The paper considers a mathematical model for dynamic price adjustment in real estate. The model is characterized by a finite number of real estate objects, a fixed sales horizon, and the presence of intermediate goals for sales and revenue. The model developed in this work addresses the case of variable total demand, incorporating the time value of money and the increase in real estate objects value as construction progresses. The general structure of pricing policy is studied, and an algorithm for determining prices under variable total demand is presented. Similar constructions are carried out for a model that accounts for the time value of money and the rising property value during construction. The case of a linear elasticity function is also examined as a basic but widely used practical scenario. Rigorous mathematical proofs of the results are provided, along with numerical simulations based on real estate data from a specific city over 3.5 years to compare different approaches to pricing policy formulation. The obtained results can be applied to effectively manage real estate pricing.
1. Araman V.F., Caldentey R. Revenue Management with Incomplete Demand Information. In: Wiley Encyclopedia of Operations Research and Management Science. Hoboken: John Wiley & Sons, Inc.; 2010. pp. 1–17. https://doi.org/10.1002/9780470400531.EORMS0728
2. Aviv Y., Vulcano G. Dynamic List Pricing. In: The Oxford Handbook of Pricing Management: Chapter 23. Oxford: Oxford University Press; 2012. pp. 522–584. https://doi.org/10.1093/oxfordhb/9780199543175.013.0023
3. Christ S. Operationalizing Dynamic Pricing Models: Bayesian Demand Forecasting and Customer Choice Modeling for Low Cost Carriers. Wiesbaden: Gabler Verlag; 2011. 351 p. https://doi.org/10.1007/978-3-8349-6184-6
4. Chen M., Chen Z.-L. Recent Developments in Dynamic Pricing Research: Multiple Products, Competition, and Limited Demand Information. Production and Operations Management. 2015;24(5):704–731. https://doi.org/10.1111/poms.12295
5. Den Boer A.V. Dynamic pricing and learning: Historical origins, current research, and new directions. Surveys in Operations Research and Management Science. 2015;20(1):1–18. https://doi.org/10.1016/j.sorms.2015.03.001
6. Bitran G., Caldentey R. An Overview of Pricing Models for Revenue Management. Manufacturing & Service Operations Management. 2003;5(3):203–229. https://doi.org/10.1287/msom.5.3.203.16031
7. Elmaghraby W., Keskinocak P. Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions. IEEE Engineering Management Review. 2003;31(4):47–47. https://doi.org/10.1109/EMR.2003.24939
8. Heching A., Leung Y.T. Product Pricing in the e-Business Era. In: Supply Chain Management on Demand: Strategies and Technologies, Applications. Berlin, Heidelberg: Springer; 2005. pp. 65–96. https://doi.org/10.1007/3-540-27354-9_4
9. Talluri K.T., Ryzin G.J. The Theory and Practice of Revenue Management. New York: Springer; 2004. 713 p. https://doi.org/10.1007/b139000
10. Gönsch J., Klein R., Steinhardt C. Dynamic Pricing – State-of-The-Art. Zeitschrift für Betriebswirtschaft, Ergänzungsheft 3 'Operations Research in der Betriebswirtschaft'. 2009;3:1–40.
11. Rao V.R. Handbook of Pricing Research in Marketing. Cheltenham: Edward Elgar Publishing Limited; 2009. 616 p.
12. Chenavaz R., Carrier L.-P., Etienne L., Paraschiv C. Dynamic Pricing in Management Science. Journal of Economics Studies and Research. 2011;2011. https://doi.org/10.5171/2011.283281
13. Deksnyte I., Lydeka Z. Dynamic Pricing and Its Forming Factors. International Journal of Business and Social Science. 2012;3(23):213–220.
14. Özer Ö., Phillips R. The Oxford Handbook of Pricing Management. Oxford: Oxford University Press; 2012. 976 p.
15. Phillips R.L. Pricing and Revenue Optimization. Stanford: Stanford University Press; 2021. 472 p.
16. Faruqui A., Sergici S. Household response to dynamic pricing of electricity: a survey of 15 experiments. Journal of Regulatory Economics. 2010;38(2):193–225. https://doi.org/10.1007/s11149-010-9127-y
17. Khan A.R., Mahmood A., Safdar A., Khan Z.A., Khan N.A. Load forecasting, dynamic pricing and DSM in smart grid: A review. Renewable and Sustainable Energy Reviews. 2016;54:1311–1322. https://doi.org/10.1016/j.rser.2015.10.117
18. Dutta G., Mitra K. A literature review on dynamic pricing of electricity. Journal of the Operational Research Society. 2017;68(10):1131–1145. https://doi.org/10.1057/s41274-016-0149-4
19. Saharan S., Bawa S., Kumar N. Dynamic pricing techniques for Intelligent Transportation System in smart cities: A systematic review. Computer Communications. 2020;150:603–625. https://doi.org/10.1016/j.comcom.2019.12.003
20. McAfee R.P., Te Velde V. Dynamic Pricing in the Airline Industry. In: Handbooks in Information Systems: Chapter 11. 2006. pp. 527–569. https://doi.org/10.1016/S1574-0145(06)01011-7
21. Kincaid W.M., Darling D.A. An inventory pricing problem. Journal of Mathematical Analysis and Applications. 1963;7(2):183–208. https://doi.org/10.1016/0022-247X(63)90047-7
22. Gallego G., Van Ryzin G. Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons. Management Science. 1994;40(8):999–1020. https://doi.org/10.1287/MNSC.40.8.999
23. Feng Y., Gallego G. Optimal Starting Times for End-of-Season Sales and Optimal Stopping Times for Promotional Fares. Management Science. 1995;41(8):1371–1391. https://doi.org/10.1287/mnsc.41.8.1371
24. Bitran G.R., Mondschein S.V. Periodic Pricing of Seasonal Products in Retailing. Management Science. 1997;43(1):64–79. https://doi.org/10.1287/MNSC.43.1.64
25. Feng Y., Xiao B. A Continuous-Time Yield Management Model with Multiple Prices and Reversible Price Changes. Management Science. 2000;46(5):644–657. https://doi.org/10.1287/mnsc.46.5.644.12050
26. Feng Y., Gallego G. Perishable Asset Revenue Management with Markovian Time Dependent Demand Intensities. Management Science. 2000;46(7):941–956. https://doi.org/10.1287/mnsc.46.7.941.12035
27. Zhao W., Zheng Y.-S. Optimal Dynamic Pricing for Perishable Assets with Nonhomogeneous Demand. Management Science. 2000;46(3):375–388. https://doi.org/10.1287/mnsc.46.3.375.12063
28. Bitran G., Caldentey R., Mondschein S. Coordinating Clearance Markdown Sales of Seasonal Products in Retail Chains. Operations Research. 1998;46(5):609–624. https://doi.org/10.1287/opre.46.5.609
29. Gallego G., Van Ryzin G. A Multiproduct Dynamic Pricing Problem and Its Applications to Network Yield Management. Operations Research. 1997;45(1):24–41. https://doi.org/10.1287/opre.45.1.24
30. Kleywegt A.J. An Optimal Control Problem of Dynamic Pricing. Working paper, School of Industrial Engineering, Georgia Institute of Technology. 2001.
31. Besbes O., Maglaras C. Dynamic Pricing with Financial Milestones: Feedback-Form Policies. Management Science. 2012;58(9):1715–1731.
