Department of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets

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1 Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0 < x < ). Thus his initial situatin can be represented as a pint in a tw-dimensinal diagram where we measure n the hrizntal axis his wealth in the bad state (lss), dented by, and n the vertical axis his wealth in the gd state (n lss), dented by 2. n insurance cntract takes the individual frm the initial pint f n insurance t sme ther pint. n insurance cntract can be described by a pair ( h, D ), where h is the premium and D is the deductible. Nte that the premium is paid in any case, that is bth in the gd state and in the bad state. Given a cntract ( h, D ), wealth in the gd state will be ( h) and wealth in the bad state will be h D. Thus if we represent a cntract in the (, 2) plane, then the premium is ( 2 ) and the deductible is ( 2 ). Fr example, in Figure ntract invlves a premium f 5,000 4,500 = $500 and a deductible equal t 4,500 3,900 = $600. and cntract B invlves a premium f $(5,000 4,00) = $900 and zer deductible, that is, full insurance. gd state: p initial situatin n insurance 45 line: full insurance 5,000 S 4,500 B 3,000 3,900 4,00 Figure Page f 8

2 I SOPRO FI T INES The expected prfit frm insurance cntract (h, D) is h p( x D) h px pd (.a) If we represent the cntract as a pint in the (, 2) plane, then the expected prfit frm the cntract can be written as px p px ( p) p h D n isprfit line is a line that jins all the cntracts that yield the same expected prfit. Suppse, and, are tw cntracts that yield the same prfit, that is, that 2 2 ( ) ( ). rise 2 2 The slpe f the line jining these tw cntracts is: run (.b). Nw, frm (.b) we get that ( ) px ( p) 2 p and ( ) px ( p) 2 p. Thus by setting we get: ( p) 2 2 p ( ) ( ) s that rise p run p 2 2 ence the isprfit line thrugh pints and is a straight line with slpe 2 gd state: p p p. prfits d nt change alng this line f slpe p/(p) 45 line 2 2 Figure 2 EXMPE. et = 6,000, x = 3,600 and p = 2. nsider a cntract, call it, with 2 premium h = $800 and deductible D = $,200. Then the expected prfit frm this cntract is Page 2 f 8

3 2 2 ( ) h px pd 800 3, 600, The slpe f the isprfit line thrugh p 2 is. Suppse that the premium is decreased by $200 but prfits remain 0 p 5 2 cnstant. Then we g frm (4000,5200) t ( y,5400) s that rise = 200. Thus, rise p 200 since, we have hence run = +,000, that is, 4,000 y =,000, run p 5 run 5 i.e. y = 3,000. ence deductible at is 5,400 3,000 = 2,400. S in the, 2 plane, (3000,5400) r, in terms f premium and deductible, cntract is given by h = 600, D = 2,400. Each isprfit line divides the plane int thee regins: () the line itself, where prfits are cnstant and equal t sme number, (2) the regin t the left f the line where prfits are larger than and (3) the regin t the right f the line where prfits are less than : 2 wealth in gd state ( ) ( B) ( ) B isprfit line fr value state Figure 3 T see this, take any cntract and draw the isprfit line that ges thrugh and take a pint S vertically belw (then S has a higher premium and a lwer deductible relative t ). e want t shw that ( S) ( ). 2 wealth in gd state 2 S 2 ( ) S isprfit line fr value S Figure 4 state Page 3 f 8

4 Nw, ( ) px ( p) 2 p and S S ( S) px ( p) 2 p (since 2 ) Thus S S p 2 2 ( ) ( ) ( ) 0, since S and p. gd state: p 45 line prfits cnstant prfits increase p/(p) Figure 5 Thus mving frm inside the shaded area prfits increase (cnvex cmbinatin f n change and increase) One particular isprfit line is the line that crrespnds t zer prfits. This is the isprfit line that ges thrugh the N Insurance pint ( x, ). In fact, we can think f this pint as a cntract with h = 0 and D = x. p 0.2 Fr example, if = 5,000, x = 2,000 and p = 0.2, then and the zerprfit cntracts are the pints n the line f equatin 2 ( p) = 5,750 (if is reduced frm 4 3,000 t 0, 2 increases by 3, frm 5,000 t 5,750) 4 T E O NSUMER Fist we want t draw the indifference curves f the cnsumer in the, 2 plane. et U ( m ) be the utility-f-mney functin and assume that U 0 and U 0, that is, the cnsumer is risk,, be a pint clse t and averse. nsider a pint 2 and let 2 suppse that and lie n the same indifference curve. Then EU ( ) pu ( ) ( p) U ( ) and EU ( ) EU ( ). If is clse t, then 2 is clse t s that Similarly, 2 is clse t 2 s that U ( ) U ( ) U ( ) (2.a) Page 4 f 8

5 ence, U ( ) U ( ) U ( ) (2.b) EU ( ) pu ( ) ( p) U ( ) p U ( ) U ( ) ( p) U ( 2 ) U ( 2 ) 2 2 EU ( ) pu ( ) ( p) U ( ) Since EU ( ) EU ( ) 2 2 2, it fllws that pu p U rise 2 2 p U( ) Thus. ence run p U( ) 2 ( ) ( ) ( ) 0. The slpe f the indifference curve at pint (, ) is p U( ) p U( ) By strict cncavity f the utility functin, if m m2 then U( m ) U( m2 ). Thus bve the 45 line (where 2 ), the slpe f the indifference curve is greater in abslute value than the slpe f the isprfit line that ges thrugh that pint, that is, p U( ) p p U( ) p 2 This is because 2 (we are abve the 45 line) and thus U( ) U ( 2 ), s that U( ) U( ) 2. lng the 45 line (where 2 ), the slpe f the indifference curve is equal t the slpe f the isprfit line that ges thrugh that pint, that is, p U ( ) p p U( ) p Putting tgether what we fund abve, namely that () at a pint abve the 45 line the slpe f the indifference curve is greater in abslute value than the slpe f the isprfit line that ges thrugh that pint, and (2) mving away frm a pint abve the 45 line in the directin between the vertical directin dwnwards and the directin f the isprfit line (which has slpe prfits increase. p ), p Thus prfits fr the insurance cmpany increase as we mve alng the indifference curve that ges thrugh pint twards the 45 line, as shwn in the fllwing figure. Page 5 f 8

6 gd state: p 45 line prfits increase in this directin full insurance indifference curve cnvex t the rigin: risk averse Figure 6 hat is the maximum premium, call it h, that an individual wuld be willing t pay fr full insurance? It is the slutin t the equatin U ( h) pu ( x) ( p) U ( ) Fr example, if =,600, x = 700, p and U ( ) then h is given by the slutin t 0 which is h = 79. 9,600 h, , Next we shw that px < h. T begin with, nte that, by definitin f risk-aversin, the expected utility f a lttery is less than the utility f the expected value: pu ( x) ( p) U( ) U px () By definitin f h, Thus frm () and (2) we get Since U is increasing, it fllws that U ( h ) pu ( x) ( p) U ( ) (2) U ( h ) U ( px). h px that is, px h Next we shw that h < x. First f all, nte that Page 6 f 8

7 U x pu x p U x pu x p U U h That is, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) because U ( x) U ( ) hich implies, since U is increasing, that U x U h ( ) ( ) This can be shwn graphically as fllws. h < x. utility pu(-x)+(-p)u() U(m) x h px -x -h mney m Tw bservatins: p(-x)+(-p)() = -px Nte: () h < x (2) h > px, i.e. risk-averse persn willing t pay a premium which is higher than the expected lss. Figure 7. e have assumed that the granting f full insurance des nt affect the value f p (the lss). This, hwever, is ften nt true. If yu are insured, yu dn t face a risk and therefre yu exert less effrt r care in trying t prevent a lss (e.g. yu are less careful abut lcking yur bike r lcking the frnt dr f yur huse). 2. The fact that h > px is what makes the sale f insurance plicies a prfitable business. If the lsses is independent acrss cnsumers and the insurance cmpany sells a large number N f plicies, then, by the law f large numbers in prbability Page 7 f 8

8 thery, it will have t make payments f $pxn; hwever, it will be cllecting $h N, thus making a prfit f $ ( h px) N (assuming n ther csts: an assumptin that we will relax later). OOSING ONTRT FROM MENU Often insurance cmpanies ffer a menu f pssible cntracts, nt just ne cntract. Typically cnsumers have a chice between a higher premium and higher cverage r a lwer premium and lwer cverage. EXMPE. nsider an individual whse initial wealth is $,000. e faces a ptential lss f $400, with prbability. Suppse that the insurance cmpany ffers the fllwing ptins: 5 premium deductible cntract $82 0 cntract 2 $62 $00 cntract 3 $40 $200 The individual s utility-f-mney functin is U ( m) m. hich cntract wuld he chse? Expected Utility N insurance: Full insurance: decuctible f 00: decuctible f 200: Thus the best ptin is cntract 3 (deductible f $200) and the next best ptin is full insurance. Nw suppse that the cnsumer can chse any cntract frm the set f cntracts that yield zer prfits. Then fr each insurance plicy the issuer cllects premium h and expects t pay ut p(x D). Thus zer prfits means that h and D are such that h = p(x D). Suppse that custmers are free t chse any premium-deductible cmbinatin (h,d) that satisfies the abve equatin. hat premium-deductible cmbinatin (h,d) wuld a risk-averse individual chse? learly, nce yu chse D, the premium is determined by the equatin h = p(x D). Thus Page 8 f 8

9 ealth if n lss: h = p(x D) = px + pd ealth if lss: h D = p(x D) D = px + pd D = px ( p)d Thus expected utility frm plicy (h,d) is f ( D) p U px ( p) D ( p) U ( px pd) The individual will chse D t maximize f(d). necessary cnditin is f ( D) 0. Nw f ( D) p U px ( p) D ( ( p)) ( p) U( px pd) ( p) Thus we need ( p) p U px ( p) D ( p) p U( px pd) 0, that is, U ( px pd) U px ( p) D px pdd Since U is strictly cncave, the slpe f U is different at any tw different pints. Thus the nly way t satisfy the equatin U( x) U( y) is t have x = y. Thus t satisfy the abve equatin we need px pd px pd D and this requires D = 0. Thus the individual wuld chse full insurance. [Of curse we already knew that, because it is nly alng the 45 line that there is a tangency between the indifference curve and the zer-prfit line: n the 45 line the slpe f bth is p p ]. ase : TE INSURNE INDUSTRY IS MONOPOY The mnplist will want t ffer a cntract that lies n the indifference curve thrugh the N Insurance pint (if it ffered a cntract abve that indifference curve it wuld nt be maximizing its prfits, because cming vertically dwn frm that pint t the indifference curve wuld yield an increase in prfits). Furthermre, as seen abve, mving alng the indifference curve twards the 45 line prfits increase. Thus the mnplist wuld want t ffer full insurance at premium h. ase 2: FREE ENTRY in the Insurance Industry leads t ZERO PROFITS cntract that yields zer prfit is called a fair cntract and the zer prfit line is called the fair dds line. Recall that the zer prfit line is the straight line that ges thrugh the N Insurance pint and has slpe p p. Page 9 f 8

10 Define an equilibrium in a cmpetitive insurance industry as a situatin where every firm makes zer prfits and n firm (existing r new) can make psitive prfits by ffering a new cntract. hat cntract(s) will be ffered in this industry? First f all, we shw that at least ne firm must ffer the full-insurance cntract (given by h = px and D = 0). Suppse nt. Then take a cntract n the zer-prfit line that is ffered and that sme cnsumers are buying. Since it is nt the full-insurance cntract, it must lie abve the 45 line, like cntract in Figure 8 belw. ence the slpe f the indifference curve thrugh is steeper than the zer-prfit line and, therefre, there is a cntract B that lies belw the zer-prfit line (hence (B) > 0) and abve the indifference curve thrugh, s that a firm that ffered cntract B wuld attract all the cnsumers wh were buying and make psitive prfit. 2 gd state: p n insurance 45 degree line B x Figure 8 Secndly, since the full-insurance cntract is ffered by at least ne firm, every cnsumer purchases that cntract (because any ther cntract that is ffered must be n the zer-prfit line and, as just shwn, it must lie n a lwer indifference curve that the full insurance cntract. dverse Selectin in Insurance Markets Suppse that there are tw types f individuals. They are all identical in terms f the initial wealth (r wealth in the gd state), dented by, and in terms f the ptential lss that they face, dented by x. They als have the same utility-f-mney functin U. hat they differ in is the lss: it is p fr type (high-risk) individuals and p fr type (lw-risk) individuals with > p > p > 0. Then type- individuals have steeper indifference curves than type- individuals. In fact, fix an arbitrary pint (, 2 ). s we saw abve, the slpe f the d2 p U ( ) indifference curve ging thrugh this pint is. Thus fr type- individuals d p U( ) 2 Page 0 f 8

11 p U( ) p U( ) it is and fr type- individuals it is. Frm p p U( 2 ) p U ( 2 ) > p we get that p p p p p. ence where the latter inequality fllws frm p p p p > p. 2 gd state: p 2 TYPE TYPE Figure 9 [Nte: if we measured wealth in gd state n the hrizntal axis, then the ppsite wuld be true: the type indifference curve wuld be steeper than the type.] Suppse that the insurance cmpanies cannt tell wh is wh. wever it is knwn that the prprtin q are f high risk and the prprtin ( q ) are f lw risk, with 0 < q <. et N be the ttal number f ptential custmers, s that q N are f type and ( q )N are f type. et h be the maximum premium that the peple wuld be willing t pay fr full insurance and h be the maximum premium that the peple wuld be willing t pay fr full insurance: ase : MONOPOY hat plicy r plicies wuld the mnplist want t ffer? There are three ptins. OPTION. Offer nly ne cntract, which is attractive nly t the type. In this case the mnplist will want t ffer a cntract which is n the indifference curve f the type that ges thrugh the N Insurance pint. Since prfits increase alng that indifference curve mving twards the 45 line, the prfit-maximizing cntract under Optin is the full insurance cntract with premium h and the crrespnding prfits will be: q N( h p x) Page f 8

12 OPTION 2. Offer nly ne cntract, which is attractive t bth types. In this case the mnplist will want t ffer a cntract which is n the indifference curve f the type that ges thrugh the N Insurance pint. wever, it is nt ptimal t ffer full insurance (at premium h ). T see this, nte that when bth types apply, prfits frm a cntract ( h, D ) are given by N h p x D where p q p ( q ) p is the average lss. learly, p p p. Thus the isprfit line that ges thrugh cntract ( h, D ) is the straight line with slpe p p and, since p p p p p,. Thus, since the slpe f the -indifference p p curve alng the 45 p line is, the isprfit line is steeper than the indifference curve at the p full insurance cntract and, therefre, there is a cntract, like B in Figure 0 belw, that yields higher prfits than the full insurance cntract. gd state: p 45 line n insurance h h h B type p line f slpe p line f slpe p p x h h Figure 0 The same argument applies t any ther pint n the indifference curve f the type that ges thrugh the N Insurance pint at which the slpe f the indifference curve is less than p. Similarly, a cntract n the indifference curve at which the slpe f the indifference curve p p is larger than p, cannt be ptimal (mving t the right twards the 45 line wuld increase prfits). Thus the best cntract under Optin 2 is that cntract n the indifference curve f the type that ges thrugh the N Insurance pint at which the slpe is equal t p p. There is n need t cmpute the ptimal cntract under Optin 2, because we will shw later that Optin 2 is never ptimal. Page 2 f 8

13 OPTION 3. Offer tw cntracts, ne targeted t the type and the ther targeted t the type. Then, by the usual argument, the cntract targeted t the type must be a full insurance cntract. Then the first cnstraint the mnplist faces is that the premium h fr the full insurance plicy targeted t the type must be h h. The secnd cnstraint is that the ther plicy must be less attractive than the full insurance plicy fr the type, that is, it must lie belw the indifference curve ging thrugh the full insurance plicy. The third cnstraint is that the plicy targeted t the type must be attractive t them, that is, it cannt lie belw their indifference curve that ges thrugh the N Insurance pint. Since prfits frm the type increase alng this indifference curve mving twards the 45 line, the cntract targeted t them must be the cntract that lies at the intersectin f the tw indifference curves (see Figure belw). gd state: p 45 line n insurance h h h h type type x h h Figure et ( h, D ) be the plicy targeted t the types and ( h,0) the plicy targeted t the types and suppse that all these cnstraints are satisfied. Then prfits will be 3 ( h p x) q N ( h px pd )( q ) N. Nw, Optin 3 yields higher prfits than Optin 2. T see this, start with the pling cntract f Optin 2 (pint B in Figure 2 belw) and draw the indifference curve fr the type that ges thrugh that cntract. et be the cntract at the intersectin f this indifference curve and the 45 line. Then prfits frm the type will be higher at than at B (prfits increase alng an indifference curve when mving twards the 45 line). If the firm ffers a full-insurance cntract with a premium slightly lwer than the premium assciated with, then the peple will switch Page 3 f 8

14 frm B t, while the peple will stay at B. Thus prfits frm the peple wn t change, but prfits frm the peple will increase. ence the riginal pling cntract B is nt ptimal. gd state: p 45 line B n insurance line f slpe type p p type x Figure 2 In cnclusin, when q is clse t, the mnplist will ffer nly the full-insurance cntract with premium h when q is nt clse t, the mnplist will ffer tw cntracts as explained under Optin 3. ase 2: OMPETITIVE INDUSTRY nsider nw a cmpetitive industry where free entry leads t zer prfits. Define an equilibrium as a set f cntracts such that () every firm makes zer prfits and (2) n (existing r new) firm culd make psitive prfits by intrducing a new cntract. Nw, culd there be a pling equilibrium where nly ne cntract is ffered (with nnnegative deductible, s that 2 ), everybdy buys it and the firms make zer prfits? The answer is N. et be such a cntract. By the crssing prperty f the indifference curves there is a cntract B which is between the tw indifference curves, s that B wuld be preferred by the type but nt by the type and therefre wuld attract nly and all the types. Page 4 f 8

15 2 gd state: p B 2 TYPE TYPE Figure 3 ntract cnsists f a premium h > 0 and deductible D 0. Since it is n the average fair dds line, h p( x D) 0. Since p < p, it fllws that h p ( x D) 0. hse a cntract B = (h B,D B ) between the tw indifference curves (as shwn in the picture abve) with h B h and DB D but small enugh s that h p ( x D ) 0 (such a cntract exists B B because the functin f ( h, D) h p ( x D) is cntinuus and f ( h, D ) h p ( x D) 0 ). Then a firm ffering such a cntract wuld attract all and nly the types and make psitive prfits. Thus if there is a zer-prfit equilibrium it must be an equilibrium with at least tw cntracts. Such an equilibrium is called a separating equilibrium if all the types buy ne cntract and all the types buy a different cntract. hat wuld such an equilibrium lk like with exactly tw cntracts? The zer-prfit equilibrium requires that the -cntract be n the fair p dds line fr the type, that is, n the line with slpe, and that the -cntract be n the p p fair dds line fr the type, that is, n the line with slpe. p Page 5 f 8

16 2 gd state: p n insurance 45 line fair dds line fr type with slpe p p average fair dds line: slpe p p x fair dds line fr p type, with slpe p Figure 4 By the argument used abve, if the type is nt ffered full insurance, then smebdy culd step in and ffer a full-insurance cntract attractive t the type and make psitive prfits (prfits frm the type increase when traveling alng an indifference curve twards the 45 line). Thus the cntract designed fr the type must be n the 45 line. It is shwn as pint in the fllwing diagram. n analgus full insurance cntract fr the type (given by the intersectin f the 45 line and the fair dds line fr the type) cannt be ffered, because such a cntract wuld be mre attractive than cntract fr the type, everybdy wuld buy it and it wuld yield negative prfits (because, when everybdy buys the same cntract the relevant fair dds line is the average ne, which is steeper than the ne). The incentive cmpatibility cnstraint fr the type requires the cntract designed fr the type t be belw r n the indifference curve f the type that ges thrugh the full insurance cntract. The zer prfit cnditin requires it t be n the fair dds line f the type as clse as pssible t the 45 line (because traveling alng the indifference curve twards the 45 line increases prfits frm the type). Such a pint is pint in the abve figure. The types prefer cntract (because f the way the and indifference curves crss at the cntract: see Figure 5 belw). Page 6 f 8

17 2 gd state: p n insurance 45 line indifference curve f type fair dds line fr type with slpe p p indifference curve f type x fair dds line fr type, with slpe p p Figure5 Is this an equilibrium? It depends n the psitin f the fair dds line. nsider cntract P in Figure 6 belw. It is mre attractive than and fr bth grups, thus a firm ffering it wuld attract bth types. Since pint P lies belw the average fair dds line, a firm ffering it wuld make psitive prfits. ence the pair and wuld nt be an equilibrium (n the ther hand we knw frm the previus analysis that P cannt be an equilibrium either, because there cannt be a pling equilibrium). Page 7 f 8

18 2 gd state: p n insurance 45 line P fair dds line fr type with slpe p p indiff curve f type x fair dds line fr p type, with slpe p indifference curve f type average fair dds line Figure 6 Thus fr a separating equilibrium it must be the case that the average fair dds line be belw (r at mst tangent t) the indifference curve f the type thrugh cntract. This amunts t saying that the fractin f type in the ppulatin is sufficiently high. Page 8 f 8