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1、CHAPTER2MATHEMATICSFORMICROECONOMICSTheproblemsinthischapterareprimarilymathematical.TheyareintendedtogivestudentssomepracticewiththeconceptsintroducedinChapter2,buttheproblemsinthemselvesofferfeweconomicinsights.Consequently,nocommentaryisprovided.Resultsfromsomeoftheanalyticalpr
2、oblemsareusedinlaterchapters,however,andinthosecasesthestudentwillbedirectedtohere.Solutions22.1Substitution:y=−1xfso==−xyxx∂f−1=2x0=∂xx=0.5,y=0.5,f=0.25Note:f′′=−<20.Thisisalocalandglobalmaximum.LagrangianMethod:£=+−−xyxλ(1y)∂£=y=0−λ∂x∂£=x−λ=0∂yso,x=y.usingtheconstraintgivesxy===
3、0.5,xy0.252.2SettinguptheLagrangian:£=x++yxλ(0.25−y).∂£=−1λy∂x∂£=−1λx∂y2So,x=y.Usingtheconstraintgivesxy==x0.25,x==y0.5.22.3a.f()tg=−0.5t+40tdf*40=gt+40=0−=,t.dtgThiseditionisintendedforuseoutsideoftheU.S.only,withcontentthatmaybedifferentfromtheU.S.Edition.Thismaynotberesold,copi
4、ed,ordistributedwithoutthepriorconsentofthepublisher.2Chapter2:MathematicsforMicroeconomics*2b.Substitutingfort*,f()tg=−0.5(40gg)+40(40)=800g.*∂ft()2=−800g.∂g∂f1*2*c.=−(t)dependsongbecausetdependsong.∂g2∂−f*2402800so=−=0.5()t−=0.5().2∂gggd.80032==25,80032.124.92areductionof.08.Not
5、icethat22−=≈800g80032−0.8soa0.1increaseingcouldbepredictedtoreduceheightby0.08fromtheenvelopetheorem.qq22qqq2.4a.TC=∫∫MCdq=(q+)1dq=+qo+C=+q+C2200Theconstantofintegrationhereisfixedcosts.b.Byprofitmaximization,p=MC(q).p=q+1;q=p-1=14Ifthefirmisjustbreakingeven,profits=totalrevenue–t
6、otalcosts=0214pq−TC(q)=15*14−(+14+C)=0298−C=0C=98=FixedCostsc.Ifp=20,q=19.Followingthesamestepsasinb.,andusingC=98,weget219π=pq−TC(q)=20*19−(+19+98)=825.2So,profitsincreaseby82.5d.Assumingprofitmaximization,wehavep=MC(q)p=q+1⇒q=p−1222q⎛(p−)1⎞(p−)1π=pq−(+q+98)=p(p−)1−⎜+(p−)1+98⎟=−9
7、8⎜⎟2⎝2⎠2e.i.Usingtheaboveequation,π(20)-π(15)=82.520202pii.π(20)−π(15)=∫(p−)1dp=−p=825.21515ThiseditionisintendedforuseoutsideoftheU.S.only,withcontentthatmaybedifferentfromtheU.S.Edition.Thismaynotberesold,copied,ordistributedwithoutthepriorconsentofthepublisher.3Chapter2:Mathema
8、ticsforMicroeconomicsThe2approach
