尼克尔森微观经济学答案.pdf

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1、CHAPTER2THEMATHEMATICSOFOPTIMIZATIONTheproblemsinthischapterareprimarilymathematical.TheyareintendedtogivestudentssomepracticewithtakingderivativesandusingtheLagrangiantechniques,buttheproblemsinthemselvesofferfeweconomicinsights.Consequently,nocommentaryisprovided.Allo

2、ftheproblemsarerelativelysimpleandinstructorsmightchoosefromamongthemonthebasisofhowtheywishtoapproachtheteachingoftheoptimizationmethodsinclass.Solutions222.1Uxy(,)4x3yUUa.=x,8=y6xyb.8,12UUc.dUdx+dy=xdx+ydy86xydyd.fordU0860xdxydydxdy8x4x==dx6y3ye.xyU1,

3、2413416dy4(1)f.2/3dx3(2)g.U=16contourlineisanellipsecenteredattheorigin.Withequation22dy4x4x3y16,slopeofthelineat(x,y)is.dx3y22.2a.ProfitsaregivenbyRC2q40q100d*4q40q10dq*22(10)40(10)1001002db.4soprofitsaremaximized2dqdRc.MRq702dqdCM

4、Cq230dq12SolutionsManualsoq*=10obeysMR=MC=50.22.3Substitution:y1xsofxyxxfx120xx=0.5,y=0.5,f=0.25Note:f20.Thisisalocalandglobalmaximum.LagrangianMethod:?xy1xy)£=y=0x£=x=0yso,x=y.usingtheconstraintgivesxy0.5,xy0.252.4SettinguptheLagrangia

5、n:?xy0.25xy).£1yx£1xy2So,x=y.Usingtheconstraintgivesxyx0.25,xy0.5.22.5a.ft()0.5gt40tdf*40gt400,t.dtg*2b.Substitutingfort*,ft()0.5(40gg)40(40g)800g.*ft()2800g.gf1*2*c.(t)dependsongbecausetdependsong.g2f*2402800so0.5()t0.5().2

6、gggd.8003225,80032.124.92,areductionof.08.Noticethat22800g800320.8soa0.1increaseingcouldbepredictedtoreduceheightby0.08fromtheenvelopetheorem.Chapter2/TheMathematicsofOptimization32.6a.Thisisthevolumeofarectangularsolidmadefromapieceofmetalwhichisxby3xwiththedefi

7、nedcornersquaresremoved.V22b.3x16xt12t0.Applyingthequadraticformulatothisexpressionyieldst2216x256x144x16x10.6xt0.225,1.11xx.Todeterminetrue24242Vmaximummustlookatsecondderivative--16x24twhichisnegativeonly2tforthefirstsolution.3333c.Ift0.225,xV0.67x

8、.04x.05x0.68xsoVincreaseswithoutlimit.d.ThiswouldrequireasolutionusingtheLagrangianmethod.Theoptimalsolution