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assignment nine
dynamic pricing (II)
integrated market ………….1
uniform pricing: no capacity constraint ………….2
customized pricing: no capacity constraint ………….3
uniform pricing: capacity constraint ………….4
customized pricing: capacity constraint ………….5
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
assignment 9
dynamic pricing (II)
the analytics of constrained optimal
decisions
rental car market
integrated market
► Notice the demand equations for the markets (represented in the two diagrams on the left)
Q1 = 3,600 – 20P, and zero when P > 180, equivalent to P = 180 – Q1/20, MR1 = 180 – Q1/10
Q2 = 3,000 – 30P, and zero when P > 100, equivalent to P = 100 – Q2/30, MR1 = 100 – Q2/15
► The aggregate demand when the same price (represented in the diagram on the right)
6,600 – 50P, when P  100
Q = Q1 + Q2 = 3,600 – 20P, when 100 < P  180
0, when P > 180
► The demand expressed as the P function Q and the corresponding marginal revenue are:
P = 180 – Q/20 for 0  Q  1,600 and P = 132 – Q/50 for 1,600  Q  6,600
MR = 180 – Q/10 for 0  Q  1,600 and MR = 132 – Q/25 for 1,600  Q  6,600
Business travelers
Integrated Market
Vacation travelers
180
180
P1 = 180 – Q1/20
P = 180 – Q/20
100
100
P2 = 100 – Q2/30
68
P = 132 – Q/50
20
0
3,600
1,800
MR1 = 180 – Q1/10
 2016 Kellogg School of Management
0
1,500
3,000
MR2 = 100 – Q1/15
assignment 9
0
1,600
3,300
6,600
page | 1
microeconomics
assignment 9
dynamic pricing (II)
the analytics of constrained optimal
decisions
rental car market
uniform pricing: no capacity constraint
► The demand expressed as the P function Q and the corresponding marginal revenue are
P = 180 – Q/20 for 0  Q  1,600 and P = 132 – Q/50 for 1,600  Q  6,600
MR = 180 – Q/10 for 0  Q  1,600 and MR = 132 – Q/25 for 1,600  Q  6,600
Setting MR = MC the optimal number of cars for the integrated market is Q* = 3,300 for a price P* = 132 – 3,300/50 = 66
► The quantity, price, marginal revenue and profit for each market are calculated below:
business travelers: Qb = 3,600 – 2066 = 2,280, Pb = 66, MRb = 180 – 2,280/10 = – 48, b = PbQb = 150,480
vacation travelers: Qv = 3,000 – 3066 = 1,020, Pv = 66, MRv = 100 – 1,020/15 = 32, v = PvQv = 67,320
► Total profit is uniform = b + v = 217,800
Business travelers
Integrated Market
Vacation travelers
180
180
P1 = 180 – Q1/20
P = 180 – Q/20
100
100
P2 = 100 – Q2/30
66
66
68
66
P = 132 – Q/50
20
0
3,600
1,800
2,280
MR1 = 180 – Q1/10
 2016 Kellogg School of Management
0
3,000
1,020 1,500
MR2 = 100 – Q1/15
assignment 9
0
1,600
3,300
6,600
page | 2
microeconomics
assignment 9
dynamic pricing (II)
the analytics of constrained optimal
decisions
rental car market
customized pricing: no capacity constraint
► If there are no capacity restrictions the optimal number of cars an each market is “priced” separately, the optimal policy for
each market is obtained by setting the corresponding marginal revenue (MR1 and MR2) equal to the marginal cost (of zero):
business travelers: Qb = 1,800 for a price Pb = 180 – 1,800/20 = 90
vacation travelers: Qv = 1,500 for a price Pv = 100 – 1,500/30 = 50
► The profit for each market:
business travelers: b = PbQb = 162,000
vacation travelers: v = PvQv = 75,000
► Total profit is uniform = b + v = 237,000 for a total number of cars: 3,300.
Business travelers
Integrated Market
Vacation travelers
180
180
P1 = 180 – Q1/20
P = 180 – Q/20
100
90
100
P2 = 100 – Q2/30
50
68
66
P = 132 – Q/50
20
0
3,600
1,800
MR1 = 180 – Q1/10
 2016 Kellogg School of Management
0
3,000
1,500
MR2 = 100 – Q1/15
assignment 9
0
1,600
3,300
6,600
page | 3
microeconomics
assignment 9
dynamic pricing (II)
the analytics of constrained optimal
decisions
rental car market
uniform pricing: capacity constraint
► The demand expressed as the P function Q and the corresponding marginal revenue are
P = 180 – Q/20 for 0  Q  1,600 and P = 132 – Q/50 for 1,600  Q  6,600
MR = 180 – Q/10 for 0  Q  1,600 and MR = 132 – Q/25 for 1,600  Q  6,600
The optimal number of cars in this case is the maximum capacity Q* = 2,400 for a price P* = 132 – 2,400/50 = 84
► The quantity, price, marginal revenue and profit for each market are calculated below:
business travelers: Qb = 3,600 – 2084 = 1,920, Pb = 84, MRb = 180 – 1,920/10 = – 12, b = PbQb = 161,280
vacation travelers: Qv = 3,000 – 3084 = 480, Pv = 84, MRv = 100 – 480/15 = 68,
v = PvQv = 48,320
► Total profit is uniform = b + v = 201,600
Business travelers
Integrated Market
Vacation travelers
180
180
P1 = 180 – Q1/20
P = 180 – Q/20
100
84
84
P2 = 100 – Q2/30
100
84
68
P = 132 – Q/50
20
0
3,600
1,800 1,920
MR1 = 180 – Q1/10
 2016 Kellogg School of Management
0
480
3,000
1,500
MR2 = 100 – Q1/15
assignment 9
0
1,600 2,400 3,300
6,600
page | 4
microeconomics
assignment 9
dynamic pricing (II)
the analytics of constrained optimal
decisions
rental car market
customized pricing: capacity constraint
Business travelers
Vacation travelers
180
MR1 = 180 – Q1/10
100
MR2 = 100 – Q2/15
36
36
► It must be the case that at the optimum
(the last vehicle used) must satisfy
MR1 = MR2
That is
180 – Q1/10 = 100 – Q2/15
► But there is a constraint on the total
quantity available to distribute between the
two stores:
Q1 + Q2 = 2400
0
1800
1440
0
1500
960
2400
► We get a system of two equations with two unknowns:
180 – Q1/10 = 100 – Q2/15
Q1 + Q2 = 2400
180 – Q1/10 = 100 – (2400 – Q1)/15
Q2 = 2400 – Q1
Q1 = 1440
Q2 = 960
► Conclusion: use 1440 vehicles for business travelers and 960 for vacation travelers, in total all 2400 vehicles available for use.
Notice that the marginal cost plays no role here (sunk cost by now).
► The prices are: P1 = 180 – Q1/20 = 108 and P2 = 100 – Q2/30 = 68 and the total profit is  = 220,800
 2016 Kellogg School of Management
assignment 9
page | 5
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