12 Adverse Selection and Insurance; The Case with a

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1 12 dverse Seecion and Insurance; The Case wih a Monopoy Suppose ha here are wo ypes of consumers. Ca hem f; Lg Type has a probabiiy of an acciden given by Type L has a probabiiy of an acciden gives by L For boh ypes, he endowmen is fm G ; m B g ; where m G > m B (i.e., \sae B is when oss occurs). Risk neura monopois seing insurance; 12.1 Benchmark; Observabe Types If he monopois knows which ypes consumer he/she deas wih one may be incined o proceed as foows. Le p be he per uni price of insurance. Le D J (p) be ype Js demand for insurance as a funcion of he price, ha is D J (p) = max J u (m B + z (1 p)) + (1 J ) u (m G z) z Then, he monopois shoud sove max D J (p) (p J ) : p One coud anayze his probem, bu in genera he monopois coud do beer! The reason is ha aowing he consumer o buy any number of unis a he same uni price necessariy eaves some consumer surpus o he consumer. Tha is, we know (assuming risk aversion) ha p = J in order for he consumer o fuy insure. Bu ha woud give no pro o he monopois. ence, whaever he pro maximizing uni price woud be, i mus invove under insurance. 86

2 Suppose insead ha he monopois proceeds as foows. The consumer may ge eiher fu insurance and consume x J unis (regardess of wheher an acciden occurs or no) or ge no insurance a a, where u (x J ) = J u (m B ) + (1 J ) u (m G ) The expeced pro of his arrangemen is J (m B x J ) + (1 J ) (m G x J ) Proposiion 1 There is no insurance conrac ha boh gives he monopois a higher pro and makes he consumer wiing o buy insurance. I.e., a pro maximizing monopois fuy insures he consumer and exracs a he consumer surpus. To see his, suppose ha x B ; are he consumpions for he consumer in a beer conrac. If u is concave/consumer is risk averse we have ha u ( J x B + (1 J ) ) J u (x B ) + (1 J ) u ( ) If he expeced pro is higher from (x B ; ) han from (x J ; x J ) hen J (m B x J ) + (1 J ) (m G x J ) < J (m B x B ) + (1 J ) (m G ) () x J > J x B + (1 J ) Bu u is sricy increasing, so J u (m B ) + (1 J ) u (m G ) = u (x J ) > u ( J x B + (1 J ) ) J u (x B ) + (1 J ) u ( ) ; meaning ha he consumer is beer o buying no insurance a a. Remark 1 Conrac specifying consumpion in each sae is wihou oss of generaiy. This is caed a reveaion principe. The idea is ha if he monopois designs any sor of 87

3 conrac, say, where he price is a highy non-inear funcion of how much insurance is bough, when he opima choice is evenuay made he agen ends up wih some CONSUMPTION in each sae. We can aways repicae his by removing from he choice se a eves of insurance ha are no purchased (excep 0 since we ake he view ha he consumer mus be wiing o buy) Non-Observabe Types (Privae Informaion) gain, for he same reasons as above, an insurance conrac can be viewed as wo numbers (x B ; ) : From hese numbers we may de ne conceps ha may be more famiiar in rea word insurance. Premium = P = m G Bene = B = x B + P m B = x B + m G m B Noaionay i is simper o perform anaysis in erms of (x B ; ) ; bu i is equivaen wih maximizing over (P; B) : The crucia aspec when he monopois canno see who is who is ha L mus be wiing o prick conrac designed for L and mus be wiing o pick conrac designed for : This yieds he foowing probem. The monopois designs wo conracs, x B ; G x and x L B ; x L G o sove max x B ;x G ;xl B ;xl G L m B xb L (1 L ) m G x L G {z } expeced pro if ype is L + (1 ) m B x B + (1 ) m G x G {z } expeced pro if ype is J u xb J + (1 J ) u xg J J u (m B ) + (1 J ) u (m G ) (2) L u xb L + (1 L ) u xg L L u x B + (1 L ) u x G (3) u x B + (1 ) u xg u xb L + (1 ) u x L G (4) (1) We wi be abe o use graphs for mos of he anaysis. Bu, o ge o his poin we need o be abe o compare sopes of he indi erence curves for ype L and : 88

4 Leing he indi erence curve be described by a funcion f L ( ) ha soves L u (f L ( )) + (1 L ) u ( ) = L u (x B) + (1 L ) u (x G) for every (in some inerva around x G ). Taking derivaives we ge d d [ L u (f L ( )) + (1 L ) u ( )] = L u 0 (f ( )) df L ( ) d + (1 L ) u 0 ( ) =, d d [ L u (x B) + (1 L ) u (x G)] = 0 Sope of indi erence curve for ype L = df L ( ) = (1 L) u 0 ( ) d L u 0 (f ( )) Finay, evauae a x G = x B ) f ( ) = x B ; Sope of indi erence curve for L a (x G; x B) = df L (x G ) = (1 L) u 0 (x G ) d L u 0 (x B ) Obviousy, we can do same hing for ype Sope of indi erence curve for a (x G; x B) = df (x G ) = (1 ) u 0 (x G ) d u 0 (f (x G )) 12.3 The Low Risk Type as Seeper Indi erence Curves Everywhere Now, jus comparing he sopes a any poin (x B ; ) we have ha Sope of indi erence curve for L a (x B ; ) Sope of indi erence curve for a (x B ; ) = = df L(x G) d = df (x G) d (1 L )u 0 (x G) L u 0 (x B) (1 )u 0 (x G) u 0 (x B) (1 L ) L = (1 L) (1 ) L (1 ) > 1 13 Monopois Indi erence Curves (Isopro s) Suppose ha he monopois ses conrac (x B ; ) o a consumer wih ow risk. Then, he expeced pro is L (x B m B ) + (1 L ) ( m G ) ; 89

5 x b igh R isk L ow R isk x G Figure 1: Reaive Sopes of Indi erence Curves where we usuay woud have ha x B m B < 0 and m G > 0: n isopro is hen simpy a ine wih consan pro s, ha is, souions o L (x B m B ) + (1 L ) ( m G ) = k x B = 1 L L + k + Lm B + (1 L ) m G L I.e., sraigh ines wih sope individua are 1 L L : Simiary, he reevan isopro ines for a high risk x B = 1 + k + m B + (1 ) m G 14 The Pro Maximizing Conrac Sep 1 If he ow risk ype isn insured (eg., if x L B ; xl G = (mb ; m G )), hen opima conrac fuy insures he high risk ype a a premium ha exracs a he consumer surpus from he high risk ype. Proof. See Picure. The sraigh ine is he isopro when seing o he ow risk ype ony ha goes hrough he fu insurance poin a indi erence curve hrough endowmen. ny 90

6 x B Low Risk igh Risk e ee sope 1 p L p L sope 1 p p \ e \\ ee \ \ e \ \\ \ \ \ \ \ \ = \\ \\ \ igher Pro s \\ \ \ \ \\ \ a a aaaaaaaaaaaaa a aaaaaaaaaaaaa / igher Pro s a Figure 2: Consan Pro Loci (Isopro s) for Low and igh Risk Type Pan ha gives a higher pro herefore vioaes he Individua raionaiy consrain for he high risk ype. Sep 2 x L G x G Proof. See he Figure. Fix x L B ; G xl arbirariy. For IC-L o be sais ed (i.e., for L o be beer o wih x L B ; G xl han wih x B ; xg ) i mus be ha x B ; xg is beow he indi erence curve for he ow risk ype. For IC- o hod (i.e., for o be beer o wih x B ; x G han wih x L B ; G xl ) i mus be ha x B ; xg is above he indi erence curve for he high risk ype. ence, ony he shaded area in he Figure remains, which proves he caim. Sep 3 x L G m G (no ani-insurance ). Proof. Suppose ha x L G > m G: Consider Fig 5, where is he hypoheica opima consrac for ype L and poin is he poin where he indi erence curve hrough he endowmen for he high risk ype inersecs he indi erence curve for he ow risk ype hough poin : To saisfy IR- and IC-L i is herefore necessary ha he conrac for ype is in he wedge beginning a poin : Noe hen ha; 91

7 m B L ower P ro s s igher P ro s s 45 o m G Figure 3: Why igh Risk Type is Fuy Insured if No Insurance Sod o Oher Type 1) If (as in Figure) ype L is a a higher indi erence curve han he one hrough he endowmen, hen i is possibe o reduce he consumpion for L in (say) he bad sae and keep everyhing he same. This increases he expeced pro for monopois. If insead ype L is a he same indi erence curve as he endowmens (redraw he picure) poin coincides wih he endowmen. Moving ype L o he endowmen wi keep IC- sais ed. Since his is a movemen in he direcion of increased insurance aong a given indi erence curve he monopois wi increase is pro. Sep 4 IC- binds Proof. See Figure. If he incenive consrain for he high risk ype is no binding he monopois may reduce he consumpion in one sae of he word for he high risk ype and keep everyhing ese he same. Because of Sep 3, poin in he graph is a eas as good as he endowmen for he high risk ype, so he movemen from h o h 0 (which corresponds o reducing he consumpion for in case of acciden) wi saisfy boh IR- and IC-. Obviousy his increases he pro s for he monopois. Noice ha his argumen uses he resu ha he ow risk ype doesn ge ani-insurance in Sep 3 o rue ou he possibiiy ha poin is worse for han he endowmen, in which case h 0 woud no saisfy IR-L. 92

8 x L B s igh L ow x L G Figure 4: Why x L G is a eas as arge as x G in Opima Souion o Conracing Probem Then, we reaize ha moving he high risk conrac o he endowmen increases pro s on he high risk ype (pro s increasing in direcion of fu insurance). For he ow risk ype, moving he conrac down o he poin on indi erence curve ha goes hrough he endowmen increases pro s (you give ess in case of a oss an keep consumpion consan when here is no acciden). Finay, moving he ow risk o he endowmen increases he pro s furher (pro s increasing in direcion of fu insurance). Sep 5 IR-L binds Proof. See Picure. Fixing he conrac for ype L (poin in graph) we know ha he conrac for mus be in he wedge. Ca ha conrac h: Now, o er 0 o ype L; where he ony di erence is ha he consumpion in he case of an acciden is reduced o make IR-L binding (consumpion when no acciden is unchanged. This is obviousy beer for monopois, bu coud possiby upse IC-. owever, by simuaneousy reducing he consumpion in case of an acciden for ype by moving from h o h 0 we see ha boh incenive consrains wi hod, and, again, reducing he consumpion in case of an acciden and keeping i consan when here is no acciden is beer for he monopoy provider. Sep 6 L is no over insured. 93

9 m B igh Low m G x L G Figure 5: No ni-insurance a he Opimum Proof. See Figure. If L is over insured i mus be ha he ges a conrac ike he poin : Since IC- binds, ges a conrac on he indi erence curve hrough and o he ef of : ence, moving from o 0 doesn change he uiiy for L and he incenive consrain for remains sais ed. The picure is a bi bad, bu he sraigh ine is supposed o be he isopro for he rm (when seing o L), which is angen o he indi erence curve a poin 0 where L ges fu insurance. ence, 0 gives a higher pro han (since i is a movemen on an indi erence curve in he direcion of fu insurance). Sep 7 Fu insurance for igh Risk Type Proof. Draw a Picure! Fix x L B ; G xl anywhere beween fu insurance poin and endowmen poin on indi erence curve going hrough he endowmen. ighes pro aong indi erence curve for high risk ype is fu insurance (jus ike he reasoning in previous picure). ence we have; Proposiion 2 The opima conrac has he foowing feaures. 1. igh risk ype fuy insured 94

10 h 0 h Figure 6: IC- Binds 2. igh risk ype indi eren beween his and ow risk ype conrac 3. Low risk ype a reservaion uiiy eve. Remarks; 1. igh risk ype earns informaiona rens. Can ge some of gains from rade due o informaiona advanage. 2. Trade-o for monopois. E ciency gains of fu insurance versus how much surpus can be exraced from igh risk ype. 3. Exampe of price discriminaion/non-inear pricing 4. so noe; any observabe variabe ha woud be correaed wih risk or wiingness o pay shoud be used by monopois. In exampe, no such observabe variabes exis. 95

11 m B h h igh Low m G Figure 7: IR-L hods wih Equaiy x B m B ) igher P ro s 45 o L ower P ro s * h 0 m G Figure 8: Monopois can improve on conrac L is overinsured 96