Chapter 3: Price Discrimination
A1. The goods produced by the monopolist are given.
Relax A2. Price discrimination (PD).
PD if 2 units of the same physical good are sold at different
prices to the same or to different consumers.
Examples: discount, airline tickets...
• PD is linked to the possibility of arbitrage
– transferability of the commodity → reduces PD
– transferability of demand → increases PD
3 kinds of PD (Pigou (1920))
1. First-degree PD - Perfect PD;
– need to have all information
2. Second-degree PD
– self-selecting device
3. Third-degree PD
– signal (age, occupation,...)
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1 First Degree Price Discrimination
• Discrete case
– Identical consumers
– unit demand
– v : consumer’s willingness to pay.
– A monopolist charges p = v in order to extract the
entire surplus.
• Continuous case
– n identical consumers
– same demand function qi = D(p)
n
– Total demand: q = D(p)
– Linear pricing schedule T (q) = pq gives a profit
pmD(pm) − C(D(pm))
– The monopolist can increase his profit if he proposes
a two-part tariff:
T (q) = A + pq.
– The monopolist sets the competitive price pc.
– The surplus of the consumers is
Z qc
Sc =
(p(q) − pc)dq
0
2
– And each consumer must pay a fixed premium
Sc
A=
n
– The monopolist proposes an affine (nonlinear)
pricing schedule (two
( part tariff):
c
pcq + Sn if q > 0
T (q) =
0
if q = 0
– The profit of the monopolist is
Π = S c + pcq − C(q c)
• Non identical consumers: they have different demand
curves.
– The optimal pricing scheme is p = MC
– and each consumer pays a personalized fixed
premium Sic(pc).
• BUT problem of information.
• and of arbitrage....
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2 Third Degree PD
•
•
•
•
•
•
•
Single product, total cost C(q)
m groups of consumers (age, sex, occupation,...).
Each group has a different demand function.
Monopolist knows these demand functions.
No arbitrage between groups, but no PD within a group.
Linear tariff: {p1, ..., pi, ..., pm}
Quantities demanded:
{q1 = D1(p1), ..., qi = Di(pi), ..., qm = Dm(pm)}
Pm
• Aggregate demand is q = i=1 Di(pi)
• Monopolist chooses prices that maximize
m
m
X
X
Di(pi)pi − C(
Di(pi))
i=1
i=1
• Remember: multiproduct pricing with independent
demands and separable costs!
pi − C 0(.)
1
⇒ $i ≡
=
pi
εi
4
Result 1 Optimal pricing implies that the monopolist
should charge more in markets with the lower elasticity.
Example: Students vs non-students...
• What can we say in terms of welfare? Does PD increase
or decrease welfare?
• What would happen if the monopolist were forced to
charge the same price? Uniform price (p)
Result 2 Consumers with high elastic demand (resp. low)
prefer discrimination (resp. uniform pricing).
• Marginal cost: c
• Under PD:
– pi in market i, Di(pi) = qi
– Aggregate CS and profit are
m
X
CSD =
Si(pi)
Πm
D =
i=1
m
X
i=1
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(pi − c)qi
• PD is prohibited
– uniform price p, and thus Di(p) = q i
– Aggregate CS and profit are
m
X
CSNo =
Si(p)
Πm
No =
i=1
m
X
i=1
(p − c)qi
• Difference in total welfare
m
∆W = CSD − CSNo + Πm
D − ΠNo
m
m
m
X
X
X
∆W =
(Si(pi) − Si(p)) +
(pi − c)qi −
(p − c)q i
i=1
i=1
• Thus
– if ∆W > 0, welfare is higher under PD,
– if ∆W < 0, welfare is lower under PD.
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i=1
• Upper and lower bounds for ∆W (to show)
m
X
∆W ≥
(pi − c)(qi − q i)
(1)
i=1
m
X
∆W ≤ (p − c)
(qi − q i)
(2)
i=1
• PD reduces welfare if it does
Pmnot increase
Pm total output
– from equation (2), if i=1 qi =
i=1 q i then
∆W ≤ 0.
• To be preferred socially, PD must raise total output.
• Case of linear demand functions:
qi = ai − bip for i
• Assume that ai > bic for any i.
• Under PD
– Monopolist chooses pi that solves
Max(pi − c)(ai − bipi)
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and thus
ai + cbi
pi =
2bi
ai − cbi
qi =
2
– Thus the sum of the output is
m
m
X
X
ai − cbi
qi =
2
i=1
i=1
• Under uniform pricing
– All markets are served at the optimum.
– Monopolist chooses p that solves
m
m
X
X
Max(p − c)(
ai − p
bi)
i=1
and thus
p =
m
X
i=1
qi =
i=1
Pm
Pm
i+c
i=1 aP
i=1 bi
2 m
i=1 bi
Pm
i=1 ai
8
−c
2
Pm
i=1 bi
• And thus, same total output
m
m
m
X
X
X
qi =
qi ⇒
(qi − qi) = 0
i=1
i=1
i=1
and according to equation (2) ∆W ≤ 0, welfare is
lower under PD.
• This outcome depends on the assumption: all markets
are served under uniform pricing (+ linear demand).
• Welfare conclusion can be reversed....
– 2 markets, i.e. m = 2
– under uniform pricing, second market is not served.
m m m
– PD: pm
1 , p2 , q1 , q2 .
m
– Uniform pricing: p = pm
1 , q 1 = q1 , q 2 = 0, and thus
m
X
(qi − q i) = q1m − q1m + q2m − 0 = q2m > 0
i=1
m
X
m
m
(pi − c)(qi − q i) = (pm
1 − c) × 0 + (p2 − c)q2 > 0
i=1
and from equation (1) ∆W ≥ 0.
– In this case: PD increases welfare.
– PD leads to Pareto improvement: monopolist makes
more profit, and CS increases in market 2.
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• Ambiguous effects of PD on welfare. Trade-off
between
– loss of consumers in low-elasticity markets
– gain of consumers in high-elasticity markets.
• If PD is prohibited: it can conduct to closure of
market....
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3 Second Degree PD
•
•
•
•
Heterogeneous consumers
Monopolist knows they are heterogeneous.
Monopolist will offer a menu of bundles to choose from.
Personal arbitrage can happen: self selection or
incentive constraints.
3.1 Two Part Tariffs
• T (q) = A + pq .
• Menu of bundles {T, q}
• Consumers’
preferences are
(
θV (q) − T if they pay T and consume q units
u=
0
otherwise
where
– V (0) = 0, V 0(q) > 0, V 00(q) < 0.
– θ taste parameter.
• 2 groups of consumers:
– a proportion λ of consumers with parameter taste θ1;
– a proportion (1 − λ) with parameter taste θ2.
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• Marginal cost c,
• Assumption:
– θ2 > θ1 > c,
1−(1−q)2
and thus V 0(q) = (1 − q) > 0
– V (q) =
2
• What is the demand function of a consumer θi facing a
price p?
– Consumer solves
Max{θiV (q) − pq}
q
– Thus demand function is
p
Di(p) = 1 −
θi
– Aggregate demand function
D(p) = λD1(p) + (1 − λ)D2(p)
D(p) = 1 − p(
where 1θ =
λ
θ1
p
λ 1−λ
+
)=1−
θ1
θ2
θ
+ 1−λ
θ2 is the “harmonic mean”.
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– Net consumers surplus is
Si(p) = θiV (Di(p)) − pDi(p)
(θi − p)2
Si(p) =
2θi
3.1.1 Perfect Price Discrimination
• If the monopolist can observe θi
• He charges
p1 = c
per unit plus a fixed premium
(θi − p1)2
Ai = Si(c) =
2θi
i
where ∂A
∂θi > 0, and thus A2 > A1 .
• Profit of the monopolist is
(θ1 − p1)2
(θ2 − p1)2
Π1 = λ
+ (1 − λ)
2θ1
2θ2
• If the monopolist cannot observe θi: arbitrage problem.
High demand consumers have an incentive to claim they
are low demand type.
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3.1.2 Monopoly Pricing
• Monopolist charges a fully linear tariff T (q) = pq
• Assumption: monopolist serves the two types of
2
consumers (Formally c+θ
2 ≤ θ 1 and λ not too small).
• Monopolist chooses p that solves
Max(p − c)D(p)
p
and thus
c+θ
p2 =
2
and the monopolist profit is
(θ − c)2
Π2 =
4θ
3.1.3 Two-part tariff
• Assumption: the monopolist serves the two-types of
consumers.
• To make them buy: A = S1(p)
• Monopolist chooses p that solves
Max{S1(p) + (p − c)D(p)}
p
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and thus the price is
p3 =
c
2 − θθ1
and the profit is
Π3 = S1(p3) + (p3 − c)D(p3)
3.1.4 Comparison
• Profits
Π1 ≥ Π3 ≥ Π2
– Maximum profit under perfect PD
– Monopolist can always duplicate a linear tariff with a
two-part tariff.
• Prices
p1 = c < p3 < p2 = pm
– A lower price induces less monopoly profit but higher
fixed part.
• Welfare
W (p3) > W (p2)
as W (p) is decreasing with p.
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W (p) = λ[S1g (p) − cD1(p)] + (1 − λ)[S2g (p) − cD2(p)]
Sig (p) = Si(p) + pDi(p)
Sig (p) = θiV (Di(p)) − pDi(p) + pDi(p)
Sig (p) = θiV (Di(p))
Thus g
∂[S1 (p) − cD1(p)]
= θiV 0(p)Di0 (p) − cDi0 (p)
∂p
c
p
= − + ≤0
θi θi
– Because p3 < p2, consumers under two part tariff
consume more (reduces distortion)
– And Π3 ≥ Π2
• Graph
• Using a graph, show that for any linear tariff T (q) = pq
e
with p > c, there exists a two-part tariff Te(q) = peq + A
such that if consumers are offered the choice between T
and Te, both types of consumers and the firm are made
better off (exercise 3.4).
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3.2 Non linear Tariffs
• If commodity arbitrage can be prevented, monopolist
can increase his profit with a more complex scheme.
• See graph
• T (q) = A + pq .
• Indifference curves for consumers: T = θV (q) − u
– concave
– θ2 > θ1, IC for θ2 is steeper than IC for θ1 when
curves cross (Spence-Mirrlees condition)
• Indifference curves for monopolist: T = cq + Π
– steeper than two-part tariff as p > c.
• Low (resp. high) demand consumers derive no (resp.
positive) net surplus.
• The binding personal arbitrage constraint is to prevent
high demand consumers from buying low demand
consumers’ bundle.
• High (resp. low) demand consumers purchase the
socially optimal quantity, q2 = D2(c), (resp. suboptimal
quantity q1 < D1(c)).
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4 Conclusion
• Here, study of PD in monopolistic situation
– often it takes place in oligopolistic markets.
• In second-degree PD, monopolist discriminates along a
single dimension (quality or quantity).
– Usually consumers can choose both quality and
quantity.
• In second-degree PD, consumers’ demands are independent.
– However they can be dependent.
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