Monopoly Problem Set

Question 1

The Royal Ontario Museum (ROM) faces inverse demand

PDres(Q) = 5 − Q from tourists and PDtou(Q) = 20 − 4Q.

from Toronto residents (Q is measured in 100 visitors per day). The marginal cost is zero.

Assume the ROM is a profit-maximizing monopolist.

(a) Assuming the ROM sets a single price for all consumers, what price does it choose? Calculate the Surplus

Answer 1

1. Rearrange the inverse demand equations

PDres(Q) = 5 − Q
Q = 5 - P

PDtou(Q) = 20 − 4Q.
Q = 5 - 1/4P

2. Find the Total Quantity Equation
   and the Total Price Equation

QT = 5 - p + 5 - 1/4P
QT = 10 - 5/4P ( The Total Quantity Equation)

The Total Price Equation: Rearrange the total quantity equation
QT = 10 - 5/4P
PT = 8 - 4/5QT

3. Find the profit, using the total price equation

π = P * Q
π = (8 - 4/5QT)* Q
π = 80 - 4/5Q^2
Take the First Order Condition
π’ = 8 - 8/5Q
8 - 8/5Q = 0
Q = 5

Find P using Q = 5, use the total P equation

Pt = 8 - 4/5QT
Pt = 8 - 4/5(5) = 4

Therefore P = 4 and Q = 5

4. Now find the ranges:
   Choose one set you want to compare: Tourist

Use the total quantity equation as one set and the other quantity equation as the other set (tourist). The range is Q = 5

Qd(P) = 5 - 1/4P P > 5
10 - 5/4P. P <= 5

Use the total price equation as one set, and then use

Pd(Q) =. 20 - 4Q Q < 15/4
8 - 4/5Q. Q >= 15/4

Now this time we need to find the range of values:

1. Set the total quantity equation equal to the quantity equation of
   the set of one (tourist), basically Qd(P)
   5 - 1/4P = 10 - 5/4P
   Solve for P
   P = 5

Plug P = 5, back into the set of one you choose (tourist)

Q = 5 - 1/4P = 5 - 1/4(5) = 15/4

5. Find Surplus

PS = P * Q = 5 * 4 = 20

Draw the Graph for CS:

Use
Pd(Q) =. 20 - 4Q Q < 15/4
8 - 4/5Q. Q >= 15/4

Graph Both: Remember P = 4 and Q = 5

CS Market= 1/2(3.75)(20-5) + (5-4)(3.75) + 1/2(5-4)(5-3.75) = 32.5

Question 2

Assuming the ROM can first-degree price discriminate, what is its producer surplus?

Background Info:

The Royal Ontario Museum (ROM) faces inverse demand

PDres(Q) = 5 − Q from tourists and PDtou(Q) = 20 − 4Q.

from Toronto residents (Q is measured in 100 visitors per day). The marginal cost is zero.

Assume the ROM is a profit-maximizing monopolist.

Answer 2

1. Use the Equations:

Use the Tourist Equation:
PDtou(Q) = 20 − 4Q

Use the Total Price Equation:
Total Q equation = 10 - 5/4P

Rearrange for P:
Pd(Q) = 8 - 0.8q

Now we are using:
PD(Q) = 20 − 4Q
Pd(Q) = 8 - 0.8q

2. Find the TR and MR through the First Order Condition and then solve for Q so that we know the range of the segments:

For Tourist:
PD(Q) = 20 − 4Q
TR = PQ
TR = (20 − 4Q)Q
TR = 20Q - 4Q^2
FOC
MR = 20 - 8Q
Q = 2.5

As a whole:
Pd(Q) = 8 - 0.8q
TR = P * Q
TR = (8 - 0.8q) * Q
TR = 8Q - 0.8Q^2
FOC
MR = 8 - 1.6Q
Q = 5

3. Now we set Q equal to each other to find the Range value of Q.

20 - 4Q = 8 - 0.8Q
-4Q + 0.8Q = 8 - 20
- 3.2Q = -12
Q = 3.75

4. Now make the Graph and the range of values using the Q = 3.75 and comparing to max Q
   PD(Q) = 20 − 4Q. Q < 3.75
   8 - 0.8q. Q >= 3.75

Draw the Graph:

Find the Intersection:

Set 3.75 into both so (3.75, 5) is the intersection

5. Find Producer Surplus:
   Since its price discrimination gets producer surplus from both categories

First Segment Residents:
5
∫ ( 5 - Q)dq
0
5
[5Q - Q^2/2 ]
0

= 5(5) - (5)^2/2 =
PS = 12.5

Secound Segment Toursit:

5
∫ ( 20 - 4Q)dq
0
5
[20Q - 2Q^2]
0
= 50
Total Producer Surplus = 12.5 + 50 = 62.5

Question 3

Assuming the ROM knows who is a tourist and who is a resident. What price should
it charge tourists and residents respectively? Is this second- or third-degree price
discrimination?

Show that the producer surplus is larger. Is the market more efficient under price
discrimination?

Background Info:

The Royal Ontario Museum (ROM) faces inverse demand

PDres(Q) = 5 − Q from tourists and PDtou(Q) = 20 − 4Q.

from Toronto residents (Q is measured in 100 visitors per day). The marginal cost is zero.

Assume the ROM is a profit-maximizing monopolist.

Answer 3

This is third degree price discrimination. In third degree price MR = MC.

PDres(Q) = 5 − Q
PDtou(Q) = 20 − 4Q.

1. Residents:

MR = MC
Find the MR
TR = P* Q
TR = (5 − Q) * Q
TR = (5Q - Q^2)
MR = 5 - 2Q
0 = 5 - 2Q
Q = 2.5

Plug Q = 2.5 Back in Demand for Residents
PDres(Q) = 5 − Q
PDres(Q) = 5 − 2.5 = 2.5

Therefore for residents:
Qres= 2.5, Pres = 2.5

2. Tourist:
   MR = MC
   TR = P* Q
   TR = (20 − 4Q) * Q
   TR = 20Q - 4Q^2
   MR = 20 - 8Q
   0 = 20 - 8Q
   Q = 2.5

Plug Q = 2.5 Back in Demand for Tourist
PDtou(Q) = 20 − 4Q.
PDtou(Q) = 20 − 4(2.5) = 10

Therefore for Tourist :
Qtou= 2.5, Ptou = 10

The ROM charges residents a low price and tourist a high price.

3. Producer Surplus and Consumer Surplus

PSres = 2.5(2.5) = 6.25
PStou = 10(2.5) = 25
Total Producer Surplus
= 25 + 6.25 = 31.25

Draw the Graph Out: Draw Both the demand equation and the MR and where MR = MC draw the line up and write the P and Q values found, and the intercept.

CSres = 1/2(5-2.5)(2.5) = 3.125
CStou = 1/2(20-10)(2.5) = 12.5
Total CS = 12.5+3.125 = 15.625

Total Surplus = 31.25 + 15.625 = 46.875

Question 4

Suppose 50% of consumers are of type H and 50% are of type L. A consumer’s type is not observed by the monopolist. The monopolist sells two versions of a product, a high- and a low-quality one.

Type. WTP High Version. WTP Low
H. $ 5 $ 2
L $2 $1.5

If the monopolist must charge a single price for both versions, what would it be?

Answer 4

Both consumers prefer the high- over the low-quality version. The monopolist’s would set P
M = 5 and only sells to the H-type. It’s expected per consumer revenue is $2.5.

Question 5

Suppose now that the monopolist sets prices P_ and P¯ for the high-quality version and
the low-quality version respectively. Write down the participation constraint for each
consumer type.

Type. WTP High Version. WTP Low
H. $ 5 $ 2
L $2 $1.5

Answer 5

The monopolist wants H-type consumers to buy the high-quality product and L-type consumers to buy the low-quality product. H-type and L-type
consumers buy if the price is less than $4 an $1 respectively.

5 − P¯ ≥. 0 P C(H)
1.5 − P_ ≥ 0 P C(L)

Question 6

(Write down the incentive constraint for each consumer type.

Type. WTP High Version. WTP Low
H. $ 5 $ 2
L $2 $1.5

Answer 6

For High Type:

P¯ and P_ must be chosen such that H-type consumers rather buy the high-quality version at P¯ than buy the low-quality version at P.

5 − P¯ ≥ 2 − P_ IC(H)

L-type consumers, in turn, must not prefer the high-quality version at price P¯ over the low-quality version at price P_.

1.5 − P_≥ 2 − P¯ IC (L)

Question 7

What are P_and P¯? What is the expected revenue per consumer?

Type. WTP High Version. WTP Low
H. $ 5 $ 2
L $2 $1.5

Answer 7

Claim: PC(H) and IC(H) are binding constraints.

1. PC(H) is redundant

Prove this: Use the IC(H)

5 - P¯ >= 2 - P_
>= 1.5 - P_
>= 0
IC(H) -> PC(H)

2. This reduces the optimization problem to the following, because PC(H) is redundant:

Max. P¯ + P_
P¯, P_

PC(L) = 1.5 - P_ >= 0
IC(L) = 1.5 - P_ >= 2 - P¯
IC(H) = 5 - P¯ >= 2 - P_

3. IC(H) Binds

If IC(H) bings it means its =. We can always increase P¯ to increase the value of the objective function, until IC(H) bings.
5 - P¯ = 2 - P_
Solve for P_
P_ = P¯ - 3 (a)

3. PC(L) binds.

We can always increase P_ to increase the value of the objective function until PC(L) binds

Given that IC(H) binds, IC(L) always holds
1.5 - P_ >= 2 - P¯
Plug P_ = P¯ - 3 (a)
1.5 - ( P¯ - 3) >= 2 - P¯
4.5 - P¯ >= 2 - P¯ always holds

4. Therfore IC(H) AND PC(L) are binding constraints

P_* = 1.5
Plug this in IC(H)
5 - P¯ = 2 - P_
5 - P¯ = 2 - 1.5
P¯ = 4.5
Consumer Revenue = 1/2(1.5 + 4.5) = 3.

Question 8

Exercise 3 There are two consumer types, 1 and 2. A type 1 consumer’s utility function is

U1(Q) = Q - 1/2Q^2
U2(Q) = 1/2Q - 1/2Q^2

The monopolist maximizes its revenue. It offers two options buy Q1 units of the good and
pay P1 or buy Q2 units of the good and pay P2.

Suppose the monopolist observes consumer types. Write down the participation constraint for each consumer type. What (Q1, P1) and (Q2, P2) does it choose?

Answer 8

The participation constraints are

Q - 1/2Q^2 - p1 >= 0 PC1
1/2Q - 1/2Q^2 - p2 >= 0 PC2

A type 1 consumer’s utility is maximized if Q1 = 1 and the utility of a type 2consumer is maximized at Q2 = 1/2. These are the quantities the monopolist will
offer type 1 and type 2 consumers because this allows it to charge them the highest
prices.

Plug back in:
U1(Q) = Q - 1/2Q^2
U1(Q) = 1- 1/21^2 = 1/2

U2(Q) = 1/2Q - 1/2Q^2
U2(Q) = 1/2(1/2) - 1/2(1/2)^2 = 1/8

The monopolist’s revenue is maximized if PC1 and PC2 bind. It offers (Q1, P1) = (1, 1/2) and
(Q2, P2) = (1/2, 1/8)