Finance Theory_Solution Manual

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1、IntermediateFinancialTheoryDanthineandDonaldsonSolutionstoExercises1Chapter11.1.Uisautilityfunction,i.e.,U(x)>U(y)Ûx
yf(.)isanincreasingmonotonetransformation,f(a)>f(b)Ûa>b;thenf(U(x))>f(U(y))ÛU(x)>U(y)Ûx
y1.2.UtilityfunctionU(cc,):12FOC:U1/U2=p1/p2Letf=f(U(.))beam

2、onotonetransformation.Applythechainruleforderivatives:FOC:f1/f2=f'U1/f'U2=p1/p2(primedenotesderivation).Economicinterpretation:fandUrepresentthesamepreferences,theymustleadtothesamechoices.1.3.Whenanagenthasverylittleofonegivengood,heiswillingtogiveupabigquantityof

3、anothergoodtoobtainabitmoreofthefirst.MRSisconstantwhentheutilityfunctionislinearadditive(thatis,theindifferencecurveisalsolinear):U(cc,)=ac+bc1212aMRS=bNotveryinteresting;forexample,theoptimalchoiceover2goodsforaconsumerisalwaystoconsumeonegoodonly(iftheslopeofthe

4、budgetlineisdifferentfromtheMRS)oranindefinitequantityofthe2goods(iftheslopesareequal).Convexpreferencescanexhibitindifferencecurveswithflatspots,strictlyconvexpreferencescannot.Theutilityfunctionisnotstrictlyquasi-concavehere.Paretoset:2cases.·Indifferencecurvesfo

5、rtheagentshavethesameslope:Paretosetistheentirebox;·Indifferencecurvesdonothavethesameslope:Paretosetisthelowersideandtherightsideofthebox,ortheuppersideandtheleftside,dependingonwhichMRSishigher.5.05.01.4.a.U1=64=4.905.05.0U2=1416=14.97¶U/¶cjjj1ac2MRS==jj()j¶U/¶c1

6、-acj21jc2witha=0.5,MRS=jjc14MRS==.06716216MRS==.114214MRS1¹MRS2,notParetoOptimal;itispossibletoreallocatethegoodsandmakeoneagent(atleast)betteroffwithouthurtingtheother.jjiib.PS={c=cj,=c:2,1+c=20i,=1,2},theParetosetisastraightline(diagonalfromlower-1212lefttoupper-

7、rightcorner).c.TheproblemoftheagentsisjjjjjMaxUs.t.pe+e=pc+c.112112TheLagrangianandtheFOC'saregivenbyLj=(cj)2/1(cj)2/1+y(pej+ej-pcj-cj)12112112/12jj¶L1c2=-yp=0¶cj2cj111jj/12¶L1c1=-y=0j2j¶cc22j¶Ljjjj=pe+e-pc-c=0112112¶yjc2RearrangingtheFOC'sleadstop=

8、.Nowweinsertthisratiointothebudgetconstraintsofagent1jc11121p6+4-2pc=0andafterrearrangingwegetc=3+.Thisexpressioncanbeinterpretedas1111p1ademandf