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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 – 2066 = 2,280, Pb = 66, MRb = 180 – 2,280/10 = – 48, b = PbQb = 150,480 vacation travelers: Qv = 3,000 – 3066 = 1,020, Pv = 66, MRv = 100 – 1,020/15 = 32, v = PvQv = 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 = PbQb = 162,000 vacation travelers: v = PvQv = 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 – 2084 = 1,920, Pb = 84, MRb = 180 – 1,920/10 = – 12, b = PbQb = 161,280 vacation travelers: Qv = 3,000 – 3084 = 480, Pv = 84, MRv = 100 – 480/15 = 68, v = PvQv = 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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