Dynamic pricing has been used by Transportation Network Companies (TNCs) to achieve a balance between the volume of ride requests with numbers of available drivers on two-sided TNC markets. Given the desire to reduce operating costs and the emergence of Autonomous Vehicles (AVs), the introduction of TNC-owned AV fleets could convert such services into one-sided markets, where operators have full control of service supply. In this paper we investigate the impact of utility-based dynamic pricing for Autonomous TNCs (ATNCs) in one-sided markets. We test the method using an Agent-Based Model (ABM) of Greater London in conditions of monopoly and competition, focusing on a statically priced ATNC service that offers a mix of private and shared ride services. Public transport is considered as an alternative mode of transportation in both scenarios. Results indicate that in monopoly, dynamic pricing provides higher revenues than static pricing at non-peak hours when average waiting times are low. On the contrary, in competition, dynamic pricing is superior at peak hours where increased waiting times are observed, thus increasing the value of low waiting time rides. Overall, in both market structures, it is found that shared trips are more popular in dynamic pricing compared to static pricing.