Both the edge and the cloud can provide computing services for mobile devices to enhance their performance. The edge can reduce the conveying delay by providing local computing services while the cloud can support enormous computing requirements. Their cooperation can improve the utilization of computing resources and ensure the QoS, and thus is critical to edge-cloud computing business models. This paper proposes an efficient framework for mobile edge-cloud computing networks, which enables the edge and the cloud to share their computing resources in the form of wholesale and buyback. To optimize the computing resource sharing process, we formulate the computing resource management problems for the edge servers to manage their wholesale and buyback scheme and the cloud to determine the wholesale price and its local computing resources. Then, we solve these problems from two perspectives: i) social welfare maximization and ii) profit maximization for the edge and the cloud. For i), we have proved the concavity of the social welfare and proposed an optimal cloud computing resource management to maximize the social welfare. For ii), since it is difficult to directly prove the convexity of the primal problem, we first proved the concavity of the wholesaled computing resources with respect to the wholesale price and designed an optimal pricing and cloud computing resource management to maximize their profits. Numerical evaluations show that the total profit can be maximized by social welfare maximization while the respective profits can be maximized by the optimal pricing and cloud computing resource management.