ECON 101- Midterm Exam Guide - Comprehensive Notes for the exam ( 81 pages long!)
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Study guide shared via c ourse h ero. co m. Economics101 problemset6solutions:ge,risk,anduncertaintytroysmithspring20171. generalequilibriuminanexchangeeconomyconsiderapureexchangeeconomywithtwoconsumers,1and2,andtwogoods,xandy. consumer1"sinitialendowmentis 1=(1,0),thatis,1unitofgoodxand0unitsofgoody. consumer2"sinitialendowmentis 2=(0,1),thatis,0unitsofgoodxand1unitofgoody. consumeri"sutilityfunction(fori=1,2)isui(xi,yi)=xiyi,where(xi,yi)representsi"sconsumptionbundle. (a)showthiseconomy(withsomeindi erencecurvesandtheinitialendow-ments)inanedgeworthboxdiagram. suggestedanswer:edgeworthbox. (b)writedowntheequationsthatdescribetheparetoe cientallocations. identifythemintheedgworthbox. istheinitialendowmentpointparetoe cient?whyorwhynot?suggestedanswer:theparetooptimalpointsinthisexamplearepointsoftangencybetweenindi erencecurvesofthetwoconsumers. tangencyrequiresthatcon-sumer1"smarginalrateofsubstitutionequalconsumer2"smarginalrateofsubstitution. consumeri"smarginalrateofsubstitutionismrsi=muixmuiy=yixisettingthetwoconsumers"marginalratesofsubstitutionequal,andthensubstituting1 x1forx2and1 y1fory2,givesy1x1=y2x2=1 y11 x1thisleadsdirectlyto1 shared via c ourse h ero. co m. T his study resource w as https://www. coursehero. com/file/24460969/problem-set-6-solutionspdf/ x1=y1thereforethesetofparetooptimalpoints,thatis,thecontractcurve,issimplytheupwardslopingdiagonaloftheboxdiagram,fromconsumer1"sorigintoconsumer2"sorigin. theinitialpointwisobviouslynote cient;it"snotonthecontractcurve. infactanymovefromwintotheinterioroftheedgeworthboxdiagramwouldmakebothconsumersbettero . 2. expectedvalueandvarianceproofsletxbearandomvariabletakingthevaluexiwithprobabilitypi,fori=1,,n. letybearandomvariabletakingthevalueyiwithprobabilitypj,forj=1,,n. xandyareindependent. letaandbberealnumbers. provethefollowing:(a)e[ax+b]=ae[x]+bsuggestedanswer:e[ax+b]=nxi=1(axi+b)pi=anxi=1xipi+bnxi=1pi=ae[x]+b(b)var[x]=e[x2] e[x]2suggestedanswer:var[x]=nxi=1(xi e[x])2pi=nxi=1(x2i 2xie[x]+e[x]2)pi=nxi=1x2ipi 2e[x]nxi=1xipi+e[x]2nxi=1pi=e[x2] 2e[x]e[x]+e[x]2=e[x2] e[x]22 shared via c ourse h ero. co m. T his study resource w as https://www. coursehero. com/file/24460969/problem-set-6-solutionspdf/ (c)var[ax+b]=a2var[x]suggestedanswer:var[ax+b]=e[(ax+b)2] e[ax+b]2=e[a2x2+2abx+b2] (ae[x]+b)2=a2e[x2]+2abe[x]+b2 (a2e[x]2+2abe[x]+b2)=a2(e[x2] e[x]2)=a2var[x](d)e[x+y]=e[x]+e[y]suggestedanswer:e[x+y]=nxi=1nxj=1(xi+yj)piqj=nxi=1nxj=1xipiqj+nxi=1nxj=1yjpiqj=nxi=1xipi nxj=1qj +nxj=1yjqj nxi=1pi!=nxi=1xipi+nxj=1yjqj=e[x]+e[y](e)var[x+y]=var[x]+var[y]suggestedanswer:var[x+y]=e[(x+y)2] e[x+y]2=e[x2+2xy+y2] (e[x]+e[y])2=e[x2]+2e[xy]+e[y2] e[x]2 2e[x]e[y] e[y]2=var[x]+var[y]+2e[xy] 2e[x]e[y]e[xy]=nxi=1nxj=1xiyjpiqj=nxi=1xipi nxj=1yjqj =e[x]e[y]3 shared via c ourse h ero. co m. 3. homeownersinsuranceahomeownerwhosehouseisworth,000,000faces oodrisk. theprobabil-ityofa oodis10%andthelossthatoccursfroma oodis,000. thehomeownerhasacrrautilityfunctionwith =3. (a)supposeaninsurancecontracto erstocovertheentirelossintheeventofa ood. whatistheactuariallyfairpremiumandwhatisthehomeowner"swillingnesstopayforthecontract?suggestedanswer:workinginunitsofmillion:0. 9p+0. 1(p 0. 2)=0p 0. 02=0actuariallyfairpremium=,000u(1 p)=0. 9u(1)+0. 1u(0. 8)u(1 p)= 0. 52811 p=0. 9730wtp=,000(b)nowsupposetheinsurancecontractonlycoversafractionsofthelosses. writetheactuariallyfairpremiumasafunctionofs. suggestedanswer:0. 9p+0. 1(p 0. 2s)=0p 0. 02s=0p=0. 02s(c)supposethepremiumfacedbythehomeowneristheactuariallyfairpre-mium. thehomeownercanchoosetoinsureanyfractionofthelossintheeventofthe ood. writedownexpectedutilityasafunctionofs. whatfractionofthelosswouldthehomeownerchoosetoinsure?suggestedanswer:0. 9u(1 p)+0. 1u(0. 8+0. 2s p)=0. 9u(1 0. 02s)+0. 1u(0. 8+0. 18s)foc[s]:0. 9( 0. 02)u (1 0. 02s)+0. 1(0. 18)u (0. 8+0. 18s)=0 0. 018u (1 0. 02s)+0. 018u (0. 8+0. 18s)=04 shared via c ourse h ero. co m. 1 0. 02s=0. 8+0. 18ss=1(d)nowsupposethattheinsurerdoesnotchargeactuariallyfairpremiums. instead,toinsureafractionsoftheloss,theinsurerchargesanadditional5%abovetheactuariallyfairpremium. whatfractionoflosseswouldthehomeownerchoosetoinsure?suggestedanswer:p=1. 05 0. 02s=0. 021seu=0. 9u(1 0. 021s)+0. 1u(0. 8+0. 179s)foc[s]:0. 9( 0. 021)u (1 0. 021s)+0. 1(0. 179)u (0. 8+0. 179s)=0 0. 0189u (1 0. 021s)+0. 0179u (0. 8+0. 179s)=00. 0189(1 0. 021s) 3=0. 0179(0. 8+0. 179s) 33. 7542(1 0. 021s)=3. 8228(0. 8+0. 179s)0. 7631s=0. 6960s=0. 914. puttingallofyoureggsinonebasketanindividualpurchasesadozeneggsandmusttakethemhome. makingtripshomeiscostless,butthereisa50%chancethatalltheeggscarriedononetripwillbebrokenduringthetrip(theindividualisveryclumsy). theindividualconsiderstwostrategies:(1)takeall12eggsinonetrip,or(2)taketwotripswith6eggsineachtrip. (a)calculatetheexpectednumberofeggsremainingunderstrategy1andstrategy2. suggestedanswer:e[x1]=0. 5(12)+0. 5(0)=6e[x2]=0. 25(12)+0. 5(6)+0. 25(0)=65 shared via c ourse h ero. co m.
