Problem Set #2
1. 1st Degree Price Discrimination
Suppose a monopolist faces demand P = 225 - Q and has marginal cost MC = 160.
a) Determine monopoly price.
b) Determine the effects of 1st degree price discrimination versus monopoly pricing on consumer surplus and deadweight loss.
Solution:
a) MR = MC:
225 – 2Q = 160
Q = 32.5
P = 192.5
P = MC:
225 – Q = 160
Q = 65
P = 160
b) Consumer surplus under monopoly pricing:
CS = ½(225 – 192.5)32.5 = 528
Consumer surplus under price discrimination = 0
Deadweight loss under monopoly pricing:
DWL = ½(192.5 – 160)(65 – 32.5) = 528
Deadweight loss under price discrimination = 0
2. Second-Degree Price Discrimination
Municipal water sales follow a declining block tariff, where lower prices are charged for additional units of consumption. Demand for water is given by
P = 200 – Q and marginal costs are MC = 20.
a) Determine monopoly price.
b) What are the second-degree discrimination prices and quantities for the first and second blocks of water?
c) Determine the effects of 2nd degree price discrimination versus monopoly pricing on consumer surplus and deadweight loss.
Solution:
a) MR = MC
200 – 2Q = 20
Q = 90
P = 110
b) Price discrimination:
Q1 = first block of consumption, so that P(1)(Q1) = 200 – Q1
Q2 = first block of consumption, so that P(2)(Q2) = 200 – Q2
Π = TR(Q1) + TR(Q2 - Q1) – TC(Q2) = Q1Q2 – Q12 – Q22 + 180 Q2
max Π = d Π/dQ1 = Q2 - 2Q1 = 0
max Π = d Π/dQ2 = Q1 - 2Q2 + 180 = 0
Substitute the first into the second equation to get:
Q1 – 2(2Q1) + 180 = 0
Q1 = 60 and Q2 = 120
P1 = 200 – 60 = 140
P2 = 200 – 120 = 80
c) Consumer surplus under uniform pricing:
CSuniform = ½(200 – 110)90 = 4,050
Consumer surplus under price discrimination:
CSprice disc = ½(200 – 140)60 + ½(140 – 80)(120 – 60) = 1,800 + 1,800 = 3,600
Deadweight loss under monopoly pricing:
DWLuniform = ½(110 – 20)(180 – 90) = 4,050
DWLprice disc = ½(80 – 20)(180 – 120) = 1,800
3. Discrimination Involving Bundling Vs. Tying
IBM’s demand for mainframe computers/services for one customer is given by
P1 = 225 – Q1 while demand for another customer is given by P2 = 225 – 2Q2. The cost of each copier is $5,000. Marginal costs of aftermarket services are $50/unit.
a) Calculate the prices for Customers 1 and 2 under a bundling approach.
b) Calculate the prices for Customers 1 and 2 under a tying approach.
c) Assuming that the customers can engage in post-sale trade in equipment, calculate the best bundled prices for each.
d) Assuming that the customers can engage in post-sale trade in equipment, calculate the best tied prices for each.
e) Fill in the table for each type of pricing scenario.
f) What is the best pricing strategy for IBM if it cannot engage in perfect price discrimination?
Solution:
a) The bundled price is obtained by calculating consumer surplus and netting out cost.
For customer 1
P1B = ½(225)(225) – 5,000
= 25,312.5 – 5,000
= 20,312.5
For customer 2
P2B = ½(225)(112.5) – 5,000
= 12,656 – 5,000
= 7,656
b) The total price under tying includes the price of the equipment and the price of the aftermarket services. At a price of $50/unit, quantity demanded for aftermarket services for customer 1 is 175 units (from the demand function) and for customer 2, it is 87 units. The total price is:
For customer 1
P1T = (175)(50) + ½(225 – 50)(175) – 5,000 =
= 8,750 + 15,312.5 – 5,000 = 19,062.5
For customer 2
P2B = (87)(50) + ½(225 – 50)(87) – 5,000
= 4,350 + 7,612.5 – 5,000 = 6,962.5
c) If customers and trade post-sale, then each should receive the lowest of the bundled prices:
P1B = 12,656 – 5,000 = 7,656
P2B = 12,656 – 5,000 = 7,656
d) If customers and trade post-sale, then each should receive the lowest of the tied prices for equipment but pay what they are willing for aftermarket services:
P1B = 8,750 + 7,612.5 – 5,000 = 11,362.5
P2B = 4,350 + 7,612.5 – 5,000 = 6,962.5
e)
Scenario 1Bundled - PPD / Scenario 2
Bundled – Uniform / Scenario 3
Tied – PPD / Scenario 4
Tied – charged lowest equip. price
Customer 1 / 20,312.5 / 7,656 / 19,062.5 / 11,362.5
Customer 2 / 7,656 / 7,656 / 6,962.5 / 6,962.5
Total / 27,968.5 / 15,312 / 26,025 / 18,325
f) The best strategy would be to charge tied prices where the firm sells at the lowest equipment cost but charges what customers are willing to pay for services.
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