(A) Monopoly

• Benchmark

• Examples:

Natural resources: Land, minerals, etc.

Markets with high entry costs: Airlines (city pairs),

grocery at the corner (small town)

• Plan

1. Solution of the Basic Monopoly Problem

2. 3rd degree price discrimination

(a) Basics

(b) Schmalensee Model

3. 1st degree price discrimination

4. 2nd degree price discrimination

5. Illustration of additional ways to price discrim-

inate:

Intertemporal Pricing: Lazaer’s model

1 Monopoly Problem and Solution

maxΠ = max [P (Q) ·Q− C(Q)]where:

Π Profit

Q Output

P Price, P (Q) inverse demand function

C(Q) Total Cost, MC = C0(Q)

R(Q) Revenue function

maxΠ = max [R(Q)− C(Q)]

• Graphical Illustration of Monopoly prob-lem I

MC

D

Q

P

• Graphical Illustration II (Solution)

MC

D

MR

Qm

Pm

Q

P

• Optimal Monopoly Solution

• One way of expressing the solution using calculusis

∂Π

∂Q=

∂R

∂Q− ∂C

∂Q= 0

Rewrite as

MR =MC

Intuition:

Benefit of an add. unit = Cost of an add. unit

• Another way: explicit solution∂Π

∂Q= P +Q · ∂P

∂Q| {z }−∂C

∂Q= 0

MR

= P ·"1 +

Q

P· ∂P∂Q

#− ∂C

∂Q= 0

Intuition: Benefits are decomposed into a price

and quantity effect

• Definition: Elasticity of Demand:

e =P

Q· ∂Q∂P

=

4QQ4PP

Interpretation: Proportional change in quantity

divided by a proportional change in price

(which % change in quantity results from a 1 %

change in price)

We say that demand is elastic if e < −1; and

inelastic if −1 ≤ e ≤ 0.

P ··1 +

1

e

¸=MC

• Why Q as a choice variable?

Can also have P the choice variable, it makes no

difference

Note: Later (Cournot, Bertrand), it makes a dif-

ference

• P the choice variable:

∂Π

∂P= Q+ P · ∂Q

∂P| {z }−∂C

∂Q· ∂Q∂P| {z } = 0 / · dP

dQ

MR MC

= P ·QP· ∂P∂Q| {z }+1

−MC = 01/e

= P ··1

e+ 1

¸−MC = 0

as before.

Comments on the Monopoly Solution

1. e < 0 implies P > MC

Note: distinct from perfect competition

2. Optimum at an elastic point, e < −1:Why?

Since MC > 0 and because:

P ··1

e+ 1

¸=MC

Now RHS is positive, implies that LHS must be

positive.

3. Lerner Index:

Measure of degree of monopoly power

P −MCP

= −1e

under perfect competition: e = −∞.

4. Second order condition satisfied if MR intersects

MC from above

MR0< MC

0

Price Discrimination

• Assumption: No arbitrage between consumers(no resale)

• First Degree: Perfect Price DiscriminationThe seller knows the consumers’ types

• Second Degree:The seller knows the distribution of consumer types

only

Example: Telephone calling plans

Idea: Menu of prices

• Third Degree:The seller knows sub-groups

Example: Skiing in Austria: price of local skiers

versus non-locals

Idea: distinct prices for distinct consumer groups

2 Third Degree Price Discrimination

2.1 Basic Model

• Assumptions:

1. Two separate markets:

P1(Q1), P2(Q2), Q = Q1 +Q2

2. Cost C(Q)

• Objective FunctionmaxQ1,Q2

Π = max [P1(Q1) ·Q1 + P2(Q2) ·Q2 − C(Q)]

• How to solve?First order condition

∂Π

∂Q1= P1 +Q1 ·

∂P1∂Q1

− ∂C

∂Q· ∂Q∂Q1

= 0

Now, since ∂Q∂Q1

= 1, this yields:

P1

"1 +

1

e1

#=MC

Similarly,

∂Π

∂Q2= P2 +Q2 ·

∂P2∂Q2

− ∂C

∂Q· ∂Q∂Q2

= 0

Now, since ∂Q∂Q2

= 1, this yields:

P2

"1 +

1

e2

#=MC

• Implication

Charge a higher price, P2 > P1, for the more inelastic

demand, e2 > e1Why? Because

P2P1=1 + 1

e1

1 + 1e2

• Illustration

D1

MR1Q1

P1

Q

P

D2

MR2

MC

MC

P2

Q2

P

2.2 Schmalensee Model

• Idea:encompass single price monopoly and price dis-

criminating monopolist

• Assumptions

1. N independent markets with demand qi(pi) in

market i

Total demand

Q =NXi=1

qi(pi)

Why independent? because qi depends on pi only.

2. Constant marginal cost c

• Objective Function

maxΠ =NXi=1

Πi(pi)

=NXi=1

(pi − c)qi(pi)

• Two extreme cases:(i) No price discrimination:

p∗ such thatNXi=1

Π0i(p∗) = 0

(ii) Third degree price discrimination:

p∗i such that Π0i(p∗i ) = 0 for all i = 1, . . . , N.

• Robinson(i) Strong Markets: p∗i > p∗(ii) Weak Markets: p∗i < p∗((iii) Intermediate Markets: p∗i = p∗.

• Notice: p∗i > p∗ if and only if Π0i(p∗) > 0

Why?

because constant marginal cost and because qidepends on pi only.

Pi*

P

Π

P* P*

• Welfare

W =NXi=1

"Z ∞Piqi(v)dv + Πi(pi)

#

• Idea: Look at an Artificial Problem

maxNXi=1

Πi(pi) s.t.NXi=1

Π0i(p∗) (pi − p∗) ≤ t

• Interpretation(i) No price discrimination: t = 0

(ii) Third degree price discrimination: t is large

• First order conditionΠ0i (pi) = λΠ

0i(p∗) with λ ∈ [0, 1]

• Solutionpi (t) smooth function with the property

pi (0) = p∗,and as t increases pi (t) −→ p∗i

• PropertyNXi=1

Π0i(pi (t)) = 0 for all t

Why?From first order condition, since

NXi=1

Π0i(pi (t)) = λ ·

NXi=1

Π0i(p∗)| {z }

equals 0

= 0

• Further, the propertyNXi=1

Π0i(pi (t)) = 0

holds for all t.

Can take the derivative with respect to t :

∂

∂t

NXi=1

Π0i(pi (t))

=NXi=1

·Π00i

¸p0i (t)

= 0

• Now, what is Π0i?

Π0i =

∂Πi∂pi

= (pi − c) q0i + qi

• Hence, we getNXi=1

·Π00i

¸p0i (t) =

NXi=1

h2q0i + (pi − c) q

00i

ip0i (t)

= 0 (1)

Results

1. Output Effect:

Q =NXi=1

qi(pi (t))

∂Q

∂t=

NXi=1

q0ip0i

= −12

NXi=1

(pi − c) q00i p0i (t)

follows from equation (1) above

• InterpretationOutput effect depends on q

00i and p

0i

• (Implicit assumption:qi(p

∗) > 0 and qi(p∗i ) > 0 for all i.

However, some weak markets may not be servedat all)

• Three cases:(i) Linear demand:

q00i = 0

implies

∂Q

∂t= 0

(ii) Strong markets:

(pi − c) p0i > 0

Sign of output effect depends on the sign of −q00i

→ concave demand: positive output effect

→ convex demand: negative output effect.

(ii) Above is reversed for weak markets

2. Welfare Effect

• Welfare (= CS +PS)

W =NXi=1

"Z ∞Pi(t)

qi(v)dv + (pi(t)− c) qi (pi(t))#

• Derivative with respect to t∂W

∂t=

NXi=1

[(pi(t)− c)] q0i · p

0i

(other terms involving p0i cancel)

= [p∗ − c] · ∂Q∂t| {z }+

NXi=1

[pi − p∗] q0i · p

0i| {z }

Welfare effect price discrimination

of output effect

• Welfare effect of output depends on ∂Q∂t

• Price discrimination effect is negativeWhy?

Look at [pi − p∗] q0i · p0i

(i) strong market: + (−) +(ii) weak market: − (−) −

• Cases:

1. Linear demand

∂Q

∂t= 0 implies

∂W

∂t< 0

2. Negative output effect

∂Q

∂t≤ 0 implies

∂W

∂t< 0

• IllustrationWelfare effect for 2 markets (strong and weak)

D1

qsqs

0

c

a’

Pw

c’

Ps

P*s

P*

qs1

P*w

qw0 qw

1 qw

b’

d’d

c

b

a

e

e’

D2

Illustration: Total effect

D1

qsqs

0

c

a’

Pw

c’

Ps

P*s

P*

qs1

P*w

qw0 qw

1 qw

b’

d’d

c

b

a

e

e’

D2

∂W

∂t=

ha0b0c0d0 − abcd

i−hb0c0e0+ bce

i= (p∗ − c)(Q1 −Q0)−

³b0c0e0+ bce

´which can be positive only if Q1 > Q0

(Note: Q1 > Q0 is not sufficient for ∂W∂t > 0)

3 First Degree Price Discrimination

• Assumptions

1. ni buyers of type i = 1, 2

2. Quasi linear utility

ui = Ui(xi) + yi

with monopolized good xi, and numeraire good

yi

3. Single crossing

U02(x) > U

01(x)

type 2 prefers good x stronger than type 1

(Assumptions continued)

4. Identical incomes M

5. Normalize

Ui(0) = 0

diminishing marginal utility

U00i (x) < 0

6. Constant marginal cost of production (equals av-

erage cost): c

• RecallDemand under quasi linear utility (from problem

set)

xi(p) = U0−1i (p)

yi = M − F − p · xi(p)here F is a fixed charge

and p is the monopoly price

• Indirect utilitylet ui denote the reservation utility (the utility

when not buying good x)

vi(p, F ) = Ui (xi (p)) +M − F − p · xi(p)| {z }equals yi

• Monopoly Profits

max(pi,Fi)

Π =2Xi=1

·ni · (pi · xi(pi) + Fi)| {z }− c · ni · xi(pi)| {z }

¸buyer’s reserv. utility cost

subject to the consumer reservation (participation)

constraint

vi(p, F ) ≥ ui for i = 1, 2

• Choose (pi, Fi)2i=1 to maximize profit subject tothe IR constraint

• Solution

pi = c

Fi s.t. vi(p, F ) = ui for i = 1, 2

• IntuitionPrice equals marginal cost

Participation fee equals the consumer surplus

• Comments

1. Solution maximizes aggregate surplus

2. All the surplus goes to the monopolist

4 Second Degree Price Discrimina-

tion

• Cannot see type or characteristics

• Knows the distribution of types, can offer options

• Examples: telephone calling plans, damaged goods,theater seating, etc.

• Consider (3rd degree) offers (c, F1) and (c, F2)Is not optimal. Why?Because type 2 would select (c, F1).

• Can do better?

• Suppose an option is(xi, Fi)

(use quantity instead of price)

Monopoly Profit

max(xi,Fi)

Π =2Xi=1

[ni · (Fi − cxi)]

subject to

1. IR (participation)

Ui (xi) +M − Fi ≥ ui for i = 1, 2

where yi = M − Fi

IC (no mimicking)

U1 (x1) +M − F1 ≥ U1 (x2) +M − F2U2 (x2) +M − F2 ≥ U2 (x1) +M − F1

Solution

• NoticeIC1 and IR2 are not binding

(proof omitted: reading material)

• Lagrangean problemFOC can be used to solve it

• Properties of the solution

1. Type 2 buyer gets the solution under first degree

price discrimination

(No distortion at the top)

2. Type 1 buyer gets less than under first degree

price discrimination

x∗1 < x1st

1

(Distortion due to IC2)

5 Illustration of additional ways to

price discriminate:

5.1 Bundling, or quantity discounts

• Examples: Microsoft Office (Word, Excel, etc.),vacation packages, mobile phone calling plans,

supermarket products, etc.

• Can offer a price for the bundle and prices foreach item separately

• Intuition: Offer lower prices for additional units(or bundles)

5.2 Intertemporal Price discrimination:

Lazaer’s fashion goods model (problem

set)

• 2 Periods

• Monopolist discounts the futureSells a product over two periods; Consumers canbuy in period one or in period two.

• SolutionP1 > P2

• High valuation consumers buy in the first period

• The remaining (low) valuation consumers buy inperiod 2

• Intuition: Fashion good

• Fashion Goods Illustration

P1

P

QQ1 QQ2

P

P2D1

D2=DR

DR