Simulation studies suggest that Shared Automated Vehicles (SAVs) could reduce the total vehicle kilometres travelled (VKT) thanks to efficiently pooling multiple users in one vehicle. However, mode choice studies indicate that SAVs would attract mostly public transport users, leading to an increase in VKT. This paper is among the first to combine these operational and behavioural expectations and the first to do so analytically. In our theoretical set-up, travellers choose between car, public transport, and SAVs, depending on their individual valuation of private travel and other attributes of each mode. We find that the introduction of SAVs lead to a VKT change in public-transport-oriented cities ranging from a small decrease to a large increase, where the latter is true for plausible parameter settings and hence is a cautionary point for SAV-introduction policies. Conversely, SAVs would attract only few travellers in private-transport-oriented cities and therefore would not significantly impact VKT.