individuals - eco.uc3m.es insurance markets.p… · market for in sunos nce 1. policies = then...
TRANSCRIPT
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ADVERSE SELECT , > w Competitive
Insurance MARKETS
( Rothschild & Stiglitz , E 1970,"
Efl . : - Cup . InaraMarkets ! )
Apop - Ceti - of individuals face The risk of
anaccident resulting in ←wealth loss ,
Not-
Li - i/
- 5We
individuals
' wealth - K-
T
p ECo
.I ) : pros . accident so ,
L E ( o , ] i monetary loss
w
O ¥w - L
Individuals ' preferences an represented by a
von Neumann - Morgenstern utterly triteh : 112 → IN , when n' so , hi ' co .
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Market For IN Sunos NCE.
1. POLICIES=
Then is - competitive insure marmot
i - whole firms compete by offering ofterret ve policies
.
A policy is a pair
( I. D ) , whenNote . Sine any insure
-a company my offer any
I i premium ingen - a policy , Theanalysis of insurance me
D : deduct 'll nets is a bit difficult . .
Thus,
an i - divided subs Ls a policyII. D ) faces The lottery :
( I. D ) o
W - I : - x,
# w - I - D= XaEu II. D) = ( I - p ) ul w - I ) t p ul w - I
.
- D ) = Ci - p ) ulxnltphlxa ) .
Ex bias :
( I,
D ) = ( o , L ) ( No insane )
= ( I,
O ) I Full inn - ) Eu II. d . - Hu. I )
=( I
, te ) ( Party intra )
Typically , p . bi - with - Cryer a verge line . ,a s - her deductible ) will involve a W get premium .
-
I-
d- teen no Curves . .- Xa
I i - p ) L C xn ) + p h l Xo ) = €
x.
-- oil )
dxn
Ex±÷,lit) h . Cx ,= ±wµ Ki - p C- IC- I #:÷=÷. ." ¥
f- -
Execs:'
Ii:E.
H
D - te .
I xn.
K ) = En I I,
D )
whenXn ⇐ w - I
Xa = w - I - D
-
z.
Preferences For Poc . cos-
Let us look at The preforms of individuals
↳ r altar - five policies . A policy isidetf.ee witha pair ( Xn , Xa ) . X
- to
+ I
( xn . xd . - Ii - plulxnlxpulxo ) .
Xa - - W - I - D
..
Xn= W - I VV - pl . . - - . - - - - - -
W - I - DI
Ths,
th expected utility Iw - L - I
of on individual who sets . I .
I FOLcrises policy ( IID ) is w.hn
,'
w - E W x✓ ( w - I. W - I - D ) . h
Ass - all agents ten Th some-
ish,
retch
is k- un by all insure co - p - ins .
In a CE, if a pot . ay
( Ii D ) is offered , The
47 I = p ( L - D ) ( ie . , putt is Zuo )
121 ¥ I E. 5) sit . I > pll - F ) and
( w - I,
w - E - I ) > it ( w - I , w - I - D ) .
THE"
Fair ODDS LINE'
( FOL ) : Set of policies ( I , D )
s - tis fyi - I =p ( L - D ) ⇒ w - xn = PIL - ( xn - x. D
⇐ xa=( ¥ - L ) - xn
-
( t - a , R slope of in Fol is
- ¥P
Let is c al - L Th slope of R i - differ .a
↳ curves of oh i -didfS,
which :S
site ,R MRSC xan, Xa ) , i. a . ,
2U/2×nMRS ( xn , × . ) = - -
2U/2×a
=( I - p )L→
pL ' ( Xc )
=- HI k¥
P L' ( Xn )
It-
, wehe R f 16--4 rslt :
N¥e : A ( E is A Ser of Policies Srstlsfsirk
(1) AND such Troost ND Polqcly SAMSFY ,N6 (2)
exists.
-
-
3 . Come ETIVE Ear , Lisa um-
Pop .
Ina CE
,Th poly ( It , Do ) = I pl , o ) .
2
is offered me se . - - bed ly way individual .Xa
If policy ( PL , o ) is x xn- Profits
-t offered , b- t
cryother policy is thot 't ,
W - pl - . . C v
offered , The.• to¥ ga co - pig , by W - L . EIoffer 't a policy A( pltc , o ) ihr Eso w - pl W Kis smell , will be able to alt satall Th ieduidnfs i - TL me reet ofnice meeee profits . b- , co - pet -will to - a E ( I he profits ) downto too .It -
,
?-
Th CE The full insurer
ter policy is offered i I all
in dividends sass cit dual policy .( we an ieplictg as -1 Tht the is a - t.me of indos )
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ADVERSE SELECTION : HETENOGENOUS Risks
.
Now essu - Rt for a fraction he ( o , 1) of
individuals Th probability of suffering Re loss
L is pH , whereas tr The new - ingtrot . - I - X , It is o< plc pH .
It- for Kn , ×c ) C- 112+2 we here
MRSL lxn ,×c)= - # k¥ )PL L' lxa )
< - 1-11 LIpH h ' ( Xo )
= MAZS ,+(×n . Xa )
x.
@
.
xn
-
If insurance competes could recognise
in
agents
of Th d- Hout types of
rises,
The navel L effectively
favs i - su - - outs,
TL - armet
f- r low rise ite u . dels are ~
- cut for high rise ones ,
I i - a CE ee - L tyre ofa a -
tw .
- he me we its - H
i - weer - fair - odds - policy , ( PiL , o)
,
if } Hill.
Thet is,
I policies offered will
↳ I p"
L, o ) tr Lal rise is . art
( pl L,
o ) her low - . see individuals .-
Ds - 12 i - a - a Companies cannot d- stir --
fish between high I how - ' see ieliuidnls .
Is ieplias Rt uh a policy it offered , in order
to alack profits one needs to e - w which types willSubscribe it ,
-
12- E- of CE remei - s 12 L - :
° - ) policies tht mere two profits would
be subscribed,
ml policies Rt
alt - at de - ml mere positive
profits , cannot exist .
A PING ( E ,is one in which
all . ' - divided of both types LsscisnR see policy .
The A pooling CE does # exist .
Pnoqf÷$0 4.D) = ( o ,L ) is
=ta pooling CE . Snply ,
a companyNt otters ×a *
H toTh policy ( p#Ltte, o ) fallfor E > o smell enoughwill attract at least •0- w.ie••L7Th high rise individuals ,ah
'
will new positiveW ×
,
profits .
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Let ( I , D ) bn a pooling policy .
In ore for it to be a CE , it ↳ he
be i - th FOL relative
toFoll ,5) Foll L )
f = Xp " -111 - d) Pt.
foilist )000
x
In This case . There is a
policy At attractsx -
Lodonly low risk indie .-
def-
- d gives positive .w Xn
profits . r
B - → A Mrs or f- I * -,
FThe snoh - policy
exists ,
.
•
*.
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A Separating GWiLiB~The following policies
( It,
D ") = ( p 'tL , o ) , ( Ih , D'
)€ Follph )
such net
in 1 W . p "L ) = p" hlw . It DL )
+ r+ ( i . pH ) ul W . IY.
Form a CE if X is
sft . a only large
gw.tw.it
's)to( W . PHL ,W-p" L ) - €±
( 1pA)
:2 ,
Note ; Pg
( i ) It :( L rise i - dividends an fll ) insured
(2) Low " a an partially u •
D) Then is = rise pool 'T amongR two groups .
( no cross - subsidies ).
-
Note Tht if X is smoke,
~
we
myhue - site at - as ilhshnked
by th following C- pl
+,
+ u
••
@F
°<
lpbyTh pooling policy is prfrd to <
as4 both types . It - is -
cap - ) oftr2 pooling policy ( Fete ,o )
destesilts 12 separate) policies .
. if
cttzafs all i - dividends I news
profits .
If its is the use , thr is -C E
.
-
^
AnEx=le . .
W=L= 1.
hlx ).
. ✓×, p± . ty , PA = 'z .
R C E with cnnlle i - to :
( I " ,D " ) = ( ÷ , ° ) , ( Ih , D' )=( f. o )
Po°L=Potcy
Told )= In + ¥ = 1¥ .
Take D= Yy , The F ( in ) = ÷ .
( E.E) = ( FL , o ) = ( ÷ , o )
U #( I. 5)=U< II. I ) = By e. 83
The poolingpolicy is destabilised by , e. g. , R
Pet 7I IL
,'D' ) = ( p 'D , D ) , whenD= 1/2
.
R If II. IL ) . 0
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The spurting policies are :( I " .D " )=( I. o ) , -1 /phk . Ds )
,Ds)=|p4
' - H ,× ) , when
X solves i
Us, ( plaxl , x )
-±0µ.
I tzio ) = t .
i. a ,
tzF¥+tzV¥x=
fzi.
e .
,
Ds = 53 - II. 732 . ( not net D'
is indp - Lt of X . )When is R 's a CE ? For values of X nottoo smell . Specifically , in poly policy anntdestabilize The sporty at lit ' - .Rat is , the pooling policy an -t a HsetLw risk a gets , WL in an wscits Thpolicy ( FK ) , ° ) . We weu< ( p4i⇒s ) ,Ds ) . TEED.it#yErs=IyE(i+r)the expected utility of R pooling policy isVL ( Ein ) . o ) = VEN =Ftf
.
1 '-
Ig(,+B)§FjI⇐s XEECTFK .* .
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Are then recalling policiestht
my ipnuThe meet outers ?
khe 2 serenity gilts is
desltttedby R pooling policy , i.e. ,ttdplli. Ds ) ,Ds ) _< ULC FW , o ) * I
Tha re - let .g policy Rt iyuns a mandatory
non . at set .int ) felli - saran policy
is Part improving . all offs or
kkr off . of course , R 's policyteuor - re high ask : - divides .
WL ineeetlg 1*7 does not hold , such
policy make know risk riski
- divide , worn
off,
Ltmy
still be justified • -
an ex -te basis : 12 the Lrph , i - R
speratyegliris :
-
Uslx ) = X U !+ ( , - d) vs = ¥ + a. NFL .ir )
www. win R poocippolicyit ison =VFsw⇐F¥
We weus(
d) < it ( x ) on 6. D.
It- ,
'
under th veil of if nose'
,
euogomshould agree on - forcing
- policy of melon non - die # irhiyLlc i - sn -
.