EconS Advanced Microeconomics II Handout on Moral Hazard

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1 EconS dvanced Microeconomics II Handout on Moral Hazard. Maco-Stadler, C. 3 #6 Consider a relationsi between a rincial and an agent in wic only two results, valued at 50,000 and 25,000 are ossible. Te agent must coose between tree ossible e orts. Te robability of eac of te results contingent on te e orts is given below Results 25,000 50,000 E orts e e e ssume tat te rincial is risk-neutral and tat te agent is risk-averse, wit teir resective references described by te following functions: (; w) = w U(w; e) = w v(e) wit v(e ) = 40, v(e 2 ) = 20, and v(e 3 ) = 5. Te reservation utility level of te agent is U= 20. (a) Write down te otimal contracts under symmetric information for eac e ort level and te ro ts obtained by te rincial in eac case. Wat e ort level does te rincial refer?. Under symmetric information, te agent receives a ed ay-o, determined by te articiation constraints. For eac tye, we ave w w w In equilibrium, all tree of tese constrants will bind, yielding solutions (w ; w 2 ; w 3 ) = (25; 600; 9; 600; 5; 625). We can ten lug eac of tese wages into te rm s eected utility (e i )(25000 w i ) + ( (e i ))(50; 000 w i ) to obtain eected utilities of te rm ( ; 2 ; 3 ) = (8; 50; 7; 900; 5; 625) Since e gives te igest eected utility for te rm, it will be te referred contract. (b) Write down te otimal contracts wen tere eists a moral azard roblem. Wat is te otimal e ort level and te contract cosen by te rincial?

2 s in art (a), to determine te otimal contract, we will want to evaluate eac e ort level seerately. Case : e = e. For ig e ort, te rm solves te following objective function ma w ;w l 4 (25000 wl ) (25000 w ) subject to wl w (P C ) 4 wl w wl + w 20 (IC 2 2) wl w wl + 4 w 5 (IC 4 3) we can rearrange te constraints to wl + 3 w 640 (P C ) w w l + 80 (IC2) w w l + 70 (IC3) It is trivial to sow tat if IC 2 olds, IC 3 de nitely olds. Tus, we can eliminate tat constraint. Taking Kun-tucker rst-order conditions yields w l 2 w l = 0 () w w = 0 (2) Now, we consider cases on te values of our lagrange multiliers. First, if ; 2 = 0 (neiter constraint binds), neiter rst-order condition can yield a solution, so at least one must bind for sure. If = 0, and 2 > 0 (i.e., only IC 3 binds), ten we would ave a contradiction in equation (), as 2 < 0. If > 0, and 2 = 0 (i.e., only P C binds) te rst-order conditions reduce to w l = w wic violates IC 3. Hence, our only otion remaining is ; 2 > 0 (bot constraints bind). Rearranging our constraints gives wl + 3 w = 640 (P C ) w = w l + 70 (IC 3) and solving, we ave (w ; w l ; ; 2 ; ) = (32; 400; 0; 000; 80; 30; 6; 950). Case 2: e = e 3 (Skiing aead, will do e 2 net). 2

3 For low e ort, te rm solves te following objective function we can rearrange te constraints to 3 ma w ;w l 4 (25000 wl ) + 4 (25000 w ) 3 subject to wl + 4 w 5 20 (P C 3 ) 4 3 wl + 4 w 5 4 wl w 40 (IC 3 4 ) 3 wl + 4 w 5 4 wl + 2 w 20 (IC 3 2 2) 3 w l + w 500 (P C 3 ) wl w 70 (IC 3 ) wl w 60 (IC 3 2) It is trivial to sow tat if IC 3 2 olds, IC 3 de nitely olds. Tus, we can eliminate tat constraint. Taking Kun-tucker rst-order conditions yields w + l 2 2 w l = 0 () w 2 w = 0 (2) Now, we consider cases on te values of our lagrange multiliers. First, if ; 2 = 0 (neiter constraint binds), neiter rst-order condition can yield a solution, so at least one must bind for sure. If = 0, and 2 > 0 (i.e., only IC 3 2 binds), ten we would ave a contradiction in equation (2), as 2 < 0. If > 0, and 2 = 0 (i.e., only P C binds) te rst-order conditions reduce to w l = w. Substituting into P C 3 gives 4 w l = 500 wic imlies tat w l = w = 5; 625, yielding a utility for te rm of 5; 625 (note tat tis is te same solution as in comlete information). Lastly, we must ceck wen ; 2 > 0 (bot constraints bind). Rearranging our rst-order conditions gives gives and combining tem yields = 3 2 w l () 2 = 2 w (2) 4 = 3 wl + w > = 3 3 w l w <

4 wic is a violation. Hence, only te solution under comlete information is valid. Case 3: e = e 2. For medium e ort, te rm solves te following objective function we can rearrange te constraints to ma w ;w l 2 (25000 wl ) + 2 (25000 w ) subject to wl + 2 w (P C 2 ) 2 wl + 2 w wl + 3 w 40 (IC 2 4 ) wl + 2 w wl + 4 w 5 (IC 2 4 3) wl + w 280 (P C 2 ) wl w 80 (IC 2 ) w w l + 60 (IC 2 3) t tis oint, it is uncertain wic of te incentive comatibility constraints binds, but it can be trivially sown tat if one constraint binds, te oter will not. Taking Kun-tucker rst-order conditions yields w + l w l 2 w l = 0 () w 2 w w = 0 (2) Since we know tat one of 2 and 3 is ositive and te oter is zero, we can use te same logic in arts (a) and (b) to sow tat > 0 (Practice: Work it out!). ll tat remains is to determine wic of te two incentive comatibility constraints binds. First, we will consider te case were 2 > 0. Our rst-order conditions become Rearranging terms gives Combining, w + l 2 2 w l = 0 () w 2 w = 0 (2) + 2 = w l () 2 = w (2) 2 = w l + w > = w l w < 0 4

5 wic is a violation. Tus, 2 = 0. Let s ceck 3 > 0. Our rst-order conditions become Rearranging terms gives w l 2 w l = 0 () w w = 0 (2) 3 = w l () + 3 = w (2) Combining, 2 = w l + w > = w wl > 0 Hence, IC 2 3 is our binding constraint. Udating our constraints, wl + w = 280 (P C 2 ) w = w l + 60 (IC 2 3) and solving, we ave (w ; w l ; ; 2 ; 3 ; 2 ) = (28; 900; 2; 00; 40; 0; 30; 7; 000). Summarizing, te table below sows te utilities tat te rm will receive by designing contracts for eac e ort level E ort Utility e 6; 950 e 2 7; 000 e 3 5; 625 Hence, te rm will refer e ort level e 2 and will design its contract accordingly. 2. Maco-Stadler C. 3 # 9 Consider a relationsi between a rincial and an agent in wic tere are only two ossible results, one ig, 2, and te oter low,. Te frequency wit wic eac result occurs deends on te agent s e ort, e 2 [0; ], and a random state variable. ssume tat te robability of te ig result is te same as te e ort, i.e., Pr( = 2 je) = e, so tat Pr( = je) = e. Te agent s utility is of te form U(w; e) = u(w) v(e), were u() is increasing and concave, and v() is increasing and conve. Te rincial s objective function is ( w), wic is increasing and concave (tat is, se could be risk averse). (a) Write down te constrained maimization roblem of te rincial, and nd te conditions tat determine te otimal contract. 5

6 Te rincial will maimize er eected utility, subject to te articiation constraint, i.e., ma w ;w l ;e e( 2 w ) + ( e)( 2 w l ) subject to e(u(w ) v(e)) + ( e)(u(w l ) v(e) U Taking rst -order conditions wit resect to w and w l, e 0 ( 2 w ) + eu 0 (w ) = 0 ( e) 0 ( w l ) + ( e)u 0 (w l ) = 0 Combining tese two equations yields our condition for te otimal contract, 0 ( 2 w ) 0 ( w l ) = u0 (w ) u 0 (w l ) (b) Now we assume tat te agent s e ort is not ublicly known. Write down te constrained maimization roblem tat de nes te otimal contract in tis case. Is te rst-order aroac valid in tis eamle? Describe te relationsi between te otimal contract s wages and te di erences in tis contract comarted to art (a). Te most callenging art of tis roblem is guring out te incentive comatibility constraints. For eac e ort tye, tere are an in nite amount of IC s, but tey reduce to te form of e 2 arg ma ^eu(w ) + ( ^e)u(w l ) v(^e) ^e taking a rst-order condition wit resect to ^e yields u(w ) u(w l ) v 0 (^e) = 0 were e solves tis equation wit equality. Since te agent s function is concave in e (since v(e) is conve) te rst-order condition is bot necessary and su cient for a maimum. Tus, te rst-order aroac is valid. We now use our derived incentive comatibility constraint as a new constraint in te objective function, wic becomes ma w ;w l ;e e( 2 w ) + ( e)( 2 w l ) subject to e(u(w ) v(e)) + ( e)(u(w l ) v(e) U u(w ) u(w l ) v 0 (^e) = 0 Taking rst -order conditions wit resect to w and w l, e 0 ( 2 w ) + eu 0 (w ) + u 0 (w ) = 0 ( e) 0 ( w l ) + ( e)u 0 (w l ) u 0 (w l ) = 0 6

7 Combining tese two equations yields our condition for te otimal contract,! 0 ( 2 w ) 0 ( w l ) = u0 (w ) + e u 0 (w l ) e {z } > Tis new condition imlies tat 0 ( 2 w ) 0 ( w l > u0 (w ) ) u 0 (w l (assuming tat bot constraints bind). ) Tis imlies tat te rincial makes te agent carry more tan te e cient level of risk. Tat is, te moral azard roblem s solution contract makes te agent more interested in te result tan wat is really otimal. Hence, tis is a generalization of wat we ave studied for a risk-neutral rincial usiness lication Cometitive Provision of Healt Insurance Consider te callenge of roviding ealt insurance to a oulation wit di erent robabilities of getting sick. : Suose tat, as in our car insurance eamle, tere are two consumer tyes-consumers of tye tat are likely to get sick, and consumers of tye 2 tat are relatively ealty. Let reresent te level of ealt insurance, wit = 0 wit no insurance and iger levels of indicating in curves (equal to marginal willingness to ay), wit d reresenting te demand curve for a single consumer of tye and d 2 reresenting te demand curve for a single consumer of tye 2. Suose furter tat te marginal cost of roviding additional ealt coverage to an individual is constant, wit MC > MC 2 : [Section ]: Tis eercise attemts to formalize a key intuition we covered in te tetbook wit a di erent tye of model for insurance. (a) For simlicity, suose trougout tat d and d 2 ave te same sloe. Suose furter, unless oterwise stated, tat d as iger intercet tan d 2. Do you tink it is reasonable to assume tat tye as iger demand for insurance? It seems reasonable to assume tat tose wo are more likely to get sick ave iger demand for ealt insurance-wic is wat we are assuming wen we assume tat te intercet of d is iger tan te intercet of d 2 : (b) egin by drawing a gra wit d ; d 2 ; MC and MC 2 assuming tat te vertical intercets of bot demand curves lie above MC : Indicate te e cient level of insurance and 2 for te two tyes. 7

8 Tis is done in gure were te e cient level of insurance for tye consumers occurs were MC intersects d and te e cient level of insurance for tye 2 consumers occurs were MC 2 intersects d 2. (Note: Tere is no articular reason for to lie te left of 2 -Had we draw te di erence between d and d 2 larger-or te di erence between MC and MC 2 smaller, te reverse would old. Noting fundamental canges in te analysis regardless of ow te gra is drawn.) MC C d 2 d MC 2 2 Figure : Equilibrium in te Insurance Market. (c) Suose te industry o ers any level of at rice = MC : Illustrate on your gra te consumer surlus tat tye individuals will get if tis were te only way to buy insurance and tey buy tere otimal olicy. How muc consumer surlus will tye 2 individuals get? Tye consumers will buy = and tus get consumer surlus (a + b + c) as sown in gure 2. Consumer of tye 2 will buy only u to te oint were MC crosses d 2 tus getting consumer surlus (a). 8

9 b a c MC d 2 d Figure 2: Only = MC is o ered. (d) Net, suose you want to o er an individual insurance contract tat earns zero ro t if bougt only by tye 2 consumers, tat is referred by tye 2 individuals to and tat makes tye consumers just as well o as tey are under te otions from art (c). Identify in your gra. Tis can be seen in gure 3. Note tat tere is no articular reason tat lies vertically underneat te intersection of MC and d 2 it could lie to te rigt or left. It must be, owever, tat lies on te MC 2 curve - oterwise rms o ering it would not make zero ro ts. In order for tye individuals to be indi erent between and, it must be tat teir consumer surlus is te same under bot contracts. Since teir consumer surlus at is (a + b + c) and teir consumer surlus as is (a + b + d), tis imlies tat (c) as to be equal to (d). Notice tat (c) gets larger and (d) gets smaller as we move to te left, wit (c) small and (d) large wen is orizontally close to. Tus, starting vertically underneat and moving it to te left, tere will come some at wic (d) is eactly equal to (c). Finally, it as to be te case tat tye 2 consumers are better o at tan would be oterwise - wic as to be te case. (It is trivial to see wen lies rigt below te intersection of d 2 and MC because ten consumer surlus simly increases from (a) to (a + d) - but it is also true if lies to te rigt or left of tat intersection oint.) 9

10 b a c MC d d 2 d MC 2 Figure 3: Contracts and are o ered. (e) Suose for a moment tat it is an equilibrium for te industry to o er only contacts and (and suose tat te actual is just sligtly to te left of te you identi ed in art (d)). True or False: Wile insurance comanies do not know wat tye consumers are wen tey walk into te insurance o ce to buy a olicy, te comanies will know wat tye of consumer tey made a contract wit after te consumer wit after te consumer leaves. Tis is true - a consumer of tye would be just sligtly better o buying wile consumers of tye 2 are better o buying. Tus, you will know tat te consumers is a tye if e bougt and a tye 2 if e bougt. (f) In order for tis to be an equilibrium, it must be te case tat it is not ossible for an insurance comany to o er a "ooling rice" tat makes at least zero ro t wile attracting bot tye and 2 consumers. (Suc a olicy as a single rice tat lies between MC and MC 2 :) Note tat te demand curves graed tus far were for only one individual of eac tye. Wat additional information would you ave to know in order to know weter te zero - ro t rice would attract bot tyes? You would need to know no additional information to know tat tye individuals would refer te ooling contract rice - because it would be below te rice at wic tey are oterwise buying. ut we don t know if suc a rice would attract consumers of tye 2. It is a iger rice, but if it allowed tye 2 consumers to buy a larger quantity, tat migt 0

11 make u for te loss in consumer surlus from te iger rice. Tis is illustrated in gures 4 and 5 were a low is graed in anel (b) and a ig is graed in anel (c). t, tye 2 consumers will buy were intersects d 2 i.e. at oint D in anel (b) and at oint E in anel (c). In anel (b), tis imlies tat consumers will lose area (g) in consumer surlus because of te increase in rice but will gain area () from being able to urcase more insurance. Since ()>(g), te consumer is better o and tus will coose te ooling contract. ut in anel (c), tye 2 consumers lose (i) and gain (j) - wit te former larger tan te latter. Tus, te iger ; te less likely it is tat a (j) - wit te former larger tan te latter. Tus, te iger ; te less likely it is tat a ooling rice could attract bot consumers. MC * g D d 2 d MC 2 D Figure 4: is relatively low.

12 * j E MC i d 2 d MC 2 E Figure 5: is relatively ig. (g) True or False: Te greater te fraction of consumers tat are of tye, te less likely it is tat suc a "ooling rice" eists. Tis is true - because te greater te fraction of tye consumers, te iger te rice will ave to be in order for rms o ering tat rice to make zero ro t. () Suose tat no suc ooling rice eists. ssuming tat ealt insurance rms cannot observe te ealt conditions of teir customers, would it be a cometitive equilibrium for te industry to o er contracts and? Would tus be a ooling or searating equilibrium? Yes, tis would be a searating equilibrium because te two tyes end u revealing wy tey are by coosing di erent contracts. In fact, te equilibrium could simly o er any insurance amount at rice = MC and any insurance amount u to at rice = MC 2 : ut no insurance above can be o ered at - oterwise tye 2 consumers will buy at - wic means would no larger be a zero ro t rice. (i) Would you still be able to identify a contract tat satis es te conditions in (d) if d = d 2? Wat if d < d 2? 2

13 Figure 6 illustrates te case were d = d 2. Te contract as to be suc tat area (a) is equal to area (b) so tat tye individuals would lose as muc (i.e. (b)) as tey would gain (i.e. (a)) from switcing from to. ut because d = d 2, tis imlies tye 2 individuals will be similarly indi erent - and not strictly refer to. In tis borderline case, it is terefore barely ossible to nd tat satis es te necessary conditions for a searating equilibrium to emerge. Te case were d 2 > d is graed in gure 7. In order for tye consumers to be indi erent between and, it as to now be tat area (c) is equal to area (d). Te tat satis es tis is graed. ut tye 2 consumers would now refer to - because teir surlus under is (c + e + f) wile teir surlus under is (d + e + f + g). Since (c) is equal to (d), te surlus under can also be written as (c + e + f + g) - imlying tat consumers of tye 2 are better o by area (g) if tey ick. Te searating equilibrium can terefore not emerge. b MC a MC 2 d = d 2 Figure 6: d = d 2. 3

14 f e d g MC c d d 2 MC 2 Figure 7: d 2 > d [Section ]: (a) Rater tan starting our analysis by distinguising between marginal costs of di erent tyes, our model from section starts by secifying te robabilities and tat tye and tye 2 individuals will nd temselves in te "bad state" tat tey are insuring against. Maing tis to our model from art of tis eercise, wit tye and 2 de ned as in art (consumers of tye 2 are relatively ealty), wat is te relationsi between and? Wit as te robability of te "bad state" for tye and te robability of te "bad state" for tye 2, it must be tat > : (b) To t te story wit te model from section, we can assume tat wat matters about bad ealt socks is only te imact tey ave on consumtion - and tat taste are state indeendent. (We will rela tis assumtion in eercise 22.8). Suose we can, for bot tyes, write taste over risky gambles as von-neumann Morgenstern eected utility functions tat emloy te same function u(y) as "utility of consumtion" (wit consumtion denoted y). Write out te eected utility functions for te two tyes. Let y be consumtion wen sick and y 2 consumtion wen ealty (wit, resumably, y < y 2 ). We would get U (y ; y 2 ) = u (y ) + ( ) u (y 2 ) ((22.)) 4

15 for tye consumers and U 2 (y ; y 2 ) = u (y ) + ( ) u (y 2 ) ((22.2)) (c) Does te fact tat we can use te same u (y) to eress eected utilities for bot tyes imly tat te two tyes ave te same taste over risky gambles - and tus te same demand for insurance? No, tey do not. Te eected utility functions U and U 2 di er because te robabilities and di er. Te eected utility functions in fact take te Cobb-Douglas form-wit U lacing eavier emasis on y tan U 2 : (d) If insurance comanies could tell wo is wat tye, tey would (in a cometitive equilibrium) simly carge a rice equal to eac tye s marginal cost. How is tis catured in te model develoed in section of te tet? Tis is catured by te zero-ro t (or actuarily fair) contract lines-wic di er for te two tyes. In articular, for tye, te zero-ro t contracts are = b (were is te insurance remium and b is te insurance bene t), and tye 2 tey are = b (a ayment to te insurance comany equals te eected value of teir future claim against tat olicy). (e) In te searating equilibrium we identi ed in art, we ad insurance comanies roviding te contract tat is e cient for tye individuals-but roviding an ine cient contract to tye 2. Draw te model from section of te tet and illustrate te same and contracts. How are tey eactly analogous to wat we derived in art? Figure 8 illustrates our model from art and our analogous model from Section of te tet in gure 9. In gure 8, ig cost tyes ave iger demand for insurance levels -and is structured so tat ig cost tyes are indi erent between teir e cient insurance coice and te otion intended for low cost tyes-. Tis is done by insuring tat te saded areas in te anel are equal to one anoter-because tat insures tat te loss in surlus from switcing between and is equal to te gain for tye consumers. Tat s eactly wat we do in gure 9 for te new model. Tere, = b reresents te zero ro t contract line for tye consumers and = b reresents te zero ro t line for tye 2 consumers wo cost less and tus ave more generous bene ts for any insurance remium. Te e cient insurance coice for tye consumers is -te oint at wic tey fully insure (given teir risk aversion) at te actuarily fair rate. Tye consumers are ten indi erent between all insurance contracts tat fall on te indi erence curve U - wic goes troug and. 5

16 Tus, is te actuarily fair insurance contract aimed at tye 2 consumers tat makes tye consumers indi erent indi erent to -just as it is in gure 8. In bot cases, tye 2 consumers are better o at tan. Te analogy etends even furter: In gure 8, all te darkened insurance ackages can be o ered - any insurance level at rice and all insurance levels u to at rice : Te analogous darkened lines in gure 9 say te same ting: ny actuarily fair (or zero-ro t) insurance contract aimed at ig cost tyes can be o ered, but only actuarily fair insurance ackages aimed at tye 2 to te left of can be o ered. If te restriction on wat can be o ered at te zero-ro t rates for tye 2 were not included, ten tye individuals would buy at te tye 2 rice in bot cases. MC d 2 d MC 2 Figure 8. Model from Section. 6

17 = θb = δb U U 2 Figure 9: ctuarily Fair Insurance Policy. b (f) In art we also investigated te ossibility of a otential ooling rice-or ooling contract-breaking te searating equilibrium in wic and are o ered. Illustrate in te di erent model ere ow te same factors are at lay in determining weter suc a ooling rice or contract eists. In te model of gure 8, te crucial factor is weter te ooling rice is suc tat it would in fact attract tye 2 consumers away from. Te closer is to ; te more likely tis is te case-and gets closer to te fewer ig cost tye s tere are. In gure 9, te zero-ro t ooling line falls between = b and = b getting closer to te former as te fraction of tye consumers increases and getting closer to te latter as te fraction of tye consumers falls. Te searating equilibrium cannot be broken in anel (b) unless te zero ro t ooling line crosses te dased indi erence curve U 2 wic is more likely to aen te fewer tye consumers tere are. Once again, te conclusion and intuition is eactly te same. (g) Evaluate again te True/False statement in art (g). Tis is true as already discussed in te revious art. gain, te two models give eactly te same unc line. 7

18 Policy lication Eanding Healt Insurance Coverage: Some countries are struggling wit te roblem of eanding te fraction of te oulation tat as good ealt insurance. [Section ]: Continue wit te set-u rst introduced in eercise 22.7 including te de nition of as te amount of insurance coverage bougt by an individual. ssume trougout tat demand for ealt insurance by te relatively ealty (tye 2) is lower tan demand for ealt insurance by te relatively sick (tye ) - i.e., d > d 2 : (a) Illustrate d ; d 2 ; MC and MC 2 and identify te contracts and from eercise Tis is done in gure 0. MC d 2 d MC 2 Figure 0: Contracts from Previous Problem. (b) Suose tat te fraction of relatively sick (tye ) consumers is su ciently ig suc tat no ooling contract can kee tis from being an equilibrium. On te MC line, indicate all te contracts tat can be o ered in tis equilibrium (even toug only is cosen). Similarly, indicate on te MC 2 line all te contracts tat can be o ered in tis equilibrium (even toug only is cosen). 8

19 Tis is done in gure by indicating te contracts tat can be o ered in te searating equilibrium troug bold lines. Te entire MC line can be o ered, and te ortion of te MC 2 line u to can be o ered. Te ooling rice is indicated su ciently ig suc tat (g) is larger tan (e) - wit (g) being te consumer surlus tat is lost by tye 2 individuals if forced to buy at rater tan get at - and (e) is ow muc tye 2 consumers would gain. MC d 2 d MC 2 Figure : Contracts tat can be o ered. (c) True or False: Insurance comanies in tis equilibrium restrict te amount of insurance tat can be bougt at te rice = MC 2 in order to kee tye consumers from buying at tat rice. Tis is true - if any olicies to te rigt of were sold, te tye consumers would want to buy at wic would make no longer a zero-ro t rice. (d) Wy is te resulting searating equilibrium ine cient? How big is te deadweigt loss? E ciency would require tat eac tye buy insurance so long as te marginal bene t is larger tan te marginal cost. For tye individuals, tis aens until we reac and for tye 2 individuals, tis aens until we get to C. t te e cient allocations, tye individuals would get consumer surlus of (a + b + c) wile tye 2 individuals would get consumer surlus (a+d+e+g+). In te searating equilibrium, tye consumers consume 9

20 eactly at te e cient level but tye 2 consumers under-consume. s a result, teir surlus is (a + d + g) or (e + ) less tan it would be at te e cient oint C. Te deadweigt loss in te searating equilibrium is ten te area (e + ) multilied by te number of tye 2 consumers. b a c MC * d g e f C d 2 d MC 2 Figure 2: Deadweigt Loss nalysis. (e) Suose tat te government regulates tis ealt insurance market in te following way: It identi es te zero-ro t ooling rice and requires insurance comanies to carge for eac unit of but does not mandate ow muc every consumer consumes. Illustrate in your gra ow muc insurance tye and tye 2 consumers will consumers under tis olicy? Does overall insurance coverage increase or decrease? Tis is illustrated in gure 3. Individual of tye initially consume insurance at and increase consumtion to wen teir rice falls from to : Tye 2 consumers initially consume and increase teir consumtion to 2 desite te fact tat rice increases because reviously tey were roibited from buying more at. Tus, bot tyes increase teir insurance levels imlying tat insurance coverage overall increases as a result of te regulation. 20

21 * MC d 2 d MC 2 2 Figure 3: Regulated Pooling Price. (f) How muc does consumer surlus for eac tye cange as a result of tis regulation? Does overall surlus increase? Using te letters labeling te areas in gure 4, consumer surlus for tye consumers increases from (a + b + c) to (a + b + c + d + e + i + j); and consumer surlus for tye 2 consumers falls from (a + d + g) to (a + d + e). (We know tat tye 2 consumer surlus falls because we know tat (g) is greater tan (e) oterwise te searating equilibrium could not ave been an equilibrium.) 2

22 b * a d c e i j MC g k l d 2 d MC 2 2 Figure 4: Consumer Surlus Canges. Overall surlus ten canges by N (d + e + i + j) N 2 (g e) were N is te number of tye consumers and N 2 is te number of tye 2 consumers. It aears likely tat tis will almost certainly be a ositive cange in overall consumer surlus desite te fact tat tye 2 consumers are made somewat worse o. (g) True or False: Tis olicy is e ciency enancing but does not lead to e ciency. Tis is likely to be true. ased on our analysis above, overall consumer surlus almost certainly increase and rms make zero ro t before and after. It is not, owever, te case tat te result is e cient. In fact, tye consumers now over consume insurance (since tey were initially consume te e cient quantity) wile tye 2 consumers are still underconsuming since tey are aying more tan teir marginal cost. () It may be di cult for te government to imlement te above rice regulation because it does not ave enoug information to do so. Some ave suggested tat te government instead set te insurance level to some and ten let insurance comanies comete on ricing tis insurance level. Could you suggest, in a new gra, a level of tat will result in greater e ciency tan regulating rice? (You need to do tis on a new gra for te following reason: If te government sets between te amounts consumed by tye and 2 under te zero ro t rice regulation, te resulting cometitive rice sould be lower tan )? 22

23 Tis is done in gure 5 were is set between 2 and from gure 3. Were eactly ends u will deend on te relationsi between te intersection of MC wit d (i.e. oint ) and te intersection of MC 2 wit d 2 (i.e. oint C) were te e cient insurance levels for tye and 2 consumers lies. It will also deend on te number of eac tye in te economy. ut you sould be able to see in te gra tat D results in substantially less deadweigt loss equal to te small darkened triangles above and below D for te gra we ave drawn. Tese emerge because, as we ave drawn tis, D as sligtly more tan te e cient insurance level for tye and sligtly less tan te e cient insurance level for tye 2. In te secial case were lies at te same level of as C i.e. were lies rigt above C, te government can set to be equal to tat level and acieve full e ciency. It is still te case tat tye consumers are subsidized by tye 2 consumers, but tat is a simle transfer from one tye to te oter. Te crucial e ciency gain comes from moving bot teir consumtions closer to te amounts tat are e cient for tem. (Te cometitive rice wen te insurance level is ed at is lower tan from anel (b) because low cost tyes now buy more insurance wile ig cost tyes buy less tus making it ceaer to service tem jointly). MC D d 2 d MC 2 Figure 5: Quantity Regulation. [Section ]: Now consider again weter we can nd analogous conclusions in te model from Section as modi ed in eercise (a) Interreting te model as in eercise 22.7, illustrate te searating equilibrium in a gra wit te insurance bene t b on te orizontal ais and te insurance remium on te vertical. Include in your gra a zero-ro t ooling contract line tat makes te searation of tyes an equilibrium outcome. 23

24 Tis is illustrated in gure 6. Te zero ro t ooling line lies to te nortwest of te dased indi erence curve U 2 for tye 2 individuals wo consume and tus would not be cosen over by tye 2 consumers. Tis imlies we ave su ciently many tye consumers to make te ooling contract line lie su ciently far u in te gra. = θb Zero rofit ooling U = δb U 2 Figure 6: Searating Equilibrium. b (b) How would you interret te rice regulation roosed in Section, art (e) in te contet of tis model? Te rice regulation ere would mean tat te government restricts te set of contracts tat can be sold to tose tat would result in zero ro t if everyone bougt at te same insurance terms. Note tat tis migt imly a zero ro t line di erent from wat we concluded in te tet were we assumed a single ooling contract was o ered in te ooling equilibrium. If we allow te two tyes to coose di erent contracts tat are structured on te same term (i.e. te same relationsi between and b), te zero ro t line will deend not only on te fraction of te oulation tat falls into eac tye category but also te insurance ackages tat are cosen. (c) Illustrate in your gra ow insurance coverage will increase if te government imlements tis olicy. 24

25 We illustrate tis in gure 7. Wit tyically saed indi erence curves (wic are wat emerges from eected utility teory), bot income and substitution e ects suggest tat tye individuals will move sout-east from to a contract like D. (Te income e ect is ositive imlying more coverage will be bougt, and te substitution e ect is laso ositive since insurance for tye consumers as become ceaer.) Individuals of tye 2 were reviously restricted to and it is because of tis restriction tat teir new coice some contract like C will ave more insurance coverage. ut it will be less tan full insurance because te terms are not actuarily fair from a tye 2 ersective. = θb Zero rofit ooling U C D U 2 = δb b Figure 7: Regulation of Prices. (d) Now consider te same roblem in a gra wit y 2 te consumtion level wen ealty on te orizontal ais and y te consumtion level wen sick on te vertical. Illustrate te "endowment oint" E = (y ; y 2 ) tat bot tyes face in te absence of insurance. Tis is done in gure 8. 25

26 y 45 o y E y 2 y 2 Figure 8: Consumtion Levels. (e) Illustrate te actuarily fair insurance contracts for tye and 2 consumers. Ten indicate were te searating equilibrium contracts and lie assuming state indeendent tastes. Tis is done in gure 9. Te sallower solid line troug E is te actuarily fair insurance line for tye and te steeer solid line troug E is te actuarily fair insurance line for tye 2. (It as to be sallower for te ig cost tyes because teir remium for te same bene t level are iger imlying tat teir increase in y for any decrease in y 2 is less.) Wit state-indeendent tastes, individuals will fully insure under actuarily fair terms wic imlies tat tye individuals will coose on te 45 degree line. To kee tye consumers from settling on te tye 2 actuarily fair contract line, insurance comanies cannot o er any contract iger tan on te tye 2 actuarily fair line because gives tye s te same utility as but anyting iger would make tem switc away from. 26

27 y 45 o U U 2 y E y 2 y 2 Figure 9. ctuarily Fair Contracts. (f) Introduce into your gra a zero - ro t ooling contract line suc tat te searating equilibrium is indeed an equilibrium. Ten illustrate ow te roosed government regulation a ects te coices of bot tyes of consumers. Tis is done in gure 20. Te ooling line is te bold dased line troug E between te actuarily fair contract lines for te two tyes. In order for te searating equilibrium to emerge, it as to be te case tat tis ooling contract line lies below te indi erence curve U 2 for tye 2 tat asses troug. If so, tye 2 consumers will not want to switc to te ooling line from. If te government regulation goes into e ect and only contracts on te ooling line are o ered, tye 2 individuals will coose some contract C to te rigt of te 45 degree line because te dased indi erence curves ave te sloe of te stee actuarily fair contract line along te 45 degree line and are sallower to te left were tey must be tangent to te sallower ooling contracts line. Tye individuals, on te oter and, will end u at some contract like D on te oosite side of te 45 degree line were teir indi erence curves are su ciently stee (relative to wat tey are along te 45 degree line). Tye individuals tus over-insure wile tye 2 individuals under-insure but bot get more insurance tan tey ad at and before te regulation. 27

28 y 45 o U D U 2 C y E y 2 y 2 Figure 20: Price Regulation. (g) Suose tat, instead of regulating rice, te government set an insurance bene t level b (as in section art ()) and ten allowed te cometitive rice to emerge. Were in your gra would te resulting contract lie if it fully insures bot tyes? Since tye consumers would now not over-insure and tye 2 consumers would not under-insure, it sould cost less to rovide tis full insurance to bot tyes wit te same contract tat is cometitively riced. Since te contract as full insurance, it must lie on te 45-degree line, and since it costs less tan te rice regulation, it must lie above te dased zero-ro t contract line in gure 2. It must also lie below te U 2 indi erence curve oterwise te searating equilibrium could not ave been an equilibrium. Contract F in te gure satis es all tese conditions. 28

29 y 45 o U F U 2 y E y 2 y 2 Figure 2: Quantity Regulation. () Suose net tat tastes were state-deendent wit u (y) and u 2 (y) te functions (for evaluating consumtion wen sick and wen ealty) tat we need to use in order to arrive at our eected utility function. If u and u 2 are te same for bot consumer tyes, does our main conclusion tat te rice regulation will cause an increase in insurance coverage cange? No te logic will be eactly as in gure 9 ecet tat no longer needs to be on te 45 degree line and C and D don t necessarily need to be were tey are relative to te 45 degree line. ut te direction of te canges is te same. 29