We study dynamic price competition between sellers offering differentiated products with limited capacity and a common sales deadline. In every period, firms simultaneously set prices, and a randomly arriving buyer decides whether to purchase a product or leave the market. Given remaining capacities, firms trade off selling today against shifting demand to competitors to obtain future market power. We provide conditions for the existence and uniqueness of pure-strategy Markov perfect equilibria. In the continuous-time limit, prices solve a system of ordinary differential equations. We derive properties of equilibrium dynamics and show that prices increase the most when the product with the lowest remaining capacity sells. Because firms do not fully internalize the social option value of future sales, equilibrium prices can be inefficiently low such that both firms and consumers would benefit if firms could commit to higher prices. We term this new welfare effect the Bertrand scarcity trap.
Authors
Jose M. Betancourt, Ali Hortaçsu, Aniko Öry, Kevin R. Williams
- Acknowledgements & Disclosure
- We thank the Yale School of Management and the Cowles Foundation for Research in Economics for financial and computational support. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
- DOI
- https://doi.org/10.3386/w32673
- Pages
- 72
- Published in
- United States of America
Table of Contents
- NBER WORKING PAPER SERIES 1
- DYNAMIC PRICE COMPETITION WITH CAPACITY CONSTRAINTS 1
- Jose M. Betancourt Ali Hortaçsu Aniko Öry Kevin R. Williams 1
- Working Paper 32673 httpwww.nber.orgpapersw32673 1
- NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge MA 02138 July 2024 1
- Dynamic Price Competition with Capacity Constraints Jose M. Betancourt Ali Hortaçsu Aniko Öry and Kevin R. Williams NBER Working Paper No. 32673 July 2024 JEL No. C7D04D6L0 2
- Jose M. Betancourt Yale University jose.betancourtvalenciayale.edu 2
- Ali Hortaçsu Kenneth C. Griffin Department of Economics University of Chicago 1126 East 59th Street Chicago IL 60637 and NBER hortacsuuchicago.edu 2
- Aniko Öry aoeryandrew.cmu.edu 2
- Carnegie Mellon University Tepper School of Business 4765 Forbes Ave Pittsburgh PA 15213 2
- Kevin R. Williams Yale School of Management 165 Whitney Avenue New Haven CT 06520 and NBER kevin.williamsyale.edu 2
- 1 Introduction 3
- 1.1 Related Literature 7
- 2 Model 9
- 3 The Monopoly Benchmark 12
- 4 Dynamic Pricing with Competition 14
- 4.1 Analysis of the Stage Game 16
- 4.2 Continuous-Time Approximation of Dynamic Markov Equilibria 22
- 4.3 Implementing Our Model as an Empirical Framework 26
- 5 Properties of Dynamic Pricing Equilibria 27
- 5.1 Strategic Equilibrium Forces 27
- 5.2 Additional Properties of the Dynamic Game 30
- 6 Welfare and the Bertrand Scarcity Trap 32
- 6.1 Welfare-Maximizing Prices 33
- 6.2 Definition of Bertrand Scarcity Trap 34
- 6.3 An Analytic Example of the Bertrand Scarcity Trap 37
- 6.4 Alleviating the Bertrand Scarcity Trap with Algorithmic Pricing 38
- 6.5 Demand Uncertainty Is Necessary for the BST to Exist 40
- 7 Conclusion 44
- References 45
- Online Appendix 48
- Dynamic Price Competition with Capacity Constraints 48
- A Proofs 48
- A.1 Technical results 48
- A.2 Proofs for Monopoly Benchmark 50
- A.3 Proofs of Dyanamic Pricing Model with Competition 55
- B Nested Logit Calculations 69
- C Additional Figures 71
