Dynamic pricing with asymmetric reference price effects in a piecewise-smooth system

Dynamic pricing is widely used in markets where consumers evaluate prices relative to reference points. While most of the literature assumes symmetric responses to price increases and decreases, asymmetric reference price effects are empirically well documented. Existing analytical research has examined special cases of such asymmetry. This paper studies dynamic pricing with asymmetric reference price effects without imposing symmetry or restricting attention to special cases. We reformulate the canonical reference-price model as a piecewise-smooth optimal control problem, reflecting regime switches when prices cross the reference price, and use this framework to characterize the qualitative structure of the resulting pricing dynamics. Asymmetry fundamentally alters long-run outcomes. Under loss aversion, the model exhibits reference-price lock–in: when the initial reference price lies between two boundary steady states, the firm becomes anchored to a continuum of constant-price outcomes. Under gain-seeking behavior, intermediate constant pricing is never optimal; instead, prices converge to one of two boundary steady states, with an indifference threshold in initial reference prices determining the selected outcome. These dynamics arise from sliding or escaping behavior at the reference-price threshold and cannot be captured by smooth or symmetric models. The results show that early pricing decisions and expectation management play a decisive and potentially irreversible role in dynamic pricing.