Chapter 3: Price Discrimination A1. The goods produced by the - - PDF document

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,...) 1

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

Sc = Z qc (p(q) − pc)dq

2

– And each consumer must pay a fixed premium

A = Sc n

– The monopolist proposes an affine (nonlinear) pricing schedule (two part tariff):

T(q) = ( pcq + Sc

n if q > 0

if q = 0 – The profit of the monopolist is

Π = Sc + pcq − C(qc)

• Non identical consumers: they have different demand

curves. – The optimal pricing scheme is p = MC – and each consumer pays a personalized fixed premium Sc

i(pc).

• BUT problem of information.
• and of arbitrage....

3

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)}

• Aggregate demand is q = Pm

i=1 Di(pi)

• Monopolist chooses prices that maximize

m

X

i=1

Di(pi)pi − C(

m

X

i=1

Di(pi))

• Remember: multiproduct pricing with independent

demands and separable costs!

⇒ $i ≡ pi − C0(.) pi = 1 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
• r 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

CSD =

m

X

i=1

Si(pi) Πm

D = m

X

i=1

(pi − c)qi

5

• PD is prohibited

– uniform price p, and thus Di(p) = qi – Aggregate CS and profit are

CSNo =

m

X

i=1

Si(p) Πm

No = m

X

i=1

(p − c)qi

• Difference in total welfare

∆W = CSD − CSNo + Πm

D − Πm No

∆W =

m

X

i=1

(Si(pi) − Si(p)) +

m

X

i=1

(pi − c)qi −

m

X

i=1

(p − c)qi

• Thus

– if ∆W > 0, welfare is higher under PD, – if ∆W < 0, welfare is lower under PD. 6

• Upper and lower bounds for ∆W (to show)

∆W ≥

m

X

i=1

(pi − c)(qi − qi)

(1)

∆W ≤ (p − c)

m

X

i=1

(qi − qi)

(2)

• PD reduces welfare if it does not increase total output

– from equation (2), if Pm

i=1 qi = Pm i=1 qi 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)

7

and thus

pi = ai + cbi 2bi qi = ai − cbi 2

– Thus the sum of the output is

m

X

i=1

qi =

m

X

i=1

ai − cbi 2

• Under uniform pricing

– All markets are served at the optimum. – Monopolist chooses p that solves

Max(p − c)(

m

X

i=1

ai − p

m

X

i=1

bi)

and thus

p = Pm

i=1 ai + c Pm i=1 bi

2 Pm

i=1 bi m

X

i=1

qi = Pm

i=1 ai − c Pm i=1 bi

2

8

• And thus, same total output

m

X

i=1

qi =

m

X

i=1

qi ⇒

m

X

i=1

(qi − qi) = 0

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. – PD: pm

1 , pm 2 , qm 1 , qm 2 .

– Uniform pricing: p = pm

1 , q1 = qm 1 , q2 = 0, and thus m

X

i=1

(qi − qi) = qm

1 − qm 1 + qm 2 − 0 = qm 2 > 0 m

X

i=1

(pi − c)(qi − qi) = (pm

1 − c) × 0 + (pm 2 − c)qm 2 > 0

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. 9

• 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.... 10

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

u = ( θV (q) − T if they pay T and consume q units

• therwise

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. 11

• Marginal cost c,
• Assumption:

– θ2 > θ1 > c, – V (q) = 1−(1−q)2

2

and thus V 0(q) = (1 − q) > 0

• What is the demand function of a consumer θi facing a

price p? – Consumer solves

Max

q

{θiV (q) − pq}

– Thus demand function is

Di(p) = 1 − p θi

– Aggregate demand function

D(p) = λD1(p) + (1 − λ)D2(p) D(p) = 1 − p( λ θ1 + 1 − λ θ2 ) = 1 − p θ

where 1

θ = λ θ1 + 1−λ θ2 is the “harmonic mean”.

12

– Net consumers surplus is

Si(p) = θiV (Di(p)) − pDi(p) Si(p) = (θi − p)2 2θi

3.1.1 Perfect Price Discrimination

• If the monopolist can observe θi
• He charges

p1 = c

per unit plus a fixed premium

Ai = Si(c) = (θi − p1)2 2θi

where ∂Ai

∂θi > 0, and thus A2 > A1.

• Profit of the monopolist is

Π1 = λ(θ1 − p1)2 2θ1 + (1 − λ)(θ2 − p1)2 2θ2

• If the monopolist cannot observe θi: arbitrage problem.

High demand consumers have an incentive to claim they are low demand type. 13

3.1.2 Monopoly Pricing

• Monopolist charges a fully linear tariff T(q) = pq
• Assumption: monopolist serves the two types of

consumers (Formally c+θ2

2

≤ θ1 and λ not too small).

• Monopolist chooses p that solves

Max

p

(p − c)D(p)

and thus

p2 = c + θ 2

and the monopolist profit is

Π2 = (θ − c)2 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

p

{S1(p) + (p − c)D(p)}

14

and thus the price is

p3 = c 2 − θ

θ1

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. 15

W(p) = λ[Sg

1(p) − cD1(p)] + (1 − λ)[Sg 2(p) − cD2(p)]

Sg

i (p) = Si(p) + pDi(p)

Sg

i (p) = θiV (Di(p)) − pDi(p) + pDi(p)

Sg

i (p) = θiV (Di(p))

Thus

∂[Sg

1(p) − cD1(p)]

∂p = θiV 0(p)D0

i(p) − cD0 i(p)

= − p θi + c θi ≤ 0

– 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

with p > c, there exists a two-part tariff e

T(q) = e pq + e A

such that if consumers are offered the choice between T and e

T, both types of consumers and the firm are made

better off (exercise 3.4). 16

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)). 17

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 indepen-

dent. – However they can be dependent. 18