# Partial equilibrium analysis: q y Monopoly

Partial equilibrium
q
analysis:
y
Monopoly
Lectures in Microeconomic Theory
Fall 2010, Part 17
07.07.2010
G.B. Asheim, ECON4230-35, #17
Pricemaking
1
price
 ( y, c)  p ( y ) y  cy
 a  by  y  cy
Inverse
demand fn.:
p ( y )  a  by
c
y y
07.07.2010
G.B. Asheim, ECON4230-35, #17
quantity
2
1
Profit maximization in special cases
P
 ( 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 )
ac
2b
p ( ym ) 
 cy
FOC : 1  1b  p  c
p ( ym ) 
ym 
Special case 2:
y  Ap  b
07.07.2010
(a  c) 2
 ( y, c) 
4b
Y
ym
 ( y, c)  A y
1
b
ac
2
1 1b
c
1  b1
G.B. Asheim, ECON4230-35, #17
3
General analysis
 ( y )  p( y ) y  c( y )

FOC :
y
 p ( y )  p( y ) y  c( y )  0
p ( y )  c( y )
y
1
  p( y )


p( y)
p( y)
where
h

1 p( y)
i the
is
h elasticity
l i i off demand.
d
d
p( y ) y
 2
SOC :
 2 p( y )  p( y ) y  c( y )  0
y 2
07.07.2010
G.B. Asheim, ECON4230-35, #17
4
2
Comparative statics
c( y )  cy
SOC
p( y )  0
2 p( y )  p( y ) y  0
Special
p
case 1:
 ( y, c)  p ( y ) y  cy  ( y, c)  0
ac
y
p( ym ) 
2
  ( y, c)
  ( y, c)
dy 
dc  0
dp  b
y
dcy

0
dc  2b
dy
1
 


 0 Special case 2:
c
d y y
dc
d
dc
2 p( y )  p( y ) y
p ( ym ) 
1  b1
p( y )
dp dp dy
1
dp


0

1
dc dy dc 2 p( y )  p( y ) y
dc 1  1b
2
2
2
2
2
2
07.07.2010
5
G.B. Asheim, ECON4230-35, #17
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.
07.07.2010
G.B. Asheim, ECON4230-35, #17
u(x)
xm
x0
x
loss
6
3
Price discrimination
Monopolist’s dilemma:
A higher quantity
price.
price
p( y)
p ( y )
The monopolist can get
out of this dilemma by
• sorting consumers
• charging different prices
t different
to
diff
t consumers
c
This requires that the monopolist can
sort, and that consumers cannot resale.
07.07.2010
G.B. Asheim, ECON4230-35, #17
y y
quantity
How can the
monopolist
sort?
7
Types of price discrimination

First-degree price discrimination
(Also called perfect discrimination) “Special
“Sp i l price
pri fforr 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
t
transportation
t ti
Third-degree price discrimination
Price differs across consumers, but does not depend on
consumed quantity. Ex: Different
Ticket price
depends
on age,
etc.
price
dom. and
07.07.2010
G.B. Asheim, ECON4230-35, #17
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4
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(x)
u ( x0 )  u (0)   p ( x)dx  0
0
x
Whole surplus to
x0
monopolist.
Why is first-degree price discrimination
difficult to implement for the monopolist?
07.07.2010
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G.B. Asheim, ECON4230-35, #17
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 ( x)  u H (0)  u L ( x)  u L (0)   0
H has higher marginal
willingness-to-pay
uL (x)
x
07.07.2010
G.B. Asheim, ECON4230-35, #17
u H ( x)  u L ( x)  0
10
5
2nd-degr. price discr. (self-selection)
Each consumer is offered a pair
of total payment and quantity: (ri , xi )
p
xL
rL   u L ( x)dx
uH (x)
Assume
no costs.
xH
rH  rL   u H ( x)dx
xL
0
Ensures participation and self-selection
uL (x)
How to determine xL and xH ?
No distortion at the top:
p
u( xH )  0  MC
xL
xL
xH
07.07.2010
x
L’s quantity is distorted:
u( xL )  0  MC Why?
11
G.B. Asheim, ECON4230-35, #17
3rd degr. price discr. (segmentation)
The monopolist is able to treat the
two consumers as separate markets.
p
uH ((xx)
uL (x)
p
pH
M
Monopoly
l price
i and
d quantity
i iin each
h market.
k
Higher elasticity leads to lower price.
What are the welfare effects of requiring
the same price in both markets?
33rdd degr.
d
price
i distr.
di t is
i welfare
lf
improving only if it leads to a
higher quantity.
pL
xL xH
07.07.2010
Assume
no costs.
x
x
G.B. Asheim, ECON4230-35, #17
x
12
6