A Con&nuous- &me Dynamic Pricing Model Knowing the Compe&tor s Pricing Strategy

Transcription

1 A Con&nuous- &me Dynamic Pricing Model Knowing the Compe&tor s Pricing Strategy Graduate School of Finance, Accoun&ng and Law, Waseda University Kimitoshi Sato Graduate School of Business Administra&on, Nanzan University Katsushige Sawaki Interna'onal Forum on Shipping, Ports and Airports May 2012 Hong Kong, China 1

2 Background Sta&c pricing is adopted by many transporta&on companies: Mul&ple prices: Full service airlines, Highway Bus: {55,45,28, 7, 3, 1day} Flat rate: High Speed Rail: 14,050 (Tokyo- Osaka) Advantage of sta&c pricing (SP): Provides the clear price to customers. Protects firm s yield levels. LCC frequently revise their online prices in response the compe&tor s price as well as market condi&ons; Low cost carrier: Peach, Ryanair, Easyjet, etc. 2

3 Change in Ticket Price During Selling Period in Compe&&ve Market From Osaka to Fukuoka,Departure day: 2012/3/ High speed rail Full service airline LCC Highway bus 0 Time 3

4 Objec&ve Compare DP with SP to inves&gate how DP is effec&ve in compe&&ve market. Consider a single leg and compe&&on between two firms (Firm1: DP, Firm2: SP). Firm 1 knows the compe&tor's pricing strategy. Develop a RM model in a con&nuous &me. Obtain an op&mal DP policy for firm 1 in closed- form. à Approximate solu&on of the discrete- &me model. à No need to be a heavy numerical computa&ons. à Analy&cal proper&es. 4

5 Related Literatures Dynamic pricing with compe&&on: Levin et al. (2009), Lin and Sibdari (2008), Sato and Sawaki (2011) à Game theory: DP v.s. DP (Compe&tor) à Require the informa&on of compe&tor s remaining capacity and selling price Dynamic pricing with compe&&on in con&nuous &me: Marcus and Anderson (2008): Determinis&c demand à Stochas&c demand. Xu and Hopp (2006): DP v.s. DP (Compe&tor) à DP v.s. Listed or flat- rate (Compe&tor) 5

6 2. Model: A Sequence of the Event Constant Firm 2 or Mul'ple Lowest available price Firm 1 Sales start Compe'tor s price Inventory level Poten'al customer Time remaining Sales end Capacity? Update Op'mal price? Time 6

7 Customer Arrival Process and U&lity Func&on Number of poten&al customer at &me t follows dx t = µ t X t dt + t X t dw t, X 0 = x U&lity of customers taking product 1 at price : X t Sales start Preference Exponen'al distribu'on with mean Sales price at 'me t Expected preference Sales end 7

8 Choice Probability and Demand Func&on Probability that customers purchase firm 1 s product Compe'tor s price Demand for firm 1 at &me Expected preference Firm 1 Firm 2 or No purchase Assump'on Non- purchase customer dose not wait un&l markdown occur. 8

9 Compe&tor s Pricing Strategy Three types of compe&tor s lowest available price 1. Flat rate p 2 (t) = p 2 2. Linear price High speed rail 3. Exponen&al price Full service airline LCC Highway bus Selling period 9

10 Objec&ve: Maximize Expected Total Profit Firm 1 applies dynamic pricing Maximal expected profit for given poten&al customer, inventory and compe&tor s price : Unit cost where inventory: is the &me when firm 1 runs out of Note that if. 10

11 Op&mal Price and Maximal Expected Profit Op&mal price at &me given poten&al customer, inventory and compe&tor s price : where Compe'tor s price Maximal expected profit for Inventory level (Op'mal price Unit cost) 11

12 Op&mal Prices and Inventory Level Poten'al customer Op'mal price Inventory level Firm2 (Compe&tor) Firm 1 12

13 Maximal Expected Profit with Op&mal Capacity Find an op&mal capacity so as to maximize the expected total profit for ; It can be obtained as Maximal expected profit under op&mal capacity: Expected preference for firm 1 Op'mal capacity 13

14 Assump&on: 3. Proper&es for Op&mal Policy - Monotone proper&es- DriX Vola'lity Selling period Unit cost Expected preferences Compe'tor s ini'al price Growth factor Constant Exponen'al 14

15 Impact of the Compe&tor s Pricing Strategy on the Maximal Expected Profit of Firm 1 Firm 1 adopts DP, then :Firm 1 is monopoly in the market :Compe&tor s pricing is constant :Compe&tor s pricing is linear Rela&onship between the expected profit of firm 1: Rela&onship between the op&mal capacity of firm 1: 15

16 Comparison Between DP and SP - Formula&on of Sta&c Model- Sta&c model Firm 1 sells at a flat rate rate or linear price. Expected profit of firm 1: and firm 2 adopts a flat is concave in. Maximal expected profit of firm 1: 16

17 Comparison Between DP and SP Firm 1 should be selected a constant pricing policy,, if Profit of selling one seat with a dynamic price Expected marginal profit of a constant pricing policy 17

18 Comparison Between DP and SP - Impact of expected preference- Firm 1: DP v.s. Flat rate Firm 2: Flat rate Firm 1: DP v.s. Flat rate Firm 2: Linear pricing When dril of the number of poten&al customer is small, DP is effec&ve. As increases, constant pricing is effec&ve. 18

19 Comparison Between DP and SP - Impact of the vola&lity- Compe&tor s pricing strategy à Flat rate Compe&tor s pricing strategy à Linear pricing When is large, DP is effec&ve for firm 1. When is small, a flat- rate pricing is effec&ve for firm 1. 19

20 4.Conclusion and Future Research Conclusion: What the compe&&ve situa&on is preferable to use DP rather than a flat- rate pricing? 1. Compe&tor adopts a flat rate. 2. Expected preference of compe&tor is higher than that of own company. 3. Number of poten&al customer is small on average and varies widely. Future research: Incorporate the strategic customer behavior Compe&tor s price follows a stochas&c process. Applica&on to supply chain management. 20

21 Deriva&on of the Op&mal Price Change the problem to find the op&mal demand Op&mal value func&on is given by the solu&on of HJB equa&on: 21

22 Airline and RM system Company SoXware Carrier Revenue Management Systems, INC. airrm AirAsia, Air Iceland, Air One, Air Tahi& Nui, Avianca Brasil, Azul, Blue Panorama, Cape Air, Condor, Garuda Ci&link, Germania, GoAir, go!mokulele, Jazeera Airways, JetBlue Airways, Jetstar, Merpa&, Monarch Airlines, OpenSkies, Peach Avia'on, Pegasus Airlines, Ryanair, Transat A.T. Transavia, Vueling Airlines, Webjet, Westjet, Zest Air PROS PROS ANA, China southern Airlines, Malév Hungarian Airlines, Southwest Airlines, China Airlines, Royal Jordanian Airlines 22