 2: Small number of insurees; Non Independent Loss
No private firm would be willing to provide insurance; but
government intervention might help
Government can appropriate money from non-LA citizens to
cover the loss of LA citizens
Suppose that there are 3 mil households in LA and the US
consists of is 300 mil households
Assume that everyone outside of LA in the US has wealth of
$1 mil.
The total loss from the earthquake to LA divided by the
number of non-LA citizens is
3000000 × $1000000
≈ $10101
297000000
So if everyone outside of LA pays $10101 in case of
earthquake, LA citizens are fully insured
What happens to the expected utility of people in the LA and
outside of LA?
53 / 191 Case 2: Small number of insurees; Non Independent Loss
Without government intervention, the expected utility of
LA citizen: 995 utils
Non-LA citizens: 1000 utils
The sum of expected utility: 995×3 mil + 1000×297 mil =
299.985 tril utils With government intervention, the expected utility of
LA citizen: 1000 utils
Non-LA citizens: 999.97
The sum of expected utility: 1000× 3 mil + 999.97× 297 mil
= 299.99109 tril utils So the sum of expected utility increased
However, this is not a Pareto improvement (why?) 54 / 191 Case 2: Small number of insurees; Non Independent Loss Examples of government insurance
9/11 Victim compensation fund
Medicaid (government insurance prpogram in the US for low
income families)
International donations for refugees
All kinds of disaster relief 55 / 191 Insurance Markets Are Welfare Improving
For every risk in life, if full insurance is offered at the
actuarially fair price, then everyone will benefit
However, in the real world things are not that simple
There are risks for which not everyone is insured
Health insurance market in the US
Life insurance (the risk of death): as of 2015, 60 percent of
Americans own life insurance There are risks for which full insurance is not available
Deductibles: even after paying the premiums, we pay certain
amount if we go see doctors
Caps on coverage: most insurance policies have caps on
coverage There are risks for which there exists no insurance
No insurance market for low income due to decline of the
industry your job belongs to
No insurance market for marrying the wrong person 56 / 191 Why Insurance Markets Are Incomplete? There can be many reasons
1. Some people cannot afford to buy insurance (e.g., health
insurance)
2. Some risks are uninsurable because everyone faces the same
risk (e.g., global warming)
3. Insurees know the likelihood of bad events while the insurer
does not; leads to adverse selection
4. Moral hazard: insurees will engage in ”bad” behavior which
makes insurers not want to offer insurance (e.g., eating junk
food only after getting full health insurance) We are going to study how adverse selection leads to
insurance market failure 57 / 191 Recap of the Lecture on March 24th
Risk averse decision makers would choose to be fully insured if
the price is fair
Some losses are large (medical expenses for serious illnesses,
accidents of expensive cars); why would anyone want to
supply insurance
When large number of insurees with independent risk: as the
pool gets large, the prob. of positive expected profit for the
insurers converges to 1
Results in Pareto improvement When small number of insurees with dependent risk: private
insurers would not be able to provide insurance; government
provides insurance
Not Pareto improvement; the sum of expected utility rises 58 / 191 Recap of the Lecture on March 24th Real world insurance markets are perfect
In some markets, there are people who are not insured at all
(e.g., some not medically insured)
In other markets, full insurances are not provided (e.g.,
deductibles, cap on coverage)
For some risks there are no insurance that can be purchased in
the market (e.g., risk of bad marriage) Reasons for imperfect insurance markets
Cannot afford insurance
Some risks are uninsurable
Moral hazard (bad behavior)
Private information leading to adverse selection 59 / 191 Model Without Private Information All insurees
have the same initial wealth $w
have the same probability of loss p
If loss, then the same amount of loss $L
have the same Bernoulli utility function v (x); are risk averse 60 / 191 Model Without Private Information An insurance company can offer an insurance policy whose
premium is $I and whose benefit is $B
The price of one dollar in the bad state would be I
B There is competition between insurance companies
Question: what are the insurance policies that are offered in
equilibrium?
Definition of the equilibrium: in equilibrium
1. No insurance policies that are offered makes negative profits
2. No insurance policies that are not offered, if offered, would be
more profitable than policies that are offered 61 / 191 Equilibrium in Model Without Private Info We have seen that when the price is actuarially fair, then risk
averse decision makers will get full insurance
So our natural ”guess” about the insurance policy (I e , B e )
that is offered in equilibrium is the full insurance at the fair
price
B e = L (”full insurance”)
I e = pL (”at fair price”) 62 / 191 Equilibrium in Model Without Private Info Indifference curve? 63 / 191 Equilibrium in Model Without Private Info 64 / 191 Equilibrium in Model Without Private Info. Let’s verify if this is an equilibrium.
Equilibrium Condition 1: Does (I e , B e ) make nonnegative
profits?
What is the expected profit by insurance companies from
offering (I e , B e )?
Firms that offer this policy make the following expected profit:
p(I e − B e ) + (1 − p)I e = p(pL − L) + (1 − p)pL = 0
So yes 65 / 191 Equilibrium in Model Without Private Info. Equilibrium Condition 2: are there...