Novel Dynamic Pricing Framework Captures Asymmetric Reference Price Effects in Piecewise-Smooth Systems

A groundbreaking study by Dr Anton Bondarev, Associate Professor of Economics at the International Business School Suzhou (IBSS) of Xi’an Jiaotong-Liverpool University, has been published in European Journal of Operational Research (EJOR), a top‑tier international journal in operations research, management science and decision support (ABS 4, ABDC A*, JCR Q1, Impact Factor = 6.0).

The paper, entitled Dynamic pricing with asymmetric reference price effects in a piecewise‑smooth system, develops a unified theoretical framework to study optimal dynamic pricing when consumers react asymmetrically to price gains and losses relative to their reference price.

Against the widespread use of dynamic pricing in ride‑hailing, ticketing, retail, electricity and platform markets, empirical evidence consistently shows consumers are far more sensitive to price increases (losses) than to equivalent price decreases (gains). Most existing models assume symmetric responses, ruling out loss aversion and gain‑seeking behavior that shape real demand. Dr Bondarev’s research breaks new ground by relaxing symmetry and using a piecewise‑smooth optimal control system combined with Filippov’s method to capture regime shifts when prices cross the reference price.

Using rigorous theoretical analysis and phase‑diagram characterization, the study identifies fundamentally different long‑run pricing dynamics:

• Under loss aversion, the model exhibits reference‑price lock‑in: if the initial reference price lies between two boundary steady states, the firm becomes anchored to a continuum of constant‑price outcomes. Early pricing decisions have irreversible long‑run effects.
• Under gain‑seeking behavior, intermediate constant pricing is never optimal; prices converge to one of two boundary steady states, with a sharp indifference threshold in initial reference prices determining the outcome.

These sliding and escaping dynamics at the reference‑price threshold cannot be captured by standard smooth or symmetric models. The research further extends the monopoly framework to an n‑firm oligopoly using hybrid differential games, showing the core structural dynamics persist under competition.

The findings offer clear managerial guidance:

• For firms facing loss‑averse consumers, introductory pricing and early price adjustments are critical strategic commitments that lock in long‑run price levels.
• For gain‑seeking markets, firms should actively steer consumer expectations across the indifference threshold to reach a more profitable steady state.
• Dynamic pricing near the reference threshold is inherently unstable and requires careful expectation management.

This work makes three key contributions:

1. Methodological: Establishes a unified piecewise‑smooth optimal control framework for asymmetric reference‑price dynamics.
2. Theoretical: Reveals distinct lock‑in and tipping effects arising from loss aversion and gain seeking.
3. Practical: Provides actionable pricing strategies for industries where reference dependence and asymmetric demand are prevalent.

About Dr Anton Bondarev

Dr Anton Bondarev is an Associate professor in Economics at IBSS. He acquired his PhD from Bielefeld University concentrating on dynamics of innovations and optimal management of RD. Prior to joining XJTLU he served as a Postdoc in different international projects, such as Climate Policy of Nations in Germany, Competence Center for Research in Energy, Society and Transition in Switzerland and others. His research is focused on endogenous growth, environment, promotion of innovations and optimal governance of large-scale technological transitions, be it renewable energy or modernization of the economy.He actively publishes in such journals as Automatica (IF=6.150), European Economic Review (ABDC: A*), Journal of Economic Dynamics and Control ( ABDC: A*), China Economic Review (ABDC: A), Energy Policy (ABDC: A), Macroeconomic Dynamics (ABDC: A), Journal of Evolutionary Economics (ABDC: A) and others.

About the Journal

European Journal of Operational Research (EJOR) is the flagship journal of the Association of European Operational Research Societies (EURO), published by Elsevier. It is rated ABS 4, ABDC A*, JCR Q1 and is globally recognized as a leading outlet for high‑quality theoretical and applied research in operations research, optimization, logistics and decision support systems.