Lecture 8: Price Discrimination

Mauricio Romero

Introduction

First Degree Price Discrimination

Two-part tariff

Two-part tariff vs 1st degree price discrimination

Third Degree Price Discrimination

Monopsony

Double Marginalization Problem

Profit Sharing and Double Marginalization

Introduction

Two-part tariff

Monopsony

I In real life, firms often have different prices for differentconsumers/units

I We will explore some of these now

I In a competitive market such exotic pricing schemes couldnever arise since p = marginal cost

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I Suppose the firm can observe all characteristics of theconsumer

I What should the firm do?

I Demand curve illustrates the willingness to pay for the q-thunit of the product

I Firm can extract all of the surplus of the consumer. How?

I Firm will price at p(q) for the q-th unit and continue toproduce until p(q) = MC (q)

I Firm gets all of the consumer surplus as his profits:

q∗∫0

(p(q)− c ′(q))dq =

q∗∫0

p(q)dq − c(q∗),

where q∗ is the quantity at which p(q∗) = c ′(q∗).

I Firm will price at p(q) for the q-th unit and continue toproduce until p(q) = MC (q)

I Firm gets all of the consumer surplus as his profits:

q∗∫0

(p(q)− c ′(q))dq =

q∗∫0

p(q)dq − c(q∗),

where q∗ is the quantity at which p(q∗) = c ′(q∗).

Profit

0 Quantity

DemandMarginal revenue

Marginal cost

Monopoly

Quantity sold

Consumer surplusDeadweight loss

(a) Monopolist with Single Price

Profit

0 Quantity

Demand

Marginal cost

(b) Monopolist with Perfect Price Discrimination

Quantity sold

I Firm can do this is because it knows the exact demand curveof each consumer

I Such activity is prohibited in many countries

I Amazon tries to estimate everyone’s demand curve

I Firm can do this is because it knows the exact demand curveof each consumer

I Such activity is prohibited in many countries

I Amazon tries to estimate everyone’s demand curve

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I Suppose that a bar has a monopoly in a community

I Each drink costs c dollars to provide

I Consumers have diminishing marginal returns on the alcoholconsumed

This bar would produce q at price p(q) such that

p′(q)q + p(q) = c

if it were only able to charge one price

I Many bars have a cover charge (an entry fee)

I Does this increase profits?

I Two quantities (f , q∗) where f is the entry fee and q is thedrinks sold

I How much are consumers willing to pay to enter the bar whenthere are q∗ units of drinks being served:

q∗∫0

(p(q)− p(q∗))dq.

I As long as

f ≤q∗∫

(p(q)− p(q∗))dq,

then all consumers will come to the barI For a fix q∗, the monopolist will always charge an entry fee of

q∗∫0

(p(q)− p(q∗))dq.

q∗∫0

(p(q)− p(q∗))dq.

I As long as

f ≤q∗∫

(p(q)− p(q∗))dq,

q∗∫0

(p(q)− p(q∗))dq.

q∗∫0

(p(q)− p(q∗))dq.

I As long as

f ≤q∗∫

(p(q)− p(q∗))dq,

then all consumers will come to the bar

I For a fix q∗, the monopolist will always charge an entry fee of

q∗∫0

(p(q)− p(q∗))dq.

q∗∫0

(p(q)− p(q∗))dq.

I As long as

f ≤q∗∫

(p(q)− p(q∗))dq,

q∗∫0

(p(q)− p(q∗))dq.

I What is then the profit maximizing price and quantity giventhis entry fee?

maxq∗

q∗∫0

(p(q)−p(q∗))dq+p(q∗)q∗−cq∗ = maxq∗

q∗∫0

(p(q)−c)dq.

I The first order condition is:

p(q)− c = 0

I Thenp(q∗) = c.

I The entry fee is:p−1(c)∫

(p(q)− c)dq

maxq∗

q∗∫0

(p(q)−c)dq.

p(q)− c = 0

I Thenp(q∗) = c.

(p(q)− c)dq

maxq∗

q∗∫0

(p(q)−c)dq.

p(q)− c = 0

I Thenp(q∗) = c.

(p(q)− c)dq

maxq∗

q∗∫0

(p(q)−c)dq.

p(q)− c = 0

I Thenp(q∗) = c.

(p(q)− c)dq

maxq∗

q∗∫0

(p(q)−c)dq.

p(q)− c = 0

I Thenp(q∗) = c.

(p(q)− c)dq

I Quantity produced is efficient

I Consumer surplus is 0

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I Under both first price discrimination and two-part tariff, thefirm is able to extract all of the consumer surplus

I What is the difference between first degree pricediscrimination and two-part tariff?

I Let’s see with an example

pA = 2− 1

pB = 8− qB

Marginal cost of production of 1

I If the monopolist knew the demand curve of each consumer

I First degree price discrimination

I Different price for each consumer and each unit, and extractall consumer surplus

I Two-part tariff

I Different fee and different price for each consumer

I Price of 1 to all consumers

I Entry fee of 2 for consumer A (consumer surplus when p = 1)

I Entry fee of 49/2 = 24.5 for consumer B (consumer surpluswhen p = 1)

I Two-part tariff

I What if monopolist doesn’t know who is who

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I What if monopolist doesn’t know who is whoI First degree price discrimination

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2

Q = qA + qb = 16− 5p

P =16− Q

I if p > 2Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2

Q = qA + qb = 16− 5p

P =16− Q

I if p > 2Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

I if p > 2Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

I Aggregate demand

pA = 2− 1

pB = 8− qB

qA = 8− 4pa

qB = 8− pB

I if p ≤ 2Q = qA + qb = 16− 5p

P =16− Q

5I if p > 2

Q = qA + qb = 8− p

P = 8− Q

Q(p) =

{16− 5p if p ≤ 2

8− p if p ≥ 2

P(Q) =

{16−Q

5 if Q ≥ 6

8− Q if Q ≤ 6

I We are unsure where the monopoly will maximize

I maxπ = qp(q)− q

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I Cannot be a solution

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I FOC: p(q) + qp′(q)− 1 = 0

I If Q ≥ 6

I FOC: 16−Q5 − Q

I Q = 115 = 2.2

I If Q ≤ 6

I FOC: 8− Q − Q = 1

I Q = 3.5

I P = 5.5

I Is the solution

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Everyone enters the bar. Tariff=2 and profit equal to 4

I Tariff ≥ 2, but ≤ 24.5

I Only B enters the bar. Tariff=24.5 and profit equal to 24.5

I Tariff ≥ 24.5

I No one enters the bar

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

I Two-part tariff

I Price equal to 1

I Tariff ≤ 2

I Tariff ≥ 24.5

I Zero profit

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I Market is segmented (no re-selling across markets)

I Firm knows the characteristics of each market (demand curve)

I Consider the following example: Two kinds of consumers:

qA(pA) = 24− pA

qB(pB) = 24− 2pB .

I constant marginal cost of production of 6

If the firm were allowed to set different prices in the differentmarkets, then he would choose:

(24− pA)(pA − 6) =⇒ p∗A = 15

(24− 2pB)(pB − 6) =⇒ p∗B = 9.

Total consumer surplus (CS) and profits of the firm in each market:

π∗A = 81, π∗B = 18,CSA = 40.5,CSB = 9.

Firm chose to set the same price in each market. Then he wouldmaximize the following:

{maxp≥12

(24− p)(p − 6),maxp<12

(24− p)(p − 6) + (24− 2p)(p − 6)

}= max{81, 75} = 81

I Price of p∗ = 15 in both markets, which leads to onlyconsumers in market A buying

I To summarize, the consumer surplus and profits in eachmarket are:

π∗A = 81, π∗B = 0,CSA = 40.5,CSB = 0.

I Prohibiting third degree price discrimination can exclude awhole market altogether

I Highly inefficient compared to the social welfare outcomegiven third degree price discrimination

I Suppose that the constant marginal cost of production is now4 instead of 6

I With third degree price discrimination, the firm sets thefollowing prices:

(24− pA)(pA − 4) =⇒ p∗A = 14,

(24− 2pB)(pB − 4) =⇒ p∗B = 8.

I In this case, the profits and consumer surplus in each marketis given by:

π∗A = 100, π∗B = 32,CSA = 50,CSB = 16,TS = 198.

I If the firm were prohibited from using third degree pricediscrimination, then:

{maxp≥12

(24− p)(p − 4),maxp<12

(48− 3p)(p − 4)

}= max{100, 108} = 108.

I p = 10

I profits in both markets and the consumer surplus in bothmarkets:

π∗A = 84, π∗B = 24,CSA = 98,CSB = 4,TS = 210.

I Consumers in region B are hurt but consumers in region Again significantly leading to an increase in consumer surplus

I The firm’s joint profits are hurt but the total surplus actuallyincreases

I Total surplus decreases

I Third degree price discrimination is considered illegal in manycountries and the European union

I It is possible to get around such allegations by claiming thatthe differential pricing comes from cost reasons

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I When someone or some firm is the sole buyer (monopoly isthe sole seller)

I Often arises in the context of firms being the sole buyers oflabor

I Let us study the profit maximization problem of a firm:

maxK ,L

pf (K , L)− rK − w(L)L.

I w is now a function of the amount of labor demanded(reflecting the power of the firm in the labor market)

I The first order condition yields:

∂L(K ∗, L∗) = w ′(L∗)L∗ + w(L∗) =⇒ pMPL = L∗w ′ + w .

I In a competitive market w ′ = 0 and so pMPL = w

I Wages and labor below the competitive level (an argument forminimum wages and union)

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I What happens when there are multiple monopolies involved inthe market?

I Firm A produces factor a at no cost

I Firm b in order to supply qb units of b must buy qa units of a

I Firm B produces according to a cost function:

C (qb) = (pa + c)qb.

I Demand equation for good b is linear:

qb(pb) = 100− pb.

I Firm B’s optimization problem becomes:

(100− qb)qb − paqb − cqb.

I The first order condition tells us:

100− 2qb = pa + c =⇒ pa = 100− 2qb − c .

I Since firm b is the only demander of commodity a, we have:

pa = 100− 2qb − c = 100− 2qa − c.

100− 2qb = pa + c =⇒ pa = 100− 2qb − c .

pa = 100− 2qb − c = 100− 2qa − c.

100− 2qb = pa + c =⇒ pa = 100− 2qb − c .

pa = 100− 2qb − c = 100− 2qa − c.

I If the price is pa then the qa that solves the above equationwould be the amount demanded of good a

I Thus firm B’s maximization problem has given us an inversedemand function for commodity a

I If the price is pa then the qa that solves the above equationwould be the amount demanded of good a

I Thus firm B’s maximization problem has given us an inversedemand function for commodity a

I Since firm A is also a monopolist in producing good a, we cansolve firm A’s maximization problem in the following way:

qa (100− 2qa − c) .

I As a result, we get:

100− 4qa − c = 0⇒ q∗a =100− c

4, p∗a = 50− c

I Firm a decides to supply the above units of a at a price50− c/2

qa (100− 2qa − c) .

100− 4qa − c = 0⇒ q∗a =100− c

4, p∗a = 50− c

qa (100− 2qa − c) .

100− 4qa − c = 0⇒ q∗a =100− c

4, p∗a = 50− c

I Firm B will produce q∗b = q∗a = 100−c4

I Then the price is given by:

p∗b = 100− 100− c

4= 75 +

I To summarize, we have:

p∗a = 50− c

q∗a =100− c

p∗b = 75 +c

q∗b =100− c

p∗b = 100− 100− c

4= 75 +

p∗a = 50− c

q∗a =100− c

p∗b = 75 +c

q∗b =100− c

p∗b = 100− 100− c

4= 75 +

p∗a = 50− c

q∗a =100− c

p∗b = 75 +c

q∗b =100− c

I Case 1: c = 0

p∗a = 50, q∗a = 25, p∗b = 75, q∗b = 25.

I If the firms were to merge so that whatever is produced byone of the firms can be used freely by that firm?

I The monopolists problem becomes:

q(100− q).

I The first order condition states that:

100− 2q∗ = 0 =⇒ q∗ = 50, p∗ = 50.

I Price of good b comes down from 75 to 50

I Production of good b goes up from 25 to 50

I This increases both the profits of the firm and the consumersurplus!

I Case 1: c = 10

p∗a = 45, q∗a = 22.5, p∗b = 77.5, q∗b = 22.5.

I If the firms were to merge so that whatever is produced byone of the firms can be used freely by that firm?

I The monopolists problem becomes:

q(100− q)− 10q

I The first order condition states that:

100− 2q = 10 =⇒ p∗ = 55, q∗ = 45.

I This increases both the profits of the firm and the consumersurplus!

I What is going on in the above examples?

I because the first firm is a monopolist, it charges a mark upabove marginal cost for its intermediate good

I This then distorts the marginal cost of firm B up additionally

I This then leads an even larger mark up on top of thisadditional marginal cost that affects the price of good b

I Essentially a markup on product a indirectly leads to an evenlarger markup on the final product b

I This is called the double marginalization problem

Introduction

Two-part tariff

Monopsony

Introduction

Two-part tariff

Monopsony

I Double marginalization can lead to inefficiently high pricesand inefficiently low levels of production

I By merging, both profits of the firm and consumer surplusmay simultaneously go up

I Difficult to tell if two firms are merging to solve a doublemarginalization problem or if they are simply merging tocreate a monopoly

I What are some potential ways to solve this problem withoutmergers?

I One possible way might be to engage in profit sharing

I Firms agree to share profits according to the following rule

I Prices charged for good a are zero

I In exchange, the profits of firm B are shared via a split of αgoing to firm A and (1− α) going to firm B

I Firm A’s decision is trivial. He simply produces qa = qb

I Firm B chooses to maximize:

(1−α) ((100− q)q − cq) = (1−α)

(100− q)q − cq

I Term inside the parentheses is just the monopoly profits if thetwo firms merged:

(1− α) maxq

Πm(q).

I Firms agree to share profits according to the following rule

I Prices charged for good a are zero

I In exchange, the profits of firm B are shared via a split of αgoing to firm A and (1− α) going to firm B

I Firm A’s decision is trivial. He simply produces qa = qb

I Firm B chooses to maximize:

(1−α) ((100− q)q − cq) = (1−α)

(100− q)q − cq

I Term inside the parentheses is just the monopoly profits if thetwo firms merged:

(1− α) maxq

Πm(q).

I The firms will produce at the monopoly quantities which wewere found were strictly greater than if the two firmsproduced completely separately without any such agreement

I The price will be the monopoly price

I For any α ∈ (0, 1), we get an increase in consumer surplusand total profits

I Really, for any α?

I Such arrangements can break down easily. Profits are hard toverify.

I Profits are usually difficult to verify. However, revenues aremuch easier to check.

I Firms enter into an arrangement where the revenues areshared according to α (firm A) and (1− α) (firm B) split

I Suppose that α = 1/2 and c = 10. Then firm 2 maximizes:

2q(100− q)− 10q.

I The first order condition gives:

2MR(q) = MC = 10 =⇒ MR(q) = 2MC = 20.

I Firm will produce below monopoly profits since it will produceat a point where MR = 2MC instead of MR = MC

2q(100− q)− 10q.

2MR(q) = MC = 10 =⇒ MR(q) = 2MC = 20.

2q(100− q)− 10q.

2MR(q) = MC = 10 =⇒ MR(q) = 2MC = 20.

2q(100− q)− 10q.

2MR(q) = MC = 10 =⇒ MR(q) = 2MC = 20.

2q(100− q)− 10q.

2MR(q) = MC = 10 =⇒ MR(q) = 2MC = 20.

I Solving, we get:

100− 2q = 20 =⇒ p∗ = 60 > pm = 55, q∗ = 40 < qm = 45.

I This does solve the double marginalization problem slightly:

p∗b = 77.5 > p∗ = 60, q∗b = 22.5 < q∗ = 40.

I Solving, we get:

100− 2q = 20 =⇒ p∗ = 60 > pm = 55, q∗ = 40 < qm = 45.

I This does solve the double marginalization problem slightly:

p∗b = 77.5 > p∗ = 60, q∗b = 22.5 < q∗ = 40.

Lecture 8: Price Discrimination - Mauricio Romero

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Transcript of Lecture 8: Price Discrimination - Mauricio Romero