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Lecture Notes: Microeoconomics
Joe Chen
132
Price Discrimination
Generally speaking, price discrimination involves selling different units of the same good at
different prices, either to the same or to different consumers. A monopoly, if possible, has
every incentive to price discriminate. The usual graph of the monopoly’s pricing decision
explains the intuition. Note that at the profit maximization output level p(xm ) = AR >
c0 (xm ), so if a monopoly can find a way to sell additional output without lowering the price
on the units that it is currently selling, it can then increase its profit.
According to Pigou, there are three kinds of price discriminations:
• First-degree (or Perfect) price discrimination: charging the maximum willingnessto-pay for each unit of the product bought;
• Second-degree price discrimination (Nonlinear Pricing): charging different
prices to different number of units of the product bought; e.g., quantity discount or
premia;
• Third-degree price discrimination: charging different prices to different purchasers; e.g., student or youth discounts.
In the case of 2nd◦ price discrimination, prices are the same for consumers who demand
the same number of units of the product. The producer sets up a price schedule for
different units of the product, consumers then “selfselect” how many units to buy. In the
case of 3rd◦ price discrimination, prices are the same for consumers of the same group, no
matter how many units they bought.
In order for price discrimination to be a viable strategy of a monopolistic firm, it must
have the ability to sort consumers and to prevent resale. In particular, it is difficult to sort
consumers. In the 3rd◦ price discrimination, a firm relies on some exogenous signal, e.g.,
age; in the 2nd◦ price discrimination, a firm relies on some endogenous category; e.g., the
amount of purchase.
Lecture Notes: Microeoconomics
Joe Chen
133
Let’s set up a framework for our study of price discrimination. For the production side,
there is a monopolist that produces at constant marginal cost so that the cost function is
c(x) = cx. Let there be two consumers with utility function: ui (xi ) + yi , i = 1, 2. Let
ui (0) = 0. So consumer’s (maximum) willingness to pay for x units of the first good (given
the quantity of the second good fixed) is: for all i = 1, 2,
ri (x) = ui (x) + yi − [ui (0) + yi ]
= ui (x).
Without loss of generality, let’s assume:
u2 (x) > u1 (x) ∀ x.
The marginal willingness to pay is given by the FOC of the consumer’s utility maximization
problem:
p = u0i (x), ∀ i = 1, 2.
Let’s also assume that the marginal willingness to pay for the second consumer is higher
than that of the first consumer; i.e.,
u02 (x) > u01 (x) ∀ x.
This condition is known as the single-crossing condition because the indifference curves
of these two consumers can cross at most once.
Given the assumptions, it is natural that we call the first consumer the low-demand
consumer and the second consumer the high-demand consumer.
Lecture Notes: Microeoconomics
Joe Chen
134
First-degree Price Discrimination
A monopolist can adopt a take-it-or-leave-it pricing–each consumer i can either pay ri and
consumer xi , or consume zero units of the first good. Put it differently, the monopolist
simply offer a package (ri , xi ) for each consumer. How should this package be determined?
The monopolist’s problem is: ∀ i = 1, 2,
max ri − cxi
ri ,xi
s.t. ui (xi ) ≥ ri .
The constraint is called the participation (or individual rationality; PC or IR) constraint. It holds with equality in optimal, hence we can re-write the problem as:
max ui (xi ) − cxi .
The FOC requires:
u0i (xi ) = c.
We can then solve the FOC for x∗i , then ri∗ = ui (x∗i ).
First, observe that (ri∗ , x∗i ) is 1st◦ price discrimination since the monopolist gets everything.
Second, x∗i is Pareto efficient! It is the same level of output if the market is a competitive
one!!! The only difference is who-gets-what.
Finally, the monopolist can achieve the same outcome without using the take-it-or-leaveit strategy. It can also sell each unit of output to the consumer at a different price. To dig
deeper, see the textbook for a mathematical construction.
Lecture Notes: Microeoconomics
Joe Chen
135
Second-degree Price Discrimination
In the 2nd◦ price discrimination, the monopolist chooses a price schedule–a function t(x)–
that indicates how much it charges when x units are demanded. Suppose consumer i
demands xi units and spends ri = t(xi )xi dollars. From the view points of both consumers
and the monopolist all that is relevant is that the consumer spends ri dollars and receives xi
units of output. Hence, the choice of the function t(x) is equivalent to the choice of (ri , xi )
for i = 1, 2.
(A graphical treatment)
As before, we have the following participation constraints:
u1 (x1 ) − r1 ≥ 0
(PC_L)
u2 (x2 ) − r2 ≥ 0.
(PC_H)
In addition, we also have the following incentive compatibility (selfselection, personal
arbitrage, or simply incentive) constraints:
u1 (x1 ) − r1 ≥ u1 (x2 ) − r2
(ICC_L)
u2 (x2 ) − r2 ≥ u2 (x1 ) − r1 .
(ICC_H)
The first intuition is that the monopolist can extract as much as possible from the lowdemand consumer. This suggests the participation constraint for the low-demand consumer
holds with equality (PC_L is binding):
r1 = u1 (x1 ).
This implies that the monopolist charges the low-demand consumer with his total (maximum) willingness to pay.
The second intuition is that the monopolist cannot ask too much from the high-demand
consumer, otherwise she will choose (r1 , x1 ) which is designed for the low-demand consumer.
This suggests the incentive compatibility constraint for the high-demand consumer holds
with equality (ICC_H is binding):
u2 (x2 ) − r2 = u2 (x1 ) − r1 ⇔ r2 = u2 (x2 ) − u2 (x1 ) + r1 .
Lecture Notes: Microeoconomics
Joe Chen
136
This implies that the monopolist charges the high-demand consumer with the highest price
just induce her to choose (r2 , x2 ) rather than (r1 , x2 ).
The rest of the problem is to determine x1 and x2 . The monopoly’s problem is now,
max r1 − cx1 + r2 − cx2
x1 ,x2
= max u1 (x1 ) − cx1 + u2 (x2 ) − u2 (x1 ) + u1 (x1 ) − cx2 .
x1 ,x2
The FOC requires:
u01 (x1 ) − c − u02 (x1 ) + u01 (x1 ) = 0
u02 (x2 ) − c = 0.
Rearrange, we have:
£
¤
u01 (x1 ) = c + u02 (x1 ) − u01 (x1 ) > c
u02 (x2 ) − c = 0.
So at optimal nonlinear pricing, the low-demand consumer receives the quantity x∗1 such that
the marginal value of the good exceeds marginal cost. Hence he consumers an inefficiently
small amount of the the good. The high-demand consumer receives the quantity x∗2 such
that the marginal value of the good equals marginal cost. Hence she consumes the socially
correct amount of the good.
We conclude:
• The low-demand consumer derives no net surplus, while the high-demand consumer
derives some positive net surplus;
• The binding ICC_H is to prevent the high-demand consumer from buying the lowdemand consumer’s bundle;
• The high-demand consumer purchases the socially optimal quantity, while the lowdemand consumers purchases a suboptimal quantity.
These results hold when there are more than two types.
Lecture Notes: Microeoconomics
Joe Chen
137
Claim 1 PC_L is binding and PC_H is not.
Proof. First, notice that it cannot be the case that both PCs are not binding. Suppose not,
the monopolist can increase r1 and r2 by some amount to increase profits without violating
the PCs. This contradicts the profit maximization assumption. Second, it cannot be the
case that both PCs are binding. Suppose not, ICC_H becomes:
0 ≥ u2 (x1 ) − u1 (x1 ).
A contradiction (since we require u2 (x) > u1 (x) ∀ x). Hence, it must be the case that one
of the two PCs is binding. Finally, we need to show that PC_L is binding while PC_H is
not. Suppose not, r2 = u2 (x2 ) and r1 < u1 (x1 ). Then, ICC_H requires:
r1 ≥ u2 (x1 ).
But u2 (x1 ) > u1 (x1 ). This contradicts r1 < u1 (x1 ).
Claim 2 ICC_H is binding and ICC_L is not.
Proof. First, notice that it cannot be the case that both ICCs are not binding. Suppose not,
the monopolist can increase r2 by some small amount to increase profits without violating
any of the constraints. This contradicts profit maximization assumption. Second, it cannot
be the case that both ICCs are binding. Suppose not, together with the fact that PC_L is
binding, r2 = u1 (x2 ), ICC_H becomes:
u2 (x2 ) − u2 (x1 ) = u1 (x2 ) − u1 (x1 ).
This is equivalent to:
Z
x2
x1
u02 (x)dx
=
Z
x2
x1
u01 (x)dx.
This contradicts u02 (x) > u01 (x) ∀ x. Hence, it must be the case that one of the two ICCs
is binding. Finally, we need to show that ICC_H is binding while ICC_L is not. Suppose
not, then again, without violating any constraints, the monopolist can increase r2 by some
small amount to increase profits. This contradicts the profit maximization assumption.
Lecture Notes: Microeoconomics
Joe Chen
138
Third-degree Price Discrimination
In the 3rd◦ price discrimination, the monopolist’s problem is:
max p1 (x1 )x1 + p2 (x2 )x2 − c(x1 + x2 ).
x1 ,x2
The FOCs gives us the following:
µ
¶
1
p1 (x1 ) 1 −
=c
| 1|
¶
µ
1
= c.
p2 (x2 ) 1 −
| 2|
Hence, p1 (x1 ) > p2 (x2 ) iff | 1 | < | 2 |.
Since the 3rd◦ price discrimination is very common, let’s take a look at the welfare
implication. Let the “aggregate” utility function be: u(x1 , x2 ) + y, where u(·) is concave.
Let c(x1 , x2 ) be the cost of providing x1 and x2 . The social welfare can be measured by:
W (x1 , x2 ) = u(x1 , x2 ) − c(x1 , x2 ).
Now consider two configurations of output, (x01 , x02 ) and (x01 , x02 ) with prices (p01 , p02 ) and
(p01 , p02 ). Since u(·) is concave, we have:
∂u(x01 , x02 ) 0
∂u(x01 , x02 ) 0
(x1 − x01 ) +
(x2 − x02 )
∂x1
∂x2
∂u(x01 , x02 ) 0
∂u(x01 , x02 ) 0
u(x01 , x02 ) ≤ u(x01 , x02 ) +
(x1 − x01 ) +
(x2 − x02 ),
∂x1
∂x2
u(x01 , x02 ) ≤ u(x01 , x02 ) +
or,
∂u(x01 , x02 ) 0
∂u(x01 , x02 ) 0
(x1 − x01 ) +
(x2 − x02 )
∂x1
∂x2
∂u(x01 , x02 ) 0
∂u(x01 , x02 ) 0
u(x01 , x02 ) − u(x01 , x02 ) ≥
(x1 − x01 ) +
(x2 − x02 ).
∂x1
∂x2
u(x01 , x02 ) − u(x01 , x02 ) ≤
Hence, (by the FOC of the consumers’ utility maximization problems)
p01 4x1 + p02 4x2 ≤ 4u ≤ p01 4x1 + p02 4x2 .
Since, 4W = 4u − 4c,
p01 4x1 + p02 4x2 − 4c ≤ 4W ≤ p01 4x1 + p02 4x2 − 4c.
Lecture Notes: Microeoconomics
Joe Chen
139
Now let 4c = c4x1 + c4x2 , we have:
(p01 − c)4x1 + (p02 − c)4x2 ≤ 4W ≤ (p01 − c)4x1 + (p02 − c)4x2 .
Suppose that the initial pair of prices are the single monopoly price so that p01 = p02 = pm ,
and p01 and p02 are the discriminatory prices. Then,
(p01 − c)4x1 + (p02 − c)4x2 ≤ 4W ≤ (pm − c)(4x1 + 4x2 ).
Hence, when 4x1 +4x2 ≤ 0 (the change in outputs due to the change in pricing), 4W ≤ 0.
For the social welfare to increase, we need the sum of the weighted change in outputs to be
positive.