the analytics of constrained optimal decisions microeco nomics spring 2016 dynamic pricing (II) ………….1integrated market ………….2 uniform pricing: no capacity constraint assignment nine ………….3 customized pricing: no capacity constraint ………….4 uniform pricing: capacity constraint ………….5 customized pricing: capacity constraint

microeconomic s the analytics of constrained optimal decisions assignment 9 dynamic pricing (II)  2016 Kellogg School of Management assignment 9 page |1 rental car market Business travelers Vacation travelers 3,6003, P 1 = 180 – Q 1 /20 P 2 = 100 – Q 2 /30 integrated market ► Notice the demand equations for the markets (represented in the two diagrams on the left) Q 1 = 3,600 – 20 P, and zero when P > 180, equivalent to P = 180 – Q 1 /20, MR 1 = 180 – Q 1 /10 Q 2 = 3,000 – 30 P, and zero when P > 100, equivalent to P = 100 – Q 2 /30, MR 1 = 100 – Q 2 /15 ► The aggregate demand when the same price (represented in the diagram on the right) 6,600 – 50 P, when P  100 Q = Q 1 + Q 2 = 3,600 – 20 P, 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 MR 1 = 180 – Q 1 /10 MR 2 = 100 – Q 1 /15 Integrated Market 6, P = 180 – Q /20 P = 132 – Q / , ,300 1,500 1,800

microeconomic s the analytics of constrained optimal decisions assignment 9 dynamic pricing (II)  2016 Kellogg School of Management assignment 9 page |2 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: Q b = 3,600 – 20  66 = 2,280, P b = 66, MR b = 180 – 2,280/10 = – 48,  b = P b  Q b = 150,480 vacation travelers: Q v = 3,000 – 30  66 = 1,020, P v = 66, MR v = 100 – 1,020/15 = 32,  v = P v  Q v = 67,320 ► Total profit is  uniform =  b +  v = 217,800 Business travelers Vacation travelers 3,6003, P 1 = 180 – Q 1 /20 P 2 = 100 – Q 2 /30 MR 2 = 100 – Q 1 /15 Integrated Market 6, P = 180 – Q /20 P = 132 – Q / , ,300 1, , , ,020 MR 1 = 180 – Q 1 /10

microeconomic s the analytics of constrained optimal decisions assignment 9 dynamic pricing (II)  2016 Kellogg School of Management assignment 9 page |3 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 ( MR 1 and MR 2 ) equal to the marginal cost (of zero): business travelers : Q b = 1,800 for a price P b = 180 – 1,800/20 = 90 vacation travelers : Q v = 1,500 for a price P v = 100 – 1,500/30 = 50 ► The profit for each market: business travelers :  b = P b  Q b = 162,000 vacation travelers :  v = P v  Q v = 75,000 ► Total profit is  uniform =  b +  v = 237,000 for a total number of cars: 3,300. Business travelers Vacation travelers 3,6003, P 1 = 180 – Q 1 /20 P 2 = 100 – Q 2 /30 MR 2 = 100 – Q 1 /15 Integrated Market 6, P = 180 – Q /20 P = 132 – Q / , , , ,800 MR 1 = 180 – Q 1 /10

microeconomic s the analytics of constrained optimal decisions assignment 9 dynamic pricing (II)  2016 Kellogg School of Management assignment 9 page |4 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: Q b = 3,600 – 20  84 = 1,920, P b = 84, MR b = 180 – 1,920/10 = – 12,  b = P b  Q b = 161,280 vacation travelers: Q v = 3,000 – 30  84 = 480, P v = 84, MR v = 100 – 480/15 = 68,  v = P v  Q v = 48,320 ► Total profit is  uniform =  b +  v = 201,600 Business travelers Vacation travelers 3,6003, P 1 = 180 – Q 1 /20 P 2 = 100 – Q 2 /30 MR 2 = 100 – Q 1 /15 Integrated Market 6, P = 180 – Q /20 P = 132 – Q / , ,300 1, , ,400 1,800 MR 1 = 180 – Q 1 /10

microeconomic s the analytics of constrained optimal decisions assignment 9 dynamic pricing (II)  2016 Kellogg School of Management assignment 9 page |5 rental car market customized pricing: capacity constraint ► We get a system of two equations with two unknowns: 180 – Q 1 /10 = 100 – Q 2 / – Q 1 /10 = 100 – (2400 – Q 1 )/15 Q 1 = 1440 Q 1 + Q 2 = 2400 Q 2 = 2400 – Q 1 Q 2 = 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: P 1 = 180 – Q 1 /20 = 108 and P 2 = 100 – Q 2 /30 = 68 and the total profit is  = 220,800 ► It must be the case that at the optimum (the last vehicle used) must satisfy MR 1 = MR 2 That is 180 – Q 1 /10 = 100 – Q 2 /15 ► But there is a constraint on the total quantity available to distribute between the two stores: Q 1 + Q 2 = Business travelers Vacation travelers MR 1 = 180 – Q 1 /10 MR 2 = 100 – Q 2 /15 36