尼科尔森《微观经济学》课后习题答案

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

义=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

1

-歹

1

-

a£

-

ax

a£

-

A

1

X

-

1

-

5y

0.5.

=y=

,x

0.25

=x~=

xy

ives

ntg

strai

econ

gth

Usin

=y.

So,x

.

+40

.5g『

=—0

/(7)

a.

2.5

2

.

800/g

/g)=

40(40

/g)+

g(40

-0.5

/*)=

,/(

for/*

ing

titut

Subs

b.

gL

OO/

一

誓

鬼

.

ong

ends

tdep

use

beca

ong

ends

dep

-a*)2

—=-

c.

2

dg

-800

0、2

CQ4

、2

/*

”

df

.

$一

)———

.5(—

二一0

.5(/

——0

so—

g'

g

鬼

at

ceth

Noti

.08.

of

tion

educ

,ar

.92

=24

32.1

800/

25,

32=

800/

d.

2

edto

edict

bepr

uld

ngco

asei

incre

a0.1

.8so

«-0

/32?

800

/g=

-800

rem.

theo

lope

enve

mthe

8fro

y0.0

ghtb

ehei

reduc

hich

etalw

ofm

iece

map

efro

mad

solid

ular

tang

farec

meo

volu

sthe

hisi

a.T

2.6

ved.

remo

uares

ersq

dcom

efine

thed

with

y3x

isxb

2