Download Monopoly Price Discrimination

Monopoly Price Discrimination
Firm must have ability to sort consumers
and prevent resale.
Definition: (due to Pigou (1920))
First degree (Perfect) price discrimination
Seller charges each consumer equal his/her
willingness to pay.
Second degree price discrimination
Price differs depending on the number of units
bought (but not across consumers)
Third degree price discrimination
Different purchasers are charged different but
constant price for each unit of the good bought.
First-Degree Price Discrimination
- Let ri(x) be consumer i’s maximum willingness to
pay for x units of good1
- Consumers have quasilinear utility ui(x), hence
demand is given by p = ui( x )
1
Solution to ui(0) + y = ui(x) - ri(x) +y. Assuming ui(0), ri(x) = ui(x)
The monopolist want to offer a specific price and
output combination (ri* xi* ) for each i that
Max ri – c(xi)
r i , xi
s.t. ui(x) ≥ ri Constraint is binding at optimal
*
( xi* )
F.O.C.: u′
i ( xi ) = c′
ri* = ui ( xi* )
The monopolist who can perfectly price discriminate is
producing at Pareto efficient allocation and as
*
( xi* ) , it is producing at the same level as
p = u′
i ( xi ) = c′
a competitive firm.
Diagram
Second-Degree Price Discrimination
- Assume that there are 2 consumers
Low-demand consumer 1 with u1(x1) + y1
High-demand consumer 2 with u2(x2) + y2
( x) < u′
Assume that u1(x) < u2(x) and u1′
2 ( x)
The monopolist knows that there are low-demand
and high-demand consumer but cannot distinguish
them.
- Suppose that consumer i demands xi and spends
ri amount of money.
For the monopolist, the expenditure and
consumption for the consumers 1 and 2 are (r1, x1)
and (r2, x2) respectively.
If different types end up choosing different
quantity and hence different expenditure levels, the
following 4 inequalities must be true
u1(x1) − r1 ≥ 0
u2(x2) − r2 ≥ 0
}
Participation constraint
u1(x1) − r1 ≥ u1(x2) − r2
u2(x2) − r2 ≥ u2(x1) − r1
}
Self selection constraint
Rearranging
(1)
r1 ≤ u1(x1)
(2)
r1 ≤ u1(x1) − u1(x2) + r2
(3)
r2 ≤ u2(x2)
(4)
r2 ≤ u2(x2) − u2(x1) + r1
Profit maximization of the monopolist implies at
least one inequality from (1) and (2) is binding and at
least one inequality from (3) and (4) is binding
One can show that from our assumptions about the
utility functions, (1) and (4) are binding.
Monopolist’s profit will then be
π = [r1– c(x1)] + [r2– c(x2)]
= [u1(x1) – c(x1)] + [u2(x2) − u2(x1) + u1(x1) – c(x2)]
At the maximum, F.O.C.
u1′
( x1 ) - c′
( x1 ) + u1′
( x1 ) - u′
2 ( x1 ) = 0
u′
( x2 ) = 0
2 ( x 2 ) - c′
Low-demand consume at a point where MU>MC.
Recall that he is already charged the maximum amount
he is willing to pay
High-demand consumes at level where MU = MC.
If this were not true then the monopolist can increase
x2 by one unit and charges consumer 2 his marginal
utility from that extra consumption. Since MU > MC,
the monopolist will make extra profit. Note that (4) is
still satisfied with equality.
Diagrammatic Example
Third Degree Price Discrimination
Example: N separate markets, i = 1,2,….,N
N
N
i =0
i =0
Monopolist max ∑pi Di ( pi ) - C ( ∑Di ( pi ))
FOC. for each i :
Di ( pi ) + pi Di′
( pi ) = C′
( X ) Di′
( pi )
N
where
X =
∑Di ( pi ) is the total quantity consumed
i =0
(X = x1 + x2 +…..+ xN)
The monopolist equates
MR(x1) = MR(x2) =………= MR(xN) = MC
In term of elasticity,
pi - C′
(X )
- Di
-1
=
=
pi
pi Di′
( pi )
εi
εi is the elasticity of demand in market i,
charges more for market with less elastic demand