monopoly

MonopolyandOligopoly

-degreepricediscrimination

Sincethemarketisparatedintotwogroups,submarket,thenwefacetwodemand

functions,)(

11

ypand)(

22

,profitmaximizationproblemcanbedescribedas

)()()(:

21222111

,

21

yycyypyypMax

yy



Hence,FOCis

)()(

2111

yyMCyMR,and)()(

2122

yyMCyMR

Othersituation:combinationofthetwoparatedmarkets

)()()(:

212121

21

yycyyyypMax

yy







Horizontalsummation:)()()(11

2

1

121

ppppppyyy

Optimalsolution:)()(yMCyMR

olybehavior

tmodel——Simultaneouslyquantitytting

Conditions:marketdemand)(

21

yybap,productsareidentical,marginalcostof

thetwofirmsareequalandconstant,cMCMC

21

,theydecidetheiroutput

decisionsimultaneouslywithoutknowingother’sdecisionadvanced.

Maximizationproblem:

)(:

111

ycypMaxs.t.)(

21

yybap

)(:

222

ycypMaxs.t.)(

12

yybap

Solution:

(1)Since

11211

)]([ycyyyba,then,itsFOCisshownas

02

12

1

1





cbyyba

y



,afterrearranging,thisyieldsthereactionfunctionoffirm1,

b

ybca

y

2

2

1





(2)Thesame,wehavethereactionfunctionoffirm2

b

ybca

y

2

1

2





(3)Combinethetworeactionfunctions,wehave



















b

ca

y

b

ca

y

3

3

*

2

*

1

ndmodel——Simultaneouslypricetting

lbergmodel(quantityleadershipmodel)

Conditions:marketdemand)(

21

yybap,productsareidentical,marginalcostof

thetwofirmsareequalandconstant,cMCMC

21

,however,firm1isquantity

leader,whomakeshisoutputdecisionfirst,firm2isfollower,whomakeshisdecisionwith

knowingfirm1’sdecision.

Maximizationproblem:

)(:

111

ycypMax

s.t.)(

21

yybap&firm2’sreactionfunction)(

122

yfy

)(:

222

ycypMaxs.t.)(

12

yybap

Solution:backwardinduction

(1)Since

22122

)]([ycyyyba,thus,

02

21

2

2





cbyyba

y



,hence,firm2’sreactionfunctionis

b

ybca

y

2

1

2





(2)And

11211

)]([ycyyyba,substitutefirm2’sreactionfunction,then

Iso-profitcurvescanbeshownas)(

2

1

2

1111

bycyay

thus,optimalchoiceappearswhen

0)2(

2

1

1

1

1byca

dy

d

Sothat



















b

ca

y

b

ca

y

4

2

*

2

*

1

eadershipmodel

Conditions:marketdemand)(

21

yybap,productsareidentical,marginalcostof

thetwofirmsare

221

2,yMCcMC,however,firm1ispriceleader,whomakes

hispricingdecisionfirst,firm2isfollower,whomakeshisdecisionwithknowingfirm1’s

decision.

TraditionalMethod:

Takinguofbackwardinduction:

Maximizationproblemsoffollower

)(:

2222

ycypMax

FOC:

22

2)(yyMCp

Thus,thereactionfunctionoffollowis

2

)(

22

p

pfy,inotherwords,it’ssupply

functionofthefollower.

Thenweconsiderthemaximizationproblemofpriceleader

)(:

1111

ycypMax

s.t.)()(

2121

yybayyDp

Substituting

2

)(

22

p

pfyintotheaboveconstraint,then,

p

bb

a

y)

2

11

(

1

,moregenerally,)(

11

pfy,whichisresidualdemandforfirm1.

Thenthemaximizationproblembecomes

))(()(:

1111

pfcpfpMax

p



Thus,FOCis

0)(1

1

11

1



dp

df

df

dc

dp

df

ppf

Hence,

22

*

c

b

a

p









,then

2

)

2

11

(

2

*

1

c

bb

a

y



,andthen

424

*

2

c

b

a

y





Comparison:eadershipmodel

Here,wetrytomakeacomparisonbetweenstackelbergmodelandpriceleadership

model,focusingontheesntialdifferencewithrespondtosolutionprocessandoutcome.

Conditionsofeconomicsituation；

marketdemand)(

21

yybap,productsareidentical,marginalcostofthetwo

firmsare

2211

,yMCyMC,however,firm1isdecisionleader,whomakeshis

decisionfirst,firm2isfollower,whomakeshisdecisionwithknowingfirm1’sdecision.

Ca(1):Stackelbergmodel

Solution:

er’sMaximizationproblemwithknowingfirm1’soutputquantitydecision

)(:

222

0

2

ycypMax

y





s.t.)(

12

yybap

Thus,FOCis02

221

2

2





ybyyba

y





Hence,therespondfunctionoffirm2is)(

122

yfy







b

yba

y

2

1

2

’sMaximizationproblemwithexpectingfirm2’sresponds

)(:

111

0

1

ycypMax

y







s.t.)(

21

yybap

&firm2’sreactionfunction)(

122

yfy

Substitutingallconstraintsintomaximizationproblem,thisyields

)()

2

(

11

1

11

ycy

b

bya

yba



























Thus,FOCis0

2

2

2

2

11

2

1

1

1













yy

b

b

b

ab

bya

y







Hence,











)(22

2

2

*

1bb

abaab

y,

))(22)(2(

22

2

2

*

2









bbb

abbaabab

y

brium

))(22)(2(

32

2

222223

*











bbb

baababababab

p

Ca(2):PriceLeadershipmodel

Solution:

er’sMaximizationproblem,knowingfirm1’soutputquantityandpricedecision

)(:

222

0

2

ycypMax

y





s.t.)(

12

yybap

Atthistime,therearetworulesthatwedon’tknowwhetheritismatched.

FOC:

22

)(yyMCp,or

12

y

b

pa

y





Hence,therearetwoprobablereactionfunctions.



p

pfy)(

22

,or

1122

),(y

b

pa

ypfy





Ifbothoftheabovereactionfunctionsareequivalent,then

1

y

b

pap









,thatisfirm1’sresidualdemand:p

bb

ap

b

pa

y

























11

1

’sMaximizationproblemwithexpectingfirm2’sresponds

)(:

111

ycypMax

p



s.t.p

bb

a

y





















11

1

FOC:0

111111

21



















































































b

p

bb

a

p

bb

a

p

Hence,

)2)((

22

*





aabbb

aabab

p







Thus,

)2(

2

*

1



aabb

aaa

y





,

)2)((

*

2



aabbb

aabab

y







Alsotherearetwoapproachestosolvingthisproblem,firstly,takingtheplaceofmaxp

andsubstitutingmaxy

1

,then,wehave









bb

a

y

2

*

1

,

)2)((

22

*





abbb

aabab

p





,

)2)((

*

2



abbb

aabab

y







Ca(3)Existingunmarketablegoodsinpriceleadership

Ontheotherhand,asgoodsinmarketareidenticalandover-suppliedisreality,

ore,unmarketablegoodsofmarketshareare

denoteunmarketableproportion,

21

21

21

1)(

yy

xx

yy

pf













,hence,quantity

offirm1’sunmarketablegoodsis

111

xyy,andthatoffirm2’sis

222

xyy.Now

wecandescribethemaximizationproblemsasfollow:

Forfirm2,)(

2222

y

2

yCxpMax

s.t.

21

1

2

2

)(

yy

pf

y

x









Thus,maximizationproblembecomesifconstraintissubstitutedintotheoriginalone.

)(

)(

222

21

1

2

yCy

yy

pf

pMax









Hence,theFOCoftheaboveproblemis

2

21

1

2)(

)(

yy

y

y

b

pap







Sothat,firm2’sreactionfunctioncanbedepictedas),(

122

ypy

Forfirm1,theleader,itsmaximizationproblemis

)(

1111

)y(p,

1

yCxpMax

s.t.

21

1

1

1

)(

yy

pf

y

x







andfollower’sreactionfunction),(

122

ypy

Thus,wehave)()(

11

21

1

1

1

)y(p,

1

yC

yy

y

pfpMax





s.t.2

21

1

2)(

)(

yy

y

y

b

pap







Solution:takinguofLagrangemethod,





























2

21

1

2

11

21

1)(

)(

)(yy

y

y

b

pap

yC

yy

y

b

pa

pL





Hence,

0)(

)(

2

21

1

2









yy

y

y

b

papL



Then,thisyield2

21

1

2)(

)(

yy

y

y

b

pap







(1)

0

22

21

1





































b

pa

yy

y

b

pa

p

L

Then,thisyield0

21

1





yy

y

(2)

0

)(

)(

2

1

3

2

21

2

21

2

1

































y

y

yy

yy

y

b

pap

y

L



Substituting(1)&(2),then,

Wehave

21

yy(1*)

0

34

)(

)1()(

1

2

221

2

1

2

21

2





































y

yyyy

yy

b

pap

y

L



Substituting(1)&(2),then,

Wehave

221

2

1

3yyyy(2*)

Combining(1*)&(2*),thus,









3

*

1

y,hence,

2

2

*

23





y

ca