Monopolyvs.Compe22on
Monopoly
TheoryoftheFirm
Monopolis2cMarkets
P
PerfectCompe,,on
Monopoly’s demand =
Market demand
(ΔQ ⇒ ∇P)
P
Firm’s demand =
Horizontal line
( Δq ⇒ P does not change)
P
d
D
Q
Monopolyvs.Compe22on
Monopoly
•  Onefirm(marketpower)
•  Barrierstoentry
PerfectCompe,,on
•  Manyprice-takingfirms
•  Freeentry/exit
CausesofMonopoly
•  Natural
1.Scarcityofaresource:onefirmownsakeyresource
(DeBeers’Diamonds)
2.Technologicalreasons:existenceoflargeeconomiesofscale
rela2vetothemarketsize--onefirmmayservethedemand
moreefficientlythanseveralfirms(water,telephone,…)
•  Legal
1.Licenses(taxis,pharmacies,notaries,…)
2.Patents(medicaldrugs,intellectualproperty,…)
q
NaturalMonopoly
ProfitMaximiza2on
Anaturalmonopolyariseswhentherealargeeconomiesof
scale:costminimiza2onwouldleadtoasinglefirmproducing
thetotaloutput.
P
D
•  FirstOrderCondi2on:
π '(q) = 0 ⇔ MR(q) = MC(q).
•  SecondOrderCondi2on:
PM
€
AC
MR
2 p' (q ) + qp ' ' (q ) − MC ' (q) ≤ 0
•  ShutdownCondi2on:π(qM)≥π(0)
MC
Q
QM
TheMonopolist’sProblem
Therevenuefunc2onofthemonopolyis
R(q)=p(q)q,
ProfitMaximiza2on
C(q)
C,R, π
R(q)
Slope=MR
wherep(q)istheINVERSEdemandfunc2on.
Thereforetheproblemofthemonopolis2cfirmis
Slope=MC
Maxq≥0π(q)=p(q)q–C(q).
π(q)
qM
q
ProfitMaximiza2on:MR
ProfitMaximiza2on:MR
Marginalrevenueisgivenby
p
d
( p(q)q ) = p(q)+ p' (q)q .
MR(q) =
dq
(+)
(−)
Thetwotermsinthisexpressionhaveaveryclearinterpreta2on:
thesaleofoneaddi2onal(infinitesimal)unitgenerates
(a)  anincreaseofrevenueequaltotheprice(quan2tyeffect),and
(b)  adecreaseofrevenueequaltothetotaloutput2mesthe
reduc2oninpricenecessarytogeneratethedemandofthis
addi2onalunit(priceeffect).
MarginalRevenue:
MR(q)=p(q)+p’(q)q.
(Sincep’(q)<0,theMR
curveisalwaysbelow
thedemandcurve.)
MR(q)
P(q)
q
ProfitMaximiza2on:MR
ProfitMaximiza2on:SOC
P
A:quan2tyeffect
B:priceeffect
MR=A–B.
B
A
q
P(q)
q +1
q
TheSecondOrdenCondi2onis:
2 p' (q) + qp' ' (q) − MC ' ≤ 0
Sincep’(q)<0,thecondi2onC’’(q)>0(requiringthatthe
firmhasdiseconomiesofscaleneartheop2mallevelof
output)isnolongerrequiredforprofitmaximiza2on:at
theop2malproduc2onlevelthefirmmayhaveeconomies
ofscale.
MonopolyEquilibrium
Monopoly
SurplusandProfits
PerfectCompe22on
p
p
p
CS
pM
MC
DWL
CS
MC
CS
AC
PS
pC
MR
qM
AC
π=PS
MC=
AC
pC
PS
D
D
q
qC
Inamonopoly,thepriceisabovethecompe22veprice,
pM>pC,
theoutputisbelowthecompe22veoutput,
qM<qC,
theconsumer(producer)surplusisbelow(above)thecompe22vesurplus
CSM<CSC(PSM>PSC),
andthetotalsurplusisbelowthecompe22vesurplus,
TSM<TSC.
Also,thereisaDeadweightLoss,DWL=TSC-TSM>0.
D
MR
MonopolyEquilibrium
DWL
pM
qM
qC
q
q
LernerIndex
FirstOrderCondi2onforprofitmaximiza2onis:
MR(q)=MC(q).
Marginalrevenueis
⎛q⎞
MR(q) = p + qp' (q) = p + p⎜⎜ ⎟⎟ p' (q).
⎝ p⎠
LernerIndex
LernerIndex
Wecanwritethemonopolypriceasafunc2onofthe
demandprice-elas2city.Thedemandprice-elas2cityis
p
p
ε = q'( p) =
.
q
Then,
€
TheLernerindexmeasuresmonopolypower:itexplainsthe
percentageofpricethatisnotduetocosts.
Itisdefinedas
p − MC(qM )
1
L= M
=− .
pM
ε
Notethat0≤L≤1.(L=0underperfectcompe22on.)
€
qp'(q)
⎛
1⎞
MR = MC ⇔ p⎜1+ ⎟ = MC.
⎝
ε⎠
€
LernerIndex
Example
Ifdemandisveryelas2c(Ɛhigh),themarginwillbesmall;
andviceversa:
p
Amonopolistfacesthedemandfunc2on
D(p)=max{12–p,0},
anditscostfunc2onis
p
D
D
MC
pM
C(q)=5+4q.
Wecalculatethemonopolyequilibrium,theconsumerand
producersurplusesandthedeadweightloss,themonopoly
profits,theLernerindex.
MC
pM
MC(qM)
MC(qM)
MR
MR
qM
q
qM
q
Example
MonopolyEquilibrium.Revenueis
R(q)=(12–q)q=12q–q2.
Hence
MR(q)=12–2q.
SinceMC(q)=4,theequilibriumoutputisgivenbytheequa2on
MR(q)=MC(q)⇒12–2q=4.
Thus,qM=4,andpM=12–qM=12–4=8.
Example
WecalculatetheLernerindex:
LI=(pM–MC(qM))/pM=(8-4)/8=0.5
p
12
D
CS
8
Monopoly
PS
DWL
4
PerfectCompe22on
MC
4
Example
Wecalculatestheconsumerandproducersurplusandthe
monopoly’sprofit:
CS=0.5*(12–pM)*qM=0.5*(12-8)*4=8;
PS=(pM–CMa)*qM=(8-4)*4=16;
π(q)=pMqM–C(qM)=8*4–(5+4*4)=11;
Sincethecompe22veequilibriumispC=4,qC=8,wehave:
DWL=0.5*(pM–pC)(qC–qM)=0.5*4*4=8.
MR
8
12
q
Regula2ngaMonopoly
Anunregulatedmonopolygeneratesadeadweightloss,DWL>0.
Canthisbeavoided?
Anobvioussolu2on(ifpossible–thatis,ifweknowthe
monopoly’scostfunc2on),istoimposearegulatedpriceequal
tomarginalcost,thusgenera2ngthecompe22veoutcome.
Thissolu2onmayleadtomonopoly’slossesthatwouldhaveto
besubsidized.
Whensubsidizingisnotpossibleorconvenient(risingthe
necessaryrevenuemayinvolveotherinefficiencies),an
alterna2veregulatedpricemaybetheaveragecost.
Regula2ngaMonopoly
P=MC(q)
p
D
Subsidy=qC*(AC(qC)–pC)
MR
AC(qC)
Inamonopoly,asinacompe22vemarket,introducing
ataxcausesanincreaseinpricesandadecreasein
output,andtherefore,adeadweightloss.
Thepropor2onofthetaxthatispaidforbyconsumers
depends,inamonopolyaswellasinacompe22ve
market,ofthepriceelas2cityofthedemand.
AC
MC
pC
q
qC
Regula2ngaMonopoly
p
P=AC(q)
D
Deadweithloss
pR
Taxes
MR
AC
MC
pC
qR
qC
q
PriceCaps
Recallthatinacompe22vemarketintroducingapricecap(i.e.,a
maximumpriceatwhichthegoodmaybetraded),reducesthe
levelofoutput(andcreatesanexcessdemandthatrequires
ra2oningthedemand),resultsinadeadweightloss(reducesthe
totalsurplus)andmayormaynotincreasetheconsumer
surplus.
Inamonopoly,however,introducingapricecapbelowthe
equilibriumpricemaybeaneffec2veinstrumenttodecreasethe
priceandincreasetheoutput.Moreover,ifthepricecapis
abovemarginalcost,thenitdoesnotcreateanexcessdemand,
andresultsinagreaterconsumerandtotalsurplus,andsmaller
deadweightloss.
TaxesandPriceCap:anExample
Consideramonopolistwhosecostfunc2onis
C(q)=5+4q
facingthedemand
D(p)=max{12–p,0}.
Themonopolyoutputisthesolu2ontotheequa2on
12-2q=4
ThereforeqM=4,andpM=8.
Monopoly:Taxes
12
pC
11
T
pM
pS
1.  SupposeweintroduceataxT=1€/unit.
4
Demand: D(p,T)=max{12–(p+T),0}.
Revenue:
R(q,T)=(12–q–T)q=12q–q2–Tq.
MarginalRevenue: MR(q)=12–2q–T.
MC
D(p,T)
D(p)
q
qT
Monopoly:Taxes
Theequa2onMR(q)=MC(q)isnow
12–2q–T=4
HenceoutputisqM(T)=4–T/2,theconsumerpriceis
pM(T)=8+T/2,and
monopolyprice=8+T/2–T=8–T/2.
Thatis,thetaxburdenissharedequallybetweentheconsumers
andthemonopoly.
p
qM
Monopoly:Taxes
MR(q)=MC(q)⇒12–2q–T=4⇒q(T)=4–T/2.
WiththetaxeT,thepricepaidbytheconsumerandtheprice
receivedbymonopolyarenotthesame:
Pricepaidbyconsumer=12–q(T)=8+T/2
Pricereceivedbymonopoly=8+T/2–T=8–T/2.
Inthiscase,thetaxburdenissharedequallybetweenthe
consumersandthemonopoly.
TaxesandPriceCap:anExample
2.Supposeweintroduceapricecapp+<8=pM.
Thepricecapchangesderevenuefunc2onofthemonopoly
⎧12 − q if 12 - q ≤ p + ⎫
⎬
R( p + ,q) = ⎨ +
if 12 − q > p + ⎭
⎩p q
TheMarginalRevenueis: ⎧12 − 2q if 12 - q ≤ p + ⎫
+
⎨ +
⎬
MR(
p
,q)
=
€
if 12 − q > p + ⎭
⎩p
PriceDiscrimina2on
Sofar,wehaveassumedthatthemonopolycharges
thesamepriceforeachunit.
Wenowdiscusshowthemonopolymayincreaseits
profitbychargingdifferentpricestodifferent
consumerswithdifferentWILLINGNESSTOPAY.
Parap+=7,theequilibriumisq=5andp=p+=7(seethegraph
inthepage).
€
Monopoly:Taxes
PriceDiscrimina2on
•  Firstdegree:themonopolysellseachinfinitesimalunitatthe
maximumpriceaconsumeriswillingtopay.Withthispricing
policyweobtain
CS=0,PS=TS=maximumsurplus,DWL=0.
p
12
pC
11
pM=8
p+=7
4
MC
D(p)
qM=4
q+=5
q
MR+
•  Seconddegree:ThemonopolyusesNON-LINEARpricing
policies:forexample,volumediscounts.Thistypeof
discrimina2onisverycommoninwater,electricity,telephone
andinternetsupplies.
ThirdDegreePriceDiscrimina2on
ThirdDegreePriceDiscrimina2on
Themonopolistchoosesq1andq2tomaximizethetotal
profits:
π(q1,q2)=R1(q1)+R2(q2)–C(q1+q2)
=p1(q1)q1+p2(q2)q2–C(q1+q2)
FOC:
MR1 ( q1 ) = MC ( q1 + q2 )
MR2 ( q2 ) = MC ( q1 + q2 ).
Themonopolytriestosegmentthemarket(by
geographicalcriteria,bycustomers’features,etc.),and
chargesdifferentpricestoeach“marketsegment.”
Example:Therearetwogroupsofconsumers,1and2,
whosedemandsareD1(p1)andD2(p2).
ThirdDegreePriceDiscrimina2on
p
Ɛ2>Ɛ1
Group1
Group2
p2
1
ThirdDegreeDiscrimina2on
D1
MR1
D2
MR2
q1
q2
Intermsofelas2ci2es:
p1(1+1/Ɛ1)=p2(1+1/Ɛ2)=MC(q1+q2)
Therefore
p1/p2=(1+1/Ɛ2)/(1+1/Ɛ1)<1;
and
p1<p2;
Thatis,monopolypriceislowerinthemarketwithamore
elas2cdemand.
ThirdDegreePriceDiscrimina2on:
Example
Supposeamonopolyproducesagoodwithcosts
C(q)=q2/2,
andfacestwomarketswithdemands
D1(p1)=max{20–p1,0}
and
D2(p2)=max{60–2p2,0}.
Determinethequan22es,pricesandtotalprofitsunder
thirddegreepricediscrimina2on,andwithoutprice
discrimina2on.
ThirdDegreePriceDiscrimina2on:
Example
(a)Thirddegreediscrimina2on:
MC(q1+q2)=q1+q2
R1=20q1–q12⇒MR1=20–2q1
R2=30q2–0.5*q22⇒MR2=30–q2
Equilibrium:
20–2q1=30–q2=q1+q2
Solving,weobtainq1=2,q2=14,p1=18andp2=23.
Monopolyprofitsareπ=18*2+14*23–C(2+14)=230.
ThirdDegreePriceDiscrimina2on:
Example
(b)Withoutpricediscrimina2on:theaggregatedemandis
⎧ 80 − q
⎫
if 20 ≤ q ≤ 80⎪
⎪ 3
⎧80 − 3 p if p ≤ 20
⎫
⎪
⎪
q
⎪
⎪
⎪
⎪
D( p) = ⎨60 − 2 p if 20 < p ≤ 30⎬ ⇒ p(q) = ⎨30 − if q < 20
⎬
2
⎪0
⎪
⎪
⎪
if
p
>
30
⎩
⎭
if q > 80
⎪0
⎪
⎪
⎪
⎩
⎭
Monopolyequilibrium:MR(q)=MC(q)
Solving,weobtainqM=15,pM=22.5.
Monopolyprofitsare:π=84.375.