ch14solutionssolved edit.docx
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ch14solutionssolvededit
SolutionstoChapter14Exercises
SOLVEDEXERCISES
S1.Someexamplesofincentiveschemesthathelpinducemorecareonthepartofpolicyholders:
1.Theinsurercanprovideamultiyearcontractorotherwiseestablishesanongoingrelationshipinwhichfutureinsurancepremiumsdependoncurrentoutcomes.Thepremiumwoulddecreaseafter,say,ayearofno(orlow)claimsbutwouldincreaseafterayearofmore(orlarger)claims.
2.Theinsurercanprovidediscountsforobservableactionsthatreducetheriskthattheinsured-againstoutcomeoccurs,forexample,installingburglaralarmsystems,takingcareful-drivingcourses,andsoon.
Insurerscanalsoimplementpracticesthathelpthemdetectfraud:
1.Insurerscanofferrewardstopeoplewhoobservefraudulentbehaviorbyneighborsorothers(forexample,recklessdrivers)andreportit.CompaniescouldgettogetherandsetupaWebsite,forexample,soindividualswhoobservecarelessbehaviorcanposttheinformationwithouthavingtoknowwhichspecificinsurancecompanytocontact.Suchaschemerequiresasystemforfilteringoutfalsereports.
2.Insurerscouldincreasetheconsequencesassociatedwithdetectedfraud.Afirmcan,ofcourse,voidthecontractandnotpayanything,butinsomecountriespenaltiesincludecriminalprosecutionandfinesorevenjailtime.
S2.(a)Hereareseveralexamplesofdiscountsbeingofferedforlargerquantities:
(i)Adailynewspapersubscriptionfor$20permonth,versusasinglecopyfor$1.
(ii)Amobile-phoneplanhasalargernumberofminutespermonth,atalowercostperminute.
(iii)Asmall(100-square-inch)pizzafor$12,versusalarge(256-square-inch)pizzafor$16.
(iv)Buytwoapples,getthethirdfree.
(v)Asingledrop-inyogaclasscosts$15,whereasanunlimitedmonthlypasscosts$100.
(vi)A10-poundfamilypackofchickenbreastscosts$0.99perpoundatthesupermarket,whereasaregular2-poundpackagecosts$1.39perpound.
Somequantitydiscountsinvolvebundlingtogetherdifferenttypesofproducts:
(vii)Getaprinterfor$50(regularprice$100)whenyoubuyacomputerfor$1,000.
(viii)Purchaseatourpackageofflightplushotelatalowerpricethanyouwouldpayfortheflightandhotelseparately.
(ix)PurchaseMicrosoftExcel,Word,andPowerPointasabundleforalowerpricethanthesumofthethreeindividualprices.
(x)Purchaseaservice/repaircontractbundledtogetherwithyournewautomobileatapricelowerthanyouwouldexpecttopayifyoupurchasedyourrepairslaterasneeded.
(xi)Payanextrafeetohavebreakfastincludedwithyourhotelroom,butlessthanyouwouldpaytopurchasebreakfastinyourhotelrestaurantasaseparateitem.
(b)Thosewhochoosealargequantityrevealthemselvestobewillingtopurchasemoreoftheproduct,withthelowerpriceinducingthemtopurchasemoreratherthanless.Forexample,alargefamilymaybewillingtopurchasea10-poundpackageofchickenbreastatadiscountedper-poundprice,becausetheylikechickenenoughtobewillingtoeatitthreemealsinarow.Bycontrast,asmallfamilyknowsitcan’teatthatmuchchickenbeforeitspoils,sotheychoosea2-poundpackageatahigherpriceperpound.
Withoutofferingthedifferentpackagesizes,thegrocerywouldnotmanagetoseparatethetwotypesofcustomers.Seethenextexerciseforanumericalexampleshowinghowthistypeofscreeningstrategymayincreasethefirm’sprofits.
S3.(a)Let’sconsiderthedifferentpricesPthatOWLcouldchooseforthe300-minuteplan.Lookingatthecustomers’values,wecanseewhowillpurchaseateachprice:
P≤$20:
Everyonewillpurchase.
P>$20andP≤$25:
OnlytheRegularuserswillpurchase.
P>$25:
Noonewillpurchase.
There’snopointchargingP<$20,becausethatwouldselltothesamenumberofpeopleasP=$20,butatalowerprice.Similarly,there’snopointchargingapricestrictlybetween$20and$25,becausethatwouldselltothesamenumberofcustomersasP=$25,butatalowerprice.
Sowecanimmediatelynarrowtheoptimalpricedowntotwochoices:
eitherP=$20orP=$25.Let’sconsidereachoftheseinturn,toseewhichoneisoptimal.
P=$25:
Sellto50%ofpotentialcustomers.
Profitperpurchase=$25–$10=$15
Expectedprofitperpotentialcustomer=(0.5)($15)=$7.50
P=$20:
Sellto100%ofpotentialcustomers.
Profitperpurchase=$20–$10=$10
Expectedprofitperpotentialcustomer=
(1)($10)=$10
Since$10>$7.50,weknowthattheoptimalpriceforthe300-minuteplan(whenofferedalone)isP=$20.
(b)Similarly,let’sconsiderthedifferentpricesPthatOWLcouldchooseforthe600-minuteplan,andwhichcustomerswillpurchaseateachprice:
P≤$30:
Everyonewillpurchase.
P>$30andP≤$70:
OnlytheRegularuserswillpurchase.
P>$70:
Noonewillpurchase.
Byanargumentsimilartothatgiveninpart(a),weknowthattheonlytwochoicesweneedtoconsiderareP=$30andP=$70.Let’sconsidereachoftheseinturn,toseewhichoneisoptimal.
P=$70:
Sellto50%ofpotentialcustomers.
Profitperpurchase=$70–$10=$60
Expectedprofitperpotentialcustomer=(0.5)($60)=$30
P=$30:
Sellto100%ofpotentialcustomers.
Profitperpurchase=$30–$10=$20
Expectedprofitperpotentialcustomer=
(1)($20)=$20
Since$30>$20,weknowthattheoptimalpriceforthe600-minuteplan(whenofferedalone)isP=$70.
(c)TheICLconstraintwillguaranteethataLightuserchoosesthe300-minuteplaninsteadofthe600-minuteplan.Inotherwords,thenetpayoff(benefitminusprice)shouldbehigherfortheLightuserwhenbuying300minutesthan600minutes:
ICL:
$20–P300≥$30–P600
(d)TheICRconstraintwillguaranteethataRegularuserchoosesthe600-minuteplaninsteadofthe300-minuteplan.Inananaloguetotheaboveinequality,wehave:
ICR:
$70–P600≥$25–P300
(e)Tomaximizeprofits,OWLshouldchargeasmuchasitcanwhilestillmakingsuretheICLandICRconstraintsaresatisfied.Inaddition,OWLshouldmakesuretheparticipationconstraintsaresatisfied,sothateachtypeofcustomeriswillingtopurchaseaplanratherthannoplanatall.Theparticipationconstraintsare:
PCL:
$20–P300≥0
PCR:
$70–P600≥0
Thesecanberewrittenas:
P300≤$20
P600≤$70
Ifallthatmatteredweretheparticipationconstraints,thenOWLwouldprefertosetpricesofP300=$20andP600=$70,justaswediscoveredinparts(a)and(b)above.However,nowwemustalsolookattheincentive-compatibilityconstraintstomakesurethateachtypepurchasestherighttypeofplanfromthemenuoftwooptions.
ICLcanberewrittenbyaddingP600-$20tobothsidesoftheinequality:
P600–P300≥$10
Thatis,thedifferenceinpricebetweenthetwoplansmustbeatleast$10,orelsetheLightuserswillbuythelargerplaninsteadofthesmallerone.Wecanseethatthisconstraintisautomaticallysatisfiedwiththepriceswechoseabovetosatisfytheparticipationconstraints.
Let’salsorewriteICR,addingP600–$25tobothsidesoftheinequality:
$45≥P600–P300
P600-P300≤$45
Inotherwords,forRegularuserstobuythe600-minuteplan,the600-minuteplanmustcostnomorethan$45morethanthe300-minuteplan:
P600≤P300+$45
Notethatwhenwejustsatisfiedtheparticipationconstraints,wewantedtocharge$70forthe600-minuteplanand$20forthe300-minuteplan,whichwillnotsatisfyICR.InordertomakesuretheRegularuserspurchasethecorrectplan,wemustchargenomorethan$20+$45=$65forit.
Sotheoptimalpricesare:
P300=$20
P600=$65
Theaverageprofitperpotentialcustomeristhen:
Expectedprofit=(0.5)($20–$10)+(0.5)($65–10)=$32.50
(f)Let’sconsiderthethreecasesseparately.
Part(a)isapoolingoutcome,becausewhenOWLchargestheoptimalpriceof$20forthe300-minuteplan,bothtypesofcustomerspurchase,sowecan’ttellwhattypeofcustomersomeoneis.
Part(b)isaseparatingoutcome,becausewhenOWLchangestheoptimalpriceof$70,onlytheRegularuserspurchase.TheLightusersdonot,sowecandistinguishthetwotypesofusersbasedontheirbehavior.
Part(e)isaseparatingoutcome,becausewhenOWLchargestheoptimalpricesof$20and$65,theLightusersbuy300minutes,andtheRegularusersbuy600minutes.Again,wecandistinguishthetwotypesofusersbytheirbehavior.
S4.(a)Tocasualusers,Mictelofferstosellalow-endmachineforapriceofx=4.Thefirmmakesaprofitof3bydoingso;itwouldmakeonly2bysellingthisuserahigh-endmachine.Tointensiveusers,Mictelofferstosellahigh-endmachineforapriceofy=8.Thefirmmakesaprofitof5bydoingso;itwouldmakeonly4bysellingthisuseralow-endmachine.
(b)Ifproducingonlylow-endmachines,Mictelcaneithersetx=4andselltoeverybody,oritcansetx=5andsellonlytointensiveusers.Sellingatx=4producesanexpectedprofitperuserof
(1)(4–1)=3.Sellingatx=5producesanexpectedprofitperuserof(4/5)(5–1)=3.2.Sothehigherpriceproducesthelargerprofit,andMictelshouldchargex=5forthelow-endmachine.
(c)Ifproducingonlyhigh-endmachines,Micteleithersetsy=5andsellstoeverybody,orsetsy=8andsellsonlytointensiveusers.Sellingaty=5producesanexpectedprofitperuserof
(1)(5–3)=2.Sellingaty=8producesanexpectedprofitperuserof(4/5)(8–3)=4.Thus,thehigherpriceproducesalargerprofit,andMictelshouldchargey=8forthehigh-endmachine.
(d)Theincentive-compatibilityconstraintsare:
Forcasualusers:
4–x≥5–y,whichsimplifiestoy–x≥1
Forintensiveusers:
8–y≥5–x,whichsimplifiestoy–x≤3
(e)Thepar
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