Monopoly Pricing Industrial Organization B · MONOPOLY PRICING Industrial Organization B THIBAUD...

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MONOPOLY P RICING Industrial Organization B THIBAUD VERGÉ Autorité de la Concurrence and CREST-LEI Master of Science in Economics - HEC Lausanne (2009-2010) THIBAUD VERGÉ (AdlC, CREST-LEI) Monopoly Pricing IO B (Ch.1) 1 / 26

Transcript of Monopoly Pricing Industrial Organization B · MONOPOLY PRICING Industrial Organization B THIBAUD...

Page 1: Monopoly Pricing Industrial Organization B · MONOPOLY PRICING Industrial Organization B THIBAUD VERGÉ Autorité de la Concurrence and CREST-LEI Master of Science in Economics -

MONOPOLY PRICINGIndustrial Organization B

THIBAUD VERGÉ

Autorité de la Concurrence and CREST-LEI

Master of Science in Economics - HEC Lausanne (2009-2010)

THIBAUD VERGÉ (AdlC, CREST-LEI) Monopoly Pricing IO B (Ch.1) 1 / 26

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Pricing Behavior A Single-Product Monopolist

Benchmark Case: Competitive Firm

Denote by p the price and by q the quantityThe firm’s profit then writes as:

π = pq − C (q)

Perfect CompetitionThe firm is then price-taker, that is, p is fixed.Chooses the quantity q so as to maximize its profitSupply is given by C′ (q) = pIn equilibrium the price is determined by: Supply = DemandThat is, the equilibrium quantity qC is such that P

(qC) = C′

(qC)

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Pricing Behavior A Single-Product Monopolist

Competitive Firm

Net Consumer Surplus

Profit

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Pricing Behavior A Single-Product Monopolist

Monopolist’s Profit Maximization

Link between price and quantityDemand function: q = D (p)

Inverse demand function: p = P (q)

Profit maximizing quantityThe monopolist’s profit writes as: π (q) = P (q) q︸ ︷︷ ︸

Revenue

−C (q)︸ ︷︷ ︸Cost

First-order condition (assuming concavity of the profit function):

P ′ (q) q + P (q)︸ ︷︷ ︸Marginal Revenue

= C′ (q)︸ ︷︷ ︸Marginal Cost

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Pricing Behavior A Single-Product Monopolist

Monopolist’s Profit Maximization

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Pricing Behavior A Single-Product Monopolist

Profit MaximizationThe linear demand case

Linear Demand, Constant Marginal CostAssume D (p) = a− p and C (q) = cqwith a > 0, c ≥ 0.

We thus have P (q) = a− qProfit writes as π (q) = q (a− q)− cq = (a− c − q) q . . .

. . . and is maximum for q = qM = a−c2

Monopoly profit is then equal to πM = (a−c)2

4

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Pricing Behavior Welfare Analysis

Welfare analysis - The deadweight loss

Two effects. . .

The monopoly price is higher than the competitive price(pM > pC)

Consumer surplus decreases

Profit increases(i.e. shareholders benefit)

No ambiguity overall: Loss > Gain

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Pricing Behavior Welfare Analysis

Welfare Analysis

Net Consumer Surplus

Profit

Welfare LossWelfare Loss

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Pricing Behavior Lerner Index

Profit Maximizing Price

Link between price and quantityDemand function: q = D (p)

Inverse demand function: p = P (q)

Profit Maximizing PriceThe monopolist’s profit writes as: pD (p)︸ ︷︷ ︸

Revenue

−C (D (p))︸ ︷︷ ︸Cost

First-order condition:

D (p) + pD′ (p)︸ ︷︷ ︸Marginal Revenue

= D′ (p) C′ (D (p))︸ ︷︷ ︸Marginal Cost

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Pricing Behavior Lerner Index

Lerner Index

Rewriting the first-order conditionD′ (p) [p − C′ (D (p))] = −D (p)

orp − C′ (D (p))

p︸ ︷︷ ︸Lerner Index

=−D (p)

pD′ (p)=

1ε︸ ︷︷ ︸

Inverse of the Price Elasticity of Demand

InterpretationThe higher the price elasticity of demand, the lower themonopolist’s mark-upLimit case: if ε −→ +∞, then pM = C′ = pC

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Pricing Behavior Multi-product Monopoly

A multi-product monopolist

Some NotationsThe monopolist sells n different goods index by i = 1,2, . . . ,n.It charges prices p = (p1,p2, . . . ,pn)

and sells quantities q = (q1,q2, . . . ,qn) where the demandfunctions are qi = Di (p).Cost of production is C (q).

Profit MaximizationThe monopolist’s profit is:

Π (p) =n∑

i=1

piDi (p)− C (D1 (p) , . . . ,Dn (p))

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Pricing Behavior Multi-product Monopoly

Dependent Demands, Separable Costs

First-order ConditionsIn the general case, the first-order conditions write as:

For any i = 1,2, . . . ,n :

(Di + pi

∂Di

∂pi

)+∑j 6=i

pj∂Dj

∂pi=

n∑j=1

∂C∂qj

∂Dj

∂pi

Separable CostsWe suppose here that C (q) =

∑ni=1 Ci (qi).

We can thus simplify the f.o.c:(Di +

(pi − C′i

) ∂Di

∂pi

)= −

∑j 6=i

(pj − C′j

) ∂Dj

∂pi

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Pricing Behavior Multi-product Monopoly

Dependent Demand, Separable Costs

First-order conditions

pi − C′ipi

=

1 +∑j 6=i

(pj − C′j

)Djεji

piDi

1εii

where:εii = − pi

Di

∂Di∂pi

is the own-price elasticity of demand; and

εji = piDj

∂Dj∂pi

is the cross-price elasticity of demand for good j withrespect to the price of good i .

Comparison with the single-product firm

If goods are substitutes i.e. εji > 0, then pi−C′i

pi> 1

εii.

If goods are complements i.e. εji < 0, then pi−C′i

pi< 1

εii.

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Pricing Behavior Multi-product Monopoly

Independent Demands, Dependent Costs

Learning by DoingA monopolist produces at dates t = 1 and t = 2.At date t , demand is qt = Dt (pt ).At date t = 1, total cost is C1 (q1).At date t = 2, total cost is C2 (q2,q1) with ∂C2

∂q1< 0.

Intertemporal profitThe monopolist thus maximizes its total profit:

p1D1 (p1)− C1 (D1 (p1)) + δ (p2D2 (p2)− C2 (D2 (p2) ,D1 (p1)))

where 0 ≤ δ ≤ 1 is the discount factor.

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Pricing Behavior Multi-product Monopoly

Learning by Doing

Optimal PricingAt date t = 2, the monopolist equalizes marginal revenue andmarginal cost, i.e.

p2 − C′2 (D2 (p2) , (p1))

p2=

1ε2

At date t = 1:

p1 − C′1 − δ∂C2∂q1

p1=

1ε1

=⇒p1 − C′1

p1<

1ε1

RemarkThe firm would underproduce (at both dates) if it were run by twoconsecutive managers each maximizing the short-term profit.

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Durable Good Introduction

Durable goods

60% of all production

QuestionsDurability of the good (endogenous: chosen by the producer).Inter-temporal competition: new vs. used good.Asymmetric information about quality.See Waldman, Michael (2003), “Durable Goods Theory of RealWorld Markets”, Journal of Economic Perspectives, Vol. 17(1),pp. 131-154.

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Durable Good Introduction

Selling over several periodsThe monopoly competes against itself

Durable goodIf bought today, the good can be consumed (i.e. creates utility) todaybut also tomorrow.

Main problemDenote by pt the price in period t = 1,2Consumers who have already bought at date t = 1 do not buyagain at date t = 2To attract new consumers at date t = 2 . . .. . . the monopolist must lower its price: p2 < p1

But if consumers anticipate this, are they still willing to buy atdate t = 1?

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Durable Good Introduction

A simple two-period model

ConsumersEach Consumer buys one unit of the good or nothingThe consumer’s (inter-temporal) net utility is:

u =

(1 + δ) v − p1 if (s)he buys in the first period,δ (v − p2) if (s)he buys in the second period,0 if (s)he does not buy.

The “valuation” v is uniformly distributed over [0,1]

δ is the discount factor

The monopolistConstant marginal cost, normalized to 0.

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Durable Good Commitment

Commitment not to change the price

Assuming p2 = p1 = pIf the price is not changed (lowered) in period 2, nobody buys atthat stage.If consumer v buys the product (at t = 1), any consumer with ahigher reservation price (v ′ > v) also buys.The indifferent consumer is v (p1) = p1

1+δ .

The demand is thus: 1− p11+δ

and the monopolist’s profit is: p1

(1− p1

1+δ

)Optimum

The optimal price is therefore: p1 = 1+δ2

And the profit is Π = 1+δ4 .

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Durable Good No commitment

Without commitment

For a given p1, find the optimal p∗2 (p1)

If v buys at t = 1, then v ′ > v does tooDenote by v1 (p1) the indifferent consumerIf v > v1 (p1), then v buys at t = 1If v < v1 (p1), then v doesn’t buy at t = 1Therefore D2 (p2; p1) = max

[v1 (p1)− p2,0

]The second period profit is then: p2

(v1 (p1)− p2

)The second period optimal price is thus

p∗2 (p1) =12

v1 (p1)

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Durable Good No commitment

First period optimal price

Determining v1

(1 + δ) v1 − p1︸ ︷︷ ︸utility if purchase at t=1

= δ(v1 − p∗2 (p1)

)︸ ︷︷ ︸utility if purchase at t=2

thereforev1 (p1) =

11 + δ

2

p1

THIBAUD VERGÉ (AdlC, CREST-LEI) Monopoly Pricing IO B (Ch.1) 21 / 26

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Durable Good No commitment

First period optimal price (II)

The total profit writes as

Π1+2 (p1) = p1(1− v1 (p1)

)︸ ︷︷ ︸First period profit

4v1 (p1)2︸ ︷︷ ︸

Second period profit

that is

Π1+2 (p1) = p1

(1 +

(− 2

2 + δ+

δ

(2 + δ)2

)p1

)= p1

(1− 4 + δ

(2 + δ)2 p1

)

Optimal price

p∗1 =(2 + δ)2

2 (4 + δ)

THIBAUD VERGÉ (AdlC, CREST-LEI) Monopoly Pricing IO B (Ch.1) 22 / 26

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Durable Good Comparison

Summary and comparison

Without commitment

p∗1 =(2 + δ)2

2 (4 + δ), p∗2 =

12

2 + δ

4 + δ< p∗1 and Π∗ =

14

(2 + δ)2

4 + δ

Comparison

Π∗ < Π and p∗2 < p∗1 < p

THIBAUD VERGÉ (AdlC, CREST-LEI) Monopoly Pricing IO B (Ch.1) 23 / 26

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Durable Good Comparison

Example: δ = 1

Buy at T=1

(at price 0.9)

Buy at T=2

(at price 0.3)Do not buy.

Do not buy.Buy at T=1

(at price 1)

Without commitmentWithout commitment

Commitment Commitment not to not to lower pricelower price

Profit = 0.9 x 0.4 + 0.3 x 0.3 = 0.45

Profit = 1 x 0.5 = 0.5 > 0.45

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Durable Good Coase conjecture

Coase Conjecture

The monopolist competes against itselfTwo variables: the number of periods and the discount factorIf the number of periods is infinite and if δ −→ 1Then: Π∗ −→ 0 !That is: p∗1 = 0 (otherwise nobody buys - waiting is costless)

Evading the Coase ProblemCommitting itself to a sequence of prices (credibility? reputation?)LeasingMoney-back guarantee (“most-favored customer” clause)New consumersPlanned obsolescence (new versions, upgrades, . . . )

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Durable Good Coase conjecture

Leasing rather than selling

LeasingOne price per periodThe consumer doesn’t own the productThe good is durable for the monopolist onlyDenote by pt the price for period tConsumer leases if pt ≤ v , demand is thus: Dt (pt ) = 1− pt

Profit for period t is thus Πt (pt ) = pt (1− pt )

Optimal prices: p1 = p2 = 1/2Total profit: ΠL = 1

4 + δ 14 = 1+δ

4 = Π

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