We consider a model of 'tenancy rent control' where landlords are not allowed to raise the rent on sitting tenants nor to evict them, though they are free to set the nominal rent when taking on a new tenant. If there is any inflation in the economy, landlords prefer to take short-staying tenants. Assuming that there is no way for landlords to tell a tenant's type, an adverse selection problem arises. If in this context, landlords have monopoly power-which, as we argue, is indeed pervasive-then the housing market equilibria can exhibit some unexpected properties. Most strikingly, landlords may prefer not to raise the rent even when there is excess demand for housing. Such rents are labeled "efficiency rents" in this paper and their existence shows that tenancy rent control can give rise to equilibria that look as if there were traditional rent control in which the rent of each unit has a flat ceiling. In other words, tenancy rent control may not achieve the flexibility, which it was expected to impart, to the system of traditional rent control. Keywords: Rent Control, Rent Regulation, Asymmetric Information, Adverse Selection, Monopoly.