This paper deals with dynamic price competition in markets in which the perception of consumers regarding the value of goods depends on the choices of other consumers in the market. In particular, we consider the case in which consumers exhibit conformist preferences, which leads them to imitate their peers. In the context of a finite horizon model, we show that conformity-based behavior creates new channels of dynamic interaction between firms, changing the nature of price competition. We focus on the case of high network effects for which we obtain V-shaped equilibrium price paths and oscillating trajectories of market shares. We provide also a new rationale for the inversion of fashion trends.