Insurance Part 3 Two types of customers (different risks)

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1 Department f Ecnmics, University f Califrnia, Davis Prfessr Giacm Bnann Ecn 103 Ecnmics f Uncertainty and Infrmatin Insurance Part 3 Tw types f custmers (different risks 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 1 > 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 ( 1 indifference curve ging thrugh this pint is =. Thus fr type- individuals d1 U ( 2 p U ( 1 p U ( 1 it is and fr type- individuals it is. Frm p U ( 2 U ( 2 > p we get that p p p 1 p < 1 p. ence < < where the latter inequality fllws frm p > p. 2 2 TYPE TYPE Figure 1 [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 (1 q are f lw risk, with 0 < q < 1. et N be the ttal number f ptential custmers, s that q N are f type and (1 q N are f type. Page 1 f 8

2 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: Case 1: MONOPOY hat plicy r plicies wuld the mnplist want t ffer? There are three ptins. OPTION 1. 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 1 is the full insurance cntract with premium h and the crrespnding prfits will be: π = 1 q N( h p x 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 ( π = N h p x D where p= q p + (1 q p is the average lss. Clearly, p < p< p. Thus the isprfit line that ges thrugh cntract ( h, D is the straight line with slpe p and, since p p p < p< p, >. Thus, since the slpe f the -indifference curve alng the 45 p line is, the isprfit line is steeper than the indifference curve at the full insurance cntract and, therefre, there is a cntract, like B in Figure 2 belw, that yields higher prfits than the full insurance cntract. n insurance h h B type p line f slpe line f slpe p x Figure 2 Page 2 f 8

3 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 1 p p is larger than, 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. There is n need t cmpute the ptimal cntract under Optin 2, because we will shw later that Optin 2 is never ptimal. 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 3 belw. n insurance h h type type x Figure 3 Page 3 f 8

4 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 π 3 = ( h p x q N+ ( h px+ pd (1 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 4 belw and draw the indifference curve fr the type that ges thrugh that cntract. et C be the cntract at the intersectin f this indifference curve and the 45 line. Then prfits frm the type will be higher at C 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 C, then the peple will switch frm B t C, 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. B n insurance C line f slpe type p type x Figure 4 In cnclusin, when q is clse t 1, the mnplist will ffer nly the full-insurance cntract with premium h when q is nt clse t 1, the mnplist will ffer tw cntracts as explained under Optin 3. Page 4 f 8

5 Case 2: COMPETITIVE INDUSTRY Cnsider nw a cmpetitive industry where free entry leads t zer prfits. Define an equilibrium as a set f cntracts such that (1 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. 2 B 2 TYPE TYPE Figure 5 Cntract 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. Chse a cntract B = (h B,D B between the tw indifference curves (as shwn in the picture abve with h h D B < and B > 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 Page 5 f 8

6 p dds line fr the type, that is, n the line with slpe, and that the -cntract be n the p fair dds line fr the type, that is, n the line with slpe. 2 n insurance fair dds line fr type with slpe p average fair dds line: slpe p x fair dds line fr p type, with slpe Figure 6 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 28 belw. Page 6 f 8

7 2 n insurance indifference curve f type fair dds line fr type with slpe p indifference curve f type x fair dds line fr type, with slpe p Figure 8 Is this an equilibrium? It depends n the psitin f the fair dds line. Cnsider cntract P in Figure 9 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

8 2 n insurance P fair dds line fr type with slpe p indiff curve f type x fair dds line fr p type, with slpe indifference curve f type average fair dds line Figure 9 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