尼科爾森《微觀經(jīng)濟學》課后習題答案

CHAPTER2

THEMATHEMATICSOFOPTIMIZATION

Theproblemsinthischapterareprimarilymathematical.Theyareintendedtogivestudents

somepracticewithtakingderivativesandusingtheLagrangiantechniques,buttheproblems

inthemselvesofferfeweconomicinsights.Consequently,nocommentaryisprovided.All

oftheproblemsarerelativelysimpleandinstructorsmightchoosefromamongthemonthe

basisofhowtheywishtoapproachtheteachingoftheoptimizationmethodsinclass.

Solutions

U(x,y)=4x2-^-3y2

8,12

dy=Sxdx-^6ydy

d.—fordU=08xdx+6ydy=0

dx

dy_-8x_-4x

dx6y3y

e.x=l,y=2U=4」+3?4=16

g.U=16contourlineisanellipsecenteredattheorigin.Withequation

4x2+3y2=16,slopeofthelineat(x,y)is—.

dx3y

2.2Profitsaregivenby7i=R-C=-2q2+40q-100

d?*1八

—二一4夕+40q=10

dq

乃*=一2(10)2+40(10)-100=100

2

b.——7=-4soprofitsaremaximized

dq

dR

c.MR=——=70-2^

dq

MC=g=2q+30

dq

so(7*=10obeysMR=MC=50.

2.3Substitution:y=\-xsof=xy=x-x2

義=l—2x=0

dx

x=05y=0.5,/=0.25

Note:/"=一2<0.Thisisalocalandglobalmaximum.

LagrangianMethod:?=xy-i-A1-x-y)

dx

Jd。

砂

so,x=y.

usingtheconstraintgivesx=y=0.5,xy=0.25

2.4SettinguptheLagrangian:?=x+y+40.25-xy).

2

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