Monopoly'vs.'Compe22on'
Monopoly'
Theory'of'the'Firm'
Monopolis2c'Markets'
''
P'
'
'
'
'
'
'
'
Perfect'Compe,,on'
''
Monopoly’s demand =
Market demand
(ΔQ ⇒ ∇P)
P
Firm’s demand =
Horizontal line
( Δq ⇒ P does not change)
P
D'
Q
Monopoly'vs.'Compe22on'
Monopoly'
'
•  One'firm'(market'power)'
'
•  Barriers'to'entry'
Perfect'Compe,,on'
•  Many'price9taking'firms'
'
•  Free'entry/exit'
d'
Causes'of'Monopoly'
•  Natural'
''''1.''Scarcity'of'a'resource:'one'firm'owns'a'key'resource
''
'''''''''(De'Beers’'Diamonds;'local'land'own)'
'
'''''2.''Technological'reasons:'existence'of'large'economies'of'scale''
''''''''''rela2ve'to'the'market'size'99'one'firm'may'serve'the'demand''
''''''''''more'efficiently'than'several'firms'(water,'telephone,'…)'
'
•  Legal'
'''''1.''Licenses'(taxis,'pharmacies,'notaries,'…)'
''''''
'''''2.''Patents'(medical'drugs,'intellectual'property,'…)'
q
Natural'Monopoly'
''
'
Profit'Maximiza2on'
A'natural'monopoly'arises'when'there'a'large'economies'of'scale:'
cost'minimiza2on'would'lead'to'a'single'firm'producing'the'total'
output.''
P
•  First'Order'Condi2on:'
π’(q)'='0,''
or,'equivalently'
MR(q)'='MC'(q).'
D
'
•  Second'Order'Condi2on:''
PM
2p’(q*)'+'qp’’(q*)'–'MC’(q*)'≤'0.'
'
•  Shutdown'Condi2on:''
π(q*)'≥'π(0)'
AC
MR
QM
MC
Q
The'Monopolist’s'Problem'
The'revenue'func2on'of'the'monopoly'is'
'
'
'
'R(q)'='p(q)q,'''
Profit'Maximiza2on'
''
'
C(q)'
C,R, π
R(q)'
Slope'='MR'
'
where'p(q)'is'the'INVERSE'demand'func2on.'''
'
Therefore'the'problem'of'the'monopolis2c'firm'is'
Slope'='MC'
'
'''''''''''''''''''' 'Maxq≥0''π(q)'='p(q)q'–'C(q).'
'
π(q)'
'
qM
q
Profit'Maximiza2on:'MR'
Profit'Maximiza2on:'MR'
Marginal'revenue'is'given'by'
MR(q)'='(p(q)q)’'='p(q)'+'p’(q)q.'
''
p
'
Marginal'Revenue:'
MR'(q)'='p(q)'+'p’(q)q.'
'
(Since'p’(q)'<'0,'the'MR'
curve'is'always'below'the'
demand'curve.)'
The'two'terms'in'this'expression'have'a'clear'interpreta2on:'saling'one'
addi2onal'(infinitesimal)'unit:'
(a)  Increases'revenue'by'an'amount'equal'to'the'price'(quan2ty'effect),'
and'
(b)  Decreases'revenue'by'an'amount'equal'to'the'total'output'2mes'the'
reduc2on'in'price'necessary'to'generate'demand'for'the'addi2onal'
unit'(price'effect).'
MR(q)'
'
P(q)
q
Profit'Maximiza2on:'MR'
'
''
Profit'Maximiza2on:'SOC'
The'Second'Orden'Condi2on'is:'
P
'
A:'quan2ty'effect'
B:'price'effect'
2p’(q*)'+'qp’’(q*)'–'MC’(q*)'≤'0.'
'
Since'p’(q)'<'0,'the'second'order'condi2on'may'hold'even'when'
MC’'(q)'='C’’(q)'<'0,'that'is,'even'when'the'firm'has'economies'of'
scale'near'the'op2mal'level'of'output.'Thus,'the'op2mal'
produc2on'level'the'firm'may'not'exhaust'the'economies'of'
scale.'
'
'
'
''''MR'='A'–'B.'
B
A
q
P(q)
q +1
q
Monopoly'Equilibrium'
'
'Monopoly'
Perfect'Compe22on'
p'
p'
p'
DWL'
CS'
MC'
pC'
qM'
MC'
CS'
PS'
MR'
CS'
AC'
AC'
pM'
Surplus'and'Profits'
MC='
AC'
PS'
q'
qC'
Monopoly'Equilibrium'
In'the'monopoly'equilibrium:''
'
9'The'price'is'above'the'compe22ve'price,'pM'>'pC'.'
'
9'The'output'is'below'the'compe22ve'output,'qM'<'qC.'
'
9'The'consumer'surplus'is'below'its'compe22ve'level,'CSM'<'CSC.''
'
9'The'producer'surplus'is'above'its'compe22ve'level,'PSM'>'PSC.'
'
9'The'total'surplus'is'below'its'compe22ve'level,'TSM'<'TSC.'
D'
MR'
D'
9'There'is'a'deadweight'loss,'DWL'='TSCMTSM'>'0.'
'''''
π"="PS'
pC'
D'
'
DWL'
pM'
qM'
qC'
q'
q'
Lerner'Index'
First'Order'Condi2on'for'profit'maximiza2on'is:'
'
'
'
''MR(q)'='MC(q).'
'
Marginal'revenue'is'
'
&q#
'
MR(q) = p + qp' (q) = p + p$$ !! p' (q).
% p"
'
'
'
'
'
'
Lerner'Index'
Lerner'Index'
We'can'write'the'monopoly'price'as'a'func2on'of'the'demand'
price9elas2city.'The'demand'price9elas2city'is'
'
p
p
ε = q'( p) =
.
'
Then,'
'
'
'
q
€
The'Lerner'index'measures'monopoly'power:'it'explains'the'
percentage'of'price'that'is'not'due'to'costs.''
'
It'is'defined'as'
p − MC(qM )
1
'
L= M
=− .
pM
ε
'
qp'(q)
$
1'
MR = MC ⇔ p&1 + ) = MC.
%
ε(
'
Since'in'the'monopoly'equilibrium'pM'>'MC'(qM),'then''
€
0'≤'L'≤'1.''
(Under'perfect'compe22on,'L'='0.)'
'
'
That'is,'the'monopolist’s'mark'cap'is'91/ε.'
€
Lerner'Index'
Example'
The'more'elas2c'is'the'demand,'the'smaller'is'the'monopolist’s'
mark'up,'and'vice'versa:'
p'
A'monopolist'faces'the'demand'func2on'
' 'D(p)'='max{12'–'p,'0},''
and'its'cost'func2on'is'
p'
D'
D'
MC'
pM'
''''''''''''''C(q)'='5'+'4q.'
'
We'calculate'the'monopoly'equilibrium,'the'consumer'and'
producer'surpluses'and'the'deadweight'loss,'the'monopoly'
profits,'the'Lerner'index.'
MC'
pM'
MC(qM)'
MC(qM)'
MR'
MR'
qM'
q'
qM'
q'
Example'
Monopoly'Equilibrium.'Revenue'is'
'
'
'
'R(q)'='(12'–'q)q'='12q'–'q2.'
Hence''
'
'
'MR(q)'='12'–'2q.'
'
Since'MC(q)'='4,'the'equilibrium'output'is'given'by'the'equa2on'
'
'
'
'MR(q)'='MC(q)'⇒'12'–'2q'='4.''
'
Thus,'qM'='4,'and'pM'='12'–'qM'='12'–'4'='8.'
Example'
We'calculate'the'Lerner'index:'
'LI'='(pM'–'MC(qM))/pM'='(8M4)/8'='0.5'
'
p'
'
12'
D'
CS'
8'
Monopoly'
PS'
DWL'
4'
Perfect'Compe22on'
MC'
'
'
''
4'
Example'
We'calculates'the'consumer'and'producer'surplus'and'the'
monopoly’s'profit:'
'CS'='0.5(12'–'pM)qM'='0.5(12M8)'4'='8,'
'
'PS'='(pM'–'Cma)'qM'='(8M4)'4'='16,'
'
'π(q)'='pMqM'–'C(qM)'='8(4)'–'(5'+'4(4))'='11,'
and,
'
'DWL'='0.5(pM'–'pC)(qC'–'qM)'='0.5(4)(4)'='8.'
'
'
MR'
8'
12'
q'
Regula2ng'a'Monopoly'
Is'it'possible'to'regulate'the'monopoly'in'order'to'avoid'the'
deadweight'loss'it'generates?'
'
A'solu2on'may'be'to'impose'a'regulated'price'equal'to'marginal'cost.'
Even'if'this'solu2on'is'possible'(it'requires'the'regulator'to'know'the'
monopoly’s'cost'func2on),'it'may'involve'subsidizing'the'monopoly.'
'
When'subsidizing'the'monopoly'is'not'possible'or'sensible'(rising'the'
necessary'revenue'may'involve'other'inefficiencies),'an'alterna2ve'
may'be'to'impose'a'regulated'price'equal'to'average'cost.'
Regula2ng'a'Monopoly'
'
P'='MC'(q)'
p'
D'
Subsidy'='qC*(AC(qC)'–'pC)'
MR'
AC'(qC)'
Taxes'
In'a'monopoly,'as'in'a'compe22ve'market,'introducing'a'tax'
causes'an'increase'in'prices'and'a'decrease'in'output,'and'
therefore,'a'deadweight'loss.''
''
The'propor2on'of'the'tax'that'is'paid'for'by'consumers'
depends,'in'a'monopoly'as'well'as'in'a'compe22ve'market,'
of'the'price'elas2city'of'the'demand.''
AC'
MC'
pC'
q'
qC'
Regula2ng'a'Monopoly'
p'
Price'Caps'
In'a'compe22ve'market,'a'price'cap'(i.e.,'a'maximum'price'
at'which'the'good'may'be'traded):''
P'='AC(q)'
'
D'
Deadweith'loss'
pR'
MR'
AC'
MC'
pC'
qR'
qC'
q'
•  reduces'the'level'of'output'(and'creates'an'excess'
demand'that'requires'ra2oning'the'demand),''
•  results'in'a'deadweight'loss'(reduces'the'total'surplus),'
and''
•  may'or'may'not'increase'the'consumer'surplus.'
'
Price'Caps'
In'a'monopoly,'however,'a'price'cap'below'the'equilibrium'
price'may'be'an'effec2ve'instrument'to'decrease'the'price'
and'increase'the'output.'A'price'cap'above'marginal'cost'
does'not'create'an'excess'demand'and'leads'to:''
'
•  an'increase'of'the'equilibrium'output,'
•  a'decrease'of'the'equilibrium'price,'
•  a'greater'consumer'and'total'surplus,'and''
•  a'smaller'deadweight'loss.''
'''
Taxes'and'Price'Cap:'an'Example'
Suppose'we'introduce'a'tax'T=1'€/unit.'Then'the'demand'is'now''''''
D(p,T)=max{12'–'(p+T),'0},'
and'for'q'≤'12'M'T'the'monopolist’s'revenue'is'
'
R(q,T)'='(12'–'q'–'T)q,'
and'its'marginal'revenue'is'
'
MR(q)'='12'–'2q'–'T.''
Taxes'and'Price'Cap:'an'Example'
Consider'a'monopolist'whose'cost'func2on'is'
'
C(q)'='5'+'4q,''
facing'the'demand'
'
D(p)'='max{12'–'p,'0}.'
The'monopoly'equilibrium'output'solves'the'equa2on'
'
'
'
'12'M'2q'='4'
'
Hence'the'monopoly'equilibrium'is'
'
(qM',pM)'=(4,8).'
'
Monopoly:'Taxes'
The'equa2on'MR(q)'='MC(q)'is'now'
'12'–'2q'–'T'='4.'
'
Hence'output'is''
qM(T)'='4'–'T/2,''
'
and'the'price'is'''''''''''''
pM(T)'='8'+'T/2.''
'
The'monopolist’s'unitary'revenue'is''
'
8'+'T/2'–'T'='8'–'T/2;'
that'is,'the'tax'burden'is'shared'equally.'
'
Monopoly:Taxes'
''
Taxes'and'Price'Cap:'an'Example'
2.'Suppose'we'introduce'a'price'cap'p+'<'8'='pM.'
The'price'cap'changes'de'revenue'func2on'of'the'monopoly'
$12 − q if 12 - q ≤ p + '
''
(
R( p + ,q) = % +
if 12 − q > p + )
&p q
'
The'Marginal'Revenue'is:' ''
$12 − 2q if 12 - q ≤ p + '
'
+
% +
(
MR(
p
,q)
=
'' €
if 12 − q > p + )
&p
p'
12'
pC'
11'
T'
pM'
pS'
4'
MC'
D(p,T)'
'
D(p)'
q'
qT'
Para'p+'='7,'the'equilibrium'is'q'='5'and'p'='p+'=7'(see'the'graph''
in'the'page).'
€
qM'
Monopoly:'Taxes'
MR(q)'='MC(q)'⇒'12'–'2q'–'T'='4'⇒'q(T)'='4'–'T/2.'
'
With'the'taxe'T,'the'price'paid'by'the'consumer'and'the'price'received'
by'monopoly'are'not'the'same:'
'
Price'paid'by'consumer'='12'–'q(T)'='8'+'T/2'
Price'received'by'monopoly'=''8'+'T/2'–'T'='8'–'T/2.'
'
In'this'case,'the'tax'burden'is'shared'equally'between'the'consumers'
and'the'monopoly.'
'
''
Monopoly:Taxes'
''
p'
12'
pC'
11'
pM=8'
p+=7'
4'
MC'
D(p)'
qM=4'
q+=5'
q'
MR+'
Price'Discrimina2on'
Third'Degree'Price'Discrimina2on'
'The'monopoly'tries'to'segment'the'market'(by'geographical'
criteria,'by'customers’'features,'etc.),'and'charges'different'
prices'to'each'“market'segment.”'
So'far,'we'have'assumed'that'the'monopoly'charges'the'
same'price'for'each'unit.'
'
We'now'discuss'how'the'monopoly'may'increase'its'profit'
by'charging'different'prices'to'different'consumers'with'
different'WILLINGNESS'TO'PAY.'
''''Example:'There'are'two'groups'of'consumers,'1'and'2,'
''''whose'demands'are'D1(p1)'and'D2(p2).'
Price'Discrimina2on'
•  First'degree:'the'monopoly'sells'each'infinitesimal'unit'at'the'
maximum'price'a'consumer'is'willing'to'pay.'With'this'pricing'policy'
we'obtain'
'
''
'CS=0,'PS'='TS'='maximum'surplus,'DWL=0.''
•  Second'degree:'The'monopoly'uses'NON9LINEAR'pricing'policies:'
for'example,'volume'discounts.'This'type'of'discrimina2on'is'very'
common'in'water,'electricity,'telephone'and'internet'supplies.'
Third'Degree'Price'Discrimina2on'
'
'
p'
'
'
'
'
'
'Ɛ2'>'Ɛ1''
'
Group'1'
'
Group'2'
p2'
1'
D1'
MR1'
D2'
MR2'
q1'
q2'
Third'Degree'Price'Discrimina2on:'
Example'
Third'Degree'Price'Discrimina2on'
The'monopolist'chooses'q1'and'q2'to'maximize'the'total'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 ).
'
'
Suppose'a'monopoly'produces'a'good'with'costs''
'
'
'C(q)'='q2/2,''
and'faces'two'markets'with'demands'
'
'D1(p1)'='max{20'–'p1,0}''
and''
'
'D2(p2)'='max{60'–'2p2,0}.''
Determine'the'quan22es,'prices'and'total'profits'under'third'
degree'price'discrimina2on,'and'without'price'discrimina2on.'
'
Third'Degree'Price'Discrimina2on:'
Example'
Third'Degree'Discrimina2on'
In'terms'of'elas2ci2es:'
'
''''''''''' '''''''p1(1+1/Ɛ1)'='p2(1+1/Ɛ2)'='MC(q1+q2)'
'
Therefore'
'
'
'p1/p2'='(1+1/Ɛ2)/(1+1/Ɛ1)'<'1;'
and'
'
'
'''''''p1'<'p2;''
'
''
That'is,'monopoly'price'is'lower'in'the'market'with'a'more'elas2c'
demand.'
(a)'Third'degree'discrimina2on:'
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,'we'obtain'q1'='2,'q2'='14,'p1'='18'and'p2'='23.''
Monopoly'profits'are'π'='18*2'+'14*23'–'C(2+14)'='230.'
'
'
'
'
'
Third'Degree'Price'Discrimina2on:'
Example'
(b)'Without'price'discrimina2on:'the'aggregate'demand'is'
'
' 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
!
'
!
!
%
"
'
'
Monopoly'equilibrium:'MR(q)'='MC(q)'
Solving,'we'obtain'qM'='15,'pM'='22.5.'
Monopoly'profits'are:'π'='84.375.'
'