Transcript Topics in Health and Education Economics

Topics in Health and
Education Economics – class2
Matilde P. Machado
[email protected]
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Summary:
 Shows the impact of imperfect information on
the equilibrium outcome of a competitive
insurance market.
 Insurance companies offer insurance contracts
that rely on a self-selection mechanism
 High risk individuals cause an externality on low
risk individuals
 Everyone would be better off (or as well off) if
risks were revealed ex-ante.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Model I – the case of a single type of consumer:
 There are two states of nature {no accident,
accident}.
 No insurance: wealth = {W,W-d}
 p ≡ probability of accident occurring
 a=(a1,a2) is the insurance vector where a1 is
the premium paid by the consumer and a2 is the
net compensation in case of accident, i.e. a2 = qa1 where q is the insurance coverage.
 With insurance: wealth = {W- a1,W-d+ a2}
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
1. Supply side of the insurance market:
 Insurers are risk-neutral, they maximize expected
profits
 Perfect competition  Expected profit = 0
 ( p, a )  (1  p)a1  pa 2  0  a 2 
Expected profit of
selling contract a to
individuals with
probability of accident
=p
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1 p
a1 (A)
p
Competitive relation
between the net
compensation and the
premium
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
This condition is equivalent to say that premium is actuarial fair:
 1- p 
a1
1- p
a 2  q  a1 
a1  q  
 1  a1  q 
 a1  pq
p
p
 p

a1 is the actuarial fair premium i.e. the premium is a percentage of
the coverage that equals the probability of accident
It is well-known that at actuarial fair premiums risk
averse individuals want to hire full insurance.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
proof:
denote Q the insurance coverage and  Q the premium.
If the premium is actuarial fair then   p.
The optimal coverage is given by:
Max pU (W  d  Q   Q )  (1  p )U (W   Q )
Q
F.O.C: (1-  ) pU ' W  d  (1   )Q    (1  p)U ' W   Q 
If   p :
(1- p) pU ' W  d  (1   )Q   p(1  p)U ' W   Q  
U ' W  d  (1   )Q   U ' W   Q  
W  d  (1   )Q  W   Q 
Q*  d
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Full insurance
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
W2
Wealth after
accident
W1=W2 Full
insurance
combinations
W2>W1
relevant
W1>W2
45º
W1 Wealth with no
accident
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)

All wealth combinations along EF are feasible (i.e. Expected profit=0). F
is the feasible point of Complete insurance
W2=W1 (Completely insured)
W2
Possible wealth
combinations with
competitive insurance, slope
= - (1-p)/p
(with accident)
W2>W1
F
Wealth without
insurance
W1>W2
W-d
E
45º
W
W1
No accident
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Derivation of EF line:
Wealth with NO accident : W1  W  a1  a1  W  W1
1 p
1 p
With accident: W2  W  d  a 2  W  d 
a1  W  d 
(W  W1 )
p
p
 1 p 
W
1 p
W2   d 
W1  EF line, slope =  

p
p
p


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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Definition of Equilibrium: The equilibrium in a
competitive insurance markets is a set of contracts
such that:
(i)
No contract in the equilibrium set makes negative
profits
(ii) There is no contract outside the set that, if offered,
would make no-negative profits.
We know that when the premium is actuarial fair, risk
averse prefer F to the rest of the possible wealth
combinations, i.e. complete insurance.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
2. Demand Side of the Insurance Market

Individuals maximize their expected utility that
depends only on their wealth

Individuals know p

Their utility function is state independent

Their expected utility function is:
pU (W2 )  (1  p)U (W1 )
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
The slope of the indifference curve at F equals the slope of
EF line. F is therefore the optimum point (highest i.c.) in EF.
0  dUE  d  (1  p)U (W1 )  pU (W2 ) 
dW2
1  p U (W1 )

1  p U (W1 )dW1  pU (W2 )dW2  0 
dW1
p U (W2 )
At F, W1  W2 (complete coverage) Thus:
dW2
dW1
W1 W2
1 p

p
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The slope of the indifference curve at
W1=W2 is independent of U(.) and the
same as the EF line. The tangency of the
indifference curve to EF shows that
individuals maximize their expected utility
at F.
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
The equilibrium:
W2=W1 (total insurance)
W2
F
a*2
Optimal net
compensation
E
W-d
45º
W
Optimal
premium=a*1
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W1
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
For a*=(a*1, a*2) to be an equilibrium we need to
check that it satisfies the two previous conditions
which amount to:

BE=0 (The wealth combination must be along
the EF line)

Any other insurance contract (a1,a2) that
consumers may prefer has negative benefits <0.
Those would be contracts that lead to wealth
combinations in higher indifference curves and
obviously those are lower premium for the same
(or higher) compensation or higher
compensations for the same (or lower) premium
which yield negative expected profits.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Model II – the case of two types of consumers:
Low Risk – probability of accident = pL
High Risk – probability of accident = pH
pH > pL

l ≡ proportion of high risks 0<l<1
p  l pH  (1  l ) pL  average prob. of accident
-
-
Individuals know their types and their probability of
accident
Individuals only differ in their risk, the insurance company
cannot distinguish them ex-ante, however the insurer knows
the values of pH , pL and l.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
The insurance company knows that for the same
premium, high risks would like to hire more
coverage. It will use this information to devise selfselection mechanisms.
Individuals cannot buy more than one insurance
contract
There can only be two types of equilibrium:
-
-
-
Pooling – both types buy the same contract
Separating – Each type buys a different contract
It can be shown that a Pooling Equilibrium never exists.
-
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Proof that a Pooling Equilibrium never exists.
By contradiction, suppose aa1, a2 is a pooling
equilibrium, in that case, it can be shown that Exp.
Profits are a function of the average prob. of
accident:
 ( p, a )  l (1  pH )a1  pH a 2   (1  l ) (1  pL )a1  pLa 2  
  l (1  pH )  (1  l )(1  pL )a1  l pH a 2  (1  l ) pLa 2
  l  l pH  (1  l )  (1  l ) pL  a1   l pH  (1  l ) pL  a 2
 (1  p )a1  pa 2
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
The rate of substitution between the two states of nature
can be derived for the high risks:
UE H  1  pH U (W1 )  pHU (W2 )
dUEH  0  1  pH U ´(W1 )dW1  pHU ´(W2 )dW2  0

dW2
dW1
H

a
(1  pH )U ´(W1 )
(1  pH )U ´(W  a1 )

pHU ´(W2 )
pHU ´(W  d  a 2 )
And similarly for the low risks:
dW2
dW1
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L

a
(1  pL )U ´(W1 )
(1  pL )U ´(W  a1 )

pLU ´(W2 )
pLU ´(W  d  a 2 )
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
So the difference of the rate of substitution for high and
risks only depends on the probabilities:
pH  pL 
1
1
<
pH pL
and
1  pH < 1  pL

1  pH 1  pL
1  pH 1  p 1  pL
<
and
<
<
pH
pL
pH
p
pL
H
dW2
dW1
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a
L
dW2
<
dW1
a
The indifference curve of the
Low risks (for the equilibrium
contract a) is steeper in
absolute terms.
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Note that a must be in the EF curve so that the expected profit is =0
W2=W1
W2
a
F
UH
W-d
UL
45º
W
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W1
Note that EF has now a
slope that depends on the
average prob. of accident
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
The existence of a contract b shows that a is not an eq.
W2=W1
W2
a
F
UH
W-d
UL
45º
W
 ( pL , b )   ( pL ,a )   ( p,a )  0
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W1
b is preferred to a by the
low risk and since it is in EF
or even slightly above can
be offered in the market
(because only low risks
would buy it).
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
If an equilibrium exists it MUST be separating. They
assume that there is no cross-subsidization, i.e.
competition forces the insurance company to break
even for every contract. The zero expected profit
conditions are now given by:
 ( pL , a L )  (1  pL )a1L  pLa 2 L  0  a 2 L 
1  pL
a1L
pL
 ( pH , a H )  (1  pH )a1H  pH a 2 H  0  a 2 H 
(B1)
1  pH
a1H
pH
(B2)
Which imply two different wealth combination lines
from the initial E point.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
For the same premium the insurance company could offer a much
higher net compensation to the low risks.
W2=W1
W2
L
Slope EL: 
H
E
45º
W1
Premium =a
Measures competitive net
compensation for high and low risks
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for a premium = a
Slope EH: 
1  pH
pH
1  pL
pL
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
However H and L do not constitute an equilibrium
U’H
W2
L
UH
q
H
aH
45º
 ( pH ,a H )  0;  ( pL ,q )  0
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E
Of all wealth
combinations along
EH, aH is the preferred
one by the high risks.
And of all those along
EL, q is the preferred
by the low risks. We
know that E=0 if aH is
sold to the high risks
and q to the low risks.
The problem is that q
is also preferred to aH
by the high risks since
it means higher wealth
in both states of
nature.
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
And the insurer will have negative expected profits if it sells q to
everyone. That is if all individuals buy the contract q then:
 ( p,q ) <  ( pL ,q )  0
W2
L
UH
q
U’H
aH
E
45º
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
The segment between EaL shows the set of wealth combinations that
could be offered to the low risks at zero expected profit and are NOT
preferred to aH by the high risks (incentive compatibility constraint)
UL
W2
L
UH
q
aL
aH
E
45º
The set of candidates to an equilibrium is aH and contracts along EaL .
Of all those along EaL , aL is the preferred by the low risk. So we will
check under which conditions {aH ,aL} is an equilibrium.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
To prove that {aH, aL} is indeed an equilibrium:
The first condition is satisfied because the
insurer has expected zero profit in both contracts.
The second condition is the difficult one. The
existence of equilibrium depends on the
percentage of high risks, l. It turns out that if l
is high enough there is an equilibrium, otherwise
there isn’t.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Suppose g is also offered. Then all individuals would prefer g to the eq. candidate. The
question is can g be offered in the market? For {aH,aL} to be an equilibrium, the exp.
profit from g should be <0.
W2
L
UH
q
g
aL
H
aH
E
UL
45º
The expected profit from g is given by:  ( p, g )
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
For a low level of l, the line of possible wealth combinations achieved with
contracts that are sold to both types and breakeven is near EL for example
EF2. If l is high then the line is somewhere close to EH e.g. EF1.
W2
q
F2
UH
F1
g
aH
E
UL
45º
For the high value of l (EF1) g cannot be offered at a profit so {aH,aL} is an equilibrium.
For a low of l (EF2) there is no equilibrium in this competitive market.
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2.2. Adverse Selection/Risk Selection
Rothschild & Stiglitz (QJE,1976)
Conclusions:
 the incomplete information may cause a
competitive market to have no equilibrium
 The high risks are a negative externality on
the low risks
 Everyone would be at least as well off if
everyone revealed their type
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
This paper is nice because:
 It is an application of R&S
 It uses data!! (The British Household Panel Survey)
 It tests adverse selection in a market where there are
private health insurers on top of a National Health
Service (NHS).
 They find evidence of adverse selection in the
private insurance market
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
Setting: The British Private Health Insurance Market.
 What’s especial?




There is a public system that covers all expenditures (no copayments
with few exceptions e.g. dental) and everyone.
If people want to they can hire a private insurance. Private and Public
are then substitutes for care. Everyone contributes to the public
system through taxes.
Other systems such as most of the American is purely private (except
for Medicare/Medicaid)
In other systems, the private is supplementary, people hire the
private to cover for copayments and services not covered by the
public (Belgium, France).
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Model:







Individuals decide whether to take private insurance after observing
their type and before becoming ill.
Individuals when become ill must choose where they want to be
treated (private/public) and private and public services cannot be
combined.
If an individual chooses the private service, the private insurer must
cover the full cost.
Everyone contributes to the financing of the public health service
regardless of whether he/she uses the public health services (PUB).
There are a large set of insurers (similar to R&S)
Individuals can be of one of two types {L,H}, 1>PH>PL>0 and they
know their type
g≡ (1-l) proportion of low risks, 0< g<1
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Model (timing):
Health
authority
chooses
package
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Private insurers
simultaneously
make their packages
conditional on the
package of the HA
Individuals decide whether
or not to take private
insurance, and which
insurer, conditional on the
packages of all providers
and their type
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)




If an individual buys private insurance, he has
double coverage and in case he gets sick must decide
if he wants the public (PUB) or private treatment
(PRI)
L0 ≡ loss if an individual gets sick and does not seek
treatment
(Lpri,q) ≡ private contract (q ≡ premium, Lpri ≡
Loss i.e. L0-Lpri ≡coverage)
(Lpub,0) ≡ is the outside option of the individual
offered by the public system (premium is paid
through taxes). Assumption Lpub<L0 the public is
effective.
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)


W ≡ individual’s initial wealth net of taxes
If the individual does NOT take private insurance:



And he does not become ill: W
And he becomes ill : W-Lpub
If the individual TAKES private insurance:
And he does not become ill: W-q
 And he becomes ill : W-q-Lpri
Note: Private contracts where Lpri>Lpub are irrelevant
because they are strictly dominated by the public package
(the assumption implies that if ill the individual goes to
the private system). The insurer commits to ensure the
individual does not suffer a loss larger than Lpri

Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
p  g pL  (1  g ) pH

Is the average probability of getting sick
If the individual does NOT take private insurance:

Expected utility for probability p is:
pu(w  Lpub)  (1  p)u(w)

If the individual TAKES private insurance:

Expected utility for probability p is:
pu(w  Lpri  q)  (1  p)u(w  q)

The expected profit of a contract (L,q) is:
 ( L, q)  q  p( L0  L)
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coverage
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)


In the case of no-illness for a contract (L,q) and wealth w
n  w  q  n  W1  q  w  n
In the case of illness :
a  w L  q

 a  W 2  L  w  q  a  n  a
The expected profit of a contract (L,q) is:
 (n, a)  q  p( L0  L)  w  n  p ( L0  (n  a ))
q
L
the slope of the isoprofit curve in (n,a) space is:
0  dn  pdn  pda 
da
1 p

dn  
p
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

We have 2 zero-isoprofit curves depending on the
individual’s type, with slopes:
da
dn

H

 0
1  pH da
;
pH dn
L

 0
1  pL
pL
The zero-isoprofit curves go through point E =
(w,w-L0) (i.e. no insurance)
 (n  w, a  w  L0 )  w  n  p( L0  (n  a))  0
0
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 L0
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
3.1. Symmetric Information
 If there is NO public system, then the equilibrium
would be just as in R&S with symmetric
information, i.e. efficient contracts or complete
coverage to both types i.e. {a*L,a*H} i.e. for J=H,L
 J (n, a)  0  w  n  pJ ( L0  (n  a))  0  n  a
 w  n  pJ L0  0  n  w  pJ L0  a
actuarial fair contracts.
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2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

In the presence of a public system the status-quo is
not E=(w,w-L0) but P=(w,w-Lpub)
W2=a
W-pLL0
L
a*L
UH
a*H
W-pHL0
W-L0
E
45º
W-pHL0
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W-pLL0
W
W1=n
Possible
positions of the
public package.
If Lpub=L0
then E is the
status quo with
public coverage
(i.e. it is as if
the public
system did not
existed)
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Assumption 1: If all individuals of type J are
indifferent between the public package P and the
best private contract for them, all these individuals
choose the public package.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
In the presence of P=(w,w-Lpub) some private contracts (in the symmetric
information eq.) may not be attractive any more
a=n
a
W-pLL0
L
UH
a*L
P3
a*H
W-pHL0
P2
W-L0
E
P1
45º
n
If P=P1 then private market not affected, the public contract is NOT ACTIVE; If
P=P2 a*H not attractive anymore; If P=P3 no private contract is attractive, the private
market is NOT ACTIVE
W-pHL0
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W-pLL0
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

The equilibrium will depend on P
W2
W-pLL0
L
UH
a*L
L0
a*H
W-pHL0
W-L0
E
H0
45º
W-pHL0
W-pLL0
Proposition 1: In case P is between E-H0 the equilibrium is {a*H,a*L}; if P lies between
H0 and L0 the equilibrium is {P,a*L}, in case P lies strictly above L0 then only P exists
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
So if both sectors are active (i.e. P is between H0 and
L0) and there is no adverse selection the probability
of illness among the ones with a private insurance is
pL i.e. the low risk, lower than the average
probability of illness.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
3.2. Asymmetric Information
 If there is NO public system, then the equilibrium
would be just as in R&S with asymmetric
information, i.e. efficient contracts or complete
coverage to the high risks and less than full
aˆ , a 
coverage to the low risks
 We know that in R&S we need the proportion of
high risks to be high enough so that there is
equilibrium i.e. g≤g*
 Assumption 2: assume g≤g*
L
Matilde Pinto Machado
*
H
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

The equilibrium will depend on P:
W2
W-pLL0
aˆ L
L
UH
a*H
W-pHL0
H0
W-L0
E
L1
45º
W-pHL0
W-pLL0
Again H0 is the public contract such that a high risk is indifferent between the public
contract and the private contract a*H. L1 is the public contract such that the low-risk is
indifferent
between P and aˆ L . Note that H0 is above L1 (Lemma 2)
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
Case 1: P lies below L1  eq is aˆ , a  P is not active
Case 2-3: P coincides with L1 or is between L1 and H0 the
equilibrium is {a*H,P} (assumption 2 is no longer necessary
for the existence of the equilibrium.) Both Public and Private
are ACTIVE.
Case 4-5: P coincides with H0 or is above H0, in the equilibrium
only the public system exists.
If both sectors are active under adverse selection then the
probability of illness of those who purchase private insurance
is pH which is larger than the average probability of illness.
L
Matilde Pinto Machado
*
H
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Their Main Theoretical Result:



When both markets are active i.e. the private and the
public then, with perfect information (i.e. no adverse
selection) only the low risk would buy private insurance
When both markets are active i.e. the private and the
public then, with asymmetric information and therefore
adverse selection only the high risks would buy private
insurance.
i.e. the sign of the correlation between the probability of
buying private insurance and individuals’ risk depends on
whether there is adverse selection. This prediction can be
tested.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
Empirical Test:
 In the UK the private system is substitute to the
public one.
 Everyone is entitled to the public system, and
people pay it through taxes regardless of utilization
 The private system offers better access, in particular
negligible waiting times, in the model’s notation it
is true that Lpub>Lpri
 Data: Bristish Household Panel Survey (BHPS)
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
Empirical Test (cont.):

In the UK, Private and Public systems are active so the
model tells us that:



In the absence of adverse selection (perfect information) only the
low risks buy private insurance
In the presence of adverse selection only the high risks buy private
insurance.
We could therefore design a test by comparing the risk of requiring
medical care of those that decide to buy medical insurance to those
that decide not to buy it. However this would have problems:
1.
2.
Matilde Pinto Machado
One does not observe whether people require medical care, only
observes whether people get medical care
Access conditions are better in the private insurance so we could
observe more people getting medical care under private insurance and
wrongly conclude they were more needy of care.
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
Empirical Test (cont.):

To avoid the bias that may be introduced because of
differences in access to care between private and public,
they restrict the sample to individuals that have the same
access to hospitalizations: restrict to individuals that have
private insurance and distinguish between:



Individuals who buy private insurance on their own
Individuals who have private insurance as a fringe benefit from their
employer (i.e. they get it for free)
The test compares the probability of hospitalization of those
who purchase private insurance to those who get it as a
fringe benefit.
Note: as we mention in the previous class hospitalizations are likely to be free of
moral hazard.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)


They restrict the sample to England and to
employees employed on a permanent basis.
The identification assumption is that the individuals
that get private insurance as a fringe benefit
(conditional on covariates) are of the same risk, on
average as the population of english permanently
employed, i.e. the health insurance is orthogonal to
health status conditional on covariates such as age,
education, gender and income.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Potential bias:

Employer driven: some jobs are more likely to offer employer-paid
health insurance then others. If health of individuals in these different
jobs differ (conditional on covariates) then we may be biasing the
results. For example, in agriculture the % of employed paid
individuals is only 15% while in the banking, finance, insurance,
business sector is of 66%. By occupation, the % of managers with
employer-paid private insurance is of 63% while for operators of
plants and machines it is only 38%. To control for this possibility they
include in the regressions industry and occupation among the
covariates. This means that conditional on being of a certain
occupation we compare the effect of buying private insurance.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Potential bias (cont.):

Employee driven: If less healthy people search
for jobs that offer private insurance.


Matilde Pinto Machado
Some evidence (no proof): Most people that changed
jobs offer reasons other than private insurance, for
example, more money or better chances of promotion.
In any case, this would imply a smaller difference in
risks between the two groups (downwards bias) 
the true adverse selection would be even higher than
what they find.
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

The idea is:
Pool of permanently
employed
Get insurance as
Fringe Benefit
Conditional on X,
same average prob
of Hospitalization,
pbar
Comparison in
the data
Do not Get
insurance as
Fringe Benefit
Different risks
Buy insurance
Matilde Pinto Machado
Don’t Buy
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)


Note again that if the difference in
probabilities of hospitalization between the
two groups were statistically the same, this
would not be evidence in favor of symmetric
information (i.e no adverse selection) since in
the absence of adverse selection the difference
should be negative not zero.
Probit model with the dependent variable as
hospitalization
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
IND=1 if the individual
pays for the health
insurance directly; 0 if
Fringe Benefit
Matilde Pinto Machado
This table shows the
coefficients not the
marginal effects i.e.
the effects on the
probabilities. The
marginal effect of IND
is 0.021 given that the
average probability of
hospitalization is
0.064, this is quite
HIGH!
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)
Probit model – review:
yi  1 if b xi   i  0 where  i ~ N (0,1) (the variance cannot be identified)
yi  0 otherwise
Pr( yi  1)  Pr( i   b xi ) 
b xi
  (t )dt  (b x )
i

E ( y )  1 Pr( yi  1)  0  (1  Pr( yi  1))  Pr( yi  1)  ( b xi )
Therefore the marginal effects for the probit are:
E ( yi )
  ( b xi ) b - usually evaluated at the average value of b x in the sample
x
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Their test of adverse selection concludes that
individuals that purchase private insurance
have a higher probability of hospitalizations
(are higher risk) than individuals that have
private insurance as a fringe benefit. This
constitutes evidence of adverse selection in
the British private medical insurance market.
Matilde Pinto Machado
2.2. Adverse Selection/Risk Selection
Olivella, Vera-Hernández (WP06/02)

Robustness:

What if what they are finding is just the result of the fact
that individual purchased policies are more generous than
firm purchased policies and this is what makes us observe
more hospitalizations? They argue that if individual
policies were more generous then the probability that
conditional on being hospitalize individuals choose the
public service (NHS) should be smaller for those having
individual purchased policies. They run a probit
(dependent variable is choose NHS, conditional on
hospitalization) and find that the coefficient of IND is not
statistically significantly different from zero.
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap2)


A decision maker n faces J alternatives
The decision maker chooses the alternative that gives him/her
the greatest utility:
choose i iff: Uni  Unj j  i


The researcher does not know U, it observes the decision and
some attributes of the alternative j xnj and some attributes of
the decision maker sn.
Rewrite the utility function as a function of observables and an
additive unobservable:
Unj  Vnj   nj , the joint density of ( n1,.....,  nJ ) is f ( )
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap2)

The probability of choosing alternative i can be written as:
Pni  Pr U ni  U nj , j  i  
 Pr Vni   ni  Vnj   nj , j  i  
 Pr   nj   ni < Vni  Vnj , j  i  
  I   nj   ni < Vni  Vnj , j  i  f ( n )d  n

This integral is
of dimension J
(i.e. the number
of alternatives)
Matilde Pinto Machado
This integral has a closed form for certain
distributions of  i.e f(.). For example if f() is iid
extreme value than it converts to logit. Or when f()
is generalized extreme value we obtain nested
logit.
Review: Discrete choice Models – Random
Utility framework (Train, chap2)
There are several things one must know about these models:
1. Only differences in utilities matter, its absolute value does not
matter – add a constant to the utility of every alternative and
the decision maker keeps choosing the same alternative:
if U ni  U nj , j  i   U ni  k  U nj  k , j  i 
alternatively notice that the decision only depends on the
difference of utilities:
Pni  Pr(U ni  U nj  0, i  j ) 
 Pr( nj   ni < Vni  Vnj , i  j )
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap2)
There are several things one must know about these models:
2. When adding an alternative-specific constant, one must
normalize one of them, i.e. they are identified up to a constant
since all ki, kj for which the difference=d are possible
estimates.


Pr Vni  ki   ni  Vnj  k j   nj   Pr   nj   ni < Vni  Vnj  ki  k j 


d


This is equivalent to normalize for example ki=0 and estimate
kj=-d
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap2)
3. Notice that the model cannot incorporate attributes of the
decision maker because they do not vary by alternative, i.e.
they cancel out. Imagine we could, take wn to be individual
n’s income. Then:
Pr  X ni b x  b w wn   ni  X nj b x  b w wn   nj   Pr  nj   ni < X ni b x  X nj b x 
4.
Only if income has a different impact on the different
alternative utilities:
Pr  X ni b x  b 0 wn   ni  X nj b x  b1wn   nj  
Pr   nj   ni < X ni b x  X nj b x   b 0  b1  wn 
but only the difference  b 0  b1  can be identified.
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap2)
5. Another way to include socio-demographic characteristics of
the individual is to interact them with the alternative’s
characteristics
6. The scale of utility is irrelevant, that is if alternative i is
preferred to alternative j then we can scale up or down utility
that the result still holds true:
if U ni  U nj , j  i    lU ni  lU nj , j  i    lVni  l ni  lVnj  l nj , j  i 
Normalizing the scale of utility means normalizing the
variance of the error term. The coefficients should be
interpreted accordingly, suppose var()=s2
b 1

U nj  xnj     nj ; var( 1nj )  1
s 
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap2)
Property 6 is important when comparing the results of the same
model estimated in different samples. For example the model:
U c  aTc  b M c   c utility from commuting by car
Ub  aTb  b M b   b utility from commuting by bus
Was estimate for Chicago and Boston:
Boston sample estimates:
aˆ  0.81
Chicago sample estimates
aˆ  0.55
bˆ  2.69
aˆ
 0.301
ˆ
b
bˆ  1.78
aˆ
 0.309
ˆ
b
Which imply a lower variance of the error term in Boston, Time and Cost explain more in
Boston than in Chicago, i.e. unobservable factors are less important in Boston
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap3)
The Logit Model is derived by assuming that the error terms are
independently and identically distributed extreme value
(var=2/6):
U nj  Vnj   nj for J alternatives where: f ( nj )  e
  nj
e
e
  nj
, F ( nj )  e
The probability of choosing alternative i is given by:
eVni
e b xni
Pni 
if Vni  b xni then Pni 
Vnj
b xnj
e
e


j
Matilde Pinto Machado
j
e
  nj
Review: Discrete choice Models – Random
Utility framework (Train, chap3)
The IIA (Independence of Irrelevant alternatives) – for any two
alternatives i and k the ratio of probabilities is given by:
eVni
Vnj
e
Pni
eVni Vni Vnk
j
 Vnk = Vnk =e
e
Pnk
e
Vnj
e
j
i.e. independent of the rest of alternatives which is quite
restrictive. No matter if there are 2 or 10 alternatives
the odds ratio is exactly the same.
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap4)
GEV- Generalized Extreme Value
Because IIA is so restrictive other models emerged to avoid or
minimize its impact. The one that interests us now is the Nested
Logit. The choice set is partitioned into subsets, called nests.
The nested Logit does not completely relax the IIA assumption:
1. IIA holds within the nest- i.e. for any two alternatives
within a nest the odds ratio does not depend on any other
alternatives.
2. IIA does not hold across nests- the odds ratio between
two alternatives of different nests depend on all alternatives in
those nests.
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap4)
Example, commuting to work, I may consider:
•car, car pooling within the same nest
•Bus, train in a separate nest
•Therefore, suppose that our car breaks down, then the
probability of all the other alternatives should rise BUT
NOT proportionately i.e. the probability of car pooling
should rise by more than the probability of train/bus.
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap4)
Suppose we have K nests, B1,….BK then the
n=(n1,…nJ) has the following distribution:
 K 
 nj

exp    e
 k 1  jBk

lk



lk




The parameters lk measure the independence within
each nest k. If lk =1 there is NO correlation in nest k,
if this is so for all k we are back to LOGIT; If lk <1
then the error terms are correlated within nest k. Still
the model assumes no correlation between the ’s
across nests.
Matilde Pinto Machado
Review: Discrete choice Models – Random
Utility framework (Train, chap4)
The probabilities are given by:


e
e

lk 1
Vnj lk
Vni lk
 jBk
K 
Vnj
e


k 1  jBl
Pni 
ll

ll



IIA holds within the nest
Pni eVni lk
 Vnm lk if i and m belong to nest k
IIA does not hold but the odds ratio
Pnm e

Vnj lk
e
 jBk
Vnj ll
ll 
e
 jBl
Vni lk
e
Pni

Pnm
eVnm
Matilde Pinto Machado
lk 1
only depends on alternatives in
nests k and l. Independence of
irrelevant nests (IIN)


 if i belong to nest k and m to nest l
ll 1



Review: Discrete choice Models – Random
Utility framework (Train, chap4)
The meaning of these lambdas. Suppose there are only
2 nests (nest1=Portugal and nest2=abroad) then we can
understand the difference in the lambdas as a strictly
preference for one of the nests:
Suppose Vn1  Vn 2  V and N1 =N 2 =N then:
l1 1


eV l1   eV l1 
l1 1
V l1
V l1 l1 1
V l1 l1
e
N
e
e
N






Pni
eV
jB1
1
1
l1  l2
l1  l2




=
=
N

N




l2 1
V
l2 1
V l2
V l2 l2 1
V l2 l2
Pnm
e


e  N 2e 
e   N2 
eV l2   eV l2 
 jB2

if i belong to nest 1 and m to nest 2.
Matilde Pinto Machado