Future wireless networks should meet heterogeneous service requirements of diverse applications, including interactive multimedia, augmented reality, and autonomous driving. The fog radio access network (Fog-RAN) is a novel architecture that enables efficient and flexible allocation of network resources to end users. However, guaranteeing application-specific service requirements while maximizing resource utilization is an open challenge in Fog-RANs. This article proposes a multi-resource Fog-RAN slicing scheme that maximizes network resource utilization and satisfies important economic properties: Paretooptimality, envy-freeness, and sharing incentive. The proposed solution considers both heterogeneous resources (i.e., bandwidth, storage and computing) and the different service levels defined in 5G networks. Accordingly, a two-level resource scheduling mechanism is devised to jointly allocate Fog-RAN resources to slices in two stages: a broker allocates resources to slices at fog nodes over a given time window; a slice hypervisor then allocates slice-specific resources at each fog node to users with a much shorter time scale. An extensive evaluation based on real-world datasets demonstrates that the proposed solution significantly increases the monetary gain of service providers, namely, by 32% to 60% compared to the state of the art, including dynamic hierarchical resource allocation and dynamic slicing with proportional allocation.