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 .

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 )

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 .

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 ,

.

•

*.

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

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 .* .

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 -

.