Airbnb has shown constant growth and it provides income and taxes to tourist destinations. However, the prevalence of a substantial number of Airbnb providers in tourist destinations may lead to bottlenecks in rental housing markets. Regional planners and policy-makers across the world are therefore imposing restrictions to regulate this hitherto unregulated business model. The present paper sheds light on the link between housing-market regulation and the growth of Airbnb, based upon Norwegian Airbnb listings and agent-based modelling. The simulation results suggest that Airbnb's current growth will not simply flatten out when the supply matches the demand, but will be followed by a series of sudden crises and subsequent quick recoveries. These instabilities will put stress on local rental markets and threaten both the local tourism industry and rental housing markets. Moderate taxation may contribute to a more even distribution of Airbnb listings in Norway, notably across the urban space.