Controlling the droplet equilibrium location and shape on a conical fiber is essential to industrial applications such as dip-pen nanolithography. In this work, the equilibrium conformations of a drop on a vertical, conical fiber has been investigated by the finite element method, Surface Evolver simulations. Similar to the morphology of a drop on a cylinder, two different types (barrel shape and clam-shell shape) can be obtained. In the absence of gravity, the droplet moves upward (lower curvature) and the total surface energy decays as the drop ascends. Whatever the initial conformation of the drop on a conical fiber, the rising drop exhibits the clam-shell shape eventually and there is no equilibrium location. However, in the presence of gravity, the drop can stop at the equilibrium location stably. For a given contact angle, the clam-shell shape is generally favored for smaller drops but the barrel shape is dominant for larger drops. In a certain range of drop volume, the coexistence of both barrel and clam-shell shapes is observed. For large enough drops, the falling-off state is seen.