Partial equilibrium analysis:
Monopoly
Lectures in Microeconomic Theory
Fall 2007, Part 2
12.09.2007
G.B. Asheim, ECON4230-35, #2
Pricemaking
1
price
π ( y, c) = p ( y ) y − cy
= (a − by ) y − cy
Inverse
demand fn.:
p ( y ) = a − by
c
quantity
y y′
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G.B. Asheim, ECON4230-35, #2
Profit maximization in special cases
P
π ( y, c) = p ( y ) y − cy
Special case 1:
= (a − by − c ) y
max (a − by − c ) y
y
FOC : a − 2by − c = 0
p( ym )
ym =
ym
Special case 2:
y = Ap
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−b
a−c
2b
π ( y, c) =
Y
π ( y, c) = A y
1
b
p( ym ) =
1− 1b
− cy
FOC : (1 − ) p = c
1
b
G.B. Asheim, ECON4230-35, #2
a+c
2
(a − c) 2
4b
p( ym ) =
c
1 − 1b
3
1
General analysis
π ( y ) = p( y ) y − c( y )
∂π
FOC :
∂y
= p ( y ) + p′( y ) y − c′( y ) = 0
p ( y ) − c′( y )
y
1
= − p′( y )
=−
p( y)
p( y)
ε
where ε =
SOC :
1 p( y)
is the elasticity of demand.
p′( y ) y
∂ 2π
= 2 p′( y ) + p′′( y ) y − c′′( y ) ≤ 0
∂y 2
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G.B. Asheim, ECON4230-35, #2
Comparative statics
c ( y ) = cy
SOC
p′( y ) < 0
2 p′( y ) + p′′( y ) y < 0
∂π ( y , c)
Special case 1:
π ( y, c) = p ( y ) y − cy
=0
a+c
∂y
p( ym ) =
2
∂ π ( y, c)
∂ π ( y, c)
dy +
dc = 0
dp − b
∂y
dc∂y
=
>0
dc − 2b
dy
1
∂π ∂π
=−
=
< 0 Special case 2:
c
dc∂y ∂y
dc
2 p′( y ) + p′′( y ) y
p( ym ) =
1 − 1b
dp dp dy
p′( y )
dp
1
=
=
>0
=
>1
dc dy dc 2 p′( y ) + p′′( y ) y
dc 1 − 1b
2
2
2
2
2
2
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G.B. Asheim, ECON4230-35, #2
Welfare and output
Welfare as a function of output: p
W ( x) = u ( x) − c ( x)
Welfare maximization:
u ′( x0 ) = p ( x0 ) = c′( x0 )
c′(x)
p ( xm )
Monopoly output satisfies:
p( x0 )
p(x)
p ( xm ) + p′( xm ) xm = c′( xm )
W ′( xm ) = u′( xm ) − c′( xm ) =
− p′( xm ) xm = −u′′( xm ) xm > 0
Monopolist’s gain is smaller
than consumers’ loss.
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G.B. Asheim, ECON4230-35, #2
u′(x)
xm
x0
x
Deadweight
loss
6
2
Price discrimination
Monopolist’s dilemma:
A higher quantity
leads to a lower price.
price
p( y)
p ( y ′)
The monopolist can get
out of this dilemma by
• sorting consumers
• charging different prices
to different consumers
c
y y′
This requires that the monopolist can
sort, and that consumers cannot resale.
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quantity
How can the
monopolist
sort?
G.B. Asheim, ECON4230-35, #2
7
Types of price discrimination
„
First-degree price discrimination
(Also called perfect discrimination) “Special price for you”
Price = maximal willingness-to-pay for each unit.
„
Second-degree price discrimination
„
Price differs according to consumed quantity (or quality),
but not across consumers. Ex: Full price/disc. tickets for
transportation
Third-degree price discrimination
Price differs across consumers, but does not depend on
consumed quantity. Ex: Ticket
price
depends
on age,
etc.
Different
price
dom. and
abroad
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G.B. Asheim, ECON4230-35, #2
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1st-degr. price discr.
p
Maximization of
total surplus
c′(x)
u ′( x0 ) = c′( x0 )
No surplus to
consumers:
p ( x0 )
p(x)
x0
u ( x0 ) − u (0) − ∫ p ( x)dx = 0
u′(x)
0
x
Whole surplus to
x0
monopolist.
Why is first-degree price discrimination
difficult to implement for the monopolist?
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G.B. Asheim, ECON4230-35, #2
9
3
Model with two consumers
p
„
Low demand consumer (L)
„
High demand consumer (H)
Assumptions:
H has higher total willingness-to-pay
(u H ( x) − u H (0) ) − (u L ( x) − u L (0) ) > 0
u′H (x)
H has higher marginal
willingness-to-pay
u′L (x)
u ′H ( x) − u ′L ( x ) > 0
x
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G.B. Asheim, ECON4230-35, #2
2nd-degr. price discr. (self-selection)
Each consumer is offered a pair
of total payment and quantity: (ri , xi )
p
xH
xL
rH = rL + ∫ u ′H ( x)dx
0
xL
rL = ∫ u′L ( x) dx
u′H ( x)
Assume
no costs.
Ensures participation and self-selection
u′L (x)
How to determine xL and xH ?
No distortion at the top:
u′( xH ) = 0 = MC
xL
xH
x′L
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x
L’s quantity is distorted:
u′( xL ) > 0 = MC Why?
11
G.B. Asheim, ECON4230-35, #2
3rd degr. price discr. (segmentation)
The monopolist is able to treat the
two consumers as separate markets.
p
u′H ((xx)
Monopoly price and quantity in each market.
u′L ((xx)
p
Higher elasticity leads to lower price.
What are the welfare effects of requiring
the same price in both markets?
pH
3rd degr. price distr. is welfare
improving only if it leads to a
higher quantity.
pL
xL x H
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Assume
no costs.
x
x
G.B. Asheim, ECON4230-35, #2
x
12
4